Finite Group Databasebeta

← #11920689

Subgroups — #11920689

Order 1152.

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

normal subgroups only

This group has 301 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
1#5827⋊C3C211yesmaximal, characteristic
2#10593660C211yesmaximal, characteristic
3C2×C3×GU(2,𝔽3)C211yesmaximal, characteristic
4#73076—31nomaximal
5#70363—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7Dic3×SL(2,𝔽3)—21no—
8C3⋊#859—21no—
9C3×GU(2,𝔽3)C412yescomplemented
10#63573—41nomaximal
11#5827D311yescharacteristic
12#520⋊C3—31no—
13#5426—31no—
14#5426—61no—
15C2×GU(2,𝔽3)D311yescharacteristic
16C2×C3×#104—31no—
17#5232—31no—
18#5121—31no—
19#4935—31no—
20#4554—31no—
21#4523—31no—
22C3×#41⋊C3C2×C411yescharacteristic
23C3×#41⋊C3Q811yescharacteristic
24C2×C3×SL(2,𝔽3)D411yescharacteristic
25C4⋊(C3⋊Dic3)—41no—
26C3×C4⋊Dic3—41no—
27C2×C3×C3⋊C8—41no—
28#1859—91noSylow
29C2×C3×#41C2×D311yescharacteristic
30C2×#41⋊C3—21no—
31C2×#41⋊C3C2×D311yescharacteristic
32C2×C3×M16—31no—
33C2×C3×C42—31no—
34Q8×Dic3—21no—
35D4×Dic3—61no—
36#926—31no—
37C2×C4⋊Dic3—31no—
38C2×C4⋊Dic3—31no—
39C2×C4⋊Dic3—62no—
40C2×#208—31no—
41C4×SL(2,𝔽3)—61no—
42GU(2,𝔽3)Dic312yescomplemented
43#859—62no—
44#859—61no—
45C3×#104—32no—
46C3×#104—32no—
47#835—61no—
48#832—61no—
49#818—61no—
50#817—61no—
51#808—121no—
52C2×C32×C4—41no—
53C2×C3⋊Dic3—81no—
54C2×C3×Dic3—81no—
55C3×SL(2,𝔽3)C4⋊C411yescharacteristic, complemented
56C3×C3⋊C8—42no—
57#520—31no—
58#428—91no—
59C2×#104—31no—
60#417—91no—
61#384—91no—
62#343—91no—
63#339—91no—
64C3×#41—31no—
65C3×#41—31no—
66C3×#41Dic611yescharacteristic
67C3×#41C4×D311yescharacteristic
68C2×C3×Q8#6711yescharacteristic
69C2×C3×D4—31no—
70C22×C3×C4—31no—
71C22×C3×C4—61no—
72C22×Dic3—32no—
73C22×Dic3—61no—
74#41⋊C3—21no—
75#41⋊C3—21no—
76#41⋊C3Dic611yescharacteristic
77#41⋊C3C2×Dic311yescharacteristic
78C2×SL(2,𝔽3)—21no—
79C2×SL(2,𝔽3)D1211yescharacteristic
80C3×M16—32no—
81C3×M16—61no—
82C2×C3×C8—61no—
83C3×C4⋊C4—121no—
84C3×C42—32no—
85C3×C42—61no—
86#217—61no—
87C4⋊Dic3—31no—
88C4⋊Dic3—61no—
89C4⋊Dic3—61no—
90C4⋊Dic3—61no—
91C4⋊Dic3—121no—
92C4⋊Dic3—124no—
93C4⋊Dic3—241no—
94C4⋊Dic3S411yescharacteristic
95C4×Dic3—61no—
96#208—61no—
97#208—61no—
98C2×C3⋊C8—41no—
99C2×C3⋊C8—61no—
100C22×C32—41no—
101C32×C4—41no—
102C32×C4—41no—
103C3⋊Dic3—161no—
104C3×Dic3—161no—
105C2×#41D3211yescharacteristic
106C2×M16—31no—
107C2×M16—91no—
108C4×Q8—61no—
109C4×D4—181no—
110#117—91no—
111C2×C4⋊C4—91no—
112C2×C4⋊C4—91no—
113C2×C4⋊C4—182no—
114C2×C42—31no—
115#107—181no—
116#106—181no—
117#104—32no—
118#104—32no—
119#103—181no—
120#102—181no—
121C23×C3—31no—
122C3×Q8—31no—
123C3×Q8Dic3⋊C411yescharacteristic
124C3×D4—31no—
125C3×D4—31no—
126C3×D4—61no—
127C2×C3×C4—31no—
128C2×C3×C4—31no—
129C2×C3×C4—31no—
130C2×C3×C4—31no—
131C2×C3×C4—31no—
132C2×C3×C4—41no—
