Finite Group Databasebeta

← #10708952

Subgroups — #10708952

Order 768.

346 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 59 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#81774C211yesmaximal, characteristic
2#71256C211yesmaximal, characteristic, complemented
3#69307C212yesmaximal, complemented
4C3×#1359C211yesmaximal, characteristic, complemented
5#67568C212yesmaximal, complemented
6#12365—31nomaximal, Sylow
7#5762—22no—
8C3×#492C2211yescharacteristic
9#5177—22no—
10#5095—21no—
11#5093—21no—
12#5077—21no—
13#5070—21no—
14#4700C2212yes—
15#4680C2211yescharacteristic, complemented
16C3×#364C2212yescomplemented
17#4518—22no—
18#4513—22no—
19C8×C3⋊C8C2211yescharacteristic, complemented
20#3291—31no—
21#2230—32no—
22#1708—31no—
23#1359D311yescharacteristic, complemented
24#1335—32no—
25#1007—21no—
26#1004—21no—
27C2×C3×SD16—22no—
28C3×C4⋊Q8D411yescharacteristic
29C3×#127D411yescharacteristic
30#942—22no—
31#932—22no—
32#916—42no—
33#915—42no—
34#914—42no—
35D3×SD16—42no—
36#904D412yes—
37#901—22no—
38#869—21no—
39C3×M32—22no—
40C3×C4×C8C2311yescharacteristic
41#836—21no—
42#833—21no—
43#832—42no—
44#819D412yes—
45#817—22no—
46#810—42no—
47#805—22no—
48C4×C3⋊C8—21no—
49#577—62no—
50#498—31no—
51#492C2×D311yescharacteristic
52#487—61no—
53#486—61no—
54#471—62no—
55#456—61no—
56#453—61no—
57#443—32no—
58#365—62no—
59#364C2×D312yescomplemented
60#360—62no—
61C82—31no—
62C2×C3×Q8—21no—
63C2×C3×D4—21no—
64C2×C3×D4—41no—
65C2×#67—81no—
66#239—42no—
67D3×Q8—42no—
68#237—42no—
69D3×D4—42no—
70#235—22no—
71#235—81no—
72C2×Dic6—41no—
73C3×SD16—42no—
74C3×SD16—42no—
75C2×C3×C8C2×D412yes—
76C3×C4⋊C4—41no—
77C3×C42C2×D411yescharacteristic
78#216—42no—
79#215—42no—
80#214—42no—
81#213—42no—
82C4⋊Dic3—41no—
83C4⋊Dic3—41no—
84#208—22no—
85C2×C3⋊C8—22no—
86Dic12SD1612yes—
87D24SD1612yes—
88#204—22no—
89ΓL(1,𝔽25)—42no—
90D3×C8—42no—
91C3×C16—42no—
922−1+4—61no—
93H(5,𝔽2)—61no—
94#137—62no—
95#137—122no—
96AGL(1,ℤ/8)—62no—
97AGL(1,ℤ/8)—122no—
98#135—32no—
99#135—122no—
100C2×SD16—22no—
101C2×SD16—122no—
102#131—62no—
103C4⋊Q8—61no—
104C4⋊Q8D1211yescharacteristic
105#127D1211yescharacteristic
106M32—22no—
107#108—32no—
108#106—62no—
109#104—62no—
110#103—122no—
111#102—122no—
112#101—61no—
113#100—61no—
114C4×C8—61no—
115C4×C8C22×D311yescharacteristic
116C23×C3—41no—
117C22×D3—81no—
118C3×Q8—42no—
119C3×D4—42no—
120C3×D4—82no—
121C2×C3×C4—21no—
122C2×C3×C4—41no—
123C2×C3×C4#12011yescharacteristic
124#67—42no—
125#67—82no—
126C2×Dic3—81no—
127C2×Dic3—81no—
128D12—22no—
129D12—81no—
130C4×D3—42no—
131C4×D3—82no—
132Dic6—22no—
133Dic6—81no—
134Dic6—82no—
135C3×C8C2×SD1612yes—
136C3×C8AGL(1,ℤ/8)12yes—
137C3⋊C8—42no—
138C3⋊C8—42no—
139#41—62no—
140#41—122no—
141#41—122no—
142#41—241no—
143C2×Q8—21no—
144C2×Q8—121no—
145C2×Q8—122no—
146C2×D4—21no—
147C2×D4—41no—
148C2×D4—122no—
149C2×D4—241no—
150Q16—32no—
151Q16—122no—
152SD16—42no—
153SD16—42no—
154SD16—62no—
155SD16—122no—
156SD16—122no—
157D8—32no—
158D8—122no—
159M16—62no—
160M16—122no—
161C2×C8—62no—
162C2×C8—122no—
163C2×C8D3×D412yes—
164C4⋊C4—41no—
165C4⋊C4—121no—
166C4⋊C4—121no—
167C42C2×D1211yescharacteristic
168C16—42no—
169C22×C3—81no—
170C22×C3—161no—
171C22×C3#45311yescharacteristic
172C2×D3—42no—
173C2×D3—321no—
174C3×C4—22no—
175C3×C4—81no—
176C3×C4#45012yes—
177Dic3—42no—
178Dic3—161no—
179C23—41no—
180C23—241no—
181Q8—42no—
182Q8—62no—
183Q8—242no—
184Q8—241no—
185D4—42no—
186D4—62no—
187D4—82no—
188D4—122no—
189D4—241no—
190D4—242no—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—122no—
194C2×C4—242no—
195C2×C4—241no—
196C2×C4—241no—
197C2×C4#88211yescharacteristic
198C8—122no—
199C8—122no—
200C8#91112yes—
201C8D3×SD1612yes—
202C2×C3—21no—
203C2×C3—161no—
204C2×C3#162911yescharacteristic
205D3—82no—
206C22—81no—
207C22—122no—
208C22—161no—
209C22—961no—
210C22#474711yescharacteristic
211C4—22no—
212C4—81no—
213C4—122no—
214C4—481no—
215C4#477612yes—
216C3#1236511yescharacteristic, complemented, Sylow
217C2—21no—
218C2—161no—
219C2—242no—
220C2#6839111yescharacteristic
subgroupG/H
centerC2#68391
commutator, cosocleC3×C4×C8C23
FrattiniC4×C8C22×D3
FittingC3×#1359C2
socleC2×C3#1629
prime radical#81774C2
hypercenter#1359D3
hypocenterC3#12365
LCSG ▷ C3×C4×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1359

Complements

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

NH ≅ G/Ntype
2. #71256C2semidirect
3. #69307C2semidirect
4. C3×#1359C2semidirect
5. #67568C2semidirect
15. #4680C22semidirect
16. C3×#364C22semidirect
19. C8×C3⋊C8C22semidirect
23. #1359D3semidirect
59. #364C2×D3semidirect
216. C3#12365semidirect

Maximal subgroups

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

subgroupindex
#817742
#712562
#693072
C3×#13592
#675682
#123653