Finite Group Databasebeta

← #11921043

Subgroups — #11921043

Order 1152.

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

normal subgroups only

This group has 340 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×D4×SL(2,𝔽3)C211yesmaximal, characteristic
2#10593620C211yesmaximal, characteristic, complemented
3C3⋊(SL(2,𝔽3)⋊C8)C211yesmaximal, characteristic, complemented
4#73087—33nomaximal
5#71917—31nomaximal
6C22×C3×SL(2,𝔽3)—21no—
7C2×#3754—41nomaximal
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9C3⋊#859—21no—
10C3×D4×Q8D311yescharacteristic
11D4×SL(2,𝔽3)D313yes—
12#5386—33no—
13#5080—31no—
14#5020—31no—
15#5019—31no—
16SL(2,𝔽3)⋊C8—33no—
17#4547—31no—
18#4535—31no—
19D4⋊(C3⋊C8)—31no—
20C2×C32×D4—41no—
21C2×#634—41no—
22C2×C3×SL(2,𝔽3)—41no—
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24#3754—44no—
25C2×C3⋊(C3⋊C8)—41no—
26#1651—91noSylow
27C22×C3×Q8—21no—
28C22×SL(2,𝔽3)—23no—
29C3×C4⋊Q8—31no—
30C3×C22⋊Q8—61no—
31C3×C4×Q8C2×D311yescharacteristic
32C3×C4×D4—31no—
33C2×#214—61no—
34C2×#214—123no—
35#869—31no—
36C4×SL(2,𝔽3)C2×D313yes—
37#859—63no—
38#835—61no—
39#810—61no—
40#808—61no—
41#808—62no—
42C4⋊(C3⋊C8)—31no—
43C4⋊(C3⋊C8)—31no—
44C4×C3⋊C8—31no—
45C23×C32—81no—
46C32×D4—42no—
47C32×D4—81no—
48C2×C32×C4—41no—
49C2×C3⋊Dic3—81no—
50#634—42no—
51#634—81no—
52C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
53C3⋊(C3⋊C8)—42no—
54D4×Q8C3⋊D311yescharacteristic
55#498—91no—
56#475—91no—
57#461—91no—
58#332—91no—
59Q8⋊C8—91no—
60D4⋊C8—91no—
61C2×C3×Q8—41no—
62C2×C3×Q8—61no—
63C2×C3×Q8—121no—
64C2×C3×Q8#6711yescharacteristic
65C2×C3×D4—43no—
66C2×C3×D4S411yescharacteristic
67C22×C3×C4—61no—
68C2×Dic6—61no—
69C2×Dic6—123no—
70C2×SL(2,𝔽3)—43no—
71C2×SL(2,𝔽3)#6713yes—
72C3×C4⋊C4—31no—
73C3×C4⋊C4—32no—
74C3×C4⋊C4—31no—
75C3×C22⋊C4—61no—
76C3×C42—31no—
77#214—1212no—
78#214—122no—
79C4⋊Dic3—61no—
80C4⋊Dic3—61no—
81C2×C3⋊C8—61no—
82C2×C3⋊C8—61no—
83C2×C3⋊C8—61no—
84C2×C3⋊C8—123no—
85C22×C32—41no—
86C22×C32—82no—
87C22×C32—161no—
88C22×C32—161no—
89C32×C4—41no—
90C32×C4—41no—
91C3⋊Dic3—82no—
92C22×Q8—21no—
93C2×SD16—181no—
94C4⋊Q8—31no—
95C4⋊Q8—91no—
96C22⋊Q8—61no—
97C4×Q8C2×C3⋊D311yescharacteristic
98C4×D4—31no—
99#106—181no—
100#106—182no—
101C4⋊C8—91no—
102C4⋊C8—91no—
103#103—181no—
104#103—181no—
105C4×C8—91no—
106C23×C3—21no—
107C23×C3—83no—
108C3×Q8—31no—
109C3×Q8—121no—
110C3×Q8—121no—
111C3×Q8#21411yescharacteristic, complemented
112C3×D4—21no—
113C3×D4—46no—
114C3×D4—83no—
115C3×D4GL(2,𝔽3)12yes—
116C2×C3×C4—31no—
117C2×C3×C4—31no—
118C2×C3×C4—43no—
