Finite Group Databasebeta

← #10591612

Subgroups — #10591612

Order 576.

259 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 66 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
1Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×C3×#216C211yesmaximal, characteristic, complemented
3C2×C3×#215C211yesmaximal, characteristic
4#63863C211yesmaximal, characteristic
5#63572C211yesmaximal, characteristic, complemented
6#63566C211yesmaximal, characteristic, complemented
7#63563C211yesmaximal, characteristic
8#5185—31nomaximal
9#5166—31nomaximal
10C2×C32×Q8C2211yescharacteristic
11C2×C3×D12C2211yescharacteristic
12C2×C3×Dic6C2211yescharacteristic
13C4⋊(C3⋊Dic3)—21no—
14C4⋊(C3⋊Dic3)C2211yescharacteristic
15C4×C3⋊Dic3—21no—
16C4×C3⋊Dic3C2211yescharacteristic
17C3×#216—22no—
18C3×#215—22no—
19C2×C3×C3⋊C8C2211yescharacteristic
20C2×C3×C3⋊C8C2211yescharacteristic
21(C2×D3)⋊Dic3—21no—
22(C2×D3)⋊Dic3—21no—
23C2×C3×Q16—31no—
24C2×C3×SD16—31no—
25#947—31no—
26Q8×Dic3—31no—
27Q8×Dic3—31no—
28Q8×Dic3—61no—
29C2×#216D311yescharacteristic, complemented
30C2×#215D311yescharacteristic
31#936—31no—
32#835—31no—
33#832—31no—
34#821—31no—
35#816—31no—
36Dic3⋊C8—31no—
37Dic3⋊C8—31no—
38C22×C3×D3—21no—
39C32×Q8—21no—
40C32×Q8D412yes—
41C2×C32×C4—21no—
42C2×C32×C4C2311yescharacteristic
43C2×C3⋊Dic3—21no—
44C2×C3⋊Dic3D411yescharacteristic
45C2×C3⋊Dic3D411yescharacteristic
46C2×C3×Dic3—21no—
47C3×D12—21no—
48C3×D12—21no—
49C3×Dic6—21no—
50C3×Dic6—21no—
51C3×C3⋊C8—21no—
52C3×C3⋊C8—21no—
53#464—91noSylow
54C2×C3×Q8—21no—
55C2×C3×Q8—31no—
56C2×C3×Q8C2×D311yescharacteristic
57C2×C3×Q8C2×D311yescharacteristic
58C2×C3×D4—31no—
59C2×D12C2×D311yescharacteristic
60C2×Dic6C2×D311yescharacteristic
61C3×Q16—62no—
62C3×SD16—62no—
63C2×C3×C8—31no—
64C2×C3×C8—31no—
65#217—61no—
66#217—61no—
67#216—22no—
68#215—22no—
69(C2×D3)⋊C4—61no—
70(C2×D3)⋊C4—61no—
71C4⋊Dic3—31no—
72C4⋊Dic3—31no—
73C4⋊Dic3—61no—
74C4⋊Dic3—61no—
75C4⋊Dic3—61no—
76C4⋊Dic3—62no—
77C4×Dic3—31no—
78C4×Dic3—31no—
79C4×Dic3—61no—
80C4×Dic3—61no—
81C4×Dic3—61no—
82C4×Dic3—62no—
83C2×C3⋊C8C2×D311yescharacteristic
84C2×C3⋊C8C2×D311yescharacteristic
85C22×C32C2×D411yescharacteristic
86C2×C3×D3—41no—
87C2×C3×D3—41no—
88C2×C3×D3—41no—
89C32×C4—22no—
90C32×C4C2×D411yescharacteristic
91C32×C4#4111yescharacteristic
92C3⋊Dic3—21no—
93C3⋊Dic3—21no—
94C3⋊Dic3—41no—
95C3×Dic3—41no—
96C2×Q16—31no—
97C2×SD16—31no—
98#124—91no—
99C4×Q8—91no—
100C4⋊C8—91no—
101#103—91no—
102#102—91no—
103C23×C3—61no—
104C22×D3—21no—
105C3×Q8—22no—
106C3×Q8—21no—
107C3×Q8—21no—
108C3×Q8—22no—
109C3×Q8—61no—
110C3×Q8—61no—
111C3×Q8#6712yes—
112C3×Q8#6712yes—
113C3×D4—61no—
114C3×D4—61no—
115C2×C3×C4—22no—
116C2×C3×C4—21no—
117C2×C3×C4—21no—
118C2×C3×C4—21no—
