Finite Group Databasebeta

← #10591314

Subgroups — #10591314

Order 576.

378 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C2×C3×(C2×D3)⋊C4C211yesmaximal, characteristic
2C2×C3×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C2×(C2×D3)⋊Dic3C212yesmaximal
4#63658C211yesmaximal, characteristic, complemented
5#63579C212yesmaximal, complemented
6#4992—31nomaximal
7#4953—31nomaximal
8C23×C3×D3C2211yescharacteristic
9C22×C32×C4C2211yescharacteristic
10C22×C3⋊Dic3C2212yes—
11C22×C3×Dic3C2212yes—
12C22×C3×Dic3C2211yescharacteristic
13C3×(C2×D3)⋊C4—22no—
14C3×C4⋊Dic3—22no—
15(C2×D3)⋊Dic3—24no—
16C2×C3×C4⋊C4—31no—
17C2×C3×C22⋊C4—31no—
18C2×#217—32no—
19C2×(C2×D3)⋊C4—32no—
20C2×(C2×D3)⋊C4D311yescharacteristic
21C2×C4⋊Dic3D311yescharacteristic, complemented
22C3⋊#95—31no—
23C3⋊#95—31no—
24C3⋊#95—32no—
25C3⋊#95—32no—
26C3⋊#95—61no—
27C23×C32C2311yescharacteristic
28C22×C3×D3—21no—
29C22×C3×D3—21no—
30C22×C3×D3—22no—
31C22×C3×D3—22no—
32C22×C3×D3D412yes—
33C2×C32×C4—22no—
34C2×C32×C4D412yes—
35C2×C3⋊Dic3—22no—
36C2×C3⋊Dic3—22no—
37C2×C3⋊Dic3—22no—
38C2×C3×Dic3—22no—
39C2×C3×Dic3—22no—
40C2×C3×Dic3—21no—
41C2×C3×Dic3—22no—
42C2×C3×Dic3—22no—
43#397—91noSylow
44C24×C3—31no—
45C23×D3C2×D311yescharacteristic
46C22×C3×C4—21no—
47C22×C3×C4—31no—
48C22×C3×C4—32no—
49C22×C3×C4C2×D311yescharacteristic
50C22×C3×C4C2×D311yescharacteristic
51C22×Dic3—32no—
52C22×Dic3—32no—
53C22×Dic3—62no—
54C22×Dic3C2×D311yescharacteristic
55C22×Dic3C2×D312yes—
56C3×C4⋊C4—62no—
57C3×C22⋊C4—62no—
58#217—64no—
59(C2×D3)⋊C4—22no—
60(C2×D3)⋊C4—64no—
61C4⋊Dic3—22no—
62C22×C32C2×D411yescharacteristic
63C22×C32C2×D411yescharacteristic
64C22×C32#4111yescharacteristic
65C22×C32#4112yes—
66C22×C32#4112yes—
67C2×C3×D3—22no—
68C2×C3×D3—24no—
69C2×C3×D3—41no—
70C2×C3×D3—41no—
71C2×C3×D3—42no—
72C32×C4—22no—
73C3⋊Dic3—42no—
74C3×Dic3—42no—
75C3×Dic3—41no—
76C2×C4⋊C4—31no—
77C2×C22⋊C4—31no—
78C2×C22⋊C4—92no—
79#95—92no—
80#95—91no—
81C23×C3—21no—
82C23×C3—32no—
83C23×C3—62no—
84C23×C3—61no—
85C23×C3—62no—
86C23×C3—61no—
87C23×C3C22×D311yescharacteristic
88C23×C3C22×D311yescharacteristic
89C22×D3—21no—
90C22×D3—22no—
91C22×D3—22no—
92C22×D3—21no—
93C22×D3D1212yes—
94C2×C3×C4—22no—
95C2×C3×C4—22no—
96C2×C3×C4—22no—
97C2×C3×C4—42no—
98C2×C3×C4—62no—
99C2×C3×C4—62no—
100C2×C3×C4—62no—
101C2×C3×C4—61no—
102C2×C3×C4—62no—
103C2×C3×C4#6712yes—
104C2×C3×C4#6712yes—
105C2×Dic3—22no—
106C2×Dic3—22no—
107C2×Dic3—22no—
108C2×Dic3—22no—
109C2×Dic3—21no—
110C2×Dic3—64no—
111C2×Dic3—62no—
112C2×Dic3—62no—
113C2×Dic3—62no—
114C2×Dic3—62no—
115C2×Dic3—62no—
116C2×Dic3—62no—
117C2×Dic3—122no—
118C2×Dic3—122no—
119C2×C32#12111yescharacteristic
