Finite Group Databasebeta

← #10591634

Subgroups — #10591634

Order 576.

269 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 269 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C3×#946C211yesmaximal, characteristic, complemented
2C2×C3×#215C211yesmaximal, characteristic, complemented
3#63908C211yesmaximal, characteristic
4C2×D3×C3⋊C8C211yesmaximal, characteristic, complemented
5#63662C211yesmaximal, characteristic, complemented
6#63573C211yesmaximal, characteristic, complemented
7#63562C211yesmaximal, characteristic, complemented
8#5171—31nomaximal
9#5048—31nomaximal
10C2×C32×Q8C2211yescharacteristic
11C2×C3×D12C2211yescharacteristic
12C2×C3×C4×D3C2211yescharacteristic
13C2×#625—21no—
14C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3⋊(C3⋊C8)C2211yescharacteristic
16C3×#215—22no—
17C3×(C2×D3)⋊C4—21no—
18C3×C4⋊Dic3C2211yescharacteristic
19C3×Dic3⋊C4—21no—
20C2×C3×C3⋊C8C2211yescharacteristic
21(C2×D3)⋊Dic3—21no—
22D3×C3⋊C8—22no—
23C2×C3×SD16—31no—
24C3×C22⋊Q8—31no—
25#946D311yescharacteristic, complemented
26C2×#215D311yescharacteristic, complemented
27#938—31no—
28#930—31no—
29C22×C3⋊C8—31no—
30C2×D3×C8—31no—
31#835—31no—
32#835—31no—
33#835—61no—
34#832—31no—
35#817—31no—
36#809—31no—
37#808—31no—
38C22×C3×D3—21no—
39C22×C3×D3D411yescharacteristic
40C32×Q8—21no—
41C32×Q8—21no—
42C2×C32×C4—21no—
43C2×C32×C4C2311yescharacteristic
44C2×C3⋊Dic3—21no—
45C2×C3×Dic3—21no—
46C2×C3×Dic3D411yescharacteristic
47C3×D12—21no—
48C3×D12—21no—
49C3×C4×D3—21no—
50C3×C4×D3—21no—
51#625—42no—
52C3⋊(C3⋊C8)—21no—
53C3×C3⋊C8D412yescomplemented
54#465—91noSylow
55C2×C3×Q8—21no—
56C2×C3×Q8C2×D311yescharacteristic
57C2×C3×Q8C2×D311yescharacteristic
58C2×C3×D4—31no—
59C22×C3×C4—31no—
60C2×#67—61no—
61C2×#67—61no—
62C2×D12C2×D311yescharacteristic
63C2×C4×D3C2×D311yescharacteristic
64C3×SD16—62no—
65C2×C3×C8—31no—
66C3×C4⋊C4—31no—
67C3×C4⋊C4—61no—
68C3×C22⋊C4—61no—
69#217—61no—
70#215—22no—
71(C2×D3)⋊C4—21no—
72(C2×D3)⋊C4—61no—
73C4⋊Dic3—31no—
74C4⋊Dic3—31no—
75C4⋊Dic3—61no—
76C4⋊Dic3C2×D311yescharacteristic
77Dic3⋊C4—21no—
78C2×C3⋊C8—31no—
79C2×C3⋊C8—31no—
80C2×C3⋊C8—61no—
81C2×C3⋊C8—62no—
82C2×C3⋊C8C2×D311yescharacteristic
83D3×C8—62no—
84C22×C32C2×D411yescharacteristic
85C2×C3×D3—21no—
86C2×C3×D3—21no—
87C2×C3×D3—41no—
88C2×C3×D3—41no—
89C2×C3×D3—41no—
90C2×C3×D3SD1612yescomplemented
91C32×C4—41no—
92C32×C4C2×D411yescharacteristic
93C32×C4#4111yescharacteristic
94C3⋊Dic3—41no—
95C3×Dic3—21no—
96C3×Dic3—41no—
97C2×SD16—31no—
98C22×C8—91no—
99C22⋊Q8—31no—
100#121—91no—
101#106—91no—
102#103—91no—
103#102—91no—
104C23×C3—31no—
105C23×C3—61no—
106C22×D3—21no—
107C22×D3#6711yescharacteristic
108C3×Q8—22no—
109C3×Q8—21no—
110C3×Q8—21no—
111C3×Q8—21no—
112C3×Q8—21no—
113C3×Q8—41no—
114C3×D4—61no—
115C3×D4—61no—
116C2×C3×C4—21no—
117C2×C3×C4—21no—
118C2×C3×C4—21no—
119C2×C3×C4—31no—
120C2×C3×C4—41no—
121C2×C3×C4—61no—
122C2×C3×C4—61no—
123C2×C3×C4—61no—
124C2×C3×C4C22×D311yescharacteristic
125C2×C3×C4C22×D311yescharacteristic
