Finite Group Databasebeta

← #29610382

Subgroups — #29610382

Order 15616.

1506 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21444090C212yesmaximal, complemented
2#21444081C212yesmaximal, complemented
3#21444064C211yesmaximal, characteristic
4#21444056C212yesmaximal
5#21444053C212yesmaximal, complemented
6#21444044C212yesmaximal, complemented
7#21444041C212yesmaximal
8#21444041C211yesmaximal, characteristic
9C2×C61×#393C211yesmaximal, characteristic
10C24×Dic61C2211yescharacteristic
11C22×C4⋊Dic61C2211yescharacteristic
12C22×C22⋊Dic61C2212yes—
13C22×C22⋊Dic61C2211yescharacteristic
14C22×C4×Dic61C2212yes—
15C2×#14519684C2212yes—
16C2×#14519684C2212yes—
17C2×#14519684C2212yes—
18C2×#14519684C2212yes—
19C2×#14519684C2212yes—
20C2×#14519684C2212yes—
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2211yescharacteristic
26C23×Q8×C61C2211yescharacteristic
27C22×C22⋊C4×C61C2212yes—
28C22×C22⋊C4×C61C2211yescharacteristic
29C61×#393C418yescomplemented
30C2×#95×C61C2211yescharacteristic
31C2×#95×C61C2212yes—
32C23×Dic61—22no—
33C23×Dic61—24no—
34C23×Dic61—41no—
35C23×Dic61—42no—
36C23×Dic61—41no—
37C23×Dic61C2312yes—
38C23×Dic61C2312yes—
39C23×Dic61C2312yes—
40C23×Dic61C2312yes—
41C2×C4⋊Dic61—24no—
42C2×C4⋊Dic61—41no—
43C2×C22⋊Dic61—24no—
44C2×C22⋊Dic61—24no—
45C2×C22⋊Dic61—41no—
46C2×C22⋊Dic61—41no—
47C2×C22⋊Dic61—42no—
48C2×C22⋊Dic61Q814yescomplemented
49C2×C4×Dic61—24no—
50C2×C4×Dic61—42no—
51C2×C4×Dic61—42no—
52#14519684—24no—
53#14519684—24no—
54#14519684—42no—
55#14519684—42no—
56#14519684—42no—
57#14519684—42no—
58#14519684—42no—
59#14519684—42no—
60#14519684—42no—
61#14519684—42no—
62#14519684—41no—
63C25×C61C2311yescharacteristic
64C22×Q8×C61—42no—
65C22×Q8×C61—41no—
66C22×Q8×C61—42no—
67C22×Q8×C61—41no—
68C22×Q8×C61C2×C414yescomplemented
69C23×C4×C61C2312yes—
70C23×C4×C61C2311yescharacteristic
71C23×C4×C61C2311yescharacteristic
72C23×C4×C61C2312yes—
73C2×C22⋊C4×C61—42no—
74C2×C22⋊C4×C61—41no—
75C2×C22⋊C4×C61—42no—
76C2×C22⋊C4×C61—41no—
77C2×C22⋊C4×C61C2×C418yes—
78C2×C22⋊C4×C61C2×C414yes—
79#95×C61C2×C414yescomplemented
80#95×C61C2×C418yescomplemented
81C24×C61—22no—
82C24×C61—22no—
83C24×C61—21no—
84C24×C61—22no—
85C24×C61—21no—
86C24×C61—21no—
87C24×C61—22no—
88C24×C61—21no—
89C24×C61C22×C412yes—
90C24×C61C2×Q812yes—
91C24×C61#4112yes—
92C24×C61C2411yescharacteristic
93C2×Q8×C61—44no—
94C2×Q8×C61—48no—
95C2×Q8×C61—48no—
96C2×Q8×C61—44no—
97C2×Q8×C61—44no—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61—21no—
106C22×C4×C61—22no—
107C22×C4×C61—21no—
108C22×C4×C61—22no—
109C22×C4×C61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61C22×C412yes—
119C22×C4×C61C22×C414yes—
120C22×C4×C61C22×C412yes—
121C22×C4×C61C22×C414yes—
122C22×Dic61—22no—
123C22×Dic61—22no—
124C22×Dic61—22no—
125C22×Dic61—22no—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—44no—
148C22×Dic61—48no—
149C22×Dic61—48no—
150C22×Dic61—44no—
151C22×Dic61—44no—
152C22×Dic61C2×Q814yes—
153C22×Dic61#4114yes—
154C22×Dic61#4114yes—
155C22⋊C4×C61—48no—
156C22⋊C4×C61—44no—
157C22⋊Dic61—44no—
158C22⋊Dic61—48no—
159C4⋊Dic61—44no—
160C4×Dic61—48no—
161C23×C61—24no—
162C23×C61—22no—
163C23×C61—24no—
164C23×C61—22no—
165C23×C61—22no—
166C23×C61—22no—
167C23×C61—22no—
168C23×C61—22no—
169C23×C61—24no—
170C23×C61—24no—
171C23×C61—24no—
172C23×C61—24no—
173C23×C61—42no—
174C23×C61—42no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61C4×Q814yes—
184C23×C61C23×C411yescharacteristic
185C23×C61C22×Q811yescharacteristic
186C23×C61C2×#4111yescharacteristic
187C23×C61C2×#4112yes—
188C23×C61C2×#4112yes—
189C23×C61H(5,𝔽2)12yes—
190C23×C61H(5,𝔽2)12yes—
191C23×C61H(5,𝔽2)11yescharacteristic
