Finite Group Databasebeta

← #29681271

Subgroups — #29681271

Order 15616.

1778 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840245C211yesmaximal, characteristic
2C2×#16839771C211yesmaximal, characteristic, complemented
3C22×#14519701C211yesmaximal, characteristic, complemented
4C2×#16839729C211yesmaximal, characteristic
5#21446132C214yesmaximal, complemented
6#21446127C214yesmaximal, complemented
7#21445540C211yesmaximal, characteristic, complemented
8#21445535C211yesmaximal, characteristic, complemented
9C2×C61×#536C211yesmaximal, characteristic
10C22×#11758194C2211yescharacteristic
11C23×#92418C2211yescharacteristic
12#16840245—24no—
13#16840245—24no—
14C2×Q8×Dic61—21no—
15C2×D4×Dic61—22no—
16C2×D4×Dic61—21no—
17C2×#14519736C2212yes—
18C2×#14519736C2211yescharacteristic
19C22×C4⋊Dic61C2212yes—
20C22×C4⋊Dic61C2211yescharacteristic
21#16839771—24no—
22C2×#14519701—24no—
23C2×#14519701C2214yes—
24C2×#14519701C2212yes—
25C2×#14519701C2212yes—
26#16839729—24no—
27C2×C22⋊#92418C2214yes—
28#16839524C2212yes—
29#16839524C2212yes—
30#16839524C2214yes—
31C22×#41×C61C2211yescharacteristic
32C23×Q8×C61C2211yescharacteristic
33C61×#536—24no—
34C61×#536C418yescomplemented
35C2×C4⋊Q8×C61—21no—
36C2×#124×C61—21no—
37C2×#123×C61—22no—
38C2×C22⋊Q8×C61C2214yes—
39C2×#117×C61C2211yescharacteristic
40C23×Dic61—22no—
41C23×Dic61—21no—
42C2×#11758194—22no—
43C2×#11758194—22no—
44C2×#11758194—22no—
45C22×#92418—22no—
46C22×#92418—22no—
47C22×#92418—22no—
48C22×#92418C2312yes—
49C22×#92418C2312yes—
50Q8×Dic61—28no—
51D4×Dic61—216no—
52D4×Dic61—28no—
53#14519736—22no—
54#14519736—24no—
55#14519736—24no—
56#14519736D414yes—
57C2×C4⋊Dic61—22no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61—24no—
61C2×C4⋊Dic61—24no—
62C2×C4⋊Dic61D414yes—
63C2×C4⋊Dic61C2311yescharacteristic
64C2×C4⋊Dic61C2312yes—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C22⋊Dic61—22no—
67C2×C22⋊Dic61—21no—
68C2×C4×Dic61—22no—
69C2×C4×Dic61—21no—
70#14519701—28no—
71#14519701—24no—
72#14519701—24no—
73C22⋊#92418—28no—
74C2×#41×C61—22no—
75C2×#41×C61—22no—
76C2×#41×C61—22no—
77C2×#41×C61—22no—
78C2×#41×C61C2×C414yes—
79C2×#41×C61D414yes—
80C22×Q8×C61—21no—
81C22×Q8×C61—22no—
82C22×Q8×C61—22no—
83C22×Q8×C61—22no—
84C22×Q8×C61—22no—
85C22×Q8×C61—22no—
86C22×Q8×C61C2×C414yescomplemented
87C22×Q8×C61D411yescharacteristic
88C22×Q8×C61C2312yes—
89C22×D4×C61D412yes—
90C22×D4×C61D411yescharacteristic
91C23×C4×C61—21no—
92C23×C4×C61C2312yes—
93C23×C4×C61C2311yescharacteristic
94C4⋊Q8×C61—28no—
95#124×C61—28no—
96#123×C61—216no—
97C22⋊Q8×C61—28no—
98C22⋊Q8×C61C2×C4116yescomplemented
99#117×C61—22no—
100#117×C61C2×C414yescomplemented
101C2×C4⋊C4×C61—24no—
102C2×C4⋊C4×C61C2312yes—
103C2×C22⋊C4×C61—21no—
104C2×C22⋊C4×C61—24no—
105C2×C42×C61—21no—
106C24×C61C2×D412yes—
107C24×C61C2×D411yescharacteristic
108#41×C61—28no—
109#41×C61—28no—
110#41×C61—28no—
111#41×C61—28no—
112C2×Q8×C61—24no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—28no—
