Finite Group Databasebeta

← #10591090

Subgroups — #10591090

Order 576.

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

normal subgroups only

This group has 237 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
1C3×#930C211yesmaximal, characteristic
2C3×#926C211yesmaximal, characteristic, complemented
3#63657C211yesmaximal, characteristic, complemented
4#63576C211yesmaximal, characteristic, complemented
5#63574C211yesmaximal, characteristic, complemented
6#63562C211yesmaximal, characteristic, complemented
7#63562C211yesmaximal, characteristic, complemented
8#5035—31nomaximal
9#5003—31nomaximal
10C22×C32×C4C2211yescharacteristic
11C2×C3×#67—21no—
12C2×C3×D12C2211yescharacteristic
13C2×C3⋊(C3⋊C8)C2211yescharacteristic
14C2×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×#217—21no—
16C3×(C2×D3)⋊C4—21no—
17C3×C4⋊Dic3C2211yescharacteristic
18C3×C4⋊Dic3C2211yescharacteristic
19C3×C4⋊Dic3C2211yescharacteristic
20C3×C4×Dic3—21no—
21C3×#121—31no—
22C3×#117—31no—
23#930D311yescharacteristic
24#926D311yescharacteristic, complemented
25#832—31no—
26#832—31no—
27#830—31no—
28#830—31no—
29#830—61no—
30#809—31no—
31#809—31no—
32#808—31no—
33#808—31no—
34#807—31no—
35#807—31no—
36C23×C32D411yescharacteristic
37C22×C3×D3—21no—
38C2×C32×C4—21no—
39C2×C32×C4—21no—
40C2×C32×C4D411yescharacteristic
41C2×C32×C4C2311yescharacteristic
42C3×#67—42no—
43C2×C3×Dic3—21no—
44C2×C3×Dic3—21no—
45C2×C3×Dic3—21no—
46C3×D12—21no—
47C3×D12—21no—
48C3⋊(C3⋊C8)—21no—
49C3⋊(C3⋊C8)—21no—
50#482—91noSylow
51C2×C3×D4—31no—
52C2×C3×D4—61no—
53C22×C3×C4—21no—
54C22×C3×C4C2×D311yescharacteristic
55C22×C3×C4C2×D311yescharacteristic
56C2×#67—21no—
57C2×D12C2×D311yescharacteristic
58C3×C4⋊C4—31no—
59C3×C4⋊C4—31no—
60C3×C4⋊C4—31no—
61C3×C22⋊C4—61no—
62C3×C22⋊C4—61no—
63C3×C42—61no—
64#217—21no—
65(C2×D3)⋊C4—21no—
66C4⋊Dic3C2×D311yescharacteristic
67C4⋊Dic3C2×D311yescharacteristic
68C4⋊Dic3C2×D311yescharacteristic
69C4×Dic3—21no—
70C2×C3⋊C8—31no—
71C2×C3⋊C8—31no—
72C2×C3⋊C8—31no—
73C2×C3⋊C8—31no—
74C2×C3⋊C8—61no—
75C2×C3⋊C8—61no—
76C22×C32—21no—
77C22×C32—21no—
78C22×C32—21no—
79C22×C32C2×D411yescharacteristic
80C2×C3×D3—41no—
81C2×C3×D3—41no—
82C2×C3×D3—41no—
83C32×C4—21no—
84C32×C4#4111yescharacteristic
85C32×C4#4111yescharacteristic
86C3×Dic3—41no—
87C3×Dic3—41no—
88C3×Dic3—41no—
89#121—31no—
90#117—31no—
91#107—91no—
92#106—91no—
93#102—91no—
94#102—91no—
95C22⋊C8—91no—
96C23×C3—21no—
97C23×C3—61no—
98C23×C3#6711yescharacteristic
99C23×C3#6711yescharacteristic
100C22×D3—21no—
101C3×D4—61no—
102C3×D4—61no—
103C3×D4—122no—
104C2×C3×C4—21no—
105C2×C3×C4—21no—
106C2×C3×C4—21no—
107C2×C3×C4—21no—
108C2×C3×C4—21no—
109C2×C3×C4—21no—
110C2×C3×C4—41no—
111C2×C3×C4—41no—
112C2×C3×C4—61no—
113C2×C3×C4—61no—
114C2×C3×C4—61no—
115C2×C3×C4#6711yescharacteristic
116C2×C3×C4#6711yescharacteristic
117C2×C3×C4C22×D311yescharacteristic