133C2×C3×C4—62no—
134C2×C3×C4—62no—
135C2×C3×C4—62no—
136C2×C3×C4—81no—
137C2×C3×C4—241no—
138C2×C3×C4C2×S411yescharacteristic
139C2×Dic3—21no—
140C2×Dic3—61no—
141C2×Dic3—122no—
142C2×Dic3—122no—
143C2×Dic3—121no—
144C2×Dic3—241no—
145C2×Dic3—242no—
146SL(2,𝔽3)—21no—
147SL(2,𝔽3)C4⋊Dic311yescharacteristic, complemented
148C3×C8—62no—
149C3⋊C8—42no—
150C3⋊C8—121no—
151C2×C32—41no—
152C2×C32—41no—
153C2×C32—41no—
154#41—31no—
155#41—31no—
156#41D3×Dic311yescharacteristic
157#41#62711yescharacteristic
158C2×Q8#62611yescharacteristic
159C2×D4—31no—
160C22×C4—31no—
161C22×C4—61no—
162C22×C4—92no—
163C22×C4—181no—
164M16—32no—
165M16—61no—
166M16—181no—
167M16—181no—
168C2×C8—61no—
169C2×C8—181no—
170C4⋊C4—31no—
171C4⋊C4—91no—
172C4⋊C4—181no—
173C4⋊C4—181no—
174C4⋊C4—181no—
175C4⋊C4—364no—
176C22⋊C4—181no—
177C42—32no—
178C42—61no—
179C42—181no—
180C22×C3—31no—
181C22×C3—31no—
182C22×C3—41no—
183C22×C3—61no—
184C22×C3—61no—
185C22×C3—81no—
186C22×C3GL(2,ℤ/4)11yescharacteristic
187C3×C4—31no—
188C3×C4—31no—
189C3×C4—41no—
190C3×C4—41no—
191C3×C4—64no—
192C3×C4—81no—
193C3×C4—81no—
194C3×C4—481no—
195C3×C4A4⋊Q811yescharacteristic
196C3×C4C4×S411yescharacteristic
197Dic3—41no—
198Dic3—121no—
199Dic3—241no—
200Dic3—481no—
201Dic3—482no—
202C32—41noSylow
203C23—31no—
204Q8—31no—
205Q8Dic3⋊Dic311yescharacteristic
206D4—31no—
207D4—31no—
208D4—61no—
209C2×C4—31no—
210C2×C4—31no—
211C2×C4—31no—
212C2×C4—31no—
213C2×C4—31no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4—62no—
217C2×C4—61no—
218C2×C4—181no—
219C2×C4—362no—
220C2×C4—362no—
221C2×C4—361no—
222C2×C4D3×S411yescharacteristic
223C8—62no—
224C8—361no—
225C2×C3—41no—
226C2×C3—41no—
227C2×C3—41no—
228C2×C3—61no—
229C2×C3—61no—
230C2×C3—81no—
231C2×C3—81no—
232C2×C3—81no—
233C2×C3SL(2,ℤ/4)⋊C411yescharacteristic
234C2×C3#542911yescharacteristic
235C2×C3#543011yescharacteristic
236C22—31no—
237C22—31no—
238C22—61no—
239C22—61no—
240C22#6421011yescharacteristic
241C4—31no—
242C4—31no—
243C4—64no—
244C4—121no—
245C4—361no—
246C4—721no—
247C4#6420411yescharacteristic
248C4Dic3×S411yescharacteristic
249C3—41no—
250C3—81no—
251C3#7307611yescharacteristic, complemented
252C2—61no—
253C2—61no—
254C2#1059323211yescharacteristic
255C2#1059323411yescharacteristic
256C2SL(2,ℤ/4)⋊Dic311yescharacteristic
subgroupG/H
centerC22#64210
commutatorC3×#41⋊C3C2×C4
Frattini, hypercenterC2×C4D3×S4
FittingC2×C3×#41C2×D3
socleC22×C3GL(2,ℤ/4)
prime radical, cosocleC2×C3×#41⋊C3C22
hypocenterC3×SL(2,𝔽3)C4⋊C4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
9. C3×GU(2,𝔽3)C4semidirect
42. GU(2,𝔽3)Dic3semidirect
55. C3×SL(2,𝔽3)C4⋊C4semidirect
147. SL(2,𝔽3)C4⋊Dic3semidirect
251. C3#73076semidirect

Maximal subgroups

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

subgroupindex
#5827⋊C32
#105936602
C2×C3×GU(2,𝔽3)2
#730763
#703633
#635734