Finite Group Databasebeta

← #10589741

Subgroups — #10589741

Order 576.

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

normal subgroups only

This group has 321 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×C22⋊Q8C211yesmaximal, characteristic, complemented
2#63508C211yesmaximal, characteristic, complemented
3Q8×Dic9C211yesmaximal, characteristic, complemented
4C4×#611C211yesmaximal, characteristic
5#63488C211yesmaximal, characteristic, complemented
6#63454C212yesmaximal, complemented
7C4⋊C4×D9C211yesmaximal, characteristic, complemented
8#63452C211yesmaximal, characteristic, complemented
9#63451C212yesmaximal, complemented
10#63445C212yesmaximal
11#63441C212yesmaximal, complemented
12#5639—31nomaximal
13C2×Q8×C9C2211yescharacteristic
14C22×C4×C9C2211yescharacteristic
15C2×#611C2211yescharacteristic
16C2×C4×D9C2211yescharacteristic
17C2×C4×D9C2212yes—
18C9×C4⋊C4C2212yes—
19C9×C4⋊C4C2211yescharacteristic
20C9×C22⋊C4C2212yes—
21#3676C2212yes—
22#3676C2211yescharacteristic
23(C2×D9)⋊C4C2211yescharacteristic
24(C2×D9)⋊C4C2212yes—
25C4⋊Dic9C2211yescharacteristic
26C4⋊Dic9C2211yescharacteristic
27C4⋊Dic9C2212yes—
28C4⋊Dic9C2212yes—
29Dic9⋊C4C2211yescharacteristic
30Dic9⋊C4C2212yes—
31Dic9⋊C4C2212yes—
32Dic9⋊C4C2212yes—
33C4×Dic9C2212yes—
34C4×Dic9C2211yescharacteristic
35C4×Dic9C2212yes—
36C3×C22⋊Q8D311yescharacteristic
37#946—31no—
38Q8×Dic3—31no—
39C4×#67—31no—
40#926—31no—
41#892—32no—
42D3×C4⋊C4—31no—
43#890—31no—
44#889—32no—
45#883—32no—
46#879—32no—
47C23×C9C2311yescharacteristic
48C22×D9C2311yescharacteristic
49Q8×C9—22no—
50C2×C4×C9—22no—
51C2×C4×C9C2311yescharacteristic
52C2×C4×C9C2312yes—
53C2×C4×C9C2312yes—
54C2×C4×C9C2311yescharacteristic
55#611—22no—
56C2×Dic9C2311yescharacteristic
57C2×Dic9C2312yes—
58C2×Dic9C2311yescharacteristic
59C2×Dic9C2312yes—
60C2×Dic9C2311yescharacteristic
61C4×D9—24no—
62C4×D9—22no—
63#552—91noSylow
64C2×C3×Q8C2×D311yescharacteristic
65C22×C3×C4C2×D311yescharacteristic
66C2×#67—31no—
67C2×C4×D3—32no—
68C2×C4×D3—31no—
69C3×C4⋊C4C2×D312yes—
70C3×C4⋊C4C2×D311yescharacteristic
71C3×C22⋊C4C2×D312yes—
72#217—31no—
73#217—32no—
74(C2×D3)⋊C4—32no—
75(C2×D3)⋊C4—31no—
76C4⋊Dic3—32no—
77C4⋊Dic3—32no—
78C4⋊Dic3—31no—
79C4⋊Dic3—31no—
80Dic3⋊C4—32no—
81Dic3⋊C4—32no—
82Dic3⋊C4—32no—
83Dic3⋊C4—31no—
84C4×Dic3—32no—
85C4×Dic3—32no—
86C4×Dic3—31no—
87C22×C9—21no—
88C22×C9—21no—
89C22×C9—21no—
90C22×C9C2411yescharacteristic
91C2×D9—21no—
92C2×D9—21no—
93C2×D9#4112yes—
94C4×C9—22no—
95C4×C9—21no—
96C4×C9—22no—
97C4×C9#4112yes—
98Dic9—21no—
99Dic9—22no—
100Dic9—21no—
101Dic9—21no—
102Dic9—22no—
103#126—92no—
104#125—92no—
105#125—91no—
106#123—92no—
107C22⋊Q8—91no—
