Finite Group Databasebeta

← #30610532

Subgroups — #30610532

Order 15616.

988 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462495C211yesmaximal, characteristic
2C2×#16840778C211yesmaximal, characteristic
3#21461141C214yesmaximal, complemented
4#21461127C214yesmaximal, complemented
5#21461123C212yesmaximal, complemented
6#21461121C212yesmaximal, complemented
7C2×#16840314C211yesmaximal, characteristic
8#16840783—21no—
9#16840782—21no—
10#16840779C2212yes—
11#16840778—21no—
12#16840778—21no—
13#16840778C2212yes—
14#16840778C2212yes—
15#16840778C2212yes—
16#16840765—21no—
17C2×C4⋊#92420C2212yes—
18#16840732—21no—
19C2×C22⋊#92420—21no—
20C4×#11758217C2212yes—
21C2×C4×#92420—21no—
22#16840612C2218yescomplemented
23Q8⋊#92420C2218yescomplemented
24#16840314—21no—
25#16840314—21no—
26#16840314—21no—
27#16840314—21no—
28#16840314C412yes—
29#16840314C412yes—
30#16840314C2212yes—
31#16840314C2212yes—
32C2×#14519776C411yescharacteristic
33C2×#14519775C411yescharacteristic
34C2×#14519767—21no—
35C2×#14519756C411yescharacteristic
36C2×#14519754—21no—
37C2×Q8×Dic61C2211yescharacteristic
38C2×#14519742—21no—
39C2×#14519742C2211yescharacteristic
40C2×#14519733—21no—
41C22×C4×Dic61C2211yescharacteristic
42C2×C22⋊Q8×C61C411yescharacteristic
43C2×#11758217—22no—
44C22×#92420—22no—
45#14519844—22no—
46C4⋊#92420—22no—
47C4⋊#92420—22no—
48C4⋊#92420C2314yes—
49C22⋊#92420—22no—
50C22⋊#92420—22no—
51C22⋊#92420—22no—
52C22⋊#92420—22no—
53#14519832—22no—
54C4×#92420—22no—
55C4×#92420—22no—
56C4×#92420C2314yes—
57C22×Dic122—21no—
58C22×Dic122C2×C411yescharacteristic
59C23×Dic61D411yescharacteristic
60C23×Dic61D411yescharacteristic
61#14519776—21no—
62#14519776—21no—
63#14519776C2×C412yes—
64#14519776C2×C412yes—
65#14519775—21no—
66#14519775—21no—
67#14519775C2×C412yes—
68#14519775C2×C412yes—
69#14519767—22no—
70#14519767—22no—
71#14519767—22no—
72#14519767—22no—
73#14519756—21no—
74#14519756—21no—
75#14519756C2×C412yes—
76#14519756C2×C412yes—
77#14519754—22no—
78#14519754—22no—
79#14519754—22no—
80#14519754—22no—
81Q8×Dic61—21no—
82Q8×Dic61—21no—
83Q8×Dic61C2×C412yes—
84Q8×Dic61C2312yes—
85#14519742—22no—
86#14519742—22no—
87#14519742—22no—
88#14519742—22no—
89#14519742—21no—
90#14519742—21no—
91#14519742C2×C412yes—
92#14519742C2312yes—
93#14519733—22no—
94#14519733—22no—
95#14519733—22no—
96#14519733—22no—
97C2×C4⋊Dic61—21no—
98C2×C4⋊Dic61C2×C411yescharacteristic
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196—21no—
103C2×#11758196—21no—
104C2×C22⋊Dic61—21no—
105C2×C4×Dic61—21no—
106C2×C4×Dic61—21no—
107C2×C4×Dic61C2×C411yescharacteristic
108C2×C4×Dic61C2×C411yescharacteristic
109C2×C4×Dic61C2×C411yescharacteristic
110C2×C4×Dic61D411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C2×C4×Dic61D411yescharacteristic
114C2×C4×Dic61D411yescharacteristic
115C2×C4×Dic61D411yescharacteristic
116C2×C4×Dic61C2311yescharacteristic
117C2×C4×Dic61C2311yescharacteristic
