Finite Group Databasebeta

← #27141444

Subgroups — #27141444

Order 12416.

404 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 137 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×M16×D97C211yesmaximal, characteristic
2C2×#15690767C211yesmaximal, characteristic
3#117×D97C211yesmaximal, characteristic
4#20260496C211yesmaximal, characteristic, complemented
5#20260495C211yesmaximal, characteristic, complemented
6#20260494C211yesmaximal, characteristic, complemented
7#339×C97C211yesmaximal, characteristic, complemented
8C22×C4×D97C2211yescharacteristic
9C2×#13095495C2211yescharacteristic
10C2×#13095494—21no—
11M16×D97—22no—
12M16×D97C414yescomplemented
13C2×C8×D97—21no—
14#15690769C2211yescharacteristic
15#15690768C412yescomplemented
16#15690767—21no—
17#15690767—22no—
18#15690767C414yescomplemented
19#15690766—21no—
20C2×#13095490—21no—
21C4⋊C4×D97C412yescomplemented
22C2×#13095488—21no—
23C22⋊C4×D97—21no—
24C2×C4×Dic97C2211yescharacteristic
25C42×D97—21no—
26C2×M16×C97C2211yescharacteristic
27#117×C97C2211yescharacteristic
28C2×C42×C97C2211yescharacteristic
29C23×D97D411yescharacteristic
30C22×Dic97—21no—
31C22×Dic97D411yescharacteristic
32C2×C4×D97—22no—
33C2×C4×D97C2×C412yes—
34C2×C4×D97C2×C411yescharacteristic
35C2×C4×D97C2×C412yes—
36C2×C4×D97C2×C411yescharacteristic
37C2×C4×D97D411yescharacteristic
38C2×C4×D97D411yescharacteristic
39C2×C4×D97D412yes—
40C2×C4×D97Q812yes—
41#13095495—21no—
42#13095495C2×C412yescomplemented
43#13095494—24no—
44C2×#11715895—21no—
45C8×D97—24no—
46C4⋊Dic97C2×C412yes—
47#13095490—22no—
48#13095490—24no—
49C22⋊Dic97—21no—
50#13095488—22no—
51#13095488—24no—
52C4×Dic97—21no—
53C4×Dic97—21no—
54C4×Dic97—22no—
55C4×Dic97C2×C412yescomplemented
56C22×C4×C97—21no—
57C22×C4×C97C2311yescharacteristic
58M16×C97—21no—
59M16×C97C2×C412yescomplemented
60C2×C8×C97—21no—
61C4⋊C4×C97C2×C412yes—
62C22⋊C4×C97—21no—
63C42×C97—21no—
64C42×C97—21no—
65C42×C97C2×C412yescomplemented
66C23×C97C2×D411yescharacteristic
67C22×D97—42no—
68C22×D97C22⋊C411yescharacteristic
69C22×D97C22⋊C411yescharacteristic
70C22×D97C22⋊C412yes—
71C22×D97C4⋊C412yes—
72C2×C4×C97—22no—
73C2×C4×C97—22no—
74C2×C4×C97—42no—
75C2×C4×C97C22×C411yescharacteristic
76C2×C4×C97C22×C411yescharacteristic
77C2×C4×C97C22×C411yescharacteristic
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97C2×D411yescharacteristic
80C2×C4×C97C2×Q811yescharacteristic
81C2×Dic97—22no—
82C2×Dic97—22no—
83C2×Dic97—42no—
84C2×Dic97C22⋊C411yescharacteristic
85C2×Dic97C22⋊C412yes—
86C2×Dic97C22⋊C411yescharacteristic
87C2×Dic97C4⋊C412yes—
88C4×D97—44no—
89C4×D97C4212yes—
90C4×D97C4212yes—
91C4×D97C22⋊C412yes—
92C4×D97C22⋊C412yes—
93C4×D97C4⋊C412yes—
94C4×D97C4⋊C412yes—
95C4×D97C4⋊C412yes—
96C4×D97C4⋊C412yes—
97C8×C97—22no—
98#11715895—22no—
99C22×C97—22no—
100C22×C97C2×C22⋊C411yescharacteristic
101C22×C97C2×C22⋊C411yescharacteristic
102C22×C97C2×C4⋊C411yescharacteristic
103C2×D97—22no—
104C2×D97—22no—
105C2×D97—42no—
106C2×D97—42no—
107C2×D97#9512yes—
108C2×D97#9512yes—
109C4×C97—42no—
110C4×C97—42no—
111C4×C97C2×C4211yescharacteristic
112C4×C97C2×C22⋊C411yescharacteristic
113C4×C97C2×C4⋊C411yescharacteristic
114C4×C97C2×C4⋊C411yescharacteristic
115Dic97—42no—
116Dic97—42no—
117Dic97#9512yes—
118Dic97#9512yes—
119C2×C97—21no—
120C2×C97—21no—
121C2×C97C2×#9511yescharacteristic
