Finite Group Databasebeta

← #29667874

Subgroups — #29667874

Order 15616.

848 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456760C211yesmaximal, characteristic, complemented
2#21456744C211yesmaximal, characteristic, complemented
3#21456378C211yesmaximal, characteristic, complemented
4#21456361C211yesmaximal, characteristic, complemented
5#21456269C211yesmaximal, characteristic
6#21445449C212yesmaximal, complemented
7#21445440C212yesmaximal, complemented
8#21445419C212yesmaximal, complemented
9#21445410C212yesmaximal, complemented
10C8×#11758194C211yesmaximal, characteristic, complemented
11C2×C61×#486C211yesmaximal, characteristic, complemented
12C2×#14519803—22no—
13#16840319C2211yescharacteristic
14#16840312C2211yescharacteristic
15#16840168—22no—
16#16840165C2212yescomplemented
17#16840160C2211yescharacteristic, complemented
18#16840158C2212yescomplemented
19#16840149—22no—
20#16840138C2212yes—
21#16840136C2212yescomplemented
22#16840135C2211yescharacteristic, complemented
23#16840134C2212yes—
24#16840134C2212yes—
25#16840113C2211yescharacteristic, complemented
26#16840112C2212yescomplemented
27#16840109C2211yescharacteristic, complemented
28#16840108C2212yes—
29#16839991—22no—
30#16839964—22no—
31#16839959—22no—
32#16839913—22no—
33#16839901—22no—
34C4×#11758190—22no—
35#16839815C2211yescharacteristic
36#16839802—22no—
37C4×#11758194C2211yescharacteristic
38#16839484C2212yescomplemented
39C8×#92418C2212yescomplemented
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×#127×C61C2211yescharacteristic
42C61×#486—22no—
43C61×#486C2212yes—
44C61×#486C2212yes—
45C2×#102×C61—22no—
46C2×C4×C8×C61C2211yescharacteristic
47C2×#11758222—21no—
48#14519803—48no—
49C2×#11758190—22no—
50C2×#11758200—24no—
51C2×#11758200—24no—
52C2×#11758201—24no—
53C2×#11758199—24no—
54C2×#11758194D412yes—
55#14519777C2311yescharacteristic
56#14519777C2311yescharacteristic
57#14519777C2311yescharacteristic
58#14519774C2311yescharacteristic
59#14519769—21no—
60#14519765—21no—
61#14519756—21no—
62#14519756—21no—
63#14519751—21no—
64#14519751—21no—
65C4×Dic122—21no—
66C4×Dic122—21no—
67C4×D244—21no—
68C4×D244—21no—
69#14519736—21no—
70#14519710—24no—
71#14519709—22no—
72#14519708—24no—
73C4⋊#92418D412yes—
74#14519702—22no—
75#14519701—24no—
76#14519699—22no—
77#14519698—24no—
78C22⋊#92418D412yes—
79#14519688D412yes—
80C4×#92418C2312yes—
81C4×#92418C2312yes—
82C22×Q8×C61—21no—
83C22×D4×C61—21no—
84C22×D4×C61—21no—
85C22×D4×C61—21no—
86C22×C8×C61D412yes—
87C4⋊Q8×C61—21no—
88C4⋊Q8×C61—21no—
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61C2312yes—
91#127×C61—21no—
92#127×C61—21no—
93#127×C61—21no—
94#127×C61C2312yes—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C42×C61C2311yescharacteristic
98#102×C61—24no—
99#102×C61—24no—
100C4×C8×C61D412yes—
101C4×C8×C61C2311yescharacteristic
102C4×C8×C61C2311yescharacteristic
103C24×C61—41no—
104C2×Q8×C61—22no—
105C2×Q8×C61—41no—
106C2×Q8×C61—82no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—44no—
111C2×D4×C61—41no—
112C2×D4×C61—41no—
113C2×D4×C61—41no—
114C2×D4×C61—44no—
115C2×D4×C61—82no—
116C22×C4×C61—41no—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C2×C22⋊D61—41no—
121#11758222—44no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—21no—
125C2×C4×D61—41no—
126C2×Dic122—21no—
127C2×Dic122—21no—
128C2×Dic122—21no—
129C2×C8×C61—42no—
130C2×C8×C61C2×D412yes—
131C2×C8×C61C2×D412yes—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—41no—
135C4⋊C4×C61—41no—
136C42×C61C2×D411yescharacteristic
137C42×C61C2×D411yescharacteristic
