Finite Group Databasebeta

← #29690291

Subgroups — #29690291

Order 15616.

1194 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519689C211yesmaximal, characteristic
2#21448201C211yesmaximal, characteristic
3#380×D61C211yesmaximal, characteristic
4#21446019C212yesmaximal, complemented
5#21446015C212yesmaximal, complemented
6#21446013C212yesmaximal, complemented
7#21446005C212yesmaximal, complemented
8#21445846C211yesmaximal, characteristic, complemented
9#21445844C211yesmaximal, characteristic, complemented
10#21445842C211yesmaximal, characteristic, complemented
11C61×#1808C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14C2×#14519735C2212yes—
15C2×C4⋊C4×D61C2212yes—
16C2×C22⋊C4×D61—21no—
17C2×#14519710C2212yescomplemented
18C2×#14519709C2212yescomplemented
19C2×#14519708C2212yescomplemented
20C2×#14519689C418yescomplemented
21C2×#14519689C2211yescharacteristic
22C2×#14519689C2211yescharacteristic
23C2×#14519689C2211yescharacteristic
24C2×#14519689C2211yescharacteristic
25#16839624—21no—
26#16839622C2211yescharacteristic
27#16839622C2211yescharacteristic
28#16839621—21no—
29#16839620—21no—
30#16839619C2212yes—
31#16839619C2212yes—
32#16839617—21no—
33C22⋊C4×Dic61—21no—
34#16839609—21no—
35#95×D61—21no—
36#16839525C2212yescomplemented
37#16839524C2212yescomplemented
38#16839524C2212yescomplemented
39C23×C8×C61C2211yescharacteristic
40C2×#107×C61C2212yescomplemented
41C61×#380C2211yescharacteristic
42C61×#380C2211yescharacteristic
43C61×#340C2212yescomplemented
44C24×D61D411yescharacteristic
45C23×Dic61D411yescharacteristic
46C22×C4×D61—22no—
47C22×C4×D61—42no—
48C22×C4×D61—42no—
49C22×C4×D61C2×C414yes—
50C22×C4×D61D411yescharacteristic
51C22×C4×D61D411yescharacteristic
52C22×C4×D61Q814yes—
53C22×C4×D61C2311yescharacteristic
54C22×C4×D61C2311yescharacteristic
55C22×#92418—42no—
56C22×#92418C2×C414yescomplemented
57C22×#92418C2311yescharacteristic
58C22×#92418C2311yescharacteristic
59#14519735—42no—
60#14519735—42no—
61C2×C4⋊Dic61C2312yes—
62C2×C4⋊Dic61C2312yes—
63C2×#11758196—22no—
64C4⋊C4×D61—42no—
65C4⋊C4×D61—42no—
66C2×C22⋊Dic61—21no—
67C2×C22⋊Dic61—21no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C22⋊C4×D61—44no—
72C2×C4×Dic61—22no—
73#14519710—42no—
74#14519709—42no—
75#14519708—42no—
76#14519689C2×C414yes—
77#14519689C2×C414yes—
78#14519689C2×C414yescomplemented
79#14519689C2×C414yescomplemented
80#14519684—21no—
81#14519684—21no—
82#14519684—22no—
83#14519684—22no—
84C23×C4×C61C2311yescharacteristic
85C22×C8×C61—42no—
86C22×C8×C61C2×C414yescomplemented
87C22×C8×C61C2311yescharacteristic
88C22×C8×C61C2311yescharacteristic
89C2×C4⋊C4×C61C2312yes—
90C2×C4⋊C4×C61C2312yes—
91C2×C22⋊C4×C61—21no—
92C2×C22⋊C4×C61—21no—
93#107×C61—42no—
94#95×C61—21no—
95#95×C61—21no—
96C24×C61C2×D411yescharacteristic
97C23×D61—22no—
98C23×D61—22no—
99C23×D61—24no—
100C23×D61—42no—
101C23×D61C4⋊C414yes—
102C23×D61C2×D411yescharacteristic
103C23×D61C2×D411yescharacteristic
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61C22×C412yes—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×Q812yes—
