Finite Group Databasebeta

← #79907⋊C3 #11921209

Subgroups — #79907⋊C3 #11921209

Order 1152.

243 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 243 conjugacy classes of subgroups — too many to draw. Only its 51 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
1C4×C3⋊(#41⋊C3)C211yesmaximal, characteristic
2C3×C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3⋊C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#79907C311yesmaximal, characteristic, complemented
5#2991⋊C3—31nomaximal
6C2×C3⋊(#41⋊C3)C411yescharacteristic
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8Dic3×SL(2,𝔽3)C411yescharacteristic
9C3×Dic3⋊C8—41nomaximal
10C4×#239C2×C311yescharacteristic
11C4×#41⋊C3—31no—
12C3×C8×Q8C2×C311yescharacteristic
13Q8×C3⋊C8C2×C311yescharacteristic
14#4838—31no—
15#4837—31no—
16#4688—31no—
17#4685—31no—
18C8×SL(2,𝔽3)—21no—
19C8×SL(2,𝔽3)—31no—
20C8×SL(2,𝔽3)D311yescharacteristic, complemented
21C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
22C3⋊(#41⋊C3)—21no—
23C2×C32×C8—41no—
24C3×C4×Dic3—41no—
25C2×C3×C3⋊C8—41no—
26#2991—31noSylow
27C2×#239C3×C411yescharacteristic
28C2×#41⋊C3—31no—
29C3×C4×Q8C22×C311yescharacteristic
30Q8×Dic3C3×C411yescharacteristic
31C2×ΓL(1,𝔽25)—32no—
32#893—31no—
33#892—31no—
34C4×D12—31no—
35C42×D3—31no—
36C4×SL(2,𝔽3)—21no—
37C4×SL(2,𝔽3)—31no—
38C4×SL(2,𝔽3)C2×D311yescharacteristic
39C3×C4⋊C8—31no—
40C3×C4⋊C8—121no—
41C3×C4×C8—31no—
42(C2×D3)⋊C8—31no—
43(C2×D3)⋊C8—31no—
44Dic3⋊C8—31no—
45Dic3⋊C8A411yescharacteristic
46C4⋊(C3⋊C8)—31no—
47C4×C3⋊C8—31no—
48C2×C32×C4—41no—
49C2×C3×Dic3—41no—
50C2×C3×Dic3—41no—
51C3×SL(2,𝔽3)M1611yescharacteristic, complemented
52C32×C8—81no—
53C3×C3⋊C8—81no—
54C4×#41—31no—
55C8×Q8—31no—
56C8×Q8C3×D311yescharacteristic
57#436—91no—
58#436—91no—
59#431—91no—
60#423—91no—
61C2×C3×Q8C2×C3×C411yescharacteristic
62#239—21no—
63#239—61no—
64C2×D12—31no—
65C2×C4×D3—31no—
66C2×C4×D3—31no—
67C2×C4×D3—31no—
68#41⋊C3—61no—
69C2×SL(2,𝔽3)—21no—
70C2×SL(2,𝔽3)C4×D311yescharacteristic
71C2×C3×C8—31no—
72C2×C3×C8—41no—
73C2×C3×C8—81no—
74C2×C3×C8—121no—
75C2×C3×C8C2×A411yescharacteristic
76C3×C4⋊C4—31no—
77C3×C42—31no—
78C3×C42—121no—
79(C2×D3)⋊C4—32no—
80C4⋊Dic3—31no—
81C4×Dic3—31no—
82C4×Dic3—31no—
83C4×Dic3C2×A411yescharacteristic
84C2×C3⋊C8—31no—
85C2×C3⋊C8C2×A411yescharacteristic
86ΓL(1,𝔽25)—122no—
87C22×C32—41no—
88C32×C4—41no—
89C32×C4—41no—
90C3×Dic3—42no—
91C3×Dic3—81no—
92C2×#41—31no—
93C2×M16—92no—
94C4×Q8—31no—
95C4×Q8C2×C3×D311yescharacteristic
96C4×D4—91no—
97C4×D4—91no—
98#117—91no—
99C2×C42—91no—
100C4⋊C8—31no—
101C4⋊C8—31no—
102C4⋊C8—91no—
103C4⋊C8—91no—
104C22⋊C8—91no—
105C22⋊C8—91no—
106C4×C8—31no—
107C4×C8—91no—
108C22×D3—31no—
109C3×Q8—31no—
110C3×Q8C3×M1611yescharacteristic
111C2×C3×C4—31no—
112C2×C3×C4—31no—
113C2×C3×C4—41no—
114C2×C3×C4—81no—
