Finite Group Databasebeta

← #10698322

Subgroups — #10698322

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×#2402C211yesmaximal, characteristic, complemented
2#69469C212yesmaximal
3#69440C211yesmaximal, characteristic
4#67620C212yesmaximal, complemented
5#67614C211yesmaximal, characteristic, complemented
6#10372—31nomaximal, Sylow
7#5477C2211yescharacteristic
8C3×#400—22no—
9C3×#398—22no—
10C3×C4×C4⋊C4—22no—
11C3×C4×C4⋊C4C2212yes—
12C3×#376C2211yescharacteristic
13C2×#805—22no—
14#4925C2212yes—
15#4924C2211yescharacteristic
16#4451—22no—
17#4451—22no—
18#2402D311yescharacteristic, complemented
19#1963—31no—
20#1502—32no—
21#1303—32no—
22#1274—31no—
23C2×#235—21no—
24C2×C3×C4⋊C4—22no—
25C2×C3×C4⋊C4—22no—
26C2×C3×C4⋊C4—22no—
27C2×C3×C42—22no—
28C2×C3×C42D412yes—
29C2×C3×C42C2311yescharacteristic
30#930—22no—
31C2×#208—21no—
32#875—21no—
33#874Q811yescharacteristic
34C4×D12—22no—
35#869Q811yescharacteristic
36C3×#95—22no—
37C3×#95—22no—
38C3×#95—22no—
39C3×#95D412yes—
40C3×#95D412yes—
41#830—22no—
42#805—44no—
43C4⋊(C3⋊C8)—22no—
44C4⋊(C3⋊C8)—22no—
45C4×C3⋊C8—22no—
46#532—31no—
47#431—32no—
48#423—31no—
49C2×#104—62no—
50#400—22no—
51#398—22no—
52C4×C4⋊C4—22no—
53C4×C4⋊C4C2×D312yes—
54#376C2×D311yescharacteristic
55#327—62no—
56#327—62no—
57C22×C3×C4—22no—
58C22×C3×C4—22no—
59C22×C3×C4—22no—
60C22×C3×C4C2×D412yes—
61C22×C3×C4C2×D411yescharacteristic
62C2×#67—41no—
63#235—42no—
64#235—42no—
65C2×D12—21no—
66C2×D12—22no—
67C2×C4×D3—41no—
68C2×Dic6—21no—
69C3×C4⋊C4—44no—
70C3×C4⋊C4—44no—
71C3×C4⋊C4—44no—
72C3×C42—22no—
73C3×C42—22no—
74C3×C42—42no—
75C3×C42—42no—
76C3×C42C2×Q811yescharacteristic
77C3×C42#4112yes—
78C3×C42#4111yescharacteristic
79(C2×D3)⋊C4—42no—
80C4⋊Dic3—22no—
81#208—42no—
82C2×C3⋊C8—42no—
83C2×C3⋊C8—41no—
84C2×#41—61no—
85C2×M16—61no—
86C4⋊Q8—31no—
87#127—31no—
88#124—61no—
89#121—62no—
90C4×D4—62no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C42—22no—
95C2×C42#6712yes—
96C2×C42C22×D311yescharacteristic
97C4⋊C8—62no—
98C4⋊C8—62no—
99#104—124no—
100C22⋊C8—62no—
101C4×C8—62no—
102#95—22no—
103#95—22no—
104#95—22no—
105#95#6712yes—
106#95#6712yes—
107C23×C3#12011yescharacteristic
108C22×D3—41no—
109C2×C3×C4—24no—
110C2×C3×C4—42no—
111C2×C3×C4—42no—
112C2×C3×C4—42no—
113C2×C3×C4—42no—
114C2×C3×C4—42no—
115C2×C3×C4—42no—
116C2×C3×C4—42no—
117C2×C3×C4—42no—
118C2×C3×C4—42no—
119C2×C3×C4C22⋊Q812yes—
120C2×C3×C4C22⋊Q811yescharacteristic
121C2×C3×C4#12412yes—
122C2×C3×C4#12411yescharacteristic
123C2×C3×C4#13512yes—
124C2×C3×C4AGL(1,ℤ/8)12yes—
125#67—82no—
126C2×Dic3—41no—
127D12—42no—
128D12—82no—
129C4×D3—81no—
130C4×D3—81no—
131Dic6—42no—
132C3⋊C8—82no—
133C3⋊C8—81no—
134#41—122no—
135#41—122no—
136C2×Q8—61no—
137C2×D4—61no—
138C2×D4—62no—
139C2×D4—121no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—121no—
144C22×C4C2×#6712yes—
145C22×C4C2×#6711yescharacteristic
146M16—122no—
147C2×C8—122no—
148C2×C8—121no—
149C4⋊C4—44no—
150C4⋊C4—44no—
151C4⋊C4—44no—
152C4⋊C4—62no—
153C22⋊C4—122no—
154C42—22no—
155C42—22no—
156C42—42no—
157C42—42no—
158C42D3×Q811yescharacteristic
159C42#23911yescharacteristic
160C42#23912yes—
161C22×C3—22no—
162C22×C3#39311yescharacteristic
163C22×C3#44912yes—
164C2×D3—81no—
165C2×D3—82no—
166C3×C4—22no—
167C3×C4—22no—
168C3×C4—44no—
169C3×C4—82no—
170C3×C4—82no—
171C3×C4#47812yes—
172C3×C4#48811yescharacteristic
173C3×C4#49011yescharacteristic
174Dic3—81no—
175C23—121no—
176C23#95311yescharacteristic
177Q8—122no—
178D4—122no—
179D4—242no—
180D4—242no—
181C2×C4—24no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—121no—
192C2×C4—241no—
193C2×C4—241no—
194C2×C4#94612yes—
195C2×C4#94611yescharacteristic
196C2×C4#94712yes—
197C2×C4#94711yescharacteristic
198C2×C4#94912yes—
199C2×C4#95012yes—
200C8—242no—
201C8—241no—
202C2×C3—22no—
203C2×C3#202511yescharacteristic
204C2×C3#203112yes—
205D3—161no—
206C22—22no—
207C22—241no—
208C22—242no—
209C22#523111yescharacteristic
210C22#523812yes—
211C4—22no—
212C4—22no—
213C4—44no—
214C4—82no—
215C4—82no—
216C4—241no—
217C4#506612yes—
218C4#506711yescharacteristic
219C4#506811yescharacteristic
220C3#1037211yescharacteristic, complemented, Sylow
221C2—22no—
222C2—481no—
223C2#7174311yescharacteristic
224C2#7184212yes—
subgroupG/H
centerC2×C4#947
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2402C2
socleC22×C3#393
prime radical#5477C22
hypercenter#2402D3
hypocenterC3#10372
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2402

Complements

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

NH ≅ G/Ntype
1. C3×#2402C2semidirect
4. #67620C2semidirect
5. #67614C2semidirect
18. #2402D3semidirect
220. C3#10372semidirect

Maximal subgroups

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

subgroupindex
C3×#24022
#694692
#694402
#676202
#676142
#103723