Finite Group Databasebeta

← #29832809

Subgroups — #29832809

Order 15616.

1252 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#21459150C211yesmaximal, characteristic
2#410×D61C212yesmaximal, complemented
3#21447493C212yesmaximal, complemented
4#21447491C211yesmaximal, characteristic, complemented
5#21447485C214yesmaximal, complemented
6#21447481C214yesmaximal, complemented
7C61×#2128C211yesmaximal, characteristic, complemented
8C22×#11758224C411yescharacteristic, complemented
9C2×#41×D61C2211yescharacteristic
10#16840341—22no—
11#16840338C411yescharacteristic, complemented
12#16840333C2211yescharacteristic
13#16840332C412yescomplemented
14#16840329C411yescharacteristic, complemented
15#16840325C2212yes—
16#16840322—22no—
17#16840311C412yescomplemented
18#16840310C2212yes—
19#16840272—22no—
20#16840268—22no—
21#117×D61—22no—
22#117×D61C411yescharacteristic, complemented
23#16839736C2212yes—
24#16839735—22no—
25#16839734—22no—
26#16839734C2214yescomplemented
27#99×D61—22no—
28#99×D61C2214yescomplemented
29#16839556C2218yescomplemented
30#16839555C2214yescomplemented
31C61×#535C2211yescharacteristic
32C61×#410C2212yes—
33C61×#353C2214yescomplemented
34C2×#11758226C2×C411yescharacteristic
35C2×#11758222C2×C411yescharacteristic
36C2×#11758222D412yes—
37C2×#11758224—21no—
38C2×#11758224—21no—
39C2×#11758224—42no—
40C2×#11758224—42no—
41C2×#11758224C2×C412yes—
42C2×#11758224C2×C412yescomplemented
43C2×#11758224D412yes—
44C2×#11758224C2311yescharacteristic
45#41×D61—42no—
46#41×D61—42no—
47C22×Dic122C2×C411yescharacteristic, complemented
48C2×Q8×D61C2×C411yescharacteristic
49C22×C22⋊D61—21no—
50C2×D4×D61D412yes—
51C2×D4×D61C2311yescharacteristic
52C23×Dic61—21no—
53C22×C4×D61C2×C411yescharacteristic
54C22×C4×D61D412yes—
55#14519777C2×C412yescomplemented
56#14519775C2×C412yescomplemented
57#14519773C2×C412yes—
58#14519765C2×C412yes—
59#14519764C2×C412yescomplemented
60#14519764C2312yes—
61#14519761—22no—
62#14519759—24no—
63#14519758—22no—
64#14519757—22no—
65#14519756—22no—
66#14519754—24no—
67#14519750—24no—
68#14519749—24no—
69#14519748—24no—
70#14519748—22no—
71#14519746—22no—
72#14519745—24no—
73D4×Dic61—24no—
74#14519737—24no—
75#14519736—22no—
76#14519736C2×C411yescharacteristic, complemented
77#14519735—21no—
78#14519735—42no—
79#14519734—42no—
80#14519734C2×C412yes—
81#14519733—21no—
82#14519733—24no—
83C4⋊C4×D61—21no—
84C4⋊C4×D61—42no—
85C22⋊C4×D61—21no—
86C22⋊C4×D61—24no—
87C42×D61—42no—
88C42×D61C2×C412yescomplemented
89#14519692—22no—
90#14519692C2314yes—
91#14519691—24no—
92#14519691—24no—
93C2×#41×C61C2311yescharacteristic
94C22×D4×C61C2×C411yescharacteristic, complemented
95#127×C61C2312yes—
96#124×C61C2×C412yescomplemented
97#121×C61—22no—
98#120×C61—22no—
99#117×C61—22no—
100#117×C61C2×C411yescharacteristic, complemented
101#99×C61—22no—
102#99×C61C2314yes—
103C24×C61—21no—
104C23×D61C22⋊C411yescharacteristic
105C23×D61C2×D412yes—
106#41×C61—42no—
107C2×Q8×C61C22×C411yescharacteristic
108C2×D4×C61—24no—
109C2×D4×C61—21no—
110C2×D4×C61—42no—
111C2×D4×C61C22×C412yes—
112C2×D4×C61C2×D412yes—
113C2×D4×C61C2411yescharacteristic
114C22×C4×C61C22×C411yescharacteristic
115C22×C4×C61C2×D412yes—
116C2×C22⋊D61—22no—
117C2×C22⋊D61—22no—
118C2×C22⋊D61—24no—
119C2×C22⋊D61—44no—
120C2×C22⋊D61C22⋊C412yes—
121C2×C22⋊D61C2×D412yes—
122C2×C22⋊D61C2×D412yes—
123C22×Dic61—21no—
124C22×Dic61—22no—
125C22×Dic61—22no—
126C22×Dic61—42no—
127C22×Dic61—42no—
128C22×Dic61C22⋊C411yescharacteristic
129C22×Dic61C2×D412yes—
130#11758226—41no—
131#11758226—41no—
132Q8×D61—42no—
133#11758224—42no—
134#11758224—42no—
135#11758224—41no—
136#11758224—41no—
137#11758224—44no—
138#11758224—44no—
139#11758224—42no—
140#11758224—42no—
141#11758224—42no—
142#11758224—42no—
143D4×D61—42no—
144D4×D61—42no—
145D4×D61—42no—
146#11758222—42no—
147#11758222—42no—
148#11758222—42no—
149C2×D244C22⋊C412yes—
150C2×D244C22×C411yescharacteristic
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—44no—
