Finite Group Databasebeta

← #27151992

Subgroups — #27151992

Order 12416.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261215C211yesmaximal, characteristic, complemented
2#20261068C211yesmaximal, characteristic, complemented
3#20261046C211yesmaximal, characteristic, complemented
4#20261027C211yesmaximal, characteristic, complemented
5#20261012C211yesmaximal, characteristic, complemented
6#20260954C211yesmaximal, characteristic, complemented
7#20260947C211yesmaximal, characteristic, complemented
8#20260895C211yesmaximal, characteristic, complemented
9#20260878C211yesmaximal, characteristic
10#20260835C211yesmaximal, characteristic, complemented
11#20260833C211yesmaximal, characteristic, complemented
12C4×#13095501C211yesmaximal, characteristic, complemented
13C4×#13095496C211yesmaximal, characteristic, complemented
14#20260796C211yesmaximal, characteristic, complemented
15#542×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C2×#13095501C2211yescharacteristic
18C2×#13095499C2211yescharacteristic
19C2×#13095498C2211yescharacteristic
20C2×#13095496C2211yescharacteristic
21C4×Dic194C2211yescharacteristic
22C4×Dic194C2211yescharacteristic
23C4×D388C2211yescharacteristic
24C4×D388C2211yescharacteristic
25#15690770—21no—
26#15690770—21no—
27#15690768—21no—
28C2×C4⋊Dic97C2211yescharacteristic
29C4⋊C4×D97—21no—
30#15690743C2211yescharacteristic
31#15690743C2211yescharacteristic
32#15690743C2211yescharacteristic
33#15690743C2211yescharacteristic
34C4⋊#11715895C2211yescharacteristic
35C4⋊#11715895C2211yescharacteristic
36#15690734C2211yescharacteristic
37#15690734C2211yescharacteristic
38#15690733C2211yescharacteristic
39#15690733C2211yescharacteristic
40#15690731C2211yescharacteristic
41#15690731C2211yescharacteristic
42#15690730C2211yescharacteristic
43#15690730C2211yescharacteristic
44C22⋊#11715895C2211yescharacteristic
45C22⋊#11715895C2211yescharacteristic
46#15690721C2211yescharacteristic
47C4×#11715895C2211yescharacteristic
48C4⋊Q8×C97C2211yescharacteristic
49#126×C97—21no—
50#124×C97—21no—
51#124×C97C2211yescharacteristic
52#123×C97—21no—
53C22⋊Q8×C97—21no—
54C22⋊Q8×C97C2211yescharacteristic
55#121×C97C2211yescharacteristic
56C4×Q8×C97C2211yescharacteristic
57C4×D4×C97C2211yescharacteristic
58#117×C97C2211yescharacteristic
59#13095511—22no—
60#13095511—22no—
61C2×Dic194C2311yescharacteristic
62C2×D388C2311yescharacteristic
63C2×C22⋊D97—21no—
64C22×Dic97—21no—
65C2×C4×D97—21no—
66C2×C4×D97—21no—
67C2×C4×D97—21no—
68#13095501—22no—
69#13095499—22no—
70#13095498—22no—
71#13095496—22no—
72C2×#11715895C2311yescharacteristic
73C2×#11715895C2311yescharacteristic
74C2×#11715895C2311yescharacteristic
75C2×#11715895C2311yescharacteristic
76C4⋊Dic97—21no—
77C4⋊Dic97C2311yescharacteristic
78C4⋊Dic97C2311yescharacteristic
79#13095490—21no—
80#13095490—21no—
81#13095488—21no—
82#13095488—21no—
83C4×Dic97—21no—
84C4×Dic97—21no—
85C2×Q8×C97—21no—
86C2×Q8×C97C2311yescharacteristic
