Finite Group Databasebeta

← #36580957

Subgroups — #36580957

Order 22784.

732 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#15667998C211yesmaximal, characteristic
2#26021192C211yesmaximal, characteristic, complemented
3#26021190C211yesmaximal, characteristic, complemented
4#26020865C211yesmaximal, characteristic, complemented
5#26020864C211yesmaximal, characteristic, complemented
6#26020862C211yesmaximal, characteristic, complemented
7#26020547C211yesmaximal, characteristic
8#26020546C211yesmaximal, characteristic
9#26020437C211yesmaximal, characteristic, complemented
10#26020436C211yesmaximal, characteristic, complemented
11#26020436C211yesmaximal, characteristic, complemented
12#26020434C211yesmaximal, characteristic, complemented
13#26020368C211yesmaximal, characteristic, complemented
14#26020367C211yesmaximal, characteristic, complemented
15C89×#1924C211yesmaximal, characteristic, complemented
16C22×#13083558C2211yescharacteristic
17C22×#13083557C2211yescharacteristic
18C2×#15667998C411yescharacteristic
19C2×C4×Dic178C2211yescharacteristic
20C2×C4×C22⋊D89C2211yescharacteristic
21C2×C4⋊#10623494C2211yescharacteristic
22C2×C4⋊#10623494C2211yescharacteristic
23C2×#15667941C2211yescharacteristic
24C2×#15667941C2211yescharacteristic
25C2×#15667941C2211yescharacteristic
26C2×C22⋊#10623494C2211yescharacteristic
27C2×#15667924C2211yescharacteristic
28C2×#15667924C2211yescharacteristic
29C2×#15667924C2211yescharacteristic
30C2×#15667922C2211yescharacteristic
31#19151502C411yescharacteristic
32#19151498C411yescharacteristic
33#19151491C411yescharacteristic
34#19151490C411yescharacteristic
35#19151487C411yescharacteristic
36#19151486C411yescharacteristic
37#19151462C411yescharacteristic
38C4×#13083551C2211yescharacteristic
39C4×#13083551C2211yescharacteristic
40C4×C22⋊Dic89C2211yescharacteristic
41C42×Dic89C2211yescharacteristic
42#19151353C2211yescharacteristic
43#19151353C2211yescharacteristic
44#19151352C2211yescharacteristic
45#19151352C2211yescharacteristic
46#19151352C2211yescharacteristic
47#19151352C2211yescharacteristic
48#19151352C2211yescharacteristic
49#19151352C2211yescharacteristic
50#19151352C2211yescharacteristic
51#19151352C2211yescharacteristic
52C2×C4⋊C8×C89C2211yescharacteristic
53C2×C22⋊C8×C89C2211yescharacteristic
54C2×C22⋊C8×C89C2211yescharacteristic
55C2×#97×C89C2211yescharacteristic
56C4×C22⋊C4×C89C2211yescharacteristic
57#336×C89C2211yescharacteristic
58#336×C89C2211yescharacteristic
59C22×Dic178C2×C411yescharacteristic
60C22×C22⋊D89C2×C411yescharacteristic
61C22×C4×D89C2311yescharacteristic
62C2×#13083558—41no—
63C2×#13083558—41no—
64C2×#13083558—41no—
65C2×#13083557—41no—
