Finite Group Databasebeta

← #29647231

Subgroups — #29647231

Order 15616.

974 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459913C211yesmaximal, characteristic
2#21454909C211yesmaximal, characteristic, complemented
3#21448183C211yesmaximal, characteristic, complemented
4#21444341C212yesmaximal
5#21444332C211yesmaximal, characteristic, complemented
6#21444329C212yesmaximal, complemented
7#21444317C211yesmaximal, characteristic, complemented
8#21444316C212yesmaximal, complemented
9#21444305C211yesmaximal, characteristic
10#21444302C212yesmaximal
11C61×#1509C211yesmaximal, characteristic
12#16840491C411yescharacteristic, complemented
13#16840490C411yescharacteristic, complemented
14#16840489C412yescomplemented
15#16840482C411yescharacteristic, complemented
16#16840481C412yescomplemented
17#16840452—21no—
18#16840449—22no—
19D4×Dic122—21no—
20D4×Dic122C411yescharacteristic, complemented
21#16840374—21no—
22#16840369—22no—
23#16840319C2211yescharacteristic
24#16840314C2212yes—
25#16840293C2212yes—
26#16840239—21no—
27C2×C4×Dic122C2211yescharacteristic
28C2×#14519742—21no—
29#16839890—22no—
30#16839889—21no—
31#16839886—21no—
32#16839816C2212yes—
33#16839815C2211yescharacteristic
34#16839812C2212yes—
35C4⋊C4×Dic61—21no—
36C4×#11758196—22no—
37#16839613C2212yes—
38#16839608C2211yescharacteristic
39#16839607C2212yes—
40#16839500C2212yes—
41#16839500C2212yes—
42#16839499C2211yescharacteristic
43#16839499C2211yescharacteristic
44#16839498C2212yes—
45#16839498C2212yes—
46#16839496C2211yescharacteristic
47#16839496C2211yescharacteristic
48C61×#529C2211yescharacteristic
49C61×#432C2211yescharacteristic
50C4×C4⋊C4×C61C2211yescharacteristic
51C61×#328C2212yes—
52C61×#327C2212yes—
53C22×Dic122—21no—
54C22×Dic122D411yescharacteristic
55#14519777—21no—
56#14519777C2×C411yescharacteristic
57#14519777C2×C411yescharacteristic
58#14519776—22no—
59#14519776C2×C412yes—
60#14519775C2×C412yes—
61#14519770—22no—
62#14519770—22no—
63#14519770—22no—
64#14519770—22no—
65#14519769—21no—
66#14519769—21no—
67#14519768—22no—
68#14519768C2×C412yes—
69#14519767—22no—
70#14519767—22no—
71#14519767—22no—
72#14519767—22no—
73#14519764C2×C412yes—
74#14519758—22no—
75#14519756—21no—
76#14519756—21no—
77#14519756—21no—
78#14519756C2312yes—
79#14519754—22no—
80#14519754—22no—
81#14519754—22no—
82Q8×Dic61C2×C412yes—
83C4×Dic122—21no—
84C4×Dic122—21no—
85C4×Dic122—21no—
86C4×Dic122C2×C412yes—
87C4×Dic122C2×C412yes—
88C4×Dic122C2×C411yescharacteristic
89C4×Dic122C2×C411yescharacteristic
90C4×Dic122D411yescharacteristic
91C4×Dic122D412yes—
92C4×Dic122D411yescharacteristic
93#14519742—28no—
94#14519742—22no—
95#14519742—22no—
96#14519742C2×C412yes—
97#14519742C2×C412yes—
98D4×Dic61—22no—
99D4×Dic61C2×C412yes—
100#14519736—21no—
101#14519736—21no—
102#14519733—22no—
103#14519733—22no—
104#14519733—22no—
105C2×C4⋊Dic61D411yescharacteristic
106C2×#11758196—22no—
107C2×#11758196D412yes—
108C2×C4×Dic61—21no—
109C2×C4×Dic61—22no—
110C2×C4×Dic61—22no—
111C2×C4×Dic61C2312yes—
112C4⋊#92418—21no—
113#14519706—22no—
114#14519706—22no—
115C22⋊#92418C2311yescharacteristic
116C22⋊#92418C2311yescharacteristic
117#14519687—22no—