122C2×C97#42112yes—
123D97—22no—
124D97—22no—
125#1733—971nomaximal, Sylow
126C97#173311yescharacteristic, complemented, Sylow
127C22×M16—971no—
128C2×#117—971no—
129C2×#117—971no—
130#339—971no—
131#339—971no—
132#339—971no—
133#339D9711yescharacteristic, complemented
134C23×C4—971no—
135C2×M16—971no—
136C2×M16—974no—
137C2×M16—1941no—
138C2×M16—1942no—
139C2×M16C2×D9711yescharacteristic
140C22×C8—1941no—
141#117—974no—
142#117—971no—
143#117—972no—
144#117—1942no—
145#117—1941no—
146#117—1941no—
147#117C2×D9711yescharacteristic
148C2×C4⋊C4—972no—
149C2×C4⋊C4—1941no—
150C2×C22⋊C4—1941no—
151C2×C22⋊C4—1941no—
152C2×C42—971no—
153C2×C42—1941no—
154C2×C42C2×D9711yescharacteristic
155C24—971no—
156C22×C4—21no—
157C22×C4—971no—
158C22×C4—971no—
159C22×C4—971no—
160C22×C4—972no—
161C22×C4—972no—
162C22×C4—972no—
163C22×C4—971no—
164C22×C4—971no—
165C22×C4—972no—
166C22×C4—1942no—
167C22×C4—1941no—
168C22×C4C22×D9711yescharacteristic
169M16—21no—
170M16—972no—
171M16—1944no—
172M16—1941no—
173M16C4×D9712yescomplemented
174C2×C8—21no—
175C2×C8—1944no—
176C2×C8—1941no—
177C4⋊C4—972no—
178C4⋊C4—1942no—
179C4⋊C4—1944no—
180C4⋊C4C4×D9712yes—
181C22⋊C4—21no—
182C22⋊C4—1942no—
183C22⋊C4—1944no—
184C22⋊C4—1941no—
185C42—21no—
186C42—21no—
187C42—972no—
188C42—1942no—
189C42—1941no—
190C42—1941no—
191C42C4×D9712yescomplemented
192C23—972no—
193C23—972no—
194C23—971no—
195C23—971no—
196C23—3882no—
197C23D4×D9711yescharacteristic
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—971no—
202C2×C4—972no—
203C2×C4—971no—
204C2×C4—972no—
205C2×C4—972no—
206C2×C4—972no—
207C2×C4—972no—
208C2×C4—972no—
209C2×C4—972no—
210C2×C4—972no—
211C2×C4—972no—
212C2×C4—972no—
213C2×C4—3884no—
214C2×C4—3882no—
215C2×C4—1942no—
216C2×C4—1942no—
217C2×C4C2×C4×D9711yescharacteristic
218C2×C4C2×C4×D9711yescharacteristic
219C2×C4C2×C4×D9711yescharacteristic
220C2×C4D4×D9711yescharacteristic
221C2×C4D4×D9711yescharacteristic
222C2×C4Q8×D9711yescharacteristic
223C8—22no—
224C8—1942no—
225C22—22no—
226C22—972no—
227C22—972no—
228C22—3882no—
229C22—3882no—
230C22—1942no—
231C22—1942no—
232C22C22⋊C4×D9711yescharacteristic
233C22C22⋊C4×D9711yescharacteristic
234C22C4⋊C4×D9711yescharacteristic
235C4—42no—
236C4—42no—
237C4—972no—
238C4—972no—
239C4—3882no—
240C4—3882no—
241C4C42×D9711yescharacteristic
242C4C22⋊C4×D9711yescharacteristic
243C4C4⋊C4×D9711yescharacteristic
244C4C4⋊C4×D9711yescharacteristic
245C2—21no—
246C2—21no—
247C2—1942no—
248C2—1942no—
249C2#95×D9711yescharacteristic
250C2#2026075012yes—
subgroupG/H
centerC2×C4D4×D97
commutatorC4×C97C2×C42
FrattiniC22×C4C22×D97
Fitting#339×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC23×D97D4
cosocleC22×C4×C97C23
hypercenter#339D97
hypocenterC97#1733
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
4. #20260496C2semidirect
5. #20260495C2semidirect
6. #20260494C2semidirect
7. #339×C97C2semidirect
12. M16×D97C4semidirect
15. #15690768C4semidirect
18. #15690767C4semidirect
21. C4⋊C4×D97C4semidirect
42. #13095495C2×C4semidirect
55. C4×Dic97C2×C4semidirect
59. M16×C97C2×C4semidirect
65. C42×C97C2×C4semidirect
126. C97#1733semidirect
133. #339D97semidirect
173. M16C4×D97semidirect
191. C42C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D972
C2×#156907672
#117×D972
#202604962
#202604952
#202604942
#339×C972
#173397