Finite Group Databasebeta

← #10594512

Subgroups — #10594512

Order 576.

631 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 631 conjugacy classes of subgroups — too many to draw. Only its 142 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×#3798C212yesmaximal, complemented
2#64102C211yesmaximal, characteristic, complemented
3C3×D3×C4⋊C4C212yesmaximal, complemented
4#63919C211yesmaximal, characteristic, complemented
5#63909C212yesmaximal, complemented
6C4×#625C211yesmaximal, characteristic, complemented
7#63865C212yesmaximal, complemented
8#63850C214yesmaximal, complemented
9#5507—32nomaximal
10C2×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C4×C3⋊D3C2212yes—
12C2×C3×C4×D3C2214yes—
13C2×C3×C4×D3C2212yes—
14C2×#627C2212yes—
15C2×#626C2214yes—
16C2×#625C2211yescharacteristic
17#3798—28no—
18C32×C4⋊C4C2211yescharacteristic
19#3752C2212yes—
20C4×C3⋊Dic3C2211yescharacteristic
21C3×C4⋊Dic3C2212yes—
22C3×Dic3⋊C4C2214yes—
23#3724C2212yes—
24#3724C2211yescharacteristic
25#3722C2214yes—
26(C2×D3)⋊Dic3C2212yes—
27C2×#235—34no—
28C2×C3×C4⋊C4—32no—
29#930—32no—
30#929—34no—
31C4×#67—32no—
32#896—34no—
33#895—32no—
34#894—34no—
35#893—32no—
36#893—61no—
37D3×C4⋊C4D312yescomplemented
38#888—32no—
39C22×C3⋊D3C2312yes—
40C22×C3×D3C2312yes—
41C2×C32×C4C2312yes—
42C2×C32×C4C2311yescharacteristic
43C2×C3⋊Dic3C2311yescharacteristic
44C2×C3⋊Dic3C2311yescharacteristic
45C4⋊(C3⋊D3)—22no—
46C4×C3⋊D3—24no—
47C2×C3×Dic3C2312yes—
48C2×C3×Dic3C2314yes—
49C3×C4×D3—28no—
50C3×C4×D3—24no—
51#627—24no—
52#626—28no—
53#625D414yescomplemented
54#547—91noSylow
55C22×C3×C4—32no—
56C22×C3×C4—34no—
57C2×#67—32no—
58C2×#67—34no—
59#235—616no—
60C2×D12—34no—
61C2×D12—32no—
62C2×D12—61no—
63C2×C4×D3—34no—
64C2×C4×D3—62no—
65C2×C4×D3C2×D314yes—
66C2×C4×D3C2×D312yes—
67C2×Dic6—34no—
68C3×C4⋊C4—21no—
69C3×C4⋊C4—32no—
70C3×C4⋊C4—34no—
71C3×C4⋊C4C2×D312yes—
72#217—32no—
73(C2×D3)⋊C4—32no—
74(C2×D3)⋊C4—34no—
75(C2×D3)⋊C4—34no—
76(C2×D3)⋊C4—34no—
77(C2×D3)⋊C4—62no—
78C4⋊Dic3C2×D312yes—
79Dic3⋊C4—34no—
80Dic3⋊C4—32no—
81Dic3⋊C4C2×D314yes—
82C4×Dic3—32no—
83C4×Dic3—61no—
84C22×C32C2411yescharacteristic
85C2×C3⋊D3—22no—
86C2×C3⋊D3—22no—
87C2×C3⋊D3—22no—
88C2×C3×D3—22no—
89C2×C3×D3—22no—
90C2×C3×D3C2×D414yes—
91C32×C4—22no—
92C32×C4#4112yescomplemented
93C3⋊Dic3—21no—
94C3⋊Dic3C2×D412yes—
95C3×Dic3—24no—
96C3×Dic3—22no—
97C2×#41—92no—
98C4⋊Q8—91no—
99#123—94no—
100C22⋊Q8—92no—
101#121—92no—
102C4×D4—91no—
103C4×D4—91no—
104C2×C4⋊C4—32no—
105C23×C3—32no—
106C22×D3—34no—
107C22×D3—62no—
108C22×D3C22×D312yes—
109C2×C3×C4—21no—
110C2×C3×C4—22no—
111C2×C3×C4—32no—
112C2×C3×C4—34no—
113C2×C3×C4—68no—
