Finite Group Databasebeta

← #82523⋊C3 #11921276

Subgroups — #82523⋊C3 #11921276

Order 1152.

251 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 251 conjugacy classes of subgroups — too many to draw. Only its 59 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×C4×#41⋊C3C211yesmaximal, characteristic
2C3⋊C8×SL(2,𝔽3)C212yesmaximal, complemented
3#82523C311yesmaximal, characteristic, complemented
4#2991⋊C3—31nomaximal
5C2×C3×#41⋊C3C411yescharacteristic
6C3×C4×SL(2,𝔽3)C411yescharacteristic
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C3×C4⋊(C3⋊C8)—41nomaximal
9C3×C4×#41C2×C311yescharacteristic
10C4×#41⋊C3—21no—
11C4×#41⋊C3D311yescharacteristic
12Q8×C3⋊C8C2×C312yes—
13#5012—32no—
14#4925—31no—
15#4924—31no—
16C8×SL(2,𝔽3)—32no—
17C3×#41⋊C3—21no—
18C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
19C32×C42—41no—
20C2×C3×C3⋊C8—42no—
21#2991—31noSylow
22C2×C3×#41C3×C411yescharacteristic
23C2×#41⋊C3—21no—
24C2×#41⋊C3Dic311yescharacteristic
25C3×C4×Q8C3×C411yescharacteristic
26C3×C4×Q8C22×C311yescharacteristic
27C3×C4×D4—31no—
28C3×C4×D4—31no—
29C3×#117—31no—
30C2×C3×C42—31no—
31C2×#208—32no—
32C4×SL(2,𝔽3)—21no—
33C4×SL(2,𝔽3)—21no—
34C4×SL(2,𝔽3)Dic311yescharacteristic
35C4×SL(2,𝔽3)C2×D311yescharacteristic
36C3×C4⋊C8—121no—
37#830—32no—
38C4⋊(C3⋊C8)—31no—
39C4⋊(C3⋊C8)—32no—
40C4⋊(C3⋊C8)A411yescharacteristic
41C4×C3⋊C8—32no—
42C2×C32×C4—41no—
43C2×C32×C4—41no—
44C2×C32×C4—41no—
45C3×SL(2,𝔽3)M1611yescharacteristic, complemented
46C3×C3⋊C8—82no—
47C4×#41C3×D311yescharacteristic
48C8×Q8—32no—
49#436—92no—
50#431—91no—
51#423—91no—
52C3×#41—21no—
53C3×#41—61no—
54C2×C3×Q8C2×C3×C411yescharacteristic
55C2×C3×D4—31no—
56C22×C3×C4—31no—
57C22×C3×C4—31no—
58C22×C3×C4—31no—
59#41⋊C3—22no—
60#41⋊C3—21no—
61C2×SL(2,𝔽3)—21no—
62C2×SL(2,𝔽3)C2×Dic311yescharacteristic
63C2×C3×C8—122no—
64C3×C4⋊C4—31no—
65C3×C4⋊C4—31no—
66C3×C22⋊C4—32no—
67C3×C42—31no—
68C3×C42—31no—
69C3×C42—31no—
70C3×C42—41no—
71C3×C42—81no—
72C3×C42C2×A411yescharacteristic
73#208—122no—
74C2×C3⋊C8—32no—
75C2×C3⋊C8C2×A412yes—
76C22×C32—41no—
77C32×C4—42no—
78C32×C4—41no—
79C32×C4—41no—
80C32×C4—81no—
81C2×#41C3×Dic311yescharacteristic
82C2×M16—92no—
83C4×Q8C3×Dic311yescharacteristic
84C4×Q8C2×C3×D311yescharacteristic
85C4×D4—31no—
86C4×D4—31no—
87#117—31no—
88C2×C42—31no—
89C4⋊C8—31no—
90C4⋊C8—92no—
91C4⋊C8—91no—
92C22⋊C8—92no—
93C4×C8—92no—
94C23×C3—31no—
95C3×Q8—31no—
96C3×Q8C3×M1611yescharacteristic
97C3×D4—61no—
98C3×D4—61no—
99C2×C3×C4—31no—
100C2×C3×C4—31no—
101C2×C3×C4—31no—
102C2×C3×C4—41no—
103C2×C3×C4—41no—
104C2×C3×C4—41no—
