Finite Group Databasebeta

← #82561⋊C3 #11921280

Subgroups — #82561⋊C3 #11921280

Order 1152.

283 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 57 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#5827⋊C3C211yesmaximal, characteristic
3C2×C3⋊(#131⋊C3)C211yesmaximal, characteristic
4#82561C311yesmaximal, characteristic, complemented
5#2912⋊C3—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7C3×C4×SL(2,𝔽3)—21no—
8C3⋊(#131⋊C3)C412yescomplemented
9Dic3×SL(2,𝔽3)—21no—
10C3×#808—41nomaximal
11C3×#520C2×C311yescharacteristic
12#5827C2×C311yescharacteristic
13C2×#948C2×C311yescharacteristic
14C2×#131⋊C3—31no—
15#520⋊C3—21no—
16#520⋊C3—31no—
17#520⋊C3D311yescharacteristic
18#4998—31no—
19#4995—31no—
20#4964—31no—
21C2×#808—31no—
22C3×#41⋊C3C2×C411yescharacteristic
23C3×#41⋊C3Q811yescharacteristic
24C2×C3×SL(2,𝔽3)D411yescharacteristic
25C32×C4⋊C4—41no—
26C3×C4⋊Dic3—41no—
27C2×C3×C3⋊C8—41no—
28#2912—31noSylow
29C2×C3×#41C22×C311yescharacteristic
30C2×#41⋊C3—21no—
31C2×#41⋊C3C2×D311yescharacteristic
32C3×C4×Q8—21no—
33C3×C4×D4—61no—
34C3×#117—31no—
35C2×C3×C4⋊C4—31no—
36#948—32no—
37#948C3×C412yescomplemented
38Q8×Dic3—21no—
39D4×Dic3—61no—
40#926—31no—
41C2×C4⋊Dic3—31no—
42C2×#208—31no—
43C22×C3⋊C8—31no—
44#131⋊C3—32no—
45C4×SL(2,𝔽3)—22no—
46C4×SL(2,𝔽3)—21no—
47C4×SL(2,𝔽3)—61no—
48C3×#106—121no—
49#808—32no—
50#808—31no—
51#808A411yescharacteristic
52#807—31no—
53#807—31no—
54C2×C32×C4—41no—
55C2×C32×C4—81no—
56C2×C3×Dic3—81no—
57C3×SL(2,𝔽3)C4⋊C411yescharacteristic
58C3×C3⋊C8—42no—
59C2×#131—31no—
60#520—31no—
61#520C3×D311yescharacteristic
62#428—91no—
63#428—91no—
64#427—91no—
65C2×#106—91no—
66C3×#41—31no—
67C3×#41—31no—
68C3×#41C2×C3×C411yescharacteristic
69C3×#41C3×Q811yescharacteristic
70C2×C3×Q8C3×D411yescharacteristic
71C2×C3×D4—31no—
72C22×C3×C4—31no—
73C22×C3×C4—61no—
74C22×Dic3—61no—
75#41⋊C3—21no—
76#41⋊C3—21no—
77#41⋊C3Dic611yescharacteristic
78#41⋊C3C4×D311yescharacteristic
79C2×SL(2,𝔽3)—21no—
80C2×SL(2,𝔽3)#6711yescharacteristic
81C2×C3×C8—121no—
82C3×C4⋊C4—31no—
83C3×C4⋊C4—41no—
84C3×C4⋊C4—61no—
85C3×C4⋊C4—81no—
86C3×C4⋊C4—121no—
87C3×C4⋊C4C2×A411yescharacteristic
88C3×C22⋊C4—61no—
89C3×C42—61no—
90#217—61no—
91C4⋊Dic3—31no—
92C4⋊Dic3—61no—
93C4⋊Dic3C2×A411yescharacteristic
94C4×Dic3—61no—
95#208—32no—
96#208—32no—
97C2×C3⋊C8—31no—
98C2×C3⋊C8—32no—
99C2×C3⋊C8—32no—
100C2×C3⋊C8C2×A411yescharacteristic
101C22×C32—41no—
102C32×C4—41no—
103C32×C4—41no—
104C32×C4—161no—
105C3×Dic3—161no—
106C2×#41C2×C3×D311yescharacteristic
107#131—32no—
108#131—92no—
109C2×M16—91no—
110C22×C8—91no—
111C4×Q8—21no—
112C4×Q8—61no—
113C4×D4—61no—
114C4×D4—181no—
115#117—31no—
116#117—91no—
117C2×C4⋊C4—31no—
118C2×C4⋊C4—91no—
119#107—91no—
120#107—91no—
121#106—31no—
122#106—91no—
123#106—92no—
124C23×C3—31no—
125C3×Q8—31no—
126C3×Q8C3×C4⋊C411yescharacteristic
127C3×D4—31no—
