Finite Group Databasebeta

← #82435⋊C3 #11921260

Subgroups — #82435⋊C3 #11921260

Order 1152.

287 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 67 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×C4×SL(2,𝔽3)C211yesmaximal, characteristic
2C3⋊C8×SL(2,𝔽3)C212yesmaximal, complemented
3#82435C311yesmaximal, characteristic, complemented
4#2982⋊C3—31nomaximal
5C22×C3×SL(2,𝔽3)C411yescharacteristic
6C3×C4×SL(2,𝔽3)C411yescharacteristic
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C3×#830—41nomaximal
9C2×C3×C4×Q8C2×C311yescharacteristic
10C2×C4×SL(2,𝔽3)—21no—
11C2×C4×SL(2,𝔽3)D311yescharacteristic
12Q8×C3⋊C8—32no—
13Q8×C3⋊C8C2×C312yescomplemented
14#4925—31no—
15#4925—31no—
16C8×SL(2,𝔽3)—32no—
17C22×C32×C4—41no—
18C2×C3×SL(2,𝔽3)C812yescomplemented
19C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
20C2×C3×C3⋊C8—42no—
21#2982—31noSylow
22C22×C3×Q8C3×C411yescharacteristic
23C22×SL(2,𝔽3)—21no—
24C22×SL(2,𝔽3)Dic311yescharacteristic
25C3×C4×Q8—31no—
26C3×C4×Q8—31no—
27C3×C4×Q8C3×C411yescharacteristic
28C3×C4×Q8C22×C311yescharacteristic
29C2×C3×C4⋊C4—31no—
30C2×C3×C42—31no—
31C4×SL(2,𝔽3)—21no—
32C4×SL(2,𝔽3)—21no—
33C4×SL(2,𝔽3)Dic311yescharacteristic
34C4×SL(2,𝔽3)C2×D311yescharacteristic
35C3×C22⋊C8—121no—
36#830—31no—
37#830A411yescharacteristic
38C4⋊(C3⋊C8)—32no—
39C4⋊(C3⋊C8)—32no—
40C4×C3⋊C8—32no—
41C4×C3⋊C8—32no—
42C23×C32—41no—
43C2×C32×C4—41no—
44C2×C32×C4—41no—
45C2×C32×C4—82no—
46C3×SL(2,𝔽3)C2×C811yescharacteristic
47C3×C3⋊C8—82no—
48C2×C4×Q8C3×D311yescharacteristic
49C8×Q8—32no—
50C8×Q8—92no—
51#431—91no—
52#431—91no—
53C2×C3×Q8—31no—
54C2×C3×Q8—32no—
55C2×C3×Q8C3×C812yescomplemented
56C2×C3×Q8C2×C3×C411yescharacteristic
57C22×C3×C4—31no—
58C22×C3×C4—31no—
59C22×C3×C4—41no—
60C22×C3×C4—81no—
61C22×C3×C4C2×A411yescharacteristic
62C2×SL(2,𝔽3)—22no—
63C2×SL(2,𝔽3)—21no—
64C2×SL(2,𝔽3)C3⋊C812yescomplemented
65C2×SL(2,𝔽3)C2×Dic311yescharacteristic
66C2×C3×C8—122no—
67C3×C4⋊C4—31no—
68C3×C4⋊C4—31no—
69C3×C4⋊C4—32no—
70C3×C42—31no—
71C3×C42—31no—
72C3×C42—31no—
73C3×C42—31no—
74C2×C3⋊C8—32no—
75C2×C3⋊C8C2×A412yes—
76C22×C32—42no—
77C22×C32—41no—
78C22×C32—81no—
79C22×C32—81no—
80C32×C4—42no—
81C32×C4—81no—
82C22×Q8C3×Dic311yescharacteristic
83C4×Q8—31no—
84C4×Q8—31no—
85C4×Q8C3×Dic311yescharacteristic
86C4×Q8C2×C3×D311yescharacteristic
87C2×C4⋊C4—31no—
88C2×C42—31no—
89C4⋊C8—92no—
90C4⋊C8—92no—
91C22⋊C8—31no—
92C22⋊C8—91no—
93C4×C8—92no—
94C4×C8—92no—
95C23×C3—41no—
96C23×C3—81no—
97C23×C3C4×A411yescharacteristic
98C3×Q8—32no—
99C3×Q8—31no—
100C3×Q8—32no—
101C3×Q8C2×C3×C811yescharacteristic
102C2×C3×C4—22no—
103C2×C3×C4—32no—
104C2×C3×C4—32no—
105C2×C3×C4—31no—
