Finite Group Databasebeta

← #10595256

Subgroups — #10595256

Order 576.

558 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 120 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#64145C211yesmaximal, characteristic, complemented
2C3×#937C212yesmaximal, complemented
3#63976C214yesmaximal, complemented
4#63959C214yesmaximal, complemented
5#63915C212yesmaximal, complemented
6#63913C212yesmaximal, complemented
7#5594—32nomaximal
8C23×C3×D3C2212yescomplemented
9C2×C32×D4C2211yescharacteristic
10C2×#634C2211yescharacteristic, complemented
11C2×C3×#67C2214yes—
12C2×#627C2212yescomplemented
13C2×#625C2212yes—
14C2×#625C2212yes—
15#3757C2214yescomplemented
16C4×C3⋊Dic3C2211yescharacteristic, complemented
17C3×#217C2212yescomplemented
18C3×(C2×D3)⋊C4C2214yescomplemented
19(C2×D3)⋊Dic3C2214yescomplemented
20(C2×D3)⋊Dic3C2214yescomplemented
21Dic32C2212yescomplemented
22C3×#120—32no—
23#953—34no—
24#937—34no—
25#937D312yescomplemented
26#936—32no—
27#936—34no—
28#936—61no—
29#885—34no—
30#885—34no—
31#885—34no—
32C23×C32C2312yes—
33C22×C3×D3—42no—
34C22×C3×D3—44no—
35C22×C3×D3C2314yes—
36C32×D4—41no—
37C2×C32×C4C2311yescharacteristic
38C2×C3⋊Dic3C2312yes—
39C2×C3⋊Dic3C2312yes—
40#634—41no—
41C3×#67—44no—
42C2×C3×Dic3C2314yes—
43#627—42no—
44#625—42no—
45#625—42no—
46H(3,𝔽4)—91noSylow
47C24×C3—32no—
48C23×D3C2×D312yescomplemented
49C2×C3×D4—21no—
50C2×C3×D4—34no—
51C2×C3×D4C2×D312yes—
52C2×#67—34no—
53C2×#67—34no—
54C2×#67C2×D314yes—
55C2×Dic6—32no—
56C2×Dic6—34no—
57C2×Dic6—61no—
58C3×C22⋊C4—32no—
59C3×C22⋊C4—34no—
60#217—34no—
61#217—34no—
62#217—34no—
63#217—34no—
64#217—64no—
65#217C2×D312yescomplemented
66(C2×D3)⋊C4—34no—
67(C2×D3)⋊C4—34no—
68(C2×D3)⋊C4C2×D314yescomplemented
69C4×Dic3—32no—
70C4×Dic3—34no—
71C4×Dic3—61no—
72C22×C32—24no—
73C22×C32—22no—
74C22×C32C2411yescharacteristic
75C2×C3×D3—24no—
76C2×C3×D3—24no—
77C2×C3×D3—24no—
78C2×C3×D3—48no—
79C32×C4—21no—
80C3⋊Dic3—22no—
81C3⋊Dic3—22no—
82C3×Dic3—24no—
83#124—92no—
84#124—92no—
85#124—94no—
86#124—91no—
87#120—32no—
88#120—94no—
89C23×C3—22no—
90C23×C3—34no—
91C23×C3—124no—
92C23×C3—122no—
93C23×C3C22×D312yes—
94C23×C3C22×D312yes—
95C22×D3—44no—
96C22×D3—42no—
97C22×D3C22×D314yes—
98C3×D4—42no—
99C3×D4—42no—
100C3×D4—124no—
101C2×C3×C4—21no—
102C2×C3×C4—34no—
103C2×C3×C4C22×D312yes—
104#67—44no—
105#67—124no—
106#67—124no—
107C2×Dic3—34no—
108C2×Dic3—34no—
109C2×Dic3—62no—
110C2×Dic3—62no—
111C2×Dic3C22×D314yes—
112Dic6—124no—
113Dic6—122no—
114Dic6—122no—
115C2×C32—42no—
116C2×C32H(5,𝔽2)12yes—
117C2×C32H(5,𝔽2)11yescharacteristic
118C3×D3—44no—
119C24—32no—
120C2×Q8—92no—
121C2×Q8—91no—
122C2×D4—34no—
123C2×D4—92no—
124C2×D4—92no—
125C2×D4D3211yescharacteristic
126C22⋊C4—32no—
127C22⋊C4—34no—
128C22⋊C4—94no—
129C22⋊C4—94no—
130C22⋊C4—94no—
131C42—92no—
132C42—91no—
133C22×C3—24no—
134C22×C3—22no—
135C22×C3—22no—
136C22×C3—24no—
137C22×C3—21no—
138C22×C3—42no—
139C22×C3—44no—
140C22×C3—64no—
141C22×C3—64no—
142C22×C3—64no—
143C22×C3—128no—
144C22×C3C23×D312yes—
145C2×D3—24no—
146C2×D3—24no—
147C2×D3—24no—
148C2×D3—48no—
149C3×C4—22no—
150C3×C4—22no—
151C3×C4—64no—
152Dic3—24no—
153Dic3—64no—
154Dic3—64no—
155Dic3—122no—
156Dic3—122no—
157C32H(3,𝔽4)11yescharacteristic, complemented, Sylow
158C23—34no—
159C23—124no—
160C23—122no—
161C23C2×D3212yes—
162Q8—361no—
163Q8—362no—
164D4—41no—
165D4—124no—
166D4—362no—
167D4—362no—
168C2×C4—34no—
169C2×C4—92no—
170C2×C4—92no—
171C2×C4C2×D3211yescharacteristic
172C2×C3—22no—
173C2×C3—21no—
174C2×C3—42no—
175C2×C3—42no—
176C2×C3—82no—
177C2×C3—124no—
178C2×C3#100414yes—
179C2×C3#100412yes—
180D3—44no—
181C22—24no—
182C22—22no—
183C22—64no—
184C22—64no—
185C22—64no—
186C22—128no—
187C22C22×D3211yescharacteristic
188C4—21no—
189C4—64no—
190C4—182no—
191C4—182no—
192C3—21no—
193C3#559412yescomplemented
194C2—42no—
195C2—124no—
196C2#6431312yes—
197C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
hypercenterC2×D4D32
hypocenterC32H(3,𝔽4)
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #64145C2semidirect
2. C3×#937C2semidirect
3. #63976C2semidirect
4. #63959C2semidirect
5. #63915C2semidirect
6. #63913C2semidirect
8. C23×C3×D3C22semidirect
10. C2×#634C22semidirect
12. C2×#627C22semidirect
15. #3757C22semidirect
16. C4×C3⋊Dic3C22semidirect
17. C3×#217C22semidirect
18. C3×(C2×D3)⋊C4C22semidirect
19. (C2×D3)⋊Dic3C22semidirect
20. (C2×D3)⋊Dic3C22semidirect
21. Dic32C22semidirect
25. #937D3semidirect
48. C23×D3C2×D3semidirect
65. #217C2×D3semidirect
68. (C2×D3)⋊C4C2×D3semidirect
157. C32H(3,𝔽4)semidirect
193. C3#5594semidirect

Maximal subgroups

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

subgroupindex
#641452
C3×#9372
#639762
#639592
#639152
#639132
#55943