Finite Group Databasebeta

← #11920699

Subgroups — #11920699

Order 1152.

309 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 55 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
1C2×C3⋊(#131⋊C3)C211yesmaximal, characteristic
2#10593658C211yesmaximal, characteristic
3C2×C3×GU(2,𝔽3)C211yesmaximal, characteristic
4#73123—31nomaximal
5#70370—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7C3⋊(#131⋊C3)C412yescomplemented
8C3⋊#859C412yescomplemented
9C3×GU(2,𝔽3)C412yescomplemented
10#63555—41nomaximal
11C2×#948D311yescharacteristic
12C2×#131⋊C3—31no—
13#5424—31no—
14#5424—61no—
15C2×GU(2,𝔽3)D311yescharacteristic
16C2×C3×#104—31no—
17#5233—31no—
18#5122—31no—
19C2×#803—31no—
20#4558—31no—
21#4523—31no—
22C3×#41⋊C3C2×C411yescharacteristic
23C3×#41⋊C3C2×C411yescharacteristic
24C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
25C4×C3⋊Dic3—41no—
26C2×C3×C3⋊C8—41no—
27C2×C3×C3⋊C8—41no—
28#1765—91noSylow
29C2×C3×#41C2×D311yescharacteristic
30C2×#41⋊C3—21no—
31C2×#41⋊C3C2×D311yescharacteristic
32C2×C3×M16—31no—
33C2×C3×C42—31no—
34#948—32no—
35#948Dic312yescomplemented
36#926—31no—
37C2×#208—31no—
38C22×C3⋊C8—31no—
39C22×C3⋊C8—31no—
40#131⋊C3—32no—
41GU(2,𝔽3)Dic312yescomplemented
42#859—32no—
43#859—62no—
44C3×#104—32no—
45C3×#104—32no—
46#835—32no—
47#832—32no—
48#815—61no—
49#815—121no—
50C8×Dic3—61no—
51#803—62no—
52C2×C32×C4—41no—
53C2×C3⋊Dic3—42no—
54C3×SL(2,𝔽3)C4211yescharacteristic
55C3×C3⋊C8—42no—
56C3×C3⋊C8—42no—
57C2×#131—31no—
58C2×#104—31no—
59#416—91no—
60#405—91no—
61C2×#97—91no—
62#341—91no—
63#339—91no—
64C3×#41—31no—
65C3×#41—31no—
66C3×#41C4×D311yescharacteristic
67C3×#41C2×Dic311yescharacteristic
68C2×C3×Q8C4×D311yescharacteristic
69C2×C3×D4—31no—
70C22×C3×C4—31no—
71C22×C3×C4—61no—
72#41⋊C3—21no—
73#41⋊C3—21no—
74#41⋊C3C4×D311yescharacteristic
75#41⋊C3C2×Dic311yescharacteristic
76C2×SL(2,𝔽3)—21no—
77C2×SL(2,𝔽3)C4×D311yescharacteristic
78C3×M16—32no—
79C3×M16—61no—
80C2×C3×C8—61no—
81C2×C3×C8—121no—
82C3×C42—32no—
83C3×C42—61no—
84#217—61no—
85C4⋊Dic3—32no—
86C4×Dic3—61no—
87C4×Dic3—121no—
88C4×Dic3—241no—
89#208—32no—
90#208—61no—
91C2×C3⋊C8—31no—
92C2×C3⋊C8—32no—
93C2×C3⋊C8—32no—
94C2×C3⋊C8—32no—
95C2×C3⋊C8—41no—
96C2×C3⋊C8—61no—
97C2×C3⋊C8—61no—
98C2×C3⋊C8S411yescharacteristic
99C22×C32—41no—
100C32×C4—41no—
101C32×C4—41no—
102C3⋊Dic3—82no—
103C2×#41D3211yescharacteristic
104#131—32no—
105#131—92no—
106C2×M16—31no—
107C2×M16—91no—
108C22×C8—91no—
109C22×C8—91no—
110#117—91no—
111C2×C42—31no—
112#104—32no—
113#104—32no—
114#103—92no—
115#102—92no—
116#97—182no—
117#97—181no—