118C2×C4×Dic61C2311yescharacteristic
119C22×Q8×C61C2×C411yescharacteristic
120C23×C4×C61C2×C411yescharacteristic
121C22⋊Q8×C61—21no—
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61C2×C412yes—
124C22⋊Q8×C61C2×C412yes—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61C2×C411yescharacteristic
127C2×C22⋊C4×C61—21no—
128C24×C61C22⋊C411yescharacteristic
129C2×Q8×C61—21no—
130C2×Q8×C61—24no—
131C2×Q8×C61C22×C412yes—
132C22×C4×C61—24no—
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×C61C22×C411yescharacteristic
141C22×Dic61—24no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—42no—
145C22×Dic61C22⋊C411yescharacteristic
146C22×Dic61C22⋊C411yescharacteristic
147C22×Dic61C22⋊C411yescharacteristic
148C22×Dic61C22⋊C411yescharacteristic
149C22×Dic61C22⋊C411yescharacteristic
150C22×Dic61C22⋊C411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C2×Dic122—24no—
158C2×Dic122—21no—
159C2×Dic122—22no—
160C2×Dic122—22no—
161C2×Dic122—44no—
162C2×Dic122C22×C412yes—
163#11758217—44no—
164C2×#92420—24no—
165C2×#92420—24no—
166C2×#92420—44no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61C22×C412yes—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊Dic61—22no—
174C22⋊Dic61—22no—
175C4⋊Dic61—21no—
176C4⋊Dic61—22no—
177C4⋊Dic61—22no—
178C4⋊Dic61C22×C412yes—
179#11758196—22no—
180#11758196—22no—
181#11758196—22no—
182#11758196—22no—
183#11758196—22no—
184#11758196—22no—
185#11758196—22no—
186#11758196—22no—
187#11758196—22no—
188#11758196—22no—
189C4×Dic61—22no—
190C4×Dic61—22no—
191C4×Dic61—22no—
192C4×Dic61—22no—
193C4×Dic61C22⋊C411yescharacteristic
194C4×Dic61C22⋊C411yescharacteristic
195C4×Dic61C22⋊C411yescharacteristic
196C4×Dic61C22⋊C411yescharacteristic
197C4×Dic61C22⋊C411yescharacteristic
198C4×Dic61C22⋊C411yescharacteristic
199C4×Dic61C22×C411yescharacteristic
200C4×Dic61C22×C411yescharacteristic
201C4×Dic61C22×C411yescharacteristic
202C4×Dic61C2×D411yescharacteristic
203C4×Dic61C2×D411yescharacteristic
204C4×Dic61C2×D411yescharacteristic
205C4×Dic61C2×D411yescharacteristic
206C4×Dic61C2×D411yescharacteristic
207C4×Dic61C2×D411yescharacteristic
208C4×Dic61C2411yescharacteristic
209C23×C61—22no—
210C23×C61—22no—
211C23×C61—41no—
212C23×C61C2×C22⋊C411yescharacteristic
213C23×C61C2×C22⋊C411yescharacteristic
214C23×C61C2×C22⋊C411yescharacteristic
215Q8×C61—24no—
216Q8×C61—42no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61C2×C22⋊C411yescharacteristic
227C2×C4×C61C2×C22⋊C411yescharacteristic
228C2×C4×C61C2×C22⋊C411yescharacteristic
229C2×C4×C61#13114yes—
230C2×C4×C61C23×C411yescharacteristic
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61C2×C22⋊C411yescharacteristic
247C2×Dic61C2×C22⋊C411yescharacteristic
248C2×Dic61C2×C22⋊C411yescharacteristic
249C2×Dic61C2×C22⋊C411yescharacteristic
250C2×Dic61C2×C22⋊C411yescharacteristic
251C2×Dic61C2×C22⋊C411yescharacteristic
252C2×Dic61#13714yes—
253C2×Dic61C22×D411yescharacteristic
