Finite Group Databasebeta

← #83685⋊C3 #11921308

Subgroups — #83685⋊C3 #11921308

Order 1152.

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

normal subgroups only

This group has 247 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, complemented
2C4⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic
3C3⋊C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#83685C311yesmaximal, characteristic, complemented
5#3392⋊C3—31nomaximal
6C2×C3⋊(#41⋊C3)—21no—
7C3×C4×SL(2,𝔽3)—21no—
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9C3×#808—41nomaximal
10C3×#557C2×C311yescharacteristic
11#5577C2×C311yescharacteristic
12#557⋊C3—21no—
13#557⋊C3D311yescharacteristic, complemented
14C4⋊(#41⋊C3)—31no—
15#5085—31no—
16#5083—31no—
17#5027—31no—
18C4×#215—31no—
19Q8×C3⋊C8C2×C311yescharacteristic
20C8×SL(2,𝔽3)—31no—
21C2×C3×SL(2,𝔽3)D411yescharacteristic
22C3⋊(#41⋊C3)—41no—
23C32×C4⋊C4—41no—
24C3×C4⋊Dic3—41no—
25C2×C3×C3⋊C8—41no—
26#3392—31noSylow
27C2×#239—21no—
28C2×#41⋊C3—61no—
29C3×C4⋊Q8—31no—
30C3×#125—61no—
31C3×C4×Q8—21no—
32C3×C4×Q8—31no—
33C3×C4×Q8C22×C311yescharacteristic
34C2×#215—32no—
35C2×#215—31no—
36#895—61no—
37#874—31no—
38C4×D12—31no—
39C4×SL(2,𝔽3)—22no—
40C4×SL(2,𝔽3)—21no—
41C4×SL(2,𝔽3)—21no—
42C4×SL(2,𝔽3)C2×D311yescharacteristic
43C3×#106—121no—
44#835—31no—
45#809—31no—
46#808A411yescharacteristic
47C4⋊(C3⋊C8)—31no—
48C4×C3⋊C8—31no—
49C2×C32×C4—41no—
50C2×C32×C4—81no—
51C2×C3×Dic3—81no—
52C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
53C3×C3⋊C8—42no—
54#557C3×D311yescharacteristic
55#550—31no—
56#492—91no—
57#461—91no—
58#460—91no—
59C8×Q8—31no—
60C4×SD16—91no—
61C2×C3×Q8—31no—
62C2×C3×Q8C3×D411yescharacteristic
63#239—41no—
64#239—121no—
65C2×D12—31no—
66C2×D12—61no—
67C2×D12—61no—
68C2×C4×D3—61no—
69#41⋊C3—121no—
70C2×SL(2,𝔽3)—21no—
71C2×SL(2,𝔽3)#6711yescharacteristic
72C2×C3×C8—121no—
73C3×C4⋊C4—31no—
74C3×C4⋊C4—41no—
75C3×C4⋊C4—61no—
76C3×C4⋊C4—61no—
77C3×C4⋊C4—61no—
78C3×C4⋊C4—81no—
79C3×C4⋊C4—121no—
80C3×C4⋊C4C2×A411yescharacteristic
81C3×C42—31no—
82C3×C42—61no—
83#215—62no—
84#215—122no—
85(C2×D3)⋊C4—61no—
86C4⋊Dic3C2×A411yescharacteristic
87C2×C3⋊C8—31no—
88C2×C3⋊C8C2×A411yescharacteristic
89C22×C32—41no—
90C32×C4—41no—
91C32×C4—41no—
92C32×C4—161no—
93C3×Dic3—161no—
94C2×#41—61no—
95C2×SD16—91no—
96C2×SD16—92no—
97C4⋊Q8—31no—
98#127—91no—
99#125—61no—
100#121—181no—
101C4×Q8—21no—
102C4×Q8—31no—
103C4×Q8C2×C3×D311yescharacteristic
104C4×D4—91no—
105#106—31no—
106C4⋊C8—91no—
107#103—91no—
108#102—91no—
109C4×C8—91no—
110C22×D3—61no—
111C3×Q8—31no—
112C3×Q8—61no—
113C3×Q8—61no—
114C3×Q8C3×SD1611yescharacteristic
115C2×C3×C4—21no—
116C2×C3×C4—31no—
117C2×C3×C4—31no—
118C2×C3×C4—41no—