117C2×Q8×C61—28no—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61—42no—
123C2×Q8×C61—44no—
124C2×Q8×C61—44no—
125C2×Q8×C61—42no—
126C2×Q8×C61C22⋊C414yes—
127C2×Q8×C61C22×C414yes—
128C2×D4×C61—24no—
129C2×D4×C61—24no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61C22⋊C418yes—
133C2×D4×C61C22⋊C414yes—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—24no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—42no—
142C22×C4×C61—42no—
143C22×C4×C61C22×C414yes—
144C22×C4×C61C22×C412yes—
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D414yes—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D412yes—
151C22×C4×C61#4112yes—
152C22×C4×C61#4112yes—
153C22×C4×C61C2411yescharacteristic
154C22×Dic61—22no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—28no—
158C22×Dic61—24no—
159C22×Dic61—82no—
160C22×Dic61—84no—
161C4⋊C4×C61—216no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61C22×C414yes—
164C22⋊C4×C61—24no—
165C22⋊C4×C61—216no—
166C42×C61—24no—
167C22⋊Dic61—28no—
168C22⋊Dic61—24no—
169C4⋊Dic61—28no—
170C4⋊Dic61—24no—
171C4⋊Dic61—22no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61C2×D412yes—
175C4⋊Dic61C2×D414yes—
176C4⋊Dic61C2×D412yes—
177C4×Dic61—28no—
178C4×Dic61—24no—
179#11758194—44no—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418—44no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61—24no—
192C23×C61—24no—
193C23×C61C2×C22⋊C414yes—
194C23×C61C2×C22⋊C412yes—
195C23×C61#12012yes—
196C23×C61#12312yes—
197C23×C61C22×D411yescharacteristic
198Q8×C61—28no—
199Q8×C61—24no—
200Q8×C61—24no—
201Q8×C61—42no—
202Q8×C61—42no—
203Q8×C61—48no—
204D4×C61—216no—
205D4×C61—28no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—24no—
209C2×C4×C61—24no—
210C2×C4×C61—24no—
211C2×C4×C61—24no—
212C2×C4×C61—24no—
213C2×C4×C61—24no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—48no—
219C2×C4×C61C2×C22⋊C414yes—
220C2×C4×C61C2×C22⋊C412yes—
221C2×C4×C61C4×D414yes—
222C2×C4×C61C4×D414yes—
223C2×C4×C61#12012yes—
224C2×C4×C61#12114yes—
225C2×C4×C61#12114yes—
226C2×C4×C61#12312yes—
227C2×C4×C61C23×C411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×Dic61—22no—
234C2×Dic61—24no—
235C2×Dic61—22no—
236C2×Dic61—44no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—84no—
240C2×Dic61—88no—
241#92418—42no—
242#92418—42no—
243#20913—611nomaximal, Sylow
244C22×C61—24no—
245C22×C61—24no—
246C22×C61—22no—
247C22×C61—22no—
248C22×C61—28no—
249C22×C61—24no—
250C22×C61—44no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61#38614yes—
254C22×C61C22×C22⋊C411yescharacteristic
255C22×C61C2×#12011yescharacteristic
256C22×C61C2×#12311yescharacteristic
257C22×C61#57712yes—
258C22×C61#57812yes—
259C4×C61—24no—
260C4×C61—44no—
261C4×C61—44no—
262C4×C61#38614yes—
263C4×C61C2×C4×D411yescharacteristic
264C4×C61C2×C4×D411yescharacteristic
265C4×C61C2×#12111yescharacteristic
266C4×C61C2×#12111yescharacteristic
267Dic61—44no—
268Dic61—42no—
269Dic61—42no—
270C2×#536D6111yescharacteristic
