Finite Group Databasebeta

← #27151364

Subgroups — #27151364

Order 12416.

496 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 135 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×#15690841C211yesmaximal, characteristic, complemented
2C2×M16×D97C211yesmaximal, characteristic, complemented
3#20261247C211yesmaximal, characteristic
4C2×#15690786C211yesmaximal, characteristic, complemented
5#20260992C212yesmaximal, complemented
6#20260984C212yesmaximal, complemented
7#20260975C212yesmaximal, complemented
8#20260967C212yesmaximal, complemented
9#20260778C211yesmaximal, characteristic, complemented
10#20260737C211yesmaximal, characteristic, complemented
11#419×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13C22×C4×D97C2211yescharacteristic
14#15690841—24no—
15C2×#13095501C2212yescomplemented
16C2×#13095498C2212yescomplemented
17C2×#13095495C2211yescharacteristic
18C2×#13095494C2212yes—
19M16×D97—24no—
20C2×C8×D97C2212yes—
21#15690794—21no—
22#15690790C2211yescharacteristic
23#15690790C2211yescharacteristic
24#15690786—21no—
25#15690786—21no—
26#15690786C2212yes—
27#15690786C2212yes—
28#15690785C2211yescharacteristic
29#15690785C2211yescharacteristic
30#15690780—21no—
31C4×D388—21no—
32C4×D388—21no—
33C2×C4⋊Dic97C2211yescharacteristic
34C2×#13095490—21no—
35C2×#13095488—21no—
36#15690743C2212yescomplemented
37#15690741C2212yescomplemented
38#15690735C2212yescomplemented
39#15690732C2212yescomplemented
40C22×Q8×C97C2211yescharacteristic
41C2×M16×C97C2211yescharacteristic
42#117×C97C2211yescharacteristic
43#103×C97C2212yescomplemented
44#103×C97C2212yescomplemented
45C23×D97D411yescharacteristic
46#13095511—22no—
47#13095511—22no—
48C2×Dic194C2311yescharacteristic
49C2×D388—21no—
50C2×D388—21no—
51C2×D388C2311yescharacteristic
52C2×C22⋊D97—21no—
53C22×Dic97—21no—
54C22×Dic97D411yescharacteristic
55C2×C4×D97—22no—
56C2×C4×D97—22no—
57C2×C4×D97—21no—
58C2×C4×D97D411yescharacteristic
59C2×C4×D97D411yescharacteristic
60C2×C4×D97C2311yescharacteristic
61C2×C4×D97C2311yescharacteristic
62#13095501—24no—
63#13095498—24no—
64#13095495D414yescomplemented
65#13095494—24no—
66C2×#11715895C2312yes—
67C8×D97—24no—
68C4⋊Dic97—21no—
69C4⋊Dic97—21no—
70C4⋊Dic97—21no—
71C4⋊Dic97C2312yes—
72#13095490—22no—
73#13095490—22no—
74#13095488—22no—
75#13095488—22no—
76#13095488—21no—
77#13095488—21no—
78#13095488—21no—
79#13095488—21no—
80C2×Q8×C97—21no—
81C2×Q8×C97—22no—
82C2×Q8×C97—22no—
83C2×Q8×C97C2312yes—
84C22×C4×C97—21no—
85C22×C4×C97C2311yescharacteristic
86M16×C97—22no—
87C2×C8×C97C2312yes—
88C4⋊C4×C97C2311yescharacteristic
89C4⋊C4×C97C2311yescharacteristic
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
99Q8×C97—22no—
100Q8×C97—22no—
101Q8×C97—42no—
102C2×C4×C97—21no—
103C2×C4×C97—21no—
104C2×C4×C97—22no—
105C2×C4×C97—42no—
106C2×C4×C97C2×D411yescharacteristic
107C2×C4×C97C2×D412yes—
108C2×C4×C97#4112yes—
109C2×C4×C97C2411yescharacteristic
110C22⋊D97—42no—
111C2×Dic97—22no—
112C2×Dic97—22no—
113C2×Dic97—21no—
114C2×Dic97—42no—
115C2×Dic97C2×D411yescharacteristic
