Finite Group Databasebeta

← #29693955

Subgroups — #29693955

Order 15616.

814 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519755C211yesmaximal, characteristic
2#21446754C211yesmaximal, characteristic, complemented
3#21446753C211yesmaximal, characteristic, complemented
4#21446534C211yesmaximal, characteristic, complemented
5#21446379C211yesmaximal, characteristic, complemented
6#21446378C211yesmaximal, characteristic, complemented
7#21446376C211yesmaximal, characteristic, complemented
8#21446373C211yesmaximal, characteristic, complemented
9#21445995C211yesmaximal, characteristic
10#21445995C211yesmaximal, characteristic
11#21445796C211yesmaximal, characteristic
12C8×#11758198C211yesmaximal, characteristic
13#21445589C211yesmaximal, characteristic
14#336×D61C211yesmaximal, characteristic
15C61×#1939C211yesmaximal, characteristic, complemented
16C22×C8×D61C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×#14519755C411yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×C42×D61C2211yescharacteristic
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519689C2211yescharacteristic
25C2×#14519689C2211yescharacteristic
26C2×C4×#92418C2211yescharacteristic
27C2×C8×Dic61C2211yescharacteristic
28C2×C8×Dic61C2211yescharacteristic
29#16839666C411yescharacteristic
30#16839657C411yescharacteristic
31#16839648C411yescharacteristic
32#16839644C411yescharacteristic
33#16839636C411yescharacteristic
34#16839633C411yescharacteristic
35#16839621C411yescharacteristic
36C4×#11758196C2211yescharacteristic
37C4×#11758196C2211yescharacteristic
38C4×#11758198C2211yescharacteristic
39C4×#11758198C2211yescharacteristic
40#16839517C2211yescharacteristic
41#16839517C2211yescharacteristic
42#16839517C2211yescharacteristic
43#16839517C2211yescharacteristic
44#16839516C2211yescharacteristic
45#16839516C2211yescharacteristic
46#16839516C2211yescharacteristic
47#16839516C2211yescharacteristic
48#16839516C2211yescharacteristic
49#16839516C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C4⋊C8×C61C2211yescharacteristic
53C2×C4⋊C8×C61C2211yescharacteristic
54C2×C4⋊C8×C61C2211yescharacteristic
55C2×C4×C8×C61C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Dic122C2×C411yescharacteristic
60C22×C4×D61C2×C411yescharacteristic
61C22×C4×D61C2×C411yescharacteristic
62C22×C4×D61C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C2×C8×D61—22no—
68C2×C8×D61—41no—
69C2×C8×D61—41no—
70C2×C8×D61—41no—
71C2×C8×D61—41no—
72C2×C8×D61Q814yescomplemented
73#14519755C818yescomplemented
74C4×Dic122—41no—
75C4×Dic122—41no—
76C2×C4⋊Dic61C2×C411yescharacteristic
77C2×#11758196C2×C411yescharacteristic
78C2×#11758196C2×C411yescharacteristic
79C2×#11758196C2×C411yescharacteristic
80C2×#11758196C2×C411yescharacteristic
81C2×#11758196C2×C411yescharacteristic
82C2×#11758196C2×C411yescharacteristic
83C2×#11758198C2×C411yescharacteristic
84C2×#11758198C2×C411yescharacteristic
85C2×#11758198C2×C411yescharacteristic
86C2×#11758198C2×C411yescharacteristic
87C2×C4×Dic61C2×C411yescharacteristic
88C2×C4×Dic61C2×C411yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92C42×D61—41no—
93C42×D61—41no—
94C4⋊#92418—41no—
95#14519706—41no—
96#14519706—41no—
97#14519689—22no—
98#14519689—22no—
99C4×#92418—41no—
100C8×Dic61—41no—
101C8×Dic61—41no—
102#14519684C2×C411yescharacteristic
103#14519684C2×C411yescharacteristic
104#14519684C2×C411yescharacteristic
