Finite Group Databasebeta

← #10589791

Subgroups — #10589791

Order 576.

309 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 125 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#63459C212yesmaximal
3#63458C212yesmaximal
4#63454C211yesmaximal, characteristic
5C4⋊C4×D9C211yesmaximal, characteristic
6#63452C211yesmaximal, characteristic, complemented
7#63451C212yesmaximal, complemented
8#63450C211yesmaximal, characteristic, complemented
9#63450C212yesmaximal, complemented
10#63434C211yesmaximal, characteristic
11#63431C211yesmaximal, characteristic, complemented
12#5689—31nomaximal
13C2×C4×D9C2212yes—
14C2×C4×D9C2211yescharacteristic
15C2×Dic18C2212yes—
16C2×Dic18C2212yes—
17C9×C4⋊C4C2211yescharacteristic
18C9×C4⋊C4C2212yes—
19C9×C4⋊C4C2212yes—
20C9×C4⋊C4C2211yescharacteristic
21C42×C9C2211yescharacteristic
22(C2×D9)⋊C4C2211yescharacteristic
23(C2×D9)⋊C4C2211yescharacteristic
24(C2×D9)⋊C4C2212yes—
25C4⋊Dic9C2211yescharacteristic
26C4⋊Dic9C2212yes—
27C4⋊Dic9C2211yescharacteristic
28C4⋊Dic9C2212yes—
29Dic9⋊C4C2211yescharacteristic
30Dic9⋊C4C2211yescharacteristic
31Dic9⋊C4C2212yes—
32Dic9⋊C4C2212yes—
33Dic9⋊C4C2212yes—
34C4×Dic9C2211yescharacteristic
35C4×Dic9C2212yes—
36C3×#125D311yescharacteristic
37#897—32no—
38#896—32no—
39#892—31no—
40D3×C4⋊C4—31no—
41#890—31no—
42#889—32no—
43#888—31no—
44#888—32no—
45#872—31no—
46#869—31no—
47C22×D9C2311yescharacteristic
48C2×C4×C9C2312yes—
49C2×C4×C9C2312yes—
50C2×C4×C9C2311yescharacteristic
51C2×C4×C9C2311yescharacteristic
52C2×C4×C9C2311yescharacteristic
53C2×Dic9C2311yescharacteristic
54C2×Dic9C2311yescharacteristic
55C2×Dic9C2312yes—
56C2×Dic9C2312yes—
57C2×Dic9C2311yescharacteristic
58C4×D9—42no—
59C4×D9Q814yescomplemented
60Dic18—42no—
61Dic18—42no—
62#544—91noSylow
63C2×C4×D3—31no—
64C2×C4×D3—32no—
65C2×Dic6—32no—
66C2×Dic6—32no—
67C3×C4⋊C4C2×D311yescharacteristic
68C3×C4⋊C4C2×D312yes—
69C3×C4⋊C4C2×D312yes—
70C3×C4⋊C4C2×D311yescharacteristic
71C3×C42C2×D311yescharacteristic
72(C2×D3)⋊C4—31no—
73(C2×D3)⋊C4—31no—
74(C2×D3)⋊C4—32no—
75C4⋊Dic3—31no—
76C4⋊Dic3—32no—
77C4⋊Dic3—31no—
78C4⋊Dic3—32no—
79Dic3⋊C4—31no—
80Dic3⋊C4—31no—
81Dic3⋊C4—32no—
82Dic3⋊C4—32no—
83Dic3⋊C4—32no—
84C4×Dic3—31no—
85C4×Dic3—32no—
86C22×C9C2411yescharacteristic
87C2×D9—21no—
88C2×D9—21no—
89C2×D9C2×Q812yes—
90C4×C9—21no—
91C4×C9—22no—
92C4×C9—22no—
93C4×C9—21no—
94C4×C9C2×Q812yes—
95Dic9—21no—
96Dic9—22no—
97Dic9—22no—
98Dic9—21no—
99Dic9C2×Q812yes—
100C4⋊Q8—92no—
101C4⋊Q8—91no—
102C4⋊Q8—91no—
103#125—92no—
104#125—91no—
105#125D911yescharacteristic, complemented
