Finite Group Databasebeta

← #10695853

Subgroups — #10695853

Order 768.

878 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 878 conjugacy classes of subgroups — too many to draw. Only its 89 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×#3192C211yesmaximal, characteristic, complemented
2C2×#5036C212yesmaximal, complemented
3#71676C212yesmaximal, complemented
4#69988C211yesmaximal, characteristic, complemented
5#67688C211yesmaximal, characteristic, complemented
6#9986—31nomaximal, Sylow
7C23×D12C2211yescharacteristic
8C2×C3×#121C2212yes—
9C22×#213—22no—
10C3×#481—22no—
11C3×#469—22no—
12C3×#447—22no—
13C3×#407C2211yescharacteristic
14#5207C2211yescharacteristic
15C2×#830C2212yes—
16C2×#823—22no—
17#5036—24no—
18#5036—24no—
19C2×#809—22no—
20#4525—24no—
21#3192D311yescharacteristic, complemented
22C2×#447—32no—
23#1783—31no—
24#1471—32no—
25#1298—31no—
26C24×D3—21no—
27C22×C3×D4—22no—
28C22×C3×D4D412yes—
29C23×C3×C4C2311yescharacteristic
30C22×D12—22no—
31C22×D12—21no—
32C22×D12—22no—
33C22×D12—44no—
34C22×D12D412yes—
35C2×C3×D8—22no—
36C2×C3×M16D412yes—
37C3×#121—22no—
38C3×#121—22no—
39C3×#121—22no—
40C3×#121—22no—
41C2×C3×C4⋊C4D412yes—
42C2×C3×C22⋊C4—22no—
43C2×#213—28no—
44C2×#213—48no—
45C2×#208—21no—
46C22×C3⋊C8—22no—
47C3×#106—22no—
48C3×#102—24no—
49C3×C22⋊C8—22no—
50#830—22no—
51#830—22no—
52#830D414yes—
53#823—44no—
54#809—28no—
55C23×D4—31no—
56C22×D8—62no—
57C2×#121C2×D312yes—
58#481—22no—
59#469—22no—
60#447—22no—
61#447—64no—
62#447—64no—
63C2×#102—62no—
64C2×#100—62no—
65#407—31no—
66#407C2×D311yescharacteristic
67C2×C22⋊C8—32no—
68#323—64no—
69C24×C3—22no—
70C24×C3#4111yescharacteristic
71C23×D3—22no—
72C23×D3—44no—
73C23×D3—42no—
74C23×D3—84no—
75C2×C3×D4—22no—
76C2×C3×D4—22no—
77C2×C3×D4—44no—
78C2×C3×D4—42no—
79C2×C3×D4—42no—
80C2×C3×D4—84no—
81C22×C3×C4—22no—
82C22×C3×C4—42no—
83C22×C3×C4C2×D411yescharacteristic
84C22×C3×C4C2×D412yes—
85C22×C3×C4C2×D412yes—
86C22×C3×C4C2×D411yescharacteristic
87C2×D12—28no—
88C2×D12—24no—
89C2×D12—24no—
90C2×D12—44no—
91C2×D12—48no—
92C2×D12—44no—
93C2×D12—42no—
94C2×D12—41no—
95C2×D12—41no—
96C3×D8—82no—
97C3×M16—42no—
98C2×C3×C8—22no—
99C3×C4⋊C4—22no—
100C3×C4⋊C4—22no—
101C3×C22⋊C4—42no—
102C3×C22⋊C4—42no—
103#213—416no—
104#208—42no—
105C2×C3⋊C8—24no—
106C2×C3⋊C8—44no—
107C2×C3⋊C8—41no—
108C25—61no—
109C22×D4—22no—
110C22×D4—32no—
111C22×D4—62no—
112C22×D4—62no—
113C22×D4—61no—
114C22×D4—124no—
115C22×D4#6712yes—
116C23×C4C22×D311yescharacteristic
117C2×D8—22no—
118C2×D8—68no—
119C2×D8—128no—
120C2×M16—61no—
121C2×M16#6712yes—
122C22×C8—62no—
123#121—22no—
124#121—22no—
125#121—22no—
126#121—22no—
127C2×C4⋊C4#6712yes—
128C2×C22⋊C4—22no—
129#106—22no—
130#102—24no—
131#102—68no—
132#100—124no—
133C22⋊C8—22no—
134C22⋊C8—34no—
135C22⋊C8—62no—
136C22⋊C8—62no—
137C23×C3—24no—
138C23×C3—41no—
139C23×C3—42no—
140C23×C3—44no—
141C23×C3—82no—
142C23×C3—82no—
143C23×C3#12111yescharacteristic
144C23×C3#12112yes—
145C22×D3—24no—
146C22×D3—48no—
147C22×D3—42no—
148C22×D3—42no—
149C22×D3—84no—
