Finite Group Databasebeta

← #10591635

Subgroups — #10591635

Order 576.

249 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 249 conjugacy classes of subgroups — too many to draw. Only its 68 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
2C2×C3×#216C211yesmaximal, characteristic, complemented
3#63856C211yesmaximal, characteristic
4C2×D3×C3⋊C8C211yesmaximal, characteristic
5#63662C211yesmaximal, characteristic, complemented
6#63575C211yesmaximal, characteristic, complemented
7#63565C211yesmaximal, characteristic, complemented
8#5188—31nomaximal
9#5051—31nomaximal
10C2×C32×Q8C2211yescharacteristic
11C2×C3×C4×D3C2211yescharacteristic
12C2×C3×Dic6C2211yescharacteristic
13C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×#216—22no—
16C3×(C2×D3)⋊C4—21no—
17C3×C4⋊Dic3C2211yescharacteristic
18C3×Dic3⋊C4—21no—
19C2×C3×C3⋊C8C2211yescharacteristic
20#3724—21no—
21(C2×D3)⋊Dic3—21no—
22D3×C3⋊C8—22no—
23C2×C3×Q16—31no—
24C3×C22⋊Q8—31no—
25#946—31no—
26#946D311yescharacteristic
27C2×#216D311yescharacteristic, complemented
28#924—31no—
29C22×C3⋊C8—31no—
30C2×D3×C8—31no—
31#835—31no—
32#835—31no—
33#835—31no—
34#835—61no—
35#818—31no—
36#810—31no—
37#807—31no—
38C22×C3×D3D411yescharacteristic
39C32×Q8—21no—
40C32×Q8—21no—
41C2×C32×C4—21no—
42C2×C32×C4C2311yescharacteristic
43C2×C3⋊Dic3—21no—
44C2×C3×Dic3—21no—
45C2×C3×Dic3—21no—
46C2×C3×Dic3D411yescharacteristic
47C3×C4×D3—21no—
48C3×C4×D3—21no—
49C3×Dic6—21no—
50C3×Dic6—21no—
51C3⋊(C3⋊C8)—21no—
52C3×C3⋊C8D412yes—
53#467—91noSylow
54C2×C3×Q8—21no—
55C2×C3×Q8—31no—
56C2×C3×Q8C2×D311yescharacteristic
57C2×C3×Q8C2×D311yescharacteristic
58C22×C3×C4—31no—
59C2×C4×D3C2×D311yescharacteristic
60C2×Dic6C2×D311yescharacteristic
61C3×Q16—62no—
62C2×C3×C8—31no—
63C3×C4⋊C4—31no—
64C3×C4⋊C4—61no—
65C3×C22⋊C4—61no—
66#217—61no—
67#216—22no—
68(C2×D3)⋊C4—21no—
69(C2×D3)⋊C4—61no—
70C4⋊Dic3—31no—
71C4⋊Dic3—31no—
72C4⋊Dic3—61no—
73C4⋊Dic3C2×D311yescharacteristic
74Dic3⋊C4—21no—
75Dic3⋊C4—61no—
76Dic3⋊C4—61no—
77C2×C3⋊C8—31no—
78C2×C3⋊C8—31no—
79C2×C3⋊C8—61no—
80C2×C3⋊C8—62no—
81C2×C3⋊C8C2×D311yescharacteristic
82D3×C8—62no—
83C22×C32C2×D411yescharacteristic
84C2×C3×D3—21no—
85C2×C3×D3—21no—
86C2×C3×D3Q1612yescomplemented
87C32×C4—41no—
88C32×C4C2×D411yescharacteristic
89C32×C4#4111yescharacteristic
90C3⋊Dic3—41no—
91C3×Dic3—21no—
92C3×Dic3—41no—
93C3×Dic3—41no—
94C2×Q16—31no—
95C22×C8—91no—
96C22⋊Q8—31no—
97C22⋊Q8—91no—
98#107—91no—
99#103—91no—
100#103—91no—
101C23×C3—31no—
102C22×D3#6711yescharacteristic
103C3×Q8—22no—
104C3×Q8—21no—
105C3×Q8—21no—
106C3×Q8—21no—
107C3×Q8—21no—
108C3×Q8—41no—
109C3×Q8—61no—
110C3×Q8—61no—
111C2×C3×C4—21no—
112C2×C3×C4—21no—
113C2×C3×C4—21no—
114C2×C3×C4—31no—
115C2×C3×C4—41no—
116C2×C3×C4—61no—
117C2×C3×C4—61no—
118C2×C3×C4—61no—
