Finite Group Databasebeta

← #10591569

Subgroups — #10591569

Order 576.

274 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 64 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
2C2×C3×#215C211yesmaximal, characteristic
3#63898C211yesmaximal, characteristic
4C2×#3716C211yesmaximal, characteristic
5C3×#835C211yesmaximal, characteristic, complemented
6#63572C211yesmaximal, characteristic, complemented
7#63567C211yesmaximal, characteristic, complemented
8#5162—31nomaximal
9#4818—31nomaximal
10C2×C32×Q8C2211yescharacteristic
11C2×#634C2211yescharacteristic
12C2×C3×D12C2211yescharacteristic
13C2×D3×Dic3—21no—
14#3750—21no—
15#3750—21no—
16C4×C3⋊Dic3C2211yescharacteristic
17C3×#215—22no—
18C3×C4⋊Dic3C2211yescharacteristic
19C2×C3×C3⋊C8C2211yescharacteristic
20C2×C3×C3⋊C8C2211yescharacteristic
21(C2×D3)⋊Dic3—21no—
22#3716—22no—
23C2×C3×SD16—31no—
24#944—31no—
25#944—31no—
26#944—61no—
27C2×#215D311yescharacteristic
28D4×Dic3—31no—
29C2×#214—31no—
30C2×#204—31no—
31#893—31no—
32C3×#103—31no—
33#835D311yescharacteristic, complemented
34#816—31no—
35Dic3⋊C8—31no—
36Dic3⋊C8—31no—
37#810—31no—
38C22×C3×D3—21no—
39C32×Q8—21no—
40C32×Q8—21no—
41C2×C32×C4—21no—
42C2×C32×C4C2311yescharacteristic
43C2×C3⋊Dic3—21no—
44C2×C3⋊Dic3D411yescharacteristic
45C2×C3⋊Dic3D411yescharacteristic
46#634—21no—
47#634—21no—
48C2×C3×Dic3—21no—
49C3×D12—21no—
50C3×D12D412yes—
51D3×Dic3—42no—
52C3×C3⋊C8—21no—
53C3×C3⋊C8—21no—
54#461—91noSylow
55C2×C3×Q8—21no—
56C2×C3×Q8C2×D311yescharacteristic
57C2×C3×Q8C2×D311yescharacteristic
58C2×C3×D4—31no—
59C22×Dic3—61no—
60C2×D12C2×D311yescharacteristic
61C2×C4×D3—61no—
62C2×Dic6—31no—
63C2×Dic6—31no—
64C2×Dic6—61no—
65C3×SD16—62no—
66C2×C3×C8—31no—
67C2×C3×C8—31no—
68C3×C4⋊C4—31no—
69#217—61no—
70#215—22no—
71#214—62no—
72(C2×D3)⋊C4—61no—
73C4⋊Dic3C2×D311yescharacteristic
74Dic3⋊C4—61no—
75Dic3⋊C4—61no—
76Dic3⋊C4—61no—
77Dic3⋊C4—61no—
78Dic3⋊C4—62no—
79Dic3⋊C4—62no—
80C4×Dic3—31no—
81C4×Dic3—31no—
82C4×Dic3—61no—
83C2×C3⋊C8C2×D311yescharacteristic
84C2×C3⋊C8C2×D311yescharacteristic
85#204—62no—
86C22×C32C2×D411yescharacteristic
87C2×C3×D3—22no—
88C2×C3×D3—41no—
89C2×C3×D3—41no—
90C32×C4—41no—
91C32×C4C2×D411yescharacteristic
92C32×C4#4111yescharacteristic
93C3⋊Dic3—21no—
94C3⋊Dic3—41no—
95C3⋊Dic3SD1612yescomplemented
96C3×Dic3—41no—
97C2×SD16—31no—
98C2×SD16—91no—
99C4⋊Q8—91no—
100C4×D4—91no—
101C4⋊C8—91no—
102#103—31no—
103#103—91no—
104C23×C3—61no—
105C22×D3—21no—
106C3×Q8—21no—
107C3×Q8—21no—
108C3×Q8—21no—
109C3×Q8—21no—
110C3×Q8—41no—
111C3×Q8—41no—
112C3×D4—32no—
113C3×D4—61no—
114C2×C3×C4—22no—
115C2×C3×C4—21no—
116C2×C3×C4—21no—
117C2×C3×C4—21no—
118C2×C3×C4—61no—
119C2×C3×C4C22×D311yescharacteristic
120C2×C3×C4C22×D311yescharacteristic
121C2×Dic3—21no—
122C2×Dic3—31no—
123C2×Dic3—31no—
