Finite Group Databasebeta

← #30610551

Subgroups — #30610551

Order 15616.

1010 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462496C211yesmaximal, characteristic, complemented
2C2×#16840781C211yesmaximal, characteristic, complemented
3#21461142C212yesmaximal, complemented
4#21461139C212yesmaximal, complemented
5#21461131C214yesmaximal, complemented
6#21461114C214yesmaximal, complemented
7#21459114C211yesmaximal, characteristic
8#16840784—21no—
9#16840783—21no—
10#16840781—21no—
11#16840781—21no—
12#16840781C2212yes—
13#16840781C2212yes—
14#16840781C2212yes—
15#16840778C2212yes—
16#16840766—21no—
17C2×C4⋊#92420C2212yes—
18#16840733—21no—
19C2×#14519834—21no—
20C4×#11758213C2212yes—
21C2×C4×#92420—21no—
22Q8⋊#92420C2218yescomplemented
23D4⋊#92420C2218yescomplemented
24#16840319C411yescharacteristic
25#16840317C411yescharacteristic
26#16840317C2211yescharacteristic
27#16840310C411yescharacteristic
28#16840310C2211yescharacteristic
29#16840294—21no—
30#16840294—21no—
31#16840293—21no—
32#16840293—21no—
33#16840291C412yes—
34#16840291C2212yes—
35#16840272—21no—
36#16840272—21no—
37#16840269C412yes—
38#41×Dic61C2212yes—
39C4×#11758224—21no—
40C4×#11758224—21no—
41C2×C42×D61C2211yescharacteristic
42C61×#532C411yescharacteristic
43C2×#11758213—22no—
44C22×#92420—22no—
45C4⋊#92420—22no—
46C4⋊#92420—22no—
47C4⋊#92420—22no—
48C4⋊#92420C2314yes—
49#14519834—22no—
50#14519834—22no—
51#14519834—22no—
52C22⋊#92420—22no—
53#14519832—22no—
54C4×#92420—22no—
55C4×#92420—22no—
56C4×#92420C2314yes—
57C2×#11758224—21no—
58C2×#11758224—21no—
59C22×C4×D61C2×C411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62#14519777C2×C411yescharacteristic
63#14519777C2×C411yescharacteristic
64#14519770—22no—
65#14519770—22no—
66#14519769C2×C412yes—
67#14519768—21no—
68#14519768—21no—
69#14519764—21no—
70#14519764—21no—
71#14519759—22no—
72#14519759—22no—
73#14519758—22no—
74#14519757—21no—
75#14519757—21no—
76#14519756—21no—
77#14519756—21no—
78#14519753C2×C412yes—
79#14519753C2×C412yes—
80#14519748—21no—
81#14519748—21no—
82#14519748C2×C412yes—
83#14519748C2×C412yes—
84#14519745—22no—
85Q8×Dic61C2×C412yes—
86Q8×Dic61C2312yes—
87C4×Dic122—21no—
88C4×Dic122—21no—
89#14519742—21no—
90#14519742—21no—
91D4×Dic61—22no—
92D4×Dic61—22no—
93D4×Dic61—21no—
94D4×Dic61—21no—
95D4×Dic61C2×C412yes—
96D4×Dic61C2312yes—
97C4×C22⋊D61—22no—
98C4×C22⋊D61—22no—
99#14519737—22no—
100#14519737—22no—
101#14519736—22no—
102#14519736C2×C412yes—
103#14519735—21no—
104#14519735—21no—
105#14519733—22no—
106#14519733—22no—
107C2×C4×Dic61—22no—
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C2×C4×Dic61C2311yescharacteristic
111C42×D61C2×C411yescharacteristic
112C42×D61C2×C411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61C2311yescharacteristic
118C42×D61C2311yescharacteristic
119C2×#41×C61C2×C412yes—
120C4⋊Q8×C61C2×C411yescharacteristic
121#127×C61C2×C411yescharacteristic
122#124×C61C2×C412yes—
123#121×C61—21no—
124#121×C61—21no—
125C4×D4×C61—21no—
126C4×D4×C61—21no—
127C2×C42×C61C2×C411yescharacteristic
