Finite Group Databasebeta

← #10695865

Subgroups — #10695865

Order 768.

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

normal subgroups only

This group has 750 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
2C2×#5040C212yesmaximal, complemented
3#71676C212yesmaximal, complemented
4#69990C211yesmaximal, characteristic, complemented
5#67688C211yesmaximal, characteristic, complemented
6#9988—31nomaximal, Sylow
7C23×Dic6C2211yescharacteristic
8C2×C3×#121C2212yes—
9C22×#214—22no—
10C3×#481—22no—
11C3×#469—22no—
12C3×#447—22no—
13C3×#407C2211yescharacteristic
14#5207C2211yescharacteristic
15C2×#830C2212yes—
16C2×#824—22no—
17#5040—24no—
18#5040—24no—
19C2×#810—22no—
20#4525—24no—
21#3192D311yescharacteristic
22C2×#448—32no—
23#1784—31no—
24#1471—32no—
25#1298—31no—
26C22×C3×D4—22no—
27C22×C3×D4D412yes—
28C23×C3×C4C2311yescharacteristic
29C23×Dic3—21no—
30C22×Dic6—22no—
31C22×Dic6—21no—
32C22×Dic6—22no—
33C22×Dic6—44no—
34C22×Dic6D412yes—
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×#214—28no—
44C2×#214—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#824—44no—
54#810—28no—
55C23×Q8—31no—
56C22×SD16—62no—
57C2×#121C2×D312yes—
58#481—22no—
59#469—22no—
60#448—64no—
61#448—64no—
62#447—22no—
63C2×#103—62no—
64C2×#101—62no—
65#407—31no—
66#407C2×D311yescharacteristic
67C2×C22⋊C8—32no—
68#323—64no—
69C24×C3—22no—
70C24×C3#4111yescharacteristic
71C2×C3×D4—22no—
72C2×C3×D4—22no—
73C2×C3×D4—44no—
74C2×C3×D4—42no—
75C2×C3×D4—42no—
76C2×C3×D4—84no—
77C22×C3×C4—22no—
78C22×C3×C4—42no—
79C22×C3×C4C2×D411yescharacteristic
80C22×C3×C4C2×D412yes—
81C22×C3×C4C2×D412yes—
82C22×C3×C4C2×D411yescharacteristic
83C22×Dic3—22no—
84C22×Dic3—44no—
85C22×Dic3—42no—
86C2×Dic6—28no—
87C2×Dic6—24no—
88C2×Dic6—24no—
89C2×Dic6—44no—
90C2×Dic6—48no—
91C2×Dic6—44no—
92C2×Dic6—42no—
93C2×Dic6—41no—
94C2×Dic6—41no—
95C3×D8—82no—
96C3×M16—42no—
97C2×C3×C8—22no—
98C3×C4⋊C4—22no—
99C3×C4⋊C4—22no—
100C3×C22⋊C4—42no—
101C3×C22⋊C4—42no—
102#214—416no—
103#208—42no—
104C2×C3⋊C8—24no—
105C2×C3⋊C8—44no—
106C2×C3⋊C8—41no—
107C22×Q8—32no—
108C22×Q8—61no—
109C22×Q8—62no—
110C22×Q8—62no—
111C22×Q8—124no—
112C22×D4—22no—
113C22×D4#6712yes—
114C23×C4—61no—
115C23×C4C22×D311yescharacteristic
116C2×SD16—68no—
117C2×SD16—128no—
118C2×D8—22no—
119C2×M16—61no—
120C2×M16#6712yes—
121C22×C8—62no—
122#121—22no—
123#121—22no—
124#121—22no—
125#121—22no—
126C2×C4⋊C4#6712yes—
127C2×C22⋊C4—22no—
128#106—22no—
129#103—68no—
130#102—24no—
131#101—124no—
132C22⋊C8—22no—
133C22⋊C8—34no—
134C22⋊C8—62no—
135C22⋊C8—62no—
136C23×C3—24no—
137C23×C3—41no—
138C23×C3—42no—
139C23×C3—44no—
140C23×C3—82no—
141C23×C3—82no—
142C23×C3#12111yescharacteristic
143C23×C3#12112yes—
144C3×D4—44no—
145C3×D4—44no—
