Finite Group Databasebeta

← #16857640

Subgroups — #16857640

Order 3968.

398 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 123 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#14521764C211yesmaximal, characteristic, complemented
2#14521704C211yesmaximal, characteristic, complemented
3#14521400C211yesmaximal, characteristic, complemented
4#14521391C212yesmaximal, complemented
5#14521340C211yesmaximal, characteristic, complemented
6#14521278C212yesmaximal, complemented
7#14521255C211yesmaximal, characteristic, complemented
8#14521196C212yesmaximal, complemented
9C4×D248C211yesmaximal, characteristic, complemented
10#14521046C212yesmaximal, complemented
11C8×D4×C31C211yesmaximal, characteristic, complemented
12C2×#92495C2212yes—
13C22×C8×C31C2212yes—
14C4×D4×C31C2211yescharacteristic
15#11758515—21no—
16#11758515—21no—
17#11758507C2212yes—
18#11758503C2212yes—
19#11758501C2212yes—
20C2×D248C2211yescharacteristic
21C2×#92470C2212yes—
22#11758469—21no—
23C4⋊C4×D31—21no—
24#11758462—22no—
25#11758460—22no—
26C4×D124C2211yescharacteristic
27C4×D124C2211yescharacteristic
28#11758446C2211yescharacteristic
29#11758446C2211yescharacteristic
30C31×C4⋊C8C2211yescharacteristic
31C31×C22⋊C8C2212yes—
32C4×C8×C31C2211yescharacteristic
33#11758409C2211yescharacteristic
34#11758409C2211yescharacteristic
35#11758409C2212yes—
36#11758406C2211yescharacteristic
37#11758405C2212yes—
38#11758404C2212yes—
39#11758404C2212yes—
40#11758404C2212yes—
41C2×D4×C31D411yescharacteristic
42C22×C4×C31C2312yes—
43C2×C22⋊D31—22no—
44C2×C22⋊D31—22no—
45#92495—24no—
46#92495—42no—
47C2×D124C2311yescharacteristic
48C2×D124C2311yescharacteristic
49C2×C4×D31—21no—
50C2×C4×D31—21no—
51C2×C4×D31—22no—
52C2×Dic62—21no—
53C2×Dic62C2312yes—
54C2×C8×C31—42no—
55C2×C8×C31C2311yescharacteristic
56C2×C8×C31C2311yescharacteristic
57C2×C8×C31C2312yes—
58C4⋊C4×C31D411yescharacteristic
59C22⋊C4×C31D412yes—
60C42×C31C2311yescharacteristic
61C22⋊Dic31—22no—
62C22⋊Dic31—22no—
63(C2×D31)⋊C4—21no—
64(C2×D31)⋊C4—21no—
65(C2×D31)⋊C4—22no—
66C4⋊Dic31—21no—
67C4⋊Dic31—21no—
68C4⋊Dic31—21no—
69C4⋊Dic31C2312yes—
70C4⋊Dic31C2311yescharacteristic
71C4⋊Dic31C2311yescharacteristic
72Dic31⋊C4—22no—
73C4×Dic31—22no—
74D248—22no—
75#92470—42no—
76C23×C31C2×D412yes—
77C22×D31—21no—
78C22×D31—21no—
79D4×C31—22no—
80C2×C4×C31—22no—
81C2×C4×C31—22no—
82C2×C4×C31C2×D412yes—
83C2×C4×C31C2×D411yescharacteristic
84C2×C4×C31C2×D411yescharacteristic
85C2×C4×C31C2411yescharacteristic
86C22⋊D31—44no—
87C22⋊D31—82no—
88C2×Dic31—21no—
89C2×Dic31—22no—
90C2×Dic31—21no—
91C2×Dic31—22no—
92D124—21no—
93D124—21no—
94D124—21no—
95D124#4112yes—
96C4×D31—44no—
97C4×D31—42no—
98C4×D31—42no—
99Dic62—22no—
100Dic62—22no—
101Dic62—81no—
102C8×C31—21no—
103C8×C31—22no—
104C8×C31—21no—
105#3303—311nomaximal, Sylow
106C22×C31—22no—
107C22×C31—22no—
108C22×C31—22no—
109C22×C31C22×D411yescharacteristic
110C2×D31—22no—
111C2×D31—41no—
112C2×D31—41no—
113C2×D31—41no—
114C2×D31—41no—
115C2×D31—41no—
116C4×C31—22no—
117C4×C31—21no—
118C4×C31#13512yes—
119C4×C31C2×#4111yescharacteristic
120C4×C31H(5,𝔽2)11yescharacteristic
121Dic31—42no—
122Dic31—41no—
123Dic31—41no—
124Dic31—42no—
125#555—311no—
126#547—311no—
127#487—311no—
128#485—312no—
129#476—311no—
