Finite Group Databasebeta

← #10595416

Subgroups — #10595416

Order 576.

426 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 120 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
1#64153C211yesmaximal, characteristic, complemented
2C3×#946C212yesmaximal, complemented
3#63913C211yesmaximal, characteristic
4#63905C212yesmaximal
5#63862C212yesmaximal, complemented
6#63858C212yesmaximal, complemented
7#63851C212yesmaximal, complemented
8#63846C212yesmaximal, complemented
9#63840C211yesmaximal, characteristic, complemented
10#5646—32nomaximal
11C2×C32×Q8C2211yescharacteristic
12C2×#634C2211yescharacteristic
13C2×C3×C4×D3C2212yes—
14C2×#627C2211yescharacteristic
15C2×#625C2211yescharacteristic
16#3750C2212yes—
17#3750C2212yes—
18C4×C3⋊Dic3C2211yescharacteristic
19C3×(C2×D3)⋊C4C2212yes—
20C3×(C2×D3)⋊C4C2212yes—
21C3×C4⋊Dic3C2212yes—
22C3×Dic3⋊C4C2212yes—
23C3×Dic3⋊C4C2212yes—
24Dic3⋊Dic3C2212yes—
25Dic3⋊Dic3C2212yes—
26Dic3⋊Dic3C2212yes—
27(C2×D3)⋊Dic3C2212yes—
28(C2×D3)⋊Dic3C2212yes—
29(C2×D3)⋊Dic3C2212yes—
30Dic32C2211yescharacteristic
31Dic32C2211yescharacteristic
32C3×C22⋊Q8—32no—
33#946D312yescomplemented
34#944—32no—
35#944—61no—
36#929—32no—
37#924—32no—
38#898—32no—
39#898—32no—
40#896—32no—
41#890—32no—
42#889—32no—
43#888—32no—
44#885—32no—
45#883—32no—
46#879—32no—
47#879—32no—
48C22×C3×D3C2312yes—
49C32×Q8—41no—
50C2×C32×C4C2311yescharacteristic
51C2×C32×C4C2312yes—
52C2×C3⋊Dic3C2311yescharacteristic
53C2×C3⋊Dic3C2311yescharacteristic
54C2×C3⋊Dic3C2312yes—
55#634—41no—
56C2×C3×Dic3C2312yes—
57C2×C3×Dic3C2312yes—
58C2×C3×Dic3C2312yes—
59C3×C4×D3—42no—
60#627—41no—
61#625—41no—
62#563—91noSylow
63C2×C3×Q8—21no—
64C2×C3×Q8C2×D312yes—
65C22×C3×C4—32no—
66C2×#67—32no—
67C2×C4×D3C2×D312yes—
68C2×Dic6—32no—
69C2×Dic6—32no—
70C2×Dic6—61no—
71C3×C4⋊C4—32no—
72C3×C4⋊C4—32no—
73C3×C4⋊C4—32no—
74C3×C22⋊C4—32no—
75C3×C22⋊C4—32no—
76#217—32no—
77#217—32no—
78#217—32no—
79(C2×D3)⋊C4—32no—
80(C2×D3)⋊C4—32no—
81(C2×D3)⋊C4—32no—
82(C2×D3)⋊C4C2×D312yes—
83(C2×D3)⋊C4C2×D312yes—
84C4⋊Dic3—32no—
85C4⋊Dic3—32no—
86C4⋊Dic3—32no—
87C4⋊Dic3C2×D312yes—
88Dic3⋊C4—32no—
89Dic3⋊C4—32no—
90Dic3⋊C4—32no—
91Dic3⋊C4—32no—
92Dic3⋊C4—32no—
93Dic3⋊C4—32no—
94Dic3⋊C4—32no—
95Dic3⋊C4—62no—
96Dic3⋊C4—62no—
97Dic3⋊C4C2×D312yes—
98Dic3⋊C4C2×D312yes—
99C4×Dic3—32no—
100C4×Dic3—32no—
101C4×Dic3—32no—
102C4×Dic3—61no—
103C22×C32C2411yescharacteristic
104C2×C3×D3—22no—
105C2×C3×D3—22no—
106C2×C3×D3—22no—
107C32×C4—21no—
108C32×C4—22no—
109C3⋊Dic3—22no—
110C3⋊Dic3—21no—
111C3⋊Dic3—21no—
112C3×Dic3—22no—
113C3×Dic3—22no—
114C3×Dic3—22no—
115C4⋊Q8—91no—
116C4⋊Q8—91no—
117#126—92no—
