Finite Group Databasebeta

← #10591175

Subgroups — #10591175

Order 576.

285 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 76 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#64133C211yesmaximal, characteristic, complemented
2C3×#930C211yesmaximal, characteristic
3C22×C3×C3⋊C8C211yesmaximal, characteristic, complemented
4C2×#3716C211yesmaximal, characteristic, complemented
5#63573C211yesmaximal, characteristic, complemented
6#63567C211yesmaximal, characteristic, complemented
7#63563C211yesmaximal, characteristic, complemented
8#5114—31nomaximal
9#5041—31nomaximal
10C22×C32×C4C2211yescharacteristic
11C2×#634C2211yescharacteristic
12C2×C3×#67—21no—
13C2×C3×D12C2211yescharacteristic
14#3757—21no—
15C4⋊(C3⋊Dic3)C2211yescharacteristic
16#3750—21no—
17C3×(C2×D3)⋊C4—21no—
18C3×C4⋊Dic3C2211yescharacteristic
19C2×C3×C3⋊C8—22no—
20C2×C3×C3⋊C8C2211yescharacteristic
21C2×C3×C3⋊C8C2211yescharacteristic
22#3716—22no—
23C22×C3×C8—31no—
24C3×#121—31no—
25C2×#214—31no—
26#930D311yescharacteristic
27#924—31no—
28#924—31no—
29#924—61no—
30C22×C3⋊C8D311yescharacteristic, complemented
31C2×#204—31no—
32#832—31no—
33#821—31no—
34#817—31no—
35#816—31no—
36#810—31no—
37#808—31no—
38C23×C32D411yescharacteristic
39C22×C3×D3—21no—
40C2×C32×C4—21no—
41C2×C32×C4—21no—
42C2×C32×C4D411yescharacteristic
43C2×C32×C4C2311yescharacteristic
44C2×C3⋊Dic3—21no—
45C2×C3⋊Dic3—21no—
46#634—21no—
47#634—21no—
48C3×#67—42no—
49C2×C3×Dic3—21no—
50C3×D12—21no—
51C3×D12—21no—
52C3×C3⋊C8—21no—
53C3×C3⋊C8D412yescomplemented
54#465—91noSylow
55C2×C3×D4—31no—
56C2×C3×D4—61no—
57C22×C3×C4—21no—
58C22×C3×C4C2×D311yescharacteristic
59C22×C3×C4C2×D311yescharacteristic
60C2×#67—21no—
61C2×D12C2×D311yescharacteristic
62C2×Dic6—31no—
63C2×Dic6—31no—
64C2×Dic6—61no—
65C2×C3×C8—31no—
66C2×C3×C8—31no—
67C2×C3×C8—62no—
68C3×C4⋊C4—31no—
69C3×C22⋊C4—61no—
70#217—62no—
71#217—61no—
72#217—61no—
73#214—62no—
74(C2×D3)⋊C4—21no—
75C4⋊Dic3—31no—
76C4⋊Dic3—31no—
77C4⋊Dic3—61no—
78C4⋊Dic3C2×D311yescharacteristic
79Dic3⋊C4—62no—
80Dic3⋊C4—61no—
81Dic3⋊C4—61no—
82C2×C3⋊C8—22no—
83C2×C3⋊C8C2×D311yescharacteristic
84C2×C3⋊C8C2×D311yescharacteristic
85#204—62no—
86C22×C32—21no—
87C22×C32—21no—
88C22×C32SD1612yescomplemented
89C22×C32C2×D411yescharacteristic
90C2×C3×D3—41no—
91C2×C3×D3—41no—
92C2×C3×D3—41no—
93C32×C4—21no—
94C32×C4C2×D411yescharacteristic
95C32×C4#4111yescharacteristic
96C3⋊Dic3—41no—
97C3⋊Dic3—41no—
98C3×Dic3—41no—
99C2×SD16—91no—
100C22×C8—31no—
101C22⋊Q8—91no—
102#121—31no—
103#106—91no—
104#103—91no—
105#102—91no—
106C23×C3—21no—
107C23×C3—61no—
108C23×C3D1211yescharacteristic
109C23×C3#6711yescharacteristic
110C22×D3—21no—
111C3×D4—61no—
112C3×D4—61no—
113C3×D4—122no—
114C2×C3×C4—21no—
115C2×C3×C4—21no—
116C2×C3×C4—21no—
117C2×C3×C4—21no—
118C2×C3×C4—21no—
119C2×C3×C4—21no—
120C2×C3×C4—41no—
121C2×C3×C4—41no—
122C2×C3×C4—61no—
123C2×C3×C4D1211yescharacteristic
124C2×C3×C4#6711yescharacteristic
125C2×C3×C4C22×D311yescharacteristic
126C2×C3×C4C22×D311yescharacteristic
127#67—42no—
128C2×Dic3—21no—
129C2×Dic3—62no—
130C2×Dic3—62no—
131C2×Dic3—61no—
132C2×Dic3—61no—
133C2×Dic3—61no—
134C2×Dic3—61no—
