Finite Group Databasebeta

← #27152000

Subgroups — #27152000

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#20261356C211yesmaximal, characteristic, complemented
2#20261215C211yesmaximal, characteristic, complemented
3#20261071C211yesmaximal, characteristic, complemented
4#20261049C211yesmaximal, characteristic, complemented
5#20261031C211yesmaximal, characteristic, complemented
6#20261016C211yesmaximal, characteristic, complemented
7#20260958C211yesmaximal, characteristic, complemented
8#20260951C211yesmaximal, characteristic, complemented
9#20260899C211yesmaximal, characteristic, complemented
10#20260882C211yesmaximal, characteristic
11#20260837C211yesmaximal, characteristic, complemented
12#20260837C211yesmaximal, characteristic, complemented
13C4×Dic388C211yesmaximal, characteristic, complemented
14C4×D776C211yesmaximal, characteristic, complemented
15#433×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C2×Dic388C2211yescharacteristic
18C2×#13095500C2211yescharacteristic
19C2×#13095500C2211yescharacteristic
20C2×D776C2211yescharacteristic
21#15690810C2211yescharacteristic
22#15690805—21no—
23#15690798C2211yescharacteristic
24#15690796—21no—
25#15690792—21no—
26#15690789C2211yescharacteristic
27#15690788—21no—
28#15690784C2211yescharacteristic
29C4×Dic194C2211yescharacteristic
30C4×Dic194C2211yescharacteristic
31C4×Dic194C2211yescharacteristic
32C4×D388C2211yescharacteristic
33C4×D388C2211yescharacteristic
34C4×D388C2211yescharacteristic
35#15690770—21no—
36#15690770—21no—
37#15690768—21no—
38C2×C4⋊Dic97C2211yescharacteristic
39C4⋊C4×D97—21no—
40#15690744C2211yescharacteristic
41#15690744C2211yescharacteristic
42#15690744C2211yescharacteristic
43#15690744C2211yescharacteristic
44#15690735C2211yescharacteristic
45#15690735C2211yescharacteristic
46#15690735C2211yescharacteristic
47#15690735C2211yescharacteristic
48#15690732C2211yescharacteristic
49#15690732C2211yescharacteristic
50#15690732C2211yescharacteristic
51#15690732C2211yescharacteristic
52#117×C97C2211yescharacteristic
53C4⋊C8×C97C2211yescharacteristic
54C4⋊C8×C97C2211yescharacteristic
55C22⋊C8×C97C2211yescharacteristic
56C22⋊C8×C97C2211yescharacteristic
57#97×C97C2211yescharacteristic
58C4×C8×C97C2211yescharacteristic
59#13095511—22no—
60#13095511—22no—
61C2×Dic194—21no—
62C2×Dic194C2311yescharacteristic
63C2×Dic194C2311yescharacteristic
64C2×D388C2311yescharacteristic
65C2×D388C2311yescharacteristic
66C2×C22⋊D97—21no—
67C2×C22⋊D97—21no—
68C22×Dic97—21no—
69C2×C4×D97—21no—
70C2×C4×D97—21no—
71C2×C4×D97—21no—
72C2×C4×D97—21no—
73Dic388—22no—
74#13095500—22no—
75#13095500—22no—
76D776—22no—
77C4⋊Dic97—21no—
78C4⋊Dic97—21no—
79C4⋊Dic97C2311yescharacteristic
80C4⋊Dic97C2311yescharacteristic
81C4⋊Dic97C2311yescharacteristic
82C4⋊Dic97C2311yescharacteristic
83#13095490—21no—
84#13095490—21no—
85#13095490—21no—
86#13095490—21no—
87C22⋊Dic97—21no—
