Finite Group Databasebeta

← #11771919

Subgroups — #11771919

Order 1152.

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

normal subgroups only

This group has 486 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
1C32×#393C211yesmaximal, characteristic
2#10589392C211yesmaximal, characteristic
3#10589391C211yesmaximal, characteristic
4#67646—34nomaximal
5C22×C32×Q8C411yescharacteristic, complemented
6C2×C32×C22⋊C4—21no—
7C2×C32×C22⋊C4C2211yescharacteristic
8C2×#3757—21no—
9C32×#95—21no—
10C32×#95C411yescharacteristic, complemented
11#63657—21no—
12C3×#393D314yes—
13#4526—34no—
14#4525—34no—
15C24×C32D411yescharacteristic
16C2×C32×Q8—42no—
17C2×C32×Q8—41no—
18C22×C32×C4—21no—
19C22×C32×C4—21no—
20C22×C32×C4C2×C411yescharacteristic
21C22×C32×C4D411yescharacteristic
22C22×C3⋊Dic3—41no—
23C32×C22⋊C4—41no—
24C32×C22⋊C4—42no—
25#3757—22no—
26#3757—41no—
27C2×C3⋊(C3⋊C8)—41no—
28#1351—91noSylow
29C22×C3×Q8Dic314yescomplemented
30C2×C3×C22⋊C4—24no—
31C2×C3×C22⋊C4C2×D314yes—
32C2×#217—64no—
33C3×#95—24no—
34C3×#95Dic314yescomplemented
35#830—64no—
36C23×C32—21no—
37C23×C32—21no—
38C23×C32—41no—
39C23×C32—41no—
40C23×C32C22⋊C411yescharacteristic
41C23×C32SD1611yescharacteristic
42C23×C32Q1611yescharacteristic
43C32×Q8—42no—
44C32×Q8—42no—
45C2×C32×C4—22no—
46C2×C32×C4—21no—
47C2×C32×C4—21no—
48C2×C32×C4—41no—
49C2×C32×C4—41no—
50C2×C32×C4—41no—
51C2×C32×C4—41no—
52C2×C32×C4—41no—
53C2×C32×C4—41no—
54C2×C32×C4—41no—
55C2×C3⋊Dic3—42no—
56C2×C3⋊Dic3—81no—
57C2×C3⋊Dic3—81no—
58C3⋊(C3⋊C8)—81no—
59#393C3⋊D311yescharacteristic
60#342—91no—
61#323—91no—
62C24×C3#6714yes—
63C2×C3×Q8—48no—
64C2×C3×Q8—44no—
65C22×C3×C4—24no—
66C22×C3×C4—24no—
67C22×C3×C4C2×Dic314yes—
68C22×C3×C4#6714yes—
69C22×Dic3—124no—
70C3×C22⋊C4—44no—
71C3×C22⋊C4—48no—
72#217—68no—
73#217—124no—
74C2×C3⋊C8—124no—
75C22×C32—21no—
76C22×C32—21no—
77C22×C32—21no—
78C22×C32—21no—
79C22×C32—41no—
80C22×C32—41no—
81C22×C32—41no—
82C22×C32—41no—
83C22×C32—81no—
84C22×C32#9911yescharacteristic
85C22×C32#10311yescharacteristic
86C22×C32#10411yescharacteristic
87C32×C4—42no—
88C32×C4—41no—
89C32×C4—41no—
90C32×C4—81no—
91C3⋊Dic3—82no—
92C22×Q8C3⋊Dic311yescharacteristic, complemented
93C2×C22⋊C4—21no—
94C2×C22⋊C4—181no—
95C2×C22⋊C4C2×C3⋊D311yescharacteristic
96C22⋊C8—181no—
97#95—21no—
98#95C3⋊Dic311yescharacteristic, complemented
99C23×C3—24no—
100C23×C3—24no—
101C23×C3—44no—
102C23×C3—44no—
103C23×C3#21514yes—
104C23×C3#21614yes—
105C23×C3#21714yes—
106C3×Q8—48no—
107C3×Q8—48no—
108C2×C3×C4—28no—
109C2×C3×C4—24no—
110C2×C3×C4—24no—
111C2×C3×C4—44no—
112C2×C3×C4—44no—
113C2×C3×C4—44no—
114C2×C3×C4—44no—
115C2×C3×C4—44no—
116C2×C3×C4—44no—
117C2×C3×C4—44no—
118C2×Dic3—128no—
119C2×Dic3—244no—
120C2×Dic3—244no—
121C3⋊C8—244no—
122C2×C32—21no—
123C2×C32—21no—
124C2×C32—41no—
125C2×C32—41no—
126C2×C32#32811yescharacteristic
127C2×C32AΣL(1,𝔽16)11yescharacteristic
128C2×C32#35411yescharacteristic
129C24#63811yescharacteristic
130C2×Q8—42no—
131C2×Q8—41no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—361no—
135C22×C4C2×C3⋊Dic311yescharacteristic
136C22×C4#63811yescharacteristic
137C2×C8—361no—
138C22⋊C4—41no—
139C22⋊C4—42no—
140C22⋊C4—182no—
141C22⋊C4—361no—
142C22×C3—24no—
143C22×C3—24no—
144C22×C3—24no—
145C22×C3—24no—
146C22×C3—44no—
147C22×C3—44no—
148C22×C3—44no—
149C22×C3—44no—
150C22×C3—84no—
151C22×C3#83414yes—
152C22×C3#83514yes—
153C22×C3#83714yes—
154C3×C4—48no—
155C3×C4—44no—
156C3×C4—44no—
157C3×C4—84no—
158Dic3—248no—
159C32#135111yescharacteristic, complemented, Sylow
160C23—21no—
161C23—21no—
162C23—41no—
163C23—41no—
164C23#375511yescharacteristic
165C23#375611yescharacteristic
166C23#375711yescharacteristic
167Q8—42no—
168Q8—42no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—362no—
180C2×C4—721no—
181C2×C4—721no—
182C8—721no—
183C2×C3—24no—
184C2×C3—24no—
185C2×C3—44no—
186C2×C3—44no—
187C2×C3#453614yes—
188C2×C3#453814yes—
189C2×C3#454114yes—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—41no—
195C22—41no—
196C22—41no—
197C22—41no—
198C22—81no—
199C22#6366111yescharacteristic
200C22#6366211yescharacteristic
201C22#6366411yescharacteristic
202C4—42no—
203C4—41no—
204C4—41no—
205C4—81no—
206C4—722no—
207C3#6764614yescomplemented
208C2—21no—
209C2—21no—
210C2—41no—
211C2—41no—
212C2#1058940211yescharacteristic
213C2#1058940411yescharacteristic
214C2#1058940711yescharacteristic
subgroupG/H
centerC22#63661
commutatorC22×C32×C4C2×C4
FrattiniC2×C22⋊C4C2×C3⋊D3
FittingC32×#393C2
socleC22×C32#99
prime radicalC24×C32D4
cosocleC2×C32×C22⋊C4C22
hypercenter#393C3⋊D3
hypocenterC32#1351
LCSG ▷ C22×C32×C4 ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #393

Complements

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

NH ≅ G/Ntype
5. C22×C32×Q8C4semidirect
10. C32×#95C4semidirect
29. C22×C3×Q8Dic3semidirect
34. C3×#95Dic3semidirect
92. C22×Q8C3⋊Dic3semidirect
98. #95C3⋊Dic3semidirect
159. C32#1351semidirect
207. C3#67646semidirect

Maximal subgroups

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

subgroupindex
C32×#3932
#105893922
#105893912
#676463