Finite Group Databasebeta

← #10591581

Subgroups — #10591581

Order 576.

255 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 84 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
2C2×C3×#216C211yesmaximal, characteristic
3#63843C211yesmaximal, characteristic
4C3×#835C211yesmaximal, characteristic
5#63575C211yesmaximal, characteristic
6#63566C211yesmaximal, characteristic
7#63553C211yesmaximal, characteristic
8Dic3×Q16—31nomaximal
9#4790—31nomaximal
10C2×C32×Q8C2211yescharacteristic
11C2×C3×Dic6C2211yescharacteristic
12C4⋊(C3⋊Dic3)—21no—
13C4⋊(C3⋊Dic3)C2211yescharacteristic
14C4×C3⋊Dic3—21no—
15C4×C3⋊Dic3C2211yescharacteristic
16C3×#216C414yescomplemented
17C3×C4⋊Dic3C2211yescharacteristic
18C2×C3×C3⋊C8C2211yescharacteristic
19C2×C3×C3⋊C8C2211yescharacteristic
20Dic3⋊Dic3—21no—
21Dic32—21no—
22C2×C3×Q16—31no—
23Q8×Dic3—31no—
24Q8×Dic3—31no—
25Q8×Dic3—31no—
26Q8×Dic3—61no—
27C2×#216D311yescharacteristic
28#887—31no—
29C3×#103—31no—
30#835—31no—
31#835D311yescharacteristic
32#818—31no—
33#816—31no—
34C8×Dic3—31no—
35C8×Dic3—31no—
36#807—31no—
37C32×Q8—21no—
38C32×Q8C2×C412yes—
39C2×C32×C4—21no—
40C2×C32×C4C2311yescharacteristic
41C2×C3⋊Dic3—21no—
42C2×C3⋊Dic3D411yescharacteristic
43C2×C3⋊Dic3D411yescharacteristic
44C2×C3×Dic3—21no—
45C2×C3×Dic3—21no—
46C3×Dic6—21no—
47C3×Dic6C2×C412yes—
48C3×C3⋊C8—21no—
49C3×C3⋊C8C2×C412yes—
50C4×Q16—91noSylow
51C2×C3×Q8—21no—
52C2×C3×Q8—31no—
53C2×C3×Q8C2×D311yescharacteristic
54C2×C3×Q8C2×D311yescharacteristic
55C2×Dic6C2×D311yescharacteristic
56C3×Q16—34no—
57C2×C3×C8—31no—
58C2×C3×C8—31no—
59C3×C4⋊C4—31no—
60#216Dic314yescomplemented
61C4⋊Dic3—31no—
62C4⋊Dic3—31no—
63C4⋊Dic3—61no—
64C4⋊Dic3—61no—
65C4⋊Dic3—61no—
66C4⋊Dic3—61no—
67C4⋊Dic3—62no—
68C4⋊Dic3C2×D311yescharacteristic
69Dic3⋊C4—61no—
70C4×Dic3—31no—
71C4×Dic3—31no—
72C4×Dic3—61no—
73C4×Dic3—61no—
74C4×Dic3—61no—
75C4×Dic3—61no—
76C4×Dic3—61no—
77C4×Dic3—62no—
78C2×C3⋊C8C2×D311yescharacteristic
79C2×C3⋊C8C2×D311yescharacteristic
80C22×C32C2×D411yescharacteristic
81C32×C4—22no—
82C32×C4C22×C411yescharacteristic
83C32×C4#4111yescharacteristic
84C3⋊Dic3—21no—
85C3⋊Dic3—41no—
86C3⋊Dic3Q1612yescomplemented
87C3×Dic3—22no—
88C3×Dic3—41no—
89C2×Q16—31no—
90C4×Q8—91no—
91C4×Q8—91no—
92#107—91no—
93#103—31no—
94#103—91no—
95C4×C8—91no—
96C3×Q8—22no—
97C3×Q8—21no—
98C3×Q8—21no—
99C3×Q8—22no—
100C3×Q8—32no—
101C3×Q8—61no—
102C3×Q8C4×D312yes—
103C3×Q8C2×Dic312yes—
104C2×C3×C4—22no—
105C2×C3×C4—21no—
106C2×C3×C4—21no—
107C2×C3×C4—21no—
108C2×C3×C4—61no—
109C2×C3×C4—61no—