119C2×C3×C4—61no—
120C2×C3×C4—81no—
121C2×C3×C4—82no—
122C2×C3×C4—81no—
123C2×C3×C4—241no—
124C2×C3×C4C22×A411yescharacteristic
125C2×Dic3—21no—
126D12—61no—
127D12—61no—
128D12—121no—
129D12—121no—
130D12—241no—
131C4×D3—121no—
132C4×D3—121no—
133SL(2,𝔽3)—21no—
134SL(2,𝔽3)#21511yescharacteristic, complemented
135C3×C8—122no—
136C3⋊C8—61no—
137C3⋊C8#41⋊C312yescomplemented
138C2×C32—41no—
139C2×C32—41no—
140C2×C32—41no—
141#41—121no—
142#41—361no—
143C2×Q8—31no—
144C2×Q8C3×#6711yescharacteristic
145C2×D4—91no—
146C2×D4—181no—
147C2×D4—181no—
148C22×C4—181no—
149SD16—182no—
150SD16—362no—
151C2×C8—31no—
152C2×C8—91no—
153C4⋊C4—31no—
154C4⋊C4—31no—
155C4⋊C4—61no—
156C4⋊C4—61no—
157C4⋊C4—61no—
158C4⋊C4D3×A411yescharacteristic
159C22⋊C4—181no—
160C42—31no—
161C42—61no—
162C22×C3—41no—
163C22×C3—81no—
164C22×C3D4×A411yescharacteristic
165C2×D3—121no—
166C2×D3—121no—
167C2×D3—121no—
168C3×C4—31no—
169C3×C4—31no—
170C3×C4—41no—
171C3×C4—41no—
172C3×C4—41no—
173C3×C4—61no—
174C3×C4—81no—
175C3×C4—81no—
176C3×C4—121no—
177C3×C4—161no—
178C3×C4—162no—
179C3×C4—481no—
180C3×C4C2×#41⋊C311yescharacteristic
181C3×C4#99411yescharacteristic
182Dic3—41no—
183C32—41noSylow
184C23—181no—
185Q8—31no—
186Q8—61no—
187Q8—61no—
188Q8C3×#21511yescharacteristic
189D4—181no—
190D4—181no—
191D4—361no—
192D4—361no—
193D4—721no—
194C2×C4—21no—
195C2×C4—31no—
196C2×C4—31no—
197C2×C4—61no—
198C2×C4—61no—
199C2×C4—361no—
200C2×C4—361no—
201C2×C4C2×D3×A411yescharacteristic
202C8—32no—
203C8—181no—
204C2×C3—41no—
205C2×C3—41no—
206C2×C3—41no—
207C2×C3—81no—
208C2×C3—81no—
209C2×C3—81no—
210C2×C3C4⋊(#41⋊C3)11yescharacteristic
211C2×C3A4×SD1611yescharacteristic
212C2×C3#577⋊C311yescharacteristic
213D3—241no—
214C22—361no—
215C22—361no—
216C22—361no—
217C22A4×#6711yescharacteristic
218C4—31no—
219C4—31no—
220C4—41no—
221C4—61no—
222C4—121no—
223C4—121no—
224C4C3⋊#99411yescharacteristic
225C4D3×#41⋊C311yescharacteristic
226C3—41no—
227C3—81no—
228C3#3392⋊C311yescharacteristic, complemented
229C2—721no—
230C2A4×#21511yescharacteristic
231C2#5834⋊C311yescharacteristic
232C2#5837⋊C311yescharacteristic
subgroupG/H
centerC22A4×#67
commutator, cosocleC3×C4×Q8C22×C3
FrattiniC2×C4C2×D3×A4
FittingC3×#557C2×C3
socleC22×C3D4×A4
prime radicalC4⋊(C3⋊(#41⋊C3))C2
hypercenterC4⋊C4D3×A4
hypocenterC3×Q8C3×SD16
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
1. C3×#557⋊C3C2semidirect
3. C3⋊C8×SL(2,𝔽3)C2semidirect
4. #83685C3semidirect
13. #557⋊C3D3semidirect
52. C3×SL(2,𝔽3)SD16semidirect
134. SL(2,𝔽3)#215semidirect
137. C3⋊C8#41⋊C3semidirect
228. C3#3392⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
C4⋊(C3⋊(#41⋊C3))2
C3⋊C8×SL(2,𝔽3)2
#836853
#3392⋊C33
C3×#8084