Finite Group Databasebeta

← #10709960

Subgroups — #10709960

Order 768.

318 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 79 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×#2248C211yesmaximal, characteristic, complemented
2#71313C211yesmaximal, characteristic, complemented
3#71296C211yesmaximal, characteristic, complemented
4#71288C211yesmaximal, characteristic, complemented
5#71276C211yesmaximal, characteristic
6#69058C211yesmaximal, characteristic, complemented
7#67499C211yesmaximal, characteristic, complemented
8#12529—31nomaximal, Sylow
9C2×C3×SD32—22no—
10C3×#494C2211yescharacteristic
11C3×#493C2211yescharacteristic
12C2×#828—22no—
13#5093—21no—
14#5080—21no—
15#5073—21no—
16#5073—21no—
17#4695C2211yescharacteristic
18#4682C2211yescharacteristic
19C3×C4×C16C2211yescharacteristic
20#4562—22no—
21#4509—22no—
22C4×C3⋊C16C2211yescharacteristic
23C8×C3⋊C8C2211yescharacteristic
24#2248—31no—
25#2248D311yescharacteristic, complemented
26#2240—31no—
27#2240—31no—
28#1712—31no—
29#1707—31no—
30#1312—31no—
31C2×C3×Q16—21no—
32C2×C3×Q16—21no—
33C2×C3×D8—21no—
34C2×C3×D8—21no—
35C3×C4⋊Q8—21no—
36C3×#127—21no—
37C2×#213—42no—
38C2×#213—42no—
39C2×D24—21no—
40C2×D24—21no—
41#874—21no—
42#869—21no—
43C3×SD32—44no—
44C2×C3×C16D412yes—
45C3×C4×C8C2311yescharacteristic
46#828—44no—
47#818—21no—
48#818—21no—
49C2×C3⋊C16D412yes—
50#810—42no—
51#808—42no—
52C4×C3⋊C8D411yescharacteristic
53C4×C3⋊C8D411yescharacteristic
54C2×SD32—22no—
55C2×SD32—62no—
56#500—31no—
57#498—61no—
58#494C2×D311yescharacteristic
59#493—31no—
60#493—61no—
61#493—61no—
62#493C2×D311yescharacteristic
63#487—61no—
64#357—62no—
65#357—62no—
66C4×C16—31no—
67C4×C16C2×D311yescharacteristic
68C82—31no—
69C2×C3×Q8—41no—
70C2×C3×D4—41no—
71C2×C3×D4—41no—
72C2×C3×D4—41no—
73C2×D12—41no—
74C2×D12—41no—
75C2×D12—41no—
76C2×Dic6—41no—
77C3×Q16—42no—
78C3×Q16—42no—
79C3×D8—42no—
80C3×D8—42no—
81C2×C3×C8C2×D411yescharacteristic
82C2×C3×C8C2×D411yescharacteristic
83C3×C4⋊C4—41no—
84C3×C4⋊C4—41no—
85C3×C42C2×D411yescharacteristic
86#213—84no—
87#213—84no—
88C4⋊Dic3—41no—
89C4⋊Dic3—41no—
90C2×C3⋊C8D812yes—
91C2×C3⋊C8D812yes—
92D24—42no—
93D24—42no—
94C3×C16—22no—
95C3⋊C16—22no—
96C2×Q16—21no—
97C2×Q16—21no—
98C2×D8—21no—
99C2×D8—21no—
100C2×D8—61no—
101C2×D8—61no—
102C2×D8—122no—
103C2×D8—122no—
104C4⋊Q8—21no—
105C4⋊Q8—61no—
106#127—21no—
107#127—61no—
108SD32—44no—
109SD32—124no—
110C2×C16—32no—
111C2×C16#6712yes—
112#107—61no—
113#107—61no—
114#106—122no—
115#103—122no—
116C4×C8—31no—
117C4×C8—31no—
118C4×C8C22×D311yescharacteristic
119C23×C3—81no—
120C22×D3—81no—
121C3×Q8—81no—
122C3×Q8—81no—
123C3×D4—81no—
124C3×D4—81no—
