Finite Group Databasebeta

← #27151367

Subgroups — #27151367

Order 12416.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690843C211yesmaximal, characteristic, complemented
2C2×M16×D97C211yesmaximal, characteristic, complemented
3#20261247C211yesmaximal, characteristic
4C2×#15690790C211yesmaximal, characteristic, complemented
5#20260994C212yesmaximal, complemented
6#20260982C212yesmaximal, complemented
7#20260977C212yesmaximal, complemented
8#20260965C212yesmaximal, complemented
9#20260743C211yesmaximal, characteristic, complemented
10#20260734C211yesmaximal, characteristic, complemented
11#428×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13C22×Dic194C2211yescharacteristic
14C22×C4×D97C2211yescharacteristic
15#15690843—24no—
16C2×Dic388C2212yescomplemented
17C2×#13095500C2212yescomplemented
18C2×#13095495C2211yescharacteristic
19C2×#13095494C2212yes—
20M16×D97—24no—
21C2×C8×D97C2212yes—
22#15690794—21no—
23#15690790—21no—
24#15690790—21no—
25#15690790C2212yes—
26#15690790C2211yescharacteristic
27#15690790C2212yes—
28#15690790C2211yescharacteristic
29#15690785C2211yescharacteristic
30#15690785C2211yescharacteristic
31#15690780—21no—
32C4×D388—21no—
33C4×D388—21no—
34C2×C4⋊Dic97—21no—
35C2×#13095488—21no—
36#15690733C2212yescomplemented
37#15690733C2212yescomplemented
38#15690733C2212yescomplemented
39#15690730C2212yescomplemented
40C2×M16×C97C2211yescharacteristic
41#117×C97C2211yescharacteristic
42C2×C4⋊C4×C97C2211yescharacteristic
43#107×C97C2212yescomplemented
44#106×C97C2212yescomplemented
45C23×D97D411yescharacteristic
46#13095511—22no—
47#13095511—22no—
48C2×Dic194—22no—
49C2×Dic194—22no—
50C2×Dic194—21no—
51C2×Dic194C2312yes—
52C2×Dic194C2311yescharacteristic
53C2×D388—21no—
54C2×D388—21no—
55C2×D388C2311yescharacteristic
56C2×C22⋊D97—21no—
57C22×Dic97—21no—
58C22×Dic97D411yescharacteristic
59C2×C4×D97—22no—
60C2×C4×D97—22no—
61C2×C4×D97—21no—
62C2×C4×D97D411yescharacteristic
63C2×C4×D97D411yescharacteristic
64C2×C4×D97C2311yescharacteristic
65C2×C4×D97C2311yescharacteristic
66Dic388—24no—
67#13095500—24no—
68#13095495—22no—
69#13095494—24no—
70C2×#11715895C2312yes—
71C8×D97—24no—
72C4⋊Dic97—22no—
73C4⋊Dic97—22no—
74C4⋊Dic97—21no—
75C4⋊Dic97—21no—
76#13095488—22no—
77#13095488—22no—
78#13095488—21no—
79#13095488—21no—
80#13095488—21no—
81#13095488—21no—
82C22×C4×C97—21no—
83C22×C4×C97C2311yescharacteristic
84M16×C97D414yescomplemented
85C2×C8×C97C2312yes—
86C4⋊C4×C97—21no—
87C4⋊C4×C97C2311yescharacteristic
88C4⋊C4×C97C2311yescharacteristic
89C4⋊C4×C97C2312yes—
90C22⋊C4×C97—21no—
91C42×C97—21no—
92C23×C97C2×D411yescharacteristic
93C22×D97—22no—
94C22×D97—22no—
95C22×D97—21no—
96C22×D97—41no—
97C22×D97C2×D411yescharacteristic