271C2×#520—611no—
272C2×#419—611no—
273C22×#103—611no—
274C2×#408—611no—
275#1873—614no—
276#1872—614no—
277#1736—611no—
278#1735—611no—
279C2×C61—24no—
280C2×C61—44no—
281C2×C61C2×#38611yescharacteristic
282C2×C61#289812yes—
283C2×C61#300911yescharacteristic
284C2×C61#301111yescharacteristic
285C2×C61#309712yes—
286C22×#41C2×D6111yescharacteristic
287C23×Q8C2×D6111yescharacteristic
288C22×M16—611no—
289C23×C8—611no—
290#536—24no—
291#536Dic6118yescomplemented
292C2×C4⋊Q8—21no—
293C2×#124—21no—
294C2×#123—22no—
295C2×C22⋊Q8C2×D6114yes—
296#520—1224no—
297#520—1224no—
298C2×C4×Q8—1221no—
299C2×C4×D4—1222no—
300C2×C4×D4—1221no—
301C2×#117—611no—
302C2×#117—612no—
303C2×#117C2×D6111yescharacteristic
304C22×C4⋊C4—611no—
305C22×C4⋊C4—612no—
306#419—1224no—
307C2×#103—614no—
308C2×#103—612no—
309C2×#103—612no—
310C2×#103—1224no—
311#408—1224no—
312C2×C22⋊C8—614no—
313#340—612no—
314#340—614no—
315#340—612no—
316C61#2091311yescharacteristic, complemented, Sylow
317C2×#41—22no—
318C2×#41—22no—
319C2×#41—22no—
320C2×#41—22no—
321C2×#41C2×Dic6114yes—
322C2×#41C22⋊D6114yes—
323C22×Q8—21no—
324C22×Q8—22no—
325C22×Q8—22no—
326C22×Q8—22no—
327C22×Q8—22no—
328C22×Q8—22no—
329C22×Q8C2×Dic6114yescomplemented
330C22×Q8C22⋊D6111yescharacteristic
331C22×Q8C22×D6112yes—
332C22×D4C22⋊D6112yes—
333C22×D4C22⋊D6111yescharacteristic
334C23×C4—21no—
335C23×C4—1222no—
336C23×C4—1221no—
337C23×C4C22×D6112yes—
338C23×C4C22×D6111yescharacteristic
339C2×M16—1222no—
340C2×M16—1222no—
341C2×M16—1222no—
342C22×C8—612no—
343C22×C8—612no—
344C22×C8—1222no—
345C22×C8—1222no—
346C22×C8—1222no—
347C4⋊Q8—28no—
348#124—28no—
349#123—216no—
350C22⋊Q8—28no—
351C22⋊Q8C2×Dic61116yescomplemented
352C4×Q8—1228no—
353C4×D4—12216no—
354C4×D4—1228no—
355#117—22no—
356#117—614no—
357#117—1224no—
358#117—1224no—
359#117—1222no—
360#117C2×Dic6114yescomplemented
361C2×C4⋊C4—24no—
362C2×C4⋊C4—614no—
363C2×C4⋊C4—612no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—1224no—
367C2×C4⋊C4—1224no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4C22×D6112yes—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1221no—
376C2×C42—21no—
377C2×C42—1222no—
378C2×C42—1221no—
379#103—1228no—
380#103—1224no—
381#103—1224no—
382C22⋊C8—1228no—
383C24C2×C22⋊D6112yes—
384C24C2×C22⋊D6111yescharacteristic
385#41—28no—
386#41—28no—
387#41—28no—
388#41—28no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—28no—
398C2×Q8—28no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8C22⋊Dic6114yes—
404C2×Q8C22×Dic6114yes—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4C22⋊Dic6118yes—
410C2×D4C22⋊Dic6114yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—1228no—
421C22×C4—1224no—
422C22×C4—1221no—
423C22×C4—1222no—
424C22×C4—1221no—
425C22×C4—4882no—
426C22×C4—4884no—
427C22×C4D4×D6112yes—
428C22×C4D4×D6112yes—
429C22×C4#1175822412yes—
430C22×C4#1175822412yes—
431C22×C4C22×Dic6114yes—
432C22×C4C22×Dic6112yes—
433C22×C4C2×C22⋊D6114yes—
