Finite Group Databasebeta

← #11921054

Subgroups — #11921054

Order 1152.

272 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 49 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×Q8×SL(2,𝔽3)C211yesmaximal, characteristic
2#10593625C211yesmaximal, characteristic
3C3⋊(SL(2,𝔽3)⋊C8)C211yesmaximal, characteristic
4#73098—33nomaximal
5#71940—31nomaximal
6C3×C4×SL(2,𝔽3)—21no—
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C3⋊#859—21no—
9#63662—41nomaximal
10C3×Q82D311yescharacteristic
11Q8×SL(2,𝔽3)D313yes—
12#5391—33no—
13#5092—31no—
14#5021—31no—
15#5021—31no—
16SL(2,𝔽3)⋊C8—33no—
17#4546—31no—
18#4535—31no—
19#4535—31no—
20C2×C32×Q8—41no—
21C2×C3×SL(2,𝔽3)D411yescharacteristic
22C4⋊(C3⋊Dic3)—41no—
23C2×C3⋊(C3⋊C8)—41no—
24#1633—91noSylow
25C3×C4⋊Q8—31no—
26C3×C4⋊Q8—61no—
27C3×C4×Q8—21no—
28C3×C4×Q8—31no—
29C3×C4×Q8C2×D311yescharacteristic
30C2×#216—62no—
31#869—31no—
32C4×SL(2,𝔽3)—23no—
33C4×SL(2,𝔽3)C2×D313yes—
34#859—63no—
35#835—61no—
36#835—61no—
37#835—123no—
38#807—61no—
39#807—61no—
40C4⋊(C3⋊C8)—31no—
41C4⋊(C3⋊C8)—31no—
42C4×C3⋊C8—31no—
43C32×Q8—42no—
44C32×Q8—81no—
45C2×C32×C4—41no—
46C2×C32×C4—81no—
47C2×C3⋊Dic3—81no—
48C3×SL(2,𝔽3)Q1611yescharacteristic, complemented
49C3⋊(C3⋊C8)—81no—
50Q82C3⋊D311yescharacteristic
51#494—91no—
52#477—91no—
53#477—91no—
54#333—91no—
55Q8⋊C8—91no—
56Q8⋊C8—91no—
57C2×C3×Q8—32no—
58C2×C3×Q8—43no—
59C2×C3×Q8#6711yescharacteristic
60C2×C3×Q8S411yescharacteristic
61C2×Dic6—61no—
62C2×SL(2,𝔽3)#6713yes—
63C3×C4⋊C4—31no—
64C3×C4⋊C4—31no—
65C3×C4⋊C4—61no—
66C3×C4⋊C4—61no—
67C3×C42—31no—
68C3×C42—61no—
69#216—124no—
70C4⋊Dic3—61no—
71C4⋊Dic3—61no—
72C4⋊Dic3—123no—
73C2×C3⋊C8—61no—
74C2×C3⋊C8—61no—
75C2×C3⋊C8—61no—
76C2×C3⋊C8—123no—
77C22×C32—41no—
78C32×C4—41no—
79C32×C4—41no—
80C32×C4—82no—
81C3⋊Dic3—161no—
82C2×Q16—182no—
83C4⋊Q8—31no—
84C4⋊Q8—61no—
85C4⋊Q8—91no—
86C4×Q8—21no—
87C4×Q8—31no—
88C4×Q8C2×C3⋊D311yescharacteristic
89#107—181no—
90#107—181no—
91C4⋊C8—91no—
92C4⋊C8—91no—
93#103—181no—
94#103—181no—
95C4×C8—91no—
96C3×Q8—21no—
97C3×Q8—31no—
98C3×Q8—46no—
99C3×Q8—83no—
100C3×Q8—122no—
101C3×Q8#21611yescharacteristic
102C3×Q8#22612yes—
103C2×C3×C4—21no—
104C2×C3×C4—31no—
105C2×C3×C4—31no—
106C2×C3×C4—43no—
107C2×C3×C4—61no—
108C2×C3×C4—83no—
109C2×C3×C4C2×S411yescharacteristic
110C2×Dic3—121no—
111C2×Dic3—243no—
112Dic6—122no—
113SL(2,𝔽3)#21613yescomplemented
114C3⋊C8—62no—
115C3⋊C8—121no—
116C3⋊C8—121no—
117C3⋊C8—243no—
118C2×C32—41no—
119C2×C32—41no—
120C2×C32—41no—
121C2×Q8—32no—
122C2×Q8—181no—
123C2×Q8#63811yescharacteristic
124C2×Q8C3⋊S411yescharacteristic
125Q16—364no—
126C2×C8—181no—
127C2×C8—181no—
128C2×C8—181no—
129C4⋊C4—31no—
130C4⋊C4—31no—
131C4⋊C4—61no—
132C4⋊C4—61no—
133C4⋊C4—181no—
134C4⋊C4—181no—
135C42—31no—
136C42—61no—
137C22×C3—43no—
138C22×C3GL(2,ℤ/4)11yescharacteristic
139C3×C4—22no—
140C3×C4—31no—
141C3×C4—31no—
142C3×C4—43no—
143C3×C4—43no—
144C3×C4—61no—
145C3×C4—86no—
146C3×C4—121no—
147C3×C4C2×#22611yescharacteristic
148C3×C4#98311yescharacteristic
149Dic3—241no—
150Dic3—483no—
151C32—41noSylow
152Q8—21no—
153Q8—31no—
154Q8—122no—
155Q8—362no—
156Q8#375611yescharacteristic
157Q8C3⋊#22612yes—
158C2×C4—21no—
159C2×C4—31no—
160C2×C4—31no—
161C2×C4—61no—
162C2×C4—361no—
163C2×C4C2×C3⋊S411yescharacteristic
164C8—182no—
165C8—361no—
166C8—361no—
167C2×C3—43no—
168C2×C3—43no—
169C2×C3—43no—
170C2×C3#541611yescharacteristic
171C2×C3#541911yescharacteristic
172C2×C3#542911yescharacteristic
173C22#6426911yescharacteristic
174C4—22no—
175C4—31no—
176C4—31no—
177C4—61no—
178C4—121no—
179C4—721no—
180C4C2×C3⋊#22611yescharacteristic
181C4#6426411yescharacteristic
182C3—43no—
183C3#7309811yescharacteristic, complemented
184C2#1059365011yescharacteristic
185C2#1059365311yescharacteristic
186C2#1059366311yescharacteristic
subgroupG/H
centerC22#64269
commutator, cosocleC3×C4×SL(2,𝔽3)C22
FrattiniC2×C4C2×C3⋊S4
FittingC3×Q82D3
socleC22×C3GL(2,ℤ/4)
prime radicalC2×C3×SL(2,𝔽3)D4
hypercenterC2×Q8C3⋊S4
hypocenterC3×SL(2,𝔽3)Q16
LCSG ▷ C3×C4×SL(2,𝔽3) ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
48. C3×SL(2,𝔽3)Q16semidirect
113. SL(2,𝔽3)#216semidirect
183. C3#73098semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×SL(2,𝔽3)2
#105936252
C3⋊(SL(2,𝔽3)⋊C8)2
#730983
#719403
#636624