254C2×Dic61C22×D411yescharacteristic
255Dic122—24no—
256Dic122—44no—
257Dic122—42no—
258Dic122—44no—
259#92420—44no—
260#92420—44no—
261#17464—611nomaximal, Sylow
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—44no—
270C22×C61#40812yes—
271C22×C61#41912yes—
272C22×C61C22×C22⋊C411yescharacteristic
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—42no—
276C4×C61—42no—
277C4×C61#40812yes—
278C4×C61C2×#13112yes—
279Dic61—21no—
280Dic61—21no—
281Dic61—21no—
282Dic61—21no—
283Dic61—21no—
284Dic61—42no—
285Dic61—42no—
286Dic61#41912yes—
287Dic61#42112yes—
288Dic61C2×#13712yes—
289C2×#533—611no—
290#2920—611no—
291C2×#431—611no—
292#1485—614no—
293#1481—614no—
294#1479—612no—
295#1479—612no—
296C2×C61—22no—
297C2×C61—22no—
298C2×C61—41no—
299C2×C61C2×#40811yescharacteristic
300C2×C61C2×#41911yescharacteristic
301C2×C61C2×#42111yescharacteristic
302#533—612no—
303#533—612no—
304#533—612no—
305#533—612no—
306#533—1221no—
307#533—1221no—
308#533—1221no—
309#533—1221no—
310C2×C4⋊Q8—611no—
311C2×C4⋊Q8—611no—
312C2×#125—1221no—
313C2×C22⋊Q8—611no—
314C2×C22⋊Q8—1221no—
315C2×C22⋊Q8#692311yescharacteristic
316C2×C4×Q8—611no—
317C2×C4×Q8—611no—
318C2×C4×Q8—1221no—
319C2×#117—1221no—
320C22×C42—611no—
321#432—1221no—
322#432—1221no—
323#431—612no—
324#431—612no—
325#431—612no—
326#431—612no—
327#431—1221no—
328#431—1221no—
329#423—1221no—
330C2×C4⋊C8—612no—
331#407—1221no—
332C2×C22⋊C8—1221no—
333C4×M16—612no—
334C2×C4×C8—1221no—
335Q8⋊C8—618no—
336Q8⋊C8—618no—
337C61#1746411yescharacteristic, complemented, Sylow
338C22×Q8—611no—
339C22×Q8—1221no—
340C22×Q8C2×#692311yescharacteristic
341C23×C4—611no—
342C23×C4—611no—
343C23×C4C2×#692311yescharacteristic
344C2×M16—1222no—
345C22×C8—1222no—
346C4⋊Q8—612no—
347C4⋊Q8—612no—
348C4⋊Q8—612no—
349C4⋊Q8—612no—
350C4⋊Q8—1221no—
351C4⋊Q8—1221no—
352C4⋊Q8—1221no—
353C4⋊Q8—1221no—
354#125—1222no—
355#125—1222no—
356#125—1222no—
357#125—1222no—
358C22⋊Q8—21no—
359C22⋊Q8—21no—
360C22⋊Q8—612no—
361C22⋊Q8—612no—
362C22⋊Q8—1222no—
363C22⋊Q8—1222no—
364C22⋊Q8—1222no—
365C22⋊Q8—1222no—
366C22⋊Q8—1221no—
367C22⋊Q8—1221no—
368C22⋊Q8C2×#692312yes—
369C22⋊Q8C2×#692312yes—
370C4×Q8—612no—
371C4×Q8—612no—
372C4×Q8—612no—
373C4×Q8—612no—
374C4×Q8—1221no—
375C4×Q8—1221no—
376C4×Q8—1222no—
377C4×Q8—1222no—
378C4×Q8—1222no—
379C4×Q8—1222no—
380C4×Q8—1221no—
381C4×Q8—1221no—
382#117—1222no—
383#117—1222no—
384#117—1222no—
385#117—1222no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4C2×#692311yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1221no—
397C2×C42—611no—
398C2×C42—611no—
399C2×C42—611no—
400C2×C42—611no—
401C2×C42—611no—
402C2×C42—611no—
403C2×C42—611no—
404C2×C42—611no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—611no—
409C2×C42—1221no—
410C2×C42—1221no—
411C4⋊C8—614no—
412C4⋊C8—1222no—
413C4⋊C8—1222no—
414C4⋊C8—1222no—
415C22⋊C8—1222no—
416C22⋊C8—1222no—
417C22⋊C8—1222no—
418C22⋊C8—1222no—
419#97—1222no—
420C4×C8—614no—
