Finite Group Databasebeta

← #10589737

Subgroups — #10589737

Order 576.

342 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 131 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×C22⋊Q8C211yesmaximal, characteristic, complemented
2#63508C211yesmaximal, characteristic, complemented
3Q8×Dic9C211yesmaximal, characteristic, complemented
4C4×#611C211yesmaximal, characteristic
5C2×C4⋊Dic9C211yesmaximal, characteristic, complemented
6#63459C212yesmaximal, complemented
7C4⋊C4×D9C211yesmaximal, characteristic, complemented
8#63451C212yesmaximal, complemented
9#63450C211yesmaximal, characteristic, complemented
10#611⋊C4C212yesmaximal
11#63440C212yesmaximal, complemented
12#5635—31nomaximal
13C2×Q8×C9C2211yescharacteristic
14C22×C4×C9C2211yescharacteristic
15C2×#611C2211yescharacteristic
16C22×Dic9C2212yes—
17C2×C4×D9C2211yescharacteristic
18C2×C4×D9C2212yes—
19C2×Dic18C2212yes—
20C9×C4⋊C4C2211yescharacteristic
21C9×C4⋊C4C2212yes—
22C9×C22⋊C4C2212yes—
23#3676C2211yescharacteristic
24(C2×D9)⋊C4C2211yescharacteristic
25(C2×D9)⋊C4C2212yes—
26C4⋊Dic9C2212yes—
27C4⋊Dic9C2211yescharacteristic
28C4⋊Dic9C2211yescharacteristic
29C4⋊Dic9C2212yes—
30Dic9⋊C4C2211yescharacteristic
31Dic9⋊C4C2212yes—
32Dic9⋊C4C2212yes—
33Dic9⋊C4C2212yes—
34C4×Dic9C2211yescharacteristic
35C4×Dic9C2212yes—
36C3×C22⋊Q8D311yescharacteristic
37#946—31no—
38Q8×Dic3—31no—
39C4×#67—31no—
40C2×C4⋊Dic3—31no—
41#897—32no—
42D3×C4⋊C4—31no—
43#889—32no—
44#888—31no—
45#67⋊C4—32no—
46#878—32no—
47C23×C9C2311yescharacteristic
48C22×D9C2311yescharacteristic
49Q8×C9—22no—
50C2×C4×C9—22no—
51C2×C4×C9C2312yes—
52C2×C4×C9C2312yes—
53C2×C4×C9C2311yescharacteristic
54C2×C4×C9C2311yescharacteristic
55#611Q814yescomplemented
56C2×Dic9—42no—
57C2×Dic9C2312yes—
58C2×Dic9C2311yescharacteristic
59C2×Dic9C2311yescharacteristic
60C2×Dic9C2312yes—
61C2×Dic9C2311yescharacteristic
62C4×D9—22no—
63C4×D9—42no—
64Dic18—42no—
65#554—91noSylow
66C2×C3×Q8C2×D311yescharacteristic
67C22×C3×C4C2×D311yescharacteristic
68C2×#67—31no—
69C22×Dic3—32no—
70C2×C4×D3—31no—
71C2×C4×D3—32no—
72C2×Dic6—32no—
73C3×C4⋊C4C2×D311yescharacteristic
74C3×C4⋊C4C2×D312yes—
75C3×C22⋊C4C2×D312yes—
76#217—31no—
77(C2×D3)⋊C4—32no—
78(C2×D3)⋊C4—31no—
79C4⋊Dic3—32no—
80C4⋊Dic3—31no—
81C4⋊Dic3—32no—
82C4⋊Dic3—31no—
83Dic3⋊C4—32no—
84Dic3⋊C4—32no—
85Dic3⋊C4—32no—
86Dic3⋊C4—31no—
87C4×Dic3—32no—
88C4×Dic3—31no—
89C22×C9—21no—
90C22×C9—21no—
91C22×C9C2×Q812yes—
92C22×C9C2411yescharacteristic
93C2×D9—21no—
94C2×D9—21no—
95C2×D9C2×Q812yes—
96C4×C9—21no—
97C4×C9—22no—
98C4×C9—22no—
99C4×C9#4112yes—
100Dic9—21no—
101Dic9—21no—
102Dic9—22no—
103Dic9—22no—
104Dic9C2×Q812yes—
105C4⋊Q8—91no—
106#125—92no—
107C22⋊Q8—91no—
108C22⋊Q8—92no—
109C22⋊Q8—92no—
110C22⋊Q8D911yescharacteristic, complemented
111C4×Q8—91no—
112C4×D4—91no—
113C4×D4—92no—