111C22×C4×C61C2411yescharacteristic
112C22×Dic61—22no—
113C22×Dic61—22no—
114C22×Dic61—42no—
115C22×Dic61C4⋊C414yes—
116C22×Dic61C2×D411yescharacteristic
117C22×Dic61C2×D411yescharacteristic
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—44no—
123C2×C4×D61—44no—
124C2×C4×D61—48no—
125C2×C4×D61—42no—
126C2×C4×D61—42no—
127C2×C4×D61—44no—
128C2×C4×D61—44no—
129C2×C4×D61—44no—
130C2×C4×D61C4⋊C414yes—
131C2×C4×D61C4⋊C414yes—
132C2×C4×D61C22×C412yes—
133C2×C4×D61C22×C412yes—
134C2×C4×D61C2×Q812yes—
135C2×C4×D61C2×Q812yes—
136C2×C8×C61—22no—
137C2×C8×C61—22no—
138C2×C8×C61—44no—
139C2×C8×C61C22×C412yes—
140C2×C8×C61C22×C412yes—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C22⋊C4×C61—42no—
146C22⋊C4×C61—42no—
147C22⋊Dic61—42no—
148C22⋊Dic61—42no—
149#11758198—44no—
150#11758198—44no—
151#11758198—44no—
152C4⋊Dic61—22no—
153C4⋊Dic61—22no—
154C4⋊Dic61—22no—
155C4⋊Dic61—22no—
156#11758196—44no—
157C4×Dic61—44no—
158C2×#92418—22no—
159C2×#92418—22no—
160C2×#92418—44no—
161C2×#92418C22×C412yes—
162C2×#92418C22×C412yes—
163C23×C61—22no—
164C23×C61—22no—
165C23×C61C2×C4⋊C412yes—
166C23×C61AGL(1,ℤ/8)12yes—
167C23×C61#13712yes—
168C23×C61C22×D411yescharacteristic
169C22×D61—22no—
170C22×D61—21no—
171C22×D61—21no—
172C22×D61—22no—
173C22×D61—22no—
174C22×D61—21no—
175C22×D61—21no—
176C22×D61—28no—
177C22×D61—28no—
178C22×D61—44no—
179C22×D61—48no—
180C22×D61—48no—
181C22×D61C2×C4⋊C412yes—
182C22×D61C2×C4⋊C412yes—
183C22×D61#13514yes—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—24no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61C2×C4⋊C412yes—
195C2×C4×C61C23×C411yescharacteristic
196C2×C4×C61C22×Q811yescharacteristic
197C2×C4×C61H(5,𝔽2)12yes—
198C2×C4×C61H(5,𝔽2)12yes—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—44no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208C2×Dic61C2×C4⋊C412yes—
209C2×Dic61C2×C4⋊C412yes—
210C4×D61—22no—
211C4×D61—22no—
212C4×D61—22no—
213C4×D61—22no—
214C4×D61—24no—
215C4×D61—24no—
216C4×D61—44no—
217C4×D61—44no—
218C4×D61—88no—
219C8×C61—22no—
220C8×C61—22no—
221#92418—22no—
222#92418—22no—
223#21941—611nomaximal, Sylow
224C22×C61—22no—
225C22×C61—22no—
226C22×C61—24no—
227C22×C61—24no—
228C22×C61#42814yes—
229C22×C61C22×C4⋊C411yescharacteristic
230C22×C61#53412yes—
231C22×C61C2×#13512yes—
232C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
233C22×C61C2×#13711yescharacteristic
234C2×D61—22no—
235C2×D61—22no—
236C2×D61—24no—
237C2×D61—24no—
238C2×D61—22no—
239C2×D61—22no—
240C2×D61—22no—
241C2×D61—416no—
242C2×D61—42no—
243C2×D61—42no—
244C2×D61#42714yes—
245C4×C61—22no—
246C4×C61—82no—
247C4×C61—82no—
248C4×C61#51811yescharacteristic
249C4×C61#51811yescharacteristic
250C4×C61#54312yes—
251Dic61—22no—
252Dic61—22no—
253Dic61—82no—
254Dic61—82no—
255C22×C22⋊C8—611no—
256#2280—611no—
257C2×#380—611no—
258#1854—612no—
259#1854—612no—
260#1853—612no—
261#1852—612no—
262#1808—611no—
263#1808D6111yescharacteristic, complemented
