Finite Group Databasebeta

← #10591322

Subgroups — #10591322

Order 576.

482 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C22×#634C211yesmaximal, characteristic, complemented
2C2×C3×(C2×D3)⋊C4C211yesmaximal, characteristic
3C2×(C2×D3)⋊Dic3C212yesmaximal
4C3×C3⋊#95C211yesmaximal, characteristic, complemented
5#63579C212yesmaximal, complemented
6#4947—31nomaximal
7#4673—31nomaximal
8C23×C3×D3C2211yescharacteristic
9C22×C32×C4C2211yescharacteristic
10C22×C3⋊Dic3C2212yes—
11C2×#634—22no—
12C2×#634—24no—
13C22×C3×Dic3C2211yescharacteristic
14C22×C3×Dic3C2212yes—
15C3×(C2×D3)⋊C4—22no—
16(C2×D3)⋊Dic3—24no—
17C22×Dic6—31no—
18C22×Dic6—31no—
19C22×Dic6—61no—
20C2×C3×C22⋊C4—31no—
21C2×#217—32no—
22C2×(C2×D3)⋊C4—32no—
23C2×(C2×D3)⋊C4D311yescharacteristic
24C3×#95—31no—
25C3⋊#95—32no—
26C3⋊#95—32no—
27C3⋊#95D311yescharacteristic, complemented
28C23×C32C2311yescharacteristic
29C22×C3×D3—22no—
30C22×C3×D3—21no—
31C22×C3×D3—21no—
32C22×C3×D3—22no—
33C22×C3×D3Q812yescomplemented
34C2×C32×C4—22no—
35C2×C32×C4D412yes—
36C2×C3⋊Dic3—22no—
37C2×C3⋊Dic3—22no—
38C2×C3⋊Dic3D414yes—
39#634—28no—
40C2×C3×Dic3—22no—
41C2×C3×Dic3—21no—
42C2×C3×Dic3—22no—
43C2×C3×Dic3—22no—
44C2×C3×Dic3—22no—
45#393—91noSylow
46C24×C3—31no—
47C23×D3C2×D311yescharacteristic
48C22×C3×C4—21no—
49C22×C3×C4—31no—
50C22×C3×C4—32no—
51C22×C3×C4C2×D311yescharacteristic
52C22×C3×C4C2×D311yescharacteristic
53C22×Dic3—32no—
54C22×Dic3—32no—
55C22×Dic3—62no—
56C22×Dic3C2×D311yescharacteristic
57C22×Dic3C2×D312yes—
58C2×Dic6—64no—
59C2×Dic6—64no—
60C2×Dic6—68no—
61C2×Dic6—64no—
62C2×Dic6—62no—
63C2×Dic6—62no—
64C3×C22⋊C4—62no—
65#217—64no—
66(C2×D3)⋊C4—22no—
67(C2×D3)⋊C4—64no—
68C22×C32C2×D411yescharacteristic
69C22×C32C2×D412yes—
70C22×C32C2×Q811yescharacteristic
71C22×C32#4112yes—
72C22×C32#4111yescharacteristic
73C2×C3×D3—24no—
74C2×C3×D3—22no—
75C2×C3×D3—41no—
76C2×C3×D3—42no—
77C2×C3×D3—41no—
78C32×C4—22no—
79C3⋊Dic3—24no—
80C3×Dic3—42no—
81C3×Dic3—41no—
82C22×Q8—91no—
83C2×C22⋊C4—31no—
84C2×C22⋊C4—92no—
85#95—31no—
86#95—92no—
87C23×C3—21no—
88C23×C3—32no—
89C23×C3—61no—
90C23×C3—62no—
91C23×C3—62no—
92C23×C3—61no—
93C23×C3C22×D311yescharacteristic
94C23×C3C22×D311yescharacteristic
95C22×D3—21no—
96C22×D3—22no—
97C22×D3—22no—
98C22×D3—21no—
99C22×D3Dic612yescomplemented
100C2×C3×C4—24no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—61no—
105C2×C3×C4—62no—
106C2×C3×C4—62no—
107C2×C3×C4—62no—
108C2×C3×C4—62no—
109C2×C3×C4D1212yes—
110C2×C3×C4#6712yes—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—21no—
116C2×Dic3—34no—
117C2×Dic3—34no—
118C2×Dic3—64no—
119C2×Dic3—64no—
120C2×Dic3—64no—
121C2×Dic3—62no—
122C2×Dic3—62no—
123C2×Dic3—62no—
124C2×Dic3—62no—
125Dic6—616no—
126Dic6—68no—
127Dic6—68no—
128C2×C32#12011yescharacteristic