128C3×D4—31no—
129C3×D4—32no—
130C2×C3×C4—21no—
131C2×C3×C4—31no—
132C2×C3×C4—31no—
133C2×C3×C4—31no—
134C2×C3×C4—31no—
135C2×C3×C4—31no—
136C2×C3×C4—41no—
137C2×C3×C4—61no—
138C2×C3×C4—81no—
139C2×C3×C4—82no—
140C2×C3×C4—81no—
141C2×C3×C4—241no—
142C2×C3×C4—241no—
143C2×C3×C4C22×A411yescharacteristic
144C2×Dic3—21no—
145C2×Dic3—61no—
146C2×Dic3—241no—
147SL(2,𝔽3)—21no—
148SL(2,𝔽3)Dic3⋊C411yescharacteristic
149C3×C8—122no—
150C3⋊C8—32no—
151C3⋊C8C4×A412yes—
152C2×C32—41no—
153C2×C32—41no—
154C2×C32—41no—
155#41—31no—
156#41—31no—
157#41C3×Dic611yescharacteristic
158#41C3×C4×D311yescharacteristic
159C2×Q8C3×#6711yescharacteristic
160C2×D4—31no—
161C22×C4—31no—
162C22×C4—61no—
163C22×C4—181no—
164M16—92no—
165M16—92no—
166C2×C8—31no—
167C2×C8—92no—
168C2×C8—92no—
169C2×C8—91no—
170C4⋊C4—31no—
171C4⋊C4—31no—
172C4⋊C4—61no—
173C4⋊C4—91no—
174C4⋊C4—181no—
175C4⋊C4D3×A411yescharacteristic
176C22⋊C4—61no—
177C22⋊C4—181no—
178C42—61no—
179C42—181no—
180C22×C3—31no—
181C22×C3—31no—
182C22×C3—41no—
183C22×C3—61no—
184C22×C3—61no—
185C22×C3—81no—
186C22×C3D4×A411yescharacteristic
187C3×C4—31no—
188C3×C4—31no—
189C3×C4—41no—
190C3×C4—41no—
191C3×C4—41no—
192C3×C4—81no—
193C3×C4—81no—
194C3×C4—121no—
195C3×C4—161no—
196C3×C4—162no—
197C3×C4—481no—
198C3×C4C2×C4×A411yescharacteristic
199C3×C4Q8×A411yescharacteristic
200Dic3—41no—
201Dic3—121no—
202C32—41noSylow
203C23—31no—
204Q8—31no—
205Q8C3×Dic3⋊C411yescharacteristic
206D4—31no—
207D4—31no—
208D4—32no—
209C2×C4—21no—
210C2×C4—31no—
211C2×C4—31no—
212C2×C4—31no—
213C2×C4—31no—
214C2×C4—31no—
215C2×C4—61no—
216C2×C4—61no—
217C2×C4—181no—
218C2×C4—241no—
219C2×C4—721no—
220C2×C4C2×D3×A411yescharacteristic
221C8—32no—
222C8—92no—
223C2×C3—41no—
224C2×C3—41no—
225C2×C3—41no—
226C2×C3—61no—
227C2×C3—61no—
228C2×C3—81no—
229C2×C3—81no—
230C2×C3—81no—
231C2×C3A4×C4⋊C411yescharacteristic
232C2×C3#577⋊C311yescharacteristic
233C2×C3#577⋊C311yescharacteristic
234C22—31no—
235C22—31no—
236C22—61no—
237C22—61no—
238C22A4×#6711yescharacteristic
239C4—31no—
240C4—31no—
241C4—41no—
242C4—121no—
243C4—121no—
244C4—361no—
245C4A4×Dic611yescharacteristic
246C4C4×D3×A411yescharacteristic
247C3—41no—
248C3—81no—
249C3#2912⋊C311yescharacteristic, complemented
250C2—61no—
251C2—61no—
252C2A4×Dic3⋊C411yescharacteristic
253C2#5836⋊C311yescharacteristic
254C2#5837⋊C311yescharacteristic
subgroupG/H
centerC22A4×#67
commutatorC3×#41C2×C3×C4
FrattiniC2×C4C2×D3×A4
FittingC3×#520C2×C3
socleC22×C3D4×A4
prime radicalC2×C3×#41⋊C3C22
cosocleC2×C3×#41C22×C3
hypercenterC4⋊C4D3×A4
hypocenterC3×Q8C3×C4⋊C4
LCSG ▷ C3×#41 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #82561C3semidirect
8. C3⋊(#131⋊C3)C4semidirect
37. #948C3×C4semidirect
249. C3#2912⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#520⋊C32
#5827⋊C32
C2×C3⋊(#131⋊C3)2
#825613
#2912⋊C33
C3×#8084