Finite Group Databasebeta

← #11921085

Subgroups — #11921085

Order 1152.

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

normal subgroups only

This group has 300 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
1C3×#568⋊C3C211yesmaximal, characteristic
2#10593657C211yesmaximal, characteristic
3C2×C3⋊GU(2,𝔽3)C211yesmaximal, characteristic
4#73129—33nomaximal
5#72028—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7C3×#131⋊C3—21no—
8C3⋊GU(2,𝔽3)—21no—
9C3⋊GU(2,𝔽3)C411yescharacteristic, complemented
10C3⋊GU(2,𝔽3)C411yescharacteristic, complemented
11#63649—41nomaximal
12C3×#568D311yescharacteristic
13#568⋊C3D313yes—
14#5423—33no—
15C2×GU(2,𝔽3)—33no—
16#5236—31no—
17C2×#837—31no—
18#5124—31no—
19#5118—31no—
20#4556—31no—
21#4555—31no—
22C3×#41⋊C3C2×C411yescharacteristic
23C3×#41⋊C3D411yescharacteristic
24C2×C3×SL(2,𝔽3)Q811yescharacteristic
25C32×M16—41no—
26#3748—41no—
27C2×C3⋊(C3⋊C8)—41no—
28#1861—91noSylow
29C2×C3×#41C2×D311yescharacteristic
30C2×#41⋊C3C2×D313yes—
31C3×#131—21no—
32C3×#131—61no—
33C2×C3×M16—31no—
34C2×C3×M16—31no—
35#926—31no—
36C2×C4×Dic3—31no—
37C2×#208—31no—
38C2×#208—31no—
39#131⋊C3—23no—
40GU(2,𝔽3)—33no—
41GU(2,𝔽3)—33no—
42GU(2,𝔽3)—63no—
43#837—31no—
44#837—31no—
45#837—31no—
46#837—31no—
47#837—61no—
48#837—61no—
49#822—61no—
50#822—61no—
51#822—123no—
52Dic3⋊C8—62no—
53C2×C32×C4—41no—
54C3×SL(2,𝔽3)C4⋊C411yescharacteristic
55C32×C8—81no—
56C3⋊(C3⋊C8)—41no—
57C3⋊(C3⋊C8)—41no—
58C3⋊(C3⋊C8)—81no—
59#568C3⋊D311yescharacteristic
60#430—91no—
61#423—91no—
62#421—91no—
63C2×#104—91no—
64#344—91no—
65#339—91no—
66C3×#41—31no—
67C3×#41—31no—
68C3×#41C4×D311yescharacteristic
69C3×#41#6711yescharacteristic
70C2×C3×Q8Dic611yescharacteristic
71C2×C3×D4—31no—
72C22×C3×C4—31no—
73C22×Dic3—61no—
74#41⋊C3C4×D313yes—
75#41⋊C3#6713yes—
76C2×SL(2,𝔽3)Dic613yes—
77C3×M16—31no—
78C3×M16—43no—
79C3×M16—61no—
80C3×M16—61no—
81C3×M16S411yescharacteristic
82C2×C3×C8—32no—
83C2×C3×C8—61no—
84#217—61no—
85C4⋊Dic3—61no—
86C4×Dic3—31no—
87C4×Dic3—31no—
88C4×Dic3—61no—
89C4×Dic3—61no—
90#208—31no—
91#208—31no—
92#208—61no—
93#208—61no—
94#208—61no—
95#208—123no—
96C2×C3⋊C8—61no—
97C2×C3⋊C8—61no—
98C2×C3⋊C8—123no—
99C22×C32—41no—
100C32×C4—41no—
101C32×C4—41no—
102C2×#41C2×C3⋊D311yescharacteristic
103#131—21no—
104#131—61no—
105C2×M16—31no—
106C2×M16—31no—
107C2×M16—91no—
108C2×M16—91no—
109#117—91no—
110C2×C42—91no—
111#108—181no—
112#108—181no—
113C4⋊C8—182no—
114#104—91no—
115#104—91no—
116#104—91no—
117#104—91no—
118#104—181no—
119#104—181no—
120C23×C3—31no—
