Finite Group Databasebeta

← #10589721

Subgroups — #10589721

Order 576.

360 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 117 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#63502C211yesmaximal, characteristic, complemented
3#63501C212yesmaximal, complemented
4#63500C212yesmaximal, complemented
5D4×Dic9C211yesmaximal, characteristic, complemented
6#63492C211yesmaximal, characteristic, complemented
7#63488C211yesmaximal, characteristic, complemented
8#63455C211yesmaximal, characteristic, complemented
9#63452C211yesmaximal, characteristic, complemented
10#63448C212yesmaximal, complemented
11#63445C212yesmaximal, complemented
12#5619—31nomaximal
13C2×D4×C9C2211yescharacteristic
14C2×D4×C9C2212yes—
15C22×C4×C9C2211yescharacteristic
16C2×#611C2212yes—
17C2×#611C2212yes—
18C2×#611C2212yes—
19C22×Dic9C2212yes—
20C2×D36C2211yescharacteristic
21C2×C4×D9C2212yes—
22C9×C4⋊C4C2211yescharacteristic
23C9×C22⋊C4C2212yes—
24#3676C2211yescharacteristic
25#3676C2211yescharacteristic
26#3676C2212yes—
27(C2×D9)⋊C4C2212yes—
28(C2×D9)⋊C4C2212yes—
29C4⋊Dic9C2211yescharacteristic
30C4⋊Dic9C2211yescharacteristic
31C4⋊Dic9C2211yescharacteristic
32C4⋊Dic9C2212yes—
33Dic9⋊C4C2212yes—
34C4×Dic9C2211yescharacteristic
35C4×Dic9C2211yescharacteristic
36C3×#121D311yescharacteristic
37#940—31no—
38#939—32no—
39#938—32no—
40D4×Dic3—31no—
41#930—31no—
42#926—31no—
43#893—31no—
44#890—31no—
45#886—32no—
46#883—32no—
47C23×C9C2311yescharacteristic
48C23×C9C2312yes—
49C22×D9C2312yes—
50D4×C9—22no—
51D4×C9—42no—
52C2×C4×C9—22no—
53C2×C4×C9C2312yes—
54C2×C4×C9C2311yescharacteristic
55C2×C4×C9C2311yescharacteristic
56#611—42no—
57#611—42no—
58#611—42no—
59C2×Dic9—42no—
60C2×Dic9C2311yescharacteristic
61C2×Dic9C2311yescharacteristic
62C2×Dic9C2311yescharacteristic
63C2×Dic9C2311yescharacteristic
64C2×Dic9C2312yes—
65D36—22no—
66C4×D9—42no—
67#540—91noSylow
68C2×C3×D4C2×D311yescharacteristic
69C2×C3×D4C2×D312yes—
70C22×C3×C4C2×D311yescharacteristic
71C2×#67—32no—
72C2×#67—32no—
73C2×#67—32no—
74C22×Dic3—32no—
75C2×D12—31no—
76C2×C4×D3—32no—
77C3×C4⋊C4C2×D311yescharacteristic
78C3×C22⋊C4C2×D312yes—
79#217—32no—
80#217—31no—
81#217—31no—
82(C2×D3)⋊C4—32no—
83(C2×D3)⋊C4—32no—
84C4⋊Dic3—31no—
85C4⋊Dic3—31no—
86C4⋊Dic3—32no—
87C4⋊Dic3—31no—
88Dic3⋊C4—32no—
89C4×Dic3—31no—
90C4×Dic3—31no—
91C22×C9—21no—
92C22×C9—22no—
93C22×C9—22no—
94C22×C9—21no—
95C22×C9—22no—
96C22×C9—21no—
97C22×C9C2411yescharacteristic
98C2×D9—22no—
99C2×D9—22no—
100C2×D9—22no—
101C4×C9—21no—
102C4×C9—22no—
103C4×C9#4112yes—
104Dic9—21no—
105Dic9—21no—
106Dic9—22no—
107Dic9—21no—
108Dic9—21no—
109#127—91no—
110#125—91no—
111#123—92no—
112#123—92no—
113#121—92no—
114#121—92no—
115#121—91no—
116#121D911yescharacteristic, complemented
117C4×D4—91no—
118C4×D4—91no—
119#117—91no—
120C23×C3C22×D311yescharacteristic
