Finite Group Databasebeta

← #10589683

Subgroups — #10589683

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
1C4×Q8×C9C211yesmaximal, characteristic, complemented
2#63508C211yesmaximal, characteristic, complemented
3#63508C211yesmaximal, characteristic, complemented
4#63456C212yesmaximal
5#63454C211yesmaximal, characteristic, complemented
6C4⋊C4×D9C211yesmaximal, characteristic, complemented
7#63451C212yesmaximal, complemented
8#63438C212yesmaximal, complemented
9C4×D36C211yesmaximal, characteristic
10#63434C212yesmaximal, complemented
11#63432C211yesmaximal, characteristic, complemented
12#5581—31nomaximal
13C2×Q8×C9C2211yescharacteristic
14C2×D36C2211yescharacteristic
15C2×C4×D9C2211yescharacteristic
16C2×C4×D9C2211yescharacteristic
17C2×C4×D9C2212yes—
18C9×C4⋊C4C2212yes—
19C9×C4⋊C4C2211yescharacteristic
20C42×C9C2211yescharacteristic
21C42×C9C2212yes—
22(C2×D9)⋊C4C2212yes—
23(C2×D9)⋊C4C2212yes—
24(C2×D9)⋊C4C2211yescharacteristic
25(C2×D9)⋊C4C2211yescharacteristic
26(C2×D9)⋊C4C2212yes—
27C4⋊Dic9C2211yescharacteristic
28C4⋊Dic9C2211yescharacteristic
29C4⋊Dic9C2211yescharacteristic
30Dic9⋊C4C2212yes—
31Dic9⋊C4C2212yes—
32Dic9⋊C4C2212yes—
33Dic9⋊C4C2212yes—
34Dic9⋊C4C2212yes—
35C4×Dic9C2212yes—
36C3×C4×Q8D311yescharacteristic
37#946—31no—
38#946—31no—
39#894—32no—
40#892—31no—
41D3×C4⋊C4—31no—
42#889—32no—
43#876—32no—
44C4×D12—31no—
45#872—32no—
46#870—31no—
47C22×D9C2311yescharacteristic
48C22×D9C2311yescharacteristic
49Q8×C9—22no—
50C2×C4×C9C2312yes—
51C2×C4×C9C2311yescharacteristic
52C2×C4×C9C2311yescharacteristic
53C2×C4×C9C2312yes—
54C2×C4×C9C2311yescharacteristic
55C2×Dic9C2311yescharacteristic
56C2×Dic9C2312yes—
57C2×Dic9C2312yes—
58C2×Dic9C2311yescharacteristic
59D36—22no—
60C4×D9—24no—
61C4×D9—22no—
62C4×D9—22no—
63#552—91noSylow
64C2×C3×Q8C2×D311yescharacteristic
65C2×D12—31no—
66C2×C4×D3—32no—
67C2×C4×D3—31no—
68C2×C4×D3—31no—
69C3×C4⋊C4C2×D312yes—
70C3×C4⋊C4C2×D311yescharacteristic
71C3×C42C2×D311yescharacteristic
72C3×C42C2×D312yes—
73(C2×D3)⋊C4—31no—
74(C2×D3)⋊C4—32no—
75(C2×D3)⋊C4—31no—
76(C2×D3)⋊C4—32no—
77(C2×D3)⋊C4—32no—
78C4⋊Dic3—31no—
79C4⋊Dic3—31no—
80C4⋊Dic3—31no—
81Dic3⋊C4—32no—
82Dic3⋊C4—32no—
83Dic3⋊C4—32no—
84Dic3⋊C4—32no—
85Dic3⋊C4—32no—
86C4×Dic3—32no—
87C22×C9C2411yescharacteristic
88C2×D9—21no—
89C2×D9—21no—
90C2×D9—21no—
91C2×D9—21no—
92C2×D9—21no—
93C2×D9#4112yes—
94C4×C9—22no—
95C4×C9—21no—
96C4×C9—22no—
97C4×C9—21no—
98C4×C9#4112yes—
99Dic9—21no—
100Dic9—21no—
101Dic9—22no—
102Dic9—22no—
103#126—92no—
104#125—92no—
105#125—91no—
106#123—92no—
107C22⋊Q8—91no—
108C22⋊Q8—91no—
109C4×Q8D911yescharacteristic, complemented
110C4×D4—91no—