119C2×C3×C4—61no—
120C2×C3×C4—121no—
121C2×C3×C4—121no—
122C2×C3×C4C2×S411yescharacteristic
123C2×Dic3—121no—
124C2×Dic3—243no—
125Dic6—126no—
126Dic6—121no—
127Dic6—121no—
128Dic6—243no—
129SL(2,𝔽3)#21413yescomplemented
130C3⋊C8—62no—
131C3⋊C8—121no—
132C3⋊C8—126no—
133C3⋊C8—121no—
134C2×C32—41no—
135C2×C32—41no—
136C2×C32—41no—
137C2×C32—162no—
138C2×Q8—41no—
139C2×Q8—61no—
140C2×Q8—121no—
141C2×Q8—181no—
142C2×Q8#63811yescharacteristic
143C2×D4C3⋊S411yescharacteristic
144C22×C4—61no—
145SD16—362no—
146C2×C8—181no—
147C2×C8—181no—
148C2×C8—181no—
149C4⋊C4—31no—
150C4⋊C4—32no—
151C4⋊C4—31no—
152C4⋊C4—181no—
153C4⋊C4—181no—
154C22⋊C4—61no—
155C42—31no—
156C22×C3—22no—
157C22×C3—41no—
158C22×C3—41no—
159C22×C3—43no—
160C22×C3—86no—
161C22×C3—163no—
162C22×C3—163no—
163C22×C3GL(2,ℤ/4)11yescharacteristic
164C3×C4—31no—
165C3×C4—31no—
166C3×C4—43no—
167C3×C4—43no—
168C3×C4—61no—
169C3×C4—121no—
170C3×C4C2×GL(2,𝔽3)11yescharacteristic
171C3×C4#98311yescharacteristic
172Dic3—246no—
173Dic3—241no—
174C32—41noSylow
175C23—21no—
176Q8—31no—
177Q8—121no—
178Q8—121no—
179Q8—361no—
180Q8—361no—
181Q8#375411yescharacteristic, complemented
182D4—21no—
183D4C3⋊GL(2,𝔽3)12yes—
184C2×C4—31no—
185C2×C4—31no—
186C2×C4—61no—
187C2×C4—121no—
188C2×C4—121no—
189C2×C4—361no—
190C2×C4C2×C3⋊S411yescharacteristic
191C8—182no—
192C8—361no—
193C8—361no—
194C2×C3—42no—
195C2×C3—43no—
196C2×C3—43no—
197C2×C3—43no—
198C2×C3—166no—
199C2×C3#541411yescharacteristic
200C2×C3#542111yescharacteristic
201C2×C3#543011yescharacteristic
202C22—22no—
203C22—41no—
204C22—41no—
205C22#6426911yescharacteristic
206C4—31no—
207C4—31no—
208C4—61no—
209C4—121no—
210C4—721no—
211C4C2×C3⋊GL(2,𝔽3)11yescharacteristic
212C4#6426411yescharacteristic
213C3—43no—
214C3#7308711yescharacteristic, complemented
215C2—42no—
216C2#1059364811yescharacteristic
217C2#1059365511yescharacteristic
218C2#1059366411yescharacteristic
subgroupG/H
centerC22#64269
commutator, cosocleC3×C4×SL(2,𝔽3)C22
FrattiniC2×C4C2×C3⋊S4
FittingC3×D4×Q8D3
socleC22×C3GL(2,ℤ/4)
prime radicalC3×D4×SL(2,𝔽3)C2
hypercenterC2×D4C3⋊S4
hypocenterC3×SL(2,𝔽3)SD16
LCSG ▷ C3×C4×SL(2,𝔽3) ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #10593620C2semidirect
3. C3⋊(SL(2,𝔽3)⋊C8)C2semidirect
52. C3×SL(2,𝔽3)SD16semidirect
111. C3×Q8#214semidirect
129. SL(2,𝔽3)#214semidirect
181. Q8#3754semidirect
214. C3#73087semidirect

Maximal subgroups

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

subgroupindex
C3×D4×SL(2,𝔽3)2
#105936202
C3⋊(SL(2,𝔽3)⋊C8)2
#730873
#719173
C2×#37544