110C2×C3×C4C22×D311yescharacteristic
111C2×C3×C4C22×D311yescharacteristic
112C2×Dic3—21no—
113C2×Dic3—21no—
114C2×Dic3—31no—
115C2×Dic3—31no—
116C2×Dic3—31no—
117C2×Dic3—31no—
118C2×Dic3—61no—
119C2×Dic3—61no—
120C2×Dic3—62no—
121C2×Dic3—61no—
122C2×Dic3—61no—
123Dic6—21no—
124Dic6C2×Dic312yes—
125C3×C8—32no—
126C3×C8—61no—
127C3⋊C8—21no—
128C3⋊C8C2×Dic312yes—
129C2×C32C4×D411yescharacteristic
130C2×C32C2×Q1611yescharacteristic
131C2×C32#13511yescharacteristic
132C2×Q8—31no—
133C2×Q8D3211yescharacteristic
134Q16—34no—
135C2×C8—31no—
136C2×C8—31no—
137C4⋊C4—31no—
138C4⋊C4—91no—
139C4⋊C4—181no—
140C4⋊C4—181no—
141C42—91no—
142C42—181no—
143C42—181no—
144C22×C3—21no—
145C22×C3D3×D411yescharacteristic
146C22×C3D3×D411yescharacteristic
147C3×C4—24no—
148C3×C4—22no—
149C3×C4—22no—
150C3×C4—21no—
151C3×C4—21no—
152C3×C4—62no—
153C3×C4—121no—
154C3×C4C2×C4×D311yescharacteristic
155C3×C4#23711yescharacteristic
156C3×C4#23711yescharacteristic
157C3×C4C22×Dic311yescharacteristic
158Dic3—22no—
159Dic3—32no—
160Dic3—32no—
161Dic3—41no—
162Dic3—62no—
163Dic3—61no—
164Dic3—61no—
165Dic3—121no—
166Dic3—122no—
167Dic3—121no—
168Dic3—121no—
169C32C4×Q1611yescharacteristic, complemented, Sylow
170Q8—21no—
171Q8—32no—
172Q8—61no—
173Q8D3×Dic312yes—
174C2×C4—21no—
175C2×C4—61no—
176C2×C4—61no—
177C2×C4—91no—
178C2×C4—91no—
179C2×C4—181no—
180C2×C4C2×D3211yescharacteristic
181C8—32no—
182C8—61no—
183C2×C3—21no—
184C2×C3—21no—
185C2×C3—21no—
186C2×C3#67⋊C411yescharacteristic
187C2×C3#91611yescharacteristic
188C2×C3D3×Q1611yescharacteristic
189C2×C3D3×Q1611yescharacteristic
190C2×C3#91911yescharacteristic
191C2×C3D4×Dic311yescharacteristic
192C22#381111yescharacteristic
193C4—22no—
194C4—62no—
195C4—92no—
196C4—121no—
197C4—181no—
198C4—361no—
199C4C2×D3×Dic311yescharacteristic
200C4#380511yescharacteristic
201C3—21no—
202C3#479011yescharacteristic, complemented
203C3Dic3×Q1611yescharacteristic, complemented
204C2#6394411yescharacteristic
205C2#6395011yescharacteristic
206C2#67⋊Dic311yescharacteristic
subgroupG/H
centerC22#3811
commutatorC32×C4C22×C4
FrattiniC2×C4C2×D32
FittingC2×C32×Q8C22
socle, prime radicalC22×C32C2×D4
cosocleC2×C32×C4C23
hypercenterC2×Q8D32
hypocenterC32C4×Q16
LCSG ▷ 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
16. C3×#216C4semidirect
60. #216Dic3semidirect
86. C3⋊Dic3Q16semidirect
169. C32C4×Q16semidirect
202. C3#4790semidirect
203. C3Dic3×Q16semidirect

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
#638432
C3×#8352
#635752
#635662
#635532
Dic3×Q163
#47903