Finite Group Databasebeta

← #10591304

Subgroups — #10591304

Order 576.

442 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 108 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×C3×Dic6C211yesmaximal, characteristic, complemented
2C2×C3×(C2×D3)⋊C4C211yesmaximal, characteristic
3C2×(C2×D3)⋊Dic3C212yesmaximal
4#63658C211yesmaximal, characteristic, complemented
5#63579C212yesmaximal, complemented
6#5231—31nomaximal
7#4947—31nomaximal
8C23×C3×D3C2211yescharacteristic
9C22×C32×C4C2211yescharacteristic
10C22×C3⋊Dic3C2212yes—
11C22×C3×Dic3C2211yescharacteristic
12C22×C3×Dic3C2212yes—
13C2×C3×Dic6—22no—
14C2×C3×Dic6—24no—
15C3×(C2×D3)⋊C4—22no—
16(C2×D3)⋊Dic3—24no—
17C22×C3×Q8—31no—
18C22×Dic6D311yescharacteristic, complemented
19C2×C3×C22⋊C4—31no—
20C2×#217—32no—
21C2×(C2×D3)⋊C4—32no—
22C2×(C2×D3)⋊C4D311yescharacteristic
23C3⋊#95—31no—
24C3⋊#95—31no—
25C3⋊#95—32no—
26C3⋊#95—32no—
27C3⋊#95—61no—
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⋊Dic3—22no—
39C2×C3×Dic3—22no—
40C2×C3×Dic3—22no—
41C2×C3×Dic3—21no—
42C2×C3×Dic3—22no—
43C2×C3×Dic3D414yes—
44C3×Dic6—28no—
45#393—91noSylow
46C24×C3—31no—
47C23×D3C2×D311yescharacteristic
48C2×C3×Q8—64no—
49C2×C3×Q8—62no—
50C22×C3×C4—21no—
51C22×C3×C4—32no—
52C22×C3×C4—31no—
53C22×C3×C4C2×D311yescharacteristic
54C22×C3×C4C2×D311yescharacteristic
55C22×Dic3—32no—
56C22×Dic3—32no—
57C22×Dic3—62no—
58C22×Dic3C2×D311yescharacteristic
59C22×Dic3C2×D312yes—
60C2×Dic6—22no—
61C2×Dic6—24no—
62C3×C22⋊C4—62no—
63#217—64no—
64(C2×D3)⋊C4—22no—
65(C2×D3)⋊C4—64no—
66C22×C32C2×D412yes—
67C22×C32C2×D411yescharacteristic
68C22×C32C2×Q811yescharacteristic
69C22×C32#4111yescharacteristic
70C22×C32#4112yes—
71C2×C3×D3—24no—
72C2×C3×D3—22no—
73C2×C3×D3—41no—
74C2×C3×D3—42no—
75C2×C3×D3—41no—
76C32×C4—22no—
77C3⋊Dic3—42no—
78C3×Dic3—24no—
79C3×Dic3—41no—
80C22×Q8—31no—
81C2×C22⋊C4—31no—
82C2×C22⋊C4—92no—
83#95—92no—
84#95—91no—
85C23×C3—21no—
86C23×C3—32no—
87C23×C3—62no—
88C23×C3—61no—
89C23×C3—62no—
90C23×C3—61no—
91C23×C3C22×D311yescharacteristic
92C23×C3C22×D311yescharacteristic
93C22×D3—21no—
94C22×D3—22no—
95C22×D3—22no—
96C22×D3—21no—
97C22×D3Dic612yescomplemented
98C3×Q8—68no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—34no—
103C2×C3×C4—42no—
104C2×C3×C4—62no—
105C2×C3×C4—62no—
106C2×C3×C4—62no—
107C2×C3×C4—61no—
108C2×C3×C4#6712yes—
109C2×C3×C4#6712yes—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—21no—
114C2×Dic3—64no—
115C2×Dic3—62no—
116C2×Dic3—62no—
117C2×Dic3—62no—
118C2×Dic3—62no—
119C2×Dic3—62no—
120C2×Dic3—62no—
121C2×Dic3—122no—
122C2×Dic3—122no—
123C2×Dic3#6714yes—