264#1807—611no—
265#1807—611no—
266C2×C61—24no—
267C2×C61#231511yescharacteristic
268C2×C61C2×#42711yescharacteristic
269C2×C61C2×#42811yescharacteristic
270C2×C61#318812yes—
271C2×C61#319012yes—
272D61—24no—
273D61—42no—
274C24×C4—611no—
275C23×C8—611no—
276C23×C8C2×D6111yescharacteristic
277C2×#117—612no—
278C22×C4⋊C4—612no—
279C22×C22⋊C4—1221no—
280C2×#107—612no—
281C2×#107C2×D6112yescomplemented
282C2×#106—612no—
283C2×#106—612no—
284C2×C22⋊C8—618no—
285C2×C22⋊C8—611no—
286C2×C22⋊C8—611no—
287C2×C22⋊C8—611no—
288C2×C22⋊C8—611no—
289#381—1221no—
290#380—611no—
291#380—611no—
292#380—612no—
293#380—612no—
294#380—1221no—
295#380—1221no—
296#380C2×D6111yescharacteristic
297#380C2×D6111yescharacteristic
298#379—1221no—
299C4×C22⋊C4—1221no—
300C4×C22⋊C4—1221no—
301C2×#95—1221no—
302#340—612no—
303#340—612no—
304#340—612no—
305#340C2×D6112yescomplemented
306C61#2194111yescharacteristic, complemented, Sylow
307C25—611no—
308C23×C4—611no—
309C23×C4—611no—
310C23×C4—611no—
311C23×C4—614no—
312C23×C4—611no—
313C23×C4—611no—
314C23×C4—614no—
315C23×C4—1222no—
316C23×C4—2442no—
317C23×C4—2442no—
318C23×C4C22×D6111yescharacteristic
319C22×C8—42no—
320C22×C8—611no—
321C22×C8—611no—
322C22×C8—614no—
323C22×C8—2442no—
324C22×C8C4×D6114yescomplemented
325C22×C8C22×D6111yescharacteristic
326C22×C8C22×D6111yescharacteristic
327#117—2442no—
328#117—2442no—
329C2×C4⋊C4—612no—
330C2×C4⋊C4—612no—
331C2×C4⋊C4—1222no—
332C2×C4⋊C4—2442no—
333C2×C4⋊C4—2442no—
334C2×C4⋊C4C22×D6112yes—
335C2×C4⋊C4C22×D6112yes—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—1222no—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1222no—
341C2×C22⋊C4—1221no—
342C2×C22⋊C4—1221no—
343C2×C22⋊C4—2444no—
344C2×C42—1222no—
345#107—42no—
346#107—2442no—
347#106—2442no—
348#106—2442no—
349C22⋊C8—614no—
350C22⋊C8—614no—
351C22⋊C8—614no—
352C22⋊C8—614no—
353#95—21no—
354#95—21no—
355#95—1222no—
356#95—1222no—
357#95—1221no—
358#95—1221no—
359C24—611no—
360C24—611no—
361C24—614no—
362C24—1224no—
363C24—1222no—
364C24—1222no—
365C24—2442no—
366C24D4×D6111yescharacteristic
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—611no—
372C22×C4—611no—
373C22×C4—614no—
374C22×C4—612no—
375C22×C4—612no—
376C22×C4—612no—
377C22×C4—612no—
378C22×C4—614no—
379C22×C4—614no—
380C22×C4—1222no—
381C22×C4—1222no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4—1222no—
385C22×C4—1222no—
386C22×C4—2444no—
387C22×C4—2444no—
388C22×C4—2448no—
389C22×C4—2442no—
390C22×C4—2442no—
391C22×C4—2444no—
392C22×C4—2444no—
393C22×C4—2444no—
394C22×C4—2442no—
395C22×C4C2×C4×D6112yes—
396C22×C4D4×D6111yescharacteristic
397C22×C4Q8×D6112yes—
398C22×C4C23×D6111yescharacteristic
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—44no—
402C2×C8—612no—
403C2×C8—612no—
404C2×C8—1222no—
405C2×C8—1222no—
406C2×C8—2444no—
407C2×C8C2×C4×D6112yes—
408C2×C8C2×C4×D6112yes—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—1222no—
414C4⋊C4—1222no—
415C4⋊C4—1222no—
416C4⋊C4—1222no—