115C2×C3×C4—121no—
116C2×C3×C4—121no—
117C2×C3×C4C22×A411yescharacteristic
118C2×Dic3—31no—
119C2×Dic3C4×A411yescharacteristic
120C2×Dic3C4×A411yescharacteristic
121D12—61no—
122D12—61no—
123C4×D3—61no—
124C4×D3—61no—
125C4×D3—61no—
126C4×D3—61no—
127C4×D3—62no—
128SL(2,𝔽3)—21no—
129SL(2,𝔽3)ΓL(1,𝔽25)11yescharacteristic, complemented
130C3×C8—21no—
131C3×C8—61no—
132C3×C8—81no—
133C3×C8—82no—
134C3×C8—241no—
135C3⋊C8—21no—
136C3⋊C8—61no—
137C2×C32—41no—
138C2×C32—41no—
139C2×C32—41no—
140#41—61no—
141#41—181no—
142C2×Q8C3×C4×D311yescharacteristic
143C2×D4—91no—
144C22×C4—91no—
145C22×C4—91no—
146C22×C4—91no—
147M16—362no—
148C2×C8—31no—
149C2×C8—31no—
150C2×C8—91no—
151C2×C8D3×A411yescharacteristic
152C4⋊C4—31no—
153C4⋊C4—91no—
154C22⋊C4—92no—
155C42—31no—
156C42—31no—
157C42—91no—
158C42—91no—
159C22×C3—41no—
160C22×C3—81no—
161C22×C3C2×C4×A411yescharacteristic
162C2×D3—61no—
163C2×D3—61no—
164C2×D3—61no—
165C3×C4—31no—
166C3×C4—31no—
167C3×C4—41no—
168C3×C4—41no—
169C3×C4—61no—
170C3×C4—81no—
171C3×C4—81no—
172C3×C4—122no—
173C3×C4—241no—
174C3×C4#99411yescharacteristic
175C3×C42−1+4⋊C311yescharacteristic
176Dic3—21no—
177Dic3—61no—
178Dic3#131⋊C312yes—
179C32—41noSylow
180C23—91no—
181Q8—31no—
182Q8C3×ΓL(1,𝔽25)11yescharacteristic
183D4—181no—
184D4—181no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—31no—
188C2×C4—31no—
189C2×C4—91no—
190C2×C4—181no—
191C2×C4—181no—
192C2×C4—182no—
193C2×C4—181no—
194C2×C4—181no—
195C2×C4C2×D3×A411yescharacteristic
196C8—21no—
197C8—61no—
198C8—61no—
199C8—181no—
200C2×C3—41no—
201C2×C3—41no—
202C2×C3—41no—
203C2×C3—81no—
204C2×C3—81no—
205C2×C3—81no—
206C2×C3#520⋊C311yescharacteristic
207C2×C3A4×M1611yescharacteristic
208C2×C3C2×#131⋊C311yescharacteristic
209D3—121no—
210C22—181no—
211C22—181no—
212C22—181no—
213C22C4×D3×A411yescharacteristic
214C4—31no—
215C4—31no—
216C4—32no—
217C4—61no—
218C4—61no—
219C4—181no—
220C4#1007⋊C311yescharacteristic
221C4C3⋊#99411yescharacteristic
222C3—41no—
223C3—81no—
224C3#2991⋊C311yescharacteristic, complemented
225C2—361no—
226C2#5574⋊C311yescharacteristic
227C2A4×ΓL(1,𝔽25)11yescharacteristic
228C2D3×#131⋊C311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×A4
commutatorC2×C3×Q8C2×C3×C4
FittingC3×C8×Q8C2×C3
socleC22×C3C2×C4×A4
prime radicalC2×C3⋊(#41⋊C3)C4
cosocleC3×C4×Q8C22×C3
hypercenterC2×C8D3×A4
hypocenterC3×Q8C3×M16
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C3×C8×SL(2,𝔽3)C2semidirect
3. C3⋊C8×SL(2,𝔽3)C2semidirect
4. #79907C3semidirect
20. C8×SL(2,𝔽3)D3semidirect
51. C3×SL(2,𝔽3)M16semidirect
129. SL(2,𝔽3)ΓL(1,𝔽25)semidirect
224. C3#2991⋊C3semidirect

Maximal subgroups

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

subgroupindex
C4×C3⋊(#41⋊C3)2
C3×C8×SL(2,𝔽3)2
C3⋊C8×SL(2,𝔽3)2
#799073
#2991⋊C33
C3×Dic3⋊C84