150C22×D3—816no—
151C22×D3—84no—
152C22×D3—84no—
153C3×D4—44no—
154C3×D4—44no—
155C3×D4—84no—
156C3×D4—84no—
157C2×C3×C4—24no—
158C2×C3×C4—22no—
159C2×C3×C4—41no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4—82no—
163C2×C3×C4#12012yes—
164C2×C3×C4#12011yescharacteristic
165C2×C3×C4#12711yescharacteristic
166D12—28no—
167D12—44no—
168D12—44no—
169D12—44no—
170C3×C8—42no—
171C3⋊C8—44no—
172C3⋊C8—81no—
173C24—22no—
174C24—62no—
175C24—124no—
176C24—122no—
177C24—244no—
178C24#23711yescharacteristic
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—44no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—68no—
185C2×D4—64no—
186C2×D4—64no—
187C2×D4—84no—
188C2×D4—124no—
189C2×D4—122no—
190C2×D4—124no—
191C2×D4—128no—
192C2×D4—121no—
193C2×D4—121no—
194C22×C4—22no—
195C22×C4—42no—
196C22×C4D3×D411yescharacteristic
197C22×C4D3×D412yes—
198C22×C4C2×#6712yes—
199C22×C4C2×#6711yescharacteristic
200D8—82no—
201D8—1216no—
202M16—42no—
203M16—122no—
204C2×C8—22no—
205C2×C8—64no—
206C2×C8—124no—
207C2×C8—121no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—22no—
215C22×C3—21no—
216C22×C3—44no—
217C22×C3—82no—
218C22×C3—82no—
219C22×C3—82no—
220C22×C3—164no—
221C22×C3#39211yescharacteristic
222C22×C3#47214yes—
223C2×D3—44no—
224C2×D3—816no—
225C2×D3—84no—
226C2×D3—84no—
227C2×D3—84no—
228C3×C4—22no—
229C3×C4—21no—
230C3×C4—82no—
231C3×C4B(4,𝔽2)12yes—
232C23—24no—
233C23—41no—
234C23—42no—
235C23—44no—
236C23—64no—
237C23—82no—
238C23—82no—
239C23—128no—
240C23—122no—
241C23—122no—
242C23—244no—
243C23—244no—
244C23—2416no—
245C23—244no—
246C23#93811yescharacteristic
247C23#93912yes—
248D4—44no—
249D4—44no—
250D4—68no—
251D4—84no—
252D4—84no—
253D4—124no—
254D4—124no—
255D4—124no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—82no—
262C2×C4#93712yes—
263C2×C4#94011yescharacteristic
264C2×C4#95311yescharacteristic
265C8—42no—
266C8—124no—
267C8—241no—
268C2×C3—24no—
269C2×C3—41no—
270C2×C3—162no—
271C2×C3#201812yes—
272C2×C3#202111yescharacteristic
273D3—84no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22—21no—
278C22—44no—
279C22—82no—
280C22—82no—
281C22—82no—
282C22—124no—
283C22—164no—
284C22—2416no—
285C22—244no—
286C22—244no—
287C22—244no—
288C22#519914yes—
289C22#522311yescharacteristic
290C4—22no—
291C4—21no—
292C4—82no—
293C4#524412yes—
294C3#998611yescharacteristic, complemented, Sylow
295C2—24no—
296C2—41no—
297C2—162no—
298C2—244no—
299C2#7185111yescharacteristic
300C2#7209512yes—
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3192C2
socleC22×C3#392
hypercenter#3192D3
hypocenterC3#9986
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3192

Complements

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

NH ≅ G/Ntype
1. C3×#3192C2semidirect
2. C2×#5036C2semidirect
3. #71676C2semidirect
4. #69988C2semidirect
5. #67688C2semidirect
21. #3192D3semidirect
294. C3#9986semidirect

Maximal subgroups

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

subgroupindex
C3×#31922
C2×#50362
#716762
#699882
#676882
#99863