159C2×C4×D61C22⋊C411yescharacteristic
160C2×C4×D61C22⋊C412yes—
161C2×C4×D61C22⋊C412yes—
162C2×C4×D61C22⋊C412yes—
163C2×C4×D61C22⋊C411yescharacteristic
164C2×C4×D61C22×C411yescharacteristic
165C2×C4×D61C22×C411yescharacteristic
166C2×Dic122—21no—
167C2×Dic122—42no—
168C2×Dic122C22⋊C412yes—
169C2×Dic122C22×C412yes—
170C2×Dic122C22×C411yescharacteristic
171C4⋊C4×C61—21no—
172C4⋊C4×C61—42no—
173C22⋊C4×C61—21no—
174C22⋊C4×C61—24no—
175C22⋊C4×C61—24no—
176C42×C61—42no—
177C42×C61C22×C412yes—
178C22⋊Dic61—21no—
179C22⋊Dic61—24no—
180C22⋊Dic61—24no—
181C22⋊Dic61—24no—
182#11758198—22no—
183#11758198—24no—
184#11758198—44no—
185C4⋊Dic61—21no—
186C4⋊Dic61—24no—
187C4⋊Dic61—42no—
188#11758196—24no—
189#11758196—22no—
190#11758196—44no—
191C4×Dic61—22no—
192C4×Dic61—44no—
193C4×Dic61—42no—
194C4×Dic61C22×C412yes—
195C23×C61—22no—
196C23×C61—42no—
197C23×C61—82no—
198C23×C61C2×C22⋊C411yescharacteristic
199C23×C61C22×D412yes—
200C22×D61—22no—
201C22×D61—22no—
202C22×D61—22no—
203C22×D61—82no—
204C22×D61—81no—
205C22×D61C2×C22⋊C412yes—
206C22×D61C2×C22⋊C411yescharacteristic
207C22×D61C2×C22⋊C411yescharacteristic
208Q8×C61—41no—
209D4×C61—42no—
210D4×C61—41no—
211D4×C61—42no—
212D4×C61—42no—
213D4×C61—84no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—44no—
219C2×C4×C61C2×C22⋊C412yes—
220C2×C4×C61C2×C22⋊C411yescharacteristic
221C2×C4×C61C23×C411yescharacteristic
222C22⋊D61—42no—
223C22⋊D61—42no—
224C22⋊D61—42no—
225C22⋊D61—44no—
226C22⋊D61—44no—
227C22⋊D61—84no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—44no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—44no—
239C2×Dic61C2×C22⋊C411yescharacteristic
240C2×Dic61C2×C22⋊C412yes—
241C2×Dic61C2×C22⋊C411yescharacteristic
242D244—42no—
243D244—41no—
244C4×D61—22no—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—22no—
248C4×D61—22no—
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—22no—
252C4×D61—21no—
253C4×D61—21no—
254C4×D61—82no—
255C4×D61—88no—
256C4×D61#9912yes—
257C4×D61#9912yes—
258Dic122—42no—
259Dic122—41no—
260Dic122—42no—
261Dic122—42no—
262#33358—611nomaximal, Sylow
263C22×C61—22no—
264C22×C61—21no—
265C22×C61—42no—
266C22×C61—41no—
267C22×C61—44no—
268C22×C61—42no—
269C22×C61—84no—
270C22×C61C22×C22⋊C411yescharacteristic
271C2×D61—21no—
272C2×D61—22no—
273C2×D61—41no—
274C2×D61—44no—
275C2×D61—41no—
276C2×D61—42no—
277C2×D61—82no—
278C2×D61—82no—
279C2×D61C2×#9912yes—
280C4×C61—21no—
281C4×C61—22no—
282C4×C61—42no—
283C4×C61—84no—
284C4×C61C2×#9912yes—
285Dic61—21no—
286Dic61—22no—
287Dic61—42no—
288Dic61—42no—
289Dic61—84no—
290Dic61C2×#9911yescharacteristic
291Dic61C2×#9911yescharacteristic
292#3449—611no—
293C2×#410—612no—
294#2129—612no—
295#2128—611no—
296#2128D6111yescharacteristic, complemented
297#2127—614no—
298#2126—614no—
299C2×C61—21no—
300C2×C61—42no—
301C2×C61—41no—
302C2×C61—82no—
303C2×C61C22×#9911yescharacteristic
304D61—22no—
305D61—42no—
306D61—41no—
307C22×#41—611no—
308C22×#41—611no—
309#537—1222no—
310#536—611no—
311#536—612no—
312#536—611no—
313#535—611no—
314#535—612no—
315#535C2×D6111yescharacteristic
316#534—1222no—
317#532—612no—
318#532—612no—
319#525—1222no—
320#525—1222no—
321C2×#117—611no—
322C2×#117—1222no—
323#410—612no—
324#410—614no—
325#410—1222no—
326#410—1222no—
327#410C2×D6112yes—
328C2×#99—614no—
329C2×#99—1222no—
330#354—618no—
331#353—614no—
332#353C2×D6114yescomplemented
333C61#3335811yescharacteristic, complemented, Sylow
334C2×#41—611no—
335C2×#41—612no—
336C2×#41—611no—
337C2×#41—611no—
338C2×#41—612no—
339C2×#41—612no—
340C2×#41—612no—
341C2×#41—1221no—
342C2×#41—1221no—
343C2×#41—2442no—
344C2×#41—2442no—
345C2×#41—2442no—
346C2×#41—2442no—
347C2×#41C22×D6111yescharacteristic
348C22×Q8—611no—
349C22×Q8—611no—
350C22×D4—612no—
351C22×D4—611no—
352C22×D4—1221no—
353C22×D4C4×D6111yescharacteristic, complemented
354C23×C4—611no—
355C23×C4—612no—
356C23×C4—1221no—
357C4⋊Q8—612no—
358C4⋊Q8—612no—
359#127—612no—