192C23×C61H(5,𝔽2)11yescharacteristic
193C23×C61H(5,𝔽2)11yescharacteristic
194C23×C612−1+411yescharacteristic
195Q8×C61—416no—
196C2×C4×C61—24no—
197C2×C4×C61—24no—
198C2×C4×C61—24no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—24no—
202C2×C4×C61—24no—
203C2×C4×C61—24no—
204C2×C4×C61—24no—
205C2×C4×C61—24no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—41no—
218C2×C4×C61—42no—
219C2×C4×C61—41no—
220C2×Dic61—24no—
221C2×Dic61—24no—
222C2×Dic61—24no—
223C2×Dic61—24no—
224C2×Dic61—24no—
225C2×Dic61—24no—
226C2×Dic61—24no—
227C2×Dic61—24no—
228C2×Dic61—24no—
229C2×Dic61—416no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—42no—
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×Dic61—42no—
248C2×Dic61—42no—
249#14596—611nomaximal, Sylow
250C22×C61—24no—
251C22×C61—28no—
252C22×C61—44no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—44no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—44no—
261C22×C61C2×C4×Q811yescharacteristic
262C22×C61C4×#4112yes—
263C22×C61#51811yescharacteristic
264C22×C61#51812yes—
265C22×C61#52011yescharacteristic
266C22×C61#53812yes—
267C22×C61#53812yes—
268C22×C61#53811yescharacteristic
269C22×C61#53911yescharacteristic
270C22×C61#54212yes—
271C22×C61#54212yes—
272C22×C61#54311yescharacteristic
273C22×C61#54411yescharacteristic
274C22×C61#55112yes—
275C22×C61#55112yes—
276C22×C61#55111yescharacteristic
277C22×C61#55111yescharacteristic
278C22×C61#55412yes—
279C22×C61#55412yes—
280C22×C61#55512yes—
281C22×C61#56012yes—
282C22×C61#56011yescharacteristic
283C22×C61H(3,𝔽4)11yescharacteristic
284C4×C61—44no—
285C4×C61—42no—
286C4×C61—42no—
287C4×C61—44no—
288Dic61—44no—
289Dic61—44no—
290Dic61—44no—
291Dic61—82no—
292C2×#393D6111yescharacteristic
293#1446—612no—
294#1443—612no—
295#1439—611no—
296#1437—612no—
297#1436—612no—
298#1434—612no—
299C22⋊#95—611no—
300C22⋊#95—612no—
301C2×C61—44no—
302C2×C61#232911yescharacteristic
303C2×C61#233712yes—
304C2×C61#236812yes—
305C2×C61#237211yescharacteristic
306C2×C61#237511yescharacteristic
307C2×C61#273611yescharacteristic
308C2×C61#273712yes—
309C2×C61#274211yescharacteristic
310C2×C61#276011yescharacteristic
311C2×C61#276311yescharacteristic
312C2×C61#278412yes—
313C23×Q8C2×D6111yescharacteristic
314C24×C4—611no—
315C22×C4⋊C4—611no—
316C22×C22⋊C4—611no—
317C22×C22⋊C4—612no—
318C22×C22⋊C4C2×D6112yes—
319C22×C22⋊C4C2×D6111yescharacteristic
320C22×C42—612no—
321#393Dic6118yescomplemented
322C2×#95—612no—
323C2×#95—612no—
324C2×#95—612no—
325C2×#95—612no—
326C2×#95—612no—
327C2×#95—612no—
328C2×#95—612no—
329C2×#95—612no—
330C2×#95—612no—
331C2×#95—612no—
332C2×#95—611no—
333C2×#95C2×D6111yescharacteristic
334C2×#95C2×D6112yes—
335C61#1459611yescharacteristic, complemented, Sylow
336C25C22×D6111yescharacteristic
337C22×Q8—42no—
338C22×Q8—41no—
339C22×Q8—42no—
340C22×Q8—41no—
341C22×Q8C2×Dic6114yescomplemented
342C23×C4—612no—
343C23×C4—612no—
344C23×C4—612no—
345C23×C4—612no—
346C23×C4—1224no—
347C23×C4—1222no—
348C23×C4—2441no—
349C23×C4—2442no—
350C23×C4—2441no—
351C23×C4C22×D6111yescharacteristic
352C23×C4C22×D6112yes—
353C23×C4C22×D6111yescharacteristic
354C23×C4C22×D6112yes—
355C2×C4⋊C4—1224no—
356C2×C4⋊C4—2441no—
357C2×C22⋊C4—41no—
358C2×C22⋊C4—42no—
359C2×C22⋊C4—42no—
360C2×C22⋊C4—41no—
361C2×C22⋊C4—614no—
362C2×C22⋊C4—1224no—
363C2×C22⋊C4—1224no—
364C2×C22⋊C4—2441no—
365C2×C22⋊C4—2441no—
366C2×C22⋊C4—2442no—
367C2×C22⋊C4C2×Dic6118yes—
368C2×C22⋊C4C2×Dic6114yes—
369C2×C42—1224no—
370C2×C42—2442no—
371C2×C42—2442no—
372#95—1224no—
373#95—1224no—
374#95—2442no—
375#95—2441no—
376#95—2442no—
377#95—2442no—
378#95—2442no—
379#95—2442no—
380#95—2442no—