87C2×D4×C97—21no—
88C2×D4×C97C2311yescharacteristic
89C22×C4×C97—21no—
90C22×C4×C97C2311yescharacteristic
91C4⋊C4×C97—21no—
92C4⋊C4×C97—21no—
93C4⋊C4×C97—21no—
94C4⋊C4×C97D411yescharacteristic
95C4⋊C4×C97D411yescharacteristic
96C4⋊C4×C97C2311yescharacteristic
97C4⋊C4×C97C2311yescharacteristic
98C22⋊C4×C97—21no—
99C22⋊C4×C97—21no—
100C22⋊C4×C97—21no—
101C22⋊C4×C97—21no—
102C22⋊C4×C97—21no—
103C22⋊C4×C97D411yescharacteristic
104C22⋊C4×C97D411yescharacteristic
105C42×C97—21no—
106C42×C97C2311yescharacteristic
107C42×C97C2311yescharacteristic
108C23×C97—21no—
109C23×C97C2×D411yescharacteristic
110C22×D97—21no—
111Q8×C97—21no—
112Q8×C97—21no—
113Q8×C97—81no—
114D4×C97—21no—
115D4×C97—21no—
116D4×C97—81no—
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—81no—
123C2×C4×C97C2×D411yescharacteristic
124C2×C4×C97C2×D411yescharacteristic
125C2×C4×C97C2×D411yescharacteristic
126C2×C4×C97C2×D411yescharacteristic
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2411yescharacteristic
129C22⋊D97—24no—
130C2×Dic97—21no—
131C2×Dic97—21no—
132C2×Dic97—21no—
133C2×Dic97—81no—
134D388—21no—
135D388#4112yes—
136C4×D97—24no—
137C4×D97—81no—
138C4×D97—81no—
139Dic194—21no—
140Dic194#4112yes—
141#11715895—21no—
142#11715895—21no—
143#11715895—21no—
144#11715895—21no—
145C22×C97—21no—
146C22×C97—21no—
147C22×C97—21no—
148C22×C97—41no—
149C22×C97—41no—
150C22×C97—41no—
151C22×C97C22×D411yescharacteristic
152C2×D97—22no—
153C2×D97—41no—
154C2×D97—41no—
155C4×C97—21no—
156C4×C97—21no—
157C4×C97—21no—
158C4×C97—21no—
159C4×C97—21no—
160C4×C97—41no—
161C4×C97—41no—
162C4×C97—41no—
163C4×C97C2×#4111yescharacteristic
164C4×C97C2×#4111yescharacteristic
165Dic97—22no—
166Dic97—41no—
167Dic97—41no—
168C2×C97—41no—
169C2×C97—81no—
170C2×C97#52511yescharacteristic
171C2×C97#57611yescharacteristic
172C2×C97#57811yescharacteristic
173D97—42no—
174#3124—971nomaximal, Sylow
175C97#312411yescharacteristic, complemented, Sylow
176#542D9711yescharacteristic, complemented
177#520—971no—
178#484—971no—
179#480—971no—
180#477—971no—
181#474—971no—
182#464—971no—
183#463—971no—
184#452—971no—
185#449—971no—
186#440—971no—
187#440—971no—
188C4×Q16—971no—
189C4×D8—971no—
190#433—971no—
191C2×#41—971no—
192C2×Q16—971no—
193C2×SD16—971no—
194C2×SD16—971no—
195C2×D8—971no—
196C4⋊Q8C2×D9711yescharacteristic
197#126—21no—
198#124—21no—
199#124C2×D9711yescharacteristic
200#123—21no—
201C22⋊Q8—21no—
202C22⋊Q8C2×D9711yescharacteristic
203#121C2×D9711yescharacteristic
204C4×Q8—971no—
205C4×Q8—971no—
206C4×Q8C2×D9711yescharacteristic
207C4×D4—971no—
208C4×D4—971no—
209C4×D4—1941no—
210C4×D4—1941no—
211C4×D4C2×D9711yescharacteristic
212#117—1941no—
213#117C2×D9711yescharacteristic
214C2×C4⋊C4—971no—
215C2×C4⋊C4—1941no—
216#107—971no—
217#107—971no—
218#107—971no—
219#107—971no—
220C4⋊C8—971no—
221C4⋊C8—971no—
222#103—971no—
223#103—971no—
224#103—971no—
225#103—971no—
226#102—971no—
227#102—971no—
228#102—971no—
