Finite Group Databasebeta

← #859⋊A4 #11921454

Subgroups — #859⋊A4 #11921454

Order 1152.

202 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 24 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
1C2×Ω+(4,𝔽3)C211yesmaximal, characteristic
2#85528C311yesmaximal, characteristic, complemented
3#2897⋊C3—31nomaximal
4Ω+(4,𝔽3)C411yescharacteristic, complemented
5Dic3×SL(2,𝔽3)—41nomaximal
6C3×#859—41nomaximal
7C2×#997—21no—
8C2×#994D311yescharacteristic
9C2×#994C2×C311yescharacteristic
10C2×#131⋊C3—31no—
11#520⋊C3—31no—
12#5426—31no—
13#5424—31no—
14C2×C3×SL(2,𝔽3)—41no—
15C2×C3×SL(2,𝔽3)—41no—
16#2897—31noSylow
17#997—21no—
18#994—31no—
19#994—31no—
20#994Dic311yescharacteristic, complemented
21#994C3×C411yescharacteristic, complemented
22C2×#41⋊C3—31no—
23C2×#41⋊C3—31no—
24Q8×Dic3—41no—
25#131⋊C3—61no—
26C4×SL(2,𝔽3)—61no—
27#859—31no—
28#859A411yescharacteristic, complemented
29C3×#103—121no—
30C2×C3×Dic3—161no—
31C3×SL(2,𝔽3)—41no—
32C3×SL(2,𝔽3)—41no—
33C2×H(5,𝔽2)C3×D311yescharacteristic
34C2×#131—31no—
35#520—31no—
36#418—91no—
37#417—91no—
38#416—91no—
39C2×#102—91no—
40C22×A4—241no—
41C2×C3×Q8—41no—
42C2×C3×Q8—41no—
43#41⋊C3—31no—
44#41⋊C3—31no—
45#41⋊C3—31no—
46#41⋊C3—31no—
47C2×SL(2,𝔽3)—81no—
48C2×SL(2,𝔽3)—81no—
49C2×SL(2,𝔽3)S411yescharacteristic
50C2×SL(2,𝔽3)C2×A411yescharacteristic
51C2×C3×C8—121no—
52C3×C4⋊C4—121no—
53C4⋊Dic3—121no—
54C4×Dic3—121no—
55C22×C32—161no—
56C3×Dic3—162no—
57H(5,𝔽2)—31no—
58H(5,𝔽2)—31no—
59H(5,𝔽2)—91no—
60H(5,𝔽2)C3×Dic311yescharacteristic, complemented
61C2×#41—31no—
62C2×#41—31no—
63C22×D4—91no—
64#131—61no—
65#131—181no—
66C2×M16—91no—
67C22×C8—91no—
68C4×Q8—61no—
69C4×D4—181no—
70#117—91no—
71C2×C4⋊C4—91no—
72#103—31no—
73#103—91no—
74#102—91no—
75#102—91no—
76#102—92no—
77C2×A4—241no—
78C2×A4—241no—
79C2×A4—241no—
80C3×Q8—41no—
81C3×Q8—41no—
82C3×Q8—121no—
83C3×Q8—121no—
84C2×C3×C4—121no—
85C2×C3×C4—121no—
86C2×C3×C4—241no—
87C2×Dic3—41no—
88C2×Dic3—121no—
89SL(2,𝔽3)—81no—
90SL(2,𝔽3)—81no—
91SL(2,𝔽3)SL(2,ℤ/4)11yescharacteristic
92SL(2,𝔽3)C4×A411yescharacteristic
93C3×C8—241no—
94C2×C32—161no—
95C2×C32—161no—
96C2×C32—161no—
97C24—61no—
98#41—31no—
99#41—31no—
100#41—31no—
101#41—31no—
102#41—91no—
103#41—91no—
104#41—91no—
105#41—91no—
106C2×Q8C3×S411yescharacteristic
107C2×Q8D3×A411yescharacteristic
108C2×D4—91no—
109C2×D4—91no—
110C2×D4—92no—
111C2×D4—91no—
112C2×D4—91no—
113C2×D4—181no—
114C2×D4—181no—
115C22×C4—91no—
116C22×C4—181no—
117M16—181no—
118M16—181no—
119C2×C8—31no—
120C2×C8—91no—
121C2×C8—181no—
122C2×C8—181no—
123C4⋊C4—31no—
124C4⋊C4—91no—
125C4⋊C4—181no—
126C22⋊C4—181no—
127C42—181no—
128C22×C3—41no—
129C22×C3—41no—
130C22×C3—321no—
131A4—241no—
132C3×C4—121no—
133C3×C4—121no—
134C3×C4—121no—
135C3×C4—121no—
136C3×C4—481no—
137Dic3—42no—
138Dic3—241no—
139C32—161noSylow
140C23—61no—
141C23—91no—
142C23—181no—
143C23—241no—
144C23—241no—
145Q8—31no—
146Q8—31no—
147Q8C3×SL(2,ℤ/4)11yescharacteristic
148Q8Dic3×A411yescharacteristic
149D4—91no—
150D4—91no—
151D4—91no—
152D4—91no—
153D4—92no—
154D4—181no—
155C2×C4—31no—
156C2×C4—31no—
157C2×C4—61no—
158C2×C4—91no—
159C2×C4—91no—
160C2×C4—91no—
161C2×C4—91no—
162C2×C4—181no—
163C2×C4—721no—
164C8—61no—
165C8—181no—
166C2×C3—41no—
167C2×C3—41no—
168C2×C3—41no—
169C2×C3—41no—
170C2×C3—41no—
171C2×C3—41no—
172C2×C3—321no—
173C2×C3—321no—
174C2×C3—321no—
175C22—91no—
176C22—91no—
177C22—181no—
178C22—181no—
179C22—241no—
180C22—721no—
181C22A4×S411yescharacteristic
182C4—31no—
183C4—31no—
184C4—31no—
185C4—31no—
186C4—121no—
187C4—361no—
188C3—41no—
189C3—41no—
190C3—321no—
191C2—181no—
192C2—181no—
193C2#226⋊A411yescharacteristic
194C2GL(2,𝔽3)⋊A411yescharacteristic
195C2A4×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, Frattini, socle, hypercenterC22A4×S4
commutator, hypocenter#994C3×C4
FittingC2×H(5,𝔽2)C3×D3
prime radicalC2×Ω+(4,𝔽3)C2
cosocleC2×#994C2×C3
LCSG ▷ #994
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. #85528C3semidirect
4. Ω+(4,𝔽3)C4semidirect
20. #994Dic3semidirect
21. #994C3×C4semidirect
28. #859A4semidirect
60. H(5,𝔽2)C3×Dic3semidirect

Maximal subgroups

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

subgroupindex
C2×Ω+(4,𝔽3)2
#855283
#2897⋊C33
Dic3×SL(2,𝔽3)4
C3×#8594