Finite Group Databasebeta

← #10714306

Subgroups — #10714306

Order 768.

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

normal subgroups only

This group has 364 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
1#78479C211yesmaximal, characteristic, complemented
2#71450C212yesmaximal
3#69845C211yesmaximal, characteristic
4C3×#1375C211yesmaximal, characteristic, complemented
5#67822C212yesmaximal, complemented
6#13373—31nomaximal, Sylow
7C2×#904C2211yescharacteristic
8#5470—21no—
9C2×C3×M32C2212yes—
10C2×C3×#97C2211yescharacteristic
11#5110—21no—
12C2×#819C2211yescharacteristic
13#5106—21no—
14#5106C2212yes—
15C2×#805—21no—
16#4702—24no—
17#4518—24no—
18#3162—31no—
19#2145—32no—
20#1975—31no—
21#1389—32no—
22#1375D311yescharacteristic, complemented
23C2×#235—21no—
24C22×C3×C8D411yescharacteristic
25C22×C3×C8C2311yescharacteristic
26C2×C3×C42D411yescharacteristic
27#926—22no—
28#924—42no—
29C2×#208—21no—
30C2×Dic12Q811yescharacteristic, complemented
31#904—22no—
32#904—22no—
33C2×D24Q811yescharacteristic, complemented
34C2×#204—21no—
35#870—22no—
36#869—22no—
37C4×Dic6—42no—
38C3×M32—22no—
39C3×M32—22no—
40C2×C3×C16—22no—
41C3×#97—22no—
42(C2×D3)⋊C8—42no—
43#819—22no—
44#818—22no—
45#818—22no—
46#817—22no—
47#817D414yes—
48#805—42no—
49C2×#135—31no—
50#533—61no—
51C2×M32C2×D312yes—
52#501—64no—
53C2×#108—31no—
54#427—32no—
55#427—61no—
56C2×#104—61no—
57#408—61no—
58C2×#97C2×D311yescharacteristic
59#360—64no—
60C22×C3×C4—21no—
61C22×C3×C4C2×D411yescharacteristic
62C2×#67—41no—
63#235—41no—
64#235—41no—
65#235—41no—
66#235—41no—
67C2×D12—21no—
68C2×C4×D3—41no—
69C2×Dic6—21no—
70C2×Dic6—42no—
71C2×C3×C8—21no—
72C2×C3×C8—21no—
73C2×C3×C8SD1612yes—
74C2×C3×C8C2×D412yes—
75C2×C3×C8C2×Q811yescharacteristic
76C2×C3×C8#4112yes—
77C2×C3×C8#4111yescharacteristic
78C3×C42—21no—
79C3×C42D812yes—
80#217—42no—
81C4⋊Dic3—24no—
82C4⋊Dic3—42no—
83Dic3⋊C4—44no—
84C4×Dic3—42no—
85#208—41no—
86#208—41no—
87C2×C3⋊C8—41no—
88Dic12—22no—
89D24—22no—
90#204—42no—
91C3×C16—42no—
92C2×#41—61no—
93#135—62no—
94#135—62no—
95C2×Q16—31no—
96C2×SD16—61no—
97C2×D8—31no—
98C2×M16—61no—
99C22×C8D1211yescharacteristic
100C22×C8C22×D311yescharacteristic
101C4⋊Q8—62no—
102#125—62no—
103C22⋊Q8—122no—
104C4×Q8—122no—
105#117—62no—
106C2×C42D1211yescharacteristic
107M32—22no—
108M32—22no—
109C2×C16—22no—
110#108—62no—
111#107—62no—
112#107—62no—
113#106—34no—
114#106—62no—
115#104—122no—
116C22⋊C8—122no—
117#97—22no—
118C23×C3AGL(1,ℤ/8)11yescharacteristic
119C22×D3—41no—
120C2×C3×C4—22no—
121C2×C3×C4—81no—
122C2×C3×C4#12011yescharacteristic
123C2×C3×C4C22⋊Q811yescharacteristic
124C2×C3×C4#12411yescharacteristic
125C2×C3×C4C2×D811yescharacteristic
126C2×C3×C4C2×SD1611yescharacteristic
127C2×C3×C4#13711yescharacteristic