128C23×D61C22⋊C411yescharacteristic
129#41×C61—28no—
130C2×Q8×C61C22×C412yes—
131C2×D4×C61—22no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61C22×C412yes—
135C22×C4×C61—22no—
136C22×C4×C61C22⋊C411yescharacteristic
137C22×C4×C61C22⋊C411yescharacteristic
138C22×C4×C61C22×C411yescharacteristic
139C2×C22⋊D61—22no—
140C2×C22⋊D61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61C22⋊C411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146#11758224—44no—
147#11758224—44no—
148C2×C4×D61—24no—
149C2×C4×D61—42no—
150C2×C4×D61—42no—
151C2×C4×D61C22⋊C411yescharacteristic
152C2×C4×D61C22⋊C411yescharacteristic
153C2×C4×D61C22⋊C411yescharacteristic
154C2×C4×D61C22⋊C411yescharacteristic
155C2×C4×D61C22⋊C411yescharacteristic
156C2×C4×D61C22⋊C411yescharacteristic
157C2×C4×D61C22×C411yescharacteristic
158C2×C4×D61C22×C411yescharacteristic
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61C2×D411yescharacteristic
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61C2×D411yescharacteristic
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×#92420—24no—
166C2×#92420—24no—
167C2×#92420—44no—
168#11758213—44no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C42×C61C22⋊C411yescharacteristic
174C42×C61C22⋊C411yescharacteristic
175C42×C61C22×C411yescharacteristic
176C42×C61C22×C411yescharacteristic
177C22⋊Dic61—22no—
178C22⋊Dic61—22no—
179C22⋊Dic61—22no—
180C22⋊Dic61—22no—
181C22⋊Dic61—22no—
182#11758198—22no—
183#11758198—22no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—22no—
187C4⋊Dic61—22no—
188C4⋊Dic61C22×C412yes—
189C4⋊Dic61C22×C412yes—
190#11758196—22no—
191#11758196—22no—
192#11758196—22no—
193#11758196—22no—
194C4×Dic61—22no—
195C4×Dic61—22no—
196C4×Dic61—22no—
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×C61C2×C22⋊C411yescharacteristic
211C22×D61—22no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61C2×C22⋊C411yescharacteristic
215C22×D61C2×C22⋊C411yescharacteristic
216Q8×C61—24no—
217D4×C61—24no—
218D4×C61—42no—
219D4×C61—44no—
220D4×C61—42no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—41no—
224C2×C4×C61—44no—
225C2×C4×C61C2×C22⋊C411yescharacteristic
226C2×C4×C61C2×C22⋊C411yescharacteristic
227C2×C4×C61C2×C22⋊C411yescharacteristic
228C2×C4×C61C2×C22⋊C411yescharacteristic
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C2×C4×C61#13114yes—
231C2×C4×C61C23×C411yescharacteristic
232C22⋊D61—44no—
233C22⋊D61—44no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—41no—
239C2×Dic61—44no—
240C2×Dic61—44no—
241C2×Dic61C2×C22⋊C411yescharacteristic
242C2×Dic61C2×C22⋊C411yescharacteristic
243C2×Dic61C2×C22⋊C411yescharacteristic
244C2×Dic61C2×C22⋊C411yescharacteristic
245C2×Dic61C22×D411yescharacteristic
246C2×Dic61C22×D411yescharacteristic
247C4×D61—22no—
248C4×D61—22no—
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—22no—
252C4×D61—22no—
253C4×D61—22no—
254C4×D61—22no—
255C4×D61—41no—
256C4×D61—41no—
257C4×D61—41no—
258C4×D61—41no—
259Dic122—42no—
260Dic122—42no—
261#92420—44no—
