Finite Group Databasebeta

← #16845729

Subgroups — #16845729

Order 3968.

312 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 95 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#41×Dic31C211yesmaximal, characteristic
2#14521072C211yesmaximal, characteristic, complemented
3C2×#11758425C211yesmaximal, characteristic
4C4×Dic31⋊C4C211yesmaximal, characteristic, complemented
5#14520810C211yesmaximal, characteristic, complemented
6#14520809C211yesmaximal, characteristic, complemented
7C2×C31×#104C211yesmaximal, characteristic
8C2×#41×C31C2211yescharacteristic
9C2×M16×C31C2211yescharacteristic
10C2×C42×C31C2211yescharacteristic
11Q8×Dic31C411yescharacteristic, complemented
12Q8×Dic31C411yescharacteristic, complemented
13D4×Dic31—21no—
14D4×Dic31—21no—
15D4×Dic31C411yescharacteristic, complemented
16D4×Dic31C411yescharacteristic, complemented
17#11758503—21no—
18#11758503C2211yescharacteristic
19C2×Dic31⋊C4—21no—
20C2×C4×Dic31—21no—
21C2×C4×Dic31C2211yescharacteristic
22C2×C4×Dic31C2211yescharacteristic
23C2×#92474C2211yescharacteristic
24C31×#104—22no—
25#11758425—22no—
26C31⋊#95—21no—
27Dic31⋊C8—21no—
28Dic31⋊C8—21no—
29#41×C31—21no—
30#41×C31—21no—
31#41×C31D412yes—
32#41×C31Q812yes—
33C2×Q8×C31C2×C411yescharacteristic
34C2×D4×C31—21no—
35C2×D4×C31C2×C411yescharacteristic
36C22×C4×C31—21no—
37C22×C4×C31—21no—
38C22×C4×C31C2311yescharacteristic
39C22×Dic31—21no—
40C22×Dic31—21no—
41C22×Dic31—21no—
42C22×Dic31D411yescharacteristic
43C22×Dic31D411yescharacteristic
44M16×C31—21no—
45M16×C31—21no—
46C2×C8×C31—21no—
47C42×C31—21no—
48C42×C31—21no—
49C22⋊Dic31—21no—
50C22⋊Dic31—21no—
51C22⋊Dic31—21no—
52C4⋊Dic31—21no—
53C4⋊Dic31—21no—
54C4⋊Dic31C2×C411yescharacteristic
55C4⋊Dic31C2×C411yescharacteristic
56Dic31⋊C4—24no—
57C4×Dic31—21no—
58C4×Dic31—21no—
59C4×Dic31—21no—
60C4×Dic31—21no—
61C4×Dic31—21no—
62C4×Dic31C2×C411yescharacteristic
63C4×Dic31C2×C411yescharacteristic
64C4×Dic31D411yescharacteristic
65C4×Dic31D411yescharacteristic
66#92474—21no—
67#92474—21no—
68C2×C31⋊C8—21no—
69C23×C31—21no—
70C23×C31C2×D411yescharacteristic
71Q8×C31—21no—
72Q8×C31C4⋊C412yescomplemented
73D4×C31—22no—
74D4×C31—22no—
75D4×C31—21no—
76D4×C31C4⋊C412yescomplemented
77C2×C4×C31—21no—
78C2×C4×C31—22no—
79C2×C4×C31—22no—
80C2×C4×C31—22no—
81C2×C4×C31—41no—
82C2×C4×C31—41no—
83C2×C4×C31C22×C411yescharacteristic
84C2×C4×C31C2×D411yescharacteristic
85C2×C4×C31C2×D411yescharacteristic
86C2×C4×C31C2×Q811yescharacteristic
87C2×C4×C31#4111yescharacteristic
88C2×C4×C31#4111yescharacteristic
89C2×Dic31—22no—
90C2×Dic31—22no—
91C2×Dic31—21no—
92C2×Dic31—21no—
93C2×Dic31—41no—
94C2×Dic31—42no—
95C2×Dic31—41no—
96C2×Dic31—41no—
97C2×Dic31—42no—
98C2×Dic31C22⋊C411yescharacteristic
99C2×Dic31C22⋊C411yescharacteristic
100C2×Dic31C22⋊C411yescharacteristic
101C2×Dic31C22⋊C411yescharacteristic
102C8×C31—41no—
103C31⋊C8—41no—
104#1806—311nomaximal, Sylow
105C22×C31—21no—
106C22×C31—21no—
107C22×C31—22no—
108C22×C31—41no—
109C22×C31—41no—
110C22×C31C2×C22⋊C411yescharacteristic
111C22×C31#12111yescharacteristic
112C22×C31C22⋊Q811yescharacteristic
113C4×C31—22no—
114C4×C31—42no—
115C4×C31C2×C4⋊C411yescharacteristic
116C4×C31#11711yescharacteristic
117C4×C31#12111yescharacteristic
118C4×C31C22⋊Q811yescharacteristic
119Dic31—21no—
120Dic31—21no—
121Dic31—21no—
122Dic31—42no—
123Dic31—41no—
124Dic31—41no—
125Dic31#10412yescomplemented
126C4×#41—311no—
