Finite Group Databasebeta

← #11772599

Subgroups — #11772599

Order 1152.

696 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 157 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×#390C211yesmaximal, characteristic
2C2×#63661C212yesmaximal
3#67641—34nomaximal
4C22×C32×D4C2211yescharacteristic
5C2×C32×C4⋊C4C411yescharacteristic, complemented
6C2×C32×C22⋊C4—22no—
7C2×C32×C42C411yescharacteristic, complemented
8C2×#3757—22no—
9#63661—22no—
10#63661C414yescomplemented
11C3×#390D314yes—
12C2×#834—38no—
13C24×C32D412yes—
14C2×C32×D4—41no—
15C2×C32×D4—41no—
16C2×C32×D4C2×C412yes—
17C2×C32×D4D411yescharacteristic
18C2×C32×D4Q811yescharacteristic
19C22×C32×C4—21no—
20C22×C32×C4—21no—
21C22×C32×C4C2×C411yescharacteristic
22C22×C3⋊Dic3—42no—
23C32×C4⋊C4—41no—
24C32×C22⋊C4—44no—
25C32×C42—41no—
26#3757—24no—
27#3757—42no—
28#1393—91noSylow
29C22×C3×D4C2×D314yes—
30C2×C3×C4⋊C4Dic314yescomplemented
31C2×C3×C22⋊C4—28no—
32C2×C3×C42Dic314yescomplemented
33C2×#217—68no—
34#834—316no—
35#834—68no—
36C23×C32—22no—
37C23×C32—22no—
38C23×C32—42no—
39C23×C32—42no—
40C23×C32C22⋊C412yes—
41C23×C32C22⋊C411yescharacteristic
42C23×C32C4⋊C412yes—
43C32×D4—42no—
44C32×D4—41no—
45C32×D4—41no—
46C2×C32×C4—21no—
47C2×C32×C4—21no—
48C2×C32×C4—22no—
49C2×C32×C4—41no—
50C2×C32×C4—41no—
51C2×C32×C4—42no—
52C2×C32×C4—41no—
53C2×C32×C4C4211yescharacteristic
54C2×C32×C4C4⋊C411yescharacteristic
55C2×C3⋊Dic3—44no—
56C2×C3⋊Dic3—82no—
57C2×C3⋊Dic3—82no—
58C2×#99—92no—
59#390C3⋊D311yescharacteristic
60C24×C3#6718yes—
61C2×C3×D4—44no—
62C2×C3×D4—44no—
63C2×C3×D4Dic614yes—
64C2×C3×D4C4×D318yes—
65C2×C3×D4D1214yes—
66C22×C3×C4—24no—
67C22×C3×C4—24no—
68C22×C3×C4C2×Dic314yes—
69C22×Dic3—128no—
70C3×C4⋊C4—44no—
71C3×C22⋊C4—416no—
72C3×C42—44no—
73#217—616no—
74#217—128no—
75C22×C32—21no—
76C22×C32—21no—
77C22×C32—22no—
78C22×C32—22no—
79C22×C32—42no—
80C22×C32—42no—
81C22×C32—42no—
82C22×C32—42no—
83C22×C32—82no—
84C22×C32#9511yescharacteristic
85C22×C32#9911yescharacteristic
86C22×C32#9911yescharacteristic
87C32×C4—21no—
88C32×C4—21no—
89C32×C4—42no—
90C32×C4—81no—
91C3⋊Dic3—84no—
92C22×D4C2×C3⋊D311yescharacteristic
93C2×C4⋊C4C3⋊Dic311yescharacteristic, complemented
94C2×C22⋊C4—22no—
95C2×C22⋊C4—182no—
96C2×C42C3⋊Dic311yescharacteristic, complemented
97#99—94no—
98#99—182no—
99C23×C3—28no—
100C23×C3—28no—
101C23×C3—48no—
102C23×C3—48no—
103C23×C3Dic3⋊C418yes—
104C23×C3(C2×D3)⋊C418yes—
105C23×C3#21714yes—
106C3×D4—44no—
107C3×D4—48no—
108C3×D4—44no—
109C2×C3×C4—24no—
110C2×C3×C4—24no—
111C2×C3×C4—28no—
112C2×C3×C4—44no—
113C2×C3×C4—44no—
