Finite Group Databasebeta

← #27153742

Subgroups — #27153742

Order 12416.

404 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261431C211yesmaximal, characteristic, complemented
2#20261099C211yesmaximal, characteristic, complemented
3#20261056C211yesmaximal, characteristic, complemented
4#20261046C211yesmaximal, characteristic, complemented
5#20261042C211yesmaximal, characteristic, complemented
6#20260986C211yesmaximal, characteristic, complemented
7#20260969C211yesmaximal, characteristic, complemented
8#20260927C211yesmaximal, characteristic, complemented
9#20260880C211yesmaximal, characteristic, complemented
10#20260880C211yesmaximal, characteristic, complemented
11#20260833C211yesmaximal, characteristic, complemented
12#20260807C211yesmaximal, characteristic, complemented
13#20260736C211yesmaximal, characteristic, complemented
14#20260731C211yesmaximal, characteristic, complemented
15#550×C97C211yesmaximal, characteristic, complemented
16C2×#13095498C2211yescharacteristic
17C2×#13095496C2211yescharacteristic
18C2×#13095496C2211yescharacteristic
19C2×#13095495C2211yescharacteristic
20C22×#11715895C2211yescharacteristic
21#15690805—21no—
22#15690802C2211yescharacteristic
23#15690792—21no—
24#15690784C2211yescharacteristic
25#15690784C2211yescharacteristic
26#15690782—21no—
27C4×D388C2211yescharacteristic
28#15690770—21no—
29#15690769C2211yescharacteristic
30C2×C4⋊Dic97C2211yescharacteristic
31#15690743C2211yescharacteristic
32#15690743C2211yescharacteristic
33#15690741C2211yescharacteristic
34C4⋊#11715895C2211yescharacteristic
35#15690734C2211yescharacteristic
36#15690734C2211yescharacteristic
37#15690734C2211yescharacteristic
38#15690731C2211yescharacteristic
39#15690731C2211yescharacteristic
40#15690731C2211yescharacteristic
41#15690731C2211yescharacteristic
42#15690731C2211yescharacteristic
43#15690731C2211yescharacteristic
44#15690730C2211yescharacteristic
45C22⋊#11715895C2211yescharacteristic
46C22⋊#11715895C2211yescharacteristic
47#15690721C2211yescharacteristic
48C2×#41×C97C2211yescharacteristic
49C2×#41×C97C2211yescharacteristic
50#127×C97—21no—
51#127×C97C2211yescharacteristic
52#121×C97—21no—
53#121×C97—21no—
54#121×C97C2211yescharacteristic
55#121×C97C2211yescharacteristic
56C4×Q8×C97C2211yescharacteristic
57C4×D4×C97—21no—
58C4×D4×C97C2211yescharacteristic
59C2×D388C2311yescharacteristic
60C2×C22⋊D97—21no—
61C2×C22⋊D97—21no—
62C22×Dic97—21no—
63C2×C4×D97—21no—
64#13095498—22no—
65#13095496—41no—
66#13095496—41no—
67#13095495—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—
84#41×C97—22no—
85#41×C97—22no—
86#41×C97—41no—
87#41×C97—41no—
88C2×Q8×C97C2311yescharacteristic
89C2×D4×C97—21no—
90C2×D4×C97—21no—
91C2×D4×C97—21no—
92C2×D4×C97—21no—
93C2×D4×C97—21no—
94C2×D4×C97—21no—
95C2×D4×C97D411yescharacteristic
96C2×D4×C97C2311yescharacteristic
97C2×D4×C97C2311yescharacteristic
98C22×C4×C97—21no—
99C22×C4×C97—21no—
100C22×C4×C97C2311yescharacteristic
101C22×C4×C97C2311yescharacteristic
102C4⋊C4×C97—21no—
103C4⋊C4×C97D411yescharacteristic
104C4⋊C4×C97C2311yescharacteristic
105C22⋊C4×C97—21no—
106C22⋊C4×C97—21no—
107C22⋊C4×C97D411yescharacteristic
108C22⋊C4×C97D411yescharacteristic
109C42×C97—21no—
110C42×C97C2311yescharacteristic
111C23×C97—21no—
112C23×C97—21no—
113C23×C97C2×D411yescharacteristic
114C23×C97C2×D411yescharacteristic
115C22×D97—21no—
116Q8×C97—21no—
117Q8×C97#4112yes—
118D4×C97—21no—
119D4×C97—21no—
120D4×C97—21no—
121D4×C97—21no—
122D4×C97—41no—
123D4×C97—42no—
124D4×C97—42no—
125D4×C97—42no—
126D4×C97—81no—
127D4×C97—81no—
128D4×C97—81no—
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—42no—
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—21no—
159C22×C97—21no—
160C22×C97—21no—
161C22×C97—41no—
162C22×C97—41no—
163C22×C97—41no—
164C22×C97—41no—
165C22×C97—41no—
166C22×C97—41no—
167C22×C97C22×D411yescharacteristic
168C2×D97—41no—
169C2×D97—41no—
170C2×D97—41no—
171C4×C97—21no—
172C4×C97—21no—
173C4×C97—22no—
174C4×C97—21no—
175C4×C97—21no—
176C4×C97—41no—
177C4×C97C2×#4111yescharacteristic
178C4×C97H(5,𝔽2)11yescharacteristic
179Dic97—41no—
180Dic97—41no—
181Dic97—41no—
182C2×C97—41no—
183C2×C97—41no—
184C2×C97—81no—
185C2×C97—81no—
186C2×C97#54611yescharacteristic