138C42×C61C2×D411yescharacteristic
139C42×C61C2411yescharacteristic
140C22⋊Dic61—41no—
141#11758201—84no—
142#11758200—48no—
143#11758200—84no—
144#11758199—48no—
145#11758198—41no—
146#11758198—41no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153#11758196—41no—
154#11758196—41no—
155C4×Dic61—41no—
156#11758194SD1618yescomplemented
157C2×#92418C2×D414yes—
158C2×#92418C2×D414yes—
159#11758190—44no—
160C23×C61—42no—
161C23×C61—82no—
162C23×C61—82no—
163C23×C61—82no—
164C23×C61#12711yescharacteristic
165C22×D61—41no—
166Q8×C61—44no—
167Q8×C61—82no—
168D4×C61—44no—
169D4×C61—44no—
170D4×C61—44no—
171D4×C61—82no—
172D4×C61—82no—
173D4×C61—82no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—42no—
177C2×C4×C61—82no—
178C2×C4×C61#12711yescharacteristic
179C2×C4×C61#12711yescharacteristic
180C2×C4×C61#12711yescharacteristic
181C2×C4×C61C2×SD1614yes—
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C22×D411yescharacteristic
184C2×C4×C61C22×D411yescharacteristic
185C22⋊D61—82no—
186C2×Dic61—41no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189D244—42no—
190D244—42no—
191D244—42no—
192C4×D61—82no—
193Dic122—42no—
194Dic122—42no—
195Dic122—42no—
196C8×C61—22no—
197C8×C61—22no—
198#92418—24no—
199#92418C2×SD1618yes—
200#19623—611nomaximal, Sylow
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—82no—
204C22×C61—82no—
205C22×C61—82no—
206C22×C61—164no—
207C22×C61#49212yes—
208C22×C61C2×#12711yescharacteristic
209C2×D61—81no—
210C2×D61—81no—
211C2×D61—81no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—82no—
215C4×C61#49212yes—
216C4×C61C22×SD1612yes—
217C4×C61#57512yes—
218C4×C61#57612yes—
219Dic61—81no—
220Dic61—81no—
221Dic61—81no—
222#3208—611no—
223#3206—611no—
224#3167—611no—
225#3165—611no—
226#3151—611no—
227C2×#486D6111yescharacteristic, complemented
228#1715—612no—
229#1714—612no—
230#1708—612no—
231#1706—612no—
232C8×M16—611no—
233C2×C61—22no—
234C2×C61—162no—
235C2×C61C2×#49211yescharacteristic
236C2×C61#315311yescharacteristic
237C2×C61#315511yescharacteristic
238D61—161no—
239C2×AGL(1,ℤ/8)—1222no—
240#533—611no—
241#532—611no—
242C2×C4⋊Q8C2×D6111yescharacteristic
243C2×#127C2×D6111yescharacteristic
244#501—1222no—
245#500—612no—
246#498—612no—
247#498—611no—
248#496—1222no—
249#493—612no—
250#492—612no—
251#492—612no—
252#492—612no—
253#492—611no—
254#487—612no—
255#487—611no—
256#486—22no—
257#486—612no—
258#486—611no—
259#486C2×D6112yes—
260#486C2×D6112yes—
261#465—1222no—
262#461—1222no—
263#460—1222no—
264#450—1222no—
265#448—1222no—
266C4×SD16—1222no—
267#431—611no—
268#428—1222no—
269C2×#102—22no—
270C4×M16—611no—
271C2×C4×C8C2×D6111yescharacteristic
272#322—612no—
273C82—612no—
274C61#1962311yescharacteristic, complemented, Sylow
275C2×#41—1221no—
276C22×Q8—21no—
277C22×D4—21no—
278C22×D4—21no—
279C22×D4—21no—
280AGL(1,ℤ/8)—2448no—
281C2×SD16—1224no—
282C2×SD16—1224no—
283C2×SD16—1224no—
284C2×SD16—1222no—
285C2×D8—1224no—
286C2×M16—612no—
287C22×C8C22⋊D6112yes—
288C4⋊Q8—21no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8—611no—
292C4⋊Q8—611no—
293C4⋊Q8—611no—
294C4⋊Q8C22×D6112yes—
295#127—21no—
296#127—21no—
297#127—21no—
298#127—611no—
299#127C22×D6112yes—
300#125—1221no—
301#124—1221no—
302C22⋊Q8—1221no—
303C22⋊Q8—1221no—
304#121—1221no—
305#121—1221no—
306C4×Q8—1221no—
307C4×Q8—1221no—
308C4×D4—1221no—
309C4×D4—1221no—
310#117—1221no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C42C22×D6111yescharacteristic
314#107—1224no—
315#106—1224no—
316#106—1222no—
317C4⋊C8—612no—
318#103—1224no—
319#103—1222no—
320#102—24no—
321#102—24no—
322#102—1224no—
