Finite Group Databasebeta

← #27153452

Subgroups — #27153452

Order 12416.

484 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 131 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×#15690829C211yesmaximal, characteristic, complemented
2#20261384C212yesmaximal, complemented
3#20261113C211yesmaximal, characteristic, complemented
4#20260976C211yesmaximal, characteristic, complemented
5#20260973C211yesmaximal, characteristic, complemented
6#20260938C212yesmaximal, complemented
7#20260897C212yesmaximal, complemented
8#20260894C212yesmaximal, complemented
9D8×Dic97C211yesmaximal, characteristic, complemented
10C8×C22⋊D97C211yesmaximal, characteristic, complemented
11C2×#135×C97C211yesmaximal, characteristic, complemented
12C2×#13095510C2212yes—
13#15690829—24no—
14C2×Dic388C2211yescharacteristic
15C2×#13095501C2212yes—
16C2×#13095499C2212yes—
17C2×C8×D97C2211yescharacteristic
18#15690808C2212yes—
19#15690793—22no—
20#15690791—22no—
21#15690789C2212yes—
22#15690786C2212yes—
23#15690782—22no—
24D4×Dic97C2212yes—
25C4×C22⋊D97C2211yescharacteristic
26C2×#13095490—22no—
27#15690744C2211yescharacteristic
28#15690739C2211yescharacteristic
29#15690735C2212yes—
30#15690734C2212yes—
31#15690731C2212yes—
32C22⋊#11715895C2211yescharacteristic
33#15690722C2211yescharacteristic
34C8×Dic97C2211yescharacteristic
35C2×#41×C97C2212yes—
36#135×C97—24no—
37C2×Q16×C97C2211yescharacteristic
38C2×SD16×C97C2212yes—
39C2×D8×C97C2211yescharacteristic
40C22×C8×C97C2211yescharacteristic
41#13095510—24no—
42#13095510—42no—
43C2×Dic194C2312yes—
44C2×C22⋊D97—22no—
45C2×C22⋊D97D411yescharacteristic
46C22×Dic97—22no—
47C22×Dic97—22no—
48C2×C4×D97C2311yescharacteristic
49Dic388—22no—
50#13095501—42no—
51#13095499—24no—
52C2×#11715895C2311yescharacteristic
53C2×#11715895C2311yescharacteristic
54C8×D97—22no—
55C4⋊Dic97C2312yes—
56#13095490—22no—
57#13095490—22no—
58#13095490—22no—
59#13095490—22no—
60#13095490D411yescharacteristic
61C22⋊Dic97—22no—
62C22⋊Dic97—22no—
63C22⋊Dic97D411yescharacteristic
64#13095488—22no—
65#13095488D411yescharacteristic
66C4×Dic97C2311yescharacteristic
67#41×C97—24no—
68#41×C97—42no—
69C2×Q8×C97C2312yes—
70C2×D4×C97—22no—
71C2×D4×C97C2312yes—
72C22×C4×C97—22no—
73C22×C4×C97C2311yescharacteristic
74Q16×C97—22no—
75SD16×C97—24no—
76D8×C97D414yescomplemented
77C2×C8×C97—22no—
78C2×C8×C97C2311yescharacteristic
79C2×C8×C97C2311yescharacteristic
80C23×C97—22no—
81C23×C97C2×D411yescharacteristic
82C22×D97C2×D411yescharacteristic
83Q8×C97—22no—
84Q8×C97—22no—
85D4×C97—22no—
86D4×C97—44no—
87D4×C97C2×D414yes—
88C2×C4×C97—21no—
89C2×C4×C97—21no—
90C2×C4×C97—22no—
91C2×C4×C97—44no—
92C2×C4×C97C2×D411yescharacteristic
93C2×C4×C97C2411yescharacteristic
94C22⋊D97—22no—
95C22⋊D97—44no—
96C2×Dic97—22no—
97C2×Dic97—22no—
98C2×Dic97—44no—
99C2×Dic97—44no—
100C2×Dic97C2×D411yescharacteristic
101C2×Dic97C2×D411yescharacteristic
102C2×Dic97C2×D411yescharacteristic
103C4×D97—21no—
104C4×D97—21no—
105Dic194—22no—
106Dic194—22no—
107C8×C97—21no—
108C8×C97C2×D412yes—
109#11715895—21no—
110#11715895—21no—
111C22×C97—24no—
112C22×C97—21no—
113C22×C97—21no—
114C22×C97—21no—
115C22×C97—42no—
116C22×C97—42no—
117C22×C97C22×D411yescharacteristic
118C2×D97—21no—
119C2×D97—21no—
120C2×D97—21no—
121C4×C97—21no—
122C4×C97—42no—
123C4×C97C22×D411yescharacteristic
124C4×C97H(5,𝔽2)11yescharacteristic
125Dic97—21no—
126Dic97—21no—
127Dic97—42no—
128Dic97—42no—
129Dic97#13512yescomplemented