135D12—21no—
136D12—21no—
137Dic6—61no—
138Dic6—61no—
139Dic6—61no—
140Dic6—61no—
141Dic6—121no—
142Dic6—121no—
143C3×C8—32no—
144C3×C8—61no—
145C3⋊C8—21no—
146C3⋊C8#6712yescomplemented
147C2×C32—22no—
148C2×C32#12111yescharacteristic
149C2×C32C2×SD1611yescharacteristic
150C2×C32#13511yescharacteristic
151C3×D3—81no—
152C2×Q8—91no—
153C2×D4—31no—
154C2×D4—61no—
155C22×C4D3211yescharacteristic
156SD16—182no—
157C2×C8—31no—
158C2×C8—31no—
159C2×C8—62no—
160C4⋊C4—31no—
161C4⋊C4—91no—
162C4⋊C4—181no—
163C22⋊C4—61no—
164C22⋊C4—181no—
165C22×C3—21no—
166C22×C3—21no—
167C22×C3—21no—
168C22×C3—21no—
169C22×C3—22no—
170C22×C3—21no—
171C22×C3—41no—
172C22×C3—41no—
173C22×C3—121no—
174C22×C3—121no—
175C22×C3—121no—
176C22×C3#20412yescomplemented
177C22×C3#21412yescomplemented
178C22×C3C2×D1211yescharacteristic
179C22×C3C2×#6711yescharacteristic
180C2×D3—41no—
181C2×D3—41no—
182C2×D3—41no—
183C3×C4—21no—
184C3×C4—21no—
185C3×C4—21no—
186C3×C4—21no—
187C3×C4—41no—
188C3×C4—121no—
189C3×C4#23511yescharacteristic
190C3×C4D3×D411yescharacteristic
191C3×C4#23711yescharacteristic
192C3×C4C2×#6711yescharacteristic
193Dic3—41no—
194Dic3—122no—
195Dic3—122no—
196Dic3—121no—
197Dic3—121no—
198Dic3—121no—
199Dic3—121no—
200C32#46511yescharacteristic, complemented, Sylow
201C23—61no—
202C23#62611yescharacteristic
203Q8—181no—
204Q8—181no—
205D4—61no—
206D4—61no—
207D4—122no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—61no—
211C2×C4—181no—
212C2×C4—181no—
213C2×C4#62611yescharacteristic
214C2×C4C2×D3211yescharacteristic
215C8—32no—
216C8—61no—
217C2×C3—22no—
218C2×C3—22no—
219C2×C3—21no—
220C2×C3—21no—
221C2×C3—21no—
222C2×C3—42no—
223C2×C3—241no—
224C2×C3C2×#20411yescharacteristic
225C2×C3#90411yescharacteristic
226C2×C3#93011yescharacteristic
227C2×C3C2×#21411yescharacteristic
228C2×C3#93911yescharacteristic
229C2×C3#95011yescharacteristic
230D3—81no—
231C22—21no—
232C22—21no—
233C22—121no—
234C22—121no—
235C22—121no—
236C22#371612yescomplemented
237C22C2×#62611yescharacteristic
238C4—21no—
239C4—121no—
240C4—361no—
241C4—361no—
242C4#380411yescharacteristic
243C4D3×#6711yescharacteristic
244C3—21no—
245C3#504111yescharacteristic, complemented
246C3#511411yescharacteristic, complemented
247C2—22no—
248C2—241no—
249C2C2×#371611yescharacteristic
250C2#6382911yescharacteristic
251C2#6397811yescharacteristic
subgroupG/H
centerC22C2×#626
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC22×C32×C4C22
socleC22×C32C2×D4
prime radicalC3×#930C2
hypercenterC22×C4D32
hypocenterC32#465
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #64133C2semidirect
3. C22×C3×C3⋊C8C2semidirect
4. C2×#3716C2semidirect
5. #63573C2semidirect
6. #63567C2semidirect
7. #63563C2semidirect
30. C22×C3⋊C8D3semidirect
53. C3×C3⋊C8D4semidirect
88. C22×C32SD16semidirect
146. C3⋊C8#67semidirect
176. C22×C3#204semidirect
177. C22×C3#214semidirect
200. C32#465semidirect
236. C22#3716semidirect
245. C3#5041semidirect
246. C3#5114semidirect

Maximal subgroups

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

subgroupindex
#641332
C3×#9302
C22×C3×C3⋊C82
C2×#37162
#635732
#635672
#635632
#51143
#50413