114C2×C4⋊C4—91no—
115C2×C4⋊C4—91no—
116C23×C3C22×D311yescharacteristic
117C22×D3—31no—
118C3×Q8—22no—
119C2×C3×C4—22no—
120C2×C3×C4C22×D312yes—
121C2×C3×C4C22×D312yes—
122C2×C3×C4C22×D311yescharacteristic
123C2×C3×C4C22×D311yescharacteristic
124#67—34no—
125C2×Dic3—31no—
126C2×Dic3—32no—
127C2×Dic3—31no—
128C2×Dic3—31no—
129C2×Dic3—32no—
130C2×Dic3—122no—
131C4×D3—62no—
132C4×D3—122no—
133Dic6—122no—
134C2×C9—22no—
135C2×C9C22×Q811yescharacteristic
136C2×C9C2×#4111yescharacteristic
137C2×C9H(5,𝔽2)11yescharacteristic
138D9—22no—
139C2×Q8—92no—
140C2×Q8C2×D911yescharacteristic
141C2×D4—91no—
142C22×C4—92no—
143C22×C4—91no—
144C22×C4—92no—
145C22×C4C2×D911yescharacteristic
146C4⋊C4—91no—
147C4⋊C4—92no—
148C4⋊C4—92no—
149C4⋊C4—91no—
150C4⋊C4—91no—
151C4⋊C4—92no—
152C4⋊C4—92no—
153C4⋊C4—92no—
154C4⋊C4C2×D911yescharacteristic
155C4⋊C4C2×D912yes—
156C22⋊C4—91no—
157C22⋊C4—91no—
158C22⋊C4—92no—
159C22⋊C4C2×D912yes—
160C42—92no—
161C42—91no—
162C22×C3—21no—
163C22×C3—21no—
164C22×C3D3×Q812yes—
165C22×C3C23×D311yescharacteristic
166C2×D3—32no—
167C2×D3—61no—
168C2×D3—61no—
169C3×C4—21no—
170C3×C4—22no—
171C3×C4—22no—
172C3×C4#23712yes—
173Dic3—32no—
174Dic3—61no—
175Dic3—61no—
176Dic3—62no—
177Dic3—62no—
178C9#55411yescharacteristic, complemented, Sylow
179C23—91no—
180C23C22×D911yescharacteristic
181Q8—22no—
182Q8—362no—
183D4—94no—
184C2×C4—22no—
185C2×C4—91no—
186C2×C4—91no—
187C2×C4—92no—
188C2×C4—91no—
189C2×C4—92no—
190C2×C4—182no—
191C2×C4—362no—
192C2×C4—362no—
193C2×C4C22×D912yes—
194C2×C4C22×D911yescharacteristic
195C2×C4C22×D912yes—
196C2×C4C22×D911yescharacteristic
197C2×C3—22no—
198C2×C3C2×#23711yescharacteristic
199C2×C3C2×D3×Q811yescharacteristic
200C2×C3#100911yescharacteristic
201D3—62no—
202C22—21no—
203C22—21no—
204C22—92no—
205C22—181no—
206C22—181no—
207C22Q8×D912yes—
208C22C23×D911yescharacteristic
209C4—21no—
210C4—22no—
211C4—22no—
212C4—92no—
213C4—181no—
214C4—182no—
215C4—182no—
216C4—181no—
217C4#369912yes—
218C3#563511yescharacteristic
219C2—22no—
220C2—182no—
221C2C2×#369911yescharacteristic
222C2C2×Q8×D911yescharacteristic
223C2#6371511yescharacteristic
subgroupG/H
centerC22C23×D9
commutator, cosocleC22×C9C24
Frattini, socleC22×C3C23×D3
FittingC9×C22⋊Q8C2
prime radicalC2×#611C22
hypercenterC22⋊Q8D9
hypocenterC9#554
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C9×C22⋊Q8C2semidirect
2. #63508C2semidirect
3. Q8×Dic9C2semidirect
5. C2×C4⋊Dic9C2semidirect
6. #63459C2semidirect
7. C4⋊C4×D9C2semidirect
8. #63451C2semidirect
9. #63450C2semidirect
11. #63440C2semidirect
55. #611Q8semidirect
110. C22⋊Q8D9semidirect
178. C9#554semidirect

Maximal subgroups

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

subgroupindex
C9×C22⋊Q82
#635082
Q8×Dic92
C4×#6112
C2×C4⋊Dic92
#634592
C4⋊C4×D92
#634512
#634502
#611⋊C42
#634402
#56353