124C2×Dic3—31no—
125C2×Dic3—31no—
126C2×Dic3—61no—
127C2×Dic3—61no—
128C2×Dic3—62no—
129C2×Dic3—61no—
130C2×Dic3—61no—
131C2×Dic3—122no—
132D12—21no—
133D12#6712yes—
134C4×D3—122no—
135Dic6—61no—
136Dic6—61no—
137Dic6—61no—
138Dic6—61no—
139Dic6—121no—
140Dic6—121no—
141C3×C8—61no—
142C3×C8—61no—
143C3⋊C8—21no—
144C3⋊C8—21no—
145C2×C32#12111yescharacteristic
146C2×C32C2×SD1611yescharacteristic
147C2×C32#13711yescharacteristic
148C3×D3—42no—
149C2×Q8—91no—
150C2×Q8D3211yescharacteristic
151C2×D4—31no—
152C22×C4—181no—
153SD16—62no—
154SD16—182no—
155C2×C8—31no—
156C2×C8—31no—
157C4⋊C4—31no—
158C4⋊C4—181no—
159C4⋊C4—181no—
160C22⋊C4—181no—
161C42—91no—
162C22×C3—21no—
163C22×C3—62no—
164C22×C3—121no—
165C22×C3—121no—
166C22×C3D3×D411yescharacteristic
167C22×C3D3×D411yescharacteristic
168C2×D3—22no—
169C2×D3—41no—
170C2×D3—41no—
171C3×C4—21no—
172C3×C4—21no—
173C3×C4—42no—
174C3×C4—41no—
175C3×C4—41no—
176C3×C4—121no—
177C3×C4#23511yescharacteristic
178C3×C4D3×D411yescharacteristic
179C3×C4#23711yescharacteristic
180C3×C4C2×#6711yescharacteristic
181Dic3—32no—
182Dic3—32no—
183Dic3—41no—
184Dic3—62no—
185Dic3—62no—
186Dic3—61no—
187Dic3—61no—
188Dic3—122no—
189Dic3—121no—
190Dic3—121no—
191C32#46111yescharacteristic, complemented, Sylow
192C23—61no—
193Q8—21no—
194Q8—21no—
195Q8—181no—
196Q8—181no—
197D4—32no—
198D4—61no—
199C2×C4—21no—
200C2×C4—61no—
201C2×C4—91no—
202C2×C4—91no—
203C2×C4—181no—
204C2×C4—362no—
205C2×C4C2×D3211yescharacteristic
206C8—61no—
207C8—61no—
208C2×C3—21no—
209C2×C3—21no—
210C2×C3—21no—
211C2×C3—122no—
212C2×C3#88411yescharacteristic
213C2×C3D3×SD1611yescharacteristic
214C2×C3D3×SD1611yescharacteristic
215C2×C3#91511yescharacteristic
216C2×C3#91811yescharacteristic
217C2×C3#93911yescharacteristic
218D3—42no—
219C22—62no—
220C22—121no—
221C22—121no—
222C22#381111yescharacteristic
223C4—41no—
224C4—92no—
225C4—121no—
226C4—181no—
227C4—361no—
228C4#380411yescharacteristic
229C4D3×#6711yescharacteristic
230C3—21no—
231C3#481811yescharacteristic, complemented
232C3#516211yescharacteristic, complemented
233C2—122no—
234C2#6394011yescharacteristic
235C2#6395111yescharacteristic
236C2#6397111yescharacteristic
subgroupG/H
centerC22#3811
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×Q8C22
socleC22×C32C2×D4
prime radicalC2×C3×D12C22
hypercenterC2×Q8D32
hypocenterC32#461
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. #64153C2semidirect
5. C3×#835C2semidirect
6. #63572C2semidirect
7. #63567C2semidirect
33. #835D3semidirect
95. C3⋊Dic3SD16semidirect
191. C32#461semidirect
231. C3#4818semidirect
232. C3#5162semidirect

Maximal subgroups

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

subgroupindex
#641532
C2×C3×#2152
#638982
C2×#37162
C3×#8352
#635722
#635672
#51623
#48183