105C2×C3×C4—61no—
106C2×C3×C4—61no—
107C2×C3×C4—62no—
108C2×C3×C4—61no—
109C2×C3×C4—61no—
110C2×C3×C4—81no—
111C2×C3×C4—81no—
112C2×C3×C4—81no—
113C2×C3×C4C4×A411yescharacteristic
114C2×C3×C4C4×A411yescharacteristic
115C2×C3×C4C22×A411yescharacteristic
116SL(2,𝔽3)—21no—
117SL(2,𝔽3)#20811yescharacteristic, complemented
118C3×C8—242no—
119C3⋊C8—22no—
120C3⋊C8—62no—
121C2×C32—41no—
122C2×C32—41no—
123C2×C32—41no—
124#41—21no—
125#41—61no—
126C2×Q8C2×C3×Dic311yescharacteristic
127C2×D4—31no—
128C22×C4—31no—
129C22×C4—31no—
130C22×C4—31no—
131M16—362no—
132C2×C8—32no—
133C2×C8—92no—
134C4⋊C4—31no—
135C4⋊C4—31no—
136C22⋊C4—32no—
137C42—31no—
138C42—31no—
139C42—31no—
140C42D3×A411yescharacteristic
141C22×C3—41no—
142C22×C3—61no—
143C22×C3—61no—
144C22×C3—61no—
145C22×C3—81no—
146C22×C3C2×C4×A411yescharacteristic
147C3×C4—21no—
148C3×C4—31no—
149C3×C4—31no—
150C3×C4—42no—
151C3×C4—41no—
152C3×C4—41no—
153C3×C4—61no—
154C3×C4—61no—
155C3×C4—81no—
156C3×C4—82no—
157C3×C4—82no—
158C3×C4—81no—
159C3×C4—81no—
160C3×C4#131⋊C312yes—
161C3×C4#99411yescharacteristic
162C3×C42−1+4⋊C311yescharacteristic
163C32—41noSylow
164C23—31no—
165Q8—31no—
166Q8C3×#20811yescharacteristic
167D4—61no—
168D4—61no—
169C2×C4—31no—
170C2×C4—31no—
171C2×C4—31no—
172C2×C4—61no—
173C2×C4—61no—
174C2×C4—62no—
175C2×C4—61no—
176C2×C4—61no—
177C2×C4Dic3×A411yescharacteristic
178C2×C4Dic3×A411yescharacteristic
179C2×C4C2×D3×A411yescharacteristic
180C8—62no—
181C8—182no—
182C2×C3—41no—
183C2×C3—41no—
184C2×C3—41no—
185C2×C3—81no—
186C2×C3—81no—
187C2×C3—81no—
188C2×C3—121no—
189C2×C3#520⋊C311yescharacteristic
190C2×C3A4×M1611yescharacteristic
191C2×C3C2×#131⋊C311yescharacteristic
192C22—61no—
193C22—61no—
194C22—61no—
195C22C2×Dic3×A411yescharacteristic
196C4—21no—
197C4—31no—
198C4—31no—
199C4—61no—
200C4—61no—
201C4C3⋊(#131⋊C3)12yes—
202C4C3⋊#99411yescharacteristic
203C4#1010⋊C311yescharacteristic
204C3—41no—
205C3—81no—
206C3#2991⋊C311yescharacteristic, complemented
207C2—121no—
208C2A4×#20811yescharacteristic
209C2C2×C3⋊(#131⋊C3)11yescharacteristic
210C2#5827⋊C311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×A4
commutatorC2×C3×Q8C2×C3×C4
FittingC3×C4×#41C2×C3
socleC22×C3C2×C4×A4
prime radicalC2×C3×#41⋊C3C4
cosocleC3×C4×Q8C22×C3
hypercenterC42D3×A4
hypocenterC3×Q8C3×M16
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3⋊C8×SL(2,𝔽3)C2semidirect
3. #82523C3semidirect
45. C3×SL(2,𝔽3)M16semidirect
117. SL(2,𝔽3)#208semidirect
206. C3#2991⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#41⋊C32
C3⋊C8×SL(2,𝔽3)2
#825233
#2991⋊C33
C3×C4⋊(C3⋊C8)4