Finite Group Databasebeta

← #10594582

Subgroups — #10594582

Order 576.

507 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 130 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
1#64096C211yesmaximal, characteristic, complemented
2C3×#893C212yesmaximal, complemented
3C4×#627C211yesmaximal, characteristic, complemented
4#63915C212yesmaximal, complemented
5#63914C211yesmaximal, characteristic
6#63905C214yesmaximal, complemented
7C4×#625C212yesmaximal, complemented
8#63860C212yesmaximal, complemented
9#5559—32nomaximal
10C2×#634C2211yescharacteristic
11C2×C3×D12C2212yes—
12C2×C3×C4×D3C2214yes—
13C2×#627C2211yescharacteristic
14C2×#625C2212yes—
15C2×#625C2212yes—
16C32×C4⋊C4C2211yescharacteristic
17#3750C2212yes—
18C4×C3⋊Dic3C2212yes—
19C4×C3⋊Dic3C2211yescharacteristic
20C3×(C2×D3)⋊C4C2214yes—
21C3×C4×Dic3C2212yes—
22#3724C2211yescharacteristic
23Dic3⋊Dic3C2212yes—
24(C2×D3)⋊Dic3C2214yes—
25(C2×D3)⋊Dic3C2214yes—
26C3×C4×D4—32no—
27#940—32no—
28#936—32no—
29#929—34no—
30C4×#67—34no—
31#893D312yescomplemented
32#887—32no—
33#887—61no—
34#885—34no—
35#883—34no—
36#875—32no—
37C4×Dic6—32no—
38C22×C3×D3C2314yes—
39C2×C32×C4C2312yes—
40C2×C32×C4C2311yescharacteristic
41C2×C3⋊Dic3C2312yes—
42C2×C3⋊Dic3C2311yescharacteristic
43C2×C3⋊Dic3C2311yescharacteristic
44#634—22no—
45C2×C3×Dic3C2312yes—
46C2×C3×Dic3C2312yes—
47C3×D12—24no—
48C3×C4×D3—44no—
49#627—22no—
50#625—24no—
51#625—42no—
52#559—91noSylow
53C2×C3×D4—32no—
54C22×C3×C4—34no—
55C2×#67—34no—
56C2×#67—34no—
57C2×D12C2×D312yes—
58C2×C4×D3C2×D314yes—
59C2×Dic6—32no—
60C2×Dic6—32no—
61C2×Dic6—61no—
62C3×C4⋊C4—21no—
63C3×C4⋊C4C2×D312yes—
64C3×C22⋊C4—34no—
65C3×C42—32no—
66#217—34no—
67#217—34no—
68(C2×D3)⋊C4—34no—
69(C2×D3)⋊C4—34no—
70(C2×D3)⋊C4C2×D314yes—
71C4⋊Dic3—32no—
72Dic3⋊C4—34no—
73Dic3⋊C4—32no—
74Dic3⋊C4—32no—
75Dic3⋊C4—62no—
76C4×Dic3—32no—
77C4×Dic3—34no—
78C4×Dic3—62no—
79C4×Dic3—61no—
80C4×Dic3C2×D312yes—
81C22×C32C2411yescharacteristic
82C2×C3×D3—24no—
83C2×C3×D3—24no—
84C2×C3×D3—24no—
85C32×C4—22no—
86C32×C4#4112yes—
87C3⋊Dic3—22no—
88C3⋊Dic3—21no—
89C3⋊Dic3#4112yes—
90C3×Dic3—22no—
91C3×Dic3—22no—
92#127—91no—
93#124—92no—
94#124—92no—
95#123—94no—
96C4×Q8—91no—
97C4×Q8—91no—
98C4×D4—32no—
99C4×D4—92no—
100C23×C3—34no—
101C22×D3C22×D314yes—
102C3×D4—64no—
103C2×C3×C4—21no—
104C2×C3×C4—22no—
105C2×C3×C4—32no—
106C2×C3×C4—32no—
107C2×C3×C4—124no—
108C2×C3×C4C22×D314yes—
109C2×C3×C4C22×D312yes—
110#67—68no—
111#67—124no—
112C2×Dic3—32no—
113C2×Dic3—32no—
114C2×Dic3—34no—
115C2×Dic3—61no—
116C2×Dic3—62no—
117C2×Dic3—61no—
118C2×Dic3C22×D312yes—
119C2×Dic3C22×D312yes—
120D12—24no—
121C4×D3—44no—
122Dic6—64no—
123Dic6—64no—
124Dic6—64no—
125C2×C32C2×#4111yescharacteristic
126C2×C32C2×#4111yescharacteristic
127C2×C32H(5,𝔽2)11yescharacteristic
128C3×D3—44no—
129C2×Q8—91no—
130C2×Q8—91no—
131C2×D4—32no—
132C2×D4—92no—
133C2×D4—92no—
134C22×C4—34no—
135C4⋊C4—91no—
136C4⋊C4—92no—
137C4⋊C4—92no—
138C4⋊C4D3211yescharacteristic
139C22⋊C4—34no—
140C22⋊C4—94no—
141C22⋊C4—94no—
142C42—32no—
143C42—92no—
144C42—91no—
145C22×C3—21no—
146C22×C3—64no—
147C22×C3—64no—
148C22×C3—64no—
149C22×C3C23×D312yes—
150C2×D3—24no—
151C2×D3—24no—
152C2×D3—24no—
153C3×C4—24no—
154C3×C4—22no—
155C3×C4—42no—
156C3×C4—62no—
157C3×C4—62no—
158C3×C4#23514yes—
159Dic3—22no—
160Dic3—22no—
161Dic3—34no—
162Dic3—62no—
163Dic3—64no—
164Dic3—64no—
165Dic3—62no—
166Dic3—121no—
167C32#55911yescharacteristic, complemented, Sylow
168C23—34no—
169Q8—182no—
170Q8—182no—
171D4—64no—
172D4—184no—
173D4—362no—
174C2×C4—32no—
175C2×C4—32no—
176C2×C4—91no—
177C2×C4—92no—
178C2×C4—91no—
179C2×C4—124no—
180C2×C4C2×D3212yes—
181C2×C4C2×D3211yescharacteristic
182C2×C3—21no—
183C2×C3—21no—
184C2×C3—21no—
185C2×C3—124no—
186C2×C3C2×#23512yes—
187C2×C3#100412yes—
188C2×C3D3×#4112yes—
189D3—44no—
190C22—64no—
191C22—64no—
192C22—64no—
193C22C22×D3211yescharacteristic
194C4—22no—
195C4—62no—
196C4—62no—
197C4—92no—
198C4—182no—
199C4—181no—
200C4#379812yes—
201C3—21no—
202C3#555912yescomplemented
203C2—124no—
204C2C2×#379811yescharacteristic
205C2#6431311yescharacteristic
206C2#6431911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radical#63914C2
hypercenterC4⋊C4D32
hypocenterC32#559
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64096C2semidirect
2. C3×#893C2semidirect
3. C4×#627C2semidirect
4. #63915C2semidirect
6. #63905C2semidirect
7. C4×#625C2semidirect
8. #63860C2semidirect
31. #893D3semidirect
167. C32#559semidirect
202. C3#5559semidirect

Maximal subgroups

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

subgroupindex
#640962
C3×#8932
C4×#6272
#639152
#639142
#639052
C4×#6252
#638602
#55593