114C2×C3×C4—64no—
115C2×C3×C4C22×D314yes—
116C2×C3×C4C22×D312yes—
117#67—38no—
118#67—68no—
119C2×Dic3—32no—
120C2×Dic3—32no—
121C2×Dic3—61no—
122C2×Dic3—61no—
123C2×Dic3C22×D312yes—
124C2×Dic3C22×D314yes—
125D12—64no—
126D12—68no—
127D12—64no—
128C4×D3—28no—
129C4×D3—24no—
130C4×D3—68no—
131C4×D3—124no—
132Dic6—68no—
133C2×C32C22×D411yescharacteristic
134C2×C32C2×#4111yescharacteristic
135C2×C322−1+411yescharacteristic
136C3⋊D3—42no—
137C3×D3—24no—
138#41—188no—
139C2×Q8—92no—
140C2×D4—94no—
141C2×D4—91no—
142C2×D4—91no—
143C22×C4—32no—
144C22×C4—34no—
145C22×C4—92no—
146C4⋊C4—32no—
147C4⋊C4—34no—
148C4⋊C4—91no—
149C4⋊C4—92no—
150C4⋊C4D3211yescharacteristic
151C22⋊C4—94no—
152C22⋊C4—92no—
153C22⋊C4—92no—
154C42—91no—
155C22×C3—21no—
156C22×C3—34no—
157C22×C3—62no—
158C22×C3—62no—
159C22×C3C23×D312yes—
160C2×D3—22no—
161C2×D3—22no—
162C2×D3—64no—
163C2×D3—64no—
164C2×D3—64no—
165C2×D3—64no—
166C2×D3—122no—
167C2×D3—122no—
168C2×D3D3×D414yes—
169C3×C4—24no—
170C3×C4—22no—
171C3×C4—42no—
172C3×C4—62no—
173C3×C4—64no—
174C3×C4#23514yescomplemented
175Dic3—22no—
176Dic3—24no—
177Dic3—34no—
178Dic3—62no—
179Dic3—62no—
180Dic3—121no—
181C32#54711yescharacteristic, complemented, Sylow
182C23—32no—
183C23—92no—
184Q8—184no—
185D4—94no—
186D4—188no—
187D4—182no—
188C2×C4—32no—
189C2×C4—34no—
190C2×C4—68no—
191C2×C4—64no—
192C2×C4—91no—
193C2×C4—91no—
194C2×C4—184no—
195C2×C4C2×D3212yes—
196C2×C4C2×D3211yescharacteristic
197C2×C3—21no—
198C2×C3—21no—
199C2×C3—21no—
200C2×C3—64no—
201C2×C3C2×#23512yes—
202C2×C3C2×D3×D412yes—
203C2×C3#100712yes—
204D3—24no—
205D3—124no—
206D3—124no—
207C22—34no—
208C22—62no—
209C22—62no—
210C22—182no—
211C22—182no—
212C22—182no—
213C22C22×D3211yescharacteristic
214C4—22no—
215C4—62no—
216C4—64no—
217C4—92no—
218C4—181no—
219C4#379812yescomplemented
220C3—21no—
221C3#550712yescomplemented
222C2—64no—
223C2—362no—
224C2C2×#379811yescharacteristic
225C2D32×D411yescharacteristic
226C2#6431611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
hypercenterC4⋊C4D32
hypocenterC32#547
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#3798C2semidirect
2. #64102C2semidirect
3. C3×D3×C4⋊C4C2semidirect
4. #63919C2semidirect
5. #63909C2semidirect
6. C4×#625C2semidirect
7. #63865C2semidirect
8. #63850C2semidirect
37. D3×C4⋊C4D3semidirect
53. #625D4semidirect
92. C32×C4#41semidirect
174. C3×C4#235semidirect
181. C32#547semidirect
219. C4#3798semidirect
221. C3#5507semidirect

Maximal subgroups

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

subgroupindex
C2×#37982
#641022
C3×D3×C4⋊C42
#639192
#639092
C4×#6252
#638652
#638502
#55073