105#14519684C2×C411yescharacteristic
106#14519684C2×C411yescharacteristic
107#14519684C2×C411yescharacteristic
108#14519684C2×C411yescharacteristic
109#14519684C2×C411yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2×C411yescharacteristic
115C2×C4⋊C4×C61C2×C411yescharacteristic
116C2×C42×C61C2311yescharacteristic
117C2×C42×C61C2311yescharacteristic
118C2×C42×C61C2311yescharacteristic
119C4⋊C8×C61—41no—
120C4⋊C8×C61—41no—
121C4⋊C8×C61—41no—
122C4×C8×C61—41no—
123#95×C61C2×C411yescharacteristic
124#95×C61C2×C411yescharacteristic
125C23×D61C22×C411yescharacteristic
126C22×C4×C61C22×C411yescharacteristic
127C22×C4×C61C22×C411yescharacteristic
128C22×C4×C61C22×C411yescharacteristic
129C22×C4×C61C22×C411yescharacteristic
130C22×C4×C61C22×C411yescharacteristic
131C22×C4×C61C22×C411yescharacteristic
132C22×C4×C61C2411yescharacteristic
133C22×Dic61C22×C411yescharacteristic
134C22×Dic61C22×C411yescharacteristic
135C22×Dic61C22×C411yescharacteristic
136C22×Dic61C22×C411yescharacteristic
137C22×Dic61C22×C411yescharacteristic
138C22×Dic61C22×C411yescharacteristic
139C22×Dic61C22×C411yescharacteristic
140C2×C4×D61—21no—
141C2×C4×D61—21no—
142C2×C4×D61—21no—
143C2×C4×D61—21no—
144C2×C4×D61—22no—
145C2×C4×D61—41no—
146C2×C4×D61—41no—
147C2×C4×D61—41no—
148C2×C4×D61—41no—
149C2×C4×D61C2×C814yes—
150C2×C4×D61C2×Q812yes—
151C2×C4×D61#4112yes—
152C2×Dic122—41no—
153C2×Dic122—41no—
154C2×Dic122C2×C814yescomplemented
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61C2×Q812yes—
167C4⋊C4×C61—41no—
168C4⋊C4×C61C2×C814yescomplemented
169C42×C61—21no—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—21no—
175#11758198—22no—
176#11758198—22no—
177#11758198C2×C814yes—
178#11758198C2×C814yes—
179C4⋊Dic61—41no—
180#11758196—41no—
181#11758196—41no—
182#11758196—41no—
183#11758196—41no—
184#11758196C2×C814yescomplemented
185#11758196C2×C814yescomplemented
186C4×Dic61—21no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—41no—
193C4×Dic61—41no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418C2×Q812yes—
206C8×D61—44no—
207C8×D61—44no—
208C23×C61C23×C411yescharacteristic
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—21no—
213C22×D61C4×Q812yes—
214C22×D61C22×C812yes—
215C22×D61#13112yes—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61C22×C812yes—
232C2×C4×C61C22×C812yes—
233C2×C4×C61C22×C812yes—
234C2×C4×C61C22×Q811yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61H(5,𝔽2)11yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C2×C4×C61H(5,𝔽2)11yescharacteristic
239C2×C4×C612−1+411yescharacteristic
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61C4×Q812yes—
258C2×Dic61C22×C812yes—
259C2×Dic61C22×C812yes—
260C2×Dic61C22×C812yes—
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—44no—
266C4×D61—44no—
267Dic122—44no—
268C8×C61—22no—
269C8×C61—41no—
270C8×C61—41no—
271C8×C61—41no—
272#92418—22no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#22287—611nomaximal, Sylow
277C22×C61C2×C4×Q811yescharacteristic
278C22×C61#51811yescharacteristic
279C22×C61#52011yescharacteristic
280C22×C61C23×C811yescharacteristic
281C22×C61C2×#13111yescharacteristic
282C22×C61#56811yescharacteristic
283C22×C61#56811yescharacteristic
284C2×D61—22no—
285C2×D61—22no—
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—22no—
289C2×D61—22no—
290C2×D61C8×Q814yes—
291C4×C61—22no—
292C4×C61—22no—
