Finite Group Databasebeta

← #11920975

Subgroups — #11920975

Order 1152.

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

normal subgroups only

This group has 296 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
1C4×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic
2C3×C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3⋊(SL(2,𝔽3)⋊C8)C211yesmaximal, characteristic, complemented
4GL(2,𝔽3)⋊C8—33nomaximal
5#70937—31nomaximal
6C2×C3⋊GL(2,𝔽3)C411yescharacteristic
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C3⋊#859C411yescharacteristic
9#63647—41nomaximal
10C4×GL(2,𝔽3)—33no—
11C3×C8×Q8D311yescharacteristic, complemented
12C4×#215—31no—
13C8×D12—31no—
14C8×SL(2,𝔽3)D313yescomplemented
15SL(2,𝔽3)⋊C8—33no—
16#4535—31no—
17#4483—31no—
18#4479—31no—
19#4454—31no—
20C2×C4×C3⋊D3—41no—
21C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
22C3⋊GL(2,𝔽3)C812yescomplemented
23C2×C32×C8—41no—
24C2×C3⋊(C3⋊C8)—41no—
25SD16⋊C8—91noSylow
26C2×GL(2,𝔽3)—33no—
27C3×C4×Q8C2×D311yescharacteristic
28C2×#215—31no—
29C2×D3×C8—61no—
30C4×D12—31no—
31C4×SL(2,𝔽3)C2×D313yes—
32#859—33no—
33C3×C4⋊C8—31no—
34C3×C4×C8—31no—
35#835—31no—
36(C2×D3)⋊C8—61no—
37(C2×D3)⋊C8—123no—
38#809—31no—
39#808—31no—
40C4⋊(C3⋊C8)—31no—
41C4×C3⋊C8—31no—
42C4×C3⋊C8—31no—
43C22×C3⋊D3—41no—
44C2×C32×C4—41no—
45C2×C3⋊Dic3—41no—
46C4×C3⋊D3—81no—
47C4×C3⋊D3—81no—
48C3×SL(2,𝔽3)C2×C811yescharacteristic, complemented
49C32×C8—81no—
50C3⋊(C3⋊C8)—81no—
51C8×Q8C3⋊D311yescharacteristic, complemented
52C4×SD16—91no—
53C8×D4—91no—
54#335—91no—
55Q8⋊C8—91no—
56D4⋊C8—91no—
57#322—91no—
58C2×C3×Q8C4×D311yescharacteristic
59C2×D12—31no—
60C2×C4×D3—61no—
61C2×C4×D3—123no—
62C2×SL(2,𝔽3)C4×D313yes—
63GL(2,𝔽3)—36no—
64C2×C3×C8—31no—
65C2×C3×C8—43no—
66C2×C3×C8S411yescharacteristic
67C3×C4⋊C4—31no—
68C3×C42—31no—
69#215—32no—
70#215—32no—
71(C2×D3)⋊C4—61no—
72C4⋊Dic3—31no—
73C2×C3⋊C8—31no—
74C2×C3⋊C8—31no—
75C2×C3⋊C8—61no—
76C2×C3⋊C8—61no—
77C2×C3⋊C8—123no—
78D3×C8—122no—
79C22×C32—41no—
80C2×C3⋊D3—42no—
81C2×C3⋊D3—81no—
82C2×C3⋊D3—81no—
83C32×C4—41no—
84C32×C4—41no—
85C3⋊Dic3—81no—
86C2×SD16—91no—
87C22×C8—181no—
88C4×Q8C2×C3⋊D311yescharacteristic
89C4×D4—91no—
90#106—91no—
91C4⋊C8—31no—
92C4⋊C8—91no—
93#103—91no—
94#102—91no—
95C22⋊C8—181no—
96C4×C8—31no—
97C4×C8—91no—
98C4×C8—91no—
99C22×D3—61no—
100C22×D3—123no—
101C3×Q8—31no—
102C3×Q8D3×C811yescharacteristic
103C2×C3×C4—31no—
104C2×C3×C4—31no—
105C2×C3×C4—43no—