66C2×#13083557—41no—
67C2×#13083557—41no—
68C22×#10623494C2311yescharacteristic
69C22×#10623494C2311yescharacteristic
70C22×#10623494C2311yescharacteristic
71C22×#10623494C2311yescharacteristic
72#15667998—24no—
73C4×Dic178—41no—
74C4×Dic178—41no—
75C4×C22⋊D89—41no—
76C4×C22⋊D89—41no—
77C2×C4⋊Dic89C2×C411yescharacteristic
78C2×#13083553C2×C411yescharacteristic
79C2×#13083553C2×C411yescharacteristic
80C2×#13083553C2×C411yescharacteristic
81C2×C22⋊Dic89C2×C411yescharacteristic
82C2×C22⋊Dic89C2×C411yescharacteristic
83C2×C22⋊Dic89C2×C411yescharacteristic
84C2×#13083551C2×C411yescharacteristic
85C2×#13083551C2×C411yescharacteristic
86C2×#13083551C2×C411yescharacteristic
87C2×#13083551C2×C411yescharacteristic
88C2×#13083551C2×C411yescharacteristic
89C2×C4×Dic89C2×C411yescharacteristic
90C2×C4×Dic89C2×C411yescharacteristic
91C2×C4×Dic89C2×C411yescharacteristic
92C2×C4×Dic89C2×C411yescharacteristic
93C2×C4×Dic89C2311yescharacteristic
94C2×C4×Dic89C2311yescharacteristic
95C2×C4×Dic89C2311yescharacteristic
96C4⋊#10623494—41no—
97C4⋊#10623494—41no—
98#15667941—41no—
99#15667941—41no—
100#15667941—41no—
101C22⋊#10623494—41no—
102#15667924—41no—
103#15667924—41no—
104#15667924—41no—
105#15667922—41no—
106#15667919C2×C411yescharacteristic
107#15667919C2×C411yescharacteristic
108#15667919C2×C411yescharacteristic
109#15667919C2×C411yescharacteristic
110#15667919C2×C411yescharacteristic
111#15667919C2×C411yescharacteristic
112C23×C4×C89C2311yescharacteristic
113C22×C8×C89C2311yescharacteristic
114C22×C8×C89C2311yescharacteristic
115C22×C8×C89C2311yescharacteristic
116C22×C8×C89C2311yescharacteristic
117C2×C22⋊C4×C89C2×C411yescharacteristic
118C2×C22⋊C4×C89C2×C411yescharacteristic
119C2×C42×C89C2311yescharacteristic
120C2×C42×C89C2311yescharacteristic
121C4⋊C8×C89—41no—
122C22⋊C8×C89—41no—
123C22⋊C8×C89—41no—
124#97×C89—41no—
125#95×C89C2×C411yescharacteristic
126#95×C89C2×C411yescharacteristic
127C23×D89C22×C411yescharacteristic
128C2×Dic178—22no—
129C2×Dic178—41no—
130C2×Dic178—41no—
131C2×C22⋊D89—22no—
132C2×C22⋊D89—41no—
133C2×C22⋊D89—41no—
134C22×Dic89C22×C411yescharacteristic
135C22×Dic89C22×C411yescharacteristic
136C22×Dic89C22×C411yescharacteristic
137C22×Dic89C22×C411yescharacteristic
138C22×Dic89C22×C411yescharacteristic
139C22×Dic89C22×C411yescharacteristic
140C22×Dic89C22×C411yescharacteristic
141C2×C4×D89—21no—
142C2×C4×D89—21no—
143C2×C4×D89—21no—
144C2×C4×D89—21no—
145C2×C4×D89—21no—
146C2×C4×D89—21no—
147#13083558—44no—
148#13083557—44no—
149C2×#10623494—21no—
150C2×#10623494—21no—
151C2×#10623494—21no—
152C2×#10623494—21no—
153C2×#10623494—21no—
154C2×#10623494—21no—
155C2×#10623494—21no—
156C2×#10623494—21no—
