Finite Group Databasebeta

← #10596105

Subgroups — #10596105

Order 576.

726 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 191 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×C22⋊Q8C211yesmaximal, characteristic, complemented
2#64155C212yesmaximal, complemented
3C2×#3752C211yesmaximal, characteristic
4#64133C211yesmaximal, characteristic, complemented
5#64106C212yesmaximal, complemented
6#64105C212yesmaximal, complemented
7#64105C212yesmaximal, complemented
8C22⋊C4×C3⋊D3C212yesmaximal
9#64087C212yesmaximal, complemented
10#5634—34nomaximal
11C23×C3⋊D3C2211yescharacteristic
12C2×C32×Q8C2211yescharacteristic, complemented
13C22×C32×C4C2211yescharacteristic
14C22×C3⋊Dic3C2211yescharacteristic
15C2×C4×C3⋊D3C2212yes—
16C2×C4×C3⋊D3C2212yes—
17C2×#634C2212yescomplemented
18C2×#634C2211yescharacteristic, complemented
19C32×C4⋊C4C2211yescharacteristic, complemented
20C32×C4⋊C4C2212yescomplemented
21C32×C22⋊C4C2212yes—
22#3757C2212yes—
23#3752C2212yes—
24#3752C2212yes—
25#3752C2212yes—
26#3752C2212yes—
27C4⋊(C3⋊Dic3)C2212yescomplemented
28C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
29#3750C2212yescomplemented
30#3750C2212yescomplemented
31#3750C2212yescomplemented
32C3×C22⋊Q8D314yescomplemented
33#946—38no—
34C2×(C2×D3)⋊C4—34no—
35#924—34no—
36#897—38no—
37#896—38no—
38#896—38no—
39D3×C22⋊C4—38no—
40#878—38no—
41C23×C32C2311yescharacteristic
42C22×C3⋊D3—41no—
43C22×C3⋊D3—41no—
44C22×C3⋊D3Q814yescomplemented
45C22×C3⋊D3C2312yes—
46C32×Q8—41no—
47C2×C32×C4—41no—
48C2×C32×C4C2311yescharacteristic
49C2×C32×C4C2312yes—
50C2×C32×C4C2312yes—
51C2×C32×C4C2311yescharacteristic
52C2×C3⋊Dic3—41no—
53C2×C3⋊Dic3C2312yes—
54C2×C3⋊Dic3C2312yes—
55C2×C3⋊Dic3C2312yes—
56C4×C3⋊D3—42no—
57C4×C3⋊D3—42no—
58#634—41no—
59#634—42no—
60#543—91noSylow
61C23×D3—34no—
62C2×C3×Q8C2×D314yescomplemented
63C22×C3×C4C2×D314yes—
64C22×Dic3—34no—
65C2×C4×D3—38no—
66C2×C4×D3—38no—
67C2×Dic6—38no—
68C2×Dic6—34no—
69C3×C4⋊C4C2×D314yescomplemented
70C3×C4⋊C4C2×D318yescomplemented
71C3×C22⋊C4C2×D318yes—
72#217—38no—
73(C2×D3)⋊C4—38no—
74(C2×D3)⋊C4—38no—
75(C2×D3)⋊C4—38no—
76(C2×D3)⋊C4—38no—
77C4⋊Dic3—38no—
78C4⋊Dic3—34no—
79Dic3⋊C4—38no—
80Dic3⋊C4—38no—
81Dic3⋊C4—38no—
82C22×C32—21no—
83C22×C32—21no—
84C22×C32C2×Q812yes—
85C22×C32C2411yescharacteristic
86C2×C3⋊D3—22no—
87C2×C3⋊D3—22no—
88C2×C3⋊D3—44no—
89C2×C3⋊D3C2×Q814yes—
90C32×C4—22no—
91C32×C4—21no—
92C32×C4—22no—
93C32×C4—21no—
94C3⋊Dic3—22no—
95C3⋊Dic3—22no—
96C3⋊Dic3—22no—
97C22⋊Q8—92no—
98C22⋊Q8—91no—
99C22⋊Q8—92no—
100C22⋊Q8—92no—
101C22⋊Q8—92no—
102C22⋊Q8—92no—
103C22⋊Q8C3⋊D311yescharacteristic, complemented
104C2×C22⋊C4—91no—
105C2×C22⋊C4—92no—
106C23×C3C22×D314yes—
107C22×D3—38no—
108C22×D3—316no—
109C22×D3—124no—
110C22×D3—124no—
111C3×Q8—44no—
112C2×C3×C4—44no—
113C2×C3×C4C22×D314yes—
114C2×C3×C4C22×D318yes—
115C2×C3×C4C22×D318yes—
116C2×C3×C4C22×D314yes—
117C2×Dic3—38no—
