Finite Group Databasebeta

← #11921032

Subgroups — #11921032

Order 1152.

342 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 59 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×#520⋊C3C211yesmaximal, characteristic
2#10593660C211yesmaximal, characteristic
3C2×C3⋊GU(2,𝔽3)C211yesmaximal, characteristic
4#73076—33nomaximal
5#71664—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7C3×C4×SL(2,𝔽3)—21no—
8C3⋊GU(2,𝔽3)C412yescomplemented
9C3⋊#859—21no—
10#63635—41nomaximal
11C3×#520D311yescharacteristic
12#520⋊C3D313yes—
13#5426—33no—
14C2×GU(2,𝔽3)—33no—
15C2×#837—31no—
16#5232—31no—
17#4995—31no—
18#4975—31no—
19#4532—31no—
20#4530—31no—
21C3×#41⋊C3C2×C411yescharacteristic
22C3×#41⋊C3Q811yescharacteristic
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24C32×C4⋊C4—41no—
25C4⋊(C3⋊Dic3)—41no—
26C2×C3⋊(C3⋊C8)—41no—
27#1859—91noSylow
28C2×C3×#41C2×D311yescharacteristic
29C2×#41⋊C3C2×D313yes—
30C3×C4×Q8—21no—
31C3×C4×D4—61no—
32C3×#117—31no—
33C2×C3×C4⋊C4—31no—
34C2×C4⋊Dic3—31no—
35C2×Dic3⋊C4—62no—
36C2×C4×Dic3—31no—
37C2×#208—31no—
38C2×#208—31no—
39C4×SL(2,𝔽3)—23no—
40GU(2,𝔽3)—36no—
41#859—63no—
42#837—32no—
43#837—32no—
44#835—61no—
45#832—61no—
46#808—61no—
47#808—123no—
48#807—61no—
49C2×C32×C4—41no—
50C2×C32×C4—81no—
51C2×C3⋊Dic3—81no—
52C3×SL(2,𝔽3)C4⋊C411yescharacteristic, complemented
53C3⋊(C3⋊C8)—42no—
54#520C3⋊D311yescharacteristic
55#428—91no—
56C2×#104—91no—
57#417—91no—
58#384—91no—
59#343—91no—
60#339—91no—
61C3×#41—31no—
62C3×#41—31no—
63C3×#41Dic611yescharacteristic
64C3×#41C4×D311yescharacteristic
65C2×C3×Q8#6711yescharacteristic
66C2×C3×D4—31no—
67C22×C3×C4—31no—
68C22×C3×C4—32no—
69C22×Dic3—61no—
70C22×Dic3—61no—
71#41⋊C3Dic613yes—
72#41⋊C3C4×D313yes—
73C2×SL(2,𝔽3)#6713yes—
74C3×C4⋊C4—31no—
75C3×C4⋊C4—43no—
76C3×C4⋊C4—61no—
77C3×C4⋊C4S411yescharacteristic
78C3×C22⋊C4—61no—
79C3×C42—61no—
80C4⋊Dic3—61no—
81C4⋊Dic3—61no—
82C4⋊Dic3—123no—
83Dic3⋊C4—124no—
84C4×Dic3—32no—
85C4×Dic3—61no—
86#208—32no—
87#208—61no—
88#208—61no—
89#208—61no—
90C2×C3⋊C8—61no—
91C2×C3⋊C8—61no—
92C2×C3⋊C8—123no—
93C22×C32—41no—
94C32×C4—41no—
95C32×C4—41no—
96C32×C4—161no—
97C3⋊Dic3—161no—
98C2×#41C2×C3⋊D311yescharacteristic
99C2×M16—91no—
100C2×M16—91no—
101C4×Q8—21no—
102C4×D4—61no—
103#117—31no—
104C2×C4⋊C4—31no—
105C2×C4⋊C4—91no—
106C2×C4⋊C4—182no—
107C2×C42—91no—
108#107—181no—
109#106—181no—
110#104—92no—
111#104—92no—
112#103—181no—
113#102—181no—
114C23×C3—31no—
115C3×Q8—31no—
116C3×Q8Dic3⋊C411yescharacteristic
117C3×D4—31no—
118C3×D4—31no—
119C3×D4—61no—
120C2×C3×C4—21no—
121C2×C3×C4—31no—
122C2×C3×C4—31no—
123C2×C3×C4—31no—
124C2×C3×C4—31no—
125C2×C3×C4—31no—
126C2×C3×C4—43no—
