Finite Group Databasebeta

← #16846009

Subgroups — #16846009

Order 3968.

408 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C22×#92474C211yesmaximal, characteristic, complemented
2C2×C22⋊(C31⋊C8)C211yesmaximal, characteristic, complemented
3C2×C22⋊(C31⋊C8)C212yesmaximal, complemented
4#14520804C212yesmaximal, complemented
5C2×C4×D4×C31C211yesmaximal, characteristic
6C22×D4×C31C411yescharacteristic
7C23×C4×C31C2211yescharacteristic
8C23×C4×C31C2211yescharacteristic
9C4×D4×C31—22no—
10C4×D4×C31—22no—
11C2×C4⋊C4×C31C411yescharacteristic
12C2×C22⋊C4×C31C411yescharacteristic
13C2×C22⋊C4×C31C411yescharacteristic
14C2×C42×C31C2211yescharacteristic
15C2×#92474—22no—
16C2×#92474—24no—
17C22×C31⋊C8C2212yes—
18C22×C31⋊C8C2212yes—
19C22⋊(C31⋊C8)—22no—
20C22⋊(C31⋊C8)—24no—
21C24×C31C2×C411yescharacteristic
22C24×C31C2×C411yescharacteristic
23C2×D4×C31—22no—
24C2×D4×C31—22no—
25C2×D4×C31—22no—
26C22×C4×C31—22no—
27C22×C4×C31—22no—
28C22×C4×C31—22no—
29C22×C4×C31—21no—
30C22×C4×C31—21no—
31C22×C4×C31—22no—
32C22×C4×C31C2×C411yescharacteristic
33C22×C4×C31C2×C411yescharacteristic
34C22×C4×C31C2×C411yescharacteristic
35C22×C4×C31C2×C411yescharacteristic
36C22×C4×C31D412yes—
37C22×C4×C31D412yes—
38C22×C4×C31C2311yescharacteristic
39C4⋊C4×C31—22no—
40C22⋊C4×C31—22no—
41C22⋊C4×C31—22no—
42C42×C31—21no—
43C42×C31—21no—
44#92474—28no—
45C2×C31⋊C8—22no—
46C2×C31⋊C8—22no—
47C2×C31⋊C8—22no—
48C2×C31⋊C8—22no—
49C2×C31⋊C8—22no—
50C2×C31⋊C8D414yescomplemented
51C23×C31—21no—
52C23×C31—22no—
53C23×C31—22no—
54C23×C31—21no—
55C23×C31—21no—
56C23×C31—21no—
57C23×C31—22no—
58C23×C31—22no—
59C23×C31C22⋊C412yes—
60C23×C31M1612yescomplemented
61C23×C31C22×C411yescharacteristic
62D4×C31—28no—
63C2×C4×C31—21no—
64C2×C4×C31—22no—
65C2×C4×C31—21no—
66C2×C4×C31—22no—
67C2×C4×C31—24no—
68C2×C4×C31—22no—
69C2×C4×C31—21no—
70C2×C4×C31—21no—
71C2×C4×C31—22no—
72C2×C4×C31—22no—
73C2×C4×C31—41no—
74C2×C4×C31—42no—
75C2×C4×C31—41no—
76C2×C4×C31C22⋊C412yes—
77C2×C4×C31M1612yescomplemented
78C2×C4×C31C2×D412yes—
79C2×C4×C31C2×D411yescharacteristic
80C2×C4×C31C2×D411yescharacteristic
81C2×C4×C31#4112yes—
82C31⋊C8—24no—
83C31⋊C8—42no—
84#1870—311nomaximal, Sylow
85C22×C31—22no—
86C22×C31—22no—
87C22×C31—22no—
88C22×C31—22no—
89C22×C31—24no—
90C22×C31—41no—
91C22×C31—41no—
92C22×C31—42no—
93C22×C31—42no—
94C22×C31—42no—
95C22×C31C2×C22⋊C411yescharacteristic
96C22×C31C4×D412yes—
97C22×C31C2×M1611yescharacteristic
98C22×C31C2×M1611yescharacteristic
99C22×C31#13112yes—
100C4×C31—22no—
101C4×C31—22no—
102C4×C31—41no—
103C4×C31—41no—
104C4×C31#12011yescharacteristic
105C4×C31#12112yes—
106C4×C31#12311yescharacteristic
107C22×M16—311no—
108C2×C4×D4D3111yescharacteristic
109C2×C22⋊C8—312no—
110C2×C22⋊C8—311no—
111#336—312no—
112C2×C31—42no—
113C2×C31—42no—
114C2×C31#38611yescharacteristic