106C22⋊Q8—92no—
107C22⋊Q8—92no—
108#117—91no—
109#117—91no—
110C2×C4⋊C4—91no—
111C22×D3—31no—
112C2×C3×C4C22×D312yes—
113C2×C3×C4C22×D312yes—
114C2×C3×C4C22×D311yescharacteristic
115C2×C3×C4C22×D311yescharacteristic
116C2×C3×C4C22×D311yescharacteristic
117C2×Dic3—31no—
118C2×Dic3—31no—
119C2×Dic3—31no—
120C2×Dic3—32no—
121C2×Dic3—32no—
122C4×D3—34no—
123C4×D3—122no—
124Dic6—122no—
125Dic6—122no—
126C2×C9C22×Q811yescharacteristic
127C2×C9H(5,𝔽2)11yescharacteristic
128C2×C92−1+411yescharacteristic
129D9—22no—
130C2×Q8—92no—
131C2×Q8—92no—
132C22×C4—91no—
133C22×C4—92no—
134C4⋊C4—92no—
135C4⋊C4—91no—
136C4⋊C4—91no—
137C4⋊C4—91no—
138C4⋊C4—91no—
139C4⋊C4—92no—
140C4⋊C4—92no—
141C4⋊C4—92no—
142C4⋊C4—92no—
143C4⋊C4C2×D911yescharacteristic
144C4⋊C4C2×D911yescharacteristic
145C4⋊C4C2×D912yes—
146C4⋊C4C2×D912yes—
147C22⋊C4—91no—
148C22⋊C4—92no—
149C22⋊C4—91no—
150C42—92no—
151C42—91no—
152C42C2×D911yescharacteristic
153C22×C3C23×D311yescharacteristic
154C2×D3—32no—
155C2×D3—61no—
156C2×D3—61no—
157C3×C4—21no—
158C3×C4—22no—
159C3×C4—22no—
160C3×C4—21no—
161C3×C4D3×Q812yes—
162Dic3—32no—
163Dic3—61no—
164Dic3—62no—
165Dic3—62no—
166Dic3—61no—
167C9#54411yescharacteristic, complemented, Sylow
168C23—91no—
169Q8—362no—
170Q8—362no—
171C2×C4—91no—
172C2×C4—94no—
173C2×C4—91no—
174C2×C4—91no—
175C2×C4—92no—
176C2×C4—92no—
177C2×C4—362no—
178C2×C4C22×D912yes—
179C2×C4C22×D911yescharacteristic
180C2×C4C22×D911yescharacteristic
181C2×C4C22×D912yes—
182C2×C4C22×D911yescharacteristic
183C2×C3C2×D3×Q811yescharacteristic
184C2×C3#100911yescharacteristic
185C2×C3#101011yescharacteristic
186D3—62no—
187C22—92no—
188C22—181no—
189C22—181no—
190C22C23×D911yescharacteristic
191C4—22no—
192C4—21no—
193C4—22no—
194C4—21no—
195C4—92no—
196C4—182no—
197C4—181no—
198C4—182no—
199C4—181no—
200C4Q8×D912yes—
201C3#568911yescharacteristic
202C2—182no—
203C2C2×Q8×D911yescharacteristic
204C2#6371511yescharacteristic
205C2#6371611yescharacteristic
subgroupG/H
centerC22C23×D9
commutator, cosocleC22×C9C24
Frattini, socleC22×C3C23×D3
FittingC9×#125C2
prime radicalC22×D9C23
hypercenter#125D9
hypocenterC9#544
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. C9×#125C2semidirect
6. #63452C2semidirect
7. #63451C2semidirect
8. #63450C2semidirect
9. #63450C2semidirect
11. #63431C2semidirect
59. C4×D9Q8semidirect
105. #125D9semidirect
167. C9#544semidirect

Maximal subgroups

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

subgroupindex
C9×#1252
#634592
#634582
#634542
C4⋊C4×D92
#634522
#634512
#634502
#634342
#634312
#56893