118C4×#92418—21no—
119C8×Dic61—22no—
120#14519684—21no—
121#14519684—22no—
122#14519684C2311yescharacteristic
123#14519684C2311yescharacteristic
124#126×C61—21no—
125#125×C61C2×C411yescharacteristic
126#124×C61C2×C411yescharacteristic
127#123×C61—21no—
128C22⋊Q8×C61C2311yescharacteristic
129#121×C61C2311yescharacteristic
130C4×Q8×C61C2×C411yescharacteristic
131C4×D4×C61—21no—
132C4×D4×C61C2×C411yescharacteristic
133#117×C61—21no—
134C2×C4⋊C4×C61—21no—
135C2×C42×C61—21no—
136C2×C42×C61C2311yescharacteristic
137C4⋊C8×C61—21no—
138C22⋊C8×C61C2312yes—
139#97×C61—21no—
140#95×C61C2312yes—
141C2×Q8×C61C22×C411yescharacteristic
142C2×D4×C61—21no—
143C2×D4×C61C22×C411yescharacteristic
144C22×C4×C61—22no—
145C22×C4×C61—21no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—22no—
152C22×Dic61C2×D412yes—
153C22×Dic61C2×D412yes—
154C2×Dic122—24no—
155C2×Dic122—21no—
156C2×Dic122—22no—
157C2×Dic122—44no—
158C2×Dic122C22⋊C412yes—
159C2×Dic122C22⋊C411yescharacteristic
160C2×Dic122C22⋊C411yescharacteristic
161C2×Dic122C22×C412yes—
162C2×C8×C61—22no—
163C4⋊C4×C61—24no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61C22×C411yescharacteristic
169C4⋊C4×C61C22×C411yescharacteristic
170C22⋊C4×C61—21no—
171C22⋊C4×C61—21no—
172C22⋊C4×C61—21no—
173C22⋊C4×C61—21no—
174C22⋊C4×C61—21no—
175C42×C61—21no—
176C42×C61—81no—
177C42×C61C22×C411yescharacteristic
178C42×C61C22×C411yescharacteristic
179C42×C61C2×D411yescharacteristic
180C42×C61C2×D411yescharacteristic
181C22⋊Dic61—22no—
182C22⋊Dic61—22no—
183C22⋊Dic61—22no—
184C22⋊Dic61—22no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—22no—
189C4⋊Dic61—22no—
190C4⋊Dic61C22⋊C411yescharacteristic
191C4⋊Dic61C22⋊C411yescharacteristic
192C4⋊Dic61C22⋊C412yes—
193C4⋊Dic61C22×C412yes—
194#11758196—28no—
195#11758196—22no—
196#11758196—22no—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196C22⋊C412yes—
204#11758196C22⋊C412yes—
205#11758196C22⋊C412yes—
206#11758196C22⋊C412yes—
207C4×Dic61—28no—
208C4×Dic61—24no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61—82no—
214C4×Dic61C22×C412yes—
215C4×Dic61C22×C412yes—
216C4×Dic61C2×D412yes—
217C4×Dic61C2×D412yes—
218C2×#92418—21no—
219C2×#92418—21no—
220C23×C61—21no—
221C23×C61C22×D411yescharacteristic
222Q8×C61—41no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—41no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—41no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C23×C411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—24no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—44no—
255C2×Dic61C2×C22⋊C412yes—
256C2×Dic61C2×C22⋊C412yes—
257C2×Dic61C2×C22⋊C412yes—
258C2×Dic61C2×C22⋊C412yes—
259Dic122—22no—
260Dic122—24no—
261Dic122—42no—
262Dic122—44no—
263Dic122—42no—
264Dic122#10414yes—
265C8×C61—42no—
266#92418—41no—
267#92418—41no—
268#17549—611nomaximal, Sylow
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61C22×C22⋊C411yescharacteristic
275C22×C61#57711yescharacteristic
276C22×C61#57811yescharacteristic
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—44no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—41no—
286C4×C61C2×#10412yes—
287Dic61—22no—
288Dic61—22no—
