Finite Group Databasebeta

← #83696⋊C3 #11921310

Subgroups — #83696⋊C3 #11921310

Order 1152.

231 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 45 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×#557⋊C3C211yesmaximal, characteristic
2SL(2,𝔽3)×Dic6C211yesmaximal, characteristic
3C3⋊C8×SL(2,𝔽3)C211yesmaximal, characteristic
4#83696C311yesmaximal, characteristic, complemented
5#3363⋊C3—31nomaximal
6C3×C4×SL(2,𝔽3)—21no—
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8Dic3×SL(2,𝔽3)—21no—
9C3×#810—41nomaximal
10C3×#557C2×C311yescharacteristic
11Q8×Dic6C2×C311yescharacteristic
12Q8×SL(2,𝔽3)—31no—
13#557⋊C3—21no—
14#557⋊C3D311yescharacteristic
15#5090—31no—
16#5079—31no—
17C4×#216—31no—
18Q8×C3⋊C8C2×C311yescharacteristic
19#5021—31no—
20C8×SL(2,𝔽3)—31no—
21C2×C3×Dic6—41no—
22C2×C3×SL(2,𝔽3)D411yescharacteristic
23C32×C4⋊C4—41no—
24C2×C3×C3⋊C8—41no—
25#3363—31noSylow
26C3×C4⋊Q8—31no—
27C3×#125—61no—
28C3×C4×Q8—21no—
29C3×C4×Q8—31no—
30C3×C4×Q8C22×C311yescharacteristic
31Q8×Dic3—21no—
32C2×#216—31no—
33#888—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#835—32no—
43#835—31no—
44#810A411yescharacteristic
45#807—31no—
46C4⋊(C3⋊C8)—31no—
47C4×C3⋊C8—31no—
48C2×C32×C4—41no—
49C2×C32×C4—81no—
50C2×C3×Dic3—81no—
51C3×Dic6—42no—
52C3×Dic6—81no—
53C3×SL(2,𝔽3)Q1611yescharacteristic, complemented
54C3×C3⋊C8—81no—
55Q82—31no—
56#557C3×D311yescharacteristic
57#487—91no—
58#477—91no—
59#462—91no—
60C8×Q8—31no—
61C4×Q16—91no—
62C2×C3×Q8—31no—
63C2×C3×Q8—121no—
64C2×C3×Q8C3×D411yescharacteristic
65C2×Dic6—61no—
66C2×Dic6C2×A411yescharacteristic
67C2×SL(2,𝔽3)—21no—
68C2×SL(2,𝔽3)#6711yescharacteristic
69C2×C3×C8—121no—
70C3×C4⋊C4—31no—
71C3×C4⋊C4—41no—
72C3×C4⋊C4—61no—
73C3×C4⋊C4—61no—
74C3×C4⋊C4—61no—
75C3×C4⋊C4—81no—
76C3×C4⋊C4C2×A411yescharacteristic
77C3×C42—31no—
78C3×C42—61no—
79#216—62no—
80C4⋊Dic3—31no—
81C4⋊Dic3—61no—
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—161no—
90C3×Dic3—82no—
91C2×Q16—91no—
92C4⋊Q8—31no—
93C4⋊Q8—91no—
94C4⋊Q8—181no—
95#125—61no—
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—31no—
108C3×Q8—61no—
109C3×Q8—61no—
110C3×Q8—122no—
111C3×Q8—241no—
112C3×Q8C3×Q1611yescharacteristic
113C2×C3×C4—21no—
114C2×C3×C4—31no—
115C2×C3×C4—31no—
116C2×C3×C4—41no—
117C2×C3×C4—61no—
118C2×C3×C4—81no—
119C2×C3×C4—82no—
120C2×C3×C4—81no—
121C2×C3×C4—241no—
122C2×C3×C4C22×A411yescharacteristic
123C2×Dic3—21no—
124C2×Dic3—61no—
125Dic6—21no—
126Dic6—241no—
127Dic6#41⋊C312yes—
128SL(2,𝔽3)—21no—
129SL(2,𝔽3)#21611yescharacteristic, complemented
130C3×C8—241no—
131C3⋊C8—21no—
132C3⋊C8—61no—
133C2×C32—41no—
134C2×C32—41no—
135C2×C32—41no—
136C2×Q8—31no—
137C2×Q8—31no—
138C2×Q8—181no—
139C2×Q8C3×#6711yescharacteristic
140Q16—182no—
141C2×C8—31no—
142C2×C8—91no—
143C4⋊C4—31no—
144C4⋊C4—61no—
145C4⋊C4—61no—
146C4⋊C4—61no—
147C4⋊C4—91no—
148C4⋊C4—181no—
149C4⋊C4—181no—
150C4⋊C4D3×A411yescharacteristic
151C42—31no—
152C42—61no—
153C42—181no—
154C22×C3—41no—
155C22×C3—81no—
156C22×C3D4×A411yescharacteristic
157C3×C4—31no—
158C3×C4—31no—
159C3×C4—41no—
160C3×C4—41no—
161C3×C4—41no—
162C3×C4—61no—
163C3×C4—81no—
164C3×C4—81no—
165C3×C4—121no—
166C3×C4—161no—
167C3×C4—162no—
168C3×C4—242no—
169C3×C4C2×#41⋊C311yescharacteristic
170C3×C42−1+4⋊C311yescharacteristic
171Dic3—22no—
172Dic3—121no—
173C32—41noSylow
174Q8—31no—
175Q8—32no—
176Q8—61no—
177Q8—61no—
178Q8—61no—
179Q8—721no—
180Q8C3×#21611yescharacteristic
181C2×C4—21no—
182C2×C4—31no—
183C2×C4—31no—
184C2×C4—61no—
185C2×C4—61no—
186C2×C4—181no—
187C2×C4C2×D3×A411yescharacteristic
188C8—61no—
189C8—181no—
190C2×C3—41no—
191C2×C3—41no—
192C2×C3—41no—
193C2×C3—81no—
194C2×C3—81no—
195C2×C3—81no—
196C2×C3#548⋊C311yescharacteristic
197C2×C3A4×Q1611yescharacteristic
198C2×C3#577⋊C311yescharacteristic
199C22A4×#6711yescharacteristic
200C4—31no—
201C4—31no—
202C4—41no—
203C4—62no—
204C4—61no—
205C4—121no—
206C4—361no—
207C4D3×#41⋊C311yescharacteristic
208C4#1010⋊C311yescharacteristic
209C3—41no—
210C3—81no—
211C3#3363⋊C311yescharacteristic, complemented
212C2A4×#21611yescharacteristic
213C2#5831⋊C311yescharacteristic
214C2#5837⋊C311yescharacteristic
subgroupG/H
centerC22A4×#67
commutator, cosocleC3×C4×Q8C22×C3
FrattiniC2×C4C2×D3×A4
FittingC3×#557C2×C3
socleC22×C3D4×A4
prime radicalC2×C3×SL(2,𝔽3)D4
hypercenterC4⋊C4D3×A4
hypocenterC3×Q8C3×Q16
LCSG ▷ C3×C4×Q8 ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #83696C3semidirect
53. C3×SL(2,𝔽3)Q16semidirect
129. SL(2,𝔽3)#216semidirect
211. C3#3363⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
SL(2,𝔽3)×Dic62
C3⋊C8×SL(2,𝔽3)2
#836963
#3363⋊C33
C3×#8104