262#92420—44no—
263#17467—611nomaximal, Sylow
264C22×C61—21no—
265C22×C61—21no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61#40812yes—
270C22×C61C22×C22⋊C411yescharacteristic
271C2×D61—21no—
272C2×D61—21no—
273C2×D61—21no—
274C2×D61—21no—
275C2×D61—21no—
276C2×D61—21no—
277C2×D61—44no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—42no—
282C4×C61#40812yes—
283C4×C61#42112yes—
284C4×C61C2×#13112yes—
285Dic61—21no—
286Dic61—21no—
287Dic61—21no—
288Dic61—21no—
289Dic61—21no—
290Dic61—42no—
291Dic61—42no—
292Dic61#57512yes—
293#3444—611no—
294#2920—611no—
295C2×#431—611no—
296#1485—612no—
297#1484—612no—
298#1482—614no—
299#1477—614no—
300C2×C61—22no—
301C2×C61—82no—
302C2×C61C2×#40811yescharacteristic
303C2×C61#289611yescharacteristic
304C2×C61C2×#42111yescharacteristic
305D61—41no—
306D61—41no—
307#533—611no—
308#533—611no—
309#533—611no—
310#532—611no—
311#532—611no—
312#532#692311yescharacteristic
313#529—612no—
314#529—612no—
315#529—1221no—
316#529—1221no—
317#529—1221no—
318#529—1221no—
319#525—612no—
320#525—1221no—
321#525—1221no—
322C4×#41—612no—
323C4×#41—1221no—
324C4×#41—1221no—
325C22×C42—611no—
326#432—1221no—
327#432—1221no—
328#431—612no—
329#431—612no—
330#431—612no—
331#431—612no—
332#431—1221no—
333#431—1221no—
334#423—1221no—
335C2×C4⋊C8—612no—
336#407—1221no—
337C2×C22⋊C8—1221no—
338C4×M16—612no—
339C2×C4×C8—1221no—
340Q8⋊C8—618no—
341D4⋊C8—618no—
342C61#1746711yescharacteristic, complemented, Sylow
343C2×#41—1221no—
344C2×#41—1221no—
345C2×#41C2×#692312yes—
346C23×C4—611no—
347C23×C4—611no—
348C23×C4—611no—
349C2×M16—1222no—
350C22×C8—1222no—
351C4⋊Q8—611no—
352C4⋊Q8—611no—
353C4⋊Q8C2×#692311yescharacteristic
354#127C2×#692311yescharacteristic
355#126—1222no—
356#126—1222no—
357#125—612no—
358#125—1221no—
359#125—1221no—
360#124—1221no—
361#124—1221no—
362#124C2×#692312yes—
363#123—1222no—
364#123—1222no—
365#123—1222no—
366C22⋊Q8—612no—
367C22⋊Q8—612no—
368C22⋊Q8—1221no—
369C22⋊Q8—1221no—
370C22⋊Q8—1221no—
371C22⋊Q8—1221no—
372#121—21no—
373#121—21no—
374#121—612no—
375#121—612no—
376#121—1221no—
377#121—1221no—
378#120—1222no—
379C4×Q8—612no—
380C4×Q8—612no—
381C4×Q8—1221no—
382C4×Q8—1221no—
383C4×Q8—1221no—
384C4×Q8—1221no—
385C4×D4—21no—
386C4×D4—21no—
387C4×D4—612no—
388C4×D4—612no—
389C4×D4—1221no—
390C4×D4—1221no—
391C4×D4—1222no—
392C4×D4—1222no—
393C4×D4—1222no—
394C4×D4—1222no—
395C4×D4—1222no—
396C4×D4—1222no—
397#117—612no—
398#117—1222no—
399#117—1222no—
400#117—1222no—
401#117—1221no—
402#117—1221no—
403C2×C42—611no—
404C2×C42—611no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—611no—
414C2×C42—1222no—
415C2×C42C2×#692311yescharacteristic
416C4⋊C8—614no—
417C4⋊C8—1222no—
418C4⋊C8—1222no—
419C4⋊C8—1222no—
420C22⋊C8—1222no—
421C22⋊C8—1222no—
422C22⋊C8—1222no—
423C22⋊C8—1222no—
424#97—1222no—
425C4×C8—614no—
426C4×C8—1222no—
427C4×C8—1222no—
428C24—611no—
429#41—28no—
430#41—2444no—
431#41—2444no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×Q8C22×#692312yes—
435C2×D4—22no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—1222no—
