Finite Group Databasebeta

← #29706546

Subgroups — #29706546

Order 15616.

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×#14519763C211yesmaximal, characteristic
2#21446758C211yesmaximal, characteristic, complemented
3#21446757C211yesmaximal, characteristic, complemented
4#21446435C211yesmaximal, characteristic, complemented
5#21446434C211yesmaximal, characteristic, complemented
6#21446433C211yesmaximal, characteristic, complemented
7#21446430C211yesmaximal, characteristic, complemented
8#21446116C211yesmaximal, characteristic
9#21446113C211yesmaximal, characteristic
10#21446002C211yesmaximal, characteristic, complemented
11#21446000C211yesmaximal, characteristic, complemented
12#21446000C211yesmaximal, characteristic, complemented
13#21445935C211yesmaximal, characteristic, complemented
14#21445934C211yesmaximal, characteristic, complemented
15C61×#1851C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C2×#14519763C411yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×C22⋊D61C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×C22⋊#92418C2211yescharacteristic
25C2×C22⋊#92418C2211yescharacteristic
26C2×#14519689C2211yescharacteristic
27C2×#14519689C2211yescharacteristic
28C2×#14519689C2211yescharacteristic
29C2×#14519688C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31#16839666C411yescharacteristic
32#16839662C411yescharacteristic
33#16839655C411yescharacteristic
34#16839654C411yescharacteristic
35#16839651C411yescharacteristic
36#16839650C411yescharacteristic
37#16839626C411yescharacteristic
38C4×#11758198C2211yescharacteristic
39C4×#11758198C2211yescharacteristic
40C4×C22⋊Dic61C2211yescharacteristic
41C42×Dic61C2211yescharacteristic
42#16839517C2211yescharacteristic
43#16839517C2211yescharacteristic
44#16839516C2211yescharacteristic
45#16839516C2211yescharacteristic
46#16839516C2211yescharacteristic
47#16839516C2211yescharacteristic
48#16839516C2211yescharacteristic
49#16839516C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×M16×C61C2211yescharacteristic
53C2×C4⋊C8×C61C2211yescharacteristic
54C2×C4⋊C8×C61C2211yescharacteristic
55C2×C22⋊C8×C61C2211yescharacteristic
56C4×C22⋊C4×C61C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Dic122C2×C411yescharacteristic
60C22×C22⋊D61C2×C411yescharacteristic
61C22×C4×D61C2311yescharacteristic
62C2×#11758189—41no—
63C2×#11758189—41no—
64C2×#11758189—41no—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519763—24no—
70C4×Dic122—41no—
71C4×Dic122—41no—
72C4×C22⋊D61—41no—
73C4×C22⋊D61—41no—
74C2×C4⋊Dic61C2×C411yescharacteristic
75C2×#11758196C2×C411yescharacteristic
76C2×#11758196C2×C411yescharacteristic
77C2×#11758196C2×C411yescharacteristic
78C2×C22⋊Dic61C2×C411yescharacteristic
79C2×C22⋊Dic61C2×C411yescharacteristic
80C2×C22⋊Dic61C2×C411yescharacteristic
81C2×#11758198C2×C411yescharacteristic
82C2×#11758198C2×C411yescharacteristic
83C2×#11758198C2×C411yescharacteristic
84C2×#11758198C2×C411yescharacteristic
85C2×#11758198C2×C411yescharacteristic
86C2×C4×Dic61C2×C411yescharacteristic
87C2×C4×Dic61C2×C411yescharacteristic
88C2×C4×Dic61C2×C411yescharacteristic
89C2×C4×Dic61C2×C411yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93C4⋊#92418—41no—
94#14519706—41no—
95#14519706—41no—
96#14519706—41no—
97C22⋊#92418—41no—
98C22⋊#92418—41no—