88#13095488—21no—
89#13095488—21no—
90#13095488—21no—
91#13095488—21no—
92#13095488—21no—
93#13095488—21no—
94C4×Dic97—21no—
95C4×Dic97—21no—
96C4×Dic97—21no—
97C22×C4×C97C2311yescharacteristic
98C2×C8×C97C2311yescharacteristic
99C2×C8×C97C2311yescharacteristic
100C2×C8×C97C2311yescharacteristic
101C2×C8×C97C2311yescharacteristic
102C4⋊C4×C97D411yescharacteristic
103C4⋊C4×C97D411yescharacteristic
104C22⋊C4×C97D411yescharacteristic
105C22⋊C4×C97D411yescharacteristic
106C42×C97C2311yescharacteristic
107C42×C97C2311yescharacteristic
108C23×C97C2×D411yescharacteristic
109C22×D97—21no—
110C22×D97—21no—
111C2×C4×C97—21no—
112C2×C4×C97—21no—
113C2×C4×C97C2×D411yescharacteristic
114C2×C4×C97C2×D411yescharacteristic
115C2×C4×C97C2×D411yescharacteristic
116C2×C4×C97C2×D411yescharacteristic
117C2×C4×C97C2×D411yescharacteristic
118C2×C4×C97C2411yescharacteristic
119C22⋊D97—24no—
120C22⋊D97—81no—
121C2×Dic97—21no—
122C2×Dic97—21no—
123C2×Dic97—21no—
124C2×Dic97—21no—
125C2×Dic97—21no—
126C2×Dic97—21no—
127C2×Dic97—81no—
128D388—21no—
129D388—21no—
130D388—21no—
131D388#4112yes—
132C4×D97—24no—
133C4×D97—81no—
134C4×D97—81no—
135C4×D97—81no—
136Dic194—21no—
137Dic194—21no—
138Dic194—21no—
139Dic194—81no—
140Dic194#4112yes—
141C8×C97—21no—
142C8×C97—21no—
143C8×C97—21no—
144C8×C97—21no—
145C22×C97—21no—
146C22×C97—21no—
147C22×C97—21no—
148C22×C97C22×D411yescharacteristic
149C2×D97—22no—
150C2×D97—41no—
151C2×D97—41no—
152C2×D97—41no—
153C2×D97—41no—
154C2×D97—41no—
155C4×C97—21no—
156C4×C97—21no—
157C4×C97—21no—
158C4×C97—21no—
159C4×C97—21no—
160C4×C97C2×#4111yescharacteristic
161C4×C97C2×#4111yescharacteristic
162Dic97—22no—
163Dic97—41no—
164Dic97—41no—
165Dic97—41no—
166Dic97—41no—
167Dic97—41no—
168C2×C97—41no—
169C2×C97#52511yescharacteristic
170C2×C97#57611yescharacteristic
171C2×C97#57811yescharacteristic
172D97—42no—
173D97—81no—
174#3124—971nomaximal, Sylow
175C97#312411yescharacteristic, complemented, Sylow
176#542—971no—
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#433D9711yescharacteristic, complemented
191C2×#41—971no—
192C2×Q16—971no—
193C2×SD16—971no—
194C2×SD16—971no—
195C2×D8—971no—
196C4⋊Q8—971no—
197#126—1941no—
198#124—971no—
199#124—1941no—
200#123—1941no—
201C22⋊Q8—971no—
202C22⋊Q8—1941no—
203#121—971no—
204C4×Q8—971no—
205C4×Q8—971no—
206C4×Q8—971no—
207C4×D4—971no—
208C4×D4—971no—
209C4×D4—971no—
210C4×D4—1941no—
211C4×D4—1941no—
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⋊C8C2×D9711yescharacteristic
221C4⋊C8C2×D9711yescharacteristic
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⋊C8C2×D9711yescharacteristic
231C22⋊C8C2×D9711yescharacteristic
232#97C2×D9711yescharacteristic
233C4×C8C2×D9711yescharacteristic
234#41—1942no—
235#41—1942no—
236C2×Q8—971no—
237C2×Q8—971no—
238C2×Q8—1941no—
239C2×D4—971no—
240C2×D4—971no—
241C2×D4—1941no—
242C2×D4—1941no—
243C22×C4—1941no—
