Finite Group Databasebeta

← #83701⋊C3 #11921307

Subgroups — #83701⋊C3 #11921307

Order 1152.

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

normal subgroups only

This group has 225 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
2#5570⋊C3C211yesmaximal, characteristic
3C3⋊C8×SL(2,𝔽3)C211yesmaximal, characteristic
4#83701C311yesmaximal, characteristic, complemented
5#3395⋊C3—31nomaximal
6C3×C4×SL(2,𝔽3)—21no—
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8Dic3×SL(2,𝔽3)—21no—
9C3×#807—41nomaximal
10C3×#557C2×C311yescharacteristic
11#5570C2×C311yescharacteristic
12#557⋊C3—21no—
13#557⋊C3—31no—
14#557⋊C3D311yescharacteristic
15#5092—31no—
16#5090—31no—
17#5031—31no—
18C4×#216—31no—
19Q8×C3⋊C8C2×C311yescharacteristic
20C8×SL(2,𝔽3)—31no—
21C2×C3×SL(2,𝔽3)D411yescharacteristic
22C32×C4⋊C4—41no—
23C3×C4⋊Dic3—41no—
24C2×C3×C3⋊C8—41no—
25#3395—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—32no—
33C2×#216—31no—
34#890—61no—
35#869—31no—
36C4×Dic6—31no—
37C4×SL(2,𝔽3)—22no—
38C4×SL(2,𝔽3)—21no—
39C4×SL(2,𝔽3)—21no—
40C4×SL(2,𝔽3)—61no—
41C4×SL(2,𝔽3)C2×D311yescharacteristic
42C3×#107—121no—
43#835—31no—
44#810—31no—
45#807A411yescharacteristic
46C4⋊(C3⋊C8)—31no—
47C4×C3⋊C8—31no—
48C2×C32×C4—41no—
49C2×C32×C4—81no—
50C2×C3×Dic3—81no—
51C3×SL(2,𝔽3)Q1611yescharacteristic, complemented
52C3×C3⋊C8—42no—
53#557—31no—
54#557C3×D311yescharacteristic
55#494—91no—
56#462—91no—
57#462—91no—
58C8×Q8—31no—
59C4×Q16—91no—
60C2×C3×Q8—31no—
61C2×C3×Q8C3×D411yescharacteristic
62C2×Dic6—31no—
63C2×SL(2,𝔽3)—21no—
64C2×SL(2,𝔽3)#6711yescharacteristic
65C2×C3×C8—121no—
66C3×C4⋊C4—31no—
67C3×C4⋊C4—41no—
68C3×C4⋊C4—61no—
69C3×C4⋊C4—61no—
70C3×C4⋊C4—61no—
71C3×C4⋊C4—81no—
72C3×C4⋊C4—121no—
73C3×C4⋊C4C2×A411yescharacteristic
74C3×C42—31no—
75C3×C42—61no—
76#216—62no—
77#216—122no—
78C4⋊Dic3—61no—
79C4⋊Dic3—61no—
80C4⋊Dic3C2×A411yescharacteristic
81Dic3⋊C4—61no—
82C4×Dic3—61no—
83C2×C3⋊C8—31no—
84C2×C3⋊C8C2×A411yescharacteristic
85C22×C32—41no—
86C32×C4—41no—
87C32×C4—41no—
88C32×C4—161no—
89C3×Dic3—161no—
90C2×Q16—91no—
91C2×Q16—92no—
92C4⋊Q8—31no—
93C4⋊Q8—91no—
94#125—61no—
95#125—181no—
96C4×Q8—21no—
97C4×Q8—31no—
98C4×Q8—61no—
99C4×Q8—91no—
100C4×Q8C2×C3×D311yescharacteristic
101#107—31no—
102C4⋊C8—91no—
103#103—91no—
104#103—91no—
105C4×C8—91no—
106C3×Q8—31no—
107C3×Q8—61no—
108C3×Q8—61no—
109C3×Q8C3×Q1611yescharacteristic
110C2×C3×C4—21no—
111C2×C3×C4—31no—
112C2×C3×C4—31no—
113C2×C3×C4—41no—
114C2×C3×C4—61no—
115C2×C3×C4—81no—
116C2×C3×C4—82no—
117C2×C3×C4—81no—
118C2×C3×C4—241no—
119C2×C3×C4C22×A411yescharacteristic
120C2×Dic3—21no—
121C2×Dic3—61no—
122Dic6—61no—
123Dic6—61no—
124SL(2,𝔽3)—21no—
125SL(2,𝔽3)#21611yescharacteristic, complemented
126C3×C8—122no—
127C3⋊C8—61no—
128C3⋊C8#41⋊C312yes—
129C2×C32—41no—
130C2×C32—41no—
131C2×C32—41no—
132C2×Q8—31no—
133C2×Q8—91no—
134C2×Q8C3×#6711yescharacteristic
135Q16—182no—
136Q16—362no—
137C2×C8—31no—
138C2×C8—91no—
139C4⋊C4—31no—
140C4⋊C4—31no—
141C4⋊C4—61no—
142C4⋊C4—61no—
143C4⋊C4—61no—
144C4⋊C4—181no—
145C4⋊C4—181no—
146C4⋊C4—181no—
147C4⋊C4D3×A411yescharacteristic
148C42—31no—
149C42—61no—
150C42—181no—
151C22×C3—41no—
152C22×C3—81no—
153C22×C3D4×A411yescharacteristic
154C3×C4—31no—
155C3×C4—31no—
156C3×C4—41no—
157C3×C4—41no—
158C3×C4—41no—
159C3×C4—61no—
160C3×C4—81no—
161C3×C4—81no—
162C3×C4—121no—
163C3×C4—161no—
164C3×C4—162no—
165C3×C4—481no—
166C3×C4C2×#41⋊C311yescharacteristic
167C3×C4#99411yescharacteristic
168Dic3—41no—
169Dic3—121no—
170C32—41noSylow
171Q8—31no—
172Q8—61no—
173Q8—61no—
174Q8—181no—
175Q8—181no—
176Q8C3×#21611yescharacteristic
177C2×C4—21no—
178C2×C4—31no—
179C2×C4—31no—
180C2×C4—61no—
181C2×C4—61no—
182C2×C4—181no—
183C2×C4C2×D3×A411yescharacteristic
184C8—32no—
185C8—181no—
186C2×C3—41no—
187C2×C3—41no—
188C2×C3—41no—
189C2×C3—81no—
190C2×C3—81no—
191C2×C3—81no—
192C2×C3C4⋊(#41⋊C3)11yescharacteristic
193C2×C3A4×Q1611yescharacteristic
194C2×C3#578⋊C311yescharacteristic
195C22A4×#6711yescharacteristic
196C4—31no—
197C4—31no—
198C4—41no—
199C4—61no—
200C4—121no—
201C4—121no—
202C4—361no—
203C4C3⋊#99411yescharacteristic
204C4D3×#41⋊C311yescharacteristic
205C3—41no—
206C3—81no—
207C3#3395⋊C311yescharacteristic, complemented
208C2A4×#21611yescharacteristic
209C2#5834⋊C311yescharacteristic
210C2#5838⋊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. #83701C3semidirect
51. C3×SL(2,𝔽3)Q16semidirect
125. SL(2,𝔽3)#216semidirect
207. C3#3395⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
#5570⋊C32
C3⋊C8×SL(2,𝔽3)2
#837013
#3395⋊C33
C3×#8074