Finite Group Databasebeta

← #10593408

Subgroups — #10593408

Order 576.

298 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C2×#3773C212yesmaximal, complemented
2C2×D32⋊C4C212yesmaximal
3#63921C211yesmaximal, characteristic
4#63900C211yesmaximal, characteristic, complemented
5#63781C211yesmaximal, characteristic, complemented
6C22×D32C2211yescharacteristic
7C22×#157C2212yes—
8C2×C4×C3⋊D3C2211yescharacteristic
9C2×#624C2212yes—
10C2×#624C2211yescharacteristic
11#3773—24no—
12D32⋊C4—24no—
13C3×(C2×D3)⋊C4—21no—
14C3×C4⋊Dic3—21no—
15#3724—21no—
16(C2×D3)⋊Dic3—21no—
17D3×C4⋊C4—61no—
18D3×C22⋊C4—61no—
19C22×C3⋊D3C2311yescharacteristic
20C22×C3×D3—21no—
21C2×D32—21no—
22C2×D32—22no—
23C2×D32—22no—
24C2×D32Q812yes—
25C2×#157—22no—
26C2×#157—22no—
27C2×#157D414yes—
28C2×C32×C4D411yescharacteristic
29C2×C3⋊Dic3D411yescharacteristic
30C4×C3⋊D3—22no—
31C2×C3×Dic3—21no—
32C2×C3×Dic3—22no—
33#624—22no—
34#624—22no—
35#624—22no—
36#399—91nomaximal, Sylow
37C23×D3—61no—
38C2×C4×D3—61no—
39C2×C4×D3—61no—
40C2×C4×D3—62no—
41C2×C4×D3—61no—
42C3×C4⋊C4—61no—
43C3×C22⋊C4—61no—
44#217—61no—
45(C2×D3)⋊C4—21no—
46(C2×D3)⋊C4—61no—
47C4⋊Dic3—21no—
48Dic3⋊C4—62no—
49C22×C32C2×D411yescharacteristic
50C2×C3⋊D3C2×D412yes—
51C2×C3⋊D3C2×Q811yescharacteristic
52C2×C3⋊D3#4112yes—
53C2×C3⋊D3#4111yescharacteristic
54C2×C3×D3—22no—
55C2×C3×D3—41no—
56C2×C3×D3—41no—
57D32—24no—
58D32—42no—
59#157—24no—
60C32×C4—21no—
61C3⋊Dic3—21no—
62C3×Dic3—42no—
63C3×Dic3—41no—
64C2×C4⋊C4—91no—
65C2×C4⋊C4—92no—
66C2×C22⋊C4—91no—
67C2×C22⋊C4—92no—
68#95—91no—
69C23×C3—61no—
70C22×D3—21no—
71C22×D3—62no—
72C22×D3—61no—
73C22×D3—61no—
74C22×D3—62no—
75C22×D3—122no—
76C22×D3—122no—
77C2×C3×C4—21no—
78C2×C3×C4—21no—
79C2×C3×C4—62no—
80C2×C3×C4—61no—
81C2×Dic3—22no—
82C2×Dic3—21no—
83C2×Dic3—61no—
84C2×Dic3—61no—
85C4×D3—64no—
86C4×D3—122no—
87C4×D3—122no—
88C4×D3—122no—
89C4×D3—122no—
90C2×C32C22⋊Q811yescharacteristic
91C2×C32#12311yescharacteristic
92C2×C32#12711yescharacteristic
93C3⋊D3C22⋊Q812yes—
94C3⋊D3#12312yes—
95C3×D3—42no—
96C24—91no—
97C22×C4—91no—
98C22×C4—92no—
99C22×C4—92no—
100C22×C4—91no—
101C4⋊C4—61no—
102C4⋊C4—181no—
103C4⋊C4—184no—
104C22⋊C4—61no—
105C22⋊C4—181no—
106C22⋊C4—184no—
107C22×C3—21no—
108C22×C3—21no—
109C22×C3—62no—
110C22×C3—121no—
111C22×C3—121no—
112C2×D3—22no—
113C2×D3—41no—
114C2×D3—41no—
115C2×D3—62no—
116C2×D3—62no—
117C2×D3—62no—
118C2×D3—62no—
119C2×D3—61no—
120C2×D3—61no—
121C2×D3—61no—
122C2×D3—61no—
123C2×D3—124no—
124C2×D3—124no—
125C3×C4—22no—
126C3×C4—41no—
127C3×C4—122no—
128C3×C4—121no—
129Dic3—42no—
130Dic3—41no—
131Dic3—62no—
132Dic3—121no—
133C32#39911yescharacteristic, complemented, Sylow
134C23—61no—
135C23—92no—
136C23—91no—
137C23—181no—
138C23—182no—
139C23—182no—
140C2×C4—61no—
141C2×C4—62no—
142C2×C4—94no—
143C2×C4—91no—
144C2×C4—182no—
145C2×C4—182no—
146C2×C4—182no—
147C2×C4—182no—
148C2×C4—182no—
149C2×C4—182no—
150C2×C4GO+(4,𝔽2)11yescharacteristic
151C2×C3—21no—
152C2×C3—21no—
153C2×C3—21no—
154C2×C3—21no—
155C2×C3—21no—
156C2×C3—21no—
157C2×C3—122no—
158D3—42no—
159D3—62no—
160D3—62no—
161D3—62no—
162D3—62no—
163C22—62no—
164C22—91no—
165C22—91no—
166C22—92no—
167C22—92no—
168C22—121no—
169C22—121no—
170C22—184no—
171C22—362no—
172C22C2×GO+(4,𝔽2)11yescharacteristic
173C4—21no—
174C4—121no—
175C4—122no—
176C4—184no—
177C4—181no—
178C3—21no—
179C3—21no—
180C2—92no—
181C2—92no—
182C2—122no—
183C2D32⋊Q811yescharacteristic
184C2#6423111yescharacteristic
185C2#6423311yescharacteristic
subgroupG/H
center, FrattiniC22C2×GO+(4,𝔽2)
commutator, cosocleC22×C3⋊D3C23
FittingC2×C32×C4D4
socleC22×C32C2×D4
prime radicalC22×D32C22
hypercenterC2×C4GO+(4,𝔽2)
hypocenterC32#399
LCSG ▷ C22×C3⋊D3 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#3773C2semidirect
4. #63900C2semidirect
5. #63781C2semidirect
133. C32#399semidirect

Maximal subgroups

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

subgroupindex
C2×#37732
C2×D32⋊C42
#639212
#639002
#637812
#3999