293C4×C61—22no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61#53811yescharacteristic
298C4×C61#53911yescharacteristic
299C4×C61#54311yescharacteristic
300C4×C61#54411yescharacteristic
301Dic61—22no—
302Dic61—22no—
303Dic61—22no—
304Dic61—22no—
305Dic61—41no—
306Dic61—41no—
307Dic61—41no—
308C4×C22⋊Q8—611no—
309#1984—611no—
310#1984—611no—
311#1939—611no—
312#1939D6111yescharacteristic, complemented
313#1916—611no—
314#1916—611no—
315#1916—611no—
316#1916—611no—
317#1850—611no—
318#1850—611no—
319#1797—611no—
320C8×C22⋊C4—611no—
321C8×C22⋊C4—611no—
322C2×#336—611no—
323C2×C61#232911yescharacteristic
324C2×C61C2×C8×Q811yescharacteristic
325C2×C61#296111yescharacteristic
326C2×C61#297211yescharacteristic
327C2×C61#297611yescharacteristic
328C2×C61#298811yescharacteristic
329C2×C61#299211yescharacteristic
330D61—24no—
331C23×C8—611no—
332C23×C8—611no—
333C2×C22⋊Q8—611no—
334C2×C4×Q8—611no—
335C22×C42—611no—
336C2×C4⋊C8—611no—
337C2×C4⋊C8—611no—
338C2×C4⋊C8—611no—
339C2×C4⋊C8C2×D6111yescharacteristic
340C2×C4⋊C8C2×D6111yescharacteristic
341C2×C4⋊C8C2×D6111yescharacteristic
342C2×C22⋊C8—611no—
343C2×C22⋊C8—611no—
344C2×C4×C8—611no—
345C2×C4×C8—611no—
346C2×C4×C8—611no—
347C2×C4×C8C2×D6111yescharacteristic
348#391—611no—
349#389—611no—
350#387—611no—
351#387—611no—
352#385—611no—
353#385—611no—
354#380—611no—
355C4×C4⋊C4—611no—
356C4×C4⋊C4—611no—
357C4×C4⋊C4C2×D6111yescharacteristic
358C4×C22⋊C4—611no—
359C4×C22⋊C4—611no—
360#336—611no—
361#336—611no—
362#336—611no—
363#336—611no—
364#336—611no—
365#336—611no—
366#336—611no—
367#336—611no—
368#336—611no—
369#336—611no—
370#336—611no—
371#336—611no—
372#336C2×D6111yescharacteristic
373#336C2×D6111yescharacteristic
374C61#2228711yescharacteristic, complemented, Sylow
375C22×Q8—611no—
376C23×C4—611no—
377C23×C4—611no—
378C23×C4—611no—
379C22×C8—611no—
380C22×C8—614no—
381C22×C8—611no—
382C22×C8—611no—
383C22×C8—611no—
384C22×C8—1222no—
385C22×C8—2441no—
386C22×C8—2441no—
387C22×C8—2441no—
388C22×C8—2441no—
389C22×C8C22×D6111yescharacteristic
390C22×C8C22×D6111yescharacteristic
391C22×C8C22×D6111yescharacteristic
392C22×C8C22×D6111yescharacteristic
393C22⋊Q8—618no—
394C4×Q8—2441no—
395C4×Q8—2441no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4C4×D6111yescharacteristic
404C2×C4⋊C4C4×D6111yescharacteristic
405C2×C22⋊C4—611no—
406C2×C22⋊C4—611no—
407C2×C22⋊C4—611no—
408C2×C22⋊C4—611no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—611no—
414C2×C42—2441no—
415C2×C42—2441no—
416C2×C42C22×D6111yescharacteristic
417C2×C42C22×D6111yescharacteristic
418C2×C42C22×D6111yescharacteristic
419C4⋊C8—41no—
420C4⋊C8—41no—
421C4⋊C8—41no—
422C4⋊C8—2441no—
423C4⋊C8—2441no—
424C4⋊C8—2441no—
425C22⋊C8—1222no—
426C22⋊C8—1222no—
427C4×C8—41no—
428C4×C8—2441no—
429C4×C8—2441no—
430C4×C8—2441no—
431#95—611no—
432#95—611no—
433#95—611no—
434#95—611no—
435#95—611no—
436#95—611no—
437#95—611no—
438#95—611no—
439#95C4×D6111yescharacteristic
440#95C4×D6111yescharacteristic
441C24—611no—
442C2×Q8—614no—
443C2×Q8—2441no—
444C2×Q8—2441no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—612no—
450C22×C4—612no—
451C22×C4—614no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—1222no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—2441no—
461C22×C4—2441no—
462C22×C4—2441no—
463C22×C4—2441no—
464C22×C4C2×C4×D6111yescharacteristic
465C22×C4C2×C4×D6111yescharacteristic
