Finite Group Databasebeta

← #83707⋊C3 #11921312

Subgroups — #83707⋊C3 #11921312

Order 1152.

237 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 47 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×Q8×SL(2,𝔽3)C211yesmaximal, characteristic
2#5570⋊C3C211yesmaximal, characteristic
3C3⋊C8×SL(2,𝔽3)C211yesmaximal, characteristic
4#83707C311yesmaximal, characteristic, complemented
5#3363⋊C3—31nomaximal
6C3×C4×SL(2,𝔽3)—21no—
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8Dic3×SL(2,𝔽3)—21no—
9C3×#835—41nomaximal
10C3×Q82C2×C311yescharacteristic
11#5570C2×C311yescharacteristic
12Q8×SL(2,𝔽3)—21no—
13Q8×SL(2,𝔽3)D311yescharacteristic
14#557⋊C3—31no—
15#5093—31no—
16#5091—31no—
17#5031—31no—
18C4×#216—31no—
19Q8×C3⋊C8C2×C311yescharacteristic
20C8×SL(2,𝔽3)—31no—
21C2×C32×Q8—41no—
22C2×C3×SL(2,𝔽3)D411yescharacteristic
23C3×C4⋊Dic3—41no—
24C2×C3×C3⋊C8—41no—
25#3363—31noSylow
26C3×C4⋊Q8—31no—
27C3×C4⋊Q8—61no—
28C3×C4×Q8—21no—
29C3×C4×Q8—31no—
30C3×C4×Q8C22×C311yescharacteristic
31Q8×Dic3—21no—
32C2×#216—31no—
33#890—61no—
34#869—31no—
35C4×Dic6—31no—
36C4×SL(2,𝔽3)—22no—
37C4×SL(2,𝔽3)—21no—
38C4×SL(2,𝔽3)—21no—
39C4×SL(2,𝔽3)—61no—
40C4×SL(2,𝔽3)C2×D311yescharacteristic
41C3×#103—121no—
42#835A411yescharacteristic
43#810—32no—
44#810—31no—
45#807—31no—
46C4⋊(C3⋊C8)—31no—
47C4×C3⋊C8—31no—
48C32×Q8—42no—
49C32×Q8—81no—
50C2×C32×C4—41no—
51C2×C32×C4—81no—
52C2×C3×Dic3—81no—
53C3×SL(2,𝔽3)Q1611yescharacteristic, complemented
54C3×C3⋊C8—81no—
55Q82C3×D311yescharacteristic
56#557—31no—
57#487—91no—
58#477—91no—
59#462—91no—
60C8×Q8—31no—
61C4×Q16—91no—
62C2×C3×Q8—41no—
63C2×C3×Q8—61no—
64C2×C3×Q8—81no—
65C2×C3×Q8C3×D411yescharacteristic
66C2×C3×Q8C2×A411yescharacteristic
67C2×Dic6—31no—
68C2×SL(2,𝔽3)—21no—
69C2×SL(2,𝔽3)#6711yescharacteristic
70C2×C3×C8—121no—
71C3×C4⋊C4—31no—
72C3×C4⋊C4—31no—
73C3×C4⋊C4—61no—
74C3×C4⋊C4—61no—
75C3×C4⋊C4—121no—
76C3×C42—31no—
77C3×C42—61no—
78#216—62no—
79C4⋊Dic3—61no—
80C4⋊Dic3—61no—
81C4⋊Dic3C2×A411yescharacteristic
82Dic3⋊C4—61no—
83C4×Dic3—61no—
84C2×C3⋊C8—31no—
85C2×C3⋊C8C2×A411yescharacteristic
86C22×C32—41no—
87C32×C4—41no—
88C32×C4—41no—
89C32×C4—82no—
90C3×Dic3—161no—
91C2×Q16—91no—
92C4⋊Q8—31no—
93C4⋊Q8—61no—
94C4⋊Q8—91no—
95#125—181no—
96C4×Q8—21no—
97C4×Q8—31no—
98C4×Q8—61no—
99C4×Q8—91no—
100C4×Q8C2×C3×D311yescharacteristic
101#107—91no—
102C4⋊C8—91no—
103#103—31no—
104#103—91no—
105#103—92no—
106C4×C8—91no—
107C3×Q8—21no—
108C3×Q8—31no—
109C3×Q8—42no—
110C3×Q8—81no—
111C3×Q8—82no—