289Dic61—22no—
290Dic61—42no—
291Dic61—42no—
292Dic61—42no—
293Dic61—42no—
294Dic61C2×#10414yes—
295#3523—611no—
296#2984—611no—
297#2276—611no—
298#1509—612no—
299#1509D6111yescharacteristic
300#1506—611no—
301#1506—612no—
302#1503—611no—
303#1503—612no—
304#1500—612no—
305#1500—611no—
306C2×C61—21no—
307C2×C61—21no—
308C2×C61—81no—
309C2×C61#288411yescharacteristic
310C2×C61#289811yescharacteristic
311C2×C61C22×#10411yescharacteristic
312#557—612no—
313#557—611no—
314#557—611no—
315#556—612no—
316#556—611no—
317#552—1221no—
318#552—1222no—
319D4×Q8—611no—
320D4×Q8—1221no—
321#541—1221no—
322#541—1222no—
323#533—612no—
324#533—611no—
325#529—612no—
326#529C2×D6111yescharacteristic
327#519—1221no—
328C2×C4×Q8—611no—
329C2×C4×Q8—1221no—
330#446—1222no—
331#446—1221no—
332C8×Q8—1221no—
333#432—612no—
334#432C2×D6111yescharacteristic
335#431—611no—
336#431—612no—
337C4×C4⋊C4—612no—
338C4×C4⋊C4—1222no—
339C4×C4⋊C4—1221no—
340C4×C4⋊C4C2×D6111yescharacteristic
341#376—611no—
342#376—612no—
343#328—611no—
344#328—611no—
345#328—612no—
346#328—612no—
347#328—612no—
348#328—612no—
349#328C2×D6112yes—
350#327—611no—
351#327—611no—
352#327C2×D6112yes—
353C61#1754911yescharacteristic, complemented, Sylow
354C22×Q8—611no—
355C22×Q8—1221no—
356C4⋊Q8—612no—
357C4⋊Q8—612no—
358C4⋊Q8—611no—
359C4⋊Q8—611no—
360C4⋊Q8—1221no—
361C4⋊Q8—1222no—
362#126—21no—
363#126—1222no—
364#126—1222no—
365#126—1222no—
366#126—1222no—
367#125—612no—
368#125—1221no—
369#125—1222no—
370#125—1222no—
371#125—1222no—
372#125—1222no—
373#125—1222no—
374#125—1221no—
375#125C4×D6111yescharacteristic
376#124—612no—
377#124C4×D6111yescharacteristic
378#123—21no—
379#123—1222no—
380C22⋊Q8—612no—
381C22⋊Q8—1222no—
382C22⋊Q8—1222no—
383C22⋊Q8—1222no—
384C22⋊Q8—1221no—
385C22⋊Q8—1221no—
386C22⋊Q8—1221no—
387C22⋊Q8C22×D6111yescharacteristic
388#121C22×D6111yescharacteristic
389C4×Q8—611no—
390C4×Q8—612no—
391C4×Q8—611no—
392C4×Q8—611no—
393C4×Q8—612no—
394C4×Q8—611no—
395C4×Q8—612no—
396C4×Q8—612no—
397C4×Q8—612no—
398C4×Q8—612no—
399C4×Q8—1221no—
400C4×Q8—1221no—
401C4×Q8—1222no—
402C4×Q8—1222no—
403C4×Q8—1228no—
404C4×Q8—1221no—
405C4×Q8C4×D6111yescharacteristic
406C4×D4—21no—
407C4×D4—612no—
408C4×D4—1222no—
409C4×D4C4×D6111yescharacteristic
410#117—21no—
411#117—1222no—
412#117—1222no—
413#117—1222no—
414#117—1221no—
415#117—1221no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—612no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—1222no—
420C2×C42—21no—
421C2×C42—612no—
422C2×C42—1222no—
423C2×C42—1221no—
424C2×C42—1222no—
425C2×C42C22×D6111yescharacteristic
426C4⋊C8—21no—
427C4⋊C8—1222no—
428C4⋊C8—1222no—
429C4⋊C8—1221no—
430C22⋊C8—611no—
431C22⋊C8—611no—
432C22⋊C8C22×D6112yes—
433#97—21no—
434#97—1222no—
435C4×C8—1222no—
436C4×C8—1221no—
437#95—611no—
438#95—611no—
439#95—1222no—
440#95—1221no—
441#95C22×D6112yes—
442C2×Q8—611no—
443C2×Q8—612no—
444C2×Q8—611no—
445C2×Q8—612no—
446C2×Q8—1222no—
447C2×Q8—1224no—
448C2×Q8—1221no—
449C2×Q8—2444no—
450C2×Q8C2×C4×D6111yescharacteristic
451C2×D4—21no—
452C2×D4C2×C4×D6111yescharacteristic
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—612no—
456C22×C4—612no—