146C3×D4—84no—
147C3×D4—84no—
148C2×C3×C4—24no—
149C2×C3×C4—22no—
150C2×C3×C4—41no—
151C2×C3×C4—42no—
152C2×C3×C4—42no—
153C2×C3×C4—82no—
154C2×C3×C4#12012yes—
155C2×C3×C4#12011yescharacteristic
156C2×C3×C4#12711yescharacteristic
157C2×Dic3—24no—
158C2×Dic3—48no—
159C2×Dic3—42no—
160C2×Dic3—42no—
161Dic6—28no—
162Dic6—44no—
163Dic6—44no—
164Dic6—44no—
165C3×C8—42no—
166C3⋊C8—44no—
167C3⋊C8—81no—
168C24—22no—
169C24#23711yescharacteristic
170C2×Q8—64no—
171C2×Q8—68no—
172C2×Q8—64no—
173C2×Q8—124no—
174C2×Q8—122no—
175C2×Q8—124no—
176C2×Q8—128no—
177C2×Q8—121no—
178C2×Q8—121no—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—44no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—84no—
185C22×C4—22no—
186C22×C4—42no—
187C22×C4—62no—
188C22×C4—124no—
189C22×C4—122no—
190C22×C4D3×D411yescharacteristic
191C22×C4D3×D412yes—
192C22×C4C2×#6712yes—
193C22×C4C2×#6711yescharacteristic
194SD16—1216no—
195D8—82no—
196M16—42no—
197M16—122no—
198C2×C8—22no—
199C2×C8—64no—
200C2×C8—124no—
201C2×C8—121no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22×C3—22no—
207C22×C3—22no—
208C22×C3—22no—
209C22×C3—21no—
210C22×C3—44no—
211C22×C3—82no—
212C22×C3—82no—
213C22×C3—82no—
214C22×C3—164no—
215C22×C3#39211yescharacteristic
216C22×C3#47114yes—
217C3×C4—22no—
218C3×C4—21no—
219C3×C4—82no—
220C3×C4B(4,𝔽2)12yes—
221Dic3—44no—
222C23—24no—
223C23—41no—
224C23—42no—
225C23—44no—
226C23—82no—
227C23—82no—
228C23#93811yescharacteristic
229C23#93912yes—
230Q8—68no—
231Q8—124no—
232Q8—124no—
233Q8—124no—
234D4—44no—
235D4—44no—
236D4—84no—
237D4—84no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—64no—
244C2×C4—82no—
245C2×C4—122no—
246C2×C4—128no—
247C2×C4—122no—
248C2×C4#93712yes—
249C2×C4#94011yescharacteristic
250C2×C4#95311yescharacteristic
251C8—42no—
252C8—124no—
253C8—241no—
254C2×C3—24no—
255C2×C3—41no—
256C2×C3—162no—
257C2×C3#201912yes—
258C2×C3#202111yescharacteristic
259C22—22no—
260C22—22no—
261C22—22no—
262C22—21no—
263C22—44no—
264C22—82no—
265C22—82no—
266C22—82no—
267C22—164no—
268C22#520014yes—
269C22#522311yescharacteristic
270C4—22no—
271C4—21no—
272C4—82no—
273C4—124no—
274C4#524412yes—
275C3#998811yescharacteristic, complemented, Sylow
276C2—24no—
277C2—41no—
278C2—162no—
279C2#7185111yescharacteristic
280C2#7209712yes—
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#3192C2
socleC22×C3#392
hypercenter#3192D3
hypocenterC3#9988
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
2. C2×#5040C2semidirect
3. #71676C2semidirect
4. #69990C2semidirect
5. #67688C2semidirect
275. C3#9988semidirect

Maximal subgroups

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

subgroupindex
C3×#31922
C2×#50402
#716762
#699902
#676882
#99883