128#67—81no—
129#67—81no—
130C2×Dic3—44no—
131C2×Dic3—41no—
132D12—41no—
133D12—41no—
134C4×D3—81no—
135C4×D3—81no—
136Dic6—41no—
137Dic6—41no—
138Dic6—84no—
139C3×C8—22no—
140C3×C8C22⋊Q812yes—
141C3×C8#12412yes—
142C3⋊C8—81no—
143#41—121no—
144#41—121no—
145#41—121no—
146#41—121no—
147C2×Q8—61no—
148C2×Q8—122no—
149C2×D4—61no—
150C2×D4—121no—
151C22×C4—21no—
152C22×C4—121no—
153C22×C4C2×D1211yescharacteristic
154Q16—62no—
155SD16—122no—
156D8—62no—
157M16—121no—
158M16—121no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—121no—
162C2×C8#20412yes—
163C2×C8#23512yes—
164C2×C8D3×D412yes—
165C2×C8#23711yescharacteristic
166C2×C8D3×Q811yescharacteristic
167C4⋊C4—64no—
168C4⋊C4—124no—
169C4⋊C4—122no—
170C22⋊C4—122no—
171C42—21no—
172C42—122no—
173C42D2412yes—
174C16—42no—
175C22×C3—41no—
176C22×C3#44711yescharacteristic
177C22×C3#47611yescharacteristic
178C22×C3#48911yescharacteristic
179C2×D3—82no—
180C2×D3—81no—
181C3×C4—42no—
182C3×C4#39311yescharacteristic
183C3×C4#44811yescharacteristic
184C3×C4#47411yescharacteristic
185C3×C4#48611yescharacteristic
186Dic3—84no—
187Dic3—81no—
188C23—121no—
189C23#90811yescharacteristic
190Q8—121no—
191Q8—121no—
192Q8—244no—
193D4—121no—
194D4—121no—
195D4—241no—
196D4—241no—
197C2×C4—22no—
198C2×C4—81no—
199C2×C4—124no—
200C2×C4—121no—
201C2×C4—241no—
202C2×C4—241no—
203C2×C4#87511yescharacteristic
204C2×C4#88211yescharacteristic
205C2×C4#89711yescharacteristic
206C2×C4C2×#20411yescharacteristic
207C2×C4C2×D2411yescharacteristic
208C2×C4#90911yescharacteristic
209C8—22no—
210C8—241no—
211C8#88512yes—
212C8#89612yes—
213C2×C3—21no—
214C2×C3—21no—
215C2×C3—21no—
216C2×C3#202311yescharacteristic
217D3—161no—
218C22—41no—
219C22—242no—
220C22—241no—
221C22#471411yescharacteristic
222C22#473211yescharacteristic
223C22#484211yescharacteristic
224C4—42no—
225C4—244no—
226C4—241no—
227C4#467311yescharacteristic
228C4#469411yescharacteristic
229C4#473811yescharacteristic
230C4#484611yescharacteristic
231C3#1337311yescharacteristic, complemented, Sylow
232C2—21no—
233C2—21no—
234C2—21no—
235C2—481no—
236C2#6867811yescharacteristic
subgroupG/H
centerC4#4694
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC3×#1375C2
socleC2×C3#2023
prime radicalC2×#904C22
hypercenter#1375D3
hypocenterC3#13373
LCSG ▷ C22×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ #1375

Complements

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

NH ≅ G/Ntype
1. #78479C2semidirect
4. C3×#1375C2semidirect
5. #67822C2semidirect
22. #1375D3semidirect
30. C2×Dic12Q8semidirect
33. C2×D24Q8semidirect
231. C3#13373semidirect

Maximal subgroups

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

subgroupindex
#784792
#714502
#698452
C3×#13752
#678222
#133733