Finite Group Databasebeta

← #10589716

Subgroups — #10589716

Order 576.

357 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 119 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×#121C211yesmaximal, characteristic, complemented
2#63500C211yesmaximal, characteristic, complemented
3#63500C212yesmaximal, complemented
4#63497C212yesmaximal, complemented
5D4×Dic9C211yesmaximal, characteristic, complemented
6C4×#611C211yesmaximal, characteristic, complemented
7#63488C211yesmaximal, characteristic, complemented
8C4⋊C4×D9C211yesmaximal, characteristic, complemented
9#63452C211yesmaximal, characteristic, complemented
10#611⋊C4C212yesmaximal, complemented
11#63441C212yesmaximal, complemented
12#5614—31nomaximal
13C2×D4×C9C2211yescharacteristic
14C2×D4×C9C2212yes—
15C22×C4×C9C2211yescharacteristic
16C2×#611C2211yescharacteristic
17C2×#611C2212yes—
18C22×Dic9C2212yes—
19C2×C4×D9C2211yescharacteristic
20C2×C4×D9C2212yes—
21C9×C4⋊C4C2211yescharacteristic
22C9×C22⋊C4C2212yes—
23#3676C2212yes—
24#3676C2211yescharacteristic
25#3676C2212yes—
26#3676C2212yes—
27(C2×D9)⋊C4C2211yescharacteristic
28C4⋊Dic9C2211yescharacteristic
29C4⋊Dic9C2211yescharacteristic
30C4⋊Dic9C2212yes—
31Dic9⋊C4C2211yescharacteristic
32Dic9⋊C4C2212yes—
33Dic9⋊C4C2212yes—
34C4×Dic9C2212yes—
35C4×Dic9C2211yescharacteristic
36C3×#121D311yescharacteristic
37#938—32no—
38#938—31no—
39#935—32no—
40D4×Dic3—31no—
41C4×#67—31no—
42#926—31no—
43D3×C4⋊C4—31no—
44#890—31no—
45#67⋊C4—32no—
46#879—32no—
47C23×C9C2311yescharacteristic
48C23×C9C2312yes—
49C22×D9C2311yescharacteristic
50D4×C9—22no—
51D4×C9—42no—
52C2×C4×C9—22no—
53C2×C4×C9C2311yescharacteristic
54C2×C4×C9C2312yes—
55C2×C4×C9C2311yescharacteristic
56#611—22no—
57#611—24no—
58C2×Dic9—42no—
59C2×Dic9C2311yescharacteristic
60C2×Dic9C2312yes—
61C2×Dic9C2311yescharacteristic
62C2×Dic9C2312yes—
63C2×Dic9C2311yescharacteristic
64C4×D9—22no—
65C4×D9—42no—
66#553—91noSylow
67C2×C3×D4C2×D311yescharacteristic
68C2×C3×D4C2×D312yes—
69C22×C3×C4C2×D311yescharacteristic
70C2×#67—32no—
71C2×#67—31no—
72C22×Dic3—32no—
73C2×C4×D3—31no—
74C2×C4×D3—32no—
75C3×C4⋊C4C2×D311yescharacteristic
76C3×C22⋊C4C2×D312yes—
77#217—32no—
78#217—32no—
79#217—32no—
80#217—31no—
81(C2×D3)⋊C4—31no—
82C4⋊Dic3—32no—
83C4⋊Dic3—31no—
84C4⋊Dic3—31no—
85Dic3⋊C4—31no—
86Dic3⋊C4—32no—
87Dic3⋊C4—32no—
88C4×Dic3—32no—
89C4×Dic3—31no—
90C22×C9—21no—
91C22×C9—22no—
92C22×C9—21no—
93C22×C9—22no—
94C22×C9—22no—
95C22×C9—21no—
96C22×C9C2411yescharacteristic
97C2×D9—21no—
98C2×D9—21no—
99C2×D9#4112yes—
100C4×C9—21no—
101C4×C9—22no—
102C4×C9#4112yes—
103Dic9—21no—
104Dic9—22no—
105Dic9—21no—
106Dic9—22no—
107Dic9—21no—
108#126—92no—
109#125—91no—
110#123—92no—
111#121—92no—
112#121—91no—
113#121D911yescharacteristic, complemented
114C4×D4—91no—
115C4×D4—91no—
116C4×D4—92no—
117#117—91no—
118C2×C4⋊C4—91no—