187C2×C97#57511yescharacteristic
188C2×C97#57611yescharacteristic
189D97—81no—
190#3309—971nomaximal, Sylow
191C97#330911yescharacteristic, complemented, Sylow
192#553—971no—
193#550D9711yescharacteristic, complemented
194#490—971no—
195#482—971no—
196#480—971no—
197#479—971no—
198#469—971no—
199#466—971no—
200#459—971no—
201#449—971no—
202#449—971no—
203#440—971no—
204#435—971no—
205#417—971no—
206#416—971no—
207C2×#41C2×D9711yescharacteristic
208C2×#41C2×D9711yescharacteristic
209C2×SD16—971no—
210C2×D8—971no—
211C2×D8—971no—
212C2×M16—971no—
213C22×C8—971no—
214#127—21no—
215#127C2×D9711yescharacteristic
216#126—1941no—
217#125—971no—
218#123—1941no—
219#121—21no—
220#121—21no—
221#121—971no—
222#121—971no—
223#121—1941no—
224#121C2×D9711yescharacteristic
225#121C2×D9711yescharacteristic
226C4×Q8C2×D9711yescharacteristic
227C4×D4—21no—
228C4×D4—971no—
229C4×D4—1941no—
230C4×D4C2×D9711yescharacteristic
231#117—971no—
232C2×C4⋊C4—971no—
233#107—971no—
234#107—971no—
235#106—971no—
236C4⋊C8—971no—
237#103—971no—
238#103—971no—
239#103—971no—
240#102—971no—
241#102—971no—
242#102—971no—
243#102—971no—
244#102—971no—
245#102—971no—
246#102—971no—
247C22⋊C8—971no—
248C22⋊C8—971no—
249#97—971no—
250#41—22no—
251#41—22no—
252#41—41no—
253#41—41no—
254C2×Q8C22×D9711yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—971no—
262C2×D4—1941no—
263C2×D4—1941no—
264C2×D4C22⋊D9711yescharacteristic
265C2×D4C22×D9711yescharacteristic
266C2×D4C22×D9711yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—1941no—
270C22×C4—1941no—
271C22×C4C22×D9711yescharacteristic
272C22×C4C22×D9711yescharacteristic
273SD16—1942no—
274D8—3881no—
275D8—3881no—
276M16—3881no—
277C2×C8—971no—
278C2×C8—971no—
279C2×C8—971no—
280C2×C8—971no—
281C2×C8—3881no—
282C4⋊C4—21no—
283C4⋊C4—971no—
284C4⋊C4—971no—
285C4⋊C4—971no—
286C4⋊C4—1941no—
287C4⋊C4—1941no—
288C4⋊C4—1941no—
289C4⋊C4C22⋊D9711yescharacteristic
290C4⋊C4C22×D9711yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—1941no—
294C22⋊C4—1941no—
295C22⋊C4—1941no—
296C22⋊C4—1941no—
297C22⋊C4C22⋊D9711yescharacteristic
298C22⋊C4C22⋊D9711yescharacteristic
299C42—21no—
300C42—1941no—
301C42C22×D9711yescharacteristic
302C23—21no—
303C23—21no—
304C23—1941no—
305C23C2×C22⋊D9711yescharacteristic
306C23C2×C22⋊D9711yescharacteristic
307Q8—21no—
308Q8#1309551012yes—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—41no—
314D4—42no—
315D4—42no—
316D4—42no—
317D4—81no—
318D4—81no—
319D4—81no—
320D4—7761no—
321D4—3882no—
322D4—1941no—
323D4—1941no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—7761no—
333C2×C4—7761no—
334C2×C4—1941no—
335C2×C4—1941no—
336C2×C4—1941no—
337C2×C4C2×C22⋊D9711yescharacteristic
338C2×C4C2×C22⋊D9711yescharacteristic
339C2×C4C2×C22⋊D9711yescharacteristic
340C2×C4C2×C22⋊D9711yescharacteristic
341C2×C4C23×D9711yescharacteristic
342C8—1941no—
343C8—1941no—
344C8—1941no—
345C8—1941no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—3881no—
359C22—3881no—
360C22—3881no—
361C22C22×C22⋊D9711yescharacteristic
362C4—21no—
363C4—21no—
364C4—22no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—3881no—
369C4—3881no—
370C4—3881no—
371C4C2×#1309551011yescharacteristic
372C4#1569085511yescharacteristic
373C2—41no—
374C2—41no—
375C2—81no—
376C2—81no—
377C2—7761no—
378C2#2026137911yescharacteristic
379C2#2026153311yescharacteristic
380C2#2026153811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#550×C97C2
socleC22×C97C22×D4
hypercenter#550D97
hypocenterC97#3309
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #20261431C2semidirect
2. #20261099C2semidirect
3. #20261056C2semidirect
4. #20261046C2semidirect
5. #20261042C2semidirect
6. #20260986C2semidirect
7. #20260969C2semidirect
8. #20260927C2semidirect
9. #20260880C2semidirect
10. #20260880C2semidirect
11. #20260833C2semidirect
12. #20260807C2semidirect
13. #20260736C2semidirect
14. #20260731C2semidirect
15. #550×C97C2semidirect
191. C97#3309semidirect
193. #550D97semidirect

Maximal subgroups

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

subgroupindex
#202614312
#202610992
#202610562
#202610462
#202610422
#202609862
#202609692
#202609272
#202608802
#202608332
#202608072
#202607362
#202607312
#550×C972
#330997