Finite Group Databasebeta

← #10590556

Subgroups — #10590556

Order 576.

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

normal subgroups only

This group has 210 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×#887C211yesmaximal, characteristic
2#63918C211yesmaximal, characteristic
3#63635C211yesmaximal, characteristic
4C3×#808C211yesmaximal, characteristic
5Dic3⋊(C3⋊C8)C211yesmaximal, characteristic
6#63566C211yesmaximal, characteristic
7#63565C211yesmaximal, characteristic
8#5020—31nomaximal
9#4854—31nomaximal
10C2×C3×Dic6C2211yescharacteristic
11C2×#627—21no—
12C32×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×C4⋊Dic3C2211yescharacteristic
16C3×Dic3⋊C4—21no—
17C3×C4×Dic3—21no—
18C3×C4×Dic3C2211yescharacteristic
19C2×C3×C3⋊C8C2211yescharacteristic
20Dic3⋊Dic3—21no—
21C3×C4×Q8—31no—
22#888—31no—
23#887D311yescharacteristic
24#869—31no—
25C3×#106—31no—
26#835—31no—
27#835—31no—
28#816—31no—
29Dic3⋊C8—31no—
30#810—31no—
31#808—31no—
32#808—31no—
33#808—61no—
34#808D311yescharacteristic
35C4⋊(C3⋊C8)—31no—
36C2×C32×C4—21no—
37C2×C32×C4C2311yescharacteristic
38C2×C3⋊Dic3—21no—
39C2×C3×Dic3—21no—
40C2×C3×Dic3—21no—
41C2×C3×Dic3D411yescharacteristic
42C2×C3×Dic3D411yescharacteristic
43C3×Dic6—21no—
44C3×Dic6Q812yescomplemented
45#627—42no—
46C3⋊(C3⋊C8)—21no—
47C3×C3⋊C8—21no—
48#475—91noSylow
49C2×C3×Q8—31no—
50C2×Dic6—61no—
51C2×Dic6—61no—
52C2×Dic6C2×D311yescharacteristic
53C2×C3×C8—31no—
54C3×C4⋊C4—21no—
55C3×C4⋊C4—31no—
56C3×C4⋊C4—61no—
57C3×C4⋊C4C2×D311yescharacteristic
58C3×C4⋊C4C2×D311yescharacteristic
59C3×C42—31no—
60C3×C42—61no—
61C4⋊Dic3—31no—
62C4⋊Dic3—31no—
63C4⋊Dic3—61no—
64C4⋊Dic3—61no—
65C4⋊Dic3C2×D311yescharacteristic
66Dic3⋊C4—21no—
67Dic3⋊C4—61no—
68C4×Dic3—21no—
69C4×Dic3C2×D311yescharacteristic
70C2×C3⋊C8—31no—
71C2×C3⋊C8—31no—
72C2×C3⋊C8—61no—
73C2×C3⋊C8C2×D311yescharacteristic
74C22×C32C2×D411yescharacteristic
75C32×C4—41no—
76C32×C4C2×Q811yescharacteristic
77C32×C4#4111yescharacteristic
78C3⋊Dic3—41no—
79C3×Dic3—22no—
80C3×Dic3—21no—
81C3×Dic3—41no—
82C3×Dic3SD1612yes—
83C4⋊Q8—91no—
84C4×Q8—31no—
85#106—31no—
86#106—91no—
87C4⋊C8—91no—
88#103—91no—
89#103—91no—
90C3×Q8—32no—
91C3×Q8—61no—
92C2×C3×C4—21no—
93C2×C3×C4—21no—
94C2×C3×C4—21no—
95C2×C3×C4—31no—
96C2×C3×C4—31no—
97C2×C3×C4—41no—
98C2×C3×C4—61no—
99C2×C3×C4—61no—
100C2×C3×C4C22×D311yescharacteristic
101C2×C3×C4C22×D311yescharacteristic
102C2×Dic3—21no—
103C2×Dic3—21no—
104C2×Dic3—61no—
105C2×Dic3—61no—
106C2×Dic3—121no—
107C2×Dic3#6711yescharacteristic
108C2×Dic3#6711yescharacteristic
109Dic6—21no—
110Dic6—122no—
111Dic6—122no—
112Dic6Dic612yescomplemented
113C3×C8—61no—
114C3⋊C8—21no—
115C3⋊C8—62no—
116C3⋊C8—61no—
117C3⋊C8—61no—
118C2×C32C22⋊Q811yescharacteristic
119C2×C32C2×SD1611yescharacteristic
120C2×C32#13711yescharacteristic
121C2×Q8—31no—
122C2×Q8—181no—
123C2×C8—31no—
124C2×C8—91no—
125C4⋊C4—31no—
126C4⋊C4—61no—
127C4⋊C4—91no—
128C4⋊C4—181no—
129C4⋊C4D3211yescharacteristic
130C42—31no—
131C42—61no—
132C22×C3—21no—
133C22×C3D3×D411yescharacteristic
134C22×C3C2×#6711yescharacteristic
135C3×C4—21no—
136C3×C4—21no—
137C3×C4—32no—
138C3×C4—41no—
139C3×C4—41no—
140C3×C4—62no—
141C3×C4—61no—
142C3×C4—81no—
143C3×C4—121no—
144C3×C4C2×Dic611yescharacteristic
145C3×C4#23511yescharacteristic
146C3×C4#23511yescharacteristic
147C3×C4D3×Q811yescharacteristic
148Dic3—22no—
149Dic3—21no—
150Dic3—41no—
151Dic3—121no—
152Dic3—121no—
153Dic3—241no—
154Dic3#21512yes—
155C32#47511yescharacteristic, complemented, Sylow
156Q8—32no—
157Q8—61no—
158Q8—362no—
159C2×C4—21no—
160C2×C4—31no—
161C2×C4—31no—
162C2×C4—61no—
163C2×C4—61no—
164C2×C4—181no—
165C2×C4C2×D3211yescharacteristic
166C8—61no—
167C8—181no—
168C2×C3—21no—
169C2×C3—21no—
170C2×C3—21no—
171C2×C3#89611yescharacteristic
172C2×C3D3×SD1611yescharacteristic
173C2×C3#91511yescharacteristic
174C2×C3#92411yescharacteristic
175C2×C3C2×#21511yescharacteristic
176C2×C3#95111yescharacteristic
177C22D3×#6711yescharacteristic
178C4—32no—
179C4—41no—
180C4—62no—
181C4—61no—
182C4—121no—
183C4—361no—
184C4D3×Dic611yescharacteristic
185C4#379811yescharacteristic
186C3—21no—
187C3#485411yescharacteristic, complemented
188C3#502011yescharacteristic, complemented
189C2#6385111yescharacteristic
190C2#6392911yescharacteristic
191C2D3×#21511yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC32×C4⋊C4C22
socle, prime radicalC22×C32C2×D4
hypercenterC4⋊C4D32
hypocenterC32#475
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
44. C3×Dic6Q8semidirect
112. Dic6Dic6semidirect
155. C32#475semidirect
187. C3#4854semidirect
188. C3#5020semidirect

Maximal subgroups

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

subgroupindex
C3×#8872
#639182
#636352
C3×#8082
Dic3⋊(C3⋊C8)2
#635662
#635652
#50203
#48543