Finite Group Databasebeta

← #29688959

Subgroups — #29688959

Order 15616.

1554 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840269C211yesmaximal, characteristic
2#21448199C211yesmaximal, characteristic, complemented
3#381×D61C211yesmaximal, characteristic, complemented
4#21446274C214yesmaximal, complemented
5#21446269C214yesmaximal, complemented
6#21445894C211yesmaximal, characteristic, complemented
7#21445894C211yesmaximal, characteristic, complemented
8#21445737C211yesmaximal, characteristic, complemented
9C61×#1782C211yesmaximal, characteristic, complemented
10C23×C4×D61C2211yescharacteristic
11#16840269—22no—
12#16840269—22no—
13#16840269—22no—
14#16840269—22no—
15C2×#14519758C2212yes—
16C2×#14519753C412yescomplemented
17C2×#14519748C412yescomplemented
18C2×#14519745C2212yes—
19C2×C4×C22⋊D61C414yescomplemented
20C2×#14519736C2211yescharacteristic
21C2×#14519733C2212yes—
22C2×C22⋊C4×D61C2212yes—
23C2×#14519692C2214yes—
24#16839625C2211yescharacteristic
25#16839625C2211yescharacteristic
26#16839624—22no—
27#16839623C2212yes—
28#16839621—22no—
29#16839617C2212yes—
30#16839609—22no—
31#95×D61—22no—
32#16839528C2214yes—
33#16839528C2214yes—
34C22×#41×C61C2211yescharacteristic
35C2×#99×C61C2214yes—
36C61×#381C2211yescharacteristic
37C61×#381C2211yescharacteristic
38C24×D61C2311yescharacteristic
39C22×C22⋊D61C2×C414yes—
40C23×Dic61C2311yescharacteristic
41C22×C4×D61—22no—
42C22×C4×D61—44no—
43C22×C4×D61C2×C412yes—
44C22×C4×D61C2×C412yes—
45C22×C4×D61D412yes—
46C22×C4×D61D412yes—
47C22×C4×D61D412yes—
48C22×C4×D61D412yes—
49#14519758—24no—
50#14519758—22no—
51#14519758—22no—
52#14519753—22no—
53#14519753—22no—
54#14519753—22no—
55#14519753—22no—
56#14519748—22no—
57#14519748—22no—
58#14519748—22no—
59#14519748—22no—
60#14519745—22no—
61#14519745—24no—
62#14519745—22no—
63C4×C22⋊D61—24no—
64C4×C22⋊D61—24no—
65C4×C22⋊D61—24no—
66C4×C22⋊D61—24no—
67#14519736—22no—
68#14519736—22no—
69#14519733—44no—
70C2×C4⋊Dic61C2×C412yescomplemented
71C2×#11758196—22no—
72C2×#11758196C2×C414yescomplemented
73C2×C22⋊Dic61C2×C414yescomplemented
74C2×C22⋊Dic61C2312yes—
75C2×C22⋊Dic61C2312yes—
76C2×C22⋊Dic61C2312yes—
77C2×#11758198—22no—
78C2×#11758198—24no—
79C2×#11758198C2×C414yes—
80C22⋊C4×D61—44no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61C2×C412yescomplemented
83#14519692—44no—
84#14519684—22no—
85#14519684—22no—
86#14519684—24no—
87C2×#41×C61—22no—
88C2×#41×C61—22no—
89C2×#41×C61—44no—
90C22×Q8×C61C2×C411yescharacteristic, complemented
91C22×D4×C61C2×C411yescharacteristic, complemented
92C22×D4×C61C2312yes—
93C23×C4×C61C2×C412yescomplemented
94C23×C4×C61C2311yescharacteristic
95C2×C22⋊C4×C61C2312yes—
96C2×C22⋊C4×C61C2312yes—
97#99×C61—44no—
98#95×C61—22no—
99#95×C61—22no—
100C24×C61C22×C412yes—
101C24×C61C2411yescharacteristic
102C23×D61—24no—
103C23×D61—44no—
104C23×D61C22×C412yes—
105C23×D61C2×D411yescharacteristic
106C23×D61C2×D412yes—
107C23×D61C2×D411yescharacteristic
108#41×C61—48no—
109#41×C61—48no—
110C2×Q8×C61—22no—
