Finite Group Databasebeta

← #30610554

Subgroups — #30610554

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#21461143C212yesmaximal, complemented
4#21461140C212yesmaximal, complemented
5#21461133C214yesmaximal, complemented
6#21461116C214yesmaximal, complemented
7#21459116C211yesmaximal, 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—
22#16840612C2218yescomplemented
23#16840610C2218yescomplemented
24#16840317C411yescharacteristic
25#16840317C2211yescharacteristic
26#16840313C411yescharacteristic
27#16840313C2211yescharacteristic
28#16840312C411yescharacteristic
29#16840299—21no—
30#16840299—21no—
31#16840298—21no—
32#16840298—21no—
33#16840297C412yes—
34#16840297C2212yes—
35#16840275—21no—
36#16840275—21no—
37#16840274C412yes—
38C4×#11758226—21no—
39C4×#11758226—21no—
40#16840230C2212yes—
41C2×C42×D61C2211yescharacteristic
42C61×#533C411yescharacteristic
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×#11758226—21no—
58C2×#11758226—21no—
59C2×#11758222C2×C412yes—
60C22×C4×D61C2×C411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61D411yescharacteristic
63#14519777C2×C411yescharacteristic
64#14519774C2×C411yescharacteristic
65#14519772—22no—
66#14519772—22no—
67#14519768—21no—
68#14519768—21no—
69#14519766—21no—
70#14519766—21no—
71#14519765C2×C412yes—
72#14519762—22no—
73#14519762—22no—
74#14519761—22no—
75#14519757C2×C412yes—
76#14519757C2×C412yes—
77#14519753—21no—
78#14519753—21no—
79#14519752—21no—
80#14519752—21no—
81#14519752C2×C412yes—
82#14519752C2×C412yes—
83#14519751—21no—
84#14519751—21no—
85#14519747—22no—
86Q8×Dic61—21no—
87Q8×Dic61—21no—
88#14519742C2×C412yes—
89#14519742C2312yes—
90C4×D244—22no—
91C4×D244—22no—
92C4×D244—21no—
93C4×D244—21no—
94#14519739—22no—
95#14519739—22no—
96#14519739—21no—
97#14519739—21no—
98#14519739C2×C412yes—
99#14519739C2312yes—
100#14519737—22no—
101#14519737—22no—
102#14519735—21no—
103#14519735—21no—
104#14519735—22no—
105#14519735—22no—
106#14519734—22no—
107C2×C4×Dic61D411yescharacteristic
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61C2311yescharacteristic
110C42×D61—22no—
111C42×D61C2×C411yescharacteristic
112C42×D61C2×C411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61C2311yescharacteristic
118C42×D61C2311yescharacteristic
119C4⋊Q8×C61C2×C411yescharacteristic
120C4⋊Q8×C61C2×C411yescharacteristic
121#125×C61C2×C412yes—
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61—21no—
124C4×Q8×C61—21no—
125C4×Q8×C61—21no—
126#117×C61C2×C412yes—
127C2×C42×C61C2×C411yescharacteristic
128C23×D61C22⋊C411yescharacteristic
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C22×C4×C61C22⋊C411yescharacteristic
132C22×C4×C61C22⋊C411yescharacteristic
133C22×C4×C61C22×C411yescharacteristic
134C2×C22⋊D61—22no—
135C22×Dic61C22⋊C411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138#11758226—44no—
139#11758226—44no—
140#11758222—28no—
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244C22×C412yes—
146C2×C4×D61—24no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—42no—
151C2×C4×D61—42no—
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×D61C22×C411yescharacteristic
160C2×C4×D61C2×D411yescharacteristic
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×Dic122C22×C412yes—
165C2×#92420—24no—
