Finite Group Databasebeta

← #10589837

Subgroups — #10589837

Order 576.

294 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C9×C4⋊Q8C211yesmaximal, characteristic, complemented
2#63509C211yesmaximal, characteristic
3#63508C212yesmaximal, complemented
4#63506C211yesmaximal, characteristic, complemented
5#63460C212yesmaximal, complemented
6#63458C212yesmaximal, complemented
7#63456C212yesmaximal
8#63451C212yesmaximal, complemented
9#63438C212yesmaximal, complemented
10#5735—31nomaximal
11C2×Q8×C9C2211yescharacteristic
12C2×Q8×C9C2211yescharacteristic
13C2×D36C2211yescharacteristic
14C2×C4×D9C2212yes—
15C2×Dic18C2211yescharacteristic
16C9×C4⋊C4C2212yes—
17C9×C4⋊C4C2212yes—
18C42×C9C2211yescharacteristic
19(C2×D9)⋊C4C2212yes—
20(C2×D9)⋊C4C2212yes—
21(C2×D9)⋊C4C2212yes—
22(C2×D9)⋊C4C2212yes—
23(C2×D9)⋊C4C2212yes—
24C4⋊Dic9C2212yes—
25Dic9⋊C4C2212yes—
26Dic9⋊C4C2212yes—
27Dic9⋊C4C2212yes—
28Dic9⋊C4C2212yes—
29Dic9⋊C4C2212yes—
30C4×Dic9C2211yescharacteristic
31C4×Dic9C2211yescharacteristic
32C3×C4⋊Q8D311yescharacteristic
33#947—31no—
34#946—32no—
35#944—31no—
36#898—32no—
37#896—32no—
38#894—32no—
39#889—32no—
40#876—32no—
41C22×D9C2312yes—
42Q8×C9—41no—
43Q8×C9—41no—
44C2×C4×C9C2311yescharacteristic
45C2×C4×C9C2311yescharacteristic
46C2×C4×C9C2312yes—
47C2×C4×C9C2311yescharacteristic
48C2×C4×C9C2312yes—
49C2×Dic9C2312yes—
50C2×Dic9C2312yes—
51C2×Dic9C2312yes—
52D36—41no—
53C4×D9—42no—
54Dic18—41no—
55#563—91noSylow
56C2×C3×Q8C2×D311yescharacteristic
57C2×C3×Q8C2×D311yescharacteristic
58C2×D12—31no—
59C2×C4×D3—32no—
60C2×Dic6—31no—
61C3×C4⋊C4C2×D312yes—
62C3×C4⋊C4C2×D312yes—
63C3×C42C2×D311yescharacteristic
64(C2×D3)⋊C4—32no—
65(C2×D3)⋊C4—32no—
66(C2×D3)⋊C4—32no—
67(C2×D3)⋊C4—32no—
68(C2×D3)⋊C4—32no—
69C4⋊Dic3—32no—
70Dic3⋊C4—32no—
71Dic3⋊C4—32no—
72Dic3⋊C4—32no—
73Dic3⋊C4—32no—
74Dic3⋊C4—32no—
75C4×Dic3—31no—
76C4×Dic3—31no—
77C22×C9C2411yescharacteristic
78C2×D9—22no—
79C2×D9—24no—
80C4×C9—21no—
81C4×C9—22no—
82C4×C9—21no—
83C4×C9—21no—
84C4×C9—22no—
85Dic9—22no—
86Dic9—22no—
87Dic9—22no—
88C4⋊Q8—91no—
89C4⋊Q8D911yescharacteristic, complemented
90#126—92no—
91#126—92no—
92#125—92no—
93#124—91no—
94#123—92no—
95C22⋊Q8—92no—
96C22⋊Q8—92no—
97C22×D3—32no—
98C3×Q8—41no—
99C3×Q8—41no—
100C2×C3×C4C22×D311yescharacteristic
101C2×C3×C4C22×D311yescharacteristic
102C2×C3×C4C22×D312yes—
103C2×C3×C4C22×D311yescharacteristic
104C2×C3×C4C22×D312yes—
105C2×Dic3—32no—
106C2×Dic3—32no—
107C2×Dic3—32no—
108D12—121no—
109C4×D3—122no—
110Dic6—121no—
111C2×C9H(5,𝔽2)11yescharacteristic
112C2×C92−1+412yes—
113D9—42no—
114C2×Q8—91no—
115C2×Q8C2×D911yescharacteristic
116C2×Q8C2×D911yescharacteristic
117C2×D4—91no—
118C22×C4—92no—
119C4⋊C4—92no—
120C4⋊C4—92no—
121C4⋊C4—92no—
122C4⋊C4—92no—
123C4⋊C4—92no—
124C4⋊C4—92no—
125C4⋊C4C2×D912yes—
126C4⋊C4C2×D912yes—
127C22⋊C4—92no—
128C22⋊C4—92no—
129C22⋊C4—92no—
130C22⋊C4—92no—
131C22⋊C4—92no—
132C42—91no—
133C42—91no—
134C42C2×D911yescharacteristic
135C22×C3C23×D311yescharacteristic
136C2×D3—62no—
137C2×D3—64no—
138C3×C4—21no—
139C3×C4—22no—
140C3×C4—21no—
141C3×C4—21no—
142C3×C4—22no—
143Dic3—62no—
144Dic3—62no—
145Dic3—62no—
146C9#56311yescharacteristic, complemented, Sylow
147C23—92no—
148Q8—41no—
149Q8—41no—
150Q8—361no—
151D4—361no—
152C2×C4—92no—
153C2×C4—92no—
154C2×C4—92no—
155C2×C4—362no—
156C2×C4C22×D911yescharacteristic
157C2×C4C22×D911yescharacteristic
158C2×C4C22×D912yes—
159C2×C4C22×D911yescharacteristic
160C2×C4C22×D912yes—
161C2×C3#100411yescharacteristic
162C2×C3#100712yes—
163D3—122no—
164C22—182no—
165C22—184no—
166C22C23×D911yescharacteristic
167C4—22no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—22no—
172C4—182no—
173C4—182no—
174C4—182no—
175C3#573511yescharacteristic
176C2—362no—
177C2#6371011yescharacteristic
178C2#6371312yes—
subgroupG/H
centerC22C23×D9
commutator, cosocleC22×C9C24
Frattini, socleC22×C3C23×D3
FittingC9×C4⋊Q8C2
prime radicalC2×D36C22
hypercenterC4⋊Q8D9
hypocenterC9#563
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C9×C4⋊Q8C2semidirect
3. #63508C2semidirect
4. #63506C2semidirect
5. #63460C2semidirect
6. #63458C2semidirect
8. #63451C2semidirect
9. #63438C2semidirect
89. C4⋊Q8D9semidirect
146. C9#563semidirect

Maximal subgroups

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

subgroupindex
C9×C4⋊Q82
#635092
#635082
#635062
#634602
#634582
#634562
#634512
#634382
#57353