Finite Group Databasebeta

← #11920568

Subgroups — #11920568

Order 1152.

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

normal subgroups only

This group has 249 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
1SL(2,𝔽3)×Dic6C211yesmaximal, characteristic
2#10593620C211yesmaximal, characteristic
3C3×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic
4#73099—31nomaximal
5#68255—31nomaximal
6C3×C4×SL(2,𝔽3)C2211yescharacteristic
7C2×#3719—41nomaximal
8Dic3×SL(2,𝔽3)—21no—
9C3⋊#859—21no—
10Q8×Dic6D311yescharacteristic
11Q8×SL(2,𝔽3)—31no—
12#5386—31no—
13#5386—61no—
14#5020—31no—
15#4849—31no—
16#4681—31no—
17SL(2,𝔽3)⋊C8D311yescharacteristic
18C3×Q8⋊C8—31no—
19#4481—31no—
20#4456—31no—
21C2×#634—41no—
22C2×C3×Dic6—41no—
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24C2×C3×C3⋊C8—41no—
25#3719—44no—
26#1645—91noSylow
27C3×C4×Q8C2×D311yescharacteristic
28Q8×Dic3—21no—
29C2×#216—121no—
30C2×Dic12—61no—
31#888—61no—
32#869—31no—
33#869—31no—
34C4×Dic6—31no—
35C4×SL(2,𝔽3)—21no—
36C4×SL(2,𝔽3)—61no—
37C4×SL(2,𝔽3)C2×D311yescharacteristic
38#859—62no—
39#859—61no—
40C3×C4⋊C8—31no—
41C3×C4×C8—31no—
42#835—61no—
43#816—61no—
44#816—62no—
45#808—61no—
46C4⋊(C3⋊C8)—31no—
47C2×C32×C4—41no—
48C2×C3⋊Dic3—81no—
49#634—42no—
50#634—81no—
51C2×C3×Dic3—81no—
52C3×Dic6—42no—
53C3×Dic6—81no—
54C3×SL(2,𝔽3)Q1611yescharacteristic, complemented
55C3×C3⋊C8—42no—
56Q82—31no—
57#487—91no—
58#475—91no—
59#462—91no—
60#333—91no—
61Q8⋊C8—31no—
62Q8⋊C8—91no—
63C2×C3×Q8—121no—
64C2×C3×Q8#6711yescharacteristic
65C2×Dic6—32no—
66C2×Dic6—61no—
67C2×Dic6—121no—
68C2×Dic6—241no—
69C2×Dic6S411yescharacteristic
70C2×SL(2,𝔽3)—21no—
71C2×SL(2,𝔽3)D1211yescharacteristic
72C2×C3×C8—61no—
73C2×C3×C8—61no—
74C3×C4⋊C4—31no—
75C3×C42—31no—
76#216—124no—
77C4⋊Dic3—31no—
78C4⋊Dic3—61no—
79C4⋊Dic3—61no—
80C4⋊Dic3—61no—
81Dic3⋊C4—61no—
82C4×Dic3—61no—
83C2×C3⋊C8—41no—
84C2×C3⋊C8—61no—
85Dic12—122no—
86C22×C32—41no—
87C32×C4—41no—
88C32×C4—41no—
89C3⋊Dic3—82no—
90C3×Dic3—82no—
91C2×Q16—181no—
92C4⋊Q8—91no—
93C4⋊Q8—91no—
94C4⋊Q8—181no—
95C4×Q8—61no—
96C4×Q8—91no—
97C4×Q8D3211yescharacteristic
98#106—181no—
99C4⋊C8—31no—
100C4⋊C8—91no—
101#103—181no—
102#103—182no—
103#103—181no—
104C4×C8—31no—
105C3×Q8—31no—
106C3×Q8—122no—
107C3×Q8—241no—
108C3×Q8#21611yescharacteristic, complemented
109C2×C3×C4—31no—
110C2×C3×C4—31no—
111C2×C3×C4—41no—
112C2×C3×C4—81no—
113C2×C3×C4—241no—
114C2×C3×C4C2×S411yescharacteristic
115C2×Dic3—21no—
116C2×Dic3—61no—
117C2×Dic3—121no—
118C2×Dic3—241no—
119C2×Dic3—242no—
120Dic6—21no—
121Dic6—122no—
122Dic6—122no—
123Dic6—121no—
124Dic6—121no—
125Dic6—241no—
126Dic6—242no—
127Dic6—242no—
128Dic6GL(2,𝔽3)12yes—
129SL(2,𝔽3)—21no—
130SL(2,𝔽3)Dic1211yescharacteristic, complemented
131C3×C8—121no—
132C3×C8—121no—
133C3⋊C8—42no—
134C3⋊C8—121no—
135C2×C32—41no—
136C2×C32—41no—
137C2×C32—41no—
138C2×Q8—31no—
139C2×Q8—92no—
140C2×Q8—181no—
141C2×Q8#62611yescharacteristic
142Q16—362no—
143C2×C8—61no—
144C2×C8—61no—
145C2×C8—181no—
146C4⋊C4—31no—
147C4⋊C4—91no—
148C4⋊C4—181no—
149C4⋊C4—181no—
150C4⋊C4—181no—
151C4⋊C4—181no—
152C42—31no—
153C42—181no—
154C22×C3—41no—
155C22×C3—81no—
156C22×C3GL(2,ℤ/4)11yescharacteristic
157C3×C4—31no—
158C3×C4—31no—
159C3×C4—41no—
160C3×C4—41no—
161C3×C4—61no—
162C3×C4—81no—
163C3×C4—81no—
164C3×C4—242no—
165C3×C4C2×GL(2,𝔽3)11yescharacteristic
166C3×C4#98311yescharacteristic
167Dic3—22no—
168Dic3—121no—
169Dic3—242no—
170Dic3—244no—
171Dic3—241no—
172C32—41noSylow
173Q8—31no—
174Q8—32no—
175Q8—61no—
176Q8—362no—
177Q8—361no—
178Q8—361no—
179Q8#371911yescharacteristic, complemented
180C2×C4—31no—
181C2×C4—31no—
182C2×C4—61no—
183C2×C4—181no—
184C2×C4—361no—
185C2×C4D3×S411yescharacteristic
186C8—121no—
187C8—121no—
188C8—361no—
189C2×C3—41no—
190C2×C3—41no—
191C2×C3—41no—
192C2×C3—81no—
193C2×C3—81no—
194C2×C3—81no—
195C2×C3#541611yescharacteristic
196C2×C3#542111yescharacteristic
197C2×C3#542811yescharacteristic
198C22#6421011yescharacteristic
199C4—31no—
200C4—31no—
201C4—62no—
202C4—61no—
203C4—361no—
204C4—721no—
205C4#6420211yescharacteristic
206C4D3×GL(2,𝔽3)11yescharacteristic
207C3—41no—
208C3—81no—
209C3#7309911yescharacteristic, complemented
210C2#1059319011yescharacteristic
211C2#1059321811yescharacteristic
212C2#1059323611yescharacteristic
subgroupG/H
centerC22#64210
commutator, cosocleC3×C4×SL(2,𝔽3)C22
Frattini, hypercenterC2×C4D3×S4
FittingC3×C4×Q8C2×D3
socleC22×C3GL(2,ℤ/4)
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
54. C3×SL(2,𝔽3)Q16semidirect
108. C3×Q8#216semidirect
130. SL(2,𝔽3)Dic12semidirect
179. Q8#3719semidirect
209. C3#73099semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×Dic62
#105936202
C3×SL(2,𝔽3)⋊C82
#730993
#682553
C2×#37194