Finite Group Databasebeta

← #10591477

Subgroups — #10591477

Order 576.

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

normal subgroups only

This group has 245 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×#940C211yesmaximal, characteristic
2#63869C211yesmaximal, characteristic, complemented
3#63659C211yesmaximal, characteristic, complemented
4C3×#832C211yesmaximal, characteristic, complemented
5#63563C211yesmaximal, characteristic, complemented
6#63562C211yesmaximal, characteristic, complemented
7#63554C211yesmaximal, characteristic, complemented
8#5071—31nomaximal
9#4764—31nomaximal
10C2×C32×D4C2211yescharacteristic
11C2×C3×#67—21no—
12C2×C3×#67—21no—
13C2×C3×D12C2211yescharacteristic
14C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3⋊(C3⋊C8)C2211yescharacteristic
16C3×C4⋊Dic3C2211yescharacteristic
17C3×C4×Dic3C2211yescharacteristic
18C2×C3×C3⋊C8C2211yescharacteristic
19#3724—21no—
20Dic3⋊Dic3—21no—
21C3×#127—31no—
22#940D311yescharacteristic
23#890—31no—
24#870—31no—
25C3×#102—31no—
26#832—31no—
27#832—31no—
28#832—31no—
29#832—31no—
30#832—61no—
31#832D311yescharacteristic, complemented
32#821—31no—
33#815—31no—
34#809—31no—
35#803—31no—
36C23×C32—21no—
37C22×C3×D3—21no—
38C32×D4—21no—
39C32×D4—21no—
40C2×C32×C4C2311yescharacteristic
41C2×C3⋊Dic3—21no—
42C3×#67—42no—
43C3×#67—42no—
44C2×C3×Dic3—21no—
45C2×C3×Dic3D411yescharacteristic
46C2×C3×Dic3D411yescharacteristic
47C3×D12—21no—
48C3×D12—21no—
49C3⋊(C3⋊C8)—21no—
50C3×C3⋊C8—21no—
51#490—91noSylow
52C2×C3×D4—21no—
53C2×C3×D4—31no—
54C2×C3×D4—61no—
55C2×C3×D4—61no—
56C2×C3×D4C2×D311yescharacteristic
57C2×C3×D4C2×D311yescharacteristic
58C2×#67—21no—
59C2×#67—21no—
60C2×D12C2×D311yescharacteristic
61C2×C3×C8—31no—
62C3×C4⋊C4—31no—
63C3×C42—31no—
64C4⋊Dic3—31no—
65C4⋊Dic3—31no—
66C4⋊Dic3—61no—
67C4⋊Dic3—61no—
68C4⋊Dic3C2×D311yescharacteristic
69Dic3⋊C4—61no—
70Dic3⋊C4—61no—
71Dic3⋊C4—61no—
72C4×Dic3C2×D311yescharacteristic
73C2×C3⋊C8—31no—
74C2×C3⋊C8—31no—
75C2×C3⋊C8—61no—
76C2×C3⋊C8C2×D311yescharacteristic
77C22×C32—41no—
78C22×C32—41no—
79C22×C32—41no—
80C22×C32C2×D411yescharacteristic
81C2×C3×D3—41no—
82C2×C3×D3—41no—
83C2×C3×D3—41no—
84C32×C4#4111yescharacteristic
85C32×C4#4111yescharacteristic
86C3⋊Dic3—41no—
87C3×Dic3—21no—
88C3×Dic3—21no—
89C3×Dic3—41no—
90#127—31no—
91#125—91no—
92#102—31no—
93#102—91no—
94#102—91no—
95#102—91no—
96#97—91no—
97C23×C3—21no—
98C23×C3—21no—
99C23×C3—41no—
100C23×C3—61no—
101C22×D3—21no—
102C3×D4—22no—
103C3×D4—21no—
104C3×D4—21no—
105C3×D4—21no—
106C3×D4—21no—
107C3×D4—41no—
108C3×D4—61no—
109C3×D4—61no—
110C3×D4—122no—
111C3×D4—122no—
112C2×C3×C4—21no—
113C2×C3×C4—31no—
114C2×C3×C4—31no—
115C2×C3×C4—61no—
116C2×C3×C4C22×D311yescharacteristic
117C2×C3×C4C22×D311yescharacteristic
118#67—42no—
119#67—42no—
