Finite Group Databasebeta

← #10698306

Subgroups — #10698306

Order 768.

402 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 89 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#73310C211yesmaximal, characteristic, complemented
2#70790C212yesmaximal
3#69395C211yesmaximal, characteristic
4C3×#1276C211yesmaximal, characteristic, complemented
5#67738C212yesmaximal, complemented
6#10368—31nomaximal, Sylow
7#5477C2211yescharacteristic
8C3×#431C2212yes—
9C3×C43C2211yescharacteristic
10#4936C2212yes—
11#4933—24no—
12#4928—22no—
13C2×#805—22no—
14#4924C2211yescharacteristic
15#4470—24no—
16#2864—31no—
17#1962—31no—
18#1517—32no—
19#1286—32no—
20#1276D311yescharacteristic, complemented
21C2×#235—21no—
22C2×C3×C42—22no—
23C2×C3×C42D412yes—
24C2×C3×C42C2311yescharacteristic
25#930—22no—
26C2×C4⋊Dic3—22no—
27C2×Dic3⋊C4—24no—
28C2×#208—21no—
29#875—21no—
30#874Q811yescharacteristic
31C4×D12—22no—
32#869Q811yescharacteristic
33C3×C4⋊C8—22no—
34C3×C22⋊C8—22no—
35C3×C4×C8—22no—
36C3⋊#95—24no—
37C3⋊#95—22no—
38C3⋊#95D414yes—
39#805—44no—
40C4⋊(C3⋊C8)—22no—
41#532—31no—
42#431C2×D312yes—
43#423—31no—
44C2×#104—62no—
45#389—62no—
46#387—64no—
47#382—32no—
48C43C2×D311yescharacteristic
49#327—64no—
50C22×C3×C4—22no—
51C22×C3×C4C2×D411yescharacteristic
52C22×C3×C4C2×D412yes—
53C2×#67—41no—
54C22×Dic3—24no—
55#235—42no—
56#235—42no—
57C2×D12—21no—
58C2×D12—22no—
59C2×C4×D3—41no—
60C2×Dic6—21no—
61C2×C3×C8—42no—
62C3×C42—22no—
63C3×C42—22no—
64C3×C42—44no—
65C3×C42C2×Q811yescharacteristic
66C3×C42#4112yes—
67C3×C42#4111yescharacteristic
68(C2×D3)⋊C4—42no—
69C4⋊Dic3—22no—
70C4⋊Dic3—42no—
71C4⋊Dic3—42no—
72Dic3⋊C4—48no—
73#208—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—64no—
83C2×C4⋊C4—62no—
84C2×C42—22no—
85C2×C42D1212yes—
86C2×C42C22×D311yescharacteristic
87C4⋊C8—22no—
88C4⋊C8—62no—
89#104—124no—
90C22⋊C8—22no—
91C4×C8—22no—
92#95—34no—
93#95—64no—
94#95—62no—
95C23×C3#12011yescharacteristic
96C22×D3—41no—
97C2×C3×C4—24no—
98C2×C3×C4—42no—
99C2×C3×C4—42no—
100C2×C3×C4—42no—
101C2×C3×C4C22⋊Q812yes—
102C2×C3×C4C22⋊Q811yescharacteristic
103C2×C3×C4#12411yescharacteristic
104C2×C3×C4#12412yes—
105C2×C3×C4#13512yes—
106C2×C3×C4AGL(1,ℤ/8)12yes—
107#67—82no—
108C2×Dic3—48no—
109C2×Dic3—41no—
110C2×Dic3—44no—
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—62no—
122C2×D4—61no—
123C2×D4—121no—
124C22×C4—22no—
125C22×C4—64no—
126C22×C4—121no—
127C22×C4C2×D1211yescharacteristic
128C22×C4D3×D412yes—
129M16—122no—
130C2×C8—42no—
131C2×C8—121no—
132C4⋊C4—62no—
133C4⋊C4—128no—
134C4⋊C4—122no—
135C4⋊C4—122no—
136C22⋊C4—122no—
137C42—22no—
138C42—22no—
139C42—44no—
140C42#23512yes—
141C42#23711yescharacteristic
142C42D3×Q811yescharacteristic
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#47812yes—
152C3×C4#48811yescharacteristic
153C3×C4#49011yescharacteristic
154Dic3—84no—
155Dic3—81no—
156C23—121no—
157C23#88211yescharacteristic
158Q8—122no—
159D4—122no—
160D4—242no—
161D4—242no—
162C2×C4—24no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—128no—
167C2×C4—121no—
168C2×C4—124no—
169C2×C4—241no—
170C2×C4—241no—
171C2×C4#87511yescharacteristic
172C2×C4#88512yes—
173C2×C4#89612yes—
174C2×C4#89711yescharacteristic
175C2×C4#90412yes—
176C2×C4#90812yes—
177C8—82no—
178C8—241no—
179C2×C3—22no—
180C2×C3#202511yescharacteristic
181C2×C3#203112yes—
182D3—161no—
183C22—22no—
184C22—241no—
185C22—242no—
186C22#467311yescharacteristic
187C22#473412yes—
188C4—22no—
189C4—22no—
190C4—44no—
191C4—244no—
192C4—241no—
193C4#469711yescharacteristic
194C4#471311yescharacteristic
195C4#484712yes—
196C3#1036811yescharacteristic, complemented, Sylow
197C2—22no—
198C2—481no—
199C2#6867911yescharacteristic
200C2#6877912yes—
subgroupG/H
centerC22#4673
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1276C2
socleC22×C3#393
prime radical#5477C22
hypercenter#1276D3
hypocenterC3#10368
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
1. #73310C2semidirect
4. C3×#1276C2semidirect
5. #67738C2semidirect
20. #1276D3semidirect
196. C3#10368semidirect

Maximal subgroups

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

subgroupindex
#733102
#707902
#693952
C3×#12762
#677382
#103683