Finite Group Databasebeta

← #11920558

Subgroups — #11920558

Order 1152.

235 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 37 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
1#10593620C211yesmaximal, characteristic
2C3×#5386C211yesmaximal, characteristic
3C3⋊C8×SL(2,𝔽3)C211yesmaximal, characteristic
4#73015—31nomaximal
5#71117—31nomaximal
6C3×C4×SL(2,𝔽3)C2211yescharacteristic
7C2×#3719—41nomaximal
8C3⋊#859—21no—
9C3×#859—21no—
10#5386—31no—
11#5386—61no—
12#5386D311yescharacteristic
13C3×#475—31no—
14#5092—31no—
15Q8×C3⋊C8D311yescharacteristic
16#5020—31no—
17C8×SL(2,𝔽3)—31no—
18#4547—31no—
19#4486—31no—
20#4479—31no—
21C2×#634—41no—
22C2×C3×Dic6—41no—
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24C2×C3×C3⋊C8—41no—
25#3719—44no—
26#1670—91noSylow
27C3×C4⋊Q8—31no—
28C3×C4×Q8C2×D311yescharacteristic
29C2×#216—61no—
30C2×#216—61no—
31C2×Dic12—121no—
32#869—31no—
33C4×SL(2,𝔽3)—21no—
34C4×SL(2,𝔽3)C2×D311yescharacteristic
35#859—21no—
36#859—62no—
37#859—61no—
38C3×#106—61no—
39C3×C4⋊C8—31no—
40C3×#103—61no—
41#835—61no—
42#808—61no—
43C4⋊(C3⋊C8)—31no—
44C4⋊(C3⋊C8)—31no—
45C4×C3⋊C8—31no—
46C2×C32×C4—41no—
47C2×C3⋊Dic3—81no—
48#634—42no—
49#634—81no—
50C2×C3×Dic3—81no—
51C3×Dic6—42no—
52C3×Dic6—81no—
53C3×SL(2,𝔽3)Q1611yescharacteristic, complemented
54C3×C3⋊C8—42no—
55#494—91no—
56#475—31no—
57#475—91no—
58C8×Q8—31no—
59#335—91no—
60#332—91no—
61#332—91no—
62C2×C3×Q8—61no—
63C2×C3×Q8D1211yescharacteristic
64C2×Dic6—41no—
65C2×Dic6—61no—
66C2×Dic6—121no—
67C2×Dic6—241no—
68C2×SL(2,𝔽3)—21no—
69C2×SL(2,𝔽3)#6711yescharacteristic
70C2×C3×C8—61no—
71C2×C3×C8—121no—
72C3×C4⋊C4—31no—
73C3×C4⋊C4—61no—
74C3×C4⋊C4—61no—
75C3×C42—31no—
76#216—122no—
77#216—122no—
78C4⋊Dic3—61no—
79C4⋊Dic3—61no—
80C2×C3⋊C8—31no—
81C2×C3⋊C8—61no—
82C2×C3⋊C8S411yescharacteristic
83Dic12—124no—
84C22×C32—41no—
85C32×C4—41no—
86C32×C4—41no—
87C3⋊Dic3—82no—
88C3×Dic3—82no—
89C2×Q16—181no—
90C2×Q16—181no—
91C4⋊Q8—31no—
92C4⋊Q8—91no—
93C4×Q8D3211yescharacteristic
94#106—61no—
95#106—181no—
96C4⋊C8—31no—
97C4⋊C8—91no—
98C4⋊C8—91no—
99#103—61no—
100#103—181no—
101C4×C8—91no—
102C3×Q8—31no—
103C3×Q8—121no—
104C3×Q8—121no—
105C3×Q8Dic1211yescharacteristic, complemented
106C2×C3×C4—31no—
107C2×C3×C4—31no—
108C2×C3×C4—41no—
109C2×C3×C4—81no—
110C2×C3×C4—121no—
111C2×C3×C4C2×S411yescharacteristic
112C2×Dic3—81no—
113C2×Dic3—121no—
114C2×Dic3—241no—
115C2×Dic3—242no—
116Dic6—42no—
117Dic6—81no—
118Dic6—122no—
119Dic6—121no—
120Dic6—121no—
121Dic6—241no—
122Dic6—242no—
123Dic6—242no—
124SL(2,𝔽3)—21no—
125SL(2,𝔽3)#21611yescharacteristic, complemented
126C3×C8—122no—
127C3×C8—121no—
128C3⋊C8—61no—
129C3⋊C8—121no—
130C3⋊C8GL(2,𝔽3)12yes—
131C2×C32—41no—
132C2×C32—41no—
133C2×C32—41no—
134C2×Q8—61no—
135C2×Q8—181no—
136C2×Q8#62611yescharacteristic
137Q16—362no—
138Q16—362no—
139C2×C8—31no—
140C2×C8—61no—
141C2×C8—91no—
142C2×C8—181no—
143C4⋊C4—31no—
144C4⋊C4—61no—
145C4⋊C4—61no—
146C4⋊C4—181no—
147C4⋊C4—181no—
148C42—31no—
149C22×C3—41no—
150C22×C3—81no—
151C22×C3C4⋊S411yescharacteristic
152C3×C4—31no—
153C3×C4—31no—
154C3×C4—41no—
155C3×C4—41no—
156C3×C4—61no—
157C3×C4—81no—
158C3×C4—81no—
159C3×C4—241no—
160C3×C4C2×GL(2,𝔽3)11yescharacteristic
161C3×C4#98611yescharacteristic
162Dic3—82no—
163Dic3—242no—
164Dic3—244no—
165Dic3—241no—
166C32—41noSylow
167Q8—31no—
168Q8—121no—
169Q8—121no—
170Q8—361no—
171Q8—361no—
172Q8#371911yescharacteristic, complemented
173C2×C4—31no—
174C2×C4—31no—
175C2×C4—121no—
176C2×C4—361no—
177C2×C4D3×S411yescharacteristic
178C8—32no—
179C8—121no—
180C8—181no—
181C8—361no—
182C2×C3—41no—
183C2×C3—41no—
184C2×C3—41no—
185C2×C3—81no—
186C2×C3—81no—
187C2×C3—81no—
188C2×C3C4⋊GL(2,𝔽3)11yescharacteristic
189C2×C3#539811yescharacteristic
190C2×C3#540311yescharacteristic
191C22#6420711yescharacteristic
192C4—31no—
193C4—31no—
194C4—61no—
195C4—241no—
196C4—721no—
197C4#6419911yescharacteristic
198C4D3×GL(2,𝔽3)11yescharacteristic
199C3—41no—
200C3—81no—
201C3#7301511yescharacteristic, complemented
202C2#1059318011yescharacteristic
203C2#1059321511yescharacteristic
204C2#1059322611yescharacteristic
subgroupG/H
centerC22#64207
commutator, cosocleC3×C4×SL(2,𝔽3)C22
Frattini, hypercenterC2×C4D3×S4
FittingC3×C4×Q8C2×D3
socleC22×C3C4⋊S4
prime radicalC2×C3×SL(2,𝔽3)D4
hypocenterC3×SL(2,𝔽3)Q16
LCSG ▷ C3×C4×SL(2,𝔽3) ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
53. C3×SL(2,𝔽3)Q16semidirect
105. C3×Q8Dic12semidirect
125. SL(2,𝔽3)#216semidirect
172. Q8#3719semidirect
201. C3#73015semidirect

Maximal subgroups

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

subgroupindex
#105936202
C3×#53862
C3⋊C8×SL(2,𝔽3)2
#730153
#711173
C2×#37194