Finite Group Databasebeta

← #76848⋊C3 #11921176

Subgroups — #76848⋊C3 #11921176

Order 1152.

230 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 27 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×#393⋊C3C211yesmaximal, characteristic, complemented
2C3⋊(#393⋊C3)C211yesmaximal, characteristic, complemented
3C3⋊(#395⋊C3)C211yesmaximal, characteristic, complemented
4#76848C311yesmaximal, characteristic, complemented
5#2785⋊C3—31nomaximal
6C3×C22⋊SL(2,𝔽3)C2211yescharacteristic
7A4×#67—41nomaximal
8C3×#393C2×C311yescharacteristic, complemented
9#5231C2×C311yescharacteristic, complemented
10#5230C2×C311yescharacteristic, complemented
11#4959—31no—
12#4945—31no—
13#4669—31no—
14#4650—31no—
15#395⋊C3—31no—
16#393⋊C3—21no—
17#393⋊C3—31no—
18#393⋊C3D311yescharacteristic, complemented
19C22×C3×A4—41no—
20C2×D3×A4—41no—
21Dic3×A4—41no—
22#2785—31noSylow
23C22×C3×Q8C22×C311yescharacteristic
24C22×#67A411yescharacteristic
25C22⋊SL(2,𝔽3)—21no—
26C22⋊SL(2,𝔽3)C2×D311yescharacteristic
27D4×A4—121no—
28C2×C3×C22⋊C4—31no—
29C2×#217—31no—
30C2×(C2×D3)⋊C4—31no—
31C2×(C2×D3)⋊C4—31no—
32C2×Dic3⋊C4—31no—
33C2×Dic3⋊C4—31no—
34C2×C4×Dic3—31no—
35C3×#95—31no—
36C3⋊#95—31no—
37C3⋊#95—31no—
38C3⋊#95—31no—
39C2×C3×A4—41no—
40C2×C3×A4—81no—
41D3×A4—81no—
42C3×#67—161no—
43C3×SL(2,𝔽3)—161no—
44#395—31no—
45#394—91no—
46#393—31no—
47#393C3×D311yescharacteristic, complemented
48#388—91no—
49#388—91no—
50#382—91no—
51C24×C3C2×A411yescharacteristic
52C23×D3C2×A411yescharacteristic
53C22×A4—41no—
54C22×A4—81no—
55C22×A4—121no—
56C2×C3×Q8—121no—
57C22×C3×C4—31no—
58C22×C3×C4—31no—
59C2×#67—121no—
60C22×Dic3—31no—
61C22×Dic3—31no—
62C22×Dic3C2×A411yescharacteristic
63C4×A4—121no—
64C3×C22⋊C4—121no—
65#217—62no—
66(C2×D3)⋊C4—62no—
67(C2×D3)⋊C4—121no—
68Dic3⋊C4—121no—
69Dic3⋊C4—121no—
70C4×Dic3—62no—
71C22×C32—161no—
72C2×C3×D3—161no—
73C3×A4—41no—
74C3×Dic3—161no—
75C22×Q8C2×C3×D311yescharacteristic
76C22×D4—31no—
77C2×C4⋊C4—91no—
78C2×C4⋊C4—91no—
79C2×C22⋊C4—31no—
80C2×C22⋊C4—91no—
81C2×C22⋊C4—91no—
82C2×C22⋊C4—91no—
83C2×C42—91no—
84#95—31no—
85#95—91no—
86#95—91no—
87#95—91no—
88C23×C3—21no—
89C23×C3—61no—
90C23×C3—61no—
91C23×C3C22×A411yescharacteristic
92C22×D3—21no—
93C22×D3—61no—
94C22×D3—61no—
95C2×A4—41no—
96C2×A4—81no—
97C2×A4—82no—
98C2×A4—81no—
99C2×A4—241no—
100C3×Q8—41no—
101C3×Q8—121no—
102C3×D4—481no—
103C2×C3×C4—61no—
104C2×C3×C4—61no—
105C2×C3×C4—61no—