120C2×C32#12312yes—
121C2×C32#12311yescharacteristic
122C2×C32#12411yescharacteristic
123C2×C32#12612yes—
124C3×D3—42no—
125C24—31no—
126C22×C4—31no—
127C22×C4—32no—
128C22×C4—92no—
129C22×C4D3211yescharacteristic
130C4⋊C4—62no—
131C22⋊C4—62no—
132C22⋊C4—184no—
133C22×C3—21no—
134C22×C3—22no—
135C22×C3—21no—
136C22×C3—22no—
137C22×C3—21no—
138C22×C3—64no—
139C22×C3—62no—
140C22×C3—122no—
141C22×C3—121no—
142C22×C3—121no—
143C22×C3C2×D1211yescharacteristic
144C22×C3#23511yescharacteristic
145C22×C3#23512yes—
146C22×C3D3×D411yescharacteristic
147C22×C3#23711yescharacteristic
148C22×C3#23712yes—
149C22×C3#23712yes—
150C22×C3#23912yes—
151C22×C3C2×#6711yescharacteristic
152C22×C3C2×#6711yescharacteristic
153C2×D3—24no—
154C2×D3—22no—
155C2×D3—42no—
156C2×D3—41no—
157C2×D3—41no—
158C3×C4—22no—
159C3×C4—22no—
160C3×C4—42no—
161C3×C4—122no—
162C3×C4—121no—
163Dic3—41no—
164Dic3—42no—
165Dic3—124no—
166Dic3—122no—
167Dic3—122no—
168C32#39711yescharacteristic, complemented, Sylow
169C23—32no—
170C23—61no—
171C23—62no—
172C23—62no—
173C23—61no—
174C23C2×D3211yescharacteristic
175C2×C4—22no—
176C2×C4—62no—
177C2×C4—62no—
178C2×C4—62no—
179C2×C4—61no—
180C2×C4—62no—
181C2×C4—182no—
182C2×C4—182no—
183C2×C4—182no—
184C2×C4#62512yes—
185C2×C3—22no—
186C2×C3—21no—
187C2×C3—22no—
188C2×C3—21no—
189C2×C3—21no—
190C2×C3—122no—
191C2×C3#87912yes—
192C2×C3#88612yes—
193C2×C3#89412yes—
194C2×C3#89812yes—
195C2×C3#92911yescharacteristic
196C2×C3#93011yescharacteristic
197C2×C3#93511yescharacteristic
198C2×C3#93611yescharacteristic
199C2×C3#93811yescharacteristic
200C2×C3#94711yescharacteristic
201D3—42no—
202C22—64no—
203C22—62no—
204C22—122no—
205C22—121no—
206C22—121no—
207C22#379511yescharacteristic
208C22#379612yes—
209C22D3×D1211yescharacteristic
210C22#380412yes—
211C22C2×#62511yescharacteristic
212C4—22no—
213C4—121no—
214C4—122no—
215C4—362no—
216C3—21no—
217C3#495311yescharacteristic, complemented
218C3#499211yescharacteristic, complemented
219C2—122no—
220C2#6385812yes—
221C2#6386311yescharacteristic
222C2#6389012yes—
223C2#6390811yescharacteristic
224C2#6396111yescharacteristic
subgroupG/H
center, FrattiniC23C2×D32
commutator, socle, cosocleC23×C32C23
FittingC22×C32×C4C22
prime radicalC23×C3×D3C22
hypercenterC22×C4D32
hypocenterC32#397
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4⋊Dic3C2semidirect
4. #63658C2semidirect
5. #63579C2semidirect
21. C2×C4⋊Dic3D3semidirect
168. C32#397semidirect
217. C3#4953semidirect
218. C3#4992semidirect

Maximal subgroups

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

subgroupindex
C2×C3×(C2×D3)⋊C42
C2×C3×C4⋊Dic32
C2×(C2×D3)⋊Dic32
#636582
#635792
#49923
#49533