Finite Group Databasebeta

← #11920698

Subgroups — #11920698

Order 1152.

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

normal subgroups only

This group has 305 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
2C2×C3⋊GU(2,𝔽3)C211yesmaximal, characteristic
3C3×#5424C211yesmaximal, characteristic
4#73123—31nomaximal
5#70290—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7C3⋊(#131⋊C3)C412yescomplemented
8C3⋊GU(2,𝔽3)C412yescomplemented
9C3×#859C412yescomplemented
10#63554—41nomaximal
11C2×#948D311yescharacteristic
12C2×#131⋊C3—31no—
13#5424D311yescharacteristic
14C2×GU(2,𝔽3)—31no—
15C2×GU(2,𝔽3)—61no—
16C3×#416—31no—
17C2×#837—31no—
18C2×#815—31no—
19#4997—31no—
20#4552—31no—
21#4532—31no—
22C3×#41⋊C3C2×C411yescharacteristic
23C3×#41⋊C3C2×C411yescharacteristic
24C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
25C2×C3⋊(C3⋊C8)—41no—
26C3×C4×Dic3—41no—
27C2×C3×C3⋊C8—41no—
28#1765—91noSylow
29C2×C3×#41C2×D311yescharacteristic
30C2×#41⋊C3—21no—
31C2×#41⋊C3C2×D311yescharacteristic
32C22×C3×C8—31no—
33C3×#117—31no—
34#948—32no—
35#948Dic312yescomplemented
36C2×C4×Dic3—31no—
37C2×#208—31no—
38C2×#208—31no—
39C22×C3⋊C8—31no—
40#131⋊C3—32no—
41GU(2,𝔽3)—32no—
42GU(2,𝔽3)—62no—
43#859Dic312yescomplemented
44C3×#103—32no—
45C3×#102—32no—
46#837—32no—
47#837—32no—
48#815—62no—
49#815—121no—
50#803—61no—
51C4×C3⋊C8—61no—
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—91no—
59#416—31no—
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×Dic3—61no—
72#41⋊C3—21no—
73#41⋊C3—21no—
74#41⋊C3C4×D311yescharacteristic
75#41⋊C3C4×D311yescharacteristic
76C2×SL(2,𝔽3)—21no—
77C2×SL(2,𝔽3)C2×Dic311yescharacteristic
78C2×C3×C8—32no—
79C2×C3×C8—61no—
80C2×C3×C8—61no—
81C2×C3×C8—121no—
82C3×C4⋊C4—32no—
83C3×C22⋊C4—61no—
84C3×C42—61no—
85C4×Dic3—32no—
86C4×Dic3—41no—
87C4×Dic3—61no—
88#208—32no—
89#208—32no—
90#208—61no—
91#208—61no—
92C2×C3⋊C8—31no—
93C2×C3⋊C8—32no—
94C2×C3⋊C8—32no—
95C2×C3⋊C8—61no—
96C2×C3⋊C8—121no—
97C2×C3⋊C8—241no—
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—91no—
107C2×M16—91no—
108C22×C8—31no—
109C22×C8—91no—
110#117—31no—
111C2×C42—91no—
112#104—92no—
113#104—92no—
114#103—32no—
115#102—32no—
116#97—181no—
117#97—182no—
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×C4C2×S411yescharacteristic
134C2×Dic3—42no—
135C2×Dic3—62no—
136C2×Dic3—122no—
137SL(2,𝔽3)—21no—
138SL(2,𝔽3)C4×Dic311yescharacteristic
139C3×C8—62no—
140C3×C8—122no—
141C3⋊C8—32no—
142C3⋊C8—62no—
143C3⋊C8—122no—
144C3⋊C8—242no—
145C3⋊C8SL(2,ℤ/4)12yes—
146C2×C32—42no—
147C2×C32—41no—
148#41—31no—
149#41—31no—
150#41D3×Dic311yescharacteristic
151#41#62411yescharacteristic
152C2×Q8D3×Dic311yescharacteristic
153C2×D4—31no—
154C22×C4—31no—
155C22×C4—181no—
156M16—92no—
157M16—92no—
158M16—181no—
159M16—181no—
160C2×C8—31no—
161C2×C8—32no—
162C2×C8—61no—
163C2×C8—61no—
164C2×C8—92no—
165C2×C8—92no—
166C2×C8—91no—
167C2×C8—181no—
168C4⋊C4—32no—
169C22⋊C4—61no—
170C42—61no—
171C42—92no—
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—82no—
189Dic3—122no—
190C32—41noSylow
191C23—31no—
192Q8—31no—
193Q8Dic3211yescharacteristic
194D4—31no—
195D4—31no—
196D4—61no—
197C2×C4—31no—
198C2×C4—31no—
199C2×C4—31no—
200C2×C4—31no—
201C2×C4—31no—
202C2×C4—62no—
203C2×C4—182no—
204C2×C4—362no—
205C2×C4D3×S411yescharacteristic
206C8—32no—
207C8—62no—
208C8—92no—
209C8—182no—
210C2×C3—42no—
211C2×C3—41no—
212C2×C3—61no—
213C2×C3—61no—
214C2×C3—82no—
215C2×C3—81no—
216C2×C3#540512yes—
217C2×C3C4×SL(2,ℤ/4)11yescharacteristic
218C22—31no—
219C22—31no—
220C22—62no—
221C22Dic3×S411yescharacteristic
222C4—31no—
223C4—31no—
224C4—122no—
225C4—362no—
226C4(C3⋊S4)⋊C411yescharacteristic
227C4D3×SL(2,ℤ/4)11yescharacteristic
228C3—41no—
229C3—81no—
230C3#7312311yescharacteristic, complemented
231C2—61no—
232C2—61no—
233C2#1059322112yes—
234C2Dic3×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⋊GU(2,𝔽3)C4semidirect
9. C3×#859C4semidirect
35. #948Dic3semidirect
43. #859Dic3semidirect
230. 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
C2×C3⋊GU(2,𝔽3)2
C3×#54242
#731233
#702903
#635544