125C3×D4—82no—
126C3×D4—82no—
127C2×C3×C4—81no—
128C2×C3×C4#12711yescharacteristic
129C2×C3×C4C2×D811yescharacteristic
130C2×C3×C4C2×D811yescharacteristic
131C2×Dic3—81no—
132D12—81no—
133D12—81no—
134D12—82no—
135D12—82no—
136Dic6—82no—
137C3×C8AGL(1,ℤ/8)12yes—
138C3×C8AGL(1,ℤ/8)12yes—
139C3⋊C8—22no—
140C3⋊C8SD3214yescomplemented
141C2×Q8—41no—
142C2×Q8—121no—
143C2×D4—41no—
144C2×D4—41no—
145C2×D4—41no—
146C2×D4—121no—
147C2×D4—121no—
148C2×D4—121no—
149Q16—42no—
150Q16—42no—
151D8—42no—
152D8—42no—
153D8—122no—
154D8—122no—
155D8—244no—
156D8—244no—
157C2×C8—32no—
158C2×C8—32no—
159C2×C8D3×D411yescharacteristic
160C2×C8C2×#6711yescharacteristic
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—121no—
164C4⋊C4—121no—
165C42D3×D411yescharacteristic
166C16—22no—
167C16—62no—
168C22×C3—161no—
169C22×C3—161no—
170C22×C3—161no—
171C22×C3#49311yescharacteristic
172C2×D3—161no—
173C2×D3—161no—
174C2×D3—161no—
175C3×C4—161no—
176C3×C4#49611yescharacteristic
177C3×C4#49611yescharacteristic
178C3×C4C2×SD3212yes—
179C3×C4#50912yes—
180Dic3—161no—
181C23—81no—
182C23—241no—
183Q8—81no—
184Q8—81no—
185Q8—242no—
186D4—81no—
187D4—82no—
188D4—82no—
189D4—81no—
190D4—241no—
191D4—242no—
192D4—242no—
193D4—241no—
194C2×C4—81no—
195C2×C4—241no—
196C2×C4D3×D811yescharacteristic
197C2×C4D3×D811yescharacteristic
198C2×C4#94011yescharacteristic
199C8—34no—
200C8—62no—
201C8#91112yes—
202C8#93212yes—
203C2×C3—321no—
204C2×C3#172111yescharacteristic
205C2×C3#224811yescharacteristic
206C2×C3#225011yescharacteristic
207D3—321no—
208C22—161no—
209C22—161no—
210C22—161no—
211C22—481no—
212C22—481no—
213C22—481no—
214C22#515111yescharacteristic
215C4—161no—
216C4—481no—
217C4D3×SD3212yes—
218C4#491412yes—
219C4#507411yescharacteristic
220C4#515411yescharacteristic
221C3#1252911yescharacteristic, complemented, Sylow
222C2—321no—
223C2—961no—
224C2#7127911yescharacteristic
225C2#7147911yescharacteristic
226C2#7148111yescharacteristic
subgroupG/H
centerC22#5151
commutator, cosocleC3×C4×C8C23
FrattiniC4×C8C22×D3
FittingC3×#2248C2
socleC22×C3#493
prime radical#71276C2
hypercenter#2248D3
hypocenterC3#12529
LCSG ▷ C3×C4×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2248

Complements

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

NH ≅ G/Ntype
1. C3×#2248C2semidirect
2. #71313C2semidirect
3. #71296C2semidirect
4. #71288C2semidirect
6. #69058C2semidirect
7. #67499C2semidirect
25. #2248D3semidirect
140. C3⋊C8SD32semidirect
221. C3#12529semidirect

Maximal subgroups

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

subgroupindex
C3×#22482
#713132
#712962
#712882
#712762
#690582
#674992
#125293