434C22×C4C2×C22⋊D6112yes—
435C22×C4C2×C22⋊D6112yes—
436C22×C4C2×C22⋊D6111yescharacteristic
437C22×C4C23×D6111yescharacteristic
438M16—2444no—
439C2×C8—1222no—
440C2×C8—1222no—
441C2×C8—1222no—
442C2×C8—1222no—
443C2×C8—1222no—
444C2×C8—1222no—
445C2×C8—2444no—
446C4⋊C4—22no—
447C4⋊C4—216no—
448C4⋊C4—614no—
449C4⋊C4—612no—
450C4⋊C4—612no—
451C4⋊C4—1228no—
452C4⋊C4—1221no—
453C4⋊C4—1222no—
454C4⋊C4—1221no—
455C4⋊C4—1224no—
456C4⋊C4C22×Dic6114yes—
457C22⋊C4—24no—
458C22⋊C4—216no—
459C22⋊C4—1228no—
460C22⋊C4—1224no—
461C42—24no—
462C42—1228no—
463C42—1224no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23C2×C22⋊Dic6114yes—
471C23C2×C22⋊Dic6112yes—
472C23#1451974512yes—
473C23#1451975812yes—
474C23C22×C22⋊D6111yescharacteristic
475Q8—28no—
476Q8—24no—
477Q8—24no—
478Q8—42no—
479Q8—42no—
480Q8—48no—
481D4—216no—
482D4—28no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—48no—
496C2×C4—1222no—
497C2×C4—1224no—
498C2×C4—1222no—
499C2×C4—4884no—
500C2×C4—4888no—
501C2×C4—2444no—
502C2×C4—2442no—
503C2×C4—2442no—
504C2×C4C2×C22⋊Dic6114yes—
505C2×C4C2×C22⋊Dic6112yes—
506C2×C4D4×Dic6114yes—
507C2×C4D4×Dic6114yes—
508C2×C4#1451974512yes—
509C2×C4#1451974914yes—
510C2×C4#1451974914yes—
511C2×C4#1451975812yes—
512C2×C4C23×Dic6111yescharacteristic
513C2×C4C2×D4×D6111yescharacteristic
514C2×C4C2×D4×D6111yescharacteristic
515C2×C4C22×C22⋊D6111yescharacteristic
516C2×C4C2×#1175822411yescharacteristic
517C2×C4C2×#1175822411yescharacteristic
518C8—2442no—
519C8—2442no—
520C22—22no—
521C22—24no—
522C22—24no—
523C22—22no—
524C22—28no—
525C22—24no—
526C22—42no—
527C22—44no—
528C22—42no—
529C22#1683964214yes—
530C22C22×C22⋊Dic6111yescharacteristic
531C22C2×#1451974511yescharacteristic
532C22C2×#1451975811yescharacteristic
533C22#1684057012yes—
534C22#1684057712yes—
535C4—24no—
536C4—44no—
537C4—44no—
538C4—2442no—
539C4—2444no—
540C4—2442no—
541C4#1683964214yes—
542C4C2×D4×Dic6111yescharacteristic
543C4C2×D4×Dic6111yescharacteristic
544C4C2×#1451974911yescharacteristic
545C4C2×#1451974911yescharacteristic
546C2—24no—
547C2—44no—
548C2C2×#1683964211yescharacteristic
549C2#2145435212yes—
550C2#2145509411yescharacteristic
551C2#2145511211yescharacteristic
552C2#2145578212yes—
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#536C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenterC2×#536D61
hypocenterC61#20913
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
2. C2×#16839771C2semidirect
3. C22×#14519701C2semidirect
5. #21446132C2semidirect
6. #21446127C2semidirect
7. #21445540C2semidirect
8. #21445535C2semidirect
34. C61×#536C4semidirect
86. C22×Q8×C61C2×C4semidirect
98. C22⋊Q8×C61C2×C4semidirect
100. #117×C61C2×C4semidirect
291. #536Dic61semidirect
316. C61#20913semidirect
329. C22×Q8C2×Dic61semidirect
351. C22⋊Q8C2×Dic61semidirect
360. #117C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402452
C2×#168397712
C22×#145197012
C2×#168397292
#214461322
#214461272
#214455402
#214455352
C2×C61×#5362
#2091361