Finite Group Databasebeta

← #10590526

Subgroups — #10590526

Order 576.

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

normal subgroups only

This group has 230 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#64101C211yesmaximal, characteristic
2#63871C211yesmaximal, characteristic
3C3×#810C211yesmaximal, characteristic, complemented
4C3×#807C211yesmaximal, characteristic, complemented
5#63573C211yesmaximal, characteristic, complemented
6#63566C211yesmaximal, characteristic, complemented
7#63557C211yesmaximal, characteristic
8#4888—31nomaximal
9#4815—31nomaximal
10C2×C4×C3⋊D3C2211yescharacteristic
11C2×C3×Dic6C2211yescharacteristic
12C32×C4⋊C4C2211yescharacteristic
13#3752—21no—
14C4⋊(C3⋊Dic3)C2211yescharacteristic
15C4×C3⋊Dic3—21no—
16C3×C4⋊Dic3C2211yescharacteristic
17C2×C3×C3⋊C8C2211yescharacteristic
18C2×C3×C3⋊C8C2211yescharacteristic
19Dic3⋊Dic3—21no—
20#3722—21no—
21#946—31no—
22#897—31no—
23#892—31no—
24#892—31no—
25#892—61no—
26C3×#107—31no—
27C3×#103—31no—
28#835—31no—
29(C2×D3)⋊C8—31no—
30(C2×D3)⋊C8—31no—
31#817—31no—
32#816—31no—
33#810D311yescharacteristic, complemented
34#808—31no—
35#807D311yescharacteristic, complemented
36C22×C3⋊D3D411yescharacteristic
37C2×C32×C4—21no—
38C2×C32×C4C2311yescharacteristic
39C2×C3⋊Dic3—21no—
40C2×C3⋊Dic3D411yescharacteristic
41C4×C3⋊D3—21no—
42C4×C3⋊D3—21no—
43C2×C3×Dic3—21no—
44C2×C3×Dic3—21no—
45C3×Dic6—21no—
46C3×Dic6—21no—
47C3×C3⋊C8—21no—
48C3×C3⋊C8—21no—
49#485—91noSylow
50C2×C3×Q8—31no—
51C2×C4×D3—31no—
52C2×C4×D3—31no—
53C2×C4×D3—61no—
54C2×Dic6C2×D311yescharacteristic
55C2×C3×C8—31no—
56C2×C3×C8—31no—
57C3×C4⋊C4—21no—
58C3×C4⋊C4—31no—
59C3×C4⋊C4C2×D311yescharacteristic
60C3×C4⋊C4C2×D311yescharacteristic
61(C2×D3)⋊C4—61no—
62(C2×D3)⋊C4—61no—
63(C2×D3)⋊C4—61no—
64(C2×D3)⋊C4—61no—
65(C2×D3)⋊C4—62no—
66C4⋊Dic3—31no—
67C4⋊Dic3—31no—
68C4⋊Dic3—61no—
69C4⋊Dic3—61no—
70C4⋊Dic3C2×D311yescharacteristic
71Dic3⋊C4—61no—
72C4×Dic3—61no—
73C4×Dic3—61no—
74C4×Dic3—62no—
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—31no—
91#106—91no—
92#103—31no—
93#103—91no—
94C22⋊C8—91no—
95C22×D3—31no—
96C22×D3—31no—
97C22×D3—61no—
98C3×Q8—61no—
99C3×Q8—61no—
100C2×C3×C4—22no—
101C2×C3×C4—21no—
102C2×C3×C4—21no—
103C2×C3×C4—21no—
104C2×C3×C4—61no—
105C2×C3×C4—61no—
106C2×C3×C4C22×D311yescharacteristic
107C2×C3×C4C22×D311yescharacteristic
108C2×Dic3—21no—
109C2×Dic3—21no—
110C2×Dic3—31no—
111C2×Dic3—31no—
112C2×Dic3—61no—
113C2×Dic3—62no—
114C2×Dic3—61no—
115C2×Dic3—61no—
116C4×D3—62no—
117C4×D3—62no—
118C4×D3—61no—
119C4×D3—61no—
120C4×D3—61no—
121C4×D3—61no—
122Dic6—21no—
123Dic6—21no—
124C3×C8—61no—
125C3×C8—61no—
126C3⋊C8—21no—
127C3⋊C8—21no—
128C2×C32#12311yescharacteristic
129C2×C32#13511yescharacteristic
130C2×C32#13711yescharacteristic
131C3⋊D3—41no—
132C2×Q8—31no—
133C22×C4—91no—
134C2×C8—31no—
135C2×C8—31no—
136C4⋊C4—31no—
137C4⋊C4—91no—
138C4⋊C4—181no—
139C4⋊C4D3211yescharacteristic
140C22⋊C4—181no—
141C22⋊C4—181no—
142C42—181no—
143C22×C3—21no—
144C22×C3D3×D411yescharacteristic
145C22×C3D3×D411yescharacteristic
146C2×D3—62no—
147C2×D3—61no—
148C2×D3—61no—
149C2×D3—61no—
150C2×D3—61no—
151C2×D3—61no—
152C2×D3—61no—
153C2×D3—121no—
154C2×D3—121no—
155C3×C4—21no—
156C3×C4—21no—
157C3×C4—42no—
158C3×C4—41no—
159C3×C4—41no—
160C3×C4—121no—
161C3×C4—121no—
162C3×C4#23511yescharacteristic
163C3×C4#23511yescharacteristic
164C3×C4#23711yescharacteristic
165C3×C4#23911yescharacteristic
166Dic3—41no—
167Dic3—41no—
168Dic3—62no—
169Dic3—61no—
170Dic3—61no—
171Dic3—122no—
172Dic3—121no—
173Dic3—121no—
174C32#48511yescharacteristic, complemented, Sylow
175C23—91no—
176Q8—61no—
177Q8—61no—
178C2×C4—21no—
179C2×C4—61no—
180C2×C4—61no—
181C2×C4—91no—
182C2×C4—181no—
183C2×C4—181no—
184C2×C4—181no—
185C2×C4C2×D3211yescharacteristic
186C8—61no—
187C8—61no—
188C2×C3—21no—
189C2×C3—21no—
190C2×C3—21no—
191C2×C3#88311yescharacteristic
192C2×C3#89411yescharacteristic
193C2×C3#91211yescharacteristic
194C2×C3#91611yescharacteristic
195C2×C3#91811yescharacteristic
196C2×C3#91811yescharacteristic
197D3—122no—
198D3—121no—
199D3—121no—
200C22—181no—
201C22—181no—
202C22—181no—
203C22#381111yescharacteristic
204C4—41no—
205C4—121no—
206C4—121no—
207C4—181no—
208C4—361no—
209C4#379611yescharacteristic
210C4#379811yescharacteristic
211C3—21no—
212C3#481511yescharacteristic, complemented
213C3#488811yescharacteristic, complemented
214C2—361no—
215C2#6385311yescharacteristic
216C2#6393611yescharacteristic
217C2#6394511yescharacteristic
subgroupG/H
centerC22#3811
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC32×C4⋊C4C22
socleC22×C32C2×D4
prime radicalC22×C3⋊D3D4
hypercenterC4⋊C4D32
hypocenterC32#485
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C3×#810C2semidirect
4. C3×#807C2semidirect
5. #63573C2semidirect
6. #63566C2semidirect
33. #810D3semidirect
35. #807D3semidirect
174. C32#485semidirect
212. C3#4815semidirect
213. C3#4888semidirect

Maximal subgroups

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

subgroupindex
#641012
#638712
C3×#8102
C3×#8072
#635732
#635662
#635572
#48883
#48153