Finite Group Databasebeta

← #83782⋊C3 #11921334

Subgroups — #83782⋊C3 #11921334

Order 1152.

321 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 321 conjugacy classes of subgroups — too many to draw. Only its 75 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⋊C3C211yesmaximal, characteristic
2Dic3×#41⋊C3C211yesmaximal, characteristic
3C2×C3⋊(#131⋊C3)C211yesmaximal, characteristic
4#83782C311yesmaximal, characteristic, complemented
5#2875⋊C3—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7C3×#131⋊C3C412yescomplemented
8C3⋊(#131⋊C3)C412yescomplemented
9Dic3×SL(2,𝔽3)C412yes—
10C3×#815—41nomaximal
11C2×C3×#131C2×C311yescharacteristic
12Dic3×#41C2×C311yescharacteristic
13C2×#948C2×C311yescharacteristic
14C2×#131⋊C3—21no—
15C2×#131⋊C3—31no—
16C2×#131⋊C3D311yescharacteristic
17C4×#41⋊C3—31no—
18#5122—31no—
19Dic3×M16—31no—
20#5101—31no—
21C2×#815—31no—
22C3×#41⋊C3C2×C411yescharacteristic
23C3×#41⋊C3C2×C411yescharacteristic
24C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
25C2×C32×C8—41no—
26C3×C4×Dic3—41no—
27C2×C3×C3⋊C8—41no—
28#2875—31noSylow
29C2×C3×#41C22×C311yescharacteristic
30C2×#41⋊C3—21no—
31C2×#41⋊C3C2×D311yescharacteristic
32C3×#131—32no—
33C3×#131C3×C412yescomplemented
34C2×C3×M16—31no—
35C22×C3×C8—31no—
36#948—32no—
37#948C3×C412yescomplemented
38Q8×Dic3C3×C412yes—
39D4×Dic3—32no—
40#926—31no—
41C2×C4×Dic3—31no—
42C2×#208—31no—
43C22×C3⋊C8—31no—
44#131⋊C3—22no—
45#131⋊C3—32no—
46#131⋊C3Dic312yescomplemented
47C4×SL(2,𝔽3)—32no—
48C3×#97—121no—
49#815—31no—
50#815—31no—
51#815—31no—
52#815A411yescharacteristic
53C8×Dic3—32no—
54C2×C32×C4—41no—
55C2×C3×Dic3—42no—
56C3×SL(2,𝔽3)C4211yescharacteristic
57C32×C8—42no—
58C3×C3⋊C8—42no—
59C2×#131—31no—
60C2×#131C3×D311yescharacteristic
61C4×#41—31no—
62#405—91no—
63#405—91no—
64C4×M16—91no—
65C2×#97—91no—
66C3×#41—31no—
67C3×#41—31no—
68C3×#41C2×C3×C411yescharacteristic
69C3×#41C2×C3×C411yescharacteristic
70C2×C3×Q8C2×C3×C411yescharacteristic
71C2×C3×D4—31no—
72C22×C3×C4—31no—
73C22×Dic3—32no—
74#41⋊C3—21no—
75#41⋊C3—21no—
76#41⋊C3C4×D311yescharacteristic
77#41⋊C3C2×Dic311yescharacteristic
78C2×SL(2,𝔽3)—21no—
79C2×SL(2,𝔽3)C4×D311yescharacteristic
80C3×M16—32no—
81C3×M16—32no—
82C2×C3×C8—31no—
83C2×C3×C8—32no—
84C2×C3×C8—32no—
85C2×C3×C8—41no—
86C2×C3×C8—81no—
87C2×C3×C8—121no—
88C2×C3×C8C2×A411yescharacteristic
89C3×C42—121no—
90#217—32no—
91C4⋊Dic3—32no—
92C4×Dic3—31no—
93C4×Dic3—32no—
94C4×Dic3C2×A411yescharacteristic
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—
104C3×Dic3—82no—
105C2×#41C2×C3×D311yescharacteristic
106#131—32no—
107#131—32no—
108#131—92no—
109#131C3×Dic312yescomplemented
110C2×M16—31no—
111C2×M16—91no—
112C22×C8—31no—
113C22×C8—91no—
114C4×Q8—32no—
115C4×D4—92no—
116#117—91no—
117C2×C42—91no—
