Finite Group Databasebeta

← #11920567

Subgroups — #11920567

Order 1152.

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

normal subgroups only

This group has 239 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
2C3×#5386C211yesmaximal, characteristic
3C3⋊(SL(2,𝔽3)⋊C8)C211yesmaximal, characteristic
4#73099—31nomaximal
5#71129—31nomaximal
6C3×C4×SL(2,𝔽3)C2211yescharacteristic
7C2×#3718—41nomaximal
8Dic3×SL(2,𝔽3)—21no—
9C3×#859—21no—
10Q8×Dic6D311yescharacteristic
11Q8×SL(2,𝔽3)—31no—
12#5386D311yescharacteristic
13C3×#475—31no—
14#5093—31no—
15#5090—31no—
16SL(2,𝔽3)⋊C8—31no—
17SL(2,𝔽3)⋊C8—61no—
18#4535—31no—
19#4484—31no—
20#4481—31no—
21C2×C3×Dic6—41no—
22C2×C3×Dic6—41no—
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24C2×C3⋊(C3⋊C8)—41no—
25#3718—44no—
26#1645—91noSylow
27C3×C4⋊Q8—31no—
28C3×C4×Q8C2×D311yescharacteristic
29Q8×Dic3—21no—
30C2×#216—61no—
31C2×#216—121no—
32#888—61no—
33#869—31no—
34C4×Dic6—31no—
35C4×SL(2,𝔽3)—21no—
36C4×SL(2,𝔽3)—61no—
37C4×SL(2,𝔽3)C2×D311yescharacteristic
38#859—21no—
39C3×#106—61no—
40C3×C4⋊C8—31no—
41C3×#103—61no—
42#835—61no—
43#810—62no—
44C4⋊(C3⋊C8)—31no—
45C4×C3⋊C8—31no—
46C2×C32×C4—41no—
47C2×C3×Dic3—81no—
48C2×C3×Dic3—81no—
49C3×Dic6—42no—
50C3×Dic6—42no—
51C3×Dic6—81no—
52C3×Dic6—81no—
53C3×SL(2,𝔽3)Q1611yescharacteristic, complemented
54C3⋊(C3⋊C8)—42no—
55Q82—31no—
56#487—91no—
57#475—31no—
58#462—91no—
59#333—91no—
60Q8⋊C8—91no—
61Q8⋊C8—91no—
62C2×C3×Q8—61no—
63C2×C3×Q8—121no—
64C2×C3×Q8#6711yescharacteristic
65C2×Dic6—32no—
66C2×Dic6—41no—
67C2×Dic6S411yescharacteristic
68C2×SL(2,𝔽3)—21no—
69C2×SL(2,𝔽3)#6711yescharacteristic
70C2×C3×C8—61no—
71C3×C4⋊C4—31no—
72C3×C4⋊C4—61no—
73C3×C4⋊C4—61no—
74C3×C42—31no—
75#216—124no—
76#216—122no—
77C4⋊Dic3—31no—
78C4⋊Dic3—61no—
79Dic3⋊C4—61no—
80C4×Dic3—61no—
81C2×C3⋊C8—61no—
82C2×C3⋊C8—61no—
83C2×C3⋊C8—121no—
84C2×C3⋊C8—241no—
85C22×C32—41no—
86C32×C4—41no—
87C32×C4—41no—
88C3×Dic3—82no—
89C3×Dic3—82no—
90C2×Q16—181no—
91C4⋊Q8—31no—
92C4⋊Q8—91no—
93C4⋊Q8—181no—
94C4×Q8—61no—
95C4×Q8—91no—
96C4×Q8D3211yescharacteristic
97#106—61no—
98C4⋊C8—31no—
99C4⋊C8—91no—
100#103—61no—
101#103—182no—
102#103—181no—
103C4×C8—91no—
104C3×Q8—31no—
105C3×Q8—122no—
106C3×Q8—121no—
107C3×Q8—121no—
108C3×Q8—241no—
109C3×Q8#21611yescharacteristic, complemented
110C2×C3×C4—31no—
111C2×C3×C4—31no—
112C2×C3×C4—41no—
113C2×C3×C4—81no—
114C2×C3×C4—121no—
115C2×C3×C4—241no—
116C2×C3×C4C2×S411yescharacteristic
117C2×Dic3—21no—
118C2×Dic3—61no—
119C2×Dic3—81no—
120Dic6—21no—
121Dic6—42no—
122Dic6—81no—
123Dic6—122no—
124Dic6GL(2,𝔽3)12yes—
125SL(2,𝔽3)—21no—
126SL(2,𝔽3)#21611yescharacteristic, complemented
127C3×C8—121no—
128C3⋊C8—122no—
129C3⋊C8—121no—
130C3⋊C8—121no—
131C3⋊C8—242no—
132C2×C32—41no—
133C2×C32—41no—
134C2×C32—41no—
135C2×Q8—31no—
136C2×Q8—61no—
137C2×Q8—92no—
138C2×Q8#62511yescharacteristic
139Q16—362no—
140C2×C8—61no—
141C2×C8—181no—
142C2×C8—181no—
143C4⋊C4—31no—
144C4⋊C4—61no—
145C4⋊C4—61no—
146C4⋊C4—91no—
147C4⋊C4—181no—
148C4⋊C4—181no—
149C42—31no—
150C42—181no—
151C22×C3—41no—
152C22×C3—81no—
153C22×C3GL(2,ℤ/4)11yescharacteristic
154C3×C4—31no—
155C3×C4—31no—
156C3×C4—41no—
157C3×C4—41no—
158C3×C4—61no—
159C3×C4—81no—
160C3×C4—81no—
161C3×C4—242no—
162C3×C4—241no—
163C3×C4C2×GL(2,𝔽3)11yescharacteristic
164C3×C4#98311yescharacteristic
165Dic3—22no—
166Dic3—82no—
167Dic3—121no—
168C32—41noSylow
169Q8—31no—
170Q8—32no—
171Q8—61no—
172Q8—121no—
173Q8—121no—
174Q8—362no—
175Q8#371811yescharacteristic, complemented
176C2×C4—31no—
177C2×C4—31no—
178C2×C4—61no—
179C2×C4—121no—
180C2×C4—181no—
181C2×C4D3×S411yescharacteristic
182C8—121no—
183C8—361no—
184C8—361no—
185C2×C3—41no—
186C2×C3—41no—
187C2×C3—41no—
188C2×C3—81no—
189C2×C3—81no—
190C2×C3—81no—
191C2×C3#541611yescharacteristic
192C2×C3#542111yescharacteristic
193C2×C3#542811yescharacteristic
194C22#6420911yescharacteristic
195C4—31no—
196C4—31no—
197C4—62no—
198C4—61no—
199C4—241no—
200C4—361no—
201C4#6420111yescharacteristic
202C4D3×GL(2,𝔽3)11yescharacteristic
203C3—41no—
204C3—81no—
205C3#7309911yescharacteristic, complemented
206C2#1059318911yescharacteristic
207C2#1059321711yescharacteristic
208C2#1059323511yescharacteristic
subgroupG/H
centerC22#64209
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
53. C3×SL(2,𝔽3)Q16semidirect
109. C3×Q8#216semidirect
126. SL(2,𝔽3)#216semidirect
175. Q8#3718semidirect
205. 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
C3×#53862
C3⋊(SL(2,𝔽3)⋊C8)2
#730993
#711293
C2×#37184