Finite Group Databasebeta

← #16846155

Subgroups — #16846155

Order 3968.

442 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 105 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×C4×Dic31C211yesmaximal, characteristic, complemented
2C2×#11758425C212yesmaximal, complemented
3#14521013C211yesmaximal, characteristic, complemented
4#14520808C212yesmaximal, complemented
5C31×#525C211yesmaximal, characteristic
6C2×#41×C31C2211yescharacteristic
7C23×C4×C31C2211yescharacteristic
8C23×Dic31—21no—
9C31×#123—21no—
10C31×C22⋊Q8C412yescomplemented
11C31×#121C412yescomplemented
12C31×#120—21no—
13C4×D4×C31—22no—
14C31×#117C2211yescharacteristic
15C2×C4×Dic31—21no—
16C2×C4×Dic31—24no—
17C2×C4×Dic31C2212yes—
18C2×#92474C2212yes—
19#11758425—24no—
20C22⋊(C31⋊C8)—22no—
21C24×C31D411yescharacteristic
22#41×C31—22no—
23#41×C31—22no—
24C2×Q8×C31C2×C411yescharacteristic, complemented
25C2×D4×C31—22no—
26C2×D4×C31—21no—
27C2×D4×C31C2×C411yescharacteristic, complemented
28C22×C4×C31—22no—
29C22×C4×C31—22no—
30C22×C4×C31—21no—
31C22×C4×C31C2×C412yes—
32C22×C4×C31D411yescharacteristic
33C22×C4×C31D412yes—
34C22×C4×C31C2311yescharacteristic
35C22×Dic31—24no—
36C22×Dic31—28no—
37C22×Dic31—22no—
38C4⋊C4×C31—22no—
39C4⋊C4×C31C2×C412yescomplemented
40C22⋊C4×C31—22no—
41C22⋊C4×C31—21no—
42C22⋊C4×C31—22no—
43C42×C31—21no—
44C4×Dic31—24no—
45C4×Dic31—22no—
46C4×Dic31D414yescomplemented
47#92474—22no—
48#92474—22no—
49C2×C31⋊C8—22no—
50C23×C31—24no—
51C23×C31—21no—
52C23×C31—21no—
53C23×C31—21no—
54C23×C31C22⋊C412yes—
55C23×C31C2×D411yescharacteristic
56Q8×C31—22no—
57D4×C31—22no—
58D4×C31—44no—
59D4×C31—41no—
60D4×C31—41no—
61C2×C4×C31—21no—
62C2×C4×C31—21no—
63C2×C4×C31—24no—
64C2×C4×C31—24no—
65C2×C4×C31—22no—
66C2×C4×C31—21no—
67C2×C4×C31—42no—
68C2×C4×C31C22⋊C412yes—
69C2×C4×C31C22×C411yescharacteristic
70C2×C4×C31C2×D412yes—
71C2×C4×C31C2×D411yescharacteristic
72C2×C4×C31#4112yes—
73C2×Dic31—216no—
74C2×Dic31—28no—
75C2×Dic31—24no—
76C31⋊C8—42no—
77#1896—311nomaximal, Sylow
78C22×C31—24no—
79C22×C31—24no—
80C22×C31—22no—
81C22×C31—22no—
82C22×C31—22no—
83C22×C31—41no—
84C22×C31—42no—
85C22×C31#10414yescomplemented
86C22×C31C2×C22⋊C411yescharacteristic
87C22×C31#12011yescharacteristic
88C22×C31#12311yescharacteristic
89C4×C31—22no—
90C4×C31—41no—
91C4×C31—42no—
92C4×C31C4×D412yes—
93C4×C31#12112yes—
94Dic31—28no—
95#525D3111yescharacteristic
96C22×C42—311no—
97C2×#104—312no—
98#407—311no—
99#339—312no—
100C2×C31—21no—
101C2×C31—24no—
102C2×C31—21no—
103C2×C31—81no—
104C2×C31#38611yescharacteristic
105C2×C31C2×#10412yes—
106C2×#41C2×D3111yescharacteristic
107C23×C4—621no—
108C23×C4C2×D3111yescharacteristic
109C2×M16—312no—
110#123—21no—
111C22⋊Q8Dic3112yescomplemented
112#121Dic3112yescomplemented
113#120—21no—
114C4×D4—22no—
115#117C2×D3111yescharacteristic
116C2×C42—312no—
117C2×C42—624no—
118C2×C42—621no—
119#104—624no—
120C22⋊C8—622no—
121C31#189611yescharacteristic, complemented, Sylow
122C24C22⋊D3111yescharacteristic
123#41—22no—
124#41—22no—
125C2×Q8C2×Dic3111yescharacteristic, complemented
126C2×D4—22no—
127C2×D4—21no—
128C2×D4C2×Dic3111yescharacteristic, complemented
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—624no—
133C22×C4—628no—
134C22×C4—622no—
135C22×C4C2×Dic3112yes—
136C22×C4C22⋊D3111yescharacteristic
137C22×C4C22⋊D3112yes—
138C22×C4C22×D3111yescharacteristic
139M16—622no—
140M16—622no—
141C2×C8—622no—
142C4⋊C4—22no—
143C4⋊C4C2×Dic3112yescomplemented
144C22⋊C4—22no—
145C22⋊C4—21no—
146C22⋊C4—22no—
147C42—21no—
148C42—314no—
149C42—624no—
150C42—622no—
151C23—24no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23C22⋊Dic3112yes—
156C23C2×C22⋊D3111yescharacteristic
157Q8—22no—
158D4—22no—
159D4—44no—
160D4—41no—
161D4—41no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—42no—
169C2×C4—6216no—
170C2×C4—628no—
171C2×C4—624no—
172C2×C4C22⋊Dic3112yes—
173C2×C4D4×D3112yes—
174C2×C4#9249712yes—
175C2×C4C22×Dic3111yescharacteristic
176C2×C4C2×C22⋊D3111yescharacteristic
177C8—1242no—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—41no—
184C22—42no—
185C22#1175842514yescomplemented
186C22#1175851211yescharacteristic
187C22#1175851411yescharacteristic
188C22C2×C22⋊Dic3111yescharacteristic
189C4—22no—
190C4—41no—
191C4—42no—
192C4—628no—
193C4D4×Dic3112yes—
194C4#1175851612yes—
195C2—21no—
196C2—24no—
197C2—21no—
198C2—81no—
199C2#1452092911yescharacteristic
200C2C2×#1175842512yes—
subgroupG/H
centerC2×C4C2×C22⋊D31
commutatorC2×C4×C31C22×C4
FrattiniC22×C4C22×D31
Fitting, prime radicalC31×#525C2
socleC22×C31C2×C22⋊C4
cosocleC22×C4×C31C23
hypercenter#525D31
hypocenterC31#1896
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic31C2semidirect
2. C2×#11758425C2semidirect
3. #14521013C2semidirect
4. #14520808C2semidirect
10. C31×C22⋊Q8C4semidirect
11. C31×#121C4semidirect
24. C2×Q8×C31C2×C4semidirect
27. C2×D4×C31C2×C4semidirect
39. C4⋊C4×C31C2×C4semidirect
46. C4×Dic31D4semidirect
85. C22×C31#104semidirect
111. C22⋊Q8Dic31semidirect
112. #121Dic31semidirect
121. C31#1896semidirect
125. C2×Q8C2×Dic31semidirect
128. C2×D4C2×Dic31semidirect
143. C4⋊C4C2×Dic31semidirect
185. C22#11758425semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic312
C2×#117584252
#145210132
#145208082
C31×#5252
#189631