98C22×D97C2×D411yescharacteristic
99C2×C4×C97—21no—
100C2×C4×C97—21no—
101C2×C4×C97—22no—
102C2×C4×C97—42no—
103C2×C4×C97C2×D411yescharacteristic
104C2×C4×C97C2×D412yes—
105C2×C4×C97#4112yes—
106C2×C4×C97C2411yescharacteristic
107C22⋊D97—42no—
108C2×Dic97—22no—
109C2×Dic97—21no—
110C2×Dic97—22no—
111C2×Dic97—42no—
112C2×Dic97C2×D411yescharacteristic
113C2×Dic97C2×D411yescharacteristic
114D388—21no—
115D388—21no—
116D388—42no—
117D388—42no—
118C4×D97—22no—
119C4×D97—22no—
120C4×D97—21no—
121C4×D97—21no—
122C4×D97—21no—
123C4×D97—21no—
124C4×D97—42no—
125Dic194—22no—
126Dic194—21no—
127Dic194—22no—
128Dic194—21no—
129Dic194—42no—
130C8×C97C2×D414yes—
131#11715895—22no—
132C22×C97—21no—
133C22×C97—21no—
134C22×C97#12112yes—
135C22×C97C22×D411yescharacteristic
136C2×D97—21no—
137C2×D97—21no—
138C2×D97—21no—
139C2×D97—21no—
140C2×D97—21no—
141C2×D97—44no—
142C2×D97—41no—
143C2×D97—41no—
144C2×D97—41no—
145C2×D97#13712yescomplemented
146C4×C97—41no—
147C4×C97—41no—
148C4×C97—42no—
149C4×C97#12112yes—
150C4×C97C22×D411yescharacteristic
151C4×C97C2×#4111yescharacteristic
152Dic97—21no—
153Dic97—21no—
154Dic97—42no—
155Dic97—41no—
156C2×C97—22no—
157C2×C97C2×#12111yescharacteristic
158C2×C97C2×#13711yescharacteristic
159C2×C97#57511yescharacteristic
160D97—22no—
161D97—41no—
162D97—81no—
163#3056—971nomaximal, Sylow
164C97#305611yescharacteristic, complemented, Sylow
165C2×#137—971no—
166C22×M16—971no—
167#525—971no—
168C2×C22⋊Q8—971no—
169#470—972no—
170#468—972no—
171#467—972no—
172#465—972no—
173#428D9711yescharacteristic, complemented
174#419—971no—
175#417—971no—
176C2×#41—971no—
177C22×Q8—971no—
178C23×C4—971no—
179#137—1944no—
180C2×Q16—972no—
181C2×SD16—972no—
182C2×M16—971no—
183C2×M16—972no—
184C2×M16—1944no—
185C2×M16C2×D9711yescharacteristic
186C22×C8—972no—
187#123—1941no—
188C22⋊Q8—972no—
189C22⋊Q8—972no—
190C22⋊Q8—971no—
191C22⋊Q8—971no—
192C22⋊Q8—1941no—
193C22⋊Q8—1941no—
194#121—971no—
195#121—971no—
196#120—1941no—
197C4×D4—1941no—
198C4×D4—1941no—
199#117C2×D9711yescharacteristic
200C2×C4⋊C4—1941no—
201C2×C4⋊C4C2×D9711yescharacteristic
202C2×C22⋊C4—1941no—
203#107C2×D9712yescomplemented
204#106C2×D9712yescomplemented
205#103—972no—
206#103—972no—
207#103—972no—
208#102—972no—
209C24—971no—
210#41—1942no—
211#41—1942no—
212C2×Q8—972no—
213C2×Q8—971no—
214C2×Q8—1942no—
215C2×Q8—1942no—
216C2×Q8—1941no—
217C2×D4—971no—
218C2×D4—1941no—
219C2×D4—1941no—
220C2×D4—1941no—
221C22×C4—21no—
222C22×C4—971no—
223C22×C4—971no—
224C22×C4—971no—
225C22×C4—971no—
226C22×C4—971no—
227C22×C4—1941no—
228C22×C4—1941no—
229C22×C4—1942no—
230C22×C4—1942no—
231C22×C4C22×D9711yescharacteristic
232Q16—1944no—
233SD16—1944no—
234M16—1942no—
235M16—1944no—