106C2×C3×C4—61no—
107C2×C3×C4—61no—
108C2×C3×C4—61no—
109#67—41no—
110#67—121no—
111C2×Dic3—32no—
112C2×Dic3—61no—
113C2×Dic3—61no—
114C2×Dic3—61no—
115C2×Dic3—61no—
116C2×Dic3—61no—
117C2×Dic3—61no—
118SL(2,𝔽3)—161no—
119SL(2,𝔽3)—162no—
120C2×C32—161no—
121C2×C32—321no—
122C3×D3—321no—
123C24—31no—
124C24D3×A411yescharacteristic
125C2×Q8—121no—
126C2×D4—361no—
127C22×C4—31no—
128C22×C4—31no—
129C22×C4—31no—
130C22×C4—91no—
131C22×C4—91no—
132C4⋊C4—361no—
133C4⋊C4—361no—
134C22⋊C4—121no—
135C22⋊C4—182no—
136C22⋊C4—182no—
137C22⋊C4—361no—
138C42—182no—
139C22×C3—31no—
140C22×C3—31no—
141C22×C3—41no—
142C22×C3—121no—
143C22×C3—121no—
144C22×C3—161no—
145C22×C3—321no—
146C22×C3—481no—
147C22×C32−1+4⋊C311yescharacteristic
148C2×D3—41no—
149C2×D3—121no—
150C2×D3—121no—
151A4—41no—
152A4—81no—
153C3×C4—121no—
154C3×C4—121no—
155C3×C4—481no—
156Dic3—41no—
157Dic3—62no—
158Dic3—121no—
159C32—161noSylow
160C23—21no—
161C23—61no—
162C23—61no—
163C23—61no—
164C23—181no—
165C23—181no—
166C23C2×D3×A411yescharacteristic
167Q8—41no—
168Q8—121no—
169D4—121no—
170D4—361no—
171C2×C4—61no—
172C2×C4—61no—
173C2×C4—61no—
174C2×C4—61no—
175C2×C4—61no—
176C2×C4—61no—
177C2×C4—92no—
178C2×C4—181no—
179C2×C4—181no—
180C2×C4—181no—
181C2×C4—181no—
182C2×C4—181no—
183C2×C4—181no—
184C2×C3—31no—
185C2×C3—31no—
186C2×C3—81no—
187C2×C3—161no—
188C2×C3—321no—
189C2×C3—322no—
190C2×C3—321no—
191C2×C3—961no—
192C2×C3#545011yescharacteristic
193D3—81no—
194C22—31no—
195C22—31no—
196C22—41no—
197C22—121no—
198C22—121no—
199C22—121no—
200C22—361no—
201C22—361no—
202C22#1007⋊C311yescharacteristic
203C4—121no—
204C4—121no—
205C4—121no—
206C4—182no—
207C4—361no—
208C3—161no—
209C3—321no—
210C3#2785⋊C311yescharacteristic, complemented
211C2—31no—
212C2—31no—
213C2—81no—
214C2—241no—
215C2C3⋊#545011yescharacteristic
subgroupG/H
center, hypercenterC2C3⋊#5450
commutator, cosocle, hypocenterC22×C3×Q8C22×C3
FrattiniC23C2×D3×A4
FittingC3×#393C2×C3
socleC23×C3C22×A4
LCSG ▷ C22×C3×Q8
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C3×#393⋊C3C2semidirect
2. C3⋊(#393⋊C3)C2semidirect
3. C3⋊(#395⋊C3)C2semidirect
4. #76848C3semidirect
8. C3×#393C2×C3semidirect
9. #5231C2×C3semidirect
10. #5230C2×C3semidirect
18. #393⋊C3D3semidirect
47. #393C3×D3semidirect
210. C3#2785⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#393⋊C32
C3⋊(#393⋊C3)2
C3⋊(#395⋊C3)2
#768483
#2785⋊C33
A4×#674