115C2×C31#40711yescharacteristic
116C2×C31#40811yescharacteristic
117C2×C31#43512yes—
118C2×C31#43612yes—
119C22×D4Dic3111yescharacteristic
120C23×C4C2×D3111yescharacteristic
121C23×C4C2×D3111yescharacteristic
122C2×M16—624no—
123C2×M16—622no—
124C22×C8—312no—
125C22×C8—312no—
126C4×D4—22no—
127C4×D4—22no—
128C2×C4⋊C4Dic3111yescharacteristic
129C2×C22⋊C4Dic3111yescharacteristic
130C2×C22⋊C4Dic3111yescharacteristic
131C2×C42C2×D3111yescharacteristic
132C22⋊C8—624no—
133C22⋊C8—622no—
134C31#187011yescharacteristic, complemented, Sylow
135C24C2×Dic3111yescharacteristic
136C24C2×Dic3111yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4C2×Dic3111yescharacteristic
147C22×C4C2×Dic3111yescharacteristic
148C22×C4C2×Dic3111yescharacteristic
149C22×C4C2×Dic3111yescharacteristic
150C22×C4C22⋊D3112yes—
151C22×C4C22⋊D3112yes—
152C22×C4C22×D3111yescharacteristic
153M16—628no—
154C2×C8—314no—
155C2×C8—622no—
156C2×C8—622no—
157C2×C8—622no—
158C2×C8—622no—
159C2×C8—622no—
160C4⋊C4—22no—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C42—21no—
164C42—21no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—22no—
169C23—22no—
170C23—22no—
171C23—22no—
172C23—21no—
173C23#9247412yescomplemented
174C23C22⋊Dic3112yes—
175C23C22×Dic3111yescharacteristic
176D4—28no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4#9247412yescomplemented
191C2×C4C22⋊Dic3112yes—
192C2×C4D4×D3112yes—
193C2×C4#9249712yes—
194C2×C4C2×C22⋊D3111yescharacteristic
195C2×C4C2×C22⋊D3111yescharacteristic
196C8—624no—
197C8—1242no—
198C22—24no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22—41no—
207C22—41no—
208C22C2×#9247411yescharacteristic
209C22C2×#9247411yescharacteristic
210C22D4×Dic3112yes—
211C22#1175852512yes—
212C22C2×C22⋊Dic3111yescharacteristic
213C4—22no—
214C4—22no—
215C4—41no—
216C4—41no—
217C4#1175851211yescharacteristic
218C4#1175851411yescharacteristic
219C4#1175851612yes—
220C2—42no—
221C2—42no—
222C2#1452092911yescharacteristic
223C2#1452101311yescharacteristic
224C2#1452101611yescharacteristic
225C2#1452112112yes—
226C2#1452112712yes—
subgroupG/H
center, FrattiniC22×C4C22×D31
commutator, socleC23×C31C22×C4
FittingC2×C4×D4×C31C2
prime radicalC22×D4×C31C4
cosocleC22×C4×C31C23
hypercenterC2×C4×D4D31
hypocenterC31#1870
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C22×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×#92474C2semidirect
2. C2×C22⋊(C31⋊C8)C2semidirect
3. C2×C22⋊(C31⋊C8)C2semidirect
4. #14520804C2semidirect
50. C2×C31⋊C8D4semidirect
60. C23×C31M16semidirect
77. C2×C4×C31M16semidirect
134. C31#1870semidirect
173. C23#92474semidirect
190. C2×C4#92474semidirect

Maximal subgroups

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

subgroupindex
C22×#924742
C2×C22⋊(C31⋊C8)2
#145208042
C2×C4×D4×C312
#187031