108C22⋊Q8D911yescharacteristic, complemented
109C4×Q8—91no—
110C4×D4—91no—
111#117—91no—
112#117—92no—
113C2×C4⋊C4—91no—
114C23×C3C22×D311yescharacteristic
115C22×D3—31no—
116C3×Q8—22no—
117C2×C3×C4—22no—
118C2×C3×C4C22×D311yescharacteristic
119C2×C3×C4C22×D312yes—
120C2×C3×C4C22×D312yes—
121C2×C3×C4C22×D311yescharacteristic
122#67—62no—
123C2×Dic3—31no—
124C2×Dic3—32no—
125C2×Dic3—31no—
126C2×Dic3—32no—
127C2×Dic3—31no—
128C4×D3—64no—
129C4×D3—62no—
130C2×C9—41no—
131C2×C9C2×#4111yescharacteristic
132C2×C9C2×#4111yescharacteristic
133C2×C92−1+411yescharacteristic
134D9—22no—
135C2×Q8C2×D911yescharacteristic
136C2×D4—91no—
137C22×C4—92no—
138C22×C4—91no—
139C22×C4C2×D911yescharacteristic
140C4⋊C4—91no—
141C4⋊C4—92no—
142C4⋊C4—91no—
143C4⋊C4—92no—
144C4⋊C4—91no—
145C4⋊C4—92no—
146C4⋊C4—92no—
147C4⋊C4—92no—
148C4⋊C4C2×D911yescharacteristic
149C4⋊C4C2×D912yes—
150C22⋊C4—92no—
151C22⋊C4—91no—
152C22⋊C4—91no—
153C22⋊C4—92no—
154C22⋊C4C2×D912yes—
155C42—92no—
156C42—92no—
157C42—91no—
158C22×C3—21no—
159C22×C3—21no—
160C22×C3—21no—
161C22×C3C23×D311yescharacteristic
162C2×D3—32no—
163C2×D3—61no—
164C2×D3—61no—
165C3×C4—21no—
166C3×C4—22no—
167C3×C4—22no—
168C3×C4#23712yes—
169Dic3—61no—
170Dic3—61no—
171Dic3—62no—
172Dic3—62no—
173Dic3—61no—
174C9#55211yescharacteristic, complemented, Sylow
175C23—91no—
176C23C22×D911yescharacteristic
177Q8—22no—
178D4—182no—
179C2×C4—22no—
180C2×C4—91no—
181C2×C4—92no—
182C2×C4—92no—
183C2×C4—91no—
184C2×C4—91no—
185C2×C4—184no—
186C2×C4—182no—
187C2×C4C22×D911yescharacteristic
188C2×C4C22×D912yes—
189C2×C4C22×D912yes—
190C2×C4C22×D911yescharacteristic
191C2×C3—41no—
192C2×C3C2×#23711yescharacteristic
193C2×C3#100711yescharacteristic
194C2×C3D3×#4111yescharacteristic
195D3—62no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—92no—
200C22—181no—
201C22—181no—
202C22C23×D911yescharacteristic
203C4—21no—
204C4—22no—
205C4—22no—
206C4—181no—
207C4—181no—
208C4—182no—
209C4—182no—
210C4—181no—
211C4#369912yes—
212C3#563911yescharacteristic
213C2—41no—
214C2—182no—
215C2C2×#369911yescharacteristic
216C2#6371311yescharacteristic
217C2#41×D911yescharacteristic
subgroupG/H
centerC22C23×D9
commutator, cosocleC22×C9C24
Frattini, socleC22×C3C23×D3
FittingC9×C22⋊Q8C2
prime radicalC2×#611C22
hypercenterC22⋊Q8D9
hypocenterC9#552
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. #63488C2semidirect
6. #63454C2semidirect
7. C4⋊C4×D9C2semidirect
8. #63452C2semidirect
9. #63451C2semidirect
11. #63441C2semidirect
108. C22⋊Q8D9semidirect
174. C9#552semidirect

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
#634882
#634542
C4⋊C4×D92
#634522
#634512
#634452
#634412
#56393