111C2×Q8×C61—42no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—24no—
115C2×D4×C61—24no—
116C2×D4×C61—22no—
117C2×D4×C61—42no—
118C22×C4×C61—24no—
119C22×C4×C61—24no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—42no—
123C22×C4×C61—42no—
124C22×C4×C61C22×C412yes—
125C22×C4×C61C22×C411yescharacteristic
126C22×C4×C61C22×C411yescharacteristic
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D412yes—
129C2×C22⋊D61—24no—
130C2×C22⋊D61—24no—
131C2×C22⋊D61—48no—
132C22×Dic61—24no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61C22×C412yes—
136C22×Dic61C22×C414yes—
137C22×Dic61C2×D412yes—
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C2×C4×D61—44no—
141C2×C4×D61—44no—
142C2×C4×D61—44no—
143C2×C4×D61—44no—
144C2×C4×D61—44no—
145C2×C4×D61—44no—
146C2×C4×D61—48no—
147C2×C4×D61—44no—
148C2×C4×D61C22⋊C412yes—
149C2×C4×D61C22⋊C412yes—
150C2×C4×D61C22⋊C412yes—
151C2×C4×D61C22⋊C412yes—
152C2×C4×D61C22⋊C412yes—
153C2×C4×D61C22⋊C412yes—
154C2×C4×D61C22⋊C412yes—
155C2×C4×D61C22⋊C412yes—
156C22⋊C4×C61—24no—
157C22⋊C4×C61—24no—
158C22⋊Dic61—22no—
159C22⋊Dic61—22no—
160C22⋊Dic61—24no—
161C22⋊Dic61—24no—
162C22⋊Dic61—24no—
163C22⋊Dic61—24no—
164#11758198—24no—
165#11758198—24no—
166#11758198—44no—
167#11758198—48no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170#11758196—24no—
171#11758196—24no—
172#11758196—44no—
173C4×Dic61—24no—
174C4×Dic61—44no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—42no—
179C23×C61—44no—
180C23×C61C23×C411yescharacteristic
181C23×C61C22×D411yescharacteristic
182C23×C61C22×D411yescharacteristic
183C23×C61H(5,𝔽2)14yes—
184C22×D61—22no—
185C22×D61—24no—
186C22×D61—24no—
187C22×D61—22no—
188C22×D61—24no—
189C22×D61—24no—
190C22×D61—24no—
191C22×D61—42no—
192C22×D61—42no—
193C22×D61—42no—
194C22×D61—416no—
195C22×D61C2×C22⋊C412yes—
196C22×D61C2×C22⋊C412yes—
197Q8×C61—44no—
198D4×C61—44no—
199D4×C61—44no—
200D4×C61—44no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—24no—
204C2×C4×C61—24no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—48no—
209C2×C4×C61—44no—
210C2×C4×C61—44no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61C2×C22⋊C412yes—
214C2×C4×C61C2×C22⋊C412yes—
215C22⋊D61—48no—
216C22⋊D61—48no—
217C2×Dic61—24no—
218C2×Dic61—22no—
219C2×Dic61—22no—
220C2×Dic61—24no—
221C2×Dic61—24no—
222C2×Dic61—44no—
223C2×Dic61—44no—
224C2×Dic61—44no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61C2×C22⋊C412yes—
228C2×Dic61C2×C22⋊C412yes—
229C4×D61—42no—
230C4×D61—42no—
231C4×D61—42no—
232C4×D61—42no—
233C4×D61—42no—
234C4×D61—42no—
235C4×D61—42no—
236C4×D61—42no—
237C4×D61—88no—
238#21748—611nomaximal, Sylow
239C22×C61—22no—
240C22×C61—22no—
241C22×C61—22no—
242C22×C61—44no—
243C22×C61—42no—
244C22×C61—44no—
245C22×C61—82no—
246C22×C61—82no—
247C22×C61C22×C22⋊C411yescharacteristic
248C22×C61#51812yes—
249C22×C61#53412yes—
250C22×C61#53412yes—
251C2×D61—28no—
252C2×D61—22no—
253C2×D61—22no—
254C2×D61—22no—
255C2×D61—22no—
256C2×D61—22no—
257C2×D61—48no—
258C2×D61—48no—
259C2×D61—44no—
260C2×D61#41014yes—
261C4×C61—22no—
262C4×C61—22no—
