Finite Group Databasebeta

← #10591598

Subgroups — #10591598

Order 576.

250 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C2×#3756C211yesmaximal, characteristic, complemented
2C3×#946C211yesmaximal, characteristic
3#63857C211yesmaximal, characteristic
4C2×#3711C211yesmaximal, characteristic
5C3×#835C211yesmaximal, characteristic, complemented
6#63574C211yesmaximal, characteristic, complemented
7#63567C211yesmaximal, characteristic, complemented
8#5052—31nomaximal
9#4816—31nomaximal
10C2×C32×Q8C2211yescharacteristic
11C2×#634C2211yescharacteristic
12C2×C3×C4×D3C2211yescharacteristic
13#3756—22no—
14C2×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×(C2×D3)⋊C4—21no—
16C3×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic3C2211yescharacteristic
18C3×Dic3⋊C4—21no—
19C2×C3×C3⋊C8C2211yescharacteristic
20Dic3⋊Dic3—21no—
21(C2×D3)⋊Dic3—21no—
22#3711—22no—
23C3×C22⋊Q8—31no—
24#946D311yescharacteristic
25C2×#216—31no—
26C2×#216—31no—
27C2×#216—61no—
28#924—31no—
29C2×#208—31no—
30C2×ΓL(1,𝔽25)—31no—
31#897—31no—
32C3×#103—31no—
33#835D311yescharacteristic, complemented
34#816—31no—
35#810—31no—
36#808—31no—
37#808—31no—
38C22×C3×D3D411yescharacteristic
39C32×Q8—21no—
40C32×Q8—21no—
41C2×C32×C4—21no—
42C2×C32×C4C2311yescharacteristic
43C2×C3⋊Dic3—21no—
44#634—21no—
45#634—21no—
46C2×C3×Dic3—21no—
47C2×C3×Dic3—21no—
48C2×C3×Dic3D411yescharacteristic
49C3×C4×D3—21no—
50C3×C4×D3—21no—
51C3⋊(C3⋊C8)D412yes—
52C3×C3⋊C8—21no—
53#470—91noSylow
54C2×C3×Q8—21no—
55C2×C3×Q8C2×D311yescharacteristic
56C2×C3×Q8C2×D311yescharacteristic
57C22×C3×C4—31no—
58C2×C4×D3C2×D311yescharacteristic
59C2×Dic6—31no—
60C2×Dic6—31no—
61C2×Dic6—61no—
62C2×C3×C8—31no—
63C3×C4⋊C4—31no—
64C3×C4⋊C4—31no—
65C3×C4⋊C4—61no—
66C3×C22⋊C4—61no—
67#217—61no—
68#216—62no—
69#216—62no—
70#216—64no—
71(C2×D3)⋊C4—21no—
72(C2×D3)⋊C4—61no—
73C4⋊Dic3—61no—
74C4⋊Dic3C2×D311yescharacteristic
75C4⋊Dic3C2×D311yescharacteristic
76Dic3⋊C4—21no—
77Dic3⋊C4—61no—
78#208—62no—
79C2×C3⋊C8—31no—
80C2×C3⋊C8—31no—
81C2×C3⋊C8—61no—
82C2×C3⋊C8C2×D311yescharacteristic
83ΓL(1,𝔽25)—62no—
84C22×C32C2×D411yescharacteristic
85C2×C3×D3—21no—
86C2×C3×D3—21no—
87C2×C3×D3—21no—
88C32×C4—41no—
89C32×C4C2×D411yescharacteristic
90C32×C4#4111yescharacteristic
91C3⋊Dic3—41no—
92C3×Dic3—21no—
93C3×Dic3—41no—
94C3×Dic3—41no—
95C2×Q16—91no—
96C2×M16—91no—
97C22⋊Q8—31no—
98C22⋊Q8—91no—
99#106—91no—
100#103—31no—
101#103—91no—
102C23×C3—31no—
103C22×D3#6711yescharacteristic
104C3×Q8—22no—
105C3×Q8—21no—
106C3×Q8—21no—
107C3×Q8—21no—
108C3×Q8—21no—
109C3×Q8—41no—
110C2×C3×C4—21no—
111C2×C3×C4—21no—
112C2×C3×C4—21no—
113C2×C3×C4—31no—
114C2×C3×C4—41no—
115C2×C3×C4—61no—
116C2×C3×C4—61no—
117C2×C3×C4—61no—
118C2×C3×C4—61no—
