Finite Group Databasebeta

← #10591561

Subgroups — #10591561

Order 576.

229 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 229 conjugacy classes of subgroups — too many to draw. Only its 58 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#64155C211yesmaximal, characteristic
2#63849C211yesmaximal, characteristic
3C3×#835C211yesmaximal, characteristic, complemented
4C3×#835C211yesmaximal, characteristic, complemented
5#63575C211yesmaximal, characteristic, complemented
6#63573C211yesmaximal, characteristic, complemented
7#63557C211yesmaximal, characteristic
8#4815—31nomaximal
9#4814—31nomaximal
10C2×C32×Q8C2211yescharacteristic
11C2×C4×C3⋊D3C2211yescharacteristic
12#3752—21no—
13C4⋊(C3⋊Dic3)C2211yescharacteristic
14#3750—21no—
15C3×C4⋊Dic3C2211yescharacteristic
16C3×C4⋊Dic3C2211yescharacteristic
17C2×C3×C3⋊C8C2211yescharacteristic
18C2×C3×C3⋊C8C2211yescharacteristic
19#3722—21no—
20Dic32—21no—
21#946—31no—
22#946—31no—
23#946—61no—
24#892—31no—
25#892—31no—
26C3×#103—31no—
27C3×#103—31no—
28#835D311yescharacteristic, complemented
29#835D311yescharacteristic, complemented
30(C2×D3)⋊C8—31no—
31(C2×D3)⋊C8—31no—
32#818—31no—
33#817—31no—
34#808—31no—
35#807—31no—
36C22×C3⋊D3D411yescharacteristic
37C32×Q8—21no—
38C32×Q8—21no—
39C2×C32×C4—21no—
40C2×C32×C4C2311yescharacteristic
41C2×C3⋊Dic3—21no—
42C2×C3⋊Dic3D411yescharacteristic
43C4×C3⋊D3—21no—
44C4×C3⋊D3—21no—
45C2×C3×Dic3—21no—
46C2×C3×Dic3—21no—
47C3×C3⋊C8—21no—
48C3×C3⋊C8—21no—
49#485—91noSylow
50C2×C3×Q8—21no—
51C2×C3×Q8C2×D311yescharacteristic
52C2×C3×Q8C2×D311yescharacteristic
53C2×C4×D3—31no—
54C2×C4×D3—31no—
55C2×C4×D3—61no—
56C2×C3×C8—31no—
57C2×C3×C8—31no—
58C3×C4⋊C4—31no—
59C3×C4⋊C4—31no—
60(C2×D3)⋊C4—61no—
61(C2×D3)⋊C4—61no—
62(C2×D3)⋊C4—61no—
63(C2×D3)⋊C4—61no—
64(C2×D3)⋊C4—62no—
65C4⋊Dic3—31no—
66C4⋊Dic3—31no—
67C4⋊Dic3—61no—
68C4⋊Dic3C2×D311yescharacteristic
69C4⋊Dic3C2×D311yescharacteristic
70Dic3⋊C4—61no—
71Dic3⋊C4—61no—
72Dic3⋊C4—62no—
73C4×Dic3—61no—
74C4×Dic3—61no—
75C2×C3⋊C8C2×D311yescharacteristic
76C2×C3⋊C8C2×D311yescharacteristic
77C22×C32C2×D411yescharacteristic
78C2×C3⋊D3—21no—
79C2×C3⋊D3—21no—
80C2×C3⋊D3—21no—
81C32×C4—41no—
82C32×C4#4111yescharacteristic
83C32×C4#4111yescharacteristic
84C3⋊Dic3—21no—
85C3⋊Dic3—41no—
86C3×Dic3—41no—
87C3×Dic3—41no—
88C22⋊Q8—91no—
89#117—91no—
90#107—91no—
91#106—91no—
92#103—31no—
93#103—31no—
94C22⋊C8—91no—
95C22×D3—31no—
96C22×D3—31no—
97C22×D3—61no—
98C3×Q8—21no—
99C3×Q8—21no—
100C3×Q8—21no—
101C3×Q8—21no—
102C3×Q8—41no—
103C3×Q8—41no—
104C2×C3×C4—22no—
105C2×C3×C4—21no—
106C2×C3×C4—21no—
107C2×C3×C4—21no—
108C2×C3×C4—61no—
109C2×C3×C4—61no—
110C2×C3×C4C22×D311yescharacteristic
111C2×C3×C4C22×D311yescharacteristic