121C3×Q8—31no—
122C3×Q8Dic3⋊C411yescharacteristic
123C3×D4—31no—
124C3×D4—31no—
125C3×D4—61no—
126C2×C3×C4—31no—
127C2×C3×C4—31no—
128C2×C3×C4—31no—
129C2×C3×C4—31no—
130C2×C3×C4—31no—
131C2×C3×C4—43no—
132C2×C3×C4C2×S411yescharacteristic
133C2×Dic3—61no—
134C2×Dic3—61no—
135C2×Dic3—122no—
136C2×Dic3—121no—
137SL(2,𝔽3)Dic3⋊C413yes—
138C3×C8—21no—
139C3×C8—61no—
140C3×C8—83no—
141C3⋊C8—61no—
142C3⋊C8—61no—
143C3⋊C8—123no—
144C3⋊C8—123no—
145C3⋊C8—121no—
146C3⋊C8—243no—
147C2×C32—41no—
148C2×C32—81no—
149#41—31no—
150#41—31no—
151#41C4×C3⋊D311yescharacteristic
152#41#63811yescharacteristic
153C2×Q8#63411yescharacteristic
154C2×D4—31no—
155C22×C4—31no—
156C22×C4—181no—
157M16—31no—
158M16—61no—
159M16—61no—
160M16—91no—
161M16—91no—
162M16—181no—
163M16—181no—
164M16—181no—
165M16C3⋊S411yescharacteristic
166C2×C8—32no—
167C2×C8—61no—
168C2×C8—181no—
169C2×C8—181no—
170C4⋊C4—181no—
171C22⋊C4—181no—
172C42—91no—
173C42—91no—
174C42—181no—
175C42—181no—
176C22×C3—31no—
177C22×C3—31no—
178C22×C3—43no—
179C22×C3—121no—
180C22×C3A4⋊Q811yescharacteristic
181C3×C4—31no—
182C3×C4—31no—
183C3×C4—43no—
184C3×C4—43no—
185C3×C4C4×S411yescharacteristic
186C3×C4GL(2,ℤ/4)11yescharacteristic
187Dic3—62no—
188Dic3—121no—
189Dic3—241no—
190C32—41noSylow
191C23—31no—
192Q8—31no—
193Q8#375011yescharacteristic
194D4—31no—
195D4—31no—
196D4—61no—
197C2×C4—31no—
198C2×C4—31no—
199C2×C4—31no—
200C2×C4—31no—
201C2×C4—31no—
202C2×C4—181no—
203C2×C4—181no—
204C2×C4—362no—
205C2×C4—361no—
206C2×C4C2×C3⋊S411yescharacteristic
207C8—21no—
208C8—61no—
209C8—181no—
210C8—181no—
211C8—361no—
212C2×C3—21no—
213C2×C3—43no—
214C2×C3—61no—
215C2×C3—61no—
216C2×C3—83no—
217C2×C3SL(2,ℤ/4)⋊C411yescharacteristic
218C22—31no—
219C22—31no—
220C22—121no—
221C22#6425911yescharacteristic
222C4—31no—
223C4—31no—
224C4—182no—
225C4—361no—
226C4—721no—
227C4C4×C3⋊S411yescharacteristic
228C4#6426911yescharacteristic
229C3—43no—
230C3#7312911yescharacteristic, complemented
231C2—21no—
232C2—61no—
233C2—61no—
234C2#1059364511yescharacteristic
subgroupG/H
centerC4#64269
commutatorC3×#41⋊C3C2×C4
FrattiniC2×C4C2×C3⋊S4
FittingC3×#568D3
socleC2×C3SL(2,ℤ/4)⋊C4
prime radical, cosocleC2×C3×#41⋊C3C22
hypercenterM16C3⋊S4
hypocenterC3×SL(2,𝔽3)C4⋊C4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
9. C3⋊GU(2,𝔽3)C4semidirect
10. C3⋊GU(2,𝔽3)C4semidirect
230. C3#73129semidirect

Maximal subgroups

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

subgroupindex
C3×#568⋊C32
#105936572
C2×C3⋊GU(2,𝔽3)2
#731293
#720283
#636494