457C22×C4—1222no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4D4×D6111yescharacteristic
461C22×C4D4×D6111yescharacteristic
462C22×C4C23×D6111yescharacteristic
463C2×C8—22no—
464C2×C8—1221no—
465C2×C8—1221no—
466C4⋊C4—24no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—612no—
472C4⋊C4—612no—
473C4⋊C4—612no—
474C4⋊C4—612no—
475C4⋊C4—611no—
476C4⋊C4—611no—
477C4⋊C4—612no—
478C4⋊C4—612no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—1228no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4C2×C4×D6111yescharacteristic
494C4⋊C4C2×C4×D6111yescharacteristic
495C22⋊C4—21no—
496C22⋊C4—21no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—21no—
500C22⋊C4—1222no—
501C22⋊C4—1222no—
502C22⋊C4—1222no—
503C22⋊C4—1222no—
504C42—21no—
505C42—81no—
506C42—612no—
507C42—612no—
508C42—612no—
509C42—612no—
510C42—1222no—
511C42—1222no—
512C42—1222no—
513C42—1224no—
514C42—1228no—
515C42—1222no—
516C42—4882no—
517C42C2×C4×D6111yescharacteristic
518C42C2×C4×D6111yescharacteristic
519C42D4×D6111yescharacteristic
520C42D4×D6111yescharacteristic
521C23—21no—
522C23C2×D4×D6111yescharacteristic
523Q8—41no—
524Q8—614no—
525Q8—1224no—
526Q8—1222no—
527Q8—2442no—
528Q8—2444no—
529Q8—2442no—
530D4—41no—
531D4—41no—
532D4—41no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—612no—
547C2×C4—612no—
548C2×C4—1224no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—1222no—
555C2×C4—2444no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4C22⋊C4×D6111yescharacteristic
561C2×C4C22⋊C4×D6111yescharacteristic
562C2×C4C22⋊C4×D6111yescharacteristic
563C2×C4C22⋊C4×D6111yescharacteristic
564C2×C4C22×C4×D6111yescharacteristic
565C2×C4C2×D4×D6111yescharacteristic
566C8—42no—
567C8—2441no—
568C8—2441no—
569C22—21no—
570C22—21no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22C2×C22⋊C4×D6111yescharacteristic
575C22#1684057211yescharacteristic
576C22#1684057811yescharacteristic
577C4—21no—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—44no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—614no—
587C4—1222no—
588C4—1222no—
589C4—1222no—
590C4—2442no—
591C4—2442no—
592C4—2442no—
593C4—2442no—
594C4#104×D6112yes—
595C2—21no—
596C2—21no—
597C2—81no—
598C2#2145428111yescharacteristic
599C2#2145435011yescharacteristic
600C2C2×#104×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1509C2
socleC22×C61C22×C22⋊C4
prime radical#121×C61C23
cosocleC22×C4×C61C24
hypercenter#1509D61
hypocenterC61#17549
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1509

Complements

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

NH ≅ G/Ntype
2. #21454909C2semidirect
3. #21448183C2semidirect
5. #21444332C2semidirect
6. #21444329C2semidirect
7. #21444317C2semidirect
8. #21444316C2semidirect
12. #16840491C4semidirect
13. #16840490C4semidirect
14. #16840489C4semidirect
15. #16840482C4semidirect
16. #16840481C4semidirect
20. D4×Dic122C4semidirect
353. C61#17549semidirect

Maximal subgroups

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

subgroupindex
#214599132
#214549092
#214481832
#214443412
#214443322
#214443292
#214443172
#214443162
#214443052
#214443022
C61×#15092
#1754961