Finite Group Databasebeta

← #10589801

Subgroups — #10589801

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×#125C211yesmaximal, characteristic, complemented
2#63460C212yesmaximal, complemented
3#63460C212yesmaximal, complemented
4#63459C212yesmaximal, complemented
5#63458C212yesmaximal, complemented
6#63456C212yesmaximal
7#63450C212yesmaximal, complemented
8#63437C211yesmaximal, characteristic
9#63432C211yesmaximal, characteristic, complemented
10#5699—31nomaximal
11C2×D36C2211yescharacteristic
12C2×C4×D9C2212yes—
13C2×Dic18C2211yescharacteristic
14C2×Dic18C2212yes—
15C9×C4⋊C4C2212yes—
16C9×C4⋊C4C2211yescharacteristic
17C9×C4⋊C4C2212yes—
18C9×C4⋊C4C2211yescharacteristic
19C42×C9C2211yescharacteristic
20(C2×D9)⋊C4C2212yes—
21(C2×D9)⋊C4C2212yes—
22(C2×D9)⋊C4C2212yes—
23(C2×D9)⋊C4C2212yes—
24(C2×D9)⋊C4C2212yes—
25C4⋊Dic9C2211yescharacteristic
26C4⋊Dic9C2211yescharacteristic
27C4⋊Dic9C2212yes—
28Dic9⋊C4C2212yes—
29Dic9⋊C4C2212yes—
30Dic9⋊C4C2212yes—
31C4×Dic9C2212yes—
32C3×#125D311yescharacteristic
33#898—32no—
34#898—32no—
35#897—32no—
36#896—32no—
37#894—32no—
38#888—32no—
39#875—31no—
40#870—31no—
41C22×D9C2312yes—
42C2×C4×C9C2312yes—
43C2×C4×C9C2311yescharacteristic
44C2×C4×C9C2311yescharacteristic
45C2×C4×C9C2312yes—
46C2×C4×C9C2311yescharacteristic
47C2×Dic9C2312yes—
48C2×Dic9C2312yes—
49C2×Dic9C2312yes—
50D36—41no—
51C4×D9—42no—
52Dic18—41no—
53Dic18—42no—
54#563—91noSylow
55C2×D12—31no—
56C2×C4×D3—32no—
57C2×Dic6—31no—
58C2×Dic6—32no—
59C3×C4⋊C4C2×D312yes—
60C3×C4⋊C4C2×D311yescharacteristic
61C3×C4⋊C4C2×D312yes—
62C3×C4⋊C4C2×D311yescharacteristic
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—31no—
70C4⋊Dic3—31no—
71C4⋊Dic3—32no—
72Dic3⋊C4—32no—
73Dic3⋊C4—32no—
74Dic3⋊C4—32no—
75C4×Dic3—32no—
76C22×C9C2411yescharacteristic
77C2×D9—22no—
78C2×D9—22no—
79C2×D9—22no—
80C4×C9—22no—
81C4×C9—21no—
82C4×C9—21no—
83C4×C9—22no—
84C4×C9—21no—
85Dic9—22no—
86Dic9—22no—
87Dic9—22no—
88C4⋊Q8—92no—
89#126—92no—
90#126—92no—
91#125—91no—
92#125D911yescharacteristic, complemented
93#124—91no—
94#123—92no—
95C22⋊Q8—92no—
96C22⋊Q8—92no—
97C22×D3—32no—
98C2×C3×C4C22×D312yes—
99C2×C3×C4C22×D311yescharacteristic
100C2×C3×C4C22×D311yescharacteristic
101C2×C3×C4C22×D312yes—
102C2×C3×C4C22×D311yescharacteristic
103C2×Dic3—32no—
104C2×Dic3—32no—
105C2×Dic3—32no—
106D12—121no—
107C4×D3—122no—
108Dic6—121no—
109Dic6—122no—
110C2×C9H(5,𝔽2)11yescharacteristic
111C2×C92−1+411yescharacteristic
112C2×C92−1+411yescharacteristic
113D9—42no—
114C2×Q8—92no—
115C2×Q8—91no—
116C2×D4—91no—
117C22×C4—92no—
118C4⋊C4—91no—
119C4⋊C4—92no—
120C4⋊C4—92no—
121C4⋊C4—92no—
122C4⋊C4—92no—
123C4⋊C4—91no—
124C4⋊C4C2×D911yescharacteristic
125C4⋊C4C2×D912yes—
126C4⋊C4C2×D912yes—
127C4⋊C4C2×D911yescharacteristic
128C22⋊C4—92no—
129C22⋊C4—92no—
130C22⋊C4—92no—
131C22⋊C4—92no—
132C22⋊C4—92no—
133C42—92no—
134C42C2×D911yescharacteristic
135C22×C3C23×D311yescharacteristic
136C2×D3—62no—
137C2×D3—62no—
138C2×D3—62no—
139C3×C4—21no—
140C3×C4—22no—
141C3×C4—21no—
142C3×C4—22no—
143C3×C4—21no—
144Dic3—62no—
145Dic3—62no—
146Dic3—62no—
147C9#56311yescharacteristic, complemented, Sylow
148C23—92no—
149Q8—362no—
150Q8—361no—
151D4—361no—
152C2×C4—92no—
153C2×C4—92no—
154C2×C4—92no—
155C2×C4—362no—
156C2×C4C22×D912yes—
157C2×C4C22×D911yescharacteristic
158C2×C4C22×D911yescharacteristic
159C2×C4C22×D911yescharacteristic
160C2×C4C22×D912yes—
161C2×C3#100711yescharacteristic
162C2×C3#100911yescharacteristic
163C2×C3#101011yescharacteristic
164D3—122no—
165C22—182no—
166C22—182no—
167C22—182no—
168C22C23×D911yescharacteristic
169C4—21no—
170C4—22no—
171C4—21no—
172C4—22no—
173C4—21no—
174C4—182no—
175C4—182no—
176C4—182no—
177C3#569911yescharacteristic
178C2—362no—
179C2#6371311yescharacteristic
180C2#6371511yescharacteristic
181C2#6371611yescharacteristic
subgroupG/H
centerC22C23×D9
commutator, cosocleC22×C9C24
Frattini, socleC22×C3C23×D3
FittingC9×#125C2
prime radicalC2×D36C22
hypercenter#125D9
hypocenterC9#563
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. C9×#125C2semidirect
2. #63460C2semidirect
3. #63460C2semidirect
4. #63459C2semidirect
5. #63458C2semidirect
7. #63450C2semidirect
9. #63432C2semidirect
92. #125D9semidirect
147. C9#563semidirect

Maximal subgroups

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

subgroupindex
C9×#1252
#634602
#634592
#634582
#634562
#634502
#634372
#634322
#56993