129C2×C32C22⋊Q812yes—
130C2×C32C22⋊Q811yescharacteristic
131C2×C32#12411yescharacteristic
132C2×C32#12412yes—
133C3×D3—42no—
134C24—31no—
135C2×Q8—184no—
136C2×Q8—182no—
137C22×C4—32no—
138C22×C4—31no—
139C22×C4—92no—
140C22×C4D3211yescharacteristic
141C22⋊C4—62no—
142C22⋊C4—184no—
143C22×C3—21no—
144C22×C3—21no—
145C22×C3—22no—
146C22×C3—21no—
147C22×C3—22no—
148C22×C3—64no—
149C22×C3—62no—
150C22×C3—121no—
151C22×C3—121no—
152C22×C3—122no—
153C22×C3C2×Dic611yescharacteristic
154C22×C3C2×D1211yescharacteristic
155C22×C3#23512yes—
156C22×C3#23511yescharacteristic
157C22×C3D3×D412yes—
158C22×C3D3×D412yes—
159C22×C3#23712yes—
160C22×C3#23711yescharacteristic
161C22×C3D3×Q811yescharacteristic
162C22×C3C2×#6711yescharacteristic
163C2×D3—24no—
164C2×D3—22no—
165C2×D3—42no—
166C2×D3—41no—
167C2×D3—41no—
168C3×C4—24no—
169C3×C4—22no—
170C3×C4—22no—
171C3×C4—121no—
172C3×C4—122no—
173Dic3—42no—
174Dic3—41no—
175Dic3—68no—
176Dic3—64no—
177Dic3—64no—
178C32#39311yescharacteristic, complemented, Sylow
179C23—32no—
180C23—61no—
181C23—62no—
182C23—61no—
183C23—62no—
184C23C2×D3211yescharacteristic
185Q8—188no—
186C2×C4—22no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—61no—
191C2×C4—62no—
192C2×C4—94no—
193C2×C4—182no—
194C2×C4—182no—
195C2×C4#62612yes—
196C2×C3—21no—
197C2×C3—21no—
198C2×C3—22no—
199C2×C3—22no—
200C2×C3—21no—
201C2×C3—122no—
202C2×C3#87511yescharacteristic
203C2×C3#87812yes—
204C2×C3#88211yescharacteristic
205C2×C3#88512yes—
206C2×C3#88512yes—
207C2×C3#89612yes—
208C2×C3#89711yescharacteristic
209C2×C3#92411yescharacteristic
210C2×C3#93611yescharacteristic
211C2×C3#93711yescharacteristic
212D3—42no—
213C22—62no—
214C22—64no—
215C22—121no—
216C22—122no—
217C22—121no—
218C22D3×Dic611yescharacteristic
219C22#379511yescharacteristic
220C22#379512yes—
221C22C2×#62611yescharacteristic
222C22#381112yes—
223C4—22no—
224C4—121no—
225C4—122no—
226C4—184no—
227C3—21no—
228C3#467311yescharacteristic, complemented
229C3#494711yescharacteristic, complemented
230C2—122no—
231C2#6385711yescharacteristic
232C2#6386011yescharacteristic
233C2#6389612yes—
234C2#6391312yes—
235C2#6398111yescharacteristic
subgroupG/H
center, FrattiniC23C2×D32
commutator, socle, cosocleC23×C32C23
FittingC22×C32×C4C22
prime radicalC23×C3×D3C22
hypercenterC22×C4D32
hypocenterC32#393
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#634C2semidirect
4. C3×C3⋊#95C2semidirect
5. #63579C2semidirect
27. C3⋊#95D3semidirect
33. C22×C3×D3Q8semidirect
99. C22×D3Dic6semidirect
178. C32#393semidirect
228. C3#4673semidirect
229. C3#4947semidirect

Maximal subgroups

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

subgroupindex
C22×#6342
C2×C3×(C2×D3)⋊C42
C2×(C2×D3)⋊Dic32
C3×C3⋊#952
#635792
#49473
#46733