Finite Group Databasebeta

← #29610328

Subgroups — #29610328

Order 15616.

1442 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#21444081C212yesmaximal, complemented
2#21444072C212yesmaximal, complemented
3#21444064C211yesmaximal, characteristic
4#21444056C212yesmaximal
5#21444051C212yesmaximal, complemented
6#21444051C212yesmaximal, complemented
7#21444041C212yesmaximal
8#21444041C211yesmaximal, characteristic
9C2×C61×#399C211yesmaximal, 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×#14519684C2211yescharacteristic
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C22×C4⋊C4×C61C2211yescharacteristic
27C22×C4⋊C4×C61C2212yes—
28C22×C22⋊C4×C61C2212yes—
29C22×C22⋊C4×C61C2211yescharacteristic
30C61×#399C418yescomplemented
31C2×#95×C61C2211yescharacteristic
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—41no—
42C2×C4⋊Dic61—42no—
43C2×C22⋊Dic61—24no—
44C2×C22⋊Dic61—24no—
45C2×C22⋊Dic61—41no—
46C2×C22⋊Dic61—41no—
47C2×C22⋊Dic61—42no—
48C2×C22⋊Dic61Q814yes—
49C2×C4×Dic61—42no—
50C2×C4×Dic61—42no—
51C2×C4×Dic61—42no—
52#14519684—24no—
53#14519684—24no—
54#14519684—24no—
55#14519684—24no—
56#14519684—42no—
57#14519684—42no—
58#14519684—42no—
59#14519684—42no—
60#14519684—41no—
61#14519684—42no—
62#14519684—42no—
63C25×C61C2311yescharacteristic
64C23×C4×C61C2312yes—
65C23×C4×C61C2311yescharacteristic
66C23×C4×C61C2311yescharacteristic
67C23×C4×C61C2312yes—
68C2×C4⋊C4×C61—41no—
69C2×C4⋊C4×C61—42no—
70C2×C4⋊C4×C61—41no—
71C2×C4⋊C4×C61—42no—
72C2×C4⋊C4×C61C2×C414yescomplemented
73C2×C4⋊C4×C61C2×C418yescomplemented
74C2×C22⋊C4×C61—42no—
75C2×C22⋊C4×C61—41no—
76C2×C22⋊C4×C61—42no—
77C2×C22⋊C4×C61—41no—
78C2×C22⋊C4×C61C2×C418yes—
79C2×C22⋊C4×C61C2×C414yes—
80#95×C61C2×C414yescomplemented
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
93C22×C4×C61—22no—
94C22×C4×C61—22no—
95C22×C4×C61—22no—
96C22×C4×C61—22no—
97C22×C4×C61—22no—
98C22×C4×C61—22no—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—21no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61C22×C412yes—
114C22×C4×C61C22×C414yes—
115C22×C4×C61C22×C414yes—
116C22×C4×C61C22×C412yes—
117C22×Dic61—22no—
118C22×Dic61—22no—
119C22×Dic61—22no—
120C22×Dic61—22no—
121C22×Dic61—22no—
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—44no—
143C22×Dic61—48no—
144C22×Dic61—48no—
145C22×Dic61—44no—
146C22×Dic61—44no—
147C22×Dic61C2×Q814yes—
148C22×Dic61#4114yes—
149C22×Dic61#4114yes—
150C4⋊C4×C61—48no—
151C4⋊C4×C61—44no—
152C22⋊C4×C61—48no—
153C22⋊C4×C61—44no—
154C22⋊Dic61—44no—
155C22⋊Dic61—48no—
156C4⋊Dic61—44no—
157C4×Dic61—48no—
158C23×C61—24no—
159C23×C61—22no—
160C23×C61—24no—
161C23×C61—22no—
162C23×C61—22no—
163C23×C61—22no—
164C23×C61—22no—
165C23×C61—22no—
166C23×C61—24no—
167C23×C61—24no—
168C23×C61—24no—
169C23×C61—24no—
170C23×C61—42no—
171C23×C61—42no—
172C23×C61—42no—
173C23×C61—42no—
174C23×C61—41no—
175C23×C61—42no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—42no—
180C23×C61C4×Q814yes—
181C23×C61C23×C411yescharacteristic
182C23×C61C22×Q811yescharacteristic
183C23×C61C2×#4111yescharacteristic
184C23×C61C2×#4112yes—
185C23×C61C2×#4112yes—
186C23×C61H(5,𝔽2)12yes—
187C23×C61H(5,𝔽2)11yescharacteristic
188C23×C61H(5,𝔽2)11yescharacteristic
189C23×C61H(5,𝔽2)12yes—
190C23×C61H(5,𝔽2)11yescharacteristic
191C23×C612−1+411yescharacteristic
192C2×C4×C61—24no—
193C2×C4×C61—24no—
194C2×C4×C61—24no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—24no—
198C2×C4×C61—24no—
199C2×C4×C61—24no—
200C2×C4×C61—24no—
201C2×C4×C61—24no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—41no—