118C2×C3×C4C22×D311yescharacteristic
119#67—42no—
120C2×Dic3—21no—
121C2×Dic3—21no—
122C2×Dic3—21no—
123D12—21no—
124D12—21no—
125C3⋊C8—61no—
126C3⋊C8—61no—
127C3⋊C8—61no—
128C3⋊C8—61no—
129C3⋊C8—121no—
130C3⋊C8—121no—
131C2×C32—41no—
132C2×C32#12311yescharacteristic
133C2×C32#13511yescharacteristic
134C2×C32AGL(1,ℤ/8)11yescharacteristic
135C3×D3—81no—
136C2×D4—31no—
137C2×D4—61no—
138C22×C4D3211yescharacteristic
139C2×C8—91no—
140C2×C8—91no—
141C4⋊C4—31no—
142C4⋊C4—31no—
143C4⋊C4—31no—
144C22⋊C4—61no—
145C22⋊C4—61no—
146C42—61no—
147C22×C3—22no—
148C22×C3—21no—
149C22×C3—21no—
150C22×C3—21no—
151C22×C3—21no—
152C22×C3—21no—
153C22×C3—21no—
154C22×C3—21no—
155C22×C3—41no—
156C22×C3—41no—
157C22×C3—121no—
158C22×C3—121no—
159C22×C3—121no—
160C22×C3C2×#6711yescharacteristic
161C22×C3C2×#6711yescharacteristic
162C2×D3—41no—
163C2×D3—41no—
164C2×D3—41no—
165C3×C4—21no—
166C3×C4—21no—
167C3×C4—21no—
168C3×C4—21no—
169C3×C4—41no—
170C3×C4—121no—
171C3×C4—121no—
172C3×C4—121no—
173C3×C4#23511yescharacteristic
174C3×C4#23511yescharacteristic
175C3×C4#23711yescharacteristic
176C3×C4#23711yescharacteristic
177Dic3—41no—
178Dic3—41no—
179Dic3—41no—
180C32#48211yescharacteristic, complemented, Sylow
181C23—61no—
182C23#62511yescharacteristic
183D4—61no—
184D4—61no—
185D4—122no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—61no—
189C2×C4—61no—
190C2×C4—61no—
191C2×C4#62511yescharacteristic
192C2×C4C2×D3211yescharacteristic
193C8—181no—
194C8—181no—
195C2×C3—21no—
196C2×C3—21no—
197C2×C3—21no—
198C2×C3—42no—
199C2×C3—41no—
200C2×C3—41no—
201C2×C3—241no—
202C2×C3#92911yescharacteristic
203C2×C3#93211yescharacteristic
204C2×C3#93511yescharacteristic
205C2×C3#94911yescharacteristic
206C2×C3#95011yescharacteristic
207C2×C3#95011yescharacteristic
208D3—81no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—121no—
213C22—121no—
214C22—121no—
215C22C2×#62511yescharacteristic
216C4—21no—
217C4—121no—
218C4—121no—
219C4—121no—
220C4#380411yescharacteristic
221C4#380411yescharacteristic
222C3—21no—
223C3#500311yescharacteristic, complemented
224C3#503511yescharacteristic, complemented
225C2—41no—
226C2—241no—
227C2#6382211yescharacteristic
228C2#6382311yescharacteristic
229C2#6396111yescharacteristic
subgroupG/H
centerC22C2×#625
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC22×C32×C4C22
socleC22×C32C2×D4
prime radicalC3×#930C2
hypercenterC22×C4D32
hypocenterC32#482
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C3×#926C2semidirect
3. #63657C2semidirect
4. #63576C2semidirect
5. #63574C2semidirect
6. #63562C2semidirect
7. #63562C2semidirect
24. #926D3semidirect
180. C32#482semidirect
223. C3#5003semidirect
224. C3#5035semidirect

Maximal subgroups

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

subgroupindex
C3×#9302
C3×#9262
#636572
#635762
#635742
#635622
#50353
#50033