112C3×Q8—82no—
113C3×Q8—241no—
114C3×Q8C3×Q1611yescharacteristic
115C3×Q8#41⋊C312yes—
116C2×C3×C4—21no—
117C2×C3×C4—31no—
118C2×C3×C4—31no—
119C2×C3×C4—41no—
120C2×C3×C4—61no—
121C2×C3×C4—81no—
122C2×C3×C4—82no—
123C2×C3×C4—81no—
124C2×C3×C4—241no—
125C2×C3×C4C22×A411yescharacteristic
126C2×Dic3—21no—
127C2×Dic3—61no—
128Dic6—61no—
129Dic6—61no—
130SL(2,𝔽3)—21no—
131SL(2,𝔽3)#21611yescharacteristic, complemented
132C3×C8—241no—
133C3⋊C8—21no—
134C3⋊C8—61no—
135C2×C32—41no—
136C2×C32—41no—
137C2×C32—41no—
138C2×Q8—61no—
139C2×Q8—91no—
140C2×Q8C3×#6711yescharacteristic
141C2×Q8D3×A411yescharacteristic
142Q16—182no—
143C2×C8—31no—
144C2×C8—91no—
145C4⋊C4—31no—
146C4⋊C4—31no—
147C4⋊C4—31no—
148C4⋊C4—61no—
149C4⋊C4—61no—
150C4⋊C4—181no—
151C4⋊C4—181no—
152C4⋊C4—181no—
153C42—31no—
154C42—61no—
155C42—181no—
156C22×C3—41no—
157C22×C3—81no—
158C22×C3D4×A411yescharacteristic
159C3×C4—22no—
160C3×C4—31no—
161C3×C4—31no—
162C3×C4—41no—
163C3×C4—41no—
164C3×C4—61no—
165C3×C4—82no—
166C3×C4—84no—
167C3×C4—81no—
168C3×C4—81no—
169C3×C4—121no—
170C3×C4—481no—
171C3×C4C2×#41⋊C311yescharacteristic
172C3×C42−1+4⋊C311yescharacteristic
173Dic3—41no—
174Dic3—121no—
175C32—41noSylow
176Q8—21no—
177Q8—31no—
178Q8—181no—
179Q8—181no—
180Q8—241no—
181Q8C3×#21611yescharacteristic
182Q8C3⋊(#41⋊C3)12yes—
183C2×C4—21no—
184C2×C4—31no—
185C2×C4—31no—
186C2×C4—61no—
187C2×C4—61no—
188C2×C4—181no—
189C2×C4C2×D3×A411yescharacteristic
190C8—61no—
191C8—181no—
192C2×C3—41no—
193C2×C3—41no—
194C2×C3—41no—
195C2×C3—81no—
196C2×C3—81no—
197C2×C3—81no—
198C2×C3#548⋊C311yescharacteristic
199C2×C3A4×Q1611yescharacteristic
200C2×C3#577⋊C311yescharacteristic
201C22A4×#6711yescharacteristic
202C4—22no—
203C4—31no—
204C4—31no—
205C4—61no—
206C4—121no—
207C4—121no—
208C4—361no—
209C4C2×C3⋊(#41⋊C3)11yescharacteristic
210C4#1007⋊C311yescharacteristic
211C3—41no—
212C3—81no—
213C3#3363⋊C311yescharacteristic, complemented
214C2A4×#21611yescharacteristic
215C2#5817⋊C311yescharacteristic
216C2#5836⋊C311yescharacteristic
subgroupG/H
centerC22A4×#67
commutator, cosocleC3×C4×Q8C22×C3
FrattiniC2×C4C2×D3×A4
FittingC3×Q82C2×C3
socleC22×C3D4×A4
prime radicalC2×C3×SL(2,𝔽3)D4
hypercenterC2×Q8D3×A4
hypocenterC3×Q8C3×Q16
LCSG ▷ C3×C4×Q8 ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
4. #83707C3semidirect
53. C3×SL(2,𝔽3)Q16semidirect
131. SL(2,𝔽3)#216semidirect
213. C3#3363⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×SL(2,𝔽3)2
#5570⋊C32
C3⋊C8×SL(2,𝔽3)2
#837073
#3363⋊C33
C3×#8354