417C4⋊C4—2444no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—2444no—
421C22⋊C4—2444no—
422C22⋊C4—2444no—
423C22⋊C4—2442no—
424C22⋊C4—2442no—
425C42—2444no—
426C23—22no—
427C23—22no—
428C23—612no—
429C23—614no—
430C23—612no—
431C23—1228no—
432C23—1221no—
433C23—1221no—
434C23—1222no—
435C23—1222no—
436C23—1221no—
437C23—1221no—
438C23—1228no—
439C23—1222no—
440C23—2444no—
441C23—2448no—
442C23—2448no—
443C23C4⋊C4×D6112yes—
444C23#1451980112yes—
445C23#1451980912yes—
446C23C2×D4×D6111yescharacteristic
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—612no—
458C2×C4—612no—
459C2×C4—1222no—
460C2×C4—1222no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—1222no—
464C2×C4—1222no—
465C2×C4—1224no—
466C2×C4—1224no—
467C2×C4—4888no—
468C2×C4—2444no—
469C2×C4—2444no—
470C2×C4—2444no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4—2442no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4C4⋊C4×D6112yes—
478C2×C4C22×C4×D6111yescharacteristic
479C2×C4C2×Q8×D6111yescharacteristic
480C2×C4#1451982212yes—
481C2×C4#1451982312yes—
482C8—22no—
483C8—22no—
484C8—1222no—
485C8—1222no—
486C22—22no—
487C22—24no—
488C22—24no—
489C22—22no—
490C22—614no—
491C22—1222no—
492C22—1224no—
493C22—1224no—
494C22—1222no—
495C22—1222no—
496C22—1222no—
497C22—1222no—
498C22—24416no—
499C22—2442no—
500C22—2442no—
501C22#1683980314yes—
502C22C2×C4⋊C4×D6111yescharacteristic
503C22#1684032012yes—
504C22#135×D6112yes—
505C22C2×#1451980111yescharacteristic
506C22C2×#1451980911yescharacteristic
507C4—22no—
508C4—82no—
509C4—82no—
510C4—1222no—
511C4—1222no—
512C4—4882no—
513C4—4882no—
514C4#1684023511yescharacteristic
515C4#1684023611yescharacteristic
516C4#1684038712yes—
517C2—24no—
518C2—1224no—
519C2—2442no—
520C2#2144843211yescharacteristic
521C2#427×D6111yescharacteristic
522C2C2×#1683980311yescharacteristic
523C2#2145657912yes—
524C2#2145660112yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1808C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenter#1808D61
hypocenterC61#21941
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1808

Complements

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

NH ≅ G/Ntype
4. #21446019C2semidirect
5. #21446015C2semidirect
6. #21446013C2semidirect
7. #21446005C2semidirect
8. #21445846C2semidirect
9. #21445844C2semidirect
10. #21445842C2semidirect
11. C61×#1808C2semidirect
17. C2×#14519710C22semidirect
18. C2×#14519709C22semidirect
19. C2×#14519708C22semidirect
20. C2×#14519689C4semidirect
36. #16839525C22semidirect
37. #16839524C22semidirect
38. #16839524C22semidirect
40. C2×#107×C61C22semidirect
43. C61×#340C22semidirect
56. C22×#92418C2×C4semidirect
78. #14519689C2×C4semidirect
79. #14519689C2×C4semidirect
86. C22×C8×C61C2×C4semidirect
263. #1808D61semidirect
281. C2×#107C2×D61semidirect
305. #340C2×D61semidirect
306. C61#21941semidirect
324. C22×C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145196892
#214482012
#380×D612
#214460192
#214460152
#214460132
#214460052
#214458462
#214458442
#214458422
C61×#18082
#2194161