Finite Group Databasebeta

← #27153586

Subgroups — #27153586

Order 12416.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261431C211yesmaximal, characteristic, complemented
2#20261089C211yesmaximal, characteristic, complemented
3#20261068C211yesmaximal, characteristic, complemented
4#20261056C211yesmaximal, characteristic, complemented
5#20261035C211yesmaximal, characteristic, complemented
6#20260969C211yesmaximal, characteristic, complemented
7#20260966C211yesmaximal, characteristic, complemented
8#20260947C211yesmaximal, characteristic, complemented
9#20260880C211yesmaximal, characteristic, complemented
10#20260869C211yesmaximal, characteristic, complemented
11C4×#13095496C211yesmaximal, characteristic, complemented
12D4×#11715895C211yesmaximal, characteristic, complemented
13#20260731C211yesmaximal, characteristic, complemented
14C2×#15690731C211yesmaximal, characteristic, complemented
15#546×C97C211yesmaximal, characteristic, complemented
16C2×#13095498C2211yescharacteristic, complemented
17C2×#13095496C2211yescharacteristic
18C2×#13095496C2211yescharacteristic
19C22×#11715895C2211yescharacteristic
20C22×#11715895C2211yescharacteristic, complemented
21#15690805—21no—
22#15690802C2211yescharacteristic, complemented
23#15690792—21no—
24#15690784C2211yescharacteristic
25#15690784C2211yescharacteristic, complemented
26#15690782—21no—
27C4×D388C2211yescharacteristic, complemented
28#15690770—21no—
29#15690769C2211yescharacteristic, complemented
30C2×C4⋊Dic97C2211yescharacteristic
31#15690743C2211yescharacteristic, complemented
32#15690743C2211yescharacteristic, complemented
33#15690741C2211yescharacteristic, complemented
34C4⋊#11715895C2211yescharacteristic, complemented
35#15690734C2211yescharacteristic, complemented
36#15690734C2211yescharacteristic, complemented
37#15690734C2211yescharacteristic, complemented
38#15690731C2211yescharacteristic
39#15690731C2211yescharacteristic
40#15690731C2211yescharacteristic
41#15690731C2211yescharacteristic
42#15690731C2211yescharacteristic
43#15690731C2211yescharacteristic
44#15690730C2211yescharacteristic, complemented
45C22⋊#11715895C2211yescharacteristic
46C22⋊#11715895C2211yescharacteristic, complemented
47C4×#11715895C2211yescharacteristic, complemented
48C2×#41×C97C2211yescharacteristic
49C22×D4×C97C2211yescharacteristic
50#124×C97C2211yescharacteristic
51#123×C97—21no—
52C22⋊Q8×C97C2211yescharacteristic
53#121×C97—21no—
54#121×C97C2211yescharacteristic
55#120×C97—21no—
56C4×D4×C97C2211yescharacteristic
57C4×D4×C97C2211yescharacteristic
58C2×C22⋊C4×C97—21no—
59C2×D388C2311yescharacteristic
60C2×C22⋊D97—21no—
61C2×C22⋊D97—21no—
62C22×Dic97—21no—
63C2×C4×D97—21no—
64#13095498—41no—
65#13095496—22no—
66#13095496—41no—
67C2×#11715895—41no—
68C2×#11715895—41no—
69C2×#11715895C2311yescharacteristic
70C2×#11715895C2311yescharacteristic
71C2×#11715895C2311yescharacteristic
72C2×#11715895C2311yescharacteristic
73C4⋊Dic97—21no—
74C4⋊Dic97C2311yescharacteristic
75C4⋊Dic97C2311yescharacteristic
76C4⋊Dic97C2311yescharacteristic
77#13095490—21no—
78#13095490—21no—
79C22⋊Dic97—21no—
80#13095488—21no—
81#13095488—21no—
82#13095488—21no—
83C4×Dic97—21no—
84C24×C97—21no—
85#41×C97—22no—
86#41×C97—41no—
87C2×Q8×C97C2311yescharacteristic
88C2×D4×C97—22no—
89C2×D4×C97—21no—
90C2×D4×C97—21no—
91C2×D4×C97—21no—
92C2×D4×C97—41no—