116C2×Dic97C2×D411yescharacteristic
117D388—21no—
118D388—21no—
119D388—42no—
120D388—42no—
121C4×D97—22no—
122C4×D97—22no—
123C4×D97—21no—
124C4×D97—21no—
125C4×D97—21no—
126C4×D97—21no—
127C4×D97—42no—
128Dic194—21no—
129Dic194—21no—
130C8×C97—22no—
131#11715895C2×D414yes—
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—42no—
147C4×C97—41no—
148C4×C97—41no—
149C4×C97#12112yes—
150C4×C97C22×D411yescharacteristic
151C4×C97C2×#4111yescharacteristic
152Dic97—21no—
153Dic97—21no—
154Dic97—41no—
155Dic97—42no—
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#428—971no—
174#419D9711yescharacteristic, complemented
175#417—971no—
176C2×#41—971no—
177C22×Q8C2×D9711yescharacteristic
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—971no—
201C2×C4⋊C4—1941no—
202C2×C22⋊C4—1941no—
203#107—972no—
204#106—972no—
205#103—972no—
206#103C2×D9712yescomplemented
207#103C2×D9712yescomplemented
208#102—972no—
209C24—971no—
210#41—1942no—
211#41—1942no—
212C2×Q8—21no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—971no—
216C2×Q8C22×D9712yes—
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—22no—
235M16—974no—
236M16—1944no—
237C2×C8—972no—
238C2×C8—1944no—
239C2×C8C22×D9712yes—
240C4⋊C4—972no—
241C4⋊C4—1942no—
242C4⋊C4—1942no—
243C4⋊C4—1941no—
244C4⋊C4—1941no—
245C4⋊C4—1941no—
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—22no—
264Q8—22no—
265Q8—42no—
266Q8—1941no—
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—1942no—
282C2×C4—1941no—
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×C4D4×D9712yes—
292C2×C4#1309551112yes—
293C2×C4C23×D9711yescharacteristic
294C8—22no—
295C8—974no—
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#1569078312yes—
309C22C2×D4×D9711yescharacteristic
310C4—42no—
311C4—41no—
312C4—41no—
313C4—3881no—
314C4—3882no—
315C4—1941no—
316C4—1941no—
317C4#1569078312yes—
318C4C2×D4×D9711yescharacteristic
319C4C2×#1309551111yescharacteristic
320C2—22no—
321C2—7761no—
322C2—3881no—
323C2—1942no—
324C2C2×#1569078311yescharacteristic
325C2#137×D9711yescharacteristic
326C2#2026153411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#419×C97C2
socleC22×C97C22×D4
prime radical#20261247C2
hypercenter#419D97
hypocenterC97#3056
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C2×#15690841C2semidirect
2. C2×M16×D97C2semidirect
4. C2×#15690786C2semidirect
5. #20260992C2semidirect
6. #20260984C2semidirect
7. #20260975C2semidirect
8. #20260967C2semidirect
9. #20260778C2semidirect
10. #20260737C2semidirect
11. #419×C97C2semidirect
15. C2×#13095501C22semidirect
16. C2×#13095498C22semidirect
36. #15690743C22semidirect
37. #15690741C22semidirect
38. #15690735C22semidirect
39. #15690732C22semidirect
43. #103×C97C22semidirect
44. #103×C97C22semidirect
64. #13095495D4semidirect
145. C2×D97#137semidirect
164. C97#3056semidirect
174. #419D97semidirect
206. #103C2×D97semidirect
207. #103C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908412
C2×M16×D972
#202612472
C2×#156907862
#202609922
#202609842
#202609752
#202609672
#202607782
#202607372
#419×C972
#305697