Finite Group Databasebeta

← #27141755

Subgroups — #27141755

Order 12416.

312 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 87 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
1#20261203C211yesmaximal, characteristic
2#20260758C211yesmaximal, characteristic, complemented
3C2×#15690738C211yesmaximal, characteristic
4C2×#15690736C211yesmaximal, characteristic
5C4×#13095490C211yesmaximal, characteristic, complemented
6#20260494C211yesmaximal, characteristic, complemented
7#339×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic
9C2×#13095495C2211yescharacteristic
10#15690775C411yescharacteristic, complemented
11#15690775C411yescharacteristic, complemented
12#15690772C411yescharacteristic, complemented
13#15690772C411yescharacteristic, complemented
14#15690770—21no—
15#15690770—21no—
16#15690767—21no—
17C2×#13095490—21no—
18C2×C4×Dic97C2211yescharacteristic
19C2×C4×Dic97C2211yescharacteristic
20C42×D97—21no—
21#15690739—21no—
22#15690739—21no—
23#15690738—22no—
24#15690736—22no—
25#15690717—21no—
26C2×M16×C97C2211yescharacteristic
27#117×C97C2211yescharacteristic
28C2×C42×C97C2211yescharacteristic
29#13095511—21no—
30#13095511—21no—
31#13095511D412yes—
32#13095511Q812yes—
33C2×Dic194C2×C411yescharacteristic
34C2×D388C2×C411yescharacteristic
35C2×C22⋊D97—21no—
36C22×Dic97—21no—
37C22×Dic97D411yescharacteristic
38C22×Dic97D411yescharacteristic
39C2×C4×D97—21no—
40C2×C4×D97—21no—
41C2×C4×D97—21no—
42#13095495—21no—
43#13095495—21no—
44C2×#11715895—21no—
45#13095490—24no—
46#13095490—21no—
47#13095490—21no—
48#13095488—21no—
49#13095488—21no—
50C4×Dic97—21no—
51C4×Dic97—21no—
52C4×Dic97—21no—
53C4×Dic97—21no—
54C4×Dic97C2×C411yescharacteristic
55C4×Dic97C2×C411yescharacteristic
56C4×Dic97D411yescharacteristic
57C4×Dic97D411yescharacteristic
58C22×C4×C97—21no—
59C22×C4×C97C2311yescharacteristic
60M16×C97—21no—
61M16×C97—21no—
62C2×C8×C97—21no—
63C4⋊C4×C97C2×C411yescharacteristic
64C4⋊C4×C97C2×C411yescharacteristic
65C22⋊C4×C97—21no—
66C42×C97—21no—
67C42×C97—21no—
68C42×C97—21no—
69C23×C97C2×D411yescharacteristic
70C22×D97—21no—
71C2×C4×C97—22no—
72C2×C4×C97—21no—
73C2×C4×C97—21no—
74C2×C4×C97—41no—
75C2×C4×C97—41no—
76C2×C4×C97C22×C411yescharacteristic
77C2×C4×C97C2×D411yescharacteristic
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97C2×Q811yescharacteristic
80C2×C4×C97#4111yescharacteristic
81C2×C4×C97#4111yescharacteristic
82C22⋊D97—22no—
83C22⋊D97—22no—
84C2×Dic97—21no—
85C2×Dic97—22no—
86C2×Dic97—22no—
87C2×Dic97—41no—
88C2×Dic97—41no—
89C2×Dic97—41no—
90C2×Dic97C22⋊C411yescharacteristic
91C2×Dic97C22⋊C411yescharacteristic
92C2×Dic97C22⋊C411yescharacteristic
93C2×Dic97C22⋊C411yescharacteristic
94D388—21no—
95D388C4⋊C412yescomplemented
96C4×D97—22no—
97C4×D97—22no—
98C4×D97—42no—
99C4×D97—42no—
100Dic194—21no—
101Dic194C4⋊C412yescomplemented
102C8×C97—41no—
103#11715895—41no—
104C22×C97—21no—
105C22×C97—21no—
106C22×C97C2×C22⋊C411yescharacteristic
107C22×C97#12111yescharacteristic
108C22×C97C22⋊Q811yescharacteristic
109C2×D97—22no—
110C2×D97—41no—
111C2×D97—41no—
112C4×C97—42no—
113C4×C97—41no—
114C4×C97—41no—
115C4×C97C2×C4⋊C411yescharacteristic
116C4×C97#11711yescharacteristic
117C4×C97#12111yescharacteristic
118C4×C97C22⋊Q811yescharacteristic
119Dic97—22no—
120Dic97—21no—
121Dic97—21no—
122Dic97—21no—
123Dic97—42no—
124Dic97#10412yescomplemented
125C2×C97—21no—
126C2×C97—21no—
127C2×C97#38011yescharacteristic
