Finite Group Databasebeta

← #27153736

Subgroups — #27153736

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#20261406C211yesmaximal, characteristic, complemented
2#20261101C211yesmaximal, characteristic, complemented
3#20261059C211yesmaximal, characteristic, complemented
4#20261048C211yesmaximal, characteristic, complemented
5#20261039C211yesmaximal, characteristic, complemented
6#20260986C211yesmaximal, characteristic, complemented
7#20260969C211yesmaximal, characteristic, complemented
8#20260930C211yesmaximal, characteristic, complemented
9#20260878C211yesmaximal, characteristic, complemented
10#20260878C211yesmaximal, characteristic, complemented
11#20260835C211yesmaximal, characteristic, complemented
12#20260807C211yesmaximal, characteristic, complemented
13#20260734C211yesmaximal, characteristic, complemented
14#20260729C211yesmaximal, characteristic, complemented
15#553×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C2×#13095511C2211yescharacteristic
18C2×#13095499C2211yescharacteristic
19C2×#13095496C2211yescharacteristic
20C2×#13095496C2211yescharacteristic
21C2×#13095495C2211yescharacteristic
22C22×#11715895C2211yescharacteristic
23#15690807C2211yescharacteristic
24#15690806—21no—
25#15690784C2211yescharacteristic
26#15690784C2211yescharacteristic
27#15690783—21no—
28#15690783—21no—
29C4×Dic194C2211yescharacteristic
30#15690772—21no—
31#15690743C2211yescharacteristic
32#15690743C2211yescharacteristic
33#15690741C2211yescharacteristic
34C4⋊#11715895C2211yescharacteristic
35#15690733C2211yescharacteristic
36#15690733C2211yescharacteristic
37#15690733C2211yescharacteristic
38#15690731C2211yescharacteristic
39#15690730C2211yescharacteristic
40#15690730C2211yescharacteristic
41#15690730C2211yescharacteristic
42#15690730C2211yescharacteristic
43#15690730C2211yescharacteristic
44#15690730C2211yescharacteristic
45C22⋊#11715895C2211yescharacteristic
46C22⋊#11715895C2211yescharacteristic
47#15690721C2211yescharacteristic
48#126×C97—21no—
49#125×C97C2211yescharacteristic
50#123×C97—21no—
51#121×C97—21no—
52#121×C97C2211yescharacteristic
53#121×C97C2211yescharacteristic
54C4×D4×C97—21no—
55C4×D4×C97C2211yescharacteristic
56C4×D4×C97C2211yescharacteristic
57#117×C97C2211yescharacteristic
58C2×C4⋊C4×C97C2211yescharacteristic
59#13095511—22no—
60#13095511—22no—
61#13095511—41no—
62#13095511—41no—
63C2×Dic194C2311yescharacteristic
64C2×D388—21no—
65C2×D388—21no—
66C2×D388C2311yescharacteristic
67C2×D388C2311yescharacteristic
68C2×C22⋊D97—21no—
69C2×C22⋊D97—21no—
70C2×C22⋊D97—21no—
71C2×C22⋊D97—21no—
72C2×C4×D97—21no—
73C2×C4×D97—21no—
74#13095499—22no—
75#13095496—41no—
76#13095496—41no—
77#13095495—41no—
78C2×#11715895—41no—
79C2×#11715895C2311yescharacteristic
80C2×#11715895C2311yescharacteristic
81C2×#11715895C2311yescharacteristic
82C2×#11715895C2311yescharacteristic
83C4⋊Dic97C2311yescharacteristic
84#13095490—21no—
85#13095488—21no—
86#13095488—21no—
87C4×Dic97—21no—
88C2×D4×C97—21no—
89C2×D4×C97—21no—
90C2×D4×C97D411yescharacteristic
91C2×D4×C97C2311yescharacteristic
92C22×C4×C97—21no—
93C22×C4×C97—21no—
94C22×C4×C97C2311yescharacteristic
95C22×C4×C97C2311yescharacteristic
96C4⋊C4×C97—21no—
97C4⋊C4×C97—21no—
98C4⋊C4×C97—21no—
99C4⋊C4×C97D411yescharacteristic
100C4⋊C4×C97C2311yescharacteristic
101C4⋊C4×C97C2311yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C22⋊C4×C97—21no—
104C22⋊C4×C97—21no—
105C22⋊C4×C97—21no—
106C22⋊C4×C97—21no—
107C22⋊C4×C97D411yescharacteristic
108C22⋊C4×C97D411yescharacteristic
109C42×C97—21no—
110C42×C97C2311yescharacteristic
111C23×C97—21no—
112C23×C97C2×D411yescharacteristic
113C23×C97C2×D411yescharacteristic
114C22×D97—21no—
115C22×D97—21no—
116D4×C97—21no—
117D4×C97—21no—
118D4×C97—41no—
119D4×C97—42no—
120D4×C97—81no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97—21no—
128C2×C4×C97—81no—
129C2×C4×C97—81no—
130C2×C4×C97C2×D411yescharacteristic
131C2×C4×C97C2×D411yescharacteristic
132C2×C4×C97C2×D411yescharacteristic
133C2×C4×C97C2×D411yescharacteristic
134C2×C4×C97C2411yescharacteristic
