Finite Group Databasebeta

← #10698081

Subgroups — #10698081

Order 768.

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

normal subgroups only

This group has 806 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
2#82654C212yesmaximal, complemented
3#71673C212yesmaximal, complemented
4#69990C211yesmaximal, characteristic, complemented
5#67683C211yesmaximal, characteristic, complemented
6#10327—31nomaximal, Sylow
7C23×Dic6C2211yescharacteristic
8C2×C3×#121C2212yes—
9C2×#936—22no—
10C3×#481—22no—
11C3×#469—22no—
12C3×#447—22no—
13C3×#407C2211yescharacteristic
14#5220—24no—
15C4×#217C2212yes—
16#5207C2211yescharacteristic
17C2×#824—22no—
18#4973—22no—
19#4947—24no—
20#4537—24no—
21#3192D311yescharacteristic
22#2436—32no—
23#1784—31no—
24#1507—32no—
25#1297—31no—
26C22×C3×D4—22no—
27C22×C3×D4D412yes—
28C23×C3×C4C2311yescharacteristic
29C23×Dic3D412yes—
30C22×Dic6—21no—
31C22×Dic6—22no—
32C22×Dic6—22no—
33C22×Dic6—21no—
34C22×Dic6—44no—
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×#217—28no—
44C2×#217—22no—
45#936—48no—
46C2×C4×Dic3—22no—
47C2×#208—21no—
48C3×#106—22no—
49C3×#102—24no—
50C3×C22⋊C8—22no—
51C3⋊#95—28no—
52C3⋊#95D414yes—
53#830—22no—
54#824—44no—
55C23×Q8—31no—
56C2×#124—62no—
57C2×#121C2×D312yes—
58#481—22no—
59#469—22no—
60#447—22no—
61C2×#101—62no—
62#407—31no—
63#407C2×D311yescharacteristic
64#393—64no—
65#391—62no—
66#386—64no—
67C4×C22⋊C4—32no—
68#327—64no—
69C24×C3—22no—
70C24×C3C2×D411yescharacteristic
71C2×C3×D4—22no—
72C2×C3×D4—22no—
73C2×C3×D4—44no—
74C2×C3×D4—48no—
75C2×C3×D4—42no—
76C2×C3×D4—42no—
77C22×C3×C4—22no—
78C22×C3×C4—42no—
79C22×C3×C4C2×D412yes—
80C22×C3×C4C2×D412yes—
81C22×C3×C4C2×D411yescharacteristic
82C22×C3×C4#4111yescharacteristic
83C22×Dic3—24no—
84C22×Dic3—22no—
85C22×Dic3—24no—
86C22×Dic3—44no—
87C2×Dic6—24no—
88C2×Dic6—28no—
89C2×Dic6—48no—
90C2×Dic6—44no—
91C2×Dic6—42no—
92C2×Dic6—44no—
93C2×Dic6—41no—
94C2×Dic6—42no—
95C2×Dic6—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#217—416no—
104#217—44no—
105C4×Dic3—44no—
106#208—42no—
107C2×C3⋊C8—41no—
108C22×Q8—62no—
109C22×Q8—61no—
110C22×Q8—62no—
111C22×Q8—61no—
112C22×Q8—124no—
113C22×D4—22no—
114C22×D4#6712yes—
115C23×C4—32no—
116C23×C4C22×D311yescharacteristic
117C2×D8—22no—
118C2×M16—61no—
119C2×M16#6712yes—
120#124—128no—
121#121—22no—
122#121—22no—
123#121—22no—
124#121—22no—
125C2×C4⋊C4#6712yes—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—68no—
128C2×C22⋊C4—62no—
129C2×C42—62no—
130#106—22no—
131#102—24no—
132#101—124no—
133C22⋊C8—22no—
134C22⋊C8—62no—
135#95—34no—
136#95—68no—
137C23×C3—24no—
138C23×C3—41no—
139C23×C3—44no—
140C23×C3—44no—
141C23×C3—42no—
142C23×C3—44no—
143C23×C3#12012yes—
144C23×C3#12111yescharacteristic
145C3×D4—44no—
146C3×D4—82no—
147C3×D4—84no—
148C3×D4—84no—
149C2×C3×C4—22no—