99#14519689—41no—
100#14519689—41no—
101#14519689—41no—
102#14519688—41no—
103#14519687—41no—
104#14519684C2×C411yescharacteristic
105#14519684C2×C411yescharacteristic
106#14519684C2×C411yescharacteristic
107#14519684C2×C411yescharacteristic
108#14519684C2×C411yescharacteristic
109#14519684C2×C411yescharacteristic
110C23×C4×C61C2311yescharacteristic
111C2×M16×C61—41no—
112C2×M16×C61—41no—
113C2×M16×C61—41no—
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C2×C22⋊C4×C61C2×C411yescharacteristic
119C2×C22⋊C4×C61C2×C411yescharacteristic
120C2×C42×C61C2311yescharacteristic
121C2×C42×C61C2311yescharacteristic
122C4⋊C8×C61—41no—
123C4⋊C8×C61—41no—
124C22⋊C8×C61—41no—
125#95×C61C2×C411yescharacteristic
126#95×C61C2×C411yescharacteristic
127C24×C61C22×C411yescharacteristic
128C23×D61C22×C411yescharacteristic
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61C22×C411yescharacteristic
136C22×C4×C61C22×C411yescharacteristic
137C22×C4×C61C22×C411yescharacteristic
138C22×C4×C61C22×C411yescharacteristic
139C22×C4×C61C22×C411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C2×C22⋊D61—22no—
142C2×C22⋊D61—41no—
143C2×C22⋊D61—41no—
144C22×Dic61C22×C411yescharacteristic
145C22×Dic61C22×C411yescharacteristic
146C22×Dic61C22×C411yescharacteristic
147C22×Dic61C22×C411yescharacteristic
148C22×Dic61C22×C411yescharacteristic
149C22×Dic61C22×C411yescharacteristic
150C22×Dic61C22×C411yescharacteristic
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×Dic122—22no—
158C2×Dic122—41no—
159C2×Dic122—41no—
160M16×C61—44no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—41no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—21no—
179C22⋊Dic61—22no—
180C22⋊Dic61—41no—
181C22⋊Dic61—41no—
182#11758198—22no—
183#11758198—22no—
184#11758198—41no—
185#11758198—41no—
186#11758198—41no—
187C4⋊Dic61—41no—
188#11758196—41no—
189#11758196—41no—
190#11758196—41no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—41no—
198C4×Dic61—41no—
199C4×Dic61—41no—
200C4×Dic61M1614yescomplemented
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213#11758189—44no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61C23×C411yescharacteristic
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61C2×M1612yes—
248C2×C4×C61H(5,𝔽2)11yescharacteristic
249C2×C4×C61H(5,𝔽2)11yescharacteristic
250C2×C4×C61H(5,𝔽2)11yescharacteristic
251C2×C4×C61H(5,𝔽2)11yescharacteristic
252C2×C4×C612−1+411yescharacteristic
253C2×C4×C612−1+411yescharacteristic
254C22⋊D61—44no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61C2×M1612yes—
275C2×Dic61C2×M1612yes—
276C4×D61—41no—
277C4×D61—41no—
278C4×D61—41no—
279C4×D61—41no—
280Dic122—44no—
281C8×C61—41no—
282C8×C61—41no—
283C8×C61—41no—
284C8×C61—41no—
285#92418—41no—
286#92418—41no—
287#92418—41no—
288#92418—41no—
289#23353—611nomaximal, Sylow
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—41no—
297C22×C61#51811yescharacteristic
298C22×C61#52011yescharacteristic
299C22×C61#52011yescharacteristic
300C22×C61C22×M1611yescharacteristic
301C22×C61#56811yescharacteristic
302C22×C61#56811yescharacteristic
303C22×C61#56811yescharacteristic
304C2×D61—41no—
305C2×D61—41no—
306C2×D61—41no—
307C2×D61—41no—
308C2×D61—41no—
309C2×D61—41no—
310C2×D61—41no—
311C4×C61—22no—
312C4×C61—41no—
313C4×C61—41no—
314C4×C61—41no—