466C22×C4C2×C4×D6111yescharacteristic
467C22×C4C2×C4×D6111yescharacteristic
468C22×C4C2×C4×D6111yescharacteristic
469C22×C4C2×C4×D6111yescharacteristic
470C22×C4C23×D6111yescharacteristic
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—612no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—2444no—
495C2×C8—2444no—
496C2×C8Q8×D6112yes—
497C4⋊C4—41no—
498C4⋊C4—614no—
499C4⋊C4—614no—
500C4⋊C4—2441no—
501C4⋊C4—2441no—
502C4⋊C4—2441no—
503C4⋊C4—2441no—
504C4⋊C4—2441no—
505C4⋊C4C8×D6114yescomplemented
506C22⋊C4—614no—
507C22⋊C4—614no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—1221no—
517C42—1221no—
518C42—1221no—
519C42—1221no—
520C42—1221no—
521C42—1221no—
522C42—2441no—
523C42—2441no—
524C23—612no—
525C23—612no—
526C23—612no—
527C23—1221no—
528C23—1221no—
529C23—1221no—
530C23—1221no—
531C23C22×C4×D6111yescharacteristic
532Q8—2444no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—612no—
549C2×C4—612no—
550C2×C4—612no—
551C2×C4—612no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—2444no—
574C2×C4—2444no—
575C2×C4C2×C8×D6112yes—
576C2×C4C2×C8×D6112yes—
577C2×C4C2×C8×D6112yes—
578C2×C4C2×Q8×D6111yescharacteristic
579C2×C4#41×D6111yescharacteristic
580C2×C4#1451982211yescharacteristic
581C2×C4#1451982211yescharacteristic
582C2×C4#1451982311yescharacteristic
583C2×C4#1451982511yescharacteristic
584C8—22no—
585C8—41no—
586C8—41no—
587C8—41no—
588C8—1222no—
589C8—2441no—
590C8—2441no—
591C8—2441no—
592C22—614no—
593C22—1222no—
594C22—1222no—
595C22—1222no—
596C22—1222no—
597C22—1222no—
598C22—1222no—
599C22C4×Q8×D6111yescharacteristic
600C22#1684023411yescharacteristic
601C22#1684024311yescharacteristic
602C22C22×C8×D6111yescharacteristic
603C22#131×D6111yescharacteristic
604C22#1684052711yescharacteristic
605C22#1684052811yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—1222no—
613C4—1222no—
614C4—1222no—
615C4—1222no—
616C4—2441no—
617C4—2441no—
618C4—2441no—
619C4#1684034411yescharacteristic
620C4#1684036011yescharacteristic
621C4#1684038711yescharacteristic
622C4#1684038911yescharacteristic
623C2—1224no—
624C2#2144855311yescharacteristic
625C2C8×Q8×D6111yescharacteristic
626C2#2145475511yescharacteristic
627C2#2145484011yescharacteristic
628C2#2145486311yescharacteristic
629C2#2145492811yescharacteristic
630C2#2145495611yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D61
commutatorC22×C61C23×C8
FittingC61×#1939C2
socleC23×C61C23×C4
prime radicalC23×D61C22×C4
cosocleC22×C4×C61C24
hypercenter#1939D61
hypocenterC61#22287
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ #1939

Complements

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

NH ≅ G/Ntype
2. #21446754C2semidirect
3. #21446753C2semidirect
4. #21446534C2semidirect
5. #21446379C2semidirect
6. #21446378C2semidirect
7. #21446376C2semidirect
8. #21446373C2semidirect
15. C61×#1939C2semidirect
72. C2×C8×D61Q8semidirect
73. #14519755C8semidirect
154. C2×Dic122C2×C8semidirect
168. C4⋊C4×C61C2×C8semidirect
184. #11758196C2×C8semidirect
185. #11758196C2×C8semidirect
312. #1939D61semidirect
374. C61#22287semidirect
505. C4⋊C4C8×D61semidirect

Maximal subgroups

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

subgroupindex
C4×#145197552
#214467542
#214467532
#214465342
#214463792
#214463782
#214463762
#214463732
#214459952
#214457962
C8×#117581982
#214455892
#336×D612
C61×#19392
#2228761