Finite Group Databasebeta

← #10695855

Subgroups — #10695855

Order 768.

878 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 878 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#5217C212yesmaximal, complemented
2#78641C211yesmaximal, characteristic, complemented
3C3×#1783C211yesmaximal, characteristic, complemented
4#69970C212yesmaximal, complemented
5#67688C211yesmaximal, characteristic, complemented
6#9986—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C22×#213—22no—
9C2×#930C2212yes—
10C2×C3×#100—22no—
11C3×#407C2211yescharacteristic
12C3×#407C2211yescharacteristic
13#5217—24no—
14#5217—24no—
15C2×#832—22no—
16C2×#830C2212yes—
17#5135—22no—
18#4735—22no—
19#4732—22no—
20#4468—24no—
21#3192—31no—
22C2×#447—32no—
23#1783D311yescharacteristic, complemented
24#1471—32no—
25#1298—31no—
26C25×C3—21no—
27C22×C3×D4—22no—
28C22×C3×D4—21no—
29C22×C3×D4—22no—
30C22×C3×D4—44no—
31C22×C3×D4D412yes—
32C23×C3×C4C2311yescharacteristic
33C22×#67—22no—
34C22×D12D412yes—
35C2×C3×M16—21no—
36C2×C3×M16D412yes—
37C2×#213—28no—
38C2×#213—48no—
39#930—22no—
40#930—22no—
41#930—22no—
42#930—22no—
43C2×(C2×D3)⋊C4—22no—
44C2×C4⋊Dic3D412yes—
45C22×C3⋊C8—22no—
46C2×D24—22no—
47C3×#100—44no—
48C3×C22⋊C8—22no—
49C3×C22⋊C8—22no—
50#832—28no—
51#830—22no—
52#830D414yes—
53#821—24no—
54#817—22no—
55C23×D4C2×D311yescharacteristic
56C22×D8—62no—
57C2×#121—32no—
58#481—62no—
59#469—62no—
60#447—62no—
61#447—64no—
62#447—64no—
63C2×#102—62no—
64C2×#100—22no—
65#407C2×D311yescharacteristic
66#407C2×D311yescharacteristic
67C2×C22⋊C8—32no—
68#323—64no—
69C24×C3—22no—
70C24×C3—44no—
71C24×C3—42no—
72C24×C3—84no—
73C24×C3#4111yescharacteristic
74C23×D3—22no—
75C2×C3×D4—28no—
76C2×C3×D4—24no—
77C2×C3×D4—24no—
78C2×C3×D4—44no—
79C2×C3×D4—42no—
80C2×C3×D4—44no—
81C2×C3×D4—48no—
82C2×C3×D4—41no—
83C2×C3×D4—41no—
84C22×C3×C4—42no—
85C22×C3×C4C2×D411yescharacteristic
86C22×C3×C4C2×D412yes—
87C22×C3×C4C2×D412yes—
88C22×C3×C4C2×D411yescharacteristic
89C2×#67—42no—
90C2×#67—42no—
91C2×#67—84no—
92C22×Dic3—22no—
93C2×D12—22no—
94C2×D12—22no—
95C2×D12—44no—
96C3×M16—42no—
97C3×M16—42no—
98C2×C3×C8—22no—
99C2×C3×C8—41no—
100#213—416no—
101(C2×D3)⋊C4—42no—
102(C2×D3)⋊C4—42no—
103C4⋊Dic3—22no—
104C4⋊Dic3—22no—
105C2×C3⋊C8—24no—
106C2×C3⋊C8—44no—
107D24—82no—
108C25—21no—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—21no—
112C22×D4—32no—
113C22×D4—44no—
114C22×D4—62no—
115C22×D4D1212yes—
116C23×C4C22×D311yescharacteristic
117C2×D8—62no—
118C2×D8—68no—
119C2×D8—128no—
120C2×M16—21no—
121C2×M16#6712yes—
122C22×C8—62no—
123#121—62no—
124#121—62no—
125#121—62no—
126#121—62no—
127C2×C4⋊C4—32no—
128C2×C22⋊C4—62no—
129#106—62no—
130#102—64no—
131#102—68no—
132#100—44no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—34no—
136C22⋊C8—62no—
137C23×C3—24no—
138C23×C3—24no—
139C23×C3—41no—
140C23×C3—48no—
141C23×C3—42no—
142C23×C3—42no—
143C23×C3—84no—
144C23×C3—84no—
145C23×C3—816no—
146C23×C3—84no—
147C23×C3#12112yes—
148C23×C3#12111yescharacteristic
149C22×D3—42no—
150C22×D3—44no—
151C22×D3—82no—