263C4×C61—42no—
264C4×C61—82no—
265C4×C61—82no—
266Dic61—22no—
267Dic61—22no—
268Dic61—44no—
269Dic61—82no—
270Dic61—82no—
271C2×#525—611no—
272#2280—611no—
273C2×#381—611no—
274#1899—614no—
275#1898—614no—
276#1820—611no—
277#1820—611no—
278#1782—611no—
279#1782D6111yescharacteristic, complemented
280C2×C61—22no—
281C2×C61—22no—
282C2×C61—22no—
283C2×C61—82no—
284C2×C61#230811yescharacteristic
285C2×C61C2×#41012yes—
286D61—24no—
287D61—42no—
288C22×#41C2×D6111yescharacteristic
289C24×C4—611no—
290#525—1222no—
291#525—1222no—
292#525—1222no—
293#525—1222no—
294C2×#123—612no—
295C2×C22⋊Q8—612no—
296C2×#121—612no—
297C2×#120—612no—
298C2×C4×D4—614no—
299C2×#117—611no—
300C2×#117—612no—
301C22×C22⋊C4—612no—
302C2×#99—614no—
303C2×#99C2×D6114yes—
304#381—611no—
305#381—611no—
306#381—612no—
307#381—1222no—
308#381C2×D6111yescharacteristic
309#381C2×D6111yescharacteristic
310#380—1222no—
311#379—612no—
312C4×C22⋊C4—1222no—
313C2×#95—1222no—
314#342—614no—
315#342—614no—
316C61#2174811yescharacteristic, complemented, Sylow
317C25—611no—
318C2×#41—22no—
319C2×#41—22no—
320C2×#41—44no—
321C22×Q8C4×D6111yescharacteristic, complemented
322C22×D4—614no—
323C22×D4C4×D6111yescharacteristic, complemented
324C22×D4C22×D6112yes—
325C23×C4—611no—
326C23×C4—612no—
327C23×C4—612no—
328C23×C4—612no—
329C23×C4—612no—
330C23×C4—612no—
331C23×C4—612no—
332C23×C4—1222no—
333C23×C4—2444no—
334C23×C4C4×D6112yescomplemented
335C23×C4C22×D6111yescharacteristic
336#123—1224no—
337#123—1222no—
338#123—1222no—
339C22⋊Q8—1222no—
340C22⋊Q8—1222no—
341C22⋊Q8—1222no—
342C22⋊Q8—1222no—
343#121—1222no—
344#121—1222no—
345#121—1222no—
346#121—1222no—
347#120—1222no—
348#120—1224no—
349#120—1222no—
350C4×D4—1224no—
351C4×D4—1224no—
352C4×D4—1224no—
353C4×D4—1224no—
354#117—1222no—
355#117—1222no—
356#117—2444no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—614no—
359C2×C4⋊C4—1222no—
360C2×C22⋊C4—614no—
361C2×C22⋊C4—612no—
362C2×C22⋊C4—614no—
363C2×C22⋊C4—612no—
364C2×C22⋊C4—612no—
365C2×C22⋊C4—1222no—
366C2×C22⋊C4—1224no—
367C2×C22⋊C4—2444no—
368C2×C22⋊C4C22×D6112yes—
369C2×C22⋊C4C22×D6112yes—
370C2×C42—612no—
371C2×C42—1222no—
372#99—44no—
373#99—2444no—
374#95—22no—
375#95—22no—
376#95—1224no—
377#95—1222no—
378#95—1222no—
379C24—611no—
380C24—612no—
381C24—612no—
382C24—611no—
383C24—1224no—
384C24—2444no—
385C24C2×C4×D6112yes—
386C24C23×D6111yescharacteristic
387#41—48no—
388#41—48no—
389C2×Q8—22no—
390C2×Q8—42no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—42no—
397C2×D4—1224no—
398C2×D4—1224no—
399C2×D4—2448no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—612no—
407C22×C4—612no—
408C22×C4—611no—
409C22×C4—611no—
410C22×C4—612no—
411C22×C4—612no—
412C22×C4—612no—
413C22×C4—612no—
414C22×C4—612no—
415C22×C4—612no—
416C22×C4—612no—
417C22×C4—612no—
418C22×C4—614no—
419C22×C4—1224no—
420C22×C4—1222no—
421C22×C4—1222no—
422C22×C4—2444no—
423C22×C4—2444no—
424C22×C4—2444no—
425C22×C4—2444no—
426C22×C4—2444no—
427C22×C4—2444no—
428C22×C4—2444no—
429C22×C4—2448no—