421C4×C8—1222no—
422C4×C8—1222no—
423C24C22⋊#692311yescharacteristic
424C2×Q8—21no—
425C2×Q8—24no—
426C2×Q8—612no—
427C2×Q8—1222no—
428C2×Q8—1222no—
429C2×Q8—1221no—
430C2×Q8—1224no—
431C2×Q8—2444no—
432C2×Q8C22×#692312yes—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—1224no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—2442no—
452C22×C4C22⋊#692311yescharacteristic
453C22×C4C22⋊#692311yescharacteristic
454C22×C4C22⋊#692311yescharacteristic
455C22×C4C22×#692311yescharacteristic
456C22×C4C22×#692311yescharacteristic
457C22×C4C22×#692311yescharacteristic
458M16—2444no—
459C2×C8—1224no—
460C2×C8—1224no—
461C2×C8—2444no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—612no—
466C4⋊C4—1222no—
467C4⋊C4—1222no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1221no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4C22×#692312yes—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—1222no—
483C22⋊C4—1222no—
484C42—611no—
485C42—611no—
486C42—611no—
487C42—611no—
488C42—611no—
489C42—611no—
490C42—611no—
491C42—611no—
492C42—611no—
493C42—611no—
494C42—611no—
495C42—611no—
496C42—611no—
497C42—611no—
498C42—611no—
499C42—611no—
500C42—1222no—
501C42—1222no—
502C42—1222no—
503C42—1222no—
504C23—22no—
505C23—22no—
506C23—41no—
507C23C2×C22⋊#692311yescharacteristic
508C23C2×C22⋊#692311yescharacteristic
509C23C2×C22⋊#692311yescharacteristic
510Q8—24no—
511Q8—42no—
512Q8—1224no—
513Q8—2444no—
514Q8—2444no—
515Q8—2442no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—614no—
526C2×C4—611no—
527C2×C4—611no—
528C2×C4—611no—
529C2×C4—611no—
530C2×C4—611no—
531C2×C4—611no—
532C2×C4—611no—
533C2×C4—611no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4C2×C22⋊#692311yescharacteristic
550C2×C4C2×C22⋊#692311yescharacteristic
551C2×C4C2×C22⋊#692311yescharacteristic
552C2×C4#1451986414yes—
553C2×C4C23×#692311yescharacteristic
554C8—2444no—
555C8—2444no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—44no—
564C22#1684073512yes—
565C22#1684075512yes—
566C22C22×C22⋊#692311yescharacteristic
567C4—21no—
568C4—21no—
569C4—42no—
570C4—42no—
571C4—612no—
572C4—612no—
573C4—612no—
574C4—1221no—
575C4—1221no—
576C4—1221no—
577C4—1221no—
578C4—1221no—
579C4—2442no—
580C4—2442no—
581C4#1684073512yes—
582C4C2×#1451986412yes—
583C2—22no—
584C2—22no—
585C2—41no—
586C2C2×#1684073511yescharacteristic
587C2C2×#1684075511yescharacteristic
588C2D61⋊#42111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC4×Dic61C24
hypercenterC2×C22⋊Q8#6923
hypocenterC61#17464
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #21461141C2semidirect
4. #21461127C2semidirect
5. #21461123C2semidirect
6. #21461121C2semidirect
22. #16840612C22semidirect
23. Q8⋊#92420C22semidirect
337. C61#17464semidirect

Maximal subgroups

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

subgroupindex
#214624952
C2×#168407782
#214611412
#214611272
#214611232
#214611212
C2×#168403142
#1746461