166C2×#92420—24no—
167C2×#92420—44no—
168#11758213—44no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61C22×C412yes—
174C4⋊C4×C61C22×C412yes—
175C22⋊C4×C61—22no—
176C42×C61—22no—
177C42×C61C22⋊C411yescharacteristic
178C42×C61C22⋊C411yescharacteristic
179C42×C61C22×C411yescharacteristic
180C42×C61C22×C411yescharacteristic
181#11758198—22no—
182#11758198—22no—
183#11758198—22no—
184#11758198—22no—
185#11758198—22no—
186#11758198—22no—
187#11758198—22no—
188#11758198—22no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193#11758196—22no—
194#11758196—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×C61C2×C22⋊C411yescharacteristic
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61C2×C22⋊C411yescharacteristic
215C22×D61C2×C22⋊C411yescharacteristic
216Q8×C61—42no—
217Q8×C61—42no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—41no—
222C2×C4×C61C2×C22⋊C411yescharacteristic
223C2×C4×C61C2×C22⋊C411yescharacteristic
224C2×C4×C61C2×C22⋊C411yescharacteristic
225C2×C4×C61C2×C22⋊C411yescharacteristic
226C2×C4×C61C2×C22⋊C411yescharacteristic
227C2×C4×C61#13114yes—
228C2×C4×C61C23×C411yescharacteristic
229C22⋊D61—44no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—41no—
234C2×Dic61C2×C22⋊C411yescharacteristic
235C2×Dic61C2×C22⋊C411yescharacteristic
236C2×Dic61C2×C22⋊C411yescharacteristic
237C2×Dic61C2×C22⋊C411yescharacteristic
238C2×Dic61C22×D411yescharacteristic
239C2×Dic61C22×D411yescharacteristic
240D244—24no—
241D244—42no—
242D244—44no—
243D244—44no—
244D244—42no—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—22no—
248C4×D61—22no—
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—22no—
252C4×D61—22no—
253C4×D61—41no—
254C4×D61—41no—
255C4×D61—41no—
256C4×D61—41no—
257C4×D61—44no—
258C4×D61—44no—
259C4×D61—44no—
260Dic122—24no—
261#92420—44no—
262#92420—44no—
263#17467—611nomaximal, Sylow
264C22×C61—21no—
265C22×C61—21no—
266C22×C61#40812yes—
267C22×C61C22×C22⋊C411yescharacteristic
268C2×D61—21no—
269C2×D61—21no—
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—21no—
273C2×D61—21no—
274C2×D61—44no—
275C2×D61—42no—
276C2×D61—42no—
277C2×D61—42no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—42no—
282C4×C61—42no—
283C4×C61#40812yes—
284C4×C61#42112yes—
285C4×C61C2×#13112yes—
286Dic61—21no—
287Dic61—21no—
288Dic61—21no—
289Dic61—21no—
290Dic61—21no—
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×C61C2×#40811yescharacteristic
302C2×C61#289611yescharacteristic
303C2×C61C2×#42111yescharacteristic
304D61—41no—
305D61—41no—
306D61—82no—
307#533—611no—
308#533—611no—
309#533#692311yescharacteristic
310#532—611no—
311#532—611no—
312#532—611no—
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—612no—
344C2×#41—1221no—
345C2×#41—1221no—
346C23×C4—611no—
347C23×C4—611no—
348C23×C4—611no—
349C2×M16—1222no—
350C22×C8—1222no—
351C4⋊Q8—611no—
352C4⋊Q8C2×#692311yescharacteristic
353C4⋊Q8C2×#692311yescharacteristic
354#127—611no—
355#126—1222no—
356#126—1222no—
357#125—1221no—
358#125—1221no—
359#125C2×#692312yes—
360#124—612no—
361#124—1221no—
362#124—1221no—
363#123—1222no—
364#123—1222no—
365#123—1222no—
366C22⋊Q8—21no—
367C22⋊Q8—21no—
368C22⋊Q8—612no—
369C22⋊Q8—612no—
370C22⋊Q8—1221no—
371C22⋊Q8—1221no—
372#121—612no—
373#121—612no—
374#121—1221no—