120C2×Dic3—21no—
121C2×Dic3—61no—
122C2×Dic3—61no—
123C2×Dic3—121no—
124C2×Dic3#6711yescharacteristic
125C2×Dic3#6711yescharacteristic
126D12—21no—
127D12—21no—
128C3×C8—61no—
129C3⋊C8—21no—
130C3⋊C8—61no—
131C3⋊C8—61no—
132C3⋊C8—121no—
133C2×C32—81no—
134C2×C32#12411yescharacteristic
135C2×C32AGL(1,ℤ/8)11yescharacteristic
136C2×C32AGL(1,ℤ/8)11yescharacteristic
137C3×D3—81no—
138C2×D4—31no—
139C2×D4—61no—
140C2×D4—61no—
141C2×D4D3211yescharacteristic
142C2×C8—31no—
143C2×C8—91no—
144C4⋊C4—31no—
145C4⋊C4—91no—
146C4⋊C4—181no—
147C4⋊C4—181no—
148C42—31no—
149C22×C3—21no—
150C22×C3—42no—
151C22×C3—41no—
152C22×C3—41no—
153C22×C3—41no—
154C22×C3—41no—
155C22×C3—41no—
156C22×C3—41no—
157C22×C3—81no—
158C22×C3—81no—
159C22×C3—121no—
160C22×C3—121no—
161C22×C3—121no—
162C22×C3D3×D411yescharacteristic
163C22×C3C2×#6711yescharacteristic
164C2×D3—41no—
165C2×D3—41no—
166C2×D3—41no—
167C3×C4—21no—
168C3×C4—21no—
169C3×C4—61no—
170C3×C4—61no—
171C3×C4—121no—
172C3×C4#23511yescharacteristic
173C3×C4#23711yescharacteristic
174C3×C4#23711yescharacteristic
175C3×C4#23711yescharacteristic
176Dic3—21no—
177Dic3—21no—
178Dic3—41no—
179Dic3—121no—
180Dic3—121no—
181Dic3—241no—
182C32#49011yescharacteristic, complemented, Sylow
183C23—21no—
184C23—61no—
185D4—21no—
186D4—21no—
187D4—61no—
188D4—61no—
189D4—122no—
190D4—122no—
191C2×C4—31no—
192C2×C4—31no—
193C2×C4—61no—
194C2×C4—181no—
195C2×C4C2×D3211yescharacteristic
196C8—61no—
197C8—181no—
198C2×C3—21no—
199C2×C3—21no—
200C2×C3—21no—
201C2×C3—82no—
202C2×C3—81no—
203C2×C3—81no—
204C2×C3—241no—
205C2×C3#88511yescharacteristic
206C2×C3#91111yescharacteristic
207C2×C3#91411yescharacteristic
208C2×C3#93211yescharacteristic
209C2×C3#93211yescharacteristic
210C2×C3#93611yescharacteristic
211D3—81no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—121no—
216C22—121no—
217C22—121no—
218C22D3×#6711yescharacteristic
219C4—61no—
220C4—61no—
221C4—121no—
222C4—361no—
223C4#380411yescharacteristic
224C4#380511yescharacteristic
225C3—21no—
226C3#476411yescharacteristic, complemented
227C3#507111yescharacteristic, complemented
228C2—81no—
229C2—241no—
230C2#6393211yescharacteristic
231C2#6393411yescharacteristic
232C2#6395911yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×D4C22
socleC22×C32C2×D4
prime radicalC3×#940C2
hypercenterC2×D4D32
hypocenterC32#490
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #63869C2semidirect
3. #63659C2semidirect
4. C3×#832C2semidirect
5. #63563C2semidirect
6. #63562C2semidirect
7. #63554C2semidirect
31. #832D3semidirect
182. C32#490semidirect
226. C3#4764semidirect
227. C3#5071semidirect

Maximal subgroups

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

subgroupindex
C3×#9402
#638692
#636592
C3×#8322
#635632
#635622
#635542
#50713
#47643