157C2×#10623494—21no—
158C2×#10623494—21no—
159C2×#10623494—21no—
160C2×#10623494—21no—
161C4⋊Dic89—41no—
162#13083553—41no—
163#13083553—41no—
164#13083553—41no—
165C22⋊Dic89—22no—
166C22⋊Dic89—41no—
167C22⋊Dic89—41no—
168#13083551—22no—
169#13083551—22no—
170#13083551—41no—
171#13083551—41no—
172#13083551—41no—
173C4×Dic89—21no—
174C4×Dic89—21no—
175C4×Dic89—21no—
176C4×Dic89—21no—
177C4×Dic89—21no—
178C4×Dic89—21no—
179C4×Dic89—41no—
180C4×Dic89—41no—
181C4×Dic89—41no—
182C4×Dic89M1614yescomplemented
183C24×C89C22×C411yescharacteristic
184C22×C4×C89—21no—
185C22×C4×C89—21no—
186C22×C4×C89—21no—
187C22×C4×C89—21no—
188C22×C4×C89—21no—
189C22×C4×C89—21no—
190C22×C4×C89C22×C411yescharacteristic
191C22×C4×C89C22×C411yescharacteristic
192C22×C4×C89C22×C411yescharacteristic
193C22×C4×C89C22×C411yescharacteristic
194C22×C4×C89C22×C411yescharacteristic
195C22×C4×C89C2411yescharacteristic
196C2×C8×C89—21no—
197C2×C8×C89—21no—
198C2×C8×C89—21no—
199C2×C8×C89—21no—
200C2×C8×C89—21no—
201C2×C8×C89—21no—
202C2×C8×C89—21no—
203C2×C8×C89—21no—
204C2×C8×C89—21no—
205C2×C8×C89—21no—
206C2×C8×C89—21no—
207C2×C8×C89—21no—
208C22⋊C4×C89—22no—
209C22⋊C4×C89—41no—
210C42×C89—21no—
211C42×C89—21no—
212C42×C89—21no—
213C42×C89—21no—
214C23×C89—21no—
215C23×C89—21no—
216C23×C89—21no—
217C23×C89—21no—
218C23×C89—21no—
219C23×C89—21no—
220C23×C89—21no—
221C23×C89C23×C411yescharacteristic
222C22×D89—21no—
223C22×D89—21no—
224C22×D89—21no—
225C22×D89—21no—
226C22×D89—21no—
227C22×D89—21no—
228C22×D89—21no—
229C2×C4×C89—21no—
230C2×C4×C89—21no—
231C2×C4×C89—21no—
232C2×C4×C89—21no—
233C2×C4×C89—21no—
234C2×C4×C89—21no—
235C2×C4×C89—21no—
236C2×C4×C89—21no—
237C2×C4×C89—21no—
238C2×C4×C89—21no—
239C2×C4×C89—21no—
240C2×C4×C89—21no—
241C2×C4×C89—21no—
242C2×C4×C89—21no—
243C2×C4×C89—41no—
244C2×C4×C89—41no—
245C2×C4×C89—41no—
246C2×C4×C89—41no—
247C2×C4×C89C2×M1612yes—
248C2×C4×C89H(5,𝔽2)11yescharacteristic
249C2×C4×C89H(5,𝔽2)11yescharacteristic
250C2×C4×C89H(5,𝔽2)11yescharacteristic
251C2×C4×C89H(5,𝔽2)11yescharacteristic
252C2×C4×C892−1+411yescharacteristic
253C2×C4×C892−1+411yescharacteristic
254C22⋊D89—44no—
255C2×Dic89—21no—
256C2×Dic89—21no—
257C2×Dic89—21no—
258C2×Dic89—21no—
259C2×Dic89—21no—
260C2×Dic89—21no—
261C2×Dic89—21no—
262C2×Dic89—21no—
263C2×Dic89—21no—
264C2×Dic89—21no—
265C2×Dic89—21no—
266C2×Dic89—21no—
267C2×Dic89—21no—
268C2×Dic89—21no—
269C2×Dic89—21no—
270C2×Dic89—21no—
271C2×Dic89—21no—
272C2×Dic89—21no—
273C2×Dic89—21no—
274C2×Dic89C2×M1612yes—
275C2×Dic89C2×M1612yes—
276C4×D89—41no—
277C4×D89—41no—
278C4×D89—41no—
279C4×D89—41no—
280Dic178—44no—
281C8×C89—41no—
282C8×C89—41no—
283C8×C89—41no—
284C8×C89—41no—