430C22×C4C2×C4×D6112yes—
431C22×C4C2×C4×D6111yescharacteristic
432C22×C4C2×C4×D6111yescharacteristic
433C22×C4D4×D6112yes—
434C22×C4D4×D6112yes—
435C4⋊C4—1224no—
436C4⋊C4—1224no—
437C4⋊C4—1222no—
438C4⋊C4—1222no—
439C4⋊C4—2444no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—1224no—
443C22⋊C4—1224no—
444C22⋊C4—1224no—
445C22⋊C4—1224no—
446C22⋊C4—1222no—
447C22⋊C4—1222no—
448C22⋊C4—1224no—
449C22⋊C4—1224no—
450C22⋊C4—2444no—
451C22⋊C4—2448no—
452C42—1224no—
453C42—2444no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—42no—
458C23—44no—
459C23—612no—
460C23—612no—
461C23—1224no—
462C23—1224no—
463C23—1224no—
464C23—1222no—
465C23—1224no—
466C23—1222no—
467C23—1224no—
468C23—2442no—
469C23—2442no—
470C23—2442no—
471C23—24416no—
472C23C22×C4×D6111yescharacteristic
473C23C2×D4×D6111yescharacteristic
474C23C2×D4×D6111yescharacteristic
475C23#1451982214yes—
476Q8—44no—
477D4—44no—
478D4—44no—
479D4—44no—
480D4—2448no—
481D4—2448no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—48no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—612no—
495C2×C4—612no—
496C2×C4—1224no—
497C2×C4—1222no—
498C2×C4—1222no—
499C2×C4—1224no—
500C2×C4—1224no—
501C2×C4—4888no—
502C2×C4—2442no—
503C2×C4—2442no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4—2444no—
511C2×C4—2444no—
512C2×C4—2442no—
513C2×C4—2442no—
514C2×C4—2444no—
515C2×C4C22⋊C4×D6112yes—
516C2×C4C22⋊C4×D6112yes—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—44no—
521C22—44no—
522C22—42no—
523C22—82no—
524C22—82no—
525C22—614no—
526C22—1228no—
527C22—1222no—
528C22—1222no—
529C22—1222no—
530C22—1222no—
531C22—1222no—
532C22—2448no—
533C22—2448no—
534C22—2444no—
535C22C2×C22⋊C4×D6111yescharacteristic
536C22#1684023512yes—
537C22#1684032212yes—
538C22#1684032212yes—
539C4—22no—
540C4—22no—
541C4—42no—
542C4—82no—
543C4—82no—
544C4—1222no—
545C4—1222no—
546C4—4882no—
547C4—4882no—
548C4—2444no—
549C2—22no—
550C2—22no—
551C2—22no—
552C2—82no—
553C2—1224no—
554C2—2442no—
555C2#2144839511yescharacteristic
556C2#410×D6112yes—
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1782C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840269C2
cosocleC24×C61C24
hypercenter#1782D61
hypocenterC61#21748
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
2. #21448199C2semidirect
3. #381×D61C2semidirect
4. #21446274C2semidirect
5. #21446269C2semidirect
6. #21445894C2semidirect
7. #21445894C2semidirect
8. #21445737C2semidirect
9. C61×#1782C2semidirect
16. C2×#14519753C4semidirect
17. C2×#14519748C4semidirect
19. C2×C4×C22⋊D61C4semidirect
70. C2×C4⋊Dic61C2×C4semidirect
72. C2×#11758196C2×C4semidirect
73. C2×C22⋊Dic61C2×C4semidirect
82. C2×C4×Dic61C2×C4semidirect
90. C22×Q8×C61C2×C4semidirect
91. C22×D4×C61C2×C4semidirect
93. C23×C4×C61C2×C4semidirect
279. #1782D61semidirect
316. C61#21748semidirect
321. C22×Q8C4×D61semidirect
323. C22×D4C4×D61semidirect
334. C23×C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402692
#214481992
#381×D612
#214462742
#214462692
#214458942
#214457372
C61×#17822
#2174861