229#102—971no—
230C22⋊C8—971no—
231C22⋊C8—971no—
232#97—971no—
233C4×C8—971no—
234#41—1942no—
235#41—1942no—
236C2×Q8—21no—
237C2×Q8—971no—
238C2×Q8C22×D9711yescharacteristic
239C2×D4—21no—
240C2×D4—971no—
241C2×D4—1941no—
242C2×D4C22×D9711yescharacteristic
243C22×C4—21no—
244C22×C4—1941no—
245C22×C4—1941no—
246C22×C4—1941no—
247C22×C4—1941no—
248C22×C4C22×D9711yescharacteristic
249Q16—1942no—
250SD16—1942no—
251SD16—1942no—
252D8—1942no—
253C2×C8—971no—
254C2×C8—971no—
255C2×C8—971no—
256C2×C8—971no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—971no—
261C4⋊C4—971no—
262C4⋊C4—1941no—
263C4⋊C4—1941no—
264C4⋊C4—1941no—
265C4⋊C4C22⋊D9711yescharacteristic
266C4⋊C4C22⋊D9711yescharacteristic
267C4⋊C4C22×D9711yescharacteristic
268C4⋊C4C22×D9711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—1941no—
275C22⋊C4—1941no—
276C22⋊C4C22⋊D9711yescharacteristic
277C22⋊C4C22⋊D9711yescharacteristic
278C42—21no—
279C42—1941no—
280C42—1941no—
281C42C22×D9711yescharacteristic
282C42C22×D9711yescharacteristic
283C23—21no—
284C23—1941no—
285C23C2×C22⋊D9711yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—81no—
289Q8—972no—
290Q8—1941no—
291D4—21no—
292D4—21no—
293D4—81no—
294D4—972no—
295D4—1944no—
296D4—1941no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—7761no—
304C2×C4—7761no—
305C2×C4—7761no—
306C2×C4—1944no—
307C2×C4—1941no—
308C2×C4—1941no—
309C2×C4—1941no—
310C2×C4C2×C22⋊D9711yescharacteristic
311C2×C4C2×C22⋊D9711yescharacteristic
312C2×C4C2×C22⋊D9711yescharacteristic
313C2×C4C2×C22⋊D9711yescharacteristic
314C2×C4C2×C22⋊D9711yescharacteristic
315C2×C4C23×D9711yescharacteristic
316C8—1941no—
317C8—1941no—
318C8—1941no—
319C8—1941no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—3881no—
327C22—3881no—
328C22—1942no—
329C22C22×C22⋊D9711yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—3881no—
339C4—3881no—
340C4—1942no—
341C4#41×D9711yescharacteristic
342C4#41×D9711yescharacteristic
343C2—41no—
344C2—81no—
345C2—3882no—
346C2#2026124211yescharacteristic
347C2#2026153811yescharacteristic
348C2#2026155011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#542×C97C2
socleC22×C97C22×D4
prime radical#20260878C2
hypercenter#542D97
hypocenterC97#3124
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #20261215C2semidirect
2. #20261068C2semidirect
3. #20261046C2semidirect
4. #20261027C2semidirect
5. #20261012C2semidirect
6. #20260954C2semidirect
7. #20260947C2semidirect
8. #20260895C2semidirect
10. #20260835C2semidirect
11. #20260833C2semidirect
12. C4×#13095501C2semidirect
13. C4×#13095496C2semidirect
14. #20260796C2semidirect
15. #542×C97C2semidirect
175. C97#3124semidirect
176. #542D97semidirect

Maximal subgroups

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

subgroupindex
#202612152
#202610682
#202610462
#202610272
#202610122
#202609542
#202609472
#202608952
#202608782
#202608352
#202608332
C4×#130955012
C4×#130954962
#202607962
#542×C972
#312497