285#10623494—41no—
286#10623494—41no—
287#10623494—41no—
288#10623494—41no—
289C22×C89—41no—
290C22×C89—41no—
291C22×C89—41no—
292C22×C89—41no—
293C22×C89—41no—
294C22×C89—41no—
295C22×C89—41no—
296C22×C89#51811yescharacteristic
297C22×C89#52011yescharacteristic
298C22×C89#52011yescharacteristic
299C22×C89C22×M1611yescharacteristic
300C22×C89#56811yescharacteristic
301C22×C89#56811yescharacteristic
302C22×C89#56811yescharacteristic
303C2×D89—41no—
304C2×D89—41no—
305C2×D89—41no—
306C2×D89—41no—
307C2×D89—41no—
308C2×D89—41no—
309C2×D89—41no—
310C4×C89—22no—
311C4×C89—41no—
312C4×C89—41no—
313C4×C89—41no—
314C4×C89—41no—
315C4×C89H(3,𝔽4)11yescharacteristic
316C4×C89#56211yescharacteristic
317C4×C89#56211yescharacteristic
318C4×C89#56311yescharacteristic
319Dic89—22no—
320Dic89—22no—
321Dic89—41no—
322Dic89—41no—
323Dic89—41no—
324Dic89—41no—
325Dic89—41no—
326#23353—891nomaximal, Sylow
327C2×C89—81no—
328C2×C89#237911yescharacteristic
329C2×C89#297511yescharacteristic
330C2×C89#298011yescharacteristic
331C2×C89#298911yescharacteristic
332C2×C89#298911yescharacteristic
333C2×C89#299511yescharacteristic
334C2×C89#299511yescharacteristic
335D89—81no—
336C4×#124—891no—
337#1985—891no—
338#1985—891no—
339#1924—891no—
340#1924—891no—
341#1924—891no—
342#1924D8911yescharacteristic, complemented
343#1870—891no—
344#1870—891no—
345#1851—891no—
346#1851—891no—
347#1851—891no—
348#1851—891no—
349#1831—891no—
350#1831—891no—
351C89#2335311yescharacteristic, complemented, Sylow
352C22×M16—891no—
353C22×M16—891no—
354C2×#124—891no—
355C2×C4×Q8—891no—
356C2×C4×D4—891no—
357C2×C4⋊C8—891no—
358C2×C4⋊C8—891no—
359C2×C4⋊C8—891no—
360C2×C4⋊C8—891no—
361C2×C4⋊C8—891no—
362C2×C4⋊C8C2×D8911yescharacteristic
363C2×C22⋊C8—891no—
364C2×C22⋊C8—891no—
365C2×C22⋊C8—891no—
366C2×C22⋊C8—891no—
367C2×C22⋊C8C2×D8911yescharacteristic
368C2×C22⋊C8C2×D8911yescharacteristic
369C2×#97—891no—
370C2×#97C2×D8911yescharacteristic
371#391—891no—
372#390—891no—
373#388—891no—
374#388—891no—
375#388—891no—
376#388—891no—
377#382—891no—
378C4×C22⋊C4—891no—
379C4×C22⋊C4—891no—
380C4×C22⋊C4—891no—
381C4×C22⋊C4C2×D8911yescharacteristic
382C43—891no—
383#336—891no—
384#336—891no—
385#336—891no—
386#336—891no—
387#336—891no—
388#336—891no—
389#336—891no—
390#336—891no—
391#336—891no—
392#336—891no—
393#336C2×D8911yescharacteristic
394#336C2×D8911yescharacteristic
395C22×Q8—891no—
396C22×D4—891no—
397C23×C4—891no—
398C23×C4C22×D8911yescharacteristic
399C2×M16—3561no—
400C2×M16—3561no—
401C2×M16—3561no—
402C2×M16—3561no—
403C2×M16—3561no—
404C2×M16—3561no—
405C22×C8—891no—
406C22×C8—891no—
407C22×C8—891no—
408C22×C8—891no—
409C22×C8C22×D8911yescharacteristic