381#95—2442no—
382#95—2442no—
383#95C2×Dic6114yescomplemented
384#95C2×Dic6118yescomplemented
385C24—22no—
386C24—22no—
387C24—22no—
388C24—21no—
389C24—22no—
390C24—21no—
391C24—21no—
392C24—21no—
393C24Q8×D6112yes—
394C24#1175822612yes—
395C24C22×Dic6112yes—
396C24C23×D6111yescharacteristic
397C2×Q8—44no—
398C2×Q8—48no—
399C2×Q8—48no—
400C2×Q8—44no—
401C2×Q8—44no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—614no—
423C22×C4—614no—
424C22×C4—614no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—2444no—
451C22×C4—2448no—
452C22×C4—2448no—
453C22×C4—2444no—
454C22×C4—2444no—
455C22×C4C22×Dic6112yes—
456C22×C4C22×Dic6114yes—
457C22×C4C22×Dic6112yes—
458C22×C4C22×Dic6114yes—
459C4⋊C4—2444no—
460C22⋊C4—48no—
461C22⋊C4—44no—
462C22⋊C4—2444no—
463C22⋊C4—2448no—
464C42—2448no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—22no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23—41no—
487C23Q8×Dic6114yes—
488C23C23×Dic6111yescharacteristic
489C23C2×Q8×D6111yescharacteristic
490C23#41×D6112yes—
491C23#41×D6112yes—
492C23C2×#1175822611yescharacteristic
493C23#1451982212yes—
494C23#1451982212yes—
495C23#1451982211yescharacteristic
496C23#1451982211yescharacteristic
497C23#1451982311yescharacteristic
498C23#1451982411yescharacteristic
499Q8—416no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—1224no—
525C2×C4—1224no—
526C2×C4—1224no—
527C2×C4—1224no—
528C2×C4—1224no—
529C2×C4—1224no—
530C2×C4—1224no—
531C2×C4—1224no—
532C2×C4—1224no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4—2442no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—24416no—
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×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C22—24no—
554C22—28no—
555C22—42no—
556C22—42no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—44no—
562C22—44no—
563C22—42no—
564C22C2×Q8×Dic6111yescharacteristic
565C22#41×Dic6112yes—
566C22#1684023711yescharacteristic
567C22#1684023712yes—
568C22#1684024511yescharacteristic
569C22#1684034612yes—
570C22#1684034612yes—
571C22#1684035211yescharacteristic
572C22#1684036111yescharacteristic
573C22#1684037612yes—
574C22#1684037612yes—
575C22#1684038711yescharacteristic
576C22#1684039111yescharacteristic
577C22#1684043912yes—
578C22#1684043912yes—
579C22#1684044211yescharacteristic
580C22#1684044211yescharacteristic
581C22#1684046612yes—
582C22#1684046612yes—
583C22#1684047412yes—
584C22#1684050112yes—
585C22#1684050111yescharacteristic
586C22#1684050511yescharacteristic
587C4—42no—
588C4—44no—
589C4—44no—
590C4—42no—
591C4—4882no—
592C4—2444no—
593C4—2444no—
594C4—2444no—
595C2—44no—
596C2#2144855411yescharacteristic
597C2#2144863412yes—
598C2#2144890412yes—
599C2#2144893111yescharacteristic
600C2#2144896011yescharacteristic
601C2#2145286311yescharacteristic
602C2#2145288012yes—
603C2#2145294211yescharacteristic
604C2#2145319411yescharacteristic
605C2#2145323711yescharacteristic
606C2#2145351412yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#393C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×#393D61
hypocenterC61#14596
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
1. #21444090C2semidirect
2. #21444081C2semidirect
5. #21444053C2semidirect
6. #21444044C2semidirect
29. C61×#393C4semidirect
48. C2×C22⋊Dic61Q8semidirect
68. C22×Q8×C61C2×C4semidirect
79. #95×C61C2×C4semidirect
80. #95×C61C2×C4semidirect
321. #393Dic61semidirect
335. C61#14596semidirect
341. C22×Q8C2×Dic61semidirect
383. #95C2×Dic61semidirect
384. #95C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214440902
#214440812
#214440642
#214440562
#214440532
#214440442
#214440412
C2×C61×#3932
#1459661