Finite Group Databasebeta

← #29689832

Subgroups — #29689832

Order 15616.

1254 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
1C2×#16840352C211yesmaximal, characteristic
2#21448432C211yesmaximal, characteristic, complemented
3#21448414C211yesmaximal, characteristic, complemented
4#21446805C211yesmaximal, characteristic, complemented
5#21446655C211yesmaximal, characteristic, complemented
6#21446654C211yesmaximal, characteristic, complemented
7#21446271C214yesmaximal, complemented
8#21445806C214yesmaximal, complemented
9C61×#1996C211yesmaximal, characteristic, complemented
10#16840352—22no—
11#16840352—22no—
12#16840352—22no—
13#16840352—22no—
14C2×#14519772C412yescomplemented
15C2×#14519766C412yescomplemented
16C2×#14519762C412yescomplemented
17C2×#14519752C412yescomplemented
18C2×#14519751C2211yescharacteristic
19C2×#14519747C2212yes—
20C2×#14519737C2212yes—
21C22×#11758198C2211yescharacteristic
22C2×C22⋊C4×D61C2212yes—
23C2×#14519691C2214yes—
24C2×#14519691C2214yes—
25#16839653—22no—
26#16839651—22no—
27#16839636—22no—
28#16839633—22no—
29#16839622C2211yescharacteristic
30#16839620C2212yes—
31#16839620C2212yes—
32#16839617C2212yes—
33C4×C22⋊Dic61C2211yescharacteristic
34#16839529C2214yes—
35C2×C22⋊Q8×C61C2211yescharacteristic
36C61×#380C2211yescharacteristic
37C4×C22⋊C4×C61C2211yescharacteristic
38C61×#342C2214yes—
39C24×D61C2311yescharacteristic
40C22×D244C2×C412yes—
41C22×D244C2×C412yes—
42C22×D244C2×C411yescharacteristic, complemented
43C22×C22⋊D61C2312yes—
44C23×Dic61C2311yescharacteristic
45C22×C4×D61—22no—
46C22×C4×D61C2×C412yescomplemented
47#14519772—24no—
48#14519772—24no—
49#14519766—24no—
50#14519766—24no—
51#14519762—24no—
52#14519762—24no—
53#14519752—24no—
54#14519752—24no—
55#14519751—21no—
56#14519751—21no—
57#14519751—21no—
58#14519751—21no—
59#14519747—24no—
60#14519747—24no—
61#14519737—24no—
62#14519737—24no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61C2×C411yescharacteristic, complemented
65C2×#11758196—22no—
66C2×#11758196—22no—
67C2×#11758196C2×C412yescomplemented
68C2×C22⋊Dic61C2312yes—
69C2×C22⋊Dic61C2312yes—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×#11758198—22no—
73C2×#11758198C2×C412yes—
74C2×#11758198C2×C412yes—
75C2×#11758198C2×C412yes—
76C2×#11758198C2×C412yes—
77C2×#11758198C2×C412yescomplemented
78C2×#11758198C2×C412yescomplemented
79C2×#11758198D412yes—
80C2×#11758198D412yes—
81C2×#11758198D412yes—
82C2×#11758198D412yes—
83C22⋊C4×D61—44no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61C2×C412yescomplemented
86#14519691—44no—
87#14519691—44no—
88#14519684—21no—
89#14519684—21no—
90#14519684—22no—
91#14519684—22no—
92C22×Q8×C61C2×C411yescharacteristic, complemented
93C23×C4×C61C2311yescharacteristic
94C22⋊Q8×C61—21no—
95C22⋊Q8×C61—21no—
96C22⋊Q8×C61—21no—
97C22⋊Q8×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61C2×C412yescomplemented
100C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
101C2×C22⋊C4×C61C2312yes—
102C2×C22⋊C4×C61C2312yes—
103C2×C22⋊C4×C61C2312yes—
104C2×C42×C61—21no—
105#95×C61—21no—
106#95×C61—21no—
107C24×C61C2411yescharacteristic
108C23×D61—22no—
109C23×D61—22no—
