Finite Group Databasebeta

← #11920553

Subgroups — #11920553

Order 1152.

245 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 41 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×C4×GL(2,𝔽3)C211yesmaximal, characteristic
2C3⋊C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3⋊(SL(2,𝔽3)⋊C8)C211yesmaximal, characteristic, complemented
4#73018—31nomaximal
5#70896—31nomaximal
6C2×C3×GL(2,𝔽3)C411yescharacteristic
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C2×#3711—41nomaximal
9C3×#859C411yescharacteristic
10C4×GL(2,𝔽3)D311yescharacteristic
11C3×C4×SD16—31no—
12Q8×C3⋊C8D311yescharacteristic, complemented
13#5012—31no—
14C8×SL(2,𝔽3)—31no—
15SL(2,𝔽3)⋊C8—31no—
16SL(2,𝔽3)⋊C8—61no—
17#4535—31no—
18D4⋊(C3⋊C8)—31no—
19#4457—31no—
20#4454—31no—
21C2×C3×C4×D3—41no—
22C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
23C3×GL(2,𝔽3)—21no—
24C2×C3⋊(C3⋊C8)—41no—
25C2×C3×C3⋊C8—41no—
26#3711—44no—
27#1583—91noSylow
28C2×GL(2,𝔽3)Dic311yescharacteristic
29C2×C3×SD16—31no—
30C3×C4×Q8C2×D311yescharacteristic
31C3×C4×D4—31no—
32C2×#208—61no—
33C2×ΓL(1,𝔽25)—121no—
34C4×SL(2,𝔽3)—21no—
35C4×SL(2,𝔽3)C2×D311yescharacteristic
36#859Dic311yescharacteristic
37C3×#106—31no—
38C3×#103—31no—
39C3×#102—31no—
40C3×C4×C8—31no—
41#830—61no—
42C4⋊(C3⋊C8)—31no—
43C4⋊(C3⋊C8)—31no—
44C4×C3⋊C8—31no—
45C4×C3⋊C8—31no—
46C22×C3×D3—41no—
47C2×C32×C4—41no—
48C2×C3×Dic3—41no—
49C3×C4×D3—42no—
50C3×C4×D3—42no—
51C3×SL(2,𝔽3)M1611yescharacteristic, complemented
52C3⋊(C3⋊C8)—42no—
53C3×C3⋊C8—42no—
54C8×Q8—31no—
55C4×SD16—31no—
56#436—91no—
57#334—91no—
58Q8⋊C8—91no—
59D4⋊C8—91no—
60#322—91no—
61C2×C3×Q8C4×D311yescharacteristic
62C2×C3×D4—31no—
63C22×C3×C4—61no—
64C2×C4×D3—41no—
65C2×SL(2,𝔽3)—21no—
66C2×SL(2,𝔽3)C2×Dic311yescharacteristic
67GL(2,𝔽3)—21no—
68C3×SD16—61no—
69C3×SD16—61no—
70C2×C3×C8—31no—
71C2×C3×C8—31no—
72C2×C3×C8—121no—
73C3×C4⋊C4—31no—
74C3×C4⋊C4—31no—
75C3×C22⋊C4—61no—
76C3×C42—31no—
77#208—122no—
78C2×C3⋊C8—31no—
79C2×C3⋊C8—61no—
80C2×C3⋊C8—61no—
81C2×C3⋊C8—121no—
82C2×C3⋊C8—241no—
83C2×C3⋊C8S411yescharacteristic
84ΓL(1,𝔽25)—124no—
85C22×C32—41no—
86C2×C3×D3—42no—
87C2×C3×D3—81no—
88C2×C3×D3—81no—
89C32×C4—41no—
90C32×C4—41no—
91C3×Dic3—42no—
92C2×SD16—31no—
93C2×M16—181no—
94C4×Q8D3211yescharacteristic
95C4×D4—31no—
96#106—31no—
97C4⋊C8—91no—
98C4⋊C8—91no—
99#103—31no—
100#102—31no—
101C22⋊C8—181no—
102C4×C8—31no—
103C4×C8—91no—
104C4×C8—91no—
105C23×C3—61no—
106C22×D3—41no—
107C3×Q8—31no—
108C3×Q8ΓL(1,𝔽25)11yescharacteristic, complemented
109C3×D4—61no—
110C3×D4—61no—
111C2×C3×C4—31no—
112C2×C3×C4—31no—
113C2×C3×C4—41no—
114C2×C3×C4—61no—
115C2×C3×C4—81no—