315C4×C61—41no—
316C4×C61H(3,𝔽4)11yescharacteristic
317C4×C61#56211yescharacteristic
318C4×C61#56211yescharacteristic
319C4×C61#56311yescharacteristic
320Dic61—22no—
321Dic61—22no—
322Dic61—41no—
323Dic61—41no—
324Dic61—41no—
325Dic61—41no—
326Dic61—41no—
327C4×#124—611no—
328#1985—611no—
329#1985—611no—
330#1924—611no—
331#1924—611no—
332#1924—611no—
333#1924—611no—
334#1870—611no—
335#1870—611no—
336#1851—611no—
337#1851—611no—
338#1851—611no—
339#1851D6111yescharacteristic, complemented
340#1831—611no—
341#1831—611no—
342C2×C61—81no—
343C2×C61#237911yescharacteristic
344C2×C61#297511yescharacteristic
345C2×C61#298011yescharacteristic
346C2×C61#298911yescharacteristic
347C2×C61#298911yescharacteristic
348C2×C61#299511yescharacteristic
349C2×C61#299511yescharacteristic
350D61—81no—
351C22×M16—611no—
352C22×M16C2×D6111yescharacteristic
353C2×#124—611no—
354C2×C4×Q8—611no—
355C2×C4×D4—611no—
356C2×C4⋊C8—611no—
357C2×C4⋊C8—611no—
358C2×C4⋊C8—611no—
359C2×C4⋊C8—611no—
360C2×C4⋊C8C2×D6111yescharacteristic
361C2×C4⋊C8C2×D6111yescharacteristic
362C2×C22⋊C8—611no—
363C2×C22⋊C8—611no—
364C2×C22⋊C8—611no—
365C2×C22⋊C8—611no—
366C2×C22⋊C8—611no—
367C2×C22⋊C8C2×D6111yescharacteristic
368C2×#97—611no—
369C2×#97—611no—
370#391—611no—
371#390—611no—
372#388—611no—
373#388—611no—
374#388—611no—
375#388—611no—
376#382—611no—
377C4×C22⋊C4—611no—
378C4×C22⋊C4—611no—
379C4×C22⋊C4—611no—
380C4×C22⋊C4C2×D6111yescharacteristic
381C43—611no—
382#336—611no—
383#336—611no—
384#336—611no—
385#336—611no—
386#336—611no—
387#336—611no—
388#336—611no—
389#336—611no—
390#336—611no—
391#336—611no—
392#336C2×D6111yescharacteristic
393#336C2×D6111yescharacteristic
394C61#2335311yescharacteristic, complemented, Sylow
395C22×Q8—611no—
396C22×D4—611no—
397C23×C4—611no—
398C23×C4C22×D6111yescharacteristic
399C2×M16—41no—
400C2×M16—41no—
401C2×M16—41no—
402C2×M16—2441no—
403C2×M16—2441no—
404C2×M16—2441no—
405C22×C8—611no—
406C22×C8—611no—
407C22×C8—611no—
408C22×C8—611no—
409C22×C8C22×D6111yescharacteristic
410C22×C8C22×D6111yescharacteristic
411C22×C8C22×D6111yescharacteristic
412C22×C8C22×D6111yescharacteristic
413#124—1224no—
414C4×Q8—2441no—
415C4×Q8—2441no—
416C4×D4—2441no—
417C4×D4—2441no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C22⋊C4—611no—
423C2×C22⋊C4—611no—
424C2×C22⋊C4—611no—
425C2×C22⋊C4—611no—
426C2×C22⋊C4—611no—
427C2×C22⋊C4—611no—
428C2×C22⋊C4—611no—
429C2×C22⋊C4—611no—
430C2×C22⋊C4C4×D6111yescharacteristic
431C2×C22⋊C4C4×D6111yescharacteristic
432C2×C42—611no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—611no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42C22×D6111yescharacteristic
440C2×C42C22×D6111yescharacteristic
441C4⋊C8—41no—
442C4⋊C8—41no—
443C4⋊C8—2441no—
444C4⋊C8—2441no—
445C4⋊C8—2441no—
446C4⋊C8—2441no—
447C22⋊C8—41no—
448C22⋊C8—2441no—
449C22⋊C8—2441no—
450C22⋊C8—2441no—
451C22⋊C8—2441no—
452C22⋊C8—2441no—
453#97—2441no—
454#97—2441no—
455#95—611no—
456#95—611no—
457#95—611no—
458#95—611no—
459#95—611no—
460#95—611no—
461#95C4×D6111yescharacteristic
462#95C4×D6111yescharacteristic
463C24—611no—
464C24C2×C4×D6111yescharacteristic
465C2×Q8—1222no—
466C2×Q8—2441no—
467C2×Q8—2441no—
468C2×D4—1222no—
469C2×D4—2441no—
470C2×D4—2441no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4C2×C4×D6111yescharacteristic