110C23×D61—81no—
111C23×D61—81no—
112C23×D61C22×C412yes—
113C23×D61C22×C412yes—
114C23×D61C2×D412yes—
115C23×D61C2×D412yes—
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C2×Q8×C61—81no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—81no—
122C22×C4×C61C22×C412yes—
123C22×C4×C61C22×C412yes—
124C22×C4×C61C22×C411yescharacteristic
125C22×C4×C61C22×C411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C2×C22⋊D61—22no—
131C2×C22⋊D61—22no—
132C2×C22⋊D61—24no—
133C2×C22⋊D61—24no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—81no—
137C22×Dic61C22×C412yes—
138C22×Dic61C22×C412yes—
139C22×Dic61C2×D412yes—
140C22×Dic61C2×D412yes—
141C2×D244—24no—
142C2×D244—24no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—82no—
146C2×D244—82no—
147C2×D244—81no—
148C2×C4×D61—24no—
149C2×C4×D61—44no—
150C2×C4×D61—82no—
151C2×C4×D61—82no—
152C2×C4×D61—82no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—81no—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—22no—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—22no—
163C42×C61—81no—
164C22⋊Dic61—22no—
165C22⋊Dic61—24no—
166C22⋊Dic61—22no—
167#11758198—24no—
168#11758198—24no—
169#11758198—22no—
170#11758198—22no—
171#11758198—24no—
172#11758198—44no—
173#11758198—82no—
174#11758198—82no—
175#11758198C22⋊C414yes—
176#11758198C22⋊C414yes—
177#11758198C22⋊C414yes—
178#11758198C22⋊C414yes—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—81no—
182#11758196—24no—
183#11758196—82no—
184#11758196—82no—
185C4×Dic61—24no—
186C4×Dic61—81no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61C23×C411yescharacteristic
190C23×C61C22×D411yescharacteristic
191C23×C61C22×D411yescharacteristic
192C23×C61H(5,𝔽2)12yes—
193C23×C612−1+412yes—
194C22×D61—22no—
195C22×D61—22no—
196C22×D61—24no—
197C22×D61—24no—
198C22×D61—22no—
199C22×D61—24no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—84no—
209C22×D61—84no—
210C22×D61C2×C22⋊C414yes—
211Q8×C61—81no—
212Q8×C61—81no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C2×C4×C61C2×C22⋊C411yescharacteristic
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C22⋊D61—42no—
234C22⋊D61—42no—
235C22⋊D61—44no—
236C2×Dic61—22no—
237C2×Dic61—24no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61C2×C22⋊C414yes—
248D244—81no—
249D244—81no—
250D244—84no—
251D244—84no—
252C4×D61—84no—
253C4×D61—88no—
254#21876—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61C22×C22⋊C411yescharacteristic
264C22×C61#52012yes—
265C22×C61#53511yescharacteristic
266C22×C61#53611yescharacteristic
267C22×C61#53712yes—
268C2×D61—24no—
269C2×D61—24no—
270C2×D61—24no—
271C2×D61—44no—
272C2×D61—44no—
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—42no—
276C2×D61—84no—
277C2×D61—84no—
278C2×D61—84no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—42no—
284C4×C61—42no—
285C4×C61—82no—
286C4×C61—82no—
287Dic61—24no—
288Dic61—42no—
289Dic61—82no—
290Dic61—82no—
291C2×#538—611no—
292#2315—611no—
293#2312—611no—
294#1996—611no—
295#1996D6111yescharacteristic, complemented
296#1964—611no—
297#1964—611no—
298#1898—614no—
299#1799—614no—
300C2×C61—21no—
301C2×C61—22no—
302C2×C61—21no—
