Finite Group Databasebeta

← #10595392

Subgroups — #10595392

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
1Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C3×#947C212yesmaximal, complemented
3C4×#627C211yesmaximal, characteristic, complemented
4#63915C212yesmaximal, complemented
5#63914C211yesmaximal, characteristic
6#63902C212yesmaximal, complemented
7Dic3×D12C212yesmaximal, complemented
8#63891C214yesmaximal, complemented
9#5681—32nomaximal
10C2×C32×Q8C2211yescharacteristic
11C2×C3×D12C2212yes—
12C2×#627C2211yescharacteristic
13C2×#625C2212yes—
14C2×#625C2212yes—
15C2×D3×Dic3C2214yes—
16C4⋊(C3⋊Dic3)C2212yes—
17C4⋊(C3⋊Dic3)C2211yescharacteristic
18C4×C3⋊Dic3C2212yes—
19C4×C3⋊Dic3C2211yescharacteristic
20C3×(C2×D3)⋊C4C2214yes—
21C3×(C2×D3)⋊C4C2214yes—
22C3×C4×Dic3C2212yes—
23#3724C2211yescharacteristic
24Dic3⋊Dic3C2212yes—
25(C2×D3)⋊Dic3C2214yes—
26C3×#124—32no—
27#947D312yescomplemented
28Q8×Dic3—32no—
29Q8×Dic3—61no—
30#940—32no—
31D4×Dic3—32no—
32#886—34no—
33#885—34no—
34#883—34no—
35#67⋊C4—34no—
36C4×D12—32no—
37C4×Dic6—32no—
38C22×C3×D3C2314yes—
39C32×Q8—22no—
40C2×C32×C4C2312yes—
41C2×C32×C4C2311yescharacteristic
42C2×C3⋊Dic3C2312yes—
43C2×C3⋊Dic3C2311yescharacteristic
44C2×C3⋊Dic3C2311yescharacteristic
45C2×C3×Dic3C2312yes—
46C2×C3×Dic3C2312yes—
47C3×D12—24no—
48#627—22no—
49#625—24no—
50#625—42no—
51D3×Dic3—44no—
52#559—91noSylow
53C2×C3×Q8—21no—
54C2×C3×Q8C2×D312yes—
55C2×C3×D4—32no—
56C2×#67—34no—
57C2×#67—34no—
58C22×Dic3—34no—
59C2×D12C2×D312yes—
60C2×C4×D3—34no—
61C2×Dic6—32no—
62C3×C22⋊C4—34no—
63C3×C22⋊C4—34no—
64C3×C42—32no—
65#217—34no—
66(C2×D3)⋊C4—34no—
67(C2×D3)⋊C4C2×D314yes—
68(C2×D3)⋊C4C2×D314yes—
69C4⋊Dic3—32no—
70C4⋊Dic3—34no—
71C4⋊Dic3—32no—
72C4⋊Dic3—61no—
73C4⋊Dic3—62no—
74Dic3⋊C4—32no—
75Dic3⋊C4—32no—
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—32no—
94#124—92no—
95#123—94no—
96C4×Q8—91no—
97C4×Q8—91no—
98C4×D4—92no—
99C4×D4—92no—
100C23×C3—34no—
101C22×D3C22×D314yes—
102C3×Q8—24no—
103C3×Q8—24no—
104C3×D4—64no—
105C2×C3×C4—22no—
106C2×C3×C4—21no—
107C2×C3×C4—32no—
108C2×C3×C4—32no—
109C2×C3×C4C22×D314yes—
110C2×C3×C4C22×D312yes—
111#67—68no—
112#67—124no—
113C2×Dic3—32no—
114C2×Dic3—32no—
115C2×Dic3—34no—
116C2×Dic3—61no—
117C2×Dic3—62no—
118C2×Dic3—61no—
119C2×Dic3—124no—
120C2×Dic3C22×D312yes—
121C2×Dic3C22×D312yes—
122D12—24no—
123C4×D3—124no—
124Dic6—64no—
125C2×C32C2×#4111yescharacteristic
126C2×C32C2×#4111yescharacteristic
127C2×C32H(5,𝔽2)11yescharacteristic
128C3×D3—44no—
129C2×Q8—91no—
130C2×Q8D3211yescharacteristic
131C2×D4—32no—
132C2×D4—92no—
133C2×D4—92no—
134C22×C4—94no—
135C4⋊C4—91no—
136C4⋊C4—92no—
137C4⋊C4—91no—
138C4⋊C4—92no—
139C22⋊C4—34no—
140C22⋊C4—34no—
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—24no—
155C3×C4—22no—
156C3×C4—62no—
157C3×C4—62no—
158C3×C4#23714yes—
159Dic3—22no—
160Dic3—22no—
161Dic3—34no—
162Dic3—62no—
163Dic3—64no—
164Dic3—62no—
165Dic3—122no—
166Dic3—121no—
167C32#55911yescharacteristic, complemented, Sylow
168C23—34no—
169Q8—22no—
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—364no—
180C2×C4C2×D3212yes—
181C2×C4C2×D3211yescharacteristic
182C2×C3—21no—
183C2×C3—21no—
184C2×C3—21no—
185C2×C3—124no—
186C2×C3C2×#23712yes—
187C2×C3D3×#4112yes—
188C2×C3#100912yes—
189D3—44no—
190C22—64no—
191C22—64no—
192C22—64no—
193C22C22×D3211yescharacteristic
194C4—22no—
195C4—62no—
196C4—62no—
197C4—92no—
198C4—182no—
199C4—181no—
200C4#380512yes—
201C3—21no—
202C3#568112yescomplemented
203C2—124no—
204C2#6431911yescharacteristic
205C2#6432011yescharacteristic
206C2C2×#380511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×Q8C22
prime radical#63914C2
hypercenterC2×Q8D32
hypocenterC32#559
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. Q8×C3⋊Dic3C2semidirect
2. C3×#947C2semidirect
3. C4×#627C2semidirect
4. #63915C2semidirect
6. #63902C2semidirect
7. Dic3×D12C2semidirect
8. #63891C2semidirect
27. #947D3semidirect
167. C32#559semidirect
202. C3#5681semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊Dic32
C3×#9472
C4×#6272
#639152
#639142
#639022
Dic3×D122
#638912
#56813