112C2×Dic3—21no—
113C2×Dic3—21no—
114C2×Dic3—31no—
115C2×Dic3—31no—
116C2×Dic3—61no—
117C2×Dic3—62no—
118C2×Dic3—61no—
119C2×Dic3—61no—
120C4×D3—61no—
121C4×D3—61no—
122C4×D3—61no—
123C4×D3—61no—
124C4×D3—121no—
125C4×D3—121no—
126C3×C8—61no—
127C3×C8—61no—
128C3⋊C8—21no—
129C3⋊C8—21no—
130C2×C32#12311yescharacteristic
131C2×C32#13511yescharacteristic
132C2×C32#13711yescharacteristic
133C3⋊D3—41no—
134C2×Q8D3211yescharacteristic
135C22×C4—91no—
136C2×C8—31no—
137C2×C8—31no—
138C4⋊C4—31no—
139C4⋊C4—31no—
140C4⋊C4—91no—
141C4⋊C4—181no—
142C22⋊C4—181no—
143C22⋊C4—181no—
144C42—181no—
145C22×C3—21no—
146C22×C3D3×D411yescharacteristic
147C22×C3D3×D411yescharacteristic
148C2×D3—62no—
149C2×D3—61no—
150C2×D3—61no—
151C2×D3—61no—
152C2×D3—61no—
153C2×D3—61no—
154C2×D3—61no—
155C2×D3—121no—
156C2×D3—121no—
157C3×C4—21no—
158C3×C4—21no—
159C3×C4—42no—
160C3×C4—41no—
161C3×C4—41no—
162C3×C4—121no—
163C3×C4—121no—
164C3×C4#23511yescharacteristic
165C3×C4#23511yescharacteristic
166C3×C4#23711yescharacteristic
167C3×C4#23711yescharacteristic
168Dic3—41no—
169Dic3—41no—
170Dic3—61no—
171Dic3—61no—
172Dic3—121no—
173Dic3—122no—
174Dic3—121no—
175Dic3—121no—
176C32#48511yescharacteristic, complemented, Sylow
177C23—91no—
178Q8—21no—
179Q8—21no—
180C2×C4—21no—
181C2×C4—61no—
182C2×C4—61no—
183C2×C4—91no—
184C2×C4—181no—
185C2×C4—181no—
186C2×C4—181no—
187C2×C4C2×D3211yescharacteristic
188C8—61no—
189C8—61no—
190C2×C3—21no—
191C2×C3—21no—
192C2×C3—21no—
193C2×C3#88311yescharacteristic
194C2×C3#88311yescharacteristic
195C2×C3#91511yescharacteristic
196C2×C3#91611yescharacteristic
197C2×C3#91811yescharacteristic
198C2×C3#91911yescharacteristic
199D3—122no—
200D3—121no—
201D3—121no—
202C22—181no—
203C22—181no—
204C22—181no—
205C22#381111yescharacteristic
206C4—41no—
207C4—121no—
208C4—121no—
209C4—181no—
210C4—361no—
211C4#380411yescharacteristic
212C4#380411yescharacteristic
213C3—21no—
214C3#481411yescharacteristic, complemented
215C3#481511yescharacteristic, complemented
216C2—361no—
217C2#6395011yescharacteristic
218C2#6395111yescharacteristic
219C2#6397511yescharacteristic
subgroupG/H
centerC22#3811
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×Q8C22
socleC22×C32C2×D4
prime radicalC22×C3⋊D3D4
hypercenterC2×Q8D32
hypocenterC32#485
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
3. C3×#835C2semidirect
4. C3×#835C2semidirect
5. #63575C2semidirect
6. #63573C2semidirect
28. #835D3semidirect
29. #835D3semidirect
176. C32#485semidirect
214. C3#4814semidirect
215. C3#4815semidirect

Maximal subgroups

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

subgroupindex
#641552
#638492
C3×#8352
#635752
#635732
#635572
#48153
#48143