303C2×C61—22no—
304C2×C61#231111yescharacteristic
305C2×C61#288311yescharacteristic
306C2×C61#288311yescharacteristic
307D61—44no—
308D61—82no—
309#538—1222no—
310#538—1222no—
311#538—1222no—
312#538—1222no—
313C2×#126—612no—
314C2×#124—612no—
315C2×#123—612no—
316C2×C22⋊Q8C2×D6111yescharacteristic
317C2×#121—611no—
318C2×#121—612no—
319C2×#120—612no—
320C2×C4×D4—612no—
321C22×C22⋊C4—611no—
322C22×C22⋊C4—612no—
323C2×#99—614no—
324C2×#99—614no—
325#388—1222no—
326#388—1222no—
327#385—1222no—
328#385—1222no—
329#380—611no—
330#380—612no—
331#380—612no—
332#380C2×D6111yescharacteristic
333#379—612no—
334C4×C22⋊C4—611no—
335C4×C22⋊C4C2×D6111yescharacteristic
336#342—614no—
337#342C2×D6114yes—
338C61#2187611yescharacteristic, complemented, Sylow
339C25—611no—
340C22×Q8C4×D6111yescharacteristic, complemented
341C22×D4—612no—
342C22×D4—612no—
343C22×D4—612no—
344C22×D4—611no—
345C23×C4—611no—
346C23×C4—612no—
347C23×C4—1222no—
348C23×C4C22×D6111yescharacteristic
349#126—1224no—
350#126—1224no—
351#124—1224no—
352#124—1224no—
353#123—1224no—
354#123—1224no—
355C22⋊Q8—21no—
356C22⋊Q8—21no—
357C22⋊Q8—21no—
358C22⋊Q8—21no—
359#121—1224no—
360#121—1224no—
361#121—1221no—
362#121—1221no—
363#121—1221no—
364#121—1221no—
365#120—1224no—
366#120—1224no—
367C4×D4—1224no—
368C4×D4—1224no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—612no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4C4×D6112yescomplemented
376C2×C4⋊C4C4×D6111yescharacteristic, complemented
377C2×C22⋊C4—612no—
378C2×C22⋊C4—612no—
379C2×C22⋊C4—612no—
380C2×C22⋊C4—612no—
381C2×C22⋊C4—612no—
382C2×C22⋊C4—612no—
383C2×C22⋊C4—612no—
384C2×C22⋊C4—612no—
385C2×C22⋊C4—612no—
386C2×C22⋊C4—612no—
387C2×C22⋊C4—612no—
388C2×C22⋊C4—612no—
389C2×C22⋊C4—1222no—
390C2×C22⋊C4—1222no—
391C2×C22⋊C4—1222no—
392C2×C22⋊C4—2444no—
393C2×C22⋊C4C22×D6112yes—
394C2×C22⋊C4C22×D6112yes—
395C2×C22⋊C4C22×D6112yes—
396C2×C42—21no—
397C2×C42—612no—
398C2×C42—1221no—
399#99—2444no—
400#99—2444no—
401#95—21no—
402#95—21no—
403#95—1222no—
404#95—1222no—
405#95—1221no—
406#95—1221no—
407C24—612no—
408C24—612no—
409C24—612no—
410C24—612no—
411C24—1222no—
412C24—1222no—
413C24—4881no—
414C24—4881no—
415C24C23×D6111yescharacteristic
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—81no—
419C2×D4—1224no—
420C2×D4—1224no—
421C2×D4—1222no—
422C2×D4—1222no—
423C2×D4—1224no—
424C2×D4—1224no—
425C2×D4—1221no—
426C2×D4—1221no—
427C2×D4—4881no—
428C2×D4—4882no—
429C2×D4—4882no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—81no—
433C22×C4—612no—
434C22×C4—612no—
435C22×C4—612no—
436C22×C4—612no—
437C22×C4—1224no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—4882no—
441C22×C4—4882no—
442C22×C4—4882no—
443C22×C4—4881no—
444C22×C4—2444no—
445C22×C4C2×C4×D6112yes—
446C22×C4C2×C4×D6112yes—
447C22×C4C2×C4×D6111yescharacteristic
448C22×C4C2×C4×D6111yescharacteristic
449C22×C4D4×D6111yescharacteristic
450C22×C4D4×D6111yescharacteristic
451C22×C4D4×D6111yescharacteristic
452C22×C4D4×D6111yescharacteristic
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—81no—
458C4⋊C4—1224no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—4882no—