93C2×D4×C97D411yescharacteristic
94C2×D4×C97C2311yescharacteristic
95C2×D4×C97C2311yescharacteristic
96C22×C4×C97—21no—
97C22×C4×C97—21no—
98C22×C4×C97C2311yescharacteristic
99C22×C4×C97C2311yescharacteristic
100C4⋊C4×C97—21no—
101C4⋊C4×C97D411yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C22⋊C4×C97—21no—
104C22⋊C4×C97—21no—
105C22⋊C4×C97—21no—
106C22⋊C4×C97—21no—
107C22⋊C4×C97—21no—
108C22⋊C4×C97D411yescharacteristic
109C22⋊C4×C97D411yescharacteristic
110C42×C97C2311yescharacteristic
111C23×C97—21no—
112C23×C97—21no—
113C23×C97—42no—
114C23×C97—42no—
115C23×C97—81no—
116C23×C97C2×D411yescharacteristic
117C23×C97C2×D411yescharacteristic
118C22×D97—21no—
119Q8×C97—21no—
120Q8×C97—21no—
121D4×C97—22no—
122D4×C97—21no—
123D4×C97—21no—
124D4×C97—21no—
125D4×C97—42no—
126D4×C97—42no—
127D4×C97—81no—
128D4×C97#4112yes—
129C2×C4×C97—21no—
130C2×C4×C97—21no—
131C2×C4×C97—21no—
132C2×C4×C97—21no—
133C2×C4×C97—21no—
134C2×C4×C97—21no—
135C2×C4×C97—42no—
136C2×C4×C97—81no—
137C2×C4×C97C2×D411yescharacteristic
138C2×C4×C97C2×D411yescharacteristic
139C2×C4×C97C2×D411yescharacteristic
140C2×C4×C97C2×D411yescharacteristic
141C2×C4×C97C2411yescharacteristic
142C22⋊D97—42no—
143C22⋊D97—81no—
144C2×Dic97—21no—
145C2×Dic97—21no—
146C2×Dic97—21no—
147C2×Dic97—81no—
148D388—21no—
149D388—21no—
150C4×D97—81no—
151#11715895—21no—
152#11715895—21no—
153#11715895—21no—
154#11715895—21no—
155C22×C97—21no—
156C22×C97—21no—
157C22×C97—21no—
158C22×C97—22no—
159C22×C97—21no—
160C22×C97—21no—
161C22×C97—41no—
162C22×C97—41no—
163C22×C97—41no—
164C22×C97—41no—
165C22×C97—41no—
166C22×C97—84no—
167C22×C97#13512yes—
168C22×C97C22×D411yescharacteristic
169C2×D97—41no—
170C2×D97—41no—
171C2×D97—41no—
172C4×C97—21no—
173C4×C97—21no—
174C4×C97—21no—
175C4×C97—21no—
176C4×C97—41no—
177C4×C97—41no—
178C4×C97C2×#4111yescharacteristic
179C4×C97H(5,𝔽2)11yescharacteristic
180Dic97—41no—
181Dic97—41no—
182Dic97—41no—
183C2×C97—22no—
184C2×C97—41no—
185C2×C97—42no—
186C2×C97—81no—
187C2×C97#54611yescharacteristic
188C2×C97C2×#13511yescharacteristic
189C2×C97#57611yescharacteristic
190D97—81no—
191#3297—971nomaximal, Sylow
192C97#329711yescharacteristic, complemented, Sylow
193#553—971no—
194#546D9711yescharacteristic, complemented
195#488—971no—
196#484—971no—
197#482—971no—
198#478—971no—
199#466—971no—
200#465—971no—
201#463—971no—
202#449—971no—
203#447—971no—
204C4×D8—971no—
205C8×D4—971no—
206#416—971no—
207C2×#102—971no—
208C2×#41C2×D9711yescharacteristic
209C22×D4C2×D9711yescharacteristic
210C2×SD16—971no—
211C2×D8—971no—
212C2×D8—971no—
213C22×C8—971no—
214C22×C8—971no—
215#126—1941no—
216#125—971no—
217#124C2×D9711yescharacteristic
218#123—21no—
219#123—1941no—
220C22⋊Q8C2×D9711yescharacteristic
221#121—21no—
222#121—971no—
223#121—971no—
224#121—1941no—
225#121C2×D9711yescharacteristic
226#120—21no—
227C4×D4—971no—
228C4×D4—1941no—
229C4×D4C2×D9711yescharacteristic
230C4×D4C2×D9711yescharacteristic
231#117—971no—
232C2×C4⋊C4—971no—
233C2×C22⋊C4—21no—
234#107—971no—
235#107—971no—
236#106—971no—
237C4⋊C8—971no—
238#103—971no—
239#103—971no—
240#103—971no—
241#102—971no—
242#102—971no—
243#102—971no—
244#102—971no—
245#102—971no—
246#102—971no—
247#102—971no—
248C22⋊C8—971no—
249C22⋊C8—971no—
250C4×C8—971no—
251C24—21no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D9711yescharacteristic