130C2×C97—41no—
131C2×C97—44no—
132C2×C97D4211yescharacteristic
133C2×C97C2×#13511yescharacteristic
134C2×C97#57811yescharacteristic
135D97—41no—
136#3283—971nomaximal, Sylow
137C97#328311yescharacteristic, complemented, Sylow
138C2×#135—971no—
139C2×#135D9711yescharacteristic, complemented
140#547—972no—
141#494—971no—
142#467—971no—
143#467—971no—
144#461—972no—
145#452—972no—
146#452—972no—
147C4×D8—971no—
148C8×D4—971no—
149C2×#41—972no—
150C2×#41C2×D9712yes—
151#135—24no—
152#135—1944no—
153C2×Q16—972no—
154C2×Q16—971no—
155C2×Q16C2×D9711yescharacteristic
156C2×SD16—972no—
157C2×SD16C2×D9712yes—
158C2×D8C2×D9711yescharacteristic
159C22×C8—971no—
160C22×C8C2×D9711yescharacteristic
161C4⋊Q8—972no—
162#123—1942no—
163#123—1942no—
164C22⋊Q8—972no—
165C22⋊Q8—972no—
166#121—1942no—
167C4×D4—971no—
168C4×D4—972no—
169C2×C4⋊C4—1942no—
170#107—971no—
171C4⋊C8—971no—
172#103—972no—
173#103—972no—
174#102—972no—
175C22⋊C8—971no—
176C22⋊C8—971no—
177C4×C8—971no—
178#41—24no—
179#41—42no—
180#41—3882no—
181#41—1944no—
182C2×Q8—972no—
183C2×Q8C22×D9712yes—
184C2×D4—22no—
185C2×D4—971no—
186C2×D4—1942no—
187C2×D4C22×D9712yes—
188C22×C4—22no—
189C22×C4—971no—
190C22×C4—1942no—
191C22×C4—1942no—
192C22×C4C22×D9711yescharacteristic
193Q16—22no—
194Q16—3882no—
195Q16—1942no—
196SD16—24no—
197SD16—1944no—
198D8C22⋊D9714yescomplemented
199C2×C8—22no—
200C2×C8—971no—
201C2×C8—971no—
202C2×C8—1942no—
203C2×C8C22×D9711yescharacteristic
204C2×C8C22×D9711yescharacteristic
205C4⋊C4—971no—
206C4⋊C4—972no—
207C4⋊C4—1942no—
208C4⋊C4—1942no—
209C4⋊C4—1942no—
210C4⋊C4—1942no—
211C22⋊C4—971no—
212C22⋊C4—971no—
213C22⋊C4—1942no—
214C22⋊C4—1942no—
215C22⋊C4—1942no—
216C42—971no—
217C23—22no—
218C23—971no—
219C23D4×D9711yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—1942no—
223Q8—1942no—
224D4—22no—
225D4—44no—
226D4—3884no—
227D4—1942no—
228D4C2×C22⋊D9714yes—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—44no—
233C2×C4—971no—
234C2×C4—971no—
235C2×C4—971no—
236C2×C4—3884no—
237C2×C4—3884no—
238C2×C4—1942no—
239C2×C4—1942no—
240C2×C4—1941no—
241C2×C4—1941no—
242C2×C4D4×D9711yescharacteristic
243C2×C4C23×D9711yescharacteristic
244C8—21no—
245C8—1941no—
246C8—1941no—
247C8C2×C22⋊D9712yes—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—24no—
252C22—42no—
253C22—42no—
254C22—1941no—
255C22—1941no—
256C22—1941no—
257C22C2×D4×D9711yescharacteristic
258C4—21no—
259C4—42no—
260C4—972no—
261C4—3882no—
262C4—3882no—
263C4—1941no—
264C4—1941no—
265C4C22×C22⋊D9711yescharacteristic
266C4#1569085511yescharacteristic
267C2—41no—
268C2—44no—
269C2—3881no—
270C2D4×C22⋊D9711yescharacteristic
271C2#135×D9711yescharacteristic
272C2#2026154911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#135×C97C2
socleC22×C97C22×D4
hypercenterC2×#135D97
hypocenterC97#3283
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#15690829C2semidirect
2. #20261384C2semidirect
3. #20261113C2semidirect
4. #20260976C2semidirect
5. #20260973C2semidirect
6. #20260938C2semidirect
7. #20260897C2semidirect
8. #20260894C2semidirect
9. D8×Dic97C2semidirect
10. C8×C22⋊D97C2semidirect
11. C2×#135×C97C2semidirect
76. D8×C97D4semidirect
129. Dic97#135semidirect
137. C97#3283semidirect
139. C2×#135D97semidirect
198. D8C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908292
#202613842
#202611132
#202609762
#202609732
#202609382
#202608972
#202608942
D8×Dic972
C8×C22⋊D972
C2×#135×C972
#328397