Finite Group Databasebeta

← #11714076

Subgroups — #11714076

Order 768.

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

normal subgroups only

This group has 375 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#85339C211yesmaximal, characteristic
2#85336C211yesmaximal, characteristic
3#2875⋊C3C211yesmaximal, characteristic
4#73124C214yesmaximal
5#33065—31nomaximal, Sylow
6C2×#984C2212yes—
7C2×#131⋊C3C2212yes—
8C4×#41⋊C3C2211yescharacteristic
9#5426C2212yes—
10C4⋊#226—22no—
11#5389—22no—
12#4703—41nomaximal
13C2×#491—31no—
14#3080—31no—
15#3039—31no—
16#2875D311yescharacteristic
17#1998—31no—
18#1966—31no—
19#1930—34no—
20#1908—34no—
21#1838—31no—
22C2×#41⋊C3C2311yescharacteristic
23#984—24no—
24C2×#226—22no—
25#870—41no—
26#869—41no—
27#131⋊C3—22no—
28C4×SL(2,𝔽3)D412yes—
29#859—22no—
30C3×#97—41no—
31#816—44no—
32C2×#137—32no—
33C2×#131C2×D312yes—
34C2×C4⋊Q8—31no—
35C2×#125—31no—
36C4×#41C2×D311yescharacteristic
37#491—62no—
38#491—61no—
39#491—61no—
40#479—62no—
41#478—62no—
42#462—62no—
43#461—62no—
44#428—32no—
45#423—31no—
46#423—31no—
47#419—32no—
48#417—32no—
49C2×#103—34no—
50C2×#101—34no—
51#405—32no—
52C4×M16—31no—
53C2×#97—31no—
54#343—34no—
55#341—34no—
56C2×Dic6—42no—
57#41⋊C3#4112yes—
58C2×SL(2,𝔽3)C2×D411yescharacteristic
59#226—42no—
60C2×C3×C8—42no—
61C3×C42—41no—
62C4⋊Dic3—42no—
63C4⋊Dic3—82no—
64Dic3⋊C4—82no—
65C2×#41C22×D311yescharacteristic
66C22×Q8—32no—
67#137—64no—
68#137—64no—
69C2×Q16—62no—
70C2×SD16—62no—
71#131—22no—
72#131—62no—
73C2×M16—32no—
74C2×M16—32no—
75C2×M16—32no—
76C22×C8—32no—
77C4⋊Q8—62no—
78C4⋊Q8—61no—
79C4⋊Q8—61no—
80#125—62no—
81#125—61no—
82#125—61no—
83C4×Q8D1212yes—
84C4×D4—32no—
85#117—31no—
86C2×C4⋊C4—32no—
87C2×C4⋊C4—62no—
88C2×C4⋊C4—62no—
89C2×C42—31no—
90#107—62no—
91#106—62no—
92C4⋊C8—62no—
93C4⋊C8—62no—
94#103—64no—
95#103—64no—
96#103—62no—
97#103—64no—
98#102—62no—
99#101—68no—
100#97—32no—
101#97—31no—
102#97S411yescharacteristic
103C4×C8—61no—
104C2×C3×C4—42no—
105C2×C3×C4—41no—
106C2×Dic3—82no—
107C2×Dic3—82no—
108Dic6—84no—
109SL(2,𝔽3)#12411yescharacteristic
110C3×C8—82no—
111#41—32no—
112#41#23512yes—
113C2×Q8—64no—
114C2×Q8—62no—
115C2×Q8—64no—
116C2×Q8—62no—
117C2×Q8C2×D1211yescharacteristic
118C2×D4—31no—
119C22×C4—31no—
120C22×C4—32no—
121C22×C4—62no—
122C22×C4—62no—
123Q16—122no—
124Q16—122no—
125SD16—124no—
126M16—34no—
127M16—62no—
128M16—64no—
129M16—64no—
130C2×C8—32no—
131C2×C8—62no—
132C2×C8—62no—
133C2×C8—62no—
134C2×C8—62no—
135C2×C8C2×S412yes—
136C4⋊C4—32no—
137C4⋊C4—62no—
138C4⋊C4—62no—
139C4⋊C4—122no—
140C4⋊C4—122no—
141C4⋊C4—122no—
142C4⋊C4—122no—
143C22⋊C4—61no—
144C42—32no—
145C42—31no—
146C42C2×S411yescharacteristic
147C22×C3—41no—
148C3×C4—42no—
149C3×C4—82no—
150Dic3—162no—
151Dic3—162no—
152C23—31no—
153Q8—31no—
154Q8—122no—
155Q8—122no—
156Q8—124no—
157Q8#87511yescharacteristic
158D4—32no—
159D4—61no—
160C2×C4—31no—
161C2×C4—32no—
162C2×C4—32no—
163C2×C4—32no—
164C2×C4—122no—
165C2×C4—124no—
166C2×C4—122no—
167C2×C4—122no—
168C2×C4—242no—
169C2×C4C4⋊S412yes—
170C2×C4C22×S411yescharacteristic
171C8—22no—
172C8—62no—
173C8—122no—
174C8—122no—
175C2×C3—41no—
176C2×C3—42no—
177C22—32no—
178C22—62no—
179C22C2×C4⋊S411yescharacteristic
180C4—22no—
181C4—31no—
182C4—31no—
183C4—62no—
184C4—242no—
185C4—242no—
186C4#591212yes—
187C3—41noSylow
188C2—62no—
189C2#8529011yescharacteristic
190C2#8545512yes—
subgroupG/H
center, socleC22C2×C4⋊S4
commutator, prime radical, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
Fitting#2875D3
hypercenter#97S4
hypocenterSL(2,𝔽3)#124
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ #97

Maximal subgroups

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

subgroupindex
#853392
#853362
#2875⋊C32
#731242
#330653
#47034