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—41no—
260C2×D4—971no—
261C2×D4—1941no—
262C2×D4—1941no—
263C2×D4C22⋊D9711yescharacteristic
264C2×D4C22×D9711yescharacteristic
265C2×D4C22×D9711yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—1941no—
269C22×C4—1941no—
270C22×C4C22×D9711yescharacteristic
271C22×C4C22×D9711yescharacteristic
272SD16—3881no—
273D8—3881no—
274D8—1942no—
275C2×C8—971no—
276C2×C8—971no—
277C2×C8—971no—
278C2×C8—971no—
279C2×C8—3881no—
280C2×C8—3881no—
281C4⋊C4—21no—
282C4⋊C4—971no—
283C4⋊C4—971no—
284C4⋊C4—971no—
285C4⋊C4—1941no—
286C4⋊C4—1941no—
287C4⋊C4—1941no—
288C4⋊C4C22⋊D9711yescharacteristic
289C4⋊C4C22×D9711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—1941no—
296C22⋊C4—1941no—
297C22⋊C4—1941no—
298C22⋊C4—1941no—
299C22⋊C4C22⋊D9711yescharacteristic
300C22⋊C4C22⋊D9711yescharacteristic
301C42—1941no—
302C42C22×D9711yescharacteristic
303C23—21no—
304C23—21no—
305C23—42no—
306C23—42no—
307C23—81no—
308C23—1941no—
309C23C2×C22⋊D9711yescharacteristic
310C23C2×C22⋊D9711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—42no—
319D4—81no—
320D4—7761no—
321D4—3882no—
322D4—1941no—
323D4—1941no—
324D4#1309551012yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—7761no—
334C2×C4—7761no—
335C2×C4—1941no—
336C2×C4—1941no—
337C2×C4—1941no—
338C2×C4C2×C22⋊D9711yescharacteristic
339C2×C4C2×C22⋊D9711yescharacteristic
340C2×C4C2×C22⋊D9711yescharacteristic
341C2×C4C2×C22⋊D9711yescharacteristic
342C2×C4C23×D9711yescharacteristic
343C8—1941no—
344C8—1941no—
345C8—1941no—
346C8—1941no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—84no—
359C22—3881no—
360C22—3881no—
361C22—3881no—
362C22#1569083012yes—
363C22C22×C22⋊D9711yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—3881no—
371C4—3881no—
372C4—3881no—
373C4C2×#1309551011yescharacteristic
374C4#1569085511yescharacteristic
375C2—22no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—7761no—
380C2#2026137911yescharacteristic
381C2C2×#1569083011yescharacteristic
382C2#2026153811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#546×C97C2
socleC22×C97C22×D4
hypercenter#546D97
hypocenterC97#3297
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #20261431C2semidirect
2. #20261089C2semidirect
3. #20261068C2semidirect
4. #20261056C2semidirect
5. #20261035C2semidirect
6. #20260969C2semidirect
7. #20260966C2semidirect
8. #20260947C2semidirect
9. #20260880C2semidirect
10. #20260869C2semidirect
11. C4×#13095496C2semidirect
12. D4×#11715895C2semidirect
13. #20260731C2semidirect
14. C2×#15690731C2semidirect
15. #546×C97C2semidirect
16. C2×#13095498C22semidirect
20. C22×#11715895C22semidirect
22. #15690802C22semidirect
25. #15690784C22semidirect
27. C4×D388C22semidirect
29. #15690769C22semidirect
31. #15690743C22semidirect
32. #15690743C22semidirect
33. #15690741C22semidirect
34. C4⋊#11715895C22semidirect
35. #15690734C22semidirect
36. #15690734C22semidirect
37. #15690734C22semidirect
44. #15690730C22semidirect
46. C22⋊#11715895C22semidirect
47. C4×#11715895C22semidirect
192. C97#3297semidirect
194. #546D97semidirect

Maximal subgroups

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

subgroupindex
#202614312
#202610892
#202610682
#202610562
#202610352
#202609692
#202609662
#202609472
#202608802
#202608692
C4×#130954962
D4×#117158952
#202607312
C2×#156907312
#546×C972
#329797