462C4⋊C4—4882no—
463C4⋊C4—4881no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—24no—
468C22⋊C4—22no—
469C22⋊C4—614no—
470C22⋊C4—614no—
471C22⋊C4—614no—
472C22⋊C4—614no—
473C22⋊C4—1224no—
474C22⋊C4—1224no—
475C22⋊C4—1224no—
476C22⋊C4—1222no—
477C22⋊C4—1222no—
478C22⋊C4—1222no—
479C22⋊C4—1224no—
480C22⋊C4—1222no—
481C22⋊C4—4882no—
482C22⋊C4—4882no—
483C22⋊C4—2444no—
484C42—81no—
485C42—1224no—
486C42—4881no—
487C23—41no—
488C23—41no—
489C23—614no—
490C23—1224no—
491C23—1224no—
492C23—1222no—
493C23—1222no—
494C23—1224no—
495C23—1222no—
496C23—4884no—
497C23—4884no—
498C23—2442no—
499C23—2442no—
500C23—2442no—
501C23—2442no—
502C23—2442no—
503C23—2442no—
504C23—2442no—
505C23—2442no—
506C23C22×C4×D6111yescharacteristic
507C23C2×D4×D6111yescharacteristic
508C23C2×D4×D6111yescharacteristic
509C23#1451982312yes—
510C23#1451982512yes—
511Q8—81no—
512Q8—81no—
513D4—4881no—
514D4—4881no—
515D4—4884no—
516D4—4884no—
517D4—2442no—
518D4—2442no—
519D4—2444no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—614no—
537C2×C4—1222no—
538C2×C4—1224no—
539C2×C4—4884no—
540C2×C4—4888no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4C22⋊C4×D6111yescharacteristic
551C2×C4C22⋊C4×D6111yescharacteristic
552C2×C4C22⋊C4×D6111yescharacteristic
553C2×C4C22⋊C4×D6111yescharacteristic
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—42no—
561C22—42no—
562C22—1224no—
563C22—1224no—
564C22—1224no—
565C22—4884no—
566C22—4884no—
567C22—4884no—
568C22—2444no—
569C22—2444no—
570C22—2442no—
571C22—2442no—
572C22—2442no—
573C22C2×C22⋊C4×D6111yescharacteristic
574C22#1684024412yes—
575C22#1684033011yescharacteristic
576C22#1684033711yescharacteristic
577C22#1684034212yes—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—42no—
583C4—42no—
584C4—82no—
585C4—82no—
586C4—1224no—
587C4—4882no—
588C4—4882no—
589C4—2442no—
590C2—21no—
591C2—22no—
592C2—22no—
593C2—21no—
594C2—4882no—
595C2—2444no—
596C2#2144841011yescharacteristic
597C2#2145427711yescharacteristic
598C2#2145427811yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1996C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840352C2
cosocleC24×C61C24
hypercenter#1996D61
hypocenterC61#21876
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
2. #21448432C2semidirect
3. #21448414C2semidirect
4. #21446805C2semidirect
5. #21446655C2semidirect
6. #21446654C2semidirect
7. #21446271C2semidirect
8. #21445806C2semidirect
9. C61×#1996C2semidirect
14. C2×#14519772C4semidirect
15. C2×#14519766C4semidirect
16. C2×#14519762C4semidirect
17. C2×#14519752C4semidirect
42. C22×D244C2×C4semidirect
46. C22×C4×D61C2×C4semidirect
64. C2×C4⋊Dic61C2×C4semidirect
67. C2×#11758196C2×C4semidirect
77. C2×#11758198C2×C4semidirect
78. C2×#11758198C2×C4semidirect
85. C2×C4×Dic61C2×C4semidirect
92. C22×Q8×C61C2×C4semidirect
99. C2×C4⋊C4×C61C2×C4semidirect
100. C2×C4⋊C4×C61C2×C4semidirect
295. #1996D61semidirect
338. C61#21876semidirect
340. C22×Q8C4×D61semidirect
375. C2×C4⋊C4C4×D61semidirect
376. C2×C4⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403522
#214484322
#214484142
#214468052
#214466552
#214466542
#214462712
#214458062
C61×#19962
#2187661