119C2×C3×C4—61no—
120C2×C3×C4C22×D311yescharacteristic
121C2×C3×C4C22×D311yescharacteristic
122C2×Dic3—21no—
123C2×Dic3—21no—
124C2×Dic3—61no—
125C2×Dic3—61no—
126C2×Dic3—121no—
127C2×Dic3#6711yescharacteristic
128C4×D3—21no—
129C4×D3—21no—
130Dic6—21no—
131Dic6—21no—
132C3×C8—32no—
133C3⋊C8—61no—
134C3⋊C8—61no—
135C3⋊C8—121no—
136C3⋊C8#6712yes—
137C2×C32#12111yescharacteristic
138C2×C32C2×Q1611yescharacteristic
139C2×C32#13511yescharacteristic
140C3×D3—22no—
141C2×Q8—31no—
142C2×Q8D3211yescharacteristic
143C22×C4—31no—
144Q16—62no—
145C2×C8—31no—
146C2×C8—91no—
147C2×C8—182no—
148C4⋊C4—31no—
149C4⋊C4—61no—
150C4⋊C4—91no—
151C4⋊C4—181no—
152C22⋊C4—61no—
153C22⋊C4—181no—
154C22×C3—21no—
155C22×C3—32no—
156C22×C3—61no—
157C22×C3—61no—
158C22×C3D3×D411yescharacteristic
159C22×C3C2×#6711yescharacteristic
160C2×D3—21no—
161C2×D3—21no—
162C2×D3#21612yescomplemented
163C3×C4—21no—
164C3×C4—21no—
165C3×C4—42no—
166C3×C4—41no—
167C3×C4—41no—
168C3×C4—61no—
169C3×C4—121no—
170C3×C4—121no—
171C3×C4D3×D411yescharacteristic
172C3×C4#23711yescharacteristic
173C3×C4#23711yescharacteristic
174C3×C4C2×#6711yescharacteristic
175Dic3—21no—
176Dic3—41no—
177Dic3—41no—
178Dic3—121no—
179Dic3—121no—
180Dic3—241no—
181C32#46711yescharacteristic, complemented, Sylow
182C23—31no—
183Q8—21no—
184Q8—21no—
185Q8—61no—
186Q8—61no—
187C2×C4—21no—
188C2×C4—31no—
189C2×C4—61no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—61no—
193C2×C4—181no—
194C2×C4C2×D3211yescharacteristic
195C8—32no—
196C8—181no—
197C2×C3—21no—
198C2×C3—21no—
199C2×C3—21no—
200C2×C3—62no—
201C2×C3D3×Q1611yescharacteristic
202C2×C3#91911yescharacteristic
203C2×C3#93811yescharacteristic
204C2×C3#93911yescharacteristic
205C2×C3C2×#21611yescharacteristic
206C2×C3#95011yescharacteristic
207D3—22no—
208C22—32no—
209C22—61no—
210C22—61no—
211C22D3×#6711yescharacteristic
212C4—41no—
213C4—61no—
214C4—121no—
215C4—121no—
216C4—361no—
217C4#380511yescharacteristic
218C4D3×#6711yescharacteristic
219C3—21no—
220C3#505111yescharacteristic, complemented
221C3#518811yescharacteristic, complemented
222C2—62no—
223C2D3×#21611yescharacteristic
224C2#6394911yescharacteristic
225C2#6397011yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×Q8C22
socleC22×C32C2×D4
prime radicalC22×C3×D3D4
hypercenterC2×Q8D32
hypocenterC32#467
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
2. C2×C3×#216C2semidirect
5. #63662C2semidirect
6. #63575C2semidirect
7. #63565C2semidirect
27. C2×#216D3semidirect
86. C2×C3×D3Q16semidirect
162. C2×D3#216semidirect
181. C32#467semidirect
220. C3#5051semidirect
221. C3#5188semidirect

Maximal subgroups

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

subgroupindex
C3×#9462
C2×C3×#2162
#638562
C2×D3×C3⋊C82
#636622
#635752
#635652
#51883
#50513