127#423—311no—
128C2×#104—311no—
129C2×#104D3111yescharacteristic
130C4×C4⋊C4—311no—
131#339—311no—
132#339—311no—
133C2×C31—21no—
134C2×C31—21no—
135C2×C31—42no—
136C2×C31#38011yescharacteristic
137C2×C31C2×#10411yescharacteristic
138C2×C31#42111yescharacteristic
139C2×#41C2×D3111yescharacteristic
140C2×M16—311no—
141C2×M16C2×D3111yescharacteristic
142C4×Q8—311no—
143C4×Q8—311no—
144C4×D4—311no—
145C4×D4—311no—
146C4×D4—621no—
147C4×D4—621no—
148#117—311no—
149#117—621no—
150C2×C4⋊C4—621no—
151C2×C42—311no—
152C2×C42—311no—
153C2×C42—621no—
154C2×C42C2×D3111yescharacteristic
155C4⋊C8—621no—
156C4⋊C8—621no—
157#104—22no—
158#104—622no—
159#95—621no—
160C31#180611yescharacteristic, complemented, Sylow
161#41—21no—
162#41—21no—
163#41Dic6212yes—
164#41C22⋊D3112yes—
165C2×Q8C4×D3111yescharacteristic
166C2×D4—21no—
167C2×D4C4×D3111yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—311no—
171C22×C4—311no—
172C22×C4—621no—
173C22×C4—621no—
174C22×C4—621no—
175C22×C4C22×D3111yescharacteristic
176M16—21no—
177M16—21no—
178M16—621no—
179M16—621no—
180C2×C8—21no—
181C2×C8—621no—
182C4⋊C4—311no—
183C4⋊C4—311no—
184C4⋊C4—621no—
185C4⋊C4—621no—
186C4⋊C4—624no—
187C22⋊C4—621no—
188C22⋊C4—621no—
189C22⋊C4—621no—
190C42—21no—
191C42—21no—
192C42—311no—
193C42—311no—
194C42—311no—
195C42—311no—
196C42—621no—
197C42—621no—
198C42—621no—
199C42—621no—
200C42—621no—
201C23—21no—
202C23D4×D3111yescharacteristic
203Q8—21no—
204Q8Dic31⋊C412yescomplemented
205D4—22no—
206D4—22no—
207D4—21no—
208D4Dic31⋊C412yescomplemented
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—311no—
216C2×C4—311no—
217C2×C4—311no—
218C2×C4—311no—
219C2×C4—622no—
220C2×C4—622no—
221C2×C4—621no—
222C2×C4—621no—
223C2×C4—1242no—
224C2×C4—1241no—
225C2×C4—1242no—
226C2×C4—1241no—
227C2×C4—1241no—
228C2×C4C2×Dic6211yescharacteristic
229C2×C4C2×C4×D3111yescharacteristic
230C2×C4D4×D3111yescharacteristic
231C2×C4#9249711yescharacteristic
232C2×C4#9249711yescharacteristic
233C2×C4C2×C22⋊D3111yescharacteristic
234C8—41no—
235C8—1241no—
236C22—21no—
237C22—21no—
238C22—22no—
239C22—41no—
240C22—41no—
241C22#1175845511yescharacteristic
242C22C22⋊C4×D3111yescharacteristic
243C22#1175851611yescharacteristic
244C4—22no—
245C4—42no—
246C4—312no—
247C4—621no—
248C4—621no—
249C4—621no—
250C4—1242no—
251C4—1241no—
252C4—1241no—
253C4#1175845411yescharacteristic
254C4#1175845511yescharacteristic
255C4C2×Dic31⋊C411yescharacteristic
256C4#1175851611yescharacteristic
257C2—21no—
258C2—21no—
259C2—42no—
260C2#1452090711yescharacteristic
261C2#104×D3111yescharacteristic
262C2#1452106411yescharacteristic
subgroupG/H
centerC2×C4D4×D31
commutatorC2×C4×C31C22×C4
FrattiniC22×C4C22×D31
FittingC2×C31×#104C2
socleC22×C31C2×C22⋊C4
prime radicalC2×#41×C31C22
cosocleC22×C4×C31C23
hypercenterC2×#104D31
hypocenterC31#1806
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. #14521072C2semidirect
4. C4×Dic31⋊C4C2semidirect
5. #14520810C2semidirect
6. #14520809C2semidirect
11. Q8×Dic31C4semidirect
12. Q8×Dic31C4semidirect
15. D4×Dic31C4semidirect
16. D4×Dic31C4semidirect
72. Q8×C31C4⋊C4semidirect
76. D4×C31C4⋊C4semidirect
125. Dic31#104semidirect
160. C31#1806semidirect
204. Q8Dic31⋊C4semidirect
208. D4Dic31⋊C4semidirect

Maximal subgroups

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

subgroupindex
#41×Dic312
#145210722
C2×#117584252
C4×Dic31⋊C42
#145208102
#145208092
C2×C31×#1042
#180631