410C22×C8C22×D8911yescharacteristic
411C22×C8C22×D8911yescharacteristic
412C22×C8C22×D8911yescharacteristic
413#124—1784no—
414C4×Q8—3561no—
415C4×Q8—3561no—
416C4×D4—3561no—
417C4×D4—3561no—
418C2×C4⋊C4—891no—
419C2×C4⋊C4—891no—
420C2×C4⋊C4—891no—
421C2×C4⋊C4—891no—
422C2×C22⋊C4—891no—
423C2×C22⋊C4—891no—
424C2×C22⋊C4—891no—
425C2×C22⋊C4—891no—
426C2×C22⋊C4—891no—
427C2×C22⋊C4—891no—
428C2×C22⋊C4—891no—
429C2×C22⋊C4—891no—
430C2×C22⋊C4C4×D8911yescharacteristic
431C2×C22⋊C4C4×D8911yescharacteristic
432C2×C42—891no—
433C2×C42—891no—
434C2×C42—891no—
435C2×C42—891no—
436C2×C42—891no—
437C2×C42—891no—
438C2×C42—891no—
439C2×C42C22×D8911yescharacteristic
440C2×C42C22×D8911yescharacteristic
441C4⋊C8—41no—
442C4⋊C8—3561no—
443C4⋊C8—3561no—
444C4⋊C8—3561no—
445C4⋊C8—3561no—
446C4⋊C8—3561no—
447C22⋊C8—41no—
448C22⋊C8—41no—
449C22⋊C8—3561no—
450C22⋊C8—3561no—
451C22⋊C8—3561no—
452C22⋊C8—3561no—
453#97—41no—
454#97—3561no—
455#95—891no—
456#95—891no—
457#95—891no—
458#95—891no—
459#95—891no—
460#95—891no—
461#95C4×D8911yescharacteristic
462#95C4×D8911yescharacteristic
463C24—891no—
464C24C2×C4×D8911yescharacteristic
465C2×Q8—3561no—
466C2×Q8—3561no—
467C2×Q8—1782no—
468C2×D4—3561no—
469C2×D4—3561no—
470C2×D4—1782no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—891no—
478C22×C4—891no—
479C22×C4—891no—
480C22×C4—891no—
481C22×C4—891no—
482C22×C4—891no—
483C22×C4—891no—
484C22×C4—1781no—
485C22×C4—1781no—
486C22×C4—1781no—
487C22×C4—1781no—
488C22×C4—1781no—
489C22×C4—1781no—
490C22×C4C2×C4×D8911yescharacteristic
491C22×C4C2×C4×D8911yescharacteristic
492C22×C4C2×C4×D8911yescharacteristic
493C22×C4C2×C4×D8911yescharacteristic
494C22×C4C2×C4×D8911yescharacteristic
495C22×C4C23×D8911yescharacteristic
496M16—3564no—
497M16—3564no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—1781no—
511C2×C8—1781no—
512C2×C8—1781no—
513C2×C8—1781no—
514C2×C8—1781no—
515C2×C8—1781no—
516C2×C8—1781no—
517C2×C8—1781no—
518C2×C8—1781no—
519C2×C8—1781no—
520C2×C8—1781no—
521C2×C8—1781no—
522C4⋊C4—3561no—
523C4⋊C4—3561no—
524C4⋊C4—3561no—
525C4⋊C4—3561no—
526C22⋊C4—22no—
527C22⋊C4—41no—
528C22⋊C4—3561no—
529C22⋊C4—3561no—
530C22⋊C4—3561no—
531C22⋊C4—3561no—
532C22⋊C4—3561no—
533C22⋊C4—1782no—
534C22⋊C4—1782no—
535C22⋊C4—1782no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—894no—
541C42—3561no—
542C42—3561no—
543C42—3561no—
544C42—1781no—
545C42—1781no—
546C42—1781no—
547C42—1781no—
548C42—1781no—
549C42—1781no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—1781no—
558C23—1781no—
559C23—1781no—
560C23—1781no—
561C23—1781no—
562C23—1781no—
563C23—1781no—
564C23C22×C4×D8911yescharacteristic
565Q8—3564no—