244C22×C4—1941no—
245C22×C4—1941no—
246C22×C4—1941no—
247C22×C4—1941no—
248C22×C4C22×D9711yescharacteristic
249Q16—1942no—
250SD16—1942no—
251SD16—1942no—
252D8—1942no—
253C2×C8C22×D9711yescharacteristic
254C2×C8C22×D9711yescharacteristic
255C2×C8C22×D9711yescharacteristic
256C2×C8C22×D9711yescharacteristic
257C4⋊C4—971no—
258C4⋊C4—971no—
259C4⋊C4—971no—
260C4⋊C4—971no—
261C4⋊C4—1941no—
262C4⋊C4—1941no—
263C4⋊C4—1941no—
264C4⋊C4—1941no—
265C4⋊C4—1941no—
266C4⋊C4—1941no—
267C4⋊C4D38811yescharacteristic
268C4⋊C4D38811yescharacteristic
269C22⋊C4—1941no—
270C22⋊C4—1941no—
271C22⋊C4—1941no—
272C22⋊C4—1941no—
273C22⋊C4—1941no—
274C22⋊C4—1941no—
275C22⋊C4—1941no—
276C22⋊C4D38811yescharacteristic
277C22⋊C4D38811yescharacteristic
278C42—1941no—
279C42—1941no—
280C42—1941no—
281C42C22×D9711yescharacteristic
282C42C22×D9711yescharacteristic
283C23—1941no—
284C23—1941no—
285C23C2×D38811yescharacteristic
286Q8—7761no—
287Q8—972no—
288Q8—1941no—
289Q8—1941no—
290Q8—1941no—
291D4—7761no—
292D4—972no—
293D4—1944no—
294D4—1941no—
295D4—1941no—
296D4—1941no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—7761no—
300C2×C4—7761no—
301C2×C4—7761no—
302C2×C4—7761no—
303C2×C4—1944no—
304C2×C4—1941no—
305C2×C4—1941no—
306C2×C4—1941no—
307C2×C4—1941no—
308C2×C4—1941no—
309C2×C4—1941no—
310C2×C4C2×D38811yescharacteristic
311C2×C4C2×D38811yescharacteristic
312C2×C4C2×D38811yescharacteristic
313C2×C4C2×D38811yescharacteristic
314C2×C4C2×D38811yescharacteristic
315C2×C4C23×D9711yescharacteristic
316C8—21no—
317C8—21no—
318C8—21no—
319C8—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—3881no—
324C22—3881no—
325C22—3881no—
326C22—3881no—
327C22—3881no—
328C22—1942no—
329C22C22×D38811yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—3881no—
336C4—3881no—
337C4—3881no—
338C4—3881no—
339C4—3881no—
340C4—1942no—
341C4#41×D9711yescharacteristic
342C4#41×D9711yescharacteristic
343C2—41no—
344C2—7761no—
345C2—3882no—
346C2#2026124711yescharacteristic
347C2#2026154011yescharacteristic
348C2#2026155211yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#433×C97C2
socleC22×C97C22×D4
prime radical#20260882C2
hypercenter#433D97
hypocenterC97#3124
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #20261356C2semidirect
2. #20261215C2semidirect
3. #20261071C2semidirect
4. #20261049C2semidirect
5. #20261031C2semidirect
6. #20261016C2semidirect
7. #20260958C2semidirect
8. #20260951C2semidirect
9. #20260899C2semidirect
11. #20260837C2semidirect
12. #20260837C2semidirect
13. C4×Dic388C2semidirect
14. C4×D776C2semidirect
15. #433×C97C2semidirect
175. C97#3124semidirect
190. #433D97semidirect

Maximal subgroups

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

subgroupindex
#202613562
#202612152
#202610712
#202610492
#202610312
#202610162
#202609582
#202609512
#202608992
#202608822
#202608372
C4×Dic3882
C4×D7762
#433×C972
#312497