375#121—1221no—
376#121—1221no—
377#121—1221no—
378#120—1222no—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8—612no—
382C4×Q8—612no—
383C4×Q8—1221no—
384C4×Q8—1221no—
385C4×D4—612no—
386C4×D4—612no—
387C4×D4—1222no—
388C4×D4—1222no—
389C4×D4—1221no—
390C4×D4—1221no—
391C4×D4—1222no—
392C4×D4—1222no—
393C4×D4—1222no—
394C4×D4—1222no—
395C4×D4—1221no—
396C4×D4—1221no—
397#117—1222no—
398#117—1222no—
399#117—1222no—
400#117—1221no—
401#117—1221no—
402#117C2×#692312yes—
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—1228no—
430#41—2444no—
431#41—2444no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—612no—
435C2×D4—612no—
436C2×D4—1222no—
437C2×D4—1222no—
438C2×D4—1222no—
439C2×D4—1221no—
440C2×D4—1221no—
441C22×C4—611no—
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—1222no—
457C22×C4—1222no—
458C22×C4—1224no—
459C22×C4—1222no—
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—22no—
472C4⋊C4—22no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4C22×#692312yes—
480C4⋊C4C22×#692312yes—
481C22⋊C4—22no—
482C22⋊C4—1222no—
483C22⋊C4—1222no—
484C22⋊C4—1222no—
485C22⋊C4—1222no—
486C22⋊C4—1222no—
487C22⋊C4—1222no—
488C22⋊C4—1222no—
489C22⋊C4—1222no—
490C42—22no—
491C42—611no—
492C42—611no—
493C42—611no—
494C42—611no—
495C42—611no—
496C42—611no—
497C42—611no—
498C42—611no—
499C42—611no—
500C42—611no—
501C42—611no—
502C42—611no—
503C42—1222no—
504C42—1222no—
505C42C22⋊#692311yescharacteristic
506C42C22⋊#692311yescharacteristic
507C42C22×#692311yescharacteristic
508C42C22×#692311yescharacteristic
509C23—611no—
510C23—611no—
511C23—1222no—
512C23—1222no—
513C23—2441no—
514C23—2441no—
515C23C2×C22⋊#692311yescharacteristic
516Q8—42no—
517Q8—42no—
518Q8—1224no—
519D4—1224no—
520D4—2442no—
521D4—2444no—
522D4—2444no—
523D4—2444no—
524D4—2442no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—41no—
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—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2444no—
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#1451986514yes—
560C2×C4C23×#692311yescharacteristic
561C8—2444no—
562C8—2444no—
563C22—21no—
564C22—21no—
565C22—1221no—
566C22—1221no—
567C22—1221no—
568C22—1221no—
569C22—1221no—
570C22—1221no—
571C22—2444no—
572C22—2442no—
573C22—2442no—
574C22—2442no—
575C22#1684073712yes—
576C22C22×C22⋊#692311yescharacteristic
577C4—21no—
578C4—21no—
579C4—21no—
580C4—42no—
581C4—42no—
582C4—612no—
583C4—1221no—
584C4—1221no—
585C4—1221no—
586C4—1221no—
587C4—1221no—
588C4—2442no—
589C4#1684073712yes—
590C4#1684075912yes—
591C4C2×#1451986512yes—
592C2—22no—
593C2—4882no—
594C2—2441no—
595C2—2441no—
596C2C2×#1684073711yescharacteristic
597C2#2146243911yescharacteristic
598C2C2×#1684075911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#533C4
socleC22×C61C22×C22⋊C4
prime radical#21459116C2
cosocleC4×Dic61C24
hypercenter#533#6923
hypocenterC61#17467
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
1. #21462496C2semidirect
2. C2×#16840781C2semidirect
3. #21461143C2semidirect
4. #21461140C2semidirect
5. #21461133C2semidirect
6. #21461116C2semidirect
22. #16840612C22semidirect
23. #16840610C22semidirect
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
#214611432
#214611402
#214611332
#214611162
#214591162
#1746761