124Dic6—28no—
125C2×C32#12011yescharacteristic
126C2×C32C22⋊Q812yes—
127C2×C32C22⋊Q811yescharacteristic
128C2×C32#12412yes—
129C2×C32#12411yescharacteristic
130C3×D3—42no—
131C24—31no—
132C2×Q8—64no—
133C2×Q8—62no—
134C22×C4—32no—
135C22×C4—31no—
136C22×C4—92no—
137C22×C4D3211yescharacteristic
138C22⋊C4—62no—
139C22⋊C4—184no—
140C22×C3—21no—
141C22×C3—22no—
142C22×C3—21no—
143C22×C3—21no—
144C22×C3—22no—
145C22×C3—62no—
146C22×C3—64no—
147C22×C3—121no—
148C22×C3—122no—
149C22×C3—121no—
150C22×C3C2×Dic611yescharacteristic
151C22×C3#23511yescharacteristic
152C22×C3D3×D412yes—
153C22×C3#23712yes—
154C22×C3D3×Q811yescharacteristic
155C22×C3#23912yes—
156C22×C3#23911yescharacteristic
157C22×C3C2×#6711yescharacteristic
158C22×C3C2×#6711yescharacteristic
159C22×C3C2×#6712yes—
160C2×D3—24no—
161C2×D3—22no—
162C2×D3—42no—
163C2×D3—41no—
164C2×D3—41no—
165C3×C4—22no—
166C3×C4—22no—
167C3×C4—42no—
168C3×C4—64no—
169C3×C4—121no—
170Dic3—24no—
171Dic3—41no—
172Dic3—124no—
173Dic3—122no—
174Dic3—122no—
175C32#39311yescharacteristic, complemented, Sylow
176C23—32no—
177C23—62no—
178C23—61no—
179C23—61no—
180C23—62no—
181C23C2×D3211yescharacteristic
182Q8—68no—
183C2×C4—22no—
184C2×C4—34no—
185C2×C4—62no—
186C2×C4—62no—
187C2×C4—62no—
188C2×C4—61no—
189C2×C4—182no—
190C2×C4—182no—
191C2×C4—182no—
192C2×C4#62512yes—
193C2×C3—21no—
194C2×C3—21no—
195C2×C3—22no—
196C2×C3—22no—
197C2×C3—21no—
198C2×C3—122no—
199C2×C3#87812yes—
200C2×C3#88512yes—
201C2×C3#92411yescharacteristic
202C2×C3#93611yescharacteristic
203C2×C3#93711yescharacteristic
204C2×C3#94611yescharacteristic
205C2×C3#94612yes—
206C2×C3#94711yescharacteristic
207C2×C3#94712yes—
208C2×C3#95311yescharacteristic
209D3—42no—
210C22—62no—
211C22—64no—
212C22—121no—
213C22—122no—
214C22—121no—
215C22D3×Dic611yescharacteristic
216C22#379612yes—
217C22#379911yescharacteristic
218C22C2×#62511yescharacteristic
219C22D3×#6712yes—
220C4—22no—
221C4—64no—
222C4—121no—
223C4—362no—
224C3—21no—
225C3#494711yescharacteristic, complemented
226C3#523111yescharacteristic, complemented
227C2—122no—
228C2#6385611yescharacteristic
229C2#6386311yescharacteristic
230C2#6388712yes—
231C2#6390712yes—
232C2#6397611yescharacteristic
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×C3×Dic6C2semidirect
4. #63658C2semidirect
5. #63579C2semidirect
18. C22×Dic6D3semidirect
33. C22×C3×D3Q8semidirect
97. C22×D3Dic6semidirect
175. C32#393semidirect
225. C3#4947semidirect
226. C3#5231semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Dic62
C2×C3×(C2×D3)⋊C42
C2×(C2×D3)⋊Dic32
#636582
#635792
#52313
#49473