121C23×C3C22×D312yes—
122C22×D3—32no—
123C3×D4—22no—
124C3×D4—42no—
125C2×C3×C4—22no—
126C2×C3×C4C22×D312yes—
127C2×C3×C4C22×D311yescharacteristic
128C2×C3×C4C22×D311yescharacteristic
129#67—122no—
130#67—122no—
131#67—122no—
132C2×Dic3—31no—
133C2×Dic3—31no—
134C2×Dic3—31no—
135C2×Dic3—31no—
136C2×Dic3—32no—
137C2×Dic3—122no—
138D12—62no—
139C4×D3—122no—
140C2×C9—41no—
141C2×C9—42no—
142C2×C9C2×#4111yescharacteristic
143C2×C9H(5,𝔽2)11yescharacteristic
144C2×C9H(5,𝔽2)11yescharacteristic
145D9—42no—
146C2×D4—92no—
147C2×D4—92no—
148C2×D4—92no—
149C2×D4—91no—
150C2×D4C2×D911yescharacteristic
151C2×D4C2×D912yes—
152C22×C4—92no—
153C22×C4—92no—
154C22×C4C2×D911yescharacteristic
155C4⋊C4—91no—
156C4⋊C4—91no—
157C4⋊C4—91no—
158C4⋊C4—92no—
159C4⋊C4—92no—
160C4⋊C4C2×D911yescharacteristic
161C22⋊C4—91no—
162C22⋊C4—91no—
163C22⋊C4—92no—
164C22⋊C4—92no—
165C22⋊C4—92no—
166C22⋊C4C2×D912yes—
167C42—91no—
168C42—91no—
169C22×C3—21no—
170C22×C3—22no—
171C22×C3—22no—
172C22×C3—21no—
173C22×C3—22no—
174C22×C3—21no—
175C22×C3C23×D311yescharacteristic
176C2×D3—62no—
177C2×D3—62no—
178C2×D3—62no—
179C3×C4—21no—
180C3×C4—22no—
181C3×C4#23712yes—
182Dic3—61no—
183Dic3—61no—
184Dic3—61no—
185Dic3—62no—
186Dic3—61no—
187C9#54011yescharacteristic, complemented, Sylow
188C23—92no—
189C23C22×D911yescharacteristic
190C23C22×D912yes—
191D4—22no—
192D4—42no—
193D4—182no—
194D4—362no—
195D4—362no—
196D4—362no—
197C2×C4—22no—
198C2×C4—91no—
199C2×C4—91no—
200C2×C4—91no—
201C2×C4—91no—
202C2×C4—92no—
203C2×C4—362no—
204C2×C4—362no—
205C2×C4C22×D911yescharacteristic
206C2×C4C22×D912yes—
207C2×C4C22×D911yescharacteristic
208C2×C3—41no—
209C2×C3—42no—
210C2×C3C2×#23711yescharacteristic
211C2×C3#100411yescharacteristic
212C2×C3#100911yescharacteristic
213D3—122no—
214C22—21no—
215C22—22no—
216C22—21no—
217C22—22no—
218C22—22no—
219C22—21no—
220C22—182no—
221C22—182no—
222C22—182no—
223C22C23×D911yescharacteristic
224C4—21no—
225C4—22no—
226C4—181no—
227C4—181no—
228C4—181no—
229C4—182no—
230C4—181no—
231C4#369912yes—
232C3#561911yescharacteristic
233C2—41no—
234C2—42no—
235C2—362no—
236C2C2×#369911yescharacteristic
237C2#6371011yescharacteristic
238C2#6371511yescharacteristic
subgroupG/H
centerC22C23×D9
commutator, cosocleC22×C9C24
Frattini, socleC22×C3C23×D3
FittingC9×#121C2
hypercenter#121D9
hypocenterC9#540
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. #63502C2semidirect
3. #63501C2semidirect
4. #63500C2semidirect
5. D4×Dic9C2semidirect
6. #63492C2semidirect
7. #63488C2semidirect
8. #63455C2semidirect
9. #63452C2semidirect
10. #63448C2semidirect
11. #63445C2semidirect
116. #121D9semidirect
187. C9#540semidirect

Maximal subgroups

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

subgroupindex
C9×#1212
#635022
#635012
#635002
D4×Dic92
#634922
#634882
#634552
#634522
#634482
#634452
#56193