Finite Group Databasebeta

← #10698302

Subgroups — #10698302

Order 768.

402 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 93 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×#2864C211yesmaximal, characteristic, complemented
2#69545C212yesmaximal
3#69395C211yesmaximal, characteristic
4#67631C212yesmaximal, complemented
5#67611C211yesmaximal, characteristic, complemented
6#10368—31nomaximal, Sylow
7#5477C2211yescharacteristic
8C3×#389—22no—
9C3×#387—24no—
10C3×#382C2212yes—
11C3×C43C2211yescharacteristic
12C2×#805—22no—
13#4925C2212yes—
14#4924C2211yescharacteristic
15#4451—24no—
16#2864D311yescharacteristic, complemented
17#1962—31no—
18#1517—32no—
19#1286—32no—
20#1276—31no—
21C2×#235—21no—
22C2×C3×C4⋊C4—24no—
23C2×C3×C4⋊C4—22no—
24C2×C3×C42—22no—
25C2×C3×C42D412yes—
26C2×C3×C42C2311yescharacteristic
27#930—22no—
28C2×#208—21no—
29#875—21no—
30#874Q811yescharacteristic
31C4×D12—22no—
32#869Q811yescharacteristic
33C3×#95—24no—
34C3×#95—22no—
35C3×#95D414yes—
36#830—22no—
37#805—44no—
38C4⋊(C3⋊C8)—22no—
39C4⋊(C3⋊C8)—22no—
40C4×C3⋊C8—22no—
41#532—31no—
42#431—32no—
43#423—31no—
44C2×#104—62no—
45#389—22no—
46#387—24no—
47#382C2×D312yes—
48C43C2×D311yescharacteristic
49#327—64no—
50C22×C3×C4—22no—
51C22×C3×C4—24no—
52C22×C3×C4C2×D412yes—
53C22×C3×C4C2×D411yescharacteristic
54C2×#67—41no—
55#235—42no—
56#235—42no—
57C2×D12—21no—
58C2×D12—22no—
59C2×C4×D3—41no—
60C2×Dic6—21no—
61C3×C4⋊C4—48no—
62C3×C4⋊C4—42no—
63C3×C4⋊C4—42no—
64C3×C42—22no—
65C3×C42—22no—
66C3×C42—44no—
67C3×C42C2×Q811yescharacteristic
68C3×C42#4111yescharacteristic
69C3×C42#4112yes—
70(C2×D3)⋊C4—42no—
71C4⋊Dic3—22no—
72#208—42no—
73C2×C3⋊C8—42no—
74C2×C3⋊C8—41no—
75C2×#41—61no—
76C2×M16—61no—
77C4⋊Q8—31no—
78#127—31no—
79#124—61no—
80#121—62no—
81C4×D4—62no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—22no—
84C2×C42—22no—
85C2×C42#6712yes—
86C2×C42C22×D311yescharacteristic
87C4⋊C8—62no—
88C4⋊C8—62no—
89#104—124no—
90C22⋊C8—62no—
91C4×C8—62no—
92#95—24no—
93#95—22no—
94#95#6714yes—
95C23×C3#12011yescharacteristic
96C22×D3—41no—
97C2×C3×C4—24no—
98C2×C3×C4—42no—
99C2×C3×C4—42no—
100C2×C3×C4—42no—
101C2×C3×C4—48no—
102C2×C3×C4—44no—
103C2×C3×C4C22⋊Q812yes—
104C2×C3×C4C22⋊Q811yescharacteristic
105C2×C3×C4#12412yes—
106C2×C3×C4#12411yescharacteristic
107C2×C3×C4#13512yes—
108C2×C3×C4AGL(1,ℤ/8)12yes—
109#67—82no—
110C2×Dic3—41no—
111D12—42no—
112D12—82no—
113C4×D3—81no—
114C4×D3—81no—
115Dic6—42no—
116C3⋊C8—82no—
117C3⋊C8—81no—
118#41—122no—
119#41—122no—
120C2×Q8—61no—
121C2×D4—61no—
122C2×D4—62no—
123C2×D4—121no—
124C22×C4—22no—
125C22×C4—24no—
126C22×C4—121no—
127C22×C4C2×#6712yes—
128C22×C4C2×#6711yescharacteristic
129M16—122no—
130C2×C8—122no—
131C2×C8—121no—
132C4⋊C4—48no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—62no—
136C22⋊C4—122no—
137C42—22no—
138C42—22no—
139C42—44no—
140C42D3×Q811yescharacteristic
141C42#23911yescharacteristic
142C42#23912yes—
143C22×C3—22no—
144C22×C3#39311yescharacteristic
145C22×C3#44912yes—
146C2×D3—81no—
147C2×D3—82no—
148C3×C4—22no—
149C3×C4—22no—
150C3×C4—44no—
151C3×C4—84no—
152C3×C4#47812yes—
153C3×C4#48811yescharacteristic
154C3×C4#49011yescharacteristic
155Dic3—81no—
156C23—121no—
157C23#95311yescharacteristic
158Q8—122no—
159D4—122no—
160D4—242no—
161D4—242no—
162C2×C4—24no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—48no—
167C2×C4—44no—
168C2×C4—121no—
169C2×C4—241no—
170C2×C4—241no—
171C2×C4#94612yes—
172C2×C4#94611yescharacteristic
173C2×C4#94712yes—
174C2×C4#94711yescharacteristic
175C2×C4#94912yes—
176C2×C4#95012yes—
177C8—242no—
178C8—241no—
179C2×C3—22no—
180C2×C3#202511yescharacteristic
181C2×C3#203112yes—
182D3—161no—
183C22—22no—
184C22—241no—
185C22—242no—
186C22#523111yescharacteristic
187C22#523812yes—
188C4—22no—
189C4—22no—
190C4—44no—
191C4—84no—
192C4—241no—
193C4#506612yes—
194C4#506711yescharacteristic
195C4#506811yescharacteristic
196C3#1036811yescharacteristic, complemented, Sylow
197C2—22no—
198C2—481no—
199C2#7174311yescharacteristic
200C2#7184212yes—
subgroupG/H
centerC22#5231
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2864C2
socleC22×C3#393
prime radical#5477C22
hypercenter#2864D3
hypocenterC3#10368
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2864

Complements

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

NH ≅ G/Ntype
1. C3×#2864C2semidirect
4. #67631C2semidirect
5. #67611C2semidirect
16. #2864D3semidirect
196. C3#10368semidirect

Maximal subgroups

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

subgroupindex
C3×#28642
#695452
#693952
#676312
#676112
#103683