491C22×C4C2×C4×D6111yescharacteristic
492C22×C4C2×C4×D6111yescharacteristic
493C22×C4C2×C4×D6111yescharacteristic
494C22×C4C2×C4×D6111yescharacteristic
495C22×C4C23×D6111yescharacteristic
496M16—44no—
497M16—2444no—
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—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C4⋊C4—2441no—
523C4⋊C4—2441no—
524C4⋊C4—2441no—
525C4⋊C4—2441no—
526C22⋊C4—22no—
527C22⋊C4—41no—
528C22⋊C4—1222no—
529C22⋊C4—1222no—
530C22⋊C4—1222no—
531C22⋊C4—2441no—
532C22⋊C4—2441no—
533C22⋊C4—2441no—
534C22⋊C4—2441no—
535C22⋊C4—2441no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—614no—
541C42—1221no—
542C42—1221no—
543C42—1221no—
544C42—1221no—
545C42—1221no—
546C42—1221no—
547C42—2441no—
548C42—2441no—
549C42—2441no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—1221no—
558C23—1221no—
559C23—1221no—
560C23—1221no—
561C23—1221no—
562C23—1221no—
563C23—1221no—
564C23C22×C4×D6111yescharacteristic
565Q8—2444no—
566D4—2444no—
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—612no—
586C2×C4—612no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4M16×D6112yes—
611C2×C4#1451982211yescharacteristic
612C2×C4#1451982211yescharacteristic
613C2×C4#1451982211yescharacteristic
614C2×C4#1451982311yescharacteristic
615C2×C4#1451982411yescharacteristic
616C2×C4#1451982511yescharacteristic
617C8—41no—
618C8—41no—
619C8—41no—
620C8—41no—
621C8—2441no—
622C8—2441no—
623C8—2441no—
624C8—2441no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—2441no—
633C22—2441no—
634C22—2441no—
635C22—2441no—
636C22—2441no—
637C22—2441no—
638C22—2441no—
639C22#1684023411yescharacteristic
640C22#1684024111yescharacteristic
641C22#1684024311yescharacteristic
642C22C2×M16×D6111yescharacteristic
643C22#1684052711yescharacteristic
644C22#1684052711yescharacteristic
645C22#1684052811yescharacteristic
646C4—22no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—1222no—
652C4—1222no—
653C4—2441no—
654C4—2441no—
655C4—2441no—
656C4—2441no—
657C4—2441no—
658C4#1684050511yescharacteristic
659C4#1684050811yescharacteristic
660C4#1684050911yescharacteristic
661C4#1684051311yescharacteristic
662C2—81no—
663C2—4881no—
664C2#2144899211yescharacteristic
665C2#2145485711yescharacteristic
666C2#2145488711yescharacteristic
667C2#2145493111yescharacteristic
668C2#2145493311yescharacteristic
669C2#2145498311yescharacteristic
670C2#2145498711yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D61
commutator, socleC23×C61C23×C4
FittingC61×#1851C2
prime radicalC22×C22⋊D61C2×C4
cosocleC22×C4×C61C24
hypercenter#1851D61
hypocenterC61#23353
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
2. #21446758C2semidirect
3. #21446757C2semidirect
4. #21446435C2semidirect
5. #21446434C2semidirect
6. #21446433C2semidirect
7. #21446430C2semidirect
10. #21446002C2semidirect
11. #21446000C2semidirect
12. #21446000C2semidirect
13. #21445935C2semidirect
14. #21445934C2semidirect
15. C61×#1851C2semidirect
200. C4×Dic61M16semidirect
339. #1851D61semidirect
394. C61#23353semidirect

Maximal subgroups

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

subgroupindex
C4×#145197632
#214467582
#214467572
#214464352
#214464342
#214464332
#214464302
#214461162
#214461132
#214460022
#214460002
#214459352
#214459342
C61×#18512
#2335361