Finite Group Databasebeta

← #16844267

Subgroups — #16844267

Order 3968.

284 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 81 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×C22⋊(C31⋊C8)C211yesmaximal, characteristic, complemented
2#14520912C211yesmaximal, characteristic, complemented
3#14520786C212yesmaximal, complemented
4#14520783C212yesmaximal, complemented
5C31×#525C211yesmaximal, characteristic
6C2×#41×C31C411yescharacteristic, complemented
7C23×C4×C31C2211yescharacteristic
8C31×#123—21no—
9C31×C22⋊Q8C411yescharacteristic, complemented
10C31×C22⋊Q8C2211yescharacteristic
11C31×#121C411yescharacteristic, complemented
12C31×#121C2211yescharacteristic
13C31×#120—21no—
14C4×D4×C31—21no—
15C4×D4×C31—21no—
16C31×#117C411yescharacteristic, complemented
17C2×C22⋊Dic31—21no—
18C22×C31⋊C8—21no—
19C31⋊#95—21no—
20C31⋊#95C2212yes—
21C22⋊(C31⋊C8)—21no—
22C22⋊(C31⋊C8)C2212yes—
23C24×C31D411yescharacteristic
24#41×C31—42no—
25C2×Q8×C31C2×C411yescharacteristic
26C2×D4×C31—21no—
27C2×D4×C31—21no—
28C2×D4×C31—21no—
29C2×D4×C31C2×C411yescharacteristic
30C22×C4×C31—21no—
31C22×C4×C31—42no—
32C22×C4×C31C2×C411yescharacteristic
33C22×C4×C31C2×C411yescharacteristic
34C22×C4×C31D411yescharacteristic
35C22×C4×C31D411yescharacteristic
36C22×C4×C31D411yescharacteristic
37C22×C4×C31C2311yescharacteristic
38C22×Dic31—22no—
39C4⋊C4×C31—21no—
40C4⋊C4×C31—21no—
41C4⋊C4×C31C2×C411yescharacteristic
42C4⋊C4×C31C2×C411yescharacteristic
43C22⋊C4×C31—21no—
44C22⋊C4×C31—21no—
45C22⋊C4×C31—21no—
46C22⋊C4×C31—21no—
47C22⋊C4×C31—21no—
48C42×C31—21no—
49C22⋊Dic31—42no—
50C2×C31⋊C8—22no—
51C2×C31⋊C8—42no—
52C23×C31—22no—
53C23×C31—22no—
54C23×C31—21no—
55C23×C31—41no—
56C23×C31C22⋊C411yescharacteristic
57C23×C31C22⋊C411yescharacteristic
58C23×C31C2×D411yescharacteristic
59Q8×C31—41no—
60D4×C31—41no—
61D4×C31—42no—
62D4×C31—42no—
63D4×C31—42no—
64C2×C4×C31—22no—
65C2×C4×C31—22no—
66C2×C4×C31—21no—
67C2×C4×C31—21no—
68C2×C4×C31—21no—
69C2×C4×C31—41no—
70C2×C4×C31—41no—
71C2×C4×C31—41no—
72C2×C4×C31—41no—
73C2×C4×C31—42no—
74C2×C4×C31C22⋊C411yescharacteristic
75C2×C4×C31C22⋊C411yescharacteristic
76C2×C4×C31C22×C411yescharacteristic
77C2×C4×C31C2×D411yescharacteristic
78C2×Dic31—42no—
79C2×Dic31—42no—
80C2×Dic31—42no—
81C31⋊C8—42no—
82#1516—311nomaximal, Sylow
83C22×C31—21no—
84C22×C31—21no—
85C22×C31—21no—
86C22×C31—21no—
87C22×C31—21no—
88C22×C31—21no—
89C22×C31—21no—
90C22×C31—44no—
91C22×C31—41no—
92C22×C31—41no—
93C22×C31—41no—
94C22×C31#9912yes—
95C22×C31C2×C22⋊C411yescharacteristic
96C22×C31#13512yes—
97C4×C31—21no—
98C4×C31—21no—
99C4×C31—21no—
100C4×C31—21no—
101C4×C31—41no—
102C4×C31—41no—
103C4×C31—41no—
104Dic31—82no—
105#525D3111yescharacteristic
106C2×C22⋊C8—311no—
107#381—311no—
108#328—312no—
109#327—312no—
110C2×C31—22no—
111C2×C31—22no—
112C2×C31—41no—
113C2×C31—81no—