106C2×C3×C4—31no—
107C2×C3×C4—31no—
108C2×C3×C4—31no—
109C2×C3×C4—41no—
110C2×C3×C4—41no—
111C2×C3×C4—82no—
112C2×C3×C4—84no—
113C2×C3×C4—81no—
114C2×C3×C4—81no—
115C2×C3×C4—121no—
116C2×C3×C4C4×A411yescharacteristic
117C2×C3×C4C22×A411yescharacteristic
118SL(2,𝔽3)—21no—
119SL(2,𝔽3)C2×C3⋊C811yescharacteristic
120C3×C8—242no—
121C3⋊C8—22no—
122C3⋊C8—62no—
123C2×C32—41no—
124C2×C32—41no—
125C2×C32—41no—
126C2×C32—82no—
127C2×Q8—31no—
128C2×Q8—32no—
129C2×Q8C3×C3⋊C812yescomplemented
130C2×Q8C2×C3×Dic311yescharacteristic
131C22×C4—31no—
132C22×C4—31no—
133C22×C4D3×A411yescharacteristic
134C2×C8—32no—
135C2×C8—92no—
136C4⋊C4—31no—
137C4⋊C4—31no—
138C4⋊C4—32no—
139C42—31no—
140C42—31no—
141C42—31no—
142C42—31no—
143C22×C3—21no—
144C22×C3—21no—
145C22×C3—42no—
146C22×C3—41no—
147C22×C3—81no—
148C22×C3—81no—
149C22×C3—82no—
150C22×C3—82no—
151C22×C3—82no—
152C22×C3—81no—
153C22×C3C8×A412yes—
154C22×C3C2×C4×A411yescharacteristic
155C3×C4—21no—
156C3×C4—31no—
157C3×C4—32no—
158C3×C4—31no—
159C3×C4—42no—
160C3×C4—61no—
161C3×C4—61no—
162C3×C4—81no—
163C3×C4—82no—
164C3×C4—82no—
165C3×C42−1+4⋊C312yes—
166C32—41noSylow
167C23Dic3×A411yescharacteristic
168Q8—32no—
169Q8—31no—
170Q8—32no—
171Q8C2×C3×C3⋊C811yescharacteristic
172C2×C4—22no—
173C2×C4—32no—
174C2×C4—32no—
175C2×C4—31no—
176C2×C4—31no—
177C2×C4—31no—
178C2×C4—31no—
179C2×C4—121no—
180C2×C4Dic3×A411yescharacteristic
181C2×C4C2×D3×A411yescharacteristic
182C8—62no—
183C8—182no—
184C2×C3—22no—
185C2×C3—41no—
186C2×C3—41no—
187C2×C3—41no—
188C2×C3—82no—
189C2×C3—84no—
190C2×C3—81no—
191C2×C3—81no—
192C2×C3—81no—
193C2×C3#519⋊C311yescharacteristic
194C2×C3C2×C8×A411yescharacteristic
195C2×C3#568⋊C311yescharacteristic
196C22—21no—
197C22—21no—
198C22A4×C3⋊C812yes—
199C22C2×Dic3×A411yescharacteristic
200C4—21no—
201C4—31no—
202C4—32no—
203C4—31no—
204C4—61no—
205C4—61no—
206C4#1007⋊C312yes—
207C3—41no—
208C3—81no—
209C3#2982⋊C311yescharacteristic, complemented
210C2—22no—
211C2C2×A4×C3⋊C811yescharacteristic
212C2#5812⋊C311yescharacteristic
213C2#5819⋊C311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×A4
commutator, hypocenterC3×Q8C2×C3×C8
FittingC2×C3×C4×Q8C2×C3
socleC22×C3C2×C4×A4
prime radicalC22×C3×SL(2,𝔽3)C4
cosocleC3×C4×Q8C22×C3
hypercenterC22×C4D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C3⋊C8×SL(2,𝔽3)C2semidirect
3. #82435C3semidirect
13. Q8×C3⋊C8C2×C3semidirect
18. C2×C3×SL(2,𝔽3)C8semidirect
55. C2×C3×Q8C3×C8semidirect
64. C2×SL(2,𝔽3)C3⋊C8semidirect
129. C2×Q8C3×C3⋊C8semidirect
209. C3#2982⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×SL(2,𝔽3)2
C3⋊C8×SL(2,𝔽3)2
#824353
#2982⋊C33
C3×#8304