Finite Group Databasebeta

← #10591315

Subgroups — #10591315

Order 576.

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

normal subgroups only

This group has 386 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×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
2C2×C3×(C2×D3)⋊C4C211yesmaximal, characteristic
3C2×(C2×D3)⋊Dic3C212yesmaximal
4C3×C3⋊#95C211yesmaximal, characteristic, complemented
5#63579C212yesmaximal, complemented
6#4953—31nomaximal
7#4674—31nomaximal
8C23×C3×D3C2211yescharacteristic
9C22×C32×C4C2211yescharacteristic
10C22×C3⋊Dic3C2212yes—
11C22×C3×Dic3C2211yescharacteristic
12C22×C3×Dic3C2212yes—
13C4⋊(C3⋊Dic3)—22no—
14C3×(C2×D3)⋊C4—22no—
15(C2×D3)⋊Dic3—24no—
16C2×C3×C22⋊C4—31no—
17C2×#217—32no—
18C2×(C2×D3)⋊C4—32no—
19C2×(C2×D3)⋊C4D311yescharacteristic
20C2×C4⋊Dic3—31no—
21C2×C4⋊Dic3—31no—
22C2×C4⋊Dic3—61no—
23C3×#95—31no—
24C3⋊#95—32no—
25C3⋊#95—32no—
26C3⋊#95D311yescharacteristic, complemented
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—21no—
40C2×C3×Dic3—22no—
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×C22⋊C4—62no—
57#217—64no—
58(C2×D3)⋊C4—22no—
59(C2×D3)⋊C4—64no—
60C4⋊Dic3—64no—
61C4⋊Dic3—62no—
62C4⋊Dic3—62no—
63C22×C32C2×D411yescharacteristic
64C22×C32C2×D411yescharacteristic
65C22×C32#4112yes—
66C22×C32#4111yescharacteristic
67C22×C32#4112yes—
68C2×C3×D3—22no—
69C2×C3×D3—24no—
70C2×C3×D3—41no—
71C2×C3×D3—41no—
72C2×C3×D3—42no—
73C32×C4—22no—
74C3⋊Dic3—42no—
75C3×Dic3—42no—
76C3×Dic3—41no—
77C2×C4⋊C4—91no—
78C2×C22⋊C4—31no—
79C2×C22⋊C4—92no—
80#95—31no—
81#95—92no—
82C23×C3—21no—
83C23×C3—32no—
84C23×C3—61no—
85C23×C3—62no—
86C23×C3—62no—
87C23×C3—61no—
88C23×C3C22×D311yescharacteristic
89C23×C3C22×D311yescharacteristic
90C22×D3—21no—
91C22×D3—22no—
92C22×D3—22no—
93C22×D3—21no—
94C22×D3D1212yes—
95C2×C3×C4—24no—
96C2×C3×C4—22no—
97C2×C3×C4—22no—
98C2×C3×C4—22no—
99C2×C3×C4—62no—
100C2×C3×C4—61no—
101C2×C3×C4—62no—
102C2×C3×C4—62no—
103C2×C3×C4—62no—
104C2×C3×C4D1212yes—
105C2×C3×C4#6712yes—
106C2×Dic3—22no—
107C2×Dic3—22no—
108C2×Dic3—22no—
109C2×Dic3—22no—
110C2×Dic3—21no—
111C2×Dic3—64no—
112C2×Dic3—62no—
113C2×Dic3—62no—
114C2×Dic3—62no—
115C2×Dic3—62no—
116C2×Dic3—62no—
117C2×Dic3—62no—
118C2×Dic3—122no—
119C2×Dic3—122no—
120C2×C32#12111yescharacteristic
121C2×C32#12312yes—
122C2×C32#12311yescharacteristic
123C2×C32#12411yescharacteristic
124C2×C32#12612yes—
125C3×D3—42no—
126C24—31no—
127C22×C4—32no—
128C22×C4—31no—
129C22×C4—92no—
130C22×C4D3211yescharacteristic
131C4⋊C4—182no—
132C22⋊C4—62no—
133C22⋊C4—184no—
134C22×C3—22no—
135C22×C3—21no—
136C22×C3—21no—
137C22×C3—22no—
138C22×C3—21no—
139C22×C3—64no—
140C22×C3—62no—
141C22×C3—121no—
142C22×C3—121no—
143C22×C3—122no—
144C22×C3C2×D1211yescharacteristic
145C22×C3C2×D1211yescharacteristic
146C22×C3#23512yes—
147C22×C3#23511yescharacteristic
148C22×C3D3×D411yescharacteristic
149C22×C3#23712yes—
150C22×C3#23712yes—
151C22×C3#23712yes—
152C22×C3#23911yescharacteristic
153C22×C3C2×#6711yescharacteristic
154C2×D3—24no—
155C2×D3—22no—
156C2×D3—42no—
157C2×D3—41no—
158C2×D3—41no—
159C3×C4—24no—
160C3×C4—22no—
161C3×C4—22no—
162C3×C4—121no—
163C3×C4—122no—
164Dic3—42no—
165Dic3—41no—
166Dic3—124no—
167Dic3—122no—
168Dic3—122no—
169C32#39711yescharacteristic, complemented, Sylow
170C23—32no—
171C23—62no—
172C23—61no—
173C23—62no—
174C23—61no—
175C23C2×D3211yescharacteristic
176C2×C4—22no—
177C2×C4—62no—
178C2×C4—62no—
179C2×C4—62no—
180C2×C4—62no—
181C2×C4—61no—
182C2×C4—182no—
183C2×C4—182no—
184C2×C4—182no—
185C2×C4#62612yes—
186C2×C3—21no—
187C2×C3—22no—
188C2×C3—22no—
189C2×C3—21no—
190C2×C3—21no—
191C2×C3—122no—
192C2×C3#87511yescharacteristic
193C2×C3#87912yes—
194C2×C3#88312yes—
195C2×C3#88612yes—
196C2×C3#88611yescharacteristic
197C2×C3#89511yescharacteristic
198C2×C3#89812yes—
199C2×C3#93011yescharacteristic
200C2×C3#93511yescharacteristic
201C2×C3#93611yescharacteristic
202D3—42no—
203C22—64no—
204C22—62no—
205C22—121no—
206C22—122no—
207C22—121no—
208C22#379512yes—
209C22#379911yescharacteristic
210C22D3×D1211yescharacteristic
211C22#380512yes—
212C22C2×#62611yescharacteristic
213C4—22no—
214C4—121no—
215C4—122no—
216C4—362no—
217C3—21no—
218C3#467411yescharacteristic, complemented
219C3#495311yescharacteristic, complemented
220C2—122no—
221C2#6386011yescharacteristic
222C2#6386112yes—
223C2#6389112yes—
224C2#6390911yescharacteristic
225C2#6396211yescharacteristic
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
1. C2×C4⋊(C3⋊Dic3)C2semidirect
4. C3×C3⋊#95C2semidirect
5. #63579C2semidirect
26. C3⋊#95D3semidirect
169. C32#397semidirect
218. C3#4674semidirect
219. C3#4953semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊Dic3)2
C2×C3×(C2×D3)⋊C42
C2×(C2×D3)⋊Dic32
C3×C3⋊#952
#635792
#49533
#46743