119C2×C3×C4—61no—
120C2×C3×C4C22×D311yescharacteristic
121C2×C3×C4C22×D311yescharacteristic
122C2×Dic3—21no—
123C2×Dic3—31no—
124C2×Dic3—31no—
125C2×Dic3—31no—
126C2×Dic3—31no—
127C2×Dic3—61no—
128C2×Dic3—61no—
129C2×Dic3—62no—
130C2×Dic3—61no—
131C2×Dic3—61no—
132D12—21no—
133D12—21no—
134Dic6—21no—
135Dic6—21no—
136C3×C8—61no—
137C3×C8—61no—
138C3⋊C8—21no—
139C3⋊C8—21no—
140C2×C32#12111yescharacteristic
141C2×C32#13511yescharacteristic
142C2×C32#13711yescharacteristic
143C3×D3—81no—
144C2×Q8—31no—
145C2×Q8D3211yescharacteristic
146C2×D4—31no—
147Q16—62no—
148SD16—62no—
149C2×C8—31no—
150C2×C8—31no—
151C4⋊C4—91no—
152C4⋊C4—181no—
153C22⋊C4—181no—
154C22⋊C4—181no—
155C42—91no—
156C42—181no—
157C22×C3—21no—
158C22×C3—121no—
159C22×C3—121no—
160C22×C3—121no—
161C22×C3D3×D411yescharacteristic
162C22×C3D3×D411yescharacteristic
163C2×D3—41no—
164C2×D3—41no—
165C2×D3—41no—
166C3×C4—24no—
167C3×C4—22no—
168C3×C4—22no—
169C3×C4—21no—
170C3×C4—21no—
171C3×C4—121no—
172C3×C4#23711yescharacteristic
173C3×C4#23711yescharacteristic
174C3×C4C2×#6711yescharacteristic
175C3×C4C2×#6711yescharacteristic
176Dic3—41no—
177Dic3—62no—
178Dic3—61no—
179Dic3—61no—
180Dic3—61no—
181Dic3—61no—
182Dic3—121no—
183Dic3—122no—
184Dic3—121no—
185Dic3—121no—
186C32#46411yescharacteristic, complemented, Sylow
187C23—61no—
188Q8—21no—
189Q8—61no—
190Q8—61no—
191Q8#62512yes—
192D4—61no—
193D4—61no—
194C2×C4—21no—
195C2×C4—61no—
196C2×C4—91no—
197C2×C4—91no—
198C2×C4—181no—
199C2×C4C2×D3211yescharacteristic
200C8—61no—
201C8—61no—
202C2×C3—21no—
203C2×C3—21no—
204C2×C3—21no—
205C2×C3—241no—
206C2×C3#91511yescharacteristic
207C2×C3#91611yescharacteristic
208C2×C3#91811yescharacteristic
209C2×C3#91911yescharacteristic
210C2×C3#93911yescharacteristic
211C2×C3#93911yescharacteristic
212D3—81no—
213C22—121no—
214C22—121no—
215C22—121no—
216C22#381111yescharacteristic
217C4—22no—
218C4—121no—
219C4—181no—
220C4—181no—
221C4—361no—
222C4#380511yescharacteristic
223C4C2×#62511yescharacteristic
224C3—21no—
225C3#516611yescharacteristic, complemented
226C3#518511yescharacteristic, complemented
227C2—241no—
228C2#6395011yescharacteristic
229C2#6395111yescharacteristic
230C2#6398011yescharacteristic
subgroupG/H
centerC22#3811
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×Q8C22
socleC22×C32C2×D4
prime radicalC2×C3×D12C22
hypercenterC2×Q8D32
hypocenterC32#464
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. Q8×C3⋊Dic3C2semidirect
2. C2×C3×#216C2semidirect
5. #63572C2semidirect
6. #63566C2semidirect
29. C2×#216D3semidirect
186. C32#464semidirect
225. C3#5166semidirect
226. C3#5185semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊Dic32
C2×C3×#2162
C2×C3×#2152
#638632
#635722
#635662
#635632
#51853
#51663