111#117—91no—
112#117—92no—
113C2×C4⋊C4—91no—
114C22×D3—31no—
115C22×D3—31no—
116C3×Q8—22no—
117C2×C3×C4C22×D312yes—
118C2×C3×C4C22×D311yescharacteristic
119C2×C3×C4C22×D311yescharacteristic
120C2×C3×C4C22×D312yes—
121C2×C3×C4C22×D311yescharacteristic
122C2×Dic3—31no—
123C2×Dic3—32no—
124C2×Dic3—31no—
125C2×Dic3—32no—
126D12—62no—
127C4×D3—64no—
128C4×D3—62no—
129C4×D3—62no—
130C2×C9C2×#4111yescharacteristic
131C2×C9C2×#4111yescharacteristic
132C2×C92−1+411yescharacteristic
133D9—22no—
134D9—41no—
135C2×Q8C2×D911yescharacteristic
136C2×D4—91no—
137C22×C4—92no—
138C22×C4—91no—
139C22×C4—91no—
140C4⋊C4—91no—
141C4⋊C4—92no—
142C4⋊C4—92no—
143C4⋊C4—91no—
144C4⋊C4—92no—
145C4⋊C4—91no—
146C4⋊C4—92no—
147C4⋊C4—92no—
148C4⋊C4C2×D911yescharacteristic
149C4⋊C4C2×D912yes—
150C22⋊C4—91no—
151C22⋊C4—92no—
152C22⋊C4—92no—
153C22⋊C4—91no—
154C22⋊C4—92no—
155C42—92no—
156C42C2×D912yes—
157C42C2×D911yescharacteristic
158C22×C3C23×D311yescharacteristic
159C2×D3—32no—
160C2×D3—61no—
161C2×D3—61no—
162C2×D3—61no—
163C2×D3—61no—
164C2×D3—61no—
165C3×C4—22no—
166C3×C4—21no—
167C3×C4—22no—
168C3×C4—21no—
169C3×C4#23512yes—
170Dic3—61no—
171Dic3—61no—
172Dic3—62no—
173Dic3—62no—
174C9#55211yescharacteristic, complemented, Sylow
175C23—91no—
176C23—91no—
177Q8—22no—
178D4—182no—
179C2×C4—91no—
180C2×C4—92no—
181C2×C4—91no—
182C2×C4—92no—
183C2×C4—184no—
184C2×C4—182no—
185C2×C4—182no—
186C2×C4C22×D912yes—
187C2×C4C22×D911yescharacteristic
188C2×C4C22×D912yes—
189C2×C4C22×D911yescharacteristic
190C2×C4C22×D911yescharacteristic
191C2×C3C2×#23511yescharacteristic
192C2×C3#100711yescharacteristic
193C2×C3D3×#4111yescharacteristic
194D3—62no—
195D3—121no—
196C22—92no—
197C22—181no—
198C22—181no—
199C22—181no—
200C22—181no—
201C22—181no—
202C22C23×D911yescharacteristic
203C4—22no—
204C4—21no—
205C4—22no—
206C4—21no—
207C4—181no—
208C4—181no—
209C4—182no—
210C4—182no—
211C4#369712yes—
212C3#558111yescharacteristic
213C2—182no—
214C2—361no—
215C2C2×#369711yescharacteristic
216C2#6371311yescharacteristic
217C2#41×D911yescharacteristic
subgroupG/H
centerC22C23×D9
commutator, cosocleC22×C9C24
Frattini, socleC22×C3C23×D3
FittingC4×Q8×C9C2
prime radicalC2×D36C22
hypercenterC4×Q8D9
hypocenterC9#552
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C4×Q8×C9C2semidirect
2. #63508C2semidirect
3. #63508C2semidirect
5. #63454C2semidirect
6. C4⋊C4×D9C2semidirect
7. #63451C2semidirect
8. #63438C2semidirect
10. #63434C2semidirect
11. #63432C2semidirect
109. C4×Q8D9semidirect
174. C9#552semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×C92
#635082
#634562
#634542
C4⋊C4×D92
#634512
#634382
C4×D362
#634342
#634322
#55813