207C2×C4×C61—42no—
208C2×C4×C61—41no—
209C2×C4×C61—42no—
210C2×C4×C61—41no—
211C2×C4×C61—42no—
212C2×C4×C61—41no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×Dic61—24no—
217C2×Dic61—24no—
218C2×Dic61—24no—
219C2×Dic61—24no—
220C2×Dic61—24no—
221C2×Dic61—24no—
222C2×Dic61—24no—
223C2×Dic61—24no—
224C2×Dic61—24no—
225C2×Dic61—416no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
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—
245#14592—611nomaximal, Sylow
246C22×C61—24no—
247C22×C61—28no—
248C22×C61—44no—
249C22×C61—42no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—44no—
254C22×C61—42no—
255C22×C61—44no—
256C22×C61—42no—
257C22×C61C2×C4×Q811yescharacteristic
258C22×C61C4×#4112yes—
259C22×C61#51811yescharacteristic
260C22×C61#51812yes—
261C22×C61#52011yescharacteristic
262C22×C61#53812yes—
263C22×C61#53811yescharacteristic
264C22×C61#53912yes—
265C22×C61#53911yescharacteristic
266C22×C61#54012yes—
267C22×C61#54212yes—
268C22×C61#54311yescharacteristic
269C22×C61#54411yescharacteristic
270C22×C61#55112yes—
271C22×C61#55112yes—
272C22×C61#55111yescharacteristic
273C22×C61#55312yes—
274C22×C61#55412yes—
275C22×C61#55412yes—
276C22×C61#55911yescharacteristic
277C22×C61#56011yescharacteristic
278C22×C61#56011yescharacteristic
279C22×C61#56012yes—
280C4×C61—44no—
281C4×C61—42no—
282C4×C61—44no—
283C4×C61—42no—
284Dic61—44no—
285Dic61—44no—
286Dic61—44no—
287Dic61—82no—
288C2×#399D6111yescharacteristic
289#1443—612no—
290#1441—612no—
291#1439—611no—
292#1437—612no—
293#1436—612no—
294#1436—612no—
295C22⋊#95—611no—
296C22⋊#95—612no—
297C2×C61—44no—
298C2×C61#232911yescharacteristic
299C2×C61#233512yes—
300C2×C61#236812yes—
301C2×C61#237411yescharacteristic
302C2×C61#238011yescharacteristic
303C2×C61#273511yescharacteristic
304C2×C61#274211yescharacteristic
305C2×C61#274612yes—
306C2×C61#275211yescharacteristic
307C2×C61#278012yes—
308C2×C61#280711yescharacteristic
309C24×C4—611no—
310C22×C4⋊C4—611no—
311C22×C4⋊C4C2×D6111yescharacteristic
312C22×C4⋊C4C2×D6112yes—
313C22×C22⋊C4—611no—
314C22×C22⋊C4—612no—
315C22×C22⋊C4C2×D6112yes—
316C22×C22⋊C4C2×D6111yescharacteristic
317C22×C42—612no—
318#399Dic6118yescomplemented
319C2×#95—612no—
320C2×#95—612no—
321C2×#95—612no—
322C2×#95—612no—
323C2×#95—612no—
324C2×#95—612no—
325C2×#95—612no—
326C2×#95—612no—
327C2×#95—611no—
328C2×#95—612no—
329C2×#95—612no—
330C2×#95C2×D6111yescharacteristic
331C61#1459211yescharacteristic, complemented, Sylow
332C25C22×D6111yescharacteristic
333C23×C4—612no—
334C23×C4—612no—
335C23×C4—612no—
336C23×C4—612no—
337C23×C4—1224no—
338C23×C4—1222no—
339C23×C4—2441no—
340C23×C4—2442no—
341C23×C4—2441no—
342C23×C4C22×D6111yescharacteristic
343C23×C4C22×D6112yes—
344C23×C4C22×D6111yescharacteristic
345C23×C4C22×D6112yes—
346C2×C4⋊C4—41no—
347C2×C4⋊C4—42no—
348C2×C4⋊C4—41no—
349C2×C4⋊C4—42no—
350C2×C4⋊C4—2441no—
351C2×C4⋊C4—2442no—
352C2×C4⋊C4C2×Dic6114yescomplemented
353C2×C4⋊C4C2×Dic6118yescomplemented
354C2×C22⋊C4—42no—
355C2×C22⋊C4—41no—
356C2×C22⋊C4—42no—
357C2×C22⋊C4—41no—
358C2×C22⋊C4—614no—
359C2×C22⋊C4—1224no—
360C2×C22⋊C4—1224no—
361C2×C22⋊C4—2441no—
362C2×C22⋊C4—2441no—
363C2×C22⋊C4—2442no—
364C2×C22⋊C4C2×Dic6118yes—
365C2×C22⋊C4C2×Dic6114yes—
366C2×C42—2442no—
367C2×C42—2442no—
368C2×C42—2442no—
369#95—1224no—
370#95—1224no—
371#95—1224no—
372#95—1224no—
373#95—2441no—
374#95—2442no—
375#95—2442no—
376#95—2442no—
377#95—2442no—
378#95—2442no—
379#95—2442no—
380#95C2×Dic6114yescomplemented
381C24—22no—
