Finite Group Databasebeta

← #11714075

Subgroups — #11714075

Order 768.

411 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 47 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
1#85346C211yesmaximal, characteristic
2#85339C211yesmaximal, characteristic, complemented
3#2875⋊C3C211yesmaximal, characteristic, complemented
4#73128C214yesmaximal, complemented
5#33064—31nomaximal, Sylow
6C2×#986C2212yes—
7C2×#131⋊C3C2212yes—
8C4×#41⋊C3C2211yescharacteristic
9#5426C2212yes—
10C4⋊GL(2,𝔽3)—22no—
11#5389—22no—
12#4713—41nomaximal
13C2×#490—31no—
14#3080—31no—
15#3038—31no—
16#2875D311yescharacteristic, complemented
17#1998—31no—
18#1965—31no—
19#1931—34no—
20#1907—34no—
21#1838—31no—
22C2×#41⋊C3C2311yescharacteristic
23#986—24no—
24C2×GL(2,𝔽3)—22no—
25#874—41no—
26#870—41no—
27#131⋊C3—22no—
28C4×SL(2,𝔽3)D412yes—
29#859—22no—
30C3×#97—41no—
31#821—44no—
32C2×AGL(1,ℤ/8)—32no—
33C2×#131C2×D312yes—
34C2×#127—31no—
35C2×#125—31no—
36C4×#41C2×D311yescharacteristic
37#490—62no—
38#490—61no—
39#490—61no—
40#479—62no—
41#478—62no—
42#460—62no—
43#459—62no—
44#428—32no—
45#423—31no—
46#423—31no—
47#418—32no—
48#417—32no—
49C2×#102—34no—
50C2×#100—34no—
51#405—32no—
52C4×M16—31no—
53C2×#97—31no—
54#343—34no—
55#341—34no—
56C2×D12—42no—
57C2×D12—82no—
58#41⋊C3#4112yes—
59C2×SL(2,𝔽3)C2×D411yescharacteristic
60GL(2,𝔽3)—42no—
61C2×C3×C8—42no—
62C3×C42—41no—
63C4⋊Dic3—42no—
64Dic3⋊C4—82no—
65C2×#41C22×D311yescharacteristic
66C22×D4—32no—
67C22×D4—62no—
68AGL(1,ℤ/8)—64no—
69AGL(1,ℤ/8)—64no—
70C2×SD16—62no—
71C2×D8—62no—
72#131—22no—
73#131—62no—
74C2×M16—32no—
75C2×M16—32no—
76C2×M16—32no—
77C22×C8—32no—
78#127—62no—
79#127—61no—
80#127—61no—
81#125—62no—
82#125—61no—
83#125—61no—
84C4×Q8D1212yes—
85C4×D4—32no—
86#117—31no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4—62no—
89C2×C42—31no—
90#107—62no—
91#106—62no—
92C4⋊C8—62no—
93C4⋊C8—62no—
94#103—62no—
95#102—64no—
96#102—64no—
97#102—62no—
98#102—64no—
99#100—68no—
100#97—32no—
101#97—31no—
102#97S411yescharacteristic
103C4×C8—61no—
104C22×D3—82no—
105C2×C3×C4—42no—
106C2×C3×C4—41no—
107C2×Dic3—82no—
108D12—84no—
109D12—164no—
110SL(2,𝔽3)#12411yescharacteristic
111C3×C8—82no—
112C24—62no—
113#41—32no—
114#41#23512yes—
115C2×Q8C2×D1211yescharacteristic
116C2×D4—31no—
117C2×D4—64no—
118C2×D4—62no—
119C2×D4—64no—
120C2×D4—62no—
121C2×D4—122no—
122C2×D4—122no—
123C2×D4—244no—
124C22×C4—31no—
125C22×C4—32no—
126C22×C4—62no—
127SD16—122no—
128SD16—122no—
129D8—124no—
130M16—34no—
131M16—62no—
132M16—64no—
133M16—64no—
134C2×C8—32no—
135C2×C8—62no—
136C2×C8—62no—
137C2×C8—62no—
138C2×C8—62no—
139C2×C8C2×S412yes—
140C4⋊C4—32no—
141C4⋊C4—62no—
142C4⋊C4—62no—
143C4⋊C4—122no—
144C4⋊C4—122no—
145C22⋊C4—61no—
146C42—32no—
147C42—31no—
148C42C2×S411yescharacteristic
149C22×C3—41no—
150C2×D3—162no—
151C2×D3—164no—
152C3×C4—42no—
153C3×C4—82no—
154Dic3—162no—
155C23—31no—
156C23—124no—
157C23—122no—
158C23—244no—
159Q8—31no—
160Q8#87511yescharacteristic
161D4—32no—
162D4—61no—
163D4—122no—
164D4—122no—
165D4—124no—
166D4—244no—
167D4—244no—
168C2×C4—31no—
169C2×C4—32no—
170C2×C4—32no—
171C2×C4—32no—
172C2×C4—122no—
173C2×C4—122no—
174C2×C4—242no—
175C2×C4C4⋊S412yes—
176C2×C4C22×S411yescharacteristic
177C8—22no—
178C8—62no—
179C8—122no—
180C8—122no—
181C2×C3—41no—
182C2×C3—42no—
183D3—322no—
184C22—32no—
185C22—62no—
186C22—242no—
187C22—244no—
188C22—484no—
189C22C2×C4⋊S411yescharacteristic
190C4—22no—
191C4—31no—
192C4—31no—
193C4—62no—
194C4—242no—
195C4#591212yes—
196C3—41noSylow
197C2—62no—
198C2—482no—
199C2#8529011yescharacteristic
200C2#8545412yes—
subgroupG/H
center, socleC22C2×C4⋊S4
commutator, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
Fitting#2875D3
prime radical#85346C2
hypercenter#97S4
hypocenterSL(2,𝔽3)#124
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ #97

Complements

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

NH ≅ G/Ntype
2. #85339C2semidirect
3. #2875⋊C3C2semidirect
4. #73128C2semidirect
16. #2875D3semidirect

Maximal subgroups

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

subgroupindex
#853462
#853392
#2875⋊C32
#731282
#330643
#47134