118C2×Dic3—38no—
119C2×Dic3—38no—
120C2×Dic3—124no—
121C4×D3—128no—
122C4×D3—128no—
123Dic6—124no—
124Dic6—128no—
125C2×C32—22no—
126C2×C32C22×Q811yescharacteristic
127C2×C32H(5,𝔽2)11yescharacteristic
128C2×C32H(5,𝔽2)11yescharacteristic
129C3⋊D3—24no—
130C24—91no—
131C2×Q8—91no—
132C2×Q8—92no—
133C2×Q8C2×C3⋊D311yescharacteristic, complemented
134C22×C4—91no—
135C22×C4—92no—
136C22×C4—92no—
137C22×C4C2×C3⋊D311yescharacteristic
138C4⋊C4—92no—
139C4⋊C4—92no—
140C4⋊C4—92no—
141C4⋊C4—91no—
142C4⋊C4—92no—
143C4⋊C4C2×C3⋊D311yescharacteristic, complemented
144C4⋊C4C2×C3⋊D312yescomplemented
145C22⋊C4—92no—
146C22⋊C4—92no—
147C22⋊C4—92no—
148C22⋊C4—92no—
149C22⋊C4—92no—
150C22⋊C4C2×C3⋊D312yes—
151C22×C3—24no—
152C22×C3—24no—
153C22×C3D3×Q818yes—
154C22×C3C23×D314yes—
155C2×D3—316no—
156C2×D3—68no—
157C2×D3—68no—
158C2×D3—1216no—
159C3×C4—24no—
160C3×C4—28no—
161C3×C4—24no—
162C3×C4—28no—
163Dic3—68no—
164Dic3—68no—
165Dic3—68no—
166C32#54311yescharacteristic, complemented, Sylow
167C23—92no—
168C23—94no—
169C23—361no—
170C23—361no—
171C23C22×C3⋊D311yescharacteristic
172Q8—41no—
173Q8—361no—
174Q8—362no—
175C2×C4—41no—
176C2×C4—92no—
177C2×C4—92no—
178C2×C4—92no—
179C2×C4—362no—
180C2×C4—362no—
181C2×C4—361no—
182C2×C4C22×C3⋊D311yescharacteristic
183C2×C4C22×C3⋊D311yescharacteristic
184C2×C4C22×C3⋊D312yes—
185C2×C4C22×C3⋊D312yes—
186C2×C3—28no—
187C2×C3#100414yes—
188C2×C3C2×D3×Q814yes—
189C2×C3#100914yes—
190D3—616no—
191C22—21no—
192C22—21no—
193C22—94no—
194C22—182no—
195C22—182no—
196C22—364no—
197C22Q8×C3⋊D312yes—
198C22C23×C3⋊D311yescharacteristic
199C4—21no—
200C4—21no—
201C4—22no—
202C4—22no—
203C4—182no—
204C4—182no—
205C4—182no—
206C3#563414yescomplemented
207C2—22no—
208C2—184no—
209C2#6436111yescharacteristic
210C2C2×Q8×C3⋊D311yescharacteristic
211C2#6436611yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊Q8C2
prime radicalC23×C3⋊D3C22
hypercenterC22⋊Q8C3⋊D3
hypocenterC32#543
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C32×C22⋊Q8C2semidirect
2. #64155C2semidirect
4. #64133C2semidirect
5. #64106C2semidirect
6. #64105C2semidirect
7. #64105C2semidirect
9. #64087C2semidirect
12. C2×C32×Q8C22semidirect
17. C2×#634C22semidirect
18. C2×#634C22semidirect
19. C32×C4⋊C4C22semidirect
20. C32×C4⋊C4C22semidirect
27. C4⋊(C3⋊Dic3)C22semidirect
28. C4⋊(C3⋊Dic3)C22semidirect
29. #3750C22semidirect
30. #3750C22semidirect
31. #3750C22semidirect
32. C3×C22⋊Q8D3semidirect
44. C22×C3⋊D3Q8semidirect
62. C2×C3×Q8C2×D3semidirect
69. C3×C4⋊C4C2×D3semidirect
70. C3×C4⋊C4C2×D3semidirect
103. C22⋊Q8C3⋊D3semidirect
133. C2×Q8C2×C3⋊D3semidirect
143. C4⋊C4C2×C3⋊D3semidirect
144. C4⋊C4C2×C3⋊D3semidirect
166. C32#543semidirect
206. C3#5634semidirect

Maximal subgroups

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

subgroupindex
C32×C22⋊Q82
#641552
C2×#37522
#641332
#641062
#641052
C22⋊C4×C3⋊D32
#640872
#56343