119C2×C3×C4C22×D311yescharacteristic
120C2×C3×C4C22×D311yescharacteristic
121C2×Dic3—21no—
122C2×Dic3—21no—
123C2×Dic3—61no—
124C2×Dic3—61no—
125C2×Dic3—121no—
126C2×Dic3#6711yescharacteristic
127C4×D3—21no—
128C4×D3—21no—
129Dic6—62no—
130Dic6—61no—
131Dic6—61no—
132Dic6—61no—
133Dic6—61no—
134Dic6—121no—
135C3×C8—61no—
136C3⋊C8—21no—
137C3⋊C8—32no—
138C3⋊C8—32no—
139C3⋊C8—62no—
140C2×C32#12111yescharacteristic
141C2×C32#13711yescharacteristic
142C2×C32#13711yescharacteristic
143C3×D3—41no—
144C2×Q8—91no—
145C2×Q8D3211yescharacteristic
146C22×C4—31no—
147Q16—182no—
148M16—182no—
149C2×C8—31no—
150C2×C8—91no—
151C4⋊C4—31no—
152C4⋊C4—31no—
153C4⋊C4—61no—
154C4⋊C4—181no—
155C22⋊C4—61no—
156C22⋊C4—181no—
157C22×C3—21no—
158C22×C3—61no—
159C22×C3—61no—
160C22×C3—61no—
161C22×C3D3×D411yescharacteristic
162C22×C3C2×#6711yescharacteristic
163C2×D3—21no—
164C2×D3—21no—
165C2×D3—21no—
166C3×C4—21no—
167C3×C4—21no—
168C3×C4—42no—
169C3×C4—41no—
170C3×C4—41no—
171C3×C4—61no—
172C3×C4—121no—
173C3×C4—121no—
174C3×C4#23511yescharacteristic
175C3×C4D3×D411yescharacteristic
176C3×C4D3×D411yescharacteristic
177C3×C4#23711yescharacteristic
178Dic3—21no—
179Dic3—41no—
180Dic3—41no—
181Dic3—122no—
182Dic3—121no—
183Dic3—121no—
184C32#47011yescharacteristic, complemented, Sylow
185C23—31no—
186Q8—21no—
187Q8—21no—
188Q8—181no—
189Q8—181no—
190C2×C4—21no—
191C2×C4—31no—
192C2×C4—61no—
193C2×C4—61no—
194C2×C4—61no—
195C2×C4—61no—
196C2×C4—181no—
197C2×C4C2×D3211yescharacteristic
198C8—61no—
199C8—92no—
200C2×C3—21no—
201C2×C3—21no—
202C2×C3—21no—
203C2×C3—121no—
204C2×C3#88411yescharacteristic
205C2×C3#91511yescharacteristic
206C2×C3#91811yescharacteristic
207C2×C3#93911yescharacteristic
208C2×C3#94211yescharacteristic
209C2×C3#95111yescharacteristic
210D3—41no—
211C22—61no—
212C22—61no—
213C22—61no—
214C22D3×#6711yescharacteristic
215C4—41no—
216C4—61no—
217C4—121no—
218C4—121no—
219C4—361no—
220C4#380411yescharacteristic
221C4#381111yescharacteristic
222C3—21no—
223C3#481611yescharacteristic, complemented
224C3#505211yescharacteristic, complemented
225C2—121no—
226C2#6394311yescharacteristic
227C2#6394611yescharacteristic
228C2#6397111yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×Q8C22
socleC22×C32C2×D4
prime radicalC22×C3×D3D4
hypercenterC2×Q8D32
hypocenterC32#470
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
1. C2×#3756C2semidirect
5. C3×#835C2semidirect
6. #63574C2semidirect
7. #63567C2semidirect
33. #835D3semidirect
184. C32#470semidirect
223. C3#4816semidirect
224. C3#5052semidirect

Maximal subgroups

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

subgroupindex
C2×#37562
C3×#9462
#638572
C2×#37112
C3×#8352
#635742
#635672
#50523
#48163