126#67—122no—
127#67—122no—
128C2×Dic3—21no—
129C2×Dic3—61no—
130C2×Dic3—61no—
131C2×Dic3—121no—
132C2×Dic3#6711yescharacteristic
133D12—21no—
134D12—21no—
135C4×D3—21no—
136C4×D3—21no—
137C3×C8—32no—
138C3⋊C8—61no—
139C3⋊C8—61no—
140C3⋊C8—121no—
141C3⋊C8#6712yescomplemented
142C2×C32#12111yescharacteristic
143C2×C32C2×SD1611yescharacteristic
144C2×C32#13511yescharacteristic
145C3×D3—22no—
146C3×D3—81no—
147C2×Q8D3211yescharacteristic
148C2×D4—31no—
149C2×D4—181no—
150C22×C4—31no—
151SD16—62no—
152C2×C8—31no—
153C2×C8—91no—
154C2×C8—182no—
155C4⋊C4—31no—
156C4⋊C4—61no—
157C4⋊C4—91no—
158C22⋊C4—61no—
159C22⋊C4—181no—
160C22×C3—21no—
161C22×C3—32no—
162C22×C3—61no—
163C22×C3—61no—
164C22×C3—121no—
165C22×C3—121no—
166C22×C3—121no—
167C22×C3D3×D411yescharacteristic
168C22×C3C2×#6711yescharacteristic
169C2×D3—21no—
170C2×D3—21no—
171C2×D3—41no—
172C2×D3—41no—
173C2×D3—41no—
174C2×D3#21512yescomplemented
175C3×C4—21no—
176C3×C4—21no—
177C3×C4—42no—
178C3×C4—41no—
179C3×C4—41no—
180C3×C4—61no—
181C3×C4—121no—
182C3×C4D3×D411yescharacteristic
183C3×C4#23711yescharacteristic
184C3×C4#23711yescharacteristic
185C3×C4C2×#6711yescharacteristic
186Dic3—21no—
187Dic3—41no—
188Dic3—121no—
189Dic3—121no—
190Dic3—241no—
191C32#46511yescharacteristic, complemented, Sylow
192C23—31no—
193C23—61no—
194Q8—21no—
195Q8—21no—
196D4—61no—
197D4—61no—
198D4—362no—
199C2×C4—21no—
200C2×C4—31no—
201C2×C4—61no—
202C2×C4—61no—
203C2×C4—61no—
204C2×C4—181no—
205C2×C4C2×D3211yescharacteristic
206C8—32no—
207C8—181no—
208C2×C3—21no—
209C2×C3—21no—
210C2×C3—21no—
211C2×C3—62no—
212C2×C3—241no—
213C2×C3D3×SD1611yescharacteristic
214C2×C3#91611yescharacteristic
215C2×C3#93811yescharacteristic
216C2×C3#93911yescharacteristic
217C2×C3C2×#21511yescharacteristic
218C2×C3#95011yescharacteristic
219D3—22no—
220D3—81no—
221C22—32no—
222C22—61no—
223C22—61no—
224C22—121no—
225C22—121no—
226C22—121no—
227C22D3×#6711yescharacteristic
228C4—41no—
229C4—61no—
230C4—121no—
231C4—361no—
232C4#380511yescharacteristic
233C4D3×#6711yescharacteristic
234C3—21no—
235C3#504811yescharacteristic, complemented
236C3#517111yescharacteristic, complemented
237C2—62no—
238C2—241no—
239C2D3×#21511yescharacteristic
240C2#6394711yescharacteristic
241C2#6397011yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×Q8C22
socleC22×C32C2×D4
prime radical#63908C2
hypercenterC2×Q8D32
hypocenterC32#465
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#946C2semidirect
2. C2×C3×#215C2semidirect
4. C2×D3×C3⋊C8C2semidirect
5. #63662C2semidirect
6. #63573C2semidirect
7. #63562C2semidirect
25. #946D3semidirect
26. C2×#215D3semidirect
53. C3×C3⋊C8D4semidirect
90. C2×C3×D3SD16semidirect
141. C3⋊C8#67semidirect
174. C2×D3#215semidirect
191. C32#465semidirect
235. C3#5048semidirect
236. C3#5171semidirect

Maximal subgroups

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

subgroupindex
C3×#9462
C2×C3×#2152
#639082
C2×D3×C3⋊C82
#636622
#635732
#635622
#51713
#50483