Finite Group Databasebeta

← #10593244

Subgroups — #10593244

Order 576.

209 conjugacy classes of subgroups; only the 15 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 15 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
1C3⋊#1020C211yesmaximal, characteristic, complemented
2C3×#1020C211yesmaximal, characteristic, complemented
3C3⋊(#120⋊C3)C211yesmaximal, characteristic, complemented
4#1020⋊C2—31nomaximal
5#5032—31nomaximal
6C3×C22⋊A4C2211yescharacteristic, complemented
7D3×S4—41nomaximal
8#1020—31no—
9#1020—61no—
10#1020D311yescharacteristic, complemented
11#1004—31no—
12C3×#120—31no—
13#953—31no—
14#953D311yescharacteristic, complemented
15#120⋊C3—31no—
16#834—31no—
17#806—31no—
18#806—31no—
19D3×A4—41no—
20C3⋊S4—41no—
21C3⋊S4—81no—
22C3×S4—41no—
23C3×S4—81no—
24B(4,𝔽2)—91noSylow
25C24×C3C2×D311yescharacteristic, complemented
26C22⋊A4—21no—
27C22⋊A4C2×D311yescharacteristic, complemented
28C2×S4—121no—
29C2×C3×D4—31no—
30C2×C3×D4—61no—
31C2×#67—31no—
32C2×#67—31no—
33C2×#67—61no—
34C2×#67—61no—
35#237—61no—
36#237—61no—
37D3×D4—61no—
38D3×D4—61no—
39#235—61no—
40C3×C22⋊C4—31no—
41C3×C22⋊C4—61no—
42#217—31no—
43#217—31no—
44#217—61no—
45C3×A4—41no—
46C3×A4—81no—
47C3×A4—81no—
48D32—161no—
49H(5,𝔽2)—91no—
50#120—31no—
51#120—31no—
52#120—91no—
53#99—91no—
54#99—91no—
55#99—91no—
56C23×C3—31no—
57C23×C3—31no—
58C23×C3—61no—
59C23×C3—61no—
60C22×D3—31no—
61C22×D3—61no—
62C22×D3S411yescharacteristic, complemented
63C2×A4—121no—
64S4—41no—
65S4—81no—
66S4—121no—
67S4—241no—
68S4—241no—
69S4—242no—
70C3×D4—61no—
71C3×D4—61no—
72C3×D4—121no—
73C3×D4—121no—
74C2×C3×C4—31no—
75C2×C3×C4—61no—
76#67—61no—
77#67—61no—
78#67—61no—
79#67—61no—
80#67—61no—
81#67—61no—
82#67—121no—
83#67—121no—
84C2×Dic3—31no—
85C2×Dic3—31no—
86C2×Dic3—61no—
87C2×Dic3—61no—
88D12—61no—
89C4×D3—61no—
90C4×D3—61no—
91Dic6—61no—
92C3⋊D3—161no—
93C3×D3—161no—
94C3×D3—161no—
95C24D3211yescharacteristic, complemented
96#41—181no—
97#41—181no—
98#41—181no—
99C2×D4—31no—
100C2×D4—61no—
101C2×D4—91no—
102C2×D4—91no—
103C2×D4—181no—
104C2×D4—181no—
105C2×D4—181no—
106C2×D4—181no—
107C22⋊C4—31no—
108C22⋊C4—61no—
109C22⋊C4—91no—
110C22⋊C4—91no—
111C22⋊C4—181no—
112C22×C3—21no—
113C22×C3—21no—
114C22×C3—31no—
115C22×C3—31no—
116C22×C3—61no—
117C22×C3—61no—
118C22×C3—61no—
119C22×C3—121no—
120C22×C3—121no—
121C22×C3C2×S411yescharacteristic
122C2×D3—61no—
123C2×D3—61no—
124C2×D3—121no—
125C2×D3—241no—
126C2×D3—481no—
127A4—41no—
128A4—81no—
129A4—81no—
130A4—82no—
131A4—81no—
132A4—161no—
133C3×C4—61no—
134C3×C4—121no—
135Dic3—61no—
136Dic3—61no—
137Dic3—121no—
138C32—161noSylow
139C23—31no—
140C23—31no—
141C23—31no—
142C23—61no—
143C23—61no—
144C23—91no—
145C23—181no—
146Q8—181no—
147D4—61no—
148D4—61no—
149D4—121no—
150D4—121no—
151D4—181no—
152D4—181no—
153D4—181no—
154D4—181no—
155D4—181no—
156D4—181no—
157D4—181no—
158D4—361no—
159D4—361no—
160C2×C4—31no—
161C2×C4—61no—
162C2×C4—91no—
163C2×C4—91no—
164C2×C4—181no—
165C2×C4—181no—
166C2×C4—181no—
167C2×C4—181no—
168C2×C3—31no—
169C2×C3—61no—
170C2×C3—61no—
171C2×C3—121no—
172C2×C3—481no—
173D3—41no—
174D3—121no—
175D3—161no—
176D3—481no—
177D3—961no—
178C22—21no—
179C22—21no—
180C22—31no—
181C22—31no—
182C22—61no—
183C22—61no—
184C22—61no—
185C22—121no—
186C22—121no—
187C22—181no—
188C22—181no—
189C22—361no—
190C22—721no—
191C22D3×S411yescharacteristic
192C4—61no—
193C4—121no—
194C4—181no—
195C4—181no—
196C4—361no—
197C3—161no—
198C3—321no—
199C3#1020⋊C211yescharacteristic, complemented
200C2—31no—
201C2—61no—
202C2—61no—
203C2—121no—
204C2—121no—
205C2—361no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC3×C22⋊A4C22
FrattiniC22D3×S4
FittingC24×C3C2×D3
socleC22×C3C2×S4
LCSG ▷ C3×C22⋊A4
UCSC1

Complements

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

NH ≅ G/Ntype
1. C3⋊#1020C2semidirect
2. C3×#1020C2semidirect
3. C3⋊(#120⋊C3)C2semidirect
6. C3×C22⋊A4C22semidirect
10. #1020D3semidirect
14. #953D3semidirect
25. C24×C3C2×D3semidirect
27. C22⋊A4C2×D3semidirect
62. C22×D3S4semidirect
95. C24D32semidirect
199. C3#1020⋊C2semidirect

Maximal subgroups

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

subgroupindex
C3⋊#10202
C3×#10202
C3⋊(#120⋊C3)2
#1020⋊C23
#50323
D3×S44