106C2×C3×C4C2×S411yescharacteristic
107C2×Dic3—61no—
108C2×Dic3—123no—
109D12—32no—
110D12—61no—
111C4×D3—121no—
112C4×D3—121no—
113C4×D3—243no—
114C4×D3—243no—
115SL(2,𝔽3)D3×C813yescomplemented
116C3×C8—21no—
117C3×C8—61no—
118C3×C8—83no—
119C3⋊C8—32no—
120C3⋊C8—61no—
121C3⋊C8—121no—
122C3⋊C8—121no—
123C3⋊C8—243no—
124C2×C32—41no—
125C2×C32—41no—
126C2×C32—41no—
127C3⋊D3—82no—
128C2×Q8C4×C3⋊D311yescharacteristic
129C2×D4—91no—
130C22×C4—181no—
131SD16—92no—
132SD16—92no—
133C2×C8—31no—
134C2×C8—91no—
135C2×C8—91no—
136C2×C8—181no—
137C2×C8—181no—
138C2×C8—362no—
139C2×C8C3⋊S411yescharacteristic
140C4⋊C4—31no—
141C4⋊C4—91no—
142C22⋊C4—181no—
143C42—31no—
144C22×C3—43no—
145C22×C3C4×S411yescharacteristic
146C2×D3—62no—
147C2×D3—126no—
148C2×D3—121no—
149C2×D3—121no—
150C2×D3—243no—
151C2×D3—243no—
152C3×C4—31no—
153C3×C4—31no—
154C3×C4—43no—
155C3×C4—43no—
156C3×C4—61no—
157C3×C4#98311yescharacteristic
158C3×C4#98611yescharacteristic
159Dic3—121no—
160Dic3—243no—
161C32—41noSylow
162C23—181no—
163Q8—31no—
164Q8C8×C3⋊D311yescharacteristic
165D4—92no—
166D4—181no—
167C2×C4—31no—
168C2×C4—31no—
169C2×C4—181no—
170C2×C4—361no—
171C2×C4—361no—
172C2×C4C2×C3⋊S411yescharacteristic
173C8—21no—
174C8—61no—
175C8—92no—
176C8—181no—
177C8—361no—
178C8—361no—
179C2×C3—43no—
180C2×C3—43no—
181C2×C3—43no—
182C2×C3#539411yescharacteristic
183C2×C3C8×S411yescharacteristic
184C2×C3#540511yescharacteristic
185D3—122no—
186D3—246no—
187C22—182no—
188C22—361no—
189C22—361no—
190C22C4×C3⋊S411yescharacteristic
191C4—31no—
192C4—31no—
193C4—61no—
194C4—361no—
195C4#6426411yescharacteristic
196C4#6426711yescharacteristic
197C3—43no—
198C3GL(2,𝔽3)⋊C811yescharacteristic, complemented
199C2—362no—
200C2#1059362811yescharacteristic
201C2C8×C3⋊S411yescharacteristic
202C2#1059363911yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C2×C8
FittingC3×C8×Q8D3
socleC22×C3C4×S4
prime radicalC2×C3⋊GL(2,𝔽3)C4
cosocleC3×C4×SL(2,𝔽3)C22
hypercenterC2×C8C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C3×C8×SL(2,𝔽3)C2semidirect
3. C3⋊(SL(2,𝔽3)⋊C8)C2semidirect
11. C3×C8×Q8D3semidirect
14. C8×SL(2,𝔽3)D3semidirect
22. C3⋊GL(2,𝔽3)C8semidirect
48. C3×SL(2,𝔽3)C2×C8semidirect
51. C8×Q8C3⋊D3semidirect
115. SL(2,𝔽3)D3×C8semidirect
198. C3GL(2,𝔽3)⋊C8semidirect

Maximal subgroups

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

subgroupindex
C4×C3⋊GL(2,𝔽3)2
C3×C8×SL(2,𝔽3)2
C3⋊(SL(2,𝔽3)⋊C8)2
GL(2,𝔽3)⋊C83
#709373
#636474