118C4×C8—181no—
119C23×C3—31no—
120C3×Q8—31no—
121C3×Q8C4×Dic311yescharacteristic
122C3×D4—31no—
123C3×D4—31no—
124C3×D4—61no—
125C2×C3×C4—31no—
126C2×C3×C4—31no—
127C2×C3×C4—31no—
128C2×C3×C4—31no—
129C2×C3×C4—31no—
130C2×C3×C4—41no—
131C2×C3×C4—62no—
132C2×C3×C4—81no—
133C2×C3×C4—122no—
134C2×C3×C4C2×S411yescharacteristic
135C2×Dic3—62no—
136C2×Dic3—122no—
137C2×Dic3—242no—
138SL(2,𝔽3)—21no—
139SL(2,𝔽3)C4×Dic311yescharacteristic
140C3×C8—62no—
141C3×C8—122no—
142C3⋊C8—32no—
143C3⋊C8—42no—
144C3⋊C8—62no—
145C3⋊C8SL(2,ℤ/4)12yes—
146C2×C32—42no—
147C2×C32—41no—
148#41—31no—
149#41—31no—
150#41D3×Dic311yescharacteristic
151#41D3×Dic311yescharacteristic
152C2×Q8#62411yescharacteristic
153C2×D4—31no—
154C22×C4—31no—
155C22×C4—61no—
156M16—32no—
157M16—61no—
158M16—92no—
159M16—181no—
160C2×C8—31no—
161C2×C8—61no—
162C2×C8—92no—
163C2×C8—92no—
164C2×C8—91no—
165C2×C8—92no—
166C2×C8—181no—
167C2×C8—181no—
168C4⋊C4—92no—
169C22⋊C4—181no—
170C42—32no—
171C42—61no—
172C42—181no—
173C22×C3—31no—
174C22×C3—31no—
175C22×C3—41no—
176C22×C3—62no—
177C22×C3—81no—
178C22×C3C4×S411yescharacteristic
179C3×C4—31no—
180C3×C4—31no—
181C3×C4—41no—
182C3×C4—41no—
183C3×C4—81no—
184C3×C4—81no—
185C3×C4—122no—
186C3×C4C4×S411yescharacteristic
187C3×C4C2×SL(2,ℤ/4)11yescharacteristic
188Dic3—122no—
189Dic3—242no—
190Dic3—244no—
191C32—41noSylow
192C23—31no—
193Q8—31no—
194Q8Dic3211yescharacteristic
195D4—31no—
196D4—31no—
197D4—61no—
198C2×C4—31no—
199C2×C4—31no—
200C2×C4—31no—
201C2×C4—31no—
202C2×C4—31no—
203C2×C4—62no—
204C2×C4—122no—
205C2×C4—182no—
206C2×C4D3×S411yescharacteristic
207C8—32no—
208C8—62no—
209C8—92no—
210C8—182no—
211C2×C3—42no—
212C2×C3—41no—
213C2×C3—61no—
214C2×C3—61no—
215C2×C3—82no—
216C2×C3—81no—
217C2×C3#540512yes—
218C2×C3C4×SL(2,ℤ/4)11yescharacteristic
219C22—31no—
220C22—31no—
221C22—62no—
222C22(C3⋊S4)⋊C411yescharacteristic
223C4—31no—
224C4—31no—
225C4—122no—
226C4—362no—
227C4Dic3×S411yescharacteristic
228C4D3×SL(2,ℤ/4)11yescharacteristic
229C3—41no—
230C3—81no—
231C3#7312311yescharacteristic, complemented
232C2—61no—
233C2—61no—
234C2#1059322212yes—
235C2Dic3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C42
FittingC2×C3×#41C2×D3
socleC22×C3C4×S4
prime radical, cosocleC2×C3×#41⋊C3C22
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
7. C3⋊(#131⋊C3)C4semidirect
8. C3⋊#859C4semidirect
9. C3×GU(2,𝔽3)C4semidirect
35. #948Dic3semidirect
41. GU(2,𝔽3)Dic3semidirect
231. C3#73123semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊(#131⋊C3)2
#105936582
C2×C3×GU(2,𝔽3)2
#731233
#703703
#635554