119C23×C3C22×D311yescharacteristic
120C23×C3C22×D312yes—
121C22×D3—31no—
122C3×D4—22no—
123C3×D4—42no—
124C2×C3×C4—22no—
125C2×C3×C4C22×D311yescharacteristic
126C2×C3×C4C22×D312yes—
127C2×C3×C4C22×D311yescharacteristic
128#67—64no—
129#67—62no—
130C2×Dic3—31no—
131C2×Dic3—32no—
132C2×Dic3—31no—
133C2×Dic3—32no—
134C2×Dic3—31no—
135C2×Dic3—122no—
136C4×D3—62no—
137C4×D3—122no—
138C2×C9—41no—
139C2×C9—42no—
140C2×C9C2×#4111yescharacteristic
141C2×C9C2×#4111yescharacteristic
142C2×C9H(5,𝔽2)11yescharacteristic
143D9—22no—
144C2×D4—91no—
145C2×D4—92no—
146C2×D4C2×D911yescharacteristic
147C2×D4C2×D912yes—
148C22×C4—92no—
149C22×C4—91no—
150C22×C4—92no—
151C22×C4C2×D911yescharacteristic
152C4⋊C4—91no—
153C4⋊C4—92no—
154C4⋊C4—91no—
155C4⋊C4—92no—
156C4⋊C4—91no—
157C4⋊C4—92no—
158C4⋊C4C2×D911yescharacteristic
159C22⋊C4—92no—
160C22⋊C4—92no—
161C22⋊C4—91no—
162C22⋊C4—92no—
163C22⋊C4—91no—
164C22⋊C4C2×D912yes—
165C42—92no—
166C42—91no—
167C22×C3—21no—
168C22×C3—22no—
169C22×C3—22no—
170C22×C3—21no—
171C22×C3—21no—
172C22×C3—22no—
173C22×C3C23×D311yescharacteristic
174C2×D3—32no—
175C2×D3—61no—
176C2×D3—61no—
177C3×C4—21no—
178C3×C4—22no—
179C3×C4#23712yes—
180Dic3—61no—
181Dic3—62no—
182Dic3—61no—
183Dic3—62no—
184Dic3—61no—
185C9#55311yescharacteristic, complemented, Sylow
186C23—91no—
187C23C22×D911yescharacteristic
188C23C22×D912yes—
189D4—22no—
190D4—42no—
191D4—182no—
192D4—184no—
193C2×C4—22no—
194C2×C4—92no—
195C2×C4—91no—
196C2×C4—92no—
197C2×C4—91no—
198C2×C4—91no—
199C2×C4—182no—
200C2×C4—362no—
201C2×C4—362no—
202C2×C4C22×D911yescharacteristic
203C2×C4C22×D912yes—
204C2×C4C22×D911yescharacteristic
205C2×C3—41no—
206C2×C3—42no—
207C2×C3C2×#23711yescharacteristic
208C2×C3#100411yescharacteristic
209C2×C3D3×#4111yescharacteristic
210D3—62no—
211C22—21no—
212C22—22no—
213C22—22no—
214C22—21no—
215C22—21no—
216C22—22no—
217C22—92no—
218C22—181no—
219C22—181no—
220C22C23×D911yescharacteristic
221C4—21no—
222C4—22no—
223C4—181no—
224C4—182no—
225C4—181no—
226C4—182no—
227C4—181no—
228C4#369912yes—
229C3#561411yescharacteristic
230C2—41no—
231C2—42no—
232C2—182no—
233C2C2×#369911yescharacteristic
234C2#6371011yescharacteristic
235C2#41×D911yescharacteristic
subgroupG/H
centerC22C23×D9
commutator, cosocleC22×C9C24
Frattini, socleC22×C3C23×D3
FittingC9×#121C2
hypercenter#121D9
hypocenterC9#553
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C9×#121C2semidirect
2. #63500C2semidirect
3. #63500C2semidirect
4. #63497C2semidirect
5. D4×Dic9C2semidirect
6. C4×#611C2semidirect
7. #63488C2semidirect
8. C4⋊C4×D9C2semidirect
9. #63452C2semidirect
10. #611⋊C4C2semidirect
11. #63441C2semidirect
113. #121D9semidirect
185. C9#553semidirect

Maximal subgroups

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

subgroupindex
C9×#1212
#635002
#634972
D4×Dic92
C4×#6112
#634882
C4⋊C4×D92
#634522
#611⋊C42
#634412
#56143