Finite Group Databasebeta

← #10594531

Subgroups — #10594531

Order 576.

491 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 491 conjugacy classes of subgroups — too many to draw. Only its 142 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
1#64096C211yesmaximal, characteristic, complemented
2C3×D3×C4⋊C4C212yesmaximal, complemented
3#63919C211yesmaximal, characteristic, complemented
4C4×#625C211yesmaximal, characteristic
5C4×#625C212yesmaximal
6#63870C212yesmaximal, complemented
7#63857C212yesmaximal, complemented
8#63851C214yesmaximal, complemented
9#5511—32nomaximal
10C2×#634C2211yescharacteristic
11C2×C3×C4×D3C2212yes—
12C2×C3×C4×D3C2214yes—
13C2×#627C2212yes—
14C2×#625C2211yescharacteristic
15C32×C4⋊C4C2211yescharacteristic
16#3750C2212yes—
17C4×C3⋊Dic3C2212yes—
18C4×C3⋊Dic3C2211yescharacteristic
19C3×C4⋊Dic3C2212yes—
20C3×Dic3⋊C4C2214yes—
21#3724C2211yescharacteristic
22#3724C2212yes—
23Dic3⋊Dic3C2214yes—
24(C2×D3)⋊Dic3C2212yes—
25(C2×D3)⋊Dic3C2214yes—
26C2×C3×C4⋊C4—32no—
27C4×#67—32no—
28C4×#67—34no—
29#924—32no—
30#924—34no—
31#897—32no—
32#896—34no—
33D3×C4⋊C4D312yescomplemented
34#889—34no—
35#888—32no—
36#887—32no—
37#887—61no—
38C22×C3×D3C2312yes—
39C2×C32×C4C2312yes—
40C2×C32×C4C2311yescharacteristic
41C2×C3⋊Dic3C2312yes—
42C2×C3⋊Dic3C2311yescharacteristic
43C2×C3⋊Dic3C2311yescharacteristic
44#634—22no—
45C2×C3×Dic3C2312yes—
46C2×C3×Dic3C2314yes—
47C3×C4×D3—24no—
48C3×C4×D3—44no—
49#627—42no—
50#625Q814yescomplemented
51#554—91noSylow
52C22×C3×C4—32no—
53C22×C3×C4—34no—
54C2×#67—32no—
55C2×C4×D3C2×D312yes—
56C2×C4×D3C2×D314yes—
57C2×Dic6—32no—
58C2×Dic6—34no—
59C2×Dic6—61no—
60C3×C4⋊C4—21no—
61C3×C4⋊C4—32no—
62C3×C4⋊C4—34no—
63C3×C4⋊C4C2×D312yes—
64#217—34no—
65#217—32no—
66(C2×D3)⋊C4—32no—
67(C2×D3)⋊C4—34no—
68C4⋊Dic3—34no—
69C4⋊Dic3C2×D312yes—
70Dic3⋊C4—34no—
71Dic3⋊C4—34no—
72Dic3⋊C4—32no—
73Dic3⋊C4—34no—
74Dic3⋊C4—62no—
75Dic3⋊C4C2×D314yes—
76C4×Dic3—32no—
77C4×Dic3—34no—
78C4×Dic3—62no—
79C4×Dic3—61no—
80C22×C32C2411yescharacteristic
81C2×C3×D3—22no—
82C2×C3×D3—22no—
83C2×C3×D3C2×Q814yes—
84C32×C4—22no—
85C32×C4#4112yes—
86C3⋊Dic3—22no—
87C3⋊Dic3—21no—
88C3⋊Dic3C2×Q812yes—
89C3×Dic3—22no—
90C3×Dic3—24no—
91C4⋊Q8—91no—
92#125—92no—
93C22⋊Q8—94no—
94C22⋊Q8—92no—
95C4×Q8—91no—
96C4×D4—92no—
97C4×D4—91no—
98C2×C4⋊C4—32no—
99C23×C3—32no—
100C22×D3C22×D312yes—
101C2×C3×C4—21no—
102C2×C3×C4—22no—
103C2×C3×C4—32no—
104C2×C3×C4—34no—
105C2×C3×C4—64no—
106C2×C3×C4—124no—
107C2×C3×C4C22×D314yes—
108C2×C3×C4C22×D312yes—
109#67—38no—
110C2×Dic3—32no—
111C2×Dic3—34no—
112C2×Dic3—32no—
113C2×Dic3—61no—
114C2×Dic3—62no—
115C2×Dic3—61no—
116C2×Dic3C22×D314yes—
117C2×Dic3C22×D312yes—
118C4×D3—24no—
119C4×D3—44no—
120Dic6—64no—
121Dic6—64no—
122Dic6—124no—
123C2×C32C22×Q811yescharacteristic
124C2×C32C2×#4111yescharacteristic
125C2×C32H(5,𝔽2)11yescharacteristic
126C3×D3—24no—
127C2×Q8—92no—
128C2×Q8—91no—
129C2×D4—91no—
130C22×C4—32no—
131C22×C4—34no—
132C4⋊C4—34no—
133C4⋊C4—32no—
134C4⋊C4—91no—
135C4⋊C4—92no—
136C4⋊C4—94no—
137C4⋊C4—92no—
138C4⋊C4D3211yescharacteristic
139C22⋊C4—92no—
140C22⋊C4—94no—
141C42—92no—
142C42—91no—
143C22×C3—21no—
144C22×C3—34no—
145C22×C3—62no—
146C22×C3—62no—
147C22×C3C23×D312yes—
148C2×D3—22no—
149C2×D3—22no—
150C2×D3D3×Q814yes—
151C3×C4—24no—
152C3×C4—22no—
153C3×C4—42no—
154C3×C4—62no—
155C3×C4—64no—
156C3×C4#23514yes—
157Dic3—22no—
158Dic3—24no—
159Dic3—34no—
160Dic3—62no—
161Dic3—64no—
162Dic3—64no—
163Dic3—62no—
164Dic3—121no—
165C32#55411yescharacteristic, complemented, Sylow
166C23—32no—
167Q8—182no—
168Q8—362no—
169D4—94no—
170C2×C4—32no—
171C2×C4—34no—
172C2×C4—64no—
173C2×C4—91no—
174C2×C4—92no—
175C2×C4—91no—
176C2×C4—124no—
177C2×C4C2×D3212yes—
178C2×C4C2×D3211yescharacteristic
179C2×C3—21no—
180C2×C3—21no—
181C2×C3—21no—
182C2×C3—64no—
183C2×C3C2×#23512yes—
184C2×C3#100412yes—
185C2×C3C2×D3×Q812yes—
186D3—24no—
187C22—34no—
188C22—62no—
189C22—62no—
190C22C22×D3211yescharacteristic
191C4—22no—
192C4—62no—
193C4—64no—
194C4—92no—
195C4—182no—
196C4—181no—
197C4#379812yes—
198C3—21no—
199C3#551112yescomplemented
200C2—64no—
201C2C2×#379811yescharacteristic
202C2#6431311yescharacteristic
203C2D32×Q811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC2×#625C22
hypercenterC4⋊C4D32
hypocenterC32#554
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64096C2semidirect
2. C3×D3×C4⋊C4C2semidirect
3. #63919C2semidirect
6. #63870C2semidirect
7. #63857C2semidirect
8. #63851C2semidirect
33. D3×C4⋊C4D3semidirect
50. #625Q8semidirect
165. C32#554semidirect
199. C3#5511semidirect

Maximal subgroups

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

subgroupindex
#640962
C3×D3×C4⋊C42
#639192
C4×#6252
#638702
#638572
#638512
#55113