Finite Group Databasebeta

← #10591559

Subgroups — #10591559

Order 576.

213 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 213 conjugacy classes of subgroups — too many to draw. Only its 60 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
2#63841C211yesmaximal, characteristic
3#63662C211yesmaximal, characteristic, complemented
4C3×#835C211yesmaximal, characteristic, complemented
5#63576C211yesmaximal, characteristic, complemented
6#63573C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic
8#5045—31nomaximal
9#4815—31nomaximal
10C2×C32×Q8C2211yescharacteristic
11C2×C3×C4×D3C2211yescharacteristic
12C4⋊(C3⋊Dic3)C2211yescharacteristic
13C2×C3⋊(C3⋊C8)C2211yescharacteristic
14C3×(C2×D3)⋊C4—21no—
15C3×C4⋊Dic3C2211yescharacteristic
16C3×C4⋊Dic3C2211yescharacteristic
17C3×Dic3⋊C4—21no—
18C2×C3×C3⋊C8C2211yescharacteristic
19(C2×D3)⋊Dic3—21no—
20Dic32—21no—
21C3×C22⋊Q8—31no—
22#946D311yescharacteristic
23#926—31no—
24#892—31no—
25C3×#103—31no—
26#835—31no—
27#835—31no—
28#835—61no—
29#835D311yescharacteristic, complemented
30#830—31no—
31(C2×D3)⋊C8—31no—
32#817—31no—
33#808—31no—
34#807—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×Dic3—21no—
43C2×C3×Dic3—21no—
44C2×C3×Dic3D411yescharacteristic
45C3×C4×D3—21no—
46C3×C4×D3—21no—
47C3⋊(C3⋊C8)—21no—
48C3×C3⋊C8—21no—
49#485—91noSylow
50C2×C3×Q8—21no—
51C2×C3×Q8C2×D311yescharacteristic
52C2×C3×Q8C2×D311yescharacteristic
53C22×C3×C4—31no—
54C2×C4×D3C2×D311yescharacteristic
55C2×C3×C8—31no—
56C3×C4⋊C4—31no—
57C3×C4⋊C4—31no—
58C3×C4⋊C4—61no—
59C3×C22⋊C4—61no—
60#217—61no—
61(C2×D3)⋊C4—21no—
62(C2×D3)⋊C4—61no—
63C4⋊Dic3—31no—
64C4⋊Dic3—31no—
65C4⋊Dic3—61no—
66C4⋊Dic3C2×D311yescharacteristic
67C4⋊Dic3C2×D311yescharacteristic
68Dic3⋊C4—21no—
69C4×Dic3—61no—
70C4×Dic3—61no—
71C2×C3⋊C8—31no—
72C2×C3⋊C8—31no—
73C2×C3⋊C8—61no—
74C2×C3⋊C8C2×D311yescharacteristic
75C22×C32C2×D411yescharacteristic
76C2×C3×D3—21no—
77C2×C3×D3—21no—
78C2×C3×D3—21no—
79C32×C4—41no—
80C32×C4#4111yescharacteristic
81C32×C4#4111yescharacteristic
82C3⋊Dic3—41no—
83C3×Dic3—21no—
84C3×Dic3—41no—
85C3×Dic3—41no—
86C22⋊Q8—31no—
87#117—91no—
88#107—91no—
89#106—91no—
90#103—31no—
91#103—91no—
92C22⋊C8—91no—
93C23×C3—31no—
94C22×D3#6711yescharacteristic
95C3×Q8—22no—
96C3×Q8—21no—
97C3×Q8—21no—
98C3×Q8—21no—
99C3×Q8—21no—
100C3×Q8—41no—
101C2×C3×C4—21no—
102C2×C3×C4—21no—
103C2×C3×C4—21no—
104C2×C3×C4—31no—
105C2×C3×C4—41no—
106C2×C3×C4—61no—
107C2×C3×C4—61no—
108C2×C3×C4—61no—
109C2×C3×C4—61no—
110C2×C3×C4C22×D311yescharacteristic
111C2×C3×C4C22×D311yescharacteristic
112C2×Dic3—21no—
113C2×Dic3—21no—
114C2×Dic3—61no—
115C2×Dic3—61no—
116C2×Dic3—121no—
117C2×Dic3#6711yescharacteristic
118C4×D3—21no—
119C4×D3—21no—
120C3×C8—61no—
121C3⋊C8—21no—
122C3⋊C8—61no—
123C3⋊C8—61no—
124C3⋊C8—121no—
125C2×C32#12311yescharacteristic
126C2×C32#13511yescharacteristic
127C2×C32#13711yescharacteristic
128C3×D3—41no—
129C2×Q8D3211yescharacteristic
130C22×C4—31no—
131C2×C8—31no—
132C2×C8—91no—
133C4⋊C4—31no—
134C4⋊C4—31no—
135C4⋊C4—61no—
136C4⋊C4—91no—
137C22⋊C4—61no—
138C22⋊C4—181no—
139C42—181no—
140C22×C3—21no—
141C22×C3—61no—
142C22×C3—61no—
143C22×C3—61no—
144C22×C3D3×D411yescharacteristic
145C22×C3C2×#6711yescharacteristic
146C2×D3—21no—
147C2×D3—21no—
148C2×D3—21no—
149C3×C4—21no—
150C3×C4—21no—
151C3×C4—42no—
152C3×C4—41no—
153C3×C4—41no—
154C3×C4—61no—
155C3×C4—121no—
156C3×C4—121no—
157C3×C4#23511yescharacteristic
158C3×C4#23711yescharacteristic
159C3×C4#23711yescharacteristic
160C3×C4#23711yescharacteristic
161Dic3—21no—
162Dic3—41no—
163Dic3—41no—
164Dic3—121no—
165Dic3—121no—
166Dic3—241no—
167C32#48511yescharacteristic, complemented, Sylow
168C23—31no—
169Q8—21no—
170Q8—21no—
171C2×C4—21no—
172C2×C4—31no—
173C2×C4—61no—
174C2×C4—61no—
175C2×C4—61no—
176C2×C4—61no—
177C2×C4—181no—
178C2×C4C2×D3211yescharacteristic
179C8—61no—
180C8—181no—
181C2×C3—21no—
182C2×C3—21no—
183C2×C3—21no—
184C2×C3—121no—
185C2×C3#88311yescharacteristic
186C2×C3#91611yescharacteristic
187C2×C3#91811yescharacteristic
188C2×C3#93511yescharacteristic
189C2×C3#94211yescharacteristic
190C2×C3#95011yescharacteristic
191D3—41no—
192C22—61no—
193C22—61no—
194C22—61no—
195C22D3×#6711yescharacteristic
196C4—41no—
197C4—61no—
198C4—121no—
199C4—121no—
200C4—361no—
201C4#380411yescharacteristic
202C4#380511yescharacteristic
203C3—21no—
204C3#481511yescharacteristic, complemented
205C3#504511yescharacteristic, complemented
206C2—121no—
207C2#6394711yescharacteristic
208C2#6394811yescharacteristic
209C2#6396911yescharacteristic
subgroupG/H
centerC22D3×#67
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. #63662C2semidirect
4. C3×#835C2semidirect
5. #63576C2semidirect
6. #63573C2semidirect
29. #835D3semidirect
167. C32#485semidirect
204. C3#4815semidirect
205. C3#5045semidirect

Maximal subgroups

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

subgroupindex
C3×#9462
#638412
#636622
C3×#8352
#635762
#635732
(C2×D3)⋊(C3⋊C8)2
#50453
#48153