150C2×C3×C4—24no—
151C2×C3×C4—42no—
152C2×C3×C4—41no—
153C2×C3×C4—44no—
154C2×C3×C4—42no—
155C2×C3×C4#12011yescharacteristic
156C2×C3×C4#12112yes—
157C2×C3×C4#12711yescharacteristic
158C2×Dic3—28no—
159C2×Dic3—24no—
160C2×Dic3—44no—
161C2×Dic3—42no—
162C2×Dic3—42no—
163C2×Dic3—44no—
164C2×Dic3—44no—
165C2×Dic3—44no—
166Dic6—44no—
167Dic6—48no—
168Dic6—44no—
169C3×C8—42no—
170C3⋊C8—81no—
171C24—22no—
172C24D3×D411yescharacteristic
173C2×Q8—64no—
174C2×Q8—68no—
175C2×Q8—124no—
176C2×Q8—122no—
177C2×Q8—128no—
178C2×Q8—124no—
179C2×Q8—122no—
180C2×Q8—121no—
181C2×Q8—121no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—44no—
185C2×D4—48no—
186C2×D4—42no—
187C2×D4—42no—
188C22×C4—22no—
189C22×C4—42no—
190C22×C4—64no—
191C22×C4—62no—
192C22×C4—64no—
193C22×C4—124no—
194C22×C4D3×D412yes—
195C22×C4#23711yescharacteristic
196C22×C4C2×#6712yes—
197C22×C4C2×#6711yescharacteristic
198D8—82no—
199M16—42no—
200M16—122no—
201C2×C8—22no—
202C2×C8—121no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—1216no—
208C22⋊C4—124no—
209C42—124no—
210C22×C3—22no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—21no—
214C22×C3—44no—
215C22×C3—88no—
216C22×C3—82no—
217C22×C3—82no—
218C22×C3—82no—
219C22×C3#39211yescharacteristic
220C22×C3#45414yes—
221C3×C4—21no—
222C3×C4—22no—
223C3×C4—82no—
224C3×C4B(4,𝔽2)12yes—
225Dic3—44no—
226Dic3—84no—
227C23—24no—
228C23—41no—
229C23—44no—
230C23—44no—
231C23—44no—
232C23—42no—
233C23#93712yes—
234C23#93811yescharacteristic
235Q8—124no—
236Q8—128no—
237Q8—124no—
238D4—44no—
239D4—82no—
240D4—84no—
241D4—84no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—44no—
247C2×C4—42no—
248C2×C4—68no—
249C2×C4—64no—
250C2×C4—124no—
251C2×C4—122no—
252C2×C4—122no—
253C2×C4—124no—
254C2×C4—124no—
255C2×C4—124no—
256C2×C4#93912yes—
257C2×C4#94011yescharacteristic
258C2×C4#95311yescharacteristic
259C8—42no—
260C8—241no—
261C2×C3—24no—
262C2×C3—41no—
263C2×C3—162no—
264C2×C3#200312yes—
265C2×C3#202111yescharacteristic
266C22—22no—
267C22—22no—
268C22—22no—
269C22—21no—
270C22—44no—
271C22—88no—
272C22—82no—
273C22—82no—
274C22—82no—
275C22#520114yes—
276C22#522311yescharacteristic
277C4—21no—
278C4—22no—
279C4—82no—
280C4—124no—
281C4—244no—
282C4#524412yes—
283C3#1032711yescharacteristic, complemented, Sylow
284C2—24no—
285C2—41no—
286C2—162no—
287C2#7185111yescharacteristic
288C2#7209612yes—
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#3192C2
socleC22×C3#392
hypercenter#3192D3
hypocenterC3#10327
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. #82654C2semidirect
3. #71673C2semidirect
4. #69990C2semidirect
5. #67683C2semidirect
283. C3#10327semidirect

Maximal subgroups

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

subgroupindex
C3×#31922
#826542
#716732
#699902
#676832
#103273