Finite Group Databasebeta

← #83810⋊C3 #11921341

Subgroups — #83810⋊C3 #11921341

Order 1152.

258 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 53 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#5819⋊C3C211yesmaximal, characteristic
3C2×C3⋊(#131⋊C3)C211yesmaximal, characteristic
4#83810C311yesmaximal, characteristic, complemented
5#2916⋊C3—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7C3×#131⋊C3—21no—
8C3⋊(#131⋊C3)—21no—
9C3⋊(#131⋊C3)C412yes—
10C3×#822—41nomaximal
11C3×#568C2×C311yescharacteristic
12#5819C2×C311yescharacteristic
13C2×#948C2×C311yescharacteristic
14#568⋊C3—21no—
15#568⋊C3—31no—
16#568⋊C3D311yescharacteristic
17C2×#131⋊C3—31no—
18#5124—31no—
19#5124—31no—
20C2×#822—31no—
21C2×#822—31no—
22C3×#41⋊C3C2×C411yescharacteristic
23C3×#41⋊C3D411yescharacteristic
24C2×C3×SL(2,𝔽3)Q811yescharacteristic
25C32×M16—41no—
26C3×#208—41no—
27C2×C3×C3⋊C8—41no—
28#2916—31noSylow
29C2×C3×#41C22×C311yescharacteristic
30C2×#41⋊C3—21no—
31C2×#41⋊C3C2×D311yescharacteristic
32C3×#131—21no—
33C3×#131—61no—
34C2×C3×M16—31no—
35C2×C3×M16—31no—
36#948—21no—
37#948—32no—
38#948—61no—
39#948C3×C412yes—
40C2×#208—31no—
41C2×#208—31no—
42C2×#208—31no—
43C22×C3⋊C8—31no—
44#131⋊C3—22no—
45#131⋊C3—21no—
46#131⋊C3—32no—
47#131⋊C3—61no—
48C3×#108—121no—
49#822—32no—
50#822—31no—
51#822—31no—
52#822—31no—
53#822A411yescharacteristic
54C2×C32×C4—41no—
55C3×SL(2,𝔽3)C4⋊C411yescharacteristic
56C32×C8—81no—
57C3×C3⋊C8—42no—
58C3×C3⋊C8—81no—
59#568—31no—
60#568C3×D311yescharacteristic
61C2×#131—31no—
62#430—91no—
63#430—91no—
64C2×#108—91no—
65C2×#108—91no—
66C3×#41—31no—
67C3×#41—31no—
68C3×#41C2×C3×C411yescharacteristic
69C3×#41C3×D411yescharacteristic
70C2×C3×Q8C3×Q811yescharacteristic
71C2×C3×D4—31no—
72C22×C3×C4—31no—
73#41⋊C3—21no—
74#41⋊C3—21no—
75#41⋊C3C4×D311yescharacteristic
76#41⋊C3#6711yescharacteristic
77C2×SL(2,𝔽3)—21no—
78C2×SL(2,𝔽3)Dic611yescharacteristic
79C3×M16—31no—
80C3×M16—41no—
81C3×M16—61no—
82C3×M16—61no—
83C3×M16—81no—
84C3×M16—121no—
85C3×M16C2×A411yescharacteristic
86C2×C3×C8—61no—
87C2×C3×C8—61no—
88C2×C3×C8—121no—
89#208—32no—
90#208—32no—
91#208—31no—
92#208—61no—
93#208—61no—
94#208C2×A411yescharacteristic
95C2×C3⋊C8—31no—
96C2×C3⋊C8—32no—
97C2×C3⋊C8—32no—
98C2×C3⋊C8—61no—
99C2×C3⋊C8—61no—
100C2×C3⋊C8C2×A411yescharacteristic
101C22×C32—41no—
102C32×C4—41no—
103C32×C4—41no—
104C2×#41C2×C3×D311yescharacteristic
105#131—21no—
106#131—32no—
107#131—61no—
108#131—61no—
109#131—92no—
110#131—181no—
111C2×M16—31no—
112C2×M16—31no—
113C2×M16—91no—
114C2×M16—91no—