116C2×C3×C4—121no—
117C2×C3×C4—121no—
118C2×C3×C4C2×S411yescharacteristic
119C2×Dic3—41no—
120C4×D3—42no—
121C4×D3—42no—
122SL(2,𝔽3)—21no—
123SL(2,𝔽3)#20811yescharacteristic, complemented
124C3×C8—61no—
125C3×C8—61no—
126C3×C8—122no—
127C3⋊C8—61no—
128C3⋊C8—122no—
129C3⋊C8—121no—
130C3⋊C8—121no—
131C3⋊C8—242no—
132C3⋊C8GL(2,𝔽3)12yescomplemented
133C2×C32—41no—
134C2×C32—41no—
135C2×C32—41no—
136C3×D3—82no—
137C2×Q8D3×Dic311yescharacteristic
138C2×D4—31no—
139C22×C4—61no—
140SD16—61no—
141SD16—61no—
142M16—362no—
143C2×C8—31no—
144C2×C8—31no—
145C2×C8—31no—
146C2×C8—91no—
147C2×C8—181no—
148C2×C8—181no—
149C4⋊C4—31no—
150C4⋊C4—31no—
151C22⋊C4—61no—
152C42—31no—
153C22×C3—41no—
154C22×C3—81no—
155C22×C3—121no—
156C22×C3—121no—
157C22×C3—121no—
158C22×C3C4×S411yescharacteristic
159C2×D3—42no—
160C2×D3—81no—
161C2×D3—81no—
162C3×C4—31no—
163C3×C4—31no—
164C3×C4—41no—
165C3×C4—41no—
166C3×C4—61no—
167C3×C4—81no—
168C3×C4—81no—
169C3×C4—121no—
170C3×C4C2×GL(2,𝔽3)11yescharacteristic
171C3×C4#98511yescharacteristic
172Dic3—42no—
173C32—41noSylow
174C23—61no—
175Q8—31no—
176Q8#371111yescharacteristic, complemented
177D4—61no—
178D4—61no—
179C2×C4—31no—
180C2×C4—31no—
181C2×C4—61no—
182C2×C4—121no—
183C2×C4—121no—
184C2×C4D3×S411yescharacteristic
185C8—32no—
186C8—61no—
187C8—61no—
188C8—181no—
189C8—361no—
190C8—361no—
191C2×C3—41no—
192C2×C3—41no—
193C2×C3—41no—
194C2×C3—81no—
195C2×C3—81no—
196C2×C3—81no—
197C2×C3—241no—
198C2×C3C4×GL(2,𝔽3)11yescharacteristic
199C2×C3#540011yescharacteristic
200C2×C3#540511yescharacteristic
201D3—82no—
202C22—121no—
203C22—121no—
204C22—121no—
205C22Dic3×S411yescharacteristic
206C4—31no—
207C4—31no—
208C4—61no—
209C4—121no—
210C4#6419711yescharacteristic
211C4D3×GL(2,𝔽3)11yescharacteristic
212C3—41no—
213C3—81no—
214C3#7301811yescharacteristic, complemented
215C2—241no—
216C2#1059317511yescharacteristic
217C2Dic3×GL(2,𝔽3)11yescharacteristic
218C2#1059322111yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D3×S4
commutatorC2×C3×SL(2,𝔽3)C2×C4
FittingC3×C4×Q8C2×D3
socleC22×C3C4×S4
prime radicalC2×C3×GL(2,𝔽3)C4
cosocleC3×C4×SL(2,𝔽3)C22
hypocenterC3×SL(2,𝔽3)M16
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3⋊C8×SL(2,𝔽3)C2semidirect
3. C3⋊(SL(2,𝔽3)⋊C8)C2semidirect
12. Q8×C3⋊C8D3semidirect
51. C3×SL(2,𝔽3)M16semidirect
108. C3×Q8ΓL(1,𝔽25)semidirect
123. SL(2,𝔽3)#208semidirect
132. C3⋊C8GL(2,𝔽3)semidirect
176. Q8#3711semidirect
214. C3#73018semidirect

Maximal subgroups

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

subgroupindex
C3×C4×GL(2,𝔽3)2
C3⋊C8×SL(2,𝔽3)2
C3⋊(SL(2,𝔽3)⋊C8)2
#730183
#708963
C2×#37114