360#127C22×D6112yes—
361#124—612no—
362#124—612no—
363#124—612no—
364#124C4×D6112yescomplemented
365#123—1224no—
366#123—1222no—
367#123—1222no—
368C22⋊Q8—1224no—
369C22⋊Q8—1222no—
370C22⋊Q8—1222no—
371#121—22no—
372#121—1224no—
373#121—1222no—
374#121—1224no—
375#121—1224no—
376#120—22no—
377#120—1224no—
378#120—1222no—
379C4×D4—1224no—
380C4×D4—1224no—
381#117—22no—
382#117—611no—
383#117—612no—
384#117—1221no—
385#117—1221no—
386#117—1224no—
387#117—1222no—
388#117—2442no—
389#117—2442no—
390#117C4×D6111yescharacteristic, complemented
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—2442no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1224no—
395C2×C42—612no—
396C2×C42—2442no—
397#99—22no—
398#99—614no—
399#99—1224no—
400#99—1224no—
401#99—1222no—
402#99C22×D6114yes—
403C24—21no—
404C24—612no—
405C24—611no—
406#41—42no—
407#41—2442no—
408#41—2442no—
409#41—2442no—
410#41—2442no—
411#41—2442no—
412#41—2442no—
413#41—2442no—
414#41—2441no—
415#41—2441no—
416#41—2441no—
417#41—2441no—
418#41—2444no—
419#41—2444no—
420#41—2442no—
421#41—2442no—
422C2×Q8—611no—
423C2×Q8—612no—
424C2×Q8—612no—
425C2×Q8—1221no—
426C2×Q8—2442no—
427C2×Q8—2442no—
428C2×Q8C2×C4×D6111yescharacteristic
429C2×D4—24no—
430C2×D4—21no—
431C2×D4—42no—
432C2×D4—612no—
433C2×D4—612no—
434C2×D4—612no—
435C2×D4—611no—
436C2×D4—612no—
437C2×D4—1224no—
438C2×D4—1222no—
439C2×D4—1222no—
440C2×D4—2442no—
441C2×D4—2442no—
442C2×D4—2442no—
443C2×D4—2444no—
444C2×D4C2×C4×D6112yes—
445C2×D4D4×D6112yes—
446C2×D4C23×D6111yescharacteristic
447C22×C4—611no—
448C22×C4—612no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—612no—
453C22×C4—612no—
454C22×C4—611no—
455C22×C4—612no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1221no—
466C22×C4—2444no—
467C22×C4—2442no—
468C22×C4—2442no—
469C22×C4C2×C4×D6111yescharacteristic
470C22×C4D4×D6112yes—
471C4⋊C4—21no—
472C4⋊C4—42no—
473C4⋊C4—1222no—
474C4⋊C4—1224no—
475C4⋊C4—1224no—
476C4⋊C4—1221no—
477C4⋊C4—2442no—
478C4⋊C4—2444no—
479C22⋊C4—21no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—1222no—
483C22⋊C4—1224no—
484C22⋊C4—1224no—
485C22⋊C4—1224no—
486C22⋊C4—1221no—
487C22⋊C4—1224no—
488C22⋊C4—2444no—
489C42—42no—
490C42—612no—
491C42—1222no—
492C42—2442no—
493C42—2444no—
494C42C2×C4×D6112yes—
495C23—22no—
496C23—42no—
497C23—82no—
498C23—611no—
499C23—612no—
500C23—611no—
501C23—1222no—
502C23—1222no—
503C23—1222no—
504C23—4881no—
505C23—4882no—
506C23C22⋊C4×D6111yescharacteristic
507C23C2×D4×D6112yes—
508Q8—41no—
509Q8—2441no—
510Q8—2442no—
511Q8—2442no—
512Q8—2442no—
513D4—42no—
514D4—41no—
515D4—42no—
516D4—42no—
517D4—84no—
518D4—4884no—
519D4—2441no—
520D4—2442no—
521D4—2442no—
522D4—2442no—
523D4—2442no—
524D4—2444no—
525D4—2444no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—44no—
531C2×C4—611no—
532C2×C4—612no—
533C2×C4—612no—
534C2×C4—612no—
535C2×C4—611no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—4882no—
553C2×C4—4888no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2444no—
557C2×C4—2444no—
558C2×C4—2442no—
559C2×C4C22⋊C4×D6112yes—
560C2×C4C22⋊C4×D6111yescharacteristic
561C2×C4C22×C4×D6111yescharacteristic
562C22—22no—
563C22—21no—
564C22—42no—
565C22—41no—
566C22—44no—
567C22—42no—
568C22—84no—
569C22—612no—
570C22—1221no—
571C22—1222no—
572C22—4882no—
573C22—4882no—
574C22—2441no—
575C22—2441no—
576C22—2442no—
577C22—2444no—
578C22C2×C22⋊C4×D6111yescharacteristic
579C4—21no—
580C4—22no—
581C4—42no—
582C4—84no—
583C4—611no—
584C4—611no—
585C4—1221no—
586C4—1222no—
587C4—4884no—
588C4—2442no—
589C4—2442no—
590C4#99×D6112yes—
591C2—21no—
592C2—42no—
593C2—41no—
594C2—82no—
595C2—1222no—
596C2—2441no—
597C2—2442no—
598C2C2×#99×D6111yescharacteristic
subgroupG/H
centerC2C2×#99×D61
commutatorC2×C4×C61C23×C4
FrattiniC2×D4C23×D61
FittingC61×#2128C2
socleC2×C61C22×#99
prime radical#21459150C2
cosocleC2×D4×C61C24
hypercenter#2128D61
hypocenterC61#33358
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #2128