127C2×C3×C4—61no—
128C2×C3×C4—83no—
129C2×C3×C4—122no—
130C2×C3×C4C2×S411yescharacteristic
131C2×Dic3—62no—
132C2×Dic3—62no—
133C2×Dic3—62no—
134C2×Dic3—122no—
135C2×Dic3—121no—
136C2×Dic3—243no—
137SL(2,𝔽3)Dic3⋊C413yescomplemented
138C3⋊C8—62no—
139C3⋊C8—126no—
140C3⋊C8—121no—
141C2×C32—41no—
142C2×C32—41no—
143C2×C32—41no—
144#41—31no—
145#41—31no—
146#41#63411yescharacteristic
147#41C4×C3⋊D311yescharacteristic
148C2×Q8#63811yescharacteristic
149C2×D4—31no—
150C22×C4—31no—
151C22×C4—32no—
152C22×C4—181no—
153C22×C4—181no—
154M16—92no—
155M16—181no—
156M16—181no—
157M16—181no—
158C2×C8—181no—
159C2×C8—181no—
160C4⋊C4—31no—
161C4⋊C4—61no—
162C4⋊C4—181no—
163C4⋊C4—181no—
164C4⋊C4—364no—
165C4⋊C4C3⋊S411yescharacteristic
166C22⋊C4—61no—
167C42—61no—
168C42—92no—
169C42—181no—
170C22×C3—31no—
171C22×C3—31no—
172C22×C3—43no—
173C22×C3—61no—
174C22×C3—61no—
175C22×C3GL(2,ℤ/4)11yescharacteristic
176C3×C4—31no—
177C3×C4—31no—
178C3×C4—41no—
179C3×C4—43no—
180C3×C4—43no—
181C3×C4—121no—
182C3×C4—163no—
183C3×C4A4⋊Q811yescharacteristic
184C3×C4C4×S411yescharacteristic
185Dic3—64no—
186Dic3—241no—
187Dic3—483no—
188C32—41noSylow
189C23—31no—
190Q8—31no—
191Q8#375011yescharacteristic
192D4—31no—
193D4—31no—
194D4—61no—
195C2×C4—21no—
196C2×C4—31no—
197C2×C4—31no—
198C2×C4—31no—
199C2×C4—31no—
200C2×C4—31no—
201C2×C4—61no—
202C2×C4—122no—
203C2×C4—182no—
204C2×C4—182no—
205C2×C4—182no—
206C2×C4—362no—
207C2×C4—361no—
208C2×C4C2×C3⋊S411yescharacteristic
209C8—182no—
210C8—361no—
211C2×C3—43no—
212C2×C3—43no—
213C2×C3—43no—
214C2×C3—61no—
215C2×C3—61no—
216C2×C3SL(2,ℤ/4)⋊C411yescharacteristic
217C2×C3#542911yescharacteristic
218C2×C3#543011yescharacteristic
219C22—31no—
220C22—31no—
221C22—61no—
222C22—61no—
223C22#6426911yescharacteristic
224C4—31no—
225C4—31no—
226C4—41no—
227C4—121no—
228C4—184no—
229C4—721no—
230C4#6425911yescharacteristic
231C4C4×C3⋊S411yescharacteristic
232C3—43no—
233C3#7307611yescharacteristic, complemented
234C2—61no—
235C2—61no—
236C2#1059364511yescharacteristic
237C2#1059366311yescharacteristic
238C2#1059366411yescharacteristic
subgroupG/H
centerC22#64269
commutatorC3×#41⋊C3C2×C4
FrattiniC2×C4C2×C3⋊S4
FittingC3×#520D3
socleC22×C3GL(2,ℤ/4)
prime radical, cosocleC2×C3×#41⋊C3C22
hypercenterC4⋊C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C4⋊C4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
8. C3⋊GU(2,𝔽3)C4semidirect
52. C3×SL(2,𝔽3)C4⋊C4semidirect
137. SL(2,𝔽3)Dic3⋊C4semidirect
233. C3#73076semidirect

Maximal subgroups

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

subgroupindex
C3×#520⋊C32
#105936602
C2×C3⋊GU(2,𝔽3)2
#730763
#716643
#636354