118#126—92no—
119#125—92no—
120#124—91no—
121#123—92no—
122C22⋊Q8—32no—
123C22⋊Q8—92no—
124C23×C3—32no—
125C22×D3C22×D312yes—
126C3×Q8—42no—
127C3×Q8—42no—
128C2×C3×C4—21no—
129C2×C3×C4—22no—
130C2×C3×C4—32no—
131C2×C3×C4—32no—
132C2×C3×C4—32no—
133C2×C3×C4—122no—
134C2×C3×C4C22×D312yes—
135C2×C3×C4C22×D312yes—
136C2×C3×C4C22×D312yes—
137#67—122no—
138C2×Dic3—32no—
139C2×Dic3—32no—
140C2×Dic3—32no—
141C2×Dic3—32no—
142C2×Dic3—62no—
143C2×Dic3—61no—
144C2×Dic3—61no—
145C2×Dic3C22×D312yes—
146C2×Dic3C22×D312yes—
147C2×Dic3C22×D312yes—
148C4×D3—42no—
149Dic6—122no—
150Dic6—122no—
151Dic6—122no—
152C2×C32H(5,𝔽2)11yescharacteristic
153C2×C322−1+412yes—
154C3×D3—42no—
155C2×Q8—91no—
156C2×Q8—91no—
157C2×Q8D3211yescharacteristic
158C2×D4—91no—
159C22×C4—32no—
160C4⋊C4—32no—
161C4⋊C4—32no—
162C4⋊C4—32no—
163C4⋊C4—92no—
164C4⋊C4—92no—
165C4⋊C4—92no—
166C4⋊C4—92no—
167C4⋊C4—92no—
168C22⋊C4—32no—
169C22⋊C4—32no—
170C22⋊C4—92no—
171C22⋊C4—92no—
172C22⋊C4—92no—
173C42—91no—
174C42—91no—
175C42—91no—
176C22×C3—21no—
177C22×C3—62no—
178C22×C3—62no—
179C22×C3—62no—
180C22×C3C23×D312yes—
181C2×D3—22no—
182C2×D3—22no—
183C2×D3—22no—
184C3×C4—22no—
185C3×C4—22no—
186C3×C4—22no—
187C3×C4—22no—
188C3×C4—42no—
189C3×C4—62no—
190C3×C4—62no—
191C3×C4—62no—
192Dic3—22no—
193Dic3—22no—
194Dic3—22no—
195Dic3—64no—
196Dic3—62no—
197Dic3—62no—
198Dic3—62no—
199Dic3—62no—
200Dic3—121no—
201Dic3—121no—
202C32#56311yescharacteristic, complemented, Sylow
203C23—32no—
204Q8—41no—
205Q8—361no—
206Q8—361no—
207D4—361no—
208C2×C4—32no—
209C2×C4—32no—
210C2×C4—32no—
211C2×C4—92no—
212C2×C4—91no—
213C2×C4—91no—
214C2×C4—122no—
215C2×C4C2×D3211yescharacteristic
216C2×C4C2×D3212yes—
217C2×C3—22no—
218C2×C3—21no—
219C2×C3—122no—
220C2×C3#100412yes—
221C2×C3#100712yes—
222C2×C3#101012yes—
223D3—42no—
224C22—62no—
225C22—62no—
226C22—62no—
227C22C22×D3211yescharacteristic
228C4—22no—
229C4—21no—
230C4—62no—
231C4—62no—
232C4—62no—
233C4—182no—
234C4—181no—
235C4—181no—
236C3—21no—
237C3#564612yescomplemented
238C2—122no—
239C2#6431512yes—
240C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×Q8C22
prime radicalC2×#625C22
hypercenterC2×Q8D32
hypocenterC32#563
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #64153C2semidirect
2. C3×#946C2semidirect
5. #63862C2semidirect
6. #63858C2semidirect
7. #63851C2semidirect
8. #63846C2semidirect
9. #63840C2semidirect
33. #946D3semidirect
202. C32#563semidirect
237. C3#5646semidirect

Maximal subgroups

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

subgroupindex
#641532
C3×#9462
#639132
#639052
#638622
#638582
#638512
#638462
#638402
#56463