Complements

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

NH ≅ G/Ntype
2. #410×D61C2semidirect
3. #21447493C2semidirect
4. #21447491C2semidirect
5. #21447485C2semidirect
6. #21447481C2semidirect
7. C61×#2128C2semidirect
8. C22×#11758224C4semidirect
11. #16840338C4semidirect
13. #16840332C4semidirect
14. #16840329C4semidirect
17. #16840311C4semidirect
22. #117×D61C4semidirect
26. #16839734C22semidirect
28. #99×D61C22semidirect
29. #16839556C22semidirect
30. #16839555C22semidirect
33. C61×#353C22semidirect
42. C2×#11758224C2×C4semidirect
47. C22×Dic122C2×C4semidirect
55. #14519777C2×C4semidirect
56. #14519775C2×C4semidirect
59. #14519764C2×C4semidirect
76. #14519736C2×C4semidirect
88. C42×D61C2×C4semidirect
94. C22×D4×C61C2×C4semidirect
96. #124×C61C2×C4semidirect
100. #117×C61C2×C4semidirect
296. #2128D61semidirect
332. #353C2×D61semidirect
333. C61#33358semidirect
353. C22×D4C4×D61semidirect
364. #124C4×D61semidirect
390. #117C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214591502
#410×D612
#214474932
#214474912
#214474852
#214474812
C61×#21282
#3335861