323#102—1222no—
324C22⋊C8—612no—
325#97—612no—
326C4×C8—612no—
327C4×C8—612no—
328C4×C8C22⋊D6112yes—
329C4×C8C22×D6111yescharacteristic
330C4×C8C22×D6111yescharacteristic
331C24—41no—
332#41—2444no—
333C2×Q8—22no—
334C2×Q8—41no—
335C2×Q8—82no—
336C2×Q8—1221no—
337C2×Q8—1221no—
338C2×Q8—1221no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—44no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—44no—
347C2×D4—82no—
348C2×D4—1221no—
349C2×D4—1221no—
350C2×D4—1221no—
351C2×D4—2441no—
352C22×C4—41no—
353C22×C4—2441no—
354C22×C4D4×D6111yescharacteristic
355C22×C4D4×D6111yescharacteristic
356C22×C4C2×C22⋊D6111yescharacteristic
357SD16—4884no—
358SD16—4884no—
359SD16—2444no—
360SD16—2448no—
361D8—2448no—
362M16—618no—
363C2×C8—42no—
364C2×C8—614no—
365C2×C8—614no—
366C2×C8C2×C22⋊D6112yes—
367C2×C8C2×C22⋊D6112yes—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—1221no—
373C4⋊C4—1221no—
374C4⋊C4—1221no—
375C4⋊C4—1221no—
376C4⋊C4—1221no—
377C4⋊C4—1221no—
378C4⋊C4—2441no—
379C4⋊C4—2441no—
380C22⋊C4—2441no—
381C22⋊C4—2441no—
382C22⋊C4—2441no—
383C42—2441no—
384C42D4×D6111yescharacteristic
385C42D4×D6111yescharacteristic
386C42C2×C22⋊D6111yescharacteristic
387C42C23×D6111yescharacteristic
388C23—42no—
389C23—82no—
390C23—82no—
391C23—82no—
392C23—2441no—
393C23#1451977311yescharacteristic
394Q8—44no—
395Q8—82no—
396Q8—2442no—
397Q8—2442no—
398Q8—2442no—
399D4—44no—
400D4—44no—
401D4—44no—
402D4—82no—
403D4—82no—
404D4—82no—
405D4—4882no—
406D4—2442no—
407D4—2442no—
408D4—2442no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—42no—
412C2×C4—82no—
413C2×C4—4882no—
414C2×C4—2441no—
415C2×C4—2441no—
416C2×C4—2441no—
417C2×C4#1451977311yescharacteristic
418C2×C4#1451977311yescharacteristic
419C2×C4#1451977311yescharacteristic
420C2×C4SD16×D6114yes—
421C2×C4C2×D4×D6111yescharacteristic
422C2×C4C2×D4×D6111yescharacteristic
423C2×C4C22×C22⋊D6111yescharacteristic
424C8—22no—
425C8—22no—
426C8—618no—
427C8—1224no—
428C22—21no—
429C22—21no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22—164no—
434C22—4881no—
435C22—4881no—
436C22—4881no—
437C22#1684013312yes—
438C22C2×#1451977311yescharacteristic
439C4—21no—
440C4—21no—
441C4—82no—
442C4—4881no—
443C4—4881no—
444C4—4881no—
445C4#1684013312yes—
446C4C2×SD16×D6112yes—
447C4#1684056012yes—
448C4#1684056312yes—
449C2—22no—
450C2—162no—
451C2—9761no—
452C2C2×#1684013311yescharacteristic
453C2#2145628311yescharacteristic
454C2#2145630011yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#486C2
socleC22×C61C2×#127
prime radical#21456269C2
hypercenterC2×#486D61
hypocenterC61#19623
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#486

Complements

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

NH ≅ G/Ntype
1. #21456760C2semidirect
2. #21456744C2semidirect
3. #21456378C2semidirect
4. #21456361C2semidirect
6. #21445449C2semidirect
7. #21445440C2semidirect
8. #21445419C2semidirect
9. #21445410C2semidirect
10. C8×#11758194C2semidirect
11. C2×C61×#486C2semidirect
16. #16840165C22semidirect
17. #16840160C22semidirect
18. #16840158C22semidirect
21. #16840136C22semidirect
22. #16840135C22semidirect
25. #16840113C22semidirect
26. #16840112C22semidirect
27. #16840109C22semidirect
38. #16839484C22semidirect
39. C8×#92418C22semidirect
156. #11758194SD16semidirect
227. C2×#486D61semidirect
274. C61#19623semidirect

Maximal subgroups

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

subgroupindex
#214567602
#214567442
#214563782
#214563612
#214562692
#214454492
#214454402
#214454192
#214454102
C8×#117581942
C2×C61×#4862
#1962361