114C2×C3×C4—48no—
115C2×C3×C4—44no—
116C2×C3×C4C4×Dic314yes—
117C2×C3×C4C4⋊Dic314yes—
118C2×Dic3—1216no—
119C2×Dic3—248no—
120C2×Dic3—248no—
121C2×C32—21no—
122C2×C32—21no—
123C2×C32—42no—
124C2×C32—42no—
125C2×C32#34211yescharacteristic
126C2×C32#35311yescharacteristic
127C2×C32#35411yescharacteristic
128C24#63812yes—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4#63411yescharacteristic
132C2×D4C4×C3⋊D312yes—
133C2×D4C4⋊(C3⋊D3)11yescharacteristic
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—362no—
137C22×C4C2×C3⋊Dic311yescharacteristic
138C4⋊C4—41no—
139C22⋊C4—44no—
140C22⋊C4—184no—
141C22⋊C4—362no—
142C42—41no—
143C22×C3—24no—
144C22×C3—24no—
145C22×C3—28no—
146C22×C3—28no—
147C22×C3—48no—
148C22×C3—48no—
149C22×C3—48no—
150C22×C3—48no—
151C22×C3—88no—
152C22×C3C3⋊#9514yes—
153C22×C3#83414yes—
154C22×C3#83414yes—
155C3×C4—24no—
156C3×C4—24no—
157C3×C4—48no—
158C3×C4—84no—
159Dic3—2416no—
160C32#139311yescharacteristic, complemented, Sylow
161C23—22no—
162C23—22no—
163C23—42no—
164C23—42no—
165C23#375012yes—
166C23#375212yes—
167C23#375711yescharacteristic
168D4—41no—
169D4—42no—
170D4—41no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—364no—
179C2×C4—722no—
180C2×C4—722no—
181C2×C4C4×C3⋊Dic311yescharacteristic
182C2×C4C4⋊(C3⋊Dic3)11yescharacteristic
183C2×C3—24no—
184C2×C3—24no—
185C2×C3—48no—
186C2×C3—48no—
187C2×C3#452614yes—
188C2×C3#454114yes—
189C2×C3#454514yes—
190C22—21no—
191C22—21no—
192C22—22no—
193C22—22no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—42no—
198C22—82no—
199C22#6365811yescharacteristic
200C22#6366111yescharacteristic
201C22#6366111yescharacteristic
202C4—21no—
203C4—21no—
204C4—42no—
205C4—81no—
206C4—724no—
207C3#6764114yescomplemented
208C2—21no—
209C2—21no—
210C2—42no—
211C2—42no—
212C2#1058939211yescharacteristic
213C2#1058940711yescharacteristic
214C2#1058941111yescharacteristic
subgroupG/H
centerC22#63661
commutatorC2×C32×C4C42
FrattiniC22×D4C2×C3⋊D3
FittingC32×#390C2
socleC22×C32#99
prime radical, cosocleC22×C32×D4C22
hypercenter#390C3⋊D3
hypocenterC32#1393
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #390

Complements

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

NH ≅ G/Ntype
5. C2×C32×C4⋊C4C4semidirect
7. C2×C32×C42C4semidirect
10. #63661C4semidirect
30. C2×C3×C4⋊C4Dic3semidirect
32. C2×C3×C42Dic3semidirect
93. C2×C4⋊C4C3⋊Dic3semidirect
96. C2×C42C3⋊Dic3semidirect
160. C32#1393semidirect
207. C3#67641semidirect

Maximal subgroups

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

subgroupindex
C32×#3902
C2×#636612
#676413