130#465—312no—
131#461—311no—
132#449—312no—
133C4×D8—311no—
134C8×D4D3111yescharacteristic, complemented
135#416—312no—
136C2×C31—42no—
137C2×C31#54611yescharacteristic
138C2×C31C2×#13511yescharacteristic
139C2×C31#57711yescharacteristic
140D31—42no—
141D31—81no—
142C2×#41—312no—
143C2×SD16—312no—
144C2×D8—311no—
145C22×C8C2×D3112yes—
146C4⋊Q8—311no—
147C4⋊Q8—311no—
148#124—622no—
149#123—622no—
150C22⋊Q8—312no—
151#121—312no—
152#121—621no—
153#121—621no—
154C4×D4—311no—
155C4×D4—311no—
156C4×D4C2×D3111yescharacteristic
157#117—312no—
158#117—621no—
159C2×C4⋊C4—621no—
160#107—311no—
161#106—312no—
162C4⋊C8C2×D3111yescharacteristic
163#103—312no—
164#103—312no—
165#103—312no—
166#102—311no—
167#102—312no—
168#102—311no—
169C22⋊C8C2×D3112yes—
170C4×C8C2×D3111yescharacteristic
171C31#330311yescharacteristic, complemented, Sylow
172#41—624no—
173#41—1242no—
174C2×Q8—312no—
175C2×Q8—621no—
176C2×D4—311no—
177C2×D4—311no—
178C2×D4—622no—
179C2×D4—622no—
180C2×D4D12411yescharacteristic
181C22×C4—621no—
182C22×C4—622no—
183C22×C4—621no—
184C22×C4C22×D3112yes—
185SD16—1242no—
186D8—622no—
187C2×C8—42no—
188C2×C8C22×D3111yescharacteristic
189C2×C8C22×D3111yescharacteristic
190C2×C8C22×D3112yes—
191C4⋊C4—312no—
192C4⋊C4—311no—
193C4⋊C4—311no—
194C4⋊C4—622no—
195C4⋊C4—621no—
196C4⋊C4—621no—
197C4⋊C4—621no—
198C4⋊C4D12411yescharacteristic
199C22⋊C4—622no—
200C22⋊C4—622no—
201C22⋊C4—622no—
202C22⋊C4—621no—
203C22⋊C4—621no—
204C22⋊C4D12412yes—
205C42—622no—
206C42C22×D3111yescharacteristic
207C23—621no—
208C23—621no—
209C23C2×D12412yes—
210Q8—622no—
211Q8—622no—
212Q8—2481no—
213D4—22no—
214D4—312no—
215D4—621no—
216D4—621no—
217D4—621no—
218D4—1244no—
219D4—2482no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—621no—
223C2×C4—622no—
224C2×C4—621no—
225C2×C4—622no—
226C2×C4—1244no—
227C2×C4—1242no—
228C2×C4—1242no—
229C2×C4C2×D12412yes—
230C2×C4C2×D12411yescharacteristic
231C2×C4C2×D12411yescharacteristic
232C2×C4C23×D3111yescharacteristic
233C8—21no—
234C8—22no—
235C8—21no—
236C22—22no—
237C22—22no—
238C22—22no—
239C22—622no—
240C22—1241no—
241C22—1241no—
242C22—1241no—
243C22—1241no—
244C22—1241no—
245C22C22×D12411yescharacteristic
246C4—21no—
247C4—22no—
248C4—1242no—
249C4—1242no—
250C4—1241no—
251C4—1241no—
252C4#1175848112yes—
253C4#1175856111yescharacteristic
254C4#41×D3111yescharacteristic
255C2—42no—
256C2—1242no—
257C2—2481no—
258C2#1452169411yescharacteristic
259C2C2×#1175848111yescharacteristic
260C2#1452186111yescharacteristic
subgroupG/H
centerC22C22×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC8×D4×C31C2
socleC22×C31C22×D4
hypercenterC8×D4D31
hypocenterC31#3303
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #14521764C2semidirect
2. #14521704C2semidirect
3. #14521400C2semidirect
4. #14521391C2semidirect
5. #14521340C2semidirect
6. #14521278C2semidirect
7. #14521255C2semidirect
8. #14521196C2semidirect
9. C4×D248C2semidirect
10. #14521046C2semidirect
11. C8×D4×C31C2semidirect
134. C8×D4D31semidirect
171. C31#3303semidirect

Maximal subgroups

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

subgroupindex
#145217642
#145217042
#145214002
#145213912
#145213402
#145212782
#145212552
#145211962
C4×D2482
#145210462
C8×D4×C312
#330331