439C2×D4—1222no—
440C2×D4C22×#692312yes—
441C22×C4—22no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1224no—
460C22×C4—2442no—
461C22×C4—2442no—
462C22×C4C22⋊#692311yescharacteristic
463C22×C4C22⋊#692311yescharacteristic
464C22×C4C22×#692311yescharacteristic
465M16—2444no—
466C2×C8—1224no—
467C2×C8—1224no—
468C2×C8—2444no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—612no—
472C4⋊C4—612no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—1222no—
484C22⋊C4—1222no—
485C22⋊C4—1222no—
486C22⋊C4—1222no—
487C22⋊C4—1222no—
488C22⋊C4—1222no—
489C22⋊C4—1222no—
490C42—611no—
491C42—611no—
492C42—611no—
493C42—611no—
494C42—611no—
495C42—611no—
496C42—611no—
497C42—611no—
498C42—611no—
499C42—611no—
500C42—611no—
501C42—611no—
502C42—1222no—
503C42—1222no—
504C42—1222no—
505C42C22⋊#692311yescharacteristic
506C42C22⋊#692311yescharacteristic
507C42C22×#692311yescharacteristic
508C42C22×#692311yescharacteristic
509C23—22no—
510C23—611no—
511C23—611no—
512C23—1222no—
513C23—2441no—
514C23—2441no—
515C23C2×C22⋊#692311yescharacteristic
516Q8—24no—
517Q8—2442no—
518Q8—2442no—
519D4—24no—
520D4—42no—
521D4—44no—
522D4—42no—
523D4—2444no—
524D4—2444no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—611no—
530C2×C4—611no—
531C2×C4—611no—
532C2×C4—611no—
533C2×C4—611no—
534C2×C4—611no—
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—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2444no—
552C2×C4—2441no—
553C2×C4—2444no—
554C2×C4C2×C22⋊#692311yescharacteristic
555C2×C4C2×C22⋊#692311yescharacteristic
556C2×C4C2×C22⋊#692311yescharacteristic
557C2×C4C2×C22⋊#692311yescharacteristic
558C2×C4C2×C22⋊#692311yescharacteristic
559C2×C4#1451986414yes—
560C2×C4C23×#692311yescharacteristic
561C8—2444no—
562C8—2444no—
563C22—21no—
564C22—21no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—1221no—
569C22—1221no—
570C22—1221no—
571C22—1221no—
572C22—1221no—
573C22—1221no—
574C22—2444no—
575C22#1684073612yes—
576C22C22×C22⋊#692311yescharacteristic
577C4—21no—
578C4—21no—
579C4—21no—
580C4—42no—
581C4—612no—
582C4—1221no—
583C4—1221no—
584C4—1221no—
585C4—1221no—
586C4—1221no—
587C4—2442no—
588C4—2442no—
589C4#1684073612yes—
590C4#1684075912yes—
591C4C2×#1451986412yes—
592C2—22no—
593C2—82no—
594C2—2441no—
595C2—2441no—
596C2C2×#1684073611yescharacteristic
597C2#2146243911yescharacteristic
598C2C2×#1684075911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#532C4
socleC22×C61C22×C22⋊C4
prime radical#21459114C2
cosocleC4×Dic61C24
hypercenter#532#6923
hypocenterC61#17467
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #21462496C2semidirect
2. C2×#16840781C2semidirect
3. #21461142C2semidirect
4. #21461139C2semidirect
5. #21461131C2semidirect
6. #21461114C2semidirect
22. Q8⋊#92420C22semidirect
23. D4⋊#92420C22semidirect
342. C61#17467semidirect

Maximal subgroups

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

subgroupindex
#214624962
C2×#168407812
#214611422
#214611392
#214611312
#214611142
#214591142
#1746761