566D4—3564no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—892no—
586C2×C4—892no—
587C2×C4—3561no—
588C2×C4—3561no—
589C2×C4—3561no—
590C2×C4—3561no—
591C2×C4—1781no—
592C2×C4—1781no—
593C2×C4—1781no—
594C2×C4—1781no—
595C2×C4—1781no—
596C2×C4—1781no—
597C2×C4—1781no—
598C2×C4—1781no—
599C2×C4—1781no—
600C2×C4—1781no—
601C2×C4—1781no—
602C2×C4—1781no—
603C2×C4—1781no—
604C2×C4—1781no—
605C2×C4—1781no—
606C2×C4—1781no—
607C2×C4—1781no—
608C2×C4—1781no—
609C2×C4—1781no—
610C2×C4M16×D8912yes—
611C2×C4#1566805711yescharacteristic
612C2×C4#1566805711yescharacteristic
613C2×C4#1566805811yescharacteristic
614C2×C4#1566805811yescharacteristic
615C2×C4#1566805911yescharacteristic
616C2×C4#1566805911yescharacteristic
617C8—41no—
618C8—41no—
619C8—41no—
620C8—41no—
621C8—3561no—
622C8—3561no—
623C8—3561no—
624C8—3561no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—3561no—
633C22—3561no—
634C22—3561no—
635C22—3561no—
636C22—3561no—
637C22—3561no—
638C22—3561no—
639C22#1915207011yescharacteristic
640C22#1915207711yescharacteristic
641C22#1915207811yescharacteristic
642C22C2×M16×D8911yescharacteristic
643C22#1915236311yescharacteristic
644C22#1915236311yescharacteristic
645C22#1915236411yescharacteristic
646C4—22no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—3561no—
652C4—3561no—
653C4—3561no—
654C4—3561no—
655C4—3561no—
656C4—1782no—
657C4—1782no—
658C4#1915234011yescharacteristic
659C4#1915234311yescharacteristic
660C4#1915234511yescharacteristic
661C4#1915235011yescharacteristic
662C2—81no—
663C2—7121no—
664C2#2602342111yescharacteristic
665C2#2602929011yescharacteristic
666C2#2602932011yescharacteristic
667C2#2602936411yescharacteristic
668C2#2602937111yescharacteristic
669C2#2602941711yescharacteristic
670C2#2602942311yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D89
commutator, socleC23×C89C23×C4
FittingC89×#1924C2
prime radicalC22×C22⋊D89C2×C4
cosocleC22×C4×C89C24
hypercenter#1924D89
hypocenterC89#23353
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C22×C4 ◁ #1924

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #26021192C2semidirect
3. #26021190C2semidirect
4. #26020865C2semidirect
5. #26020864C2semidirect
6. #26020862C2semidirect
9. #26020437C2semidirect
10. #26020436C2semidirect
11. #26020436C2semidirect
12. #26020434C2semidirect
13. #26020368C2semidirect
14. #26020367C2semidirect
15. C89×#1924C2semidirect
182. C4×Dic89M16semidirect
342. #1924D89semidirect
351. C89#23353semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C4×#156679982
#260211922
#260211902
#260208652
#260208642
#260208622
#260205472
#260205462
#260204372
#260204362
#260204342
#260203682
#260203672
C89×#19242
#2335389