152C22×D3—82no—
153C3×D4—28no—
154C3×D4—44no—
155C3×D4—44no—
156C3×D4—44no—
157C2×C3×C4—24no—
158C2×C3×C4—22no—
159C2×C3×C4—41no—
160C2×C3×C4—42no—
161C2×C3×C4#12012yes—
162C2×C3×C4#12011yescharacteristic
163C2×C3×C4#12711yescharacteristic
164#67—84no—
165#67—84no—
166C2×Dic3—42no—
167C2×Dic3—82no—
168D12—44no—
169D12—44no—
170C3×C8—42no—
171C3×C8—81no—
172C3⋊C8—44no—
173C24—22no—
174C24—44no—
175C24—42no—
176C24—62no—
177C24—84no—
178C24#23511yescharacteristic
179C2×D4—28no—
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—44no—
183C2×D4—42no—
184C2×D4—44no—
185C2×D4—48no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—62no—
189C2×D4—62no—
190C2×D4—124no—
191C2×D4—122no—
192C2×D4—122no—
193C2×D4—244no—
194C22×C4—42no—
195C22×C4—62no—
196C22×C4C2×D1211yescharacteristic
197C22×C4D3×D412yes—
198C22×C4D3×D412yes—
199C22×C4C2×#6711yescharacteristic
200D8—1216no—
201D8—242no—
202M16—42no—
203M16—42no—
204C2×C8—22no—
205C2×C8—41no—
206C2×C8—64no—
207C2×C8—124no—
208C4⋊C4—62no—
209C4⋊C4—62no—
210C22⋊C4—122no—
211C22⋊C4—122no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—22no—
215C22×C3—21no—
216C22×C3—44no—
217C22×C3—44no—
218C22×C3—816no—
219C22×C3—84no—
220C22×C3—84no—
221C22×C3—84no—
222C22×C3#39211yescharacteristic
223C22×C3#47214yes—
224C2×D3—82no—
225C2×D3—82no—
226C2×D3—82no—
227C2×D3—164no—
228C3×C4—22no—
229C3×C4—21no—
230C3×C4B(4,𝔽2)12yes—
231Dic3—82no—
232C23—24no—
233C23—24no—
234C23—41no—
235C23—48no—
236C23—42no—
237C23—42no—
238C23—84no—
239C23—84no—
240C23—816no—
241C23—84no—
242C23—122no—
243C23—124no—
244C23—242no—
245C23—242no—
246C23#88412yes—
247C23#93011yescharacteristic
248D4—28no—
249D4—44no—
250D4—44no—
251D4—44no—
252D4—124no—
253D4—124no—
254D4—244no—
255D4—244no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—122no—
261C2×C4—242no—
262C2×C4#88212yes—
263C2×C4#93711yescharacteristic
264C2×C4#94011yescharacteristic
265C8—42no—
266C8—81no—
267C8—124no—
268C2×C3—24no—
269C2×C3—41no—
270C2×C3—84no—
271C2×C3#201812yes—
272C2×C3#202111yescharacteristic
273D3—162no—
274C22—22no—
275C22—22no—
276C22—21no—
277C22—22no—
278C22—44no—
279C22—44no—
280C22—816no—
281C22—84no—
282C22—84no—
283C22—84no—
284C22—242no—
285C22—242no—
286C22—242no—
287C22—484no—
288C22#474914yes—
289C22#496011yescharacteristic
290C4—21no—
291C4—22no—
292C4—242no—
293C4#503212yes—
294C3#998611yescharacteristic, complemented, Sylow
295C2—24no—
296C2—41no—
297C2—84no—
298C2—482no—
299C2#6950011yescharacteristic
300C2#7009012yes—
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1783C2
socleC22×C3#392
hypercenter#1783D3
hypocenterC3#9986
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1783

Complements

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

NH ≅ G/Ntype
1. C2×#5217C2semidirect
2. #78641C2semidirect
3. C3×#1783C2semidirect
4. #69970C2semidirect
5. #67688C2semidirect
23. #1783D3semidirect
294. C3#9986semidirect

Maximal subgroups

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

subgroupindex
C2×#52172
#786412
C3×#17832
#699702
#676882
#99863