Finite Group Databasebeta

← #10709596

Subgroups — #10709596

Order 768.

186 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 53 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#71317C211yesmaximal, characteristic
2#71311C211yesmaximal, characteristic, complemented
3#69308C211yesmaximal, characteristic
4#69301C211yesmaximal, characteristic
5C3×#1358C211yesmaximal, characteristic, complemented
6#67574C211yesmaximal, characteristic, complemented
7#67567C211yesmaximal, characteristic, complemented
8#12466—31nomaximal, Sylow
9C3×#494C2211yescharacteristic
10#5179—21no—
11#4907—21no—
12#4700C2211yescharacteristic
13#4683C2211yescharacteristic
14C3×#364C2211yescharacteristic
15C3×#364C2211yescharacteristic
16#4565—21no—
17#4564—21no—
18#4564—21no—
19#4518—21no—
20#4515—21no—
21#4513—21no—
22#4463C2211yescharacteristic
23#4461C2211yescharacteristic
24#2245—31no—
25#2231—31no—
26#2214—31no—
27#2171—31no—
28#1369—31no—
29#1366—31no—
30#1358D311yescharacteristic, complemented
31C2×C3×Q16—21no—
32C2×C3×Q16—21no—
33C3×C4⋊Q8—21no—
34#904D411yescharacteristic
35#901—21no—
36#870—21no—
37C3×M32—21no—
38C3×M32—21no—
39C3×C4×C8C2311yescharacteristic
40#829—41no—
41#828—41no—
42#819D411yescharacteristic
43#818—21no—
44#817—21no—
45#812—21no—
46C2×C3⋊C16D411yescharacteristic
47C2×C3⋊C16D411yescharacteristic
48#805—21no—
49#798—41no—
50D3×C16—41no—
51#510—61no—
52#504—61no—
53#499—31no—
54#494C2×D311yescharacteristic
55#443—31no—
56#365—61no—
57#364—31no—
58#364C2×D311yescharacteristic
59#364C2×D311yescharacteristic
60#362—61no—
61#360—61no—
62#358—61no—
63#358—61no—
64#350—61no—
65#346—31no—
66C2×C3×Q8—41no—
67#235—21no—
68C3×Q16—41no—
69C3×Q16—41no—
70C3×Q16—41no—
71C3×Q16—41no—
72C2×C3×C8C2×D411yescharacteristic
73C2×C3×C8C2×D411yescharacteristic
74C3×C4⋊C4—81no—
75C3×C42#4111yescharacteristic
76C4⋊Dic3—41no—
77Dic3⋊C4—81no—
78#208—21no—
79Dic12SD1611yescharacteristic
80D24SD1611yescharacteristic
81#204—21no—
82ΓL(1,𝔽25)—41no—
83D3×C8—41no—
84C3×C16—41no—
85C3×C16—41no—
86C3⋊C16—21no—
87C3⋊C16—21no—
88C3⋊C16—41no—
89#135—31no—
90C2×Q16—21no—
91C2×Q16—21no—
92#131—61no—
93C4⋊Q8—21no—
94#125—61no—
95Q32—121no—
96SD32—121no—
97M32—21no—
98M32—21no—
99M32—61no—
100M32—121no—
101C2×C16—31no—
102C2×C16—31no—
103C2×C16—121no—
104#108—31no—
105#107—61no—
106#106—61no—
107#104—61no—
108C4×C8C22×D311yescharacteristic
109C3×Q8—81no—
110C3×Q8—81no—
111C2×C3×C4—21no—
112C2×C3×C4—81no—
113C2×C3×C4#12111yescharacteristic
114#67—41no—
115C2×Dic3—81no—
116D12—21no—
117C4×D3—41no—
118Dic6—21no—
119C3×C8C2×SD1611yescharacteristic
120C3×C8#13511yescharacteristic
121C3×C8#13511yescharacteristic
122C3×C8AGL(1,ℤ/8)11yescharacteristic
123C3⋊C8—41no—
124#41—61no—
125C2×Q8—41no—
126Q16—31no—
127Q16—41no—
128Q16—41no—
129Q16—41no—
130Q16—41no—
131SD16—61no—
132D8—31no—
133M16—61no—
134M16—121no—
135C2×C8—121no—
136C2×C8D3×D411yescharacteristic
137C2×C8D3×D411yescharacteristic
138C4⋊C4—81no—
139C4⋊C4—121no—
140C4⋊C4—241no—
141C42#23511yescharacteristic
142C16—41no—
143C16—41no—
144C16—61no—
145C16—61no—
146C16—121no—
147C22×C3#47211yescharacteristic
148C2×D3—41no—
149C3×C4—41no—
150C3×C4—161no—
151C3×C4#46311yescharacteristic
152C3×C4#46511yescharacteristic
153Dic3—41no—
154Dic3—161no—
155Q8—61no—
156Q8—81no—
157Q8—81no—
158D4—61no—
159D4—121no—
160C2×C4—21no—
161C2×C4—81no—
162C2×C4—121no—
163C2×C4—241no—
164C2×C4#88411yescharacteristic
165C8—121no—
166C8#91111yescharacteristic
167C8#91211yescharacteristic
168C8D3×SD1611yescharacteristic
169C8#91611yescharacteristic
170C2×C3—21no—
171C2×C3#167711yescharacteristic
172D3—81no—
173C22—121no—
174C22#474911yescharacteristic
175C4—41no—
176C4—121no—
177C4—161no—
178C4—481no—
179C4#477811yescharacteristic
180C4#478711yescharacteristic
181C3#1246611yescharacteristic, complemented, Sylow
182C2—21no—
183C2—241no—
184C2#6839611yescharacteristic
subgroupG/H
centerC2#68396
commutator, cosocleC3×C4×C8C23
FrattiniC4×C8C22×D3
FittingC3×#1358C2
socleC2×C3#1677
prime radical#904D4
hypercenter#1358D3
hypocenterC3#12466
LCSG ▷ C3×C4×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1358

Complements

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

NH ≅ G/Ntype
2. #71311C2semidirect
5. C3×#1358C2semidirect
6. #67574C2semidirect
7. #67567C2semidirect
30. #1358D3semidirect
181. C3#12466semidirect

Maximal subgroups

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

subgroupindex
#713172
#713112
#693082
#693012
C3×#13582
#675742
#675672
#124663