382C24—22no—
383C24—22no—
384C24—21no—
385C24—22no—
386C24—21no—
387C24—21no—
388C24—21no—
389C24Q8×D6112yes—
390C24#1175822612yes—
391C24C22×Dic6112yes—
392C24C23×D6111yescharacteristic
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—614no—
414C22×C4—614no—
415C22×C4—614no—
416C22×C4—1222no—
417C22×C4—1222no—
418C22×C4—1222no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
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—2444no—
442C22×C4—2448no—
443C22×C4—2448no—
444C22×C4—2444no—
445C22×C4—2444no—
446C22×C4C22×Dic6112yes—
447C22×C4C22×Dic6114yes—
448C22×C4C22×Dic6112yes—
449C22×C4C22×Dic6114yes—
450C4⋊C4—48no—
451C4⋊C4—44no—
452C4⋊C4—2444no—
453C22⋊C4—48no—
454C22⋊C4—44no—
455C22⋊C4—2444no—
456C22⋊C4—2448no—
457C42—2448no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—22no—
462C23—24no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—42no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23Q8×Dic6114yes—
481C23C23×Dic6111yescharacteristic
482C23C2×Q8×D6111yescharacteristic
483C23#41×D6112yes—
484C23#41×D6112yes—
485C23C2×#1175822611yescharacteristic
486C23#1451982211yescharacteristic
487C23#1451982212yes—
488C23#1451982212yes—
489C23#1451982211yescharacteristic
490C23#1451982311yescharacteristic
491C23#1451982411yescharacteristic
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—1224no—
517C2×C4—1224no—
518C2×C4—1224no—
519C2×C4—1224no—
520C2×C4—1224no—
521C2×C4—1224no—
522C2×C4—1224no—
523C2×C4—1224no—
524C2×C4—1224no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—24416no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4—2442no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C22—24no—
546C22—28no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—42no—
555C22—42no—
556C22C2×Q8×Dic6111yescharacteristic
557C22#41×Dic6112yes—
558C22#1684023711yescharacteristic
559C22#1684023712yes—
560C22#1684024511yescharacteristic
561C22#1684034812yes—
562C22#1684035211yescharacteristic
563C22#1684035512yes—
564C22#1684036111yescharacteristic
565C22#1684036412yes—
566C22#1684038012yes—
567C22#1684038711yescharacteristic
568C22#1684039111yescharacteristic
569C22#1684043912yes—
570C22#1684043912yes—
571C22#1684044011yescharacteristic
572C22#1684045712yes—
573C22#1684046612yes—
574C22#1684046612yes—
575C22#1684049611yescharacteristic
576C22#1684049911yescharacteristic
577C22#1684049911yescharacteristic
578C22#1684049912yes—
579C4—44no—
580C4—42no—
581C4—44no—
582C4—42no—
583C4—4882no—
584C4—2444no—
585C4—2444no—
586C4—2444no—
587C2—44no—
588C2#2144855411yescharacteristic
589C2#2144861312yes—
590C2#2144890412yes—
591C2#2144895511yescharacteristic
592C2#2144899611yescharacteristic
593C2#2145286111yescharacteristic
594C2#2145294911yescharacteristic
595C2#2145300912yes—
596C2#2145308811yescharacteristic
597C2#2145347212yes—
598C2#2145378411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#399C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×#399D61
hypocenterC61#14592
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. #21444081C2semidirect
2. #21444072C2semidirect
5. #21444051C2semidirect
6. #21444051C2semidirect
30. C61×#399C4semidirect
72. C2×C4⋊C4×C61C2×C4semidirect
73. C2×C4⋊C4×C61C2×C4semidirect
80. #95×C61C2×C4semidirect
318. #399Dic61semidirect
331. C61#14592semidirect
352. C2×C4⋊C4C2×Dic61semidirect
353. C2×C4⋊C4C2×Dic61semidirect
380. #95C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214440812
#214440722
#214440642
#214440562
#214440512
#214440412
C2×C61×#3992
#1459261