115C2×M16—91no—
116C22×C8—91no—
117#108—31no—
118#108—91no—
119#108—91no—
120#108—92no—
121#108—91no—
122C23×C3—31no—
123C3×Q8—31no—
124C3×Q8C3×C4⋊C411yescharacteristic
125C3×D4—31no—
126C3×D4—31no—
127C3×D4—32no—
128C2×C3×C4—31no—
129C2×C3×C4—31no—
130C2×C3×C4—31no—
131C2×C3×C4—31no—
132C2×C3×C4—31no—
133C2×C3×C4—41no—
134C2×C3×C4—81no—
135C2×C3×C4C22×A411yescharacteristic
136SL(2,𝔽3)—21no—
137SL(2,𝔽3)Dic3⋊C411yescharacteristic
138C3×C8—21no—
139C3×C8—61no—
140C3×C8—81no—
141C3×C8—82no—
142C3×C8—122no—
143C3×C8—241no—
144C3⋊C8—21no—
145C3⋊C8—32no—
146C3⋊C8—61no—
147C3⋊C8C4×A412yes—
148C2×C32—41no—
149C2×C32—81no—
150#41—31no—
151#41—31no—
152#41C3×C4×D311yescharacteristic
153#41C3×#6711yescharacteristic
154C2×Q8C3×Dic611yescharacteristic
155C2×D4—31no—
156C22×C4—31no—
157M16—31no—
158M16—31no—
159M16—61no—
160M16—61no—
161M16—92no—
162M16—92no—
163M16—91no—
164M16—181no—
165M16—181no—
166M16D3×A411yescharacteristic
167C2×C8—31no—
168C2×C8—61no—
169C2×C8—61no—
170C2×C8—92no—
171C2×C8—92no—
172C2×C8—91no—
173C2×C8—181no—
174C2×C8—181no—
175C22×C3—31no—
176C22×C3—31no—
177C22×C3—41no—
178C22×C3—81no—
179C22×C3—121no—
180C22×C3Q8×A411yescharacteristic
181C3×C4—31no—
182C3×C4—31no—
183C3×C4—41no—
184C3×C4—41no—
185C3×C4—81no—
186C3×C4—81no—
187C3×C4C2×C4×A411yescharacteristic
188C3×C4D4×A411yescharacteristic
189C32—41noSylow
190C23—31no—
191Q8—31no—
192Q8C3×Dic3⋊C411yescharacteristic
193D4—31no—
194D4—31no—
195D4—32no—
196C2×C4—31no—
197C2×C4—31no—
198C2×C4—31no—
199C2×C4—31no—
200C2×C4—31no—
201C2×C4C2×D3×A411yescharacteristic
202C8—21no—
203C8—32no—
204C8—61no—
205C8—61no—
206C8—92no—
207C8—181no—
208C2×C3—21no—
209C2×C3—41no—
210C2×C3—61no—
211C2×C3—61no—
212C2×C3—81no—
213C2×C3—81no—
214C2×C3—161no—
215C2×C3A4×C4⋊C411yescharacteristic
216C22—31no—
217C22—31no—
218C22—121no—
219C22A4×Dic611yescharacteristic
220C4—31no—
221C4—31no—
222C4C4×D3×A411yescharacteristic
223C4A4×#6711yescharacteristic
224C3—41no—
225C3—81no—
226C3#2916⋊C311yescharacteristic, complemented
227C2—21no—
228C2—61no—
229C2—61no—
230C2A4×Dic3⋊C411yescharacteristic
subgroupG/H
centerC4A4×#67
commutatorC3×#41C2×C3×C4
FrattiniC2×C4C2×D3×A4
FittingC3×#568C2×C3
socleC2×C3A4×C4⋊C4
prime radicalC2×C3×#41⋊C3C22
cosocleC2×C3×#41C22×C3
hypercenterM16D3×A4
hypocenterC3×Q8C3×C4⋊C4
LCSG ▷ C3×#41 ▷ C3×Q8
UCSC1 ◁ C4 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
4. #83810C3semidirect
226. C3#2916⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#568⋊C32
#5819⋊C32
C2×C3⋊(#131⋊C3)2
#838103
#2916⋊C33
C3×#8224