114C2×C31C2×#9911yescharacteristic
115C2×C31#41611yescharacteristic
116C2×C31#42111yescharacteristic
117C2×#41Dic3111yescharacteristic, complemented
118C23×C4C2×D3111yescharacteristic
119C22×C8—621no—
120#123—21no—
121C22⋊Q8Dic3111yescharacteristic, complemented
122C22⋊Q8C2×D3111yescharacteristic
123#121Dic3111yescharacteristic, complemented
124#121C2×D3111yescharacteristic
125#120—21no—
126C4×D4—21no—
127C4×D4—21no—
128#117Dic3111yescharacteristic, complemented
129C2×C22⋊C4—621no—
130C22⋊C8—312no—
131C22⋊C8—621no—
132#95—312no—
133#95—621no—
134C31#151611yescharacteristic, complemented, Sylow
135C24C22⋊D3111yescharacteristic
136#41—42no—
137C2×Q8C2×Dic3111yescharacteristic
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4C2×Dic3111yescharacteristic
142C22×C4—21no—
143C22×C4—42no—
144C22×C4—622no—
145C22×C4C2×Dic3111yescharacteristic
146C22×C4C2×Dic3111yescharacteristic
147C22×C4C22⋊D3111yescharacteristic
148C22×C4C22⋊D3111yescharacteristic
149C22×C4C22⋊D3111yescharacteristic
150C22×C4C22×D3111yescharacteristic
151C2×C8—622no—
152C2×C8—1242no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4C2×Dic3111yescharacteristic
156C4⋊C4C2×Dic3111yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—1242no—
163C42—21no—
164C23—22no—
165C23—22no—
166C23—21no—
167C23—41no—
168C23C22⋊Dic3111yescharacteristic
169C23C22⋊Dic3111yescharacteristic
170C23C2×C22⋊D3111yescharacteristic
171Q8—41no—
172D4—41no—
173D4—42no—
174D4—42no—
175D4—42no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—1242no—
187C2×C4—1242no—
188C2×C4—1242no—
189C2×C4C22⋊Dic3111yescharacteristic
190C2×C4C22⋊Dic3111yescharacteristic
191C2×C4C22×Dic3111yescharacteristic
192C2×C4C2×C22⋊D3111yescharacteristic
193C8—1242no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22#1175842212yes—
206C22#1175852712yes—
207C22C2×C22⋊Dic3111yescharacteristic
208C4—21no—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—41no—
213C4—41no—
214C4—41no—
215C4—2482no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—81no—
220C2C2×#1175842211yescharacteristic
221C2#1452104511yescharacteristic
222C2#1452106611yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic31
commutatorC2×C4×C31C22×C4
FrattiniC22×C4C22×D31
Fitting, prime radicalC31×#525C2
socleC22×C31C2×C22⋊C4
cosocleC22×C4×C31C23
hypercenter#525D31
hypocenterC31#1516
LCSG ▷ C2×C4×C31 ▷ C22×C31 ▷ C31
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊(C31⋊C8)C2semidirect
2. #14520912C2semidirect
3. #14520786C2semidirect
4. #14520783C2semidirect
6. C2×#41×C31C4semidirect
9. C31×C22⋊Q8C4semidirect
11. C31×#121C4semidirect
16. C31×#117C4semidirect
117. C2×#41Dic31semidirect
121. C22⋊Q8Dic31semidirect
123. #121Dic31semidirect
128. #117Dic31semidirect
134. C31#1516semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊(C31⋊C8)2
#145209122
#145207862
#145207832
C31×#5252
#151631