236M16D38814yescomplemented
237C2×C8—972no—
238C2×C8—1944no—
239C2×C8C22×D9712yes—
240C4⋊C4—21no—
241C4⋊C4—1942no—
242C4⋊C4—1942no—
243C4⋊C4—1941no—
244C4⋊C4—1941no—
245C4⋊C4C22×D9712yes—
246C4⋊C4C22×D9711yescharacteristic
247C4⋊C4C22×D9711yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—1941no—
250C22⋊C4—1941no—
251C22⋊C4—1942no—
252C22⋊C4—1942no—
253C22⋊C4—1941no—
254C22⋊C4—1941no—
255C42—21no—
256C23—971no—
257C23—971no—
258C23—3881no—
259C23—1942no—
260C23—1941no—
261C23—1942no—
262C23D4×D9711yescharacteristic
263Q8—3882no—
264Q8—1942no—
265Q8—1941no—
266Q8—1942no—
267Q8—1941no—
268D4—3882no—
269D4—3882no—
270D4—3882no—
271D4—1941no—
272D4—1941no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—971no—
278C2×C4—971no—
279C2×C4—3882no—
280C2×C4—3882no—
281C2×C4—1941no—
282C2×C4—1942no—
283C2×C4—1942no—
284C2×C4—1942no—
285C2×C4—1942no—
286C2×C4—1941no—
287C2×C4—1941no—
288C2×C4—1941no—
289C2×C4—1941no—
290C2×C4D4×D9711yescharacteristic
291C2×C4C2×D38812yes—
292C2×C4#1309551212yes—
293C2×C4C23×D9711yescharacteristic
294C8—1942no—
295C8C2×D38814yes—
296C22—21no—
297C22—21no—
298C22—972no—
299C22—3884no—
300C22—3881no—
301C22—3881no—
302C22—3881no—
303C22—1941no—
304C22—1941no—
305C22—1941no—
306C22—1941no—
307C22—1941no—
308C22#1569078512yes—
309C22C2×D4×D9711yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—3881no—
314C4—3882no—
315C4—1941no—
316C4—1941no—
317C4#1569078512yes—
318C4C22×D38811yescharacteristic
319C4C2×#1309551211yescharacteristic
320C2—22no—
321C2—7761no—
322C2—3881no—
323C2—1942no—
324C2C2×#1569078511yescharacteristic
325C2#137×D9711yescharacteristic
326C2#2026153111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#428×C97C2
socleC22×C97C22×D4
prime radical#20261247C2
hypercenter#428D97
hypocenterC97#3056
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#15690843C2semidirect
2. C2×M16×D97C2semidirect
4. C2×#15690790C2semidirect
5. #20260994C2semidirect
6. #20260982C2semidirect
7. #20260977C2semidirect
8. #20260965C2semidirect
9. #20260743C2semidirect
10. #20260734C2semidirect
11. #428×C97C2semidirect
16. C2×Dic388C22semidirect
17. C2×#13095500C22semidirect
36. #15690733C22semidirect
37. #15690733C22semidirect
38. #15690733C22semidirect
39. #15690730C22semidirect
43. #107×C97C22semidirect
44. #106×C97C22semidirect
84. M16×C97D4semidirect
145. C2×D97#137semidirect
164. C97#3056semidirect
173. #428D97semidirect
203. #107C2×D97semidirect
204. #106C2×D97semidirect
236. M16D388semidirect

Maximal subgroups

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

subgroupindex
C2×#156908432
C2×M16×D972
#202612472
C2×#156907902
#202609942
#202609822
#202609772
#202609652
#202607432
#202607342
#428×C972
#305697