118#97—31no—
119#97—91no—
120#97—91no—
121#97—91no—
122C4×C8—92no—
123C23×C3—31no—
124C3×Q8—31no—
125C3×Q8C3×C4211yescharacteristic
126C3×D4—31no—
127C3×D4—31no—
128C3×D4—32no—
129C2×C3×C4—31no—
130C2×C3×C4—31no—
131C2×C3×C4—31no—
132C2×C3×C4—31no—
133C2×C3×C4—31no—
134C2×C3×C4—41no—
135C2×C3×C4—81no—
136C2×C3×C4—122no—
137C2×C3×C4C22×A411yescharacteristic
138C2×Dic3—32no—
139C2×Dic3—122no—
140C2×Dic3C4×A412yes—
141SL(2,𝔽3)—21no—
142SL(2,𝔽3)C4×Dic311yescharacteristic
143C3×C8—32no—
144C3×C8—42no—
145C3×C8—82no—
146C3×C8—122no—
147C3×C8C4×A412yes—
148C3⋊C8—32no—
149C3⋊C8C4×A412yes—
150C2×C32—41no—
151C2×C32—42no—
152#41—31no—
153#41—31no—
154#41C3×C4×D311yescharacteristic
155#41C2×C3×Dic311yescharacteristic
156C2×Q8C3×C4×D311yescharacteristic
157C2×D4—31no—
158C22×C4—31no—
159C22×C4—92no—
160M16—32no—
161M16—32no—
162M16—92no—
163M16—92no—
164C2×C8—31no—
165C2×C8—31no—
166C2×C8—32no—
167C2×C8—32no—
168C2×C8—92no—
169C2×C8—92no—
170C2×C8—91no—
171C2×C8D3×A411yescharacteristic
172C4⋊C4—92no—
173C22⋊C4—92no—
174C42—31no—
175C42—92no—
176C42—91no—
177C22×C3—31no—
178C22×C3—31no—
179C22×C3—41no—
180C22×C3—62no—
181C22×C3—81no—
182C22×C3C2×C4×A411yescharacteristic
183C3×C4—31no—
184C3×C4—31no—
185C3×C4—41no—
186C3×C4—41no—
187C3×C4—81no—
188C3×C4—81no—
189C3×C4—242no—
190C3×C4C2×C4×A411yescharacteristic
191C3×C4C2×C4×A411yescharacteristic
192Dic3—22no—
193Dic3—62no—
194C32—41noSylow
195C23—31no—
196Q8—31no—
197Q8C3×C4×Dic311yescharacteristic
198D4—31no—
199D4—31no—
200D4—32no—
201C2×C4—31no—
202C2×C4—31no—
203C2×C4—31no—
204C2×C4—31no—
205C2×C4—31no—
206C2×C4—32no—
207C2×C4—92no—
208C2×C4—362no—
209C2×C4C2×D3×A411yescharacteristic
210C8—32no—
211C8—32no—
212C8—92no—
213C8Dic3×A412yes—
214C2×C3—42no—
215C2×C3—41no—
216C2×C3—61no—
217C2×C3—61no—
218C2×C3—82no—
219C2×C3—81no—
220C2×C3C42×A411yescharacteristic
221C2×C3#568⋊C312yes—
222C22—31no—
223C22—31no—
224C22—62no—
225C22C4×D3×A411yescharacteristic
226C4—31no—
227C4—31no—
228C4—62no—
229C4—182no—
230C4C4×D3×A411yescharacteristic
231C4C2×Dic3×A411yescharacteristic
232C3—41no—
233C3—81no—
234C3#2875⋊C311yescharacteristic, complemented
235C2—61no—
236C2—61no—
237C2#5750⋊C312yes—
238C2C4×Dic3×A411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×A4
commutator, hypocenterC3×Q8C3×C42
FittingC2×C3×#131C2×C3
socleC22×C3C2×C4×A4
prime radicalC2×C3×#41⋊C3C22
cosocleC2×C3×#41C22×C3
hypercenterC2×C8D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. #83782C3semidirect
7. C3×#131⋊C3C4semidirect
8. C3⋊(#131⋊C3)C4semidirect
33. C3×#131C3×C4semidirect
37. #948C3×C4semidirect
46. #131⋊C3Dic3semidirect
109. #131C3×Dic3semidirect
234. C3#2875⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#131⋊C32
Dic3×#41⋊C32
C2×C3⋊(#131⋊C3)2
#837823
#2875⋊C33
C3×#8154