128C2×C97C2×#10411yescharacteristic
129C2×C97#42111yescharacteristic
130D97—42no—
131#1806—971nomaximal, Sylow
132C97#180611yescharacteristic, complemented, Sylow
133C4×#41—971no—
134#423—971no—
135C2×#104—971no—
136C2×#104—971no—
137C4×C4⋊C4—971no—
138#339—971no—
139#339D9711yescharacteristic, complemented
140C2×#41—971no—
141C2×M16—971no—
142C2×M16C2×D9711yescharacteristic
143C4×Q8—971no—
144C4×Q8—971no—
145C4×D4—971no—
146C4×D4—971no—
147C4×D4—1941no—
148C4×D4—1941no—
149#117—1941no—
150#117C2×D9711yescharacteristic
151C2×C4⋊C4—1941no—
152C2×C42—971no—
153C2×C42—971no—
154C2×C42—1941no—
155C2×C42C2×D9711yescharacteristic
156C4⋊C8—1941no—
157C4⋊C8—1941no—
158#104—1942no—
159#104—1942no—
160#95—1941no—
161#41—972no—
162#41—972no—
163#41—1941no—
164#41—1941no—
165C2×Q8—971no—
166C2×D4—971no—
167C2×D4—1941no—
168C22×C4—21no—
169C22×C4—971no—
170C22×C4—971no—
171C22×C4—1941no—
172C22×C4—1941no—
173C22×C4—1941no—
174C22×C4—1941no—
175C22×C4C22×D9711yescharacteristic
176M16—21no—
177M16—21no—
178M16—1941no—
179M16—1941no—
180C2×C8—21no—
181C2×C8—1941no—
182C4⋊C4—1941no—
183C4⋊C4—1941no—
184C4⋊C4—1944no—
185C4⋊C4C4×D9711yescharacteristic
186C4⋊C4C4×D9711yescharacteristic
187C22⋊C4—21no—
188C22⋊C4—1941no—
189C22⋊C4—1941no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—971no—
194C42—971no—
195C42—971no—
196C42—971no—
197C42—1941no—
198C42—1941no—
199C42—1941no—
200C42—1941no—
201C23—1941no—
202C23D4×D9711yescharacteristic
203Q8—972no—
204Q8—1941no—
205D4—972no—
206D4—1942no—
207D4—1942no—
208D4—1941no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—971no—
215C2×C4—971no—
216C2×C4—971no—
217C2×C4—971no—
218C2×C4—3882no—
219C2×C4—3882no—
220C2×C4—3881no—
221C2×C4—3881no—
222C2×C4—3881no—
223C2×C4—1942no—
224C2×C4—1942no—
225C2×C4—1941no—
226C2×C4—1942no—
227C2×C4—1942no—
228C2×C4C2×C4×D9711yescharacteristic
229C2×C4D4×D9711yescharacteristic
230C2×C4D4×D9711yescharacteristic
231C2×C4Q8×D9711yescharacteristic
232C2×C4#1309551111yescharacteristic
233C2×C4#1309551111yescharacteristic
234C8—41no—
235C8—3881no—
236C22—21no—
237C22—21no—
238C22—3881no—
239C22—3881no—
240C22—1942no—
241C22C22⋊C4×D9711yescharacteristic
242C22#1569078311yescharacteristic
243C22#1569078811yescharacteristic
244C4—42no—
245C4—41no—
246C4—41no—
247C4—972no—
248C4—3882no—
249C4—1941no—
250C4—1941no—
251C4—1941no—
252C4—1942no—
253C4C4⋊C4×D9711yescharacteristic
254C4#1569076711yescharacteristic
255C4#1569078311yescharacteristic
256C4#1569078811yescharacteristic
257C2—21no—
258C2—21no—
259C2—3882no—
260C2#2026059411yescharacteristic
261C2#104×D9711yescharacteristic
262C2#2026075011yescharacteristic
subgroupG/H
centerC2×C4D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#339×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC2×#13095511C22
cosocleC22×C4×C97C23
hypercenter#339D97
hypocenterC97#1806
LCSG ▷ C2×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
2. #20260758C2semidirect
5. C4×#13095490C2semidirect
6. #20260494C2semidirect
7. #339×C97C2semidirect
10. #15690775C4semidirect
11. #15690775C4semidirect
12. #15690772C4semidirect
13. #15690772C4semidirect
95. D388C4⋊C4semidirect
101. Dic194C4⋊C4semidirect
124. Dic97#104semidirect
132. C97#1806semidirect
139. #339D97semidirect

Maximal subgroups

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

subgroupindex
#202612032
#202607582
C2×#156907382
C2×#156907362
C4×#130954902
#202604942
#339×C972
#180697