135C22⋊D97—42no—
136C22⋊D97—42no—
137C22⋊D97—81no—
138C22⋊D97—81no—
139C2×Dic97—21no—
140C2×Dic97—21no—
141D388—21no—
142D388—21no—
143D388—21no—
144D388—21no—
145D388—42no—
146D388—81no—
147C4×D97—42no—
148C4×D97—42no—
149Dic194—21no—
150Dic194#4112yes—
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×C97C22×D411yescharacteristic
165C2×D97—41no—
166C2×D97—41no—
167C2×D97—41no—
168C2×D97—41no—
169C2×D97—41no—
170C2×D97—41no—
171C4×C97—21no—
172C4×C97—21no—
173C4×C97—21no—
174C4×C97—21no—
175C4×C97—41no—
176C4×C97—41no—
177C4×C97—41no—
178C4×C97C2×#4111yescharacteristic
179C4×C97H(5,𝔽2)11yescharacteristic
180Dic97—22no—
181Dic97—41no—
182C2×C97—41no—
183C2×C97—41no—
184C2×C97—81no—
185C2×C97#54611yescharacteristic
186C2×C97#57511yescharacteristic
187C2×C97#57611yescharacteristic
188D97—81no—
189D97—81no—
190#3309—971nomaximal, Sylow
191C97#330911yescharacteristic, complemented, Sylow
192#553D9711yescharacteristic, complemented
193#550—971no—
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×#41—971no—
208C2×#41—971no—
209C2×SD16—971no—
210C2×D8—971no—
211C2×D8—971no—
212C2×M16—971no—
213C22×C8—971no—
214#127—971no—
215#127—1941no—
216#126—21no—
217#125C2×D9711yescharacteristic
218#123—21no—
219#121—21no—
220#121—971no—
221#121—971no—
222#121—1941no—
223#121—1941no—
224#121C2×D9711yescharacteristic
225#121C2×D9711yescharacteristic
226C4×Q8—971no—
227C4×D4—21no—
228C4×D4—1941no—
229C4×D4C2×D9711yescharacteristic
230C4×D4C2×D9711yescharacteristic
231#117C2×D9711yescharacteristic
232C2×C4⋊C4C2×D9711yescharacteristic
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—3881no—
251#41—3881no—
252#41—1942no—
253#41—1942no—
254C2×Q8—971no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—971no—
258C2×D4—971no—
259C2×D4—1941no—
260C2×D4—1941no—
261C2×D4—1941no—
262C2×D4—1941no—
263C2×D4—1941no—
264C2×D4—1941no—
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—21no—
284C4⋊C4—21no—
285C4⋊C4—971no—
286C4⋊C4—1941no—
287C4⋊C4C22⋊D9711yescharacteristic
288C4⋊C4C22×D9711yescharacteristic
289C4⋊C4C22×D9711yescharacteristic
290C4⋊C4C22×D9711yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—1941no—
296C22⋊C4—1941no—
297C22⋊C4C22⋊D9711yescharacteristic
298C22⋊C4C22⋊D9711yescharacteristic
299C42—21no—
300C42—1941no—
301C42C22×D9711yescharacteristic
302C23—21no—
303C23—1941no—
304C23—1941no—
305C23C2×C22⋊D9711yescharacteristic
306C23C2×C22⋊D9711yescharacteristic
307Q8—972no—
308Q8—1941no—
309D4—21no—
310D4—21no—
311D4—41no—
312D4—42no—
313D4—81no—
314D4—7761no—
315D4—7761no—
316D4—7761no—
317D4—3882no—
318D4—3882no—
319D4—3882no—
320D4—1941no—
321D4—1941no—
322D4—1941no—
323D4—1941no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—3882no—
334C2×C4—3882no—
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—3881no—
356C22—3881no—
357C22—3881no—
358C22—3881no—
359C22—3881no—
360C22—3881no—
361C22C22×C22⋊D9711yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—3881no—
370C4—1942no—
371C4#41×D9711yescharacteristic
372C4#1569085611yescharacteristic
373C2—41no—
374C2—41no—
375C2—81no—
376C2—7761no—
377C2—7761no—
378C2#2026137311yescharacteristic
379C2#2026153311yescharacteristic
380C2#2026153811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#553×C97C2
socleC22×C97C22×D4
hypercenter#553D97
hypocenterC97#3309
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #20261406C2semidirect
2. #20261101C2semidirect
3. #20261059C2semidirect
4. #20261048C2semidirect
5. #20261039C2semidirect
6. #20260986C2semidirect
7. #20260969C2semidirect
8. #20260930C2semidirect
9. #20260878C2semidirect
10. #20260878C2semidirect
11. #20260835C2semidirect
12. #20260807C2semidirect
13. #20260734C2semidirect
14. #20260729C2semidirect
15. #553×C97C2semidirect
191. C97#3309semidirect
192. #553D97semidirect

Maximal subgroups

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

subgroupindex
#202614062
#202611012
#202610592
#202610482
#202610392
#202609862
#202609692
#202609302
#202608782
#202608352
#202608072
#202607342
#202607292
#553×C972
#330997