Finite Group Databasebeta

← #10698076

Subgroups — #10698076

Order 768.

1174 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 1174 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
2#82653C212yesmaximal, complemented
3#71673C212yesmaximal, complemented
4#69988C211yesmaximal, characteristic, complemented
5#67683C211yesmaximal, characteristic, complemented
6#10326—31nomaximal, Sylow
7C23×D12C2211yescharacteristic
8C2×C3×#121C2212yes—
9C2×#940—22no—
10C2×#937—24no—
11C3×#481—22no—
12C3×#469—22no—
13C3×#447—22no—
14C3×#407C2211yescharacteristic
15C4×#217C2212yes—
16#5207C2211yescharacteristic
17C2×#823—22no—
18#4988—22no—
19#4960—24no—
20#4537—24no—
21#3192D311yescharacteristic, complemented
22#2433—32no—
23#1783—31no—
24#1507—32no—
25#1297—31no—
26C24×D3D412yes—
27C22×C3×D4—22no—
28C22×C3×D4D412yescomplemented
29C23×C3×C4C2311yescharacteristic
30C22×#67—28no—
31C22×D12—21no—
32C22×D12—22no—
33C22×D12—22no—
34C22×D12—21no—
35C22×D12—44no—
36C2×C3×D8—22no—
37C2×C3×M16D412yescomplemented
38C3×#121—22no—
39C3×#121—22no—
40C3×#121—22no—
41C3×#121—22no—
42C2×C3×C4⋊C4D412yescomplemented
43C2×C3×C22⋊C4—22no—
44C2×#217—22no—
45#940—48no—
46#937—416no—
47C2×(C2×D3)⋊C4—28no—
48C2×C4×Dic3—22no—
49C2×#208—21no—
50C3×#106—22no—
51C3×#102—24no—
52C3×C22⋊C8—22no—
53C3⋊#95D414yescomplemented
54#830—22no—
55#823—44no—
56C23×D4—31no—
57C2×#127—62no—
58C2×#121C2×D312yes—
59C2×#120—64no—
60#481—22no—
61#469—22no—
62#447—22no—
63C2×#100—62no—
64#407—31no—
65#407C2×D311yescharacteristic
66#392—64no—
67#390—62no—
68C4×C22⋊C4—32no—
69#327—64no—
70C24×C3—22no—
71C24×C3C2×D411yescharacteristic
72C23×D3—24no—
73C23×D3—22no—
74C23×D3—44no—
75C23×D3—48no—
76C2×C3×D4—22no—
77C2×C3×D4—22no—
78C2×C3×D4—44no—
79C2×C3×D4—48no—
80C2×C3×D4—42no—
81C2×C3×D4—42no—
82C22×C3×C4—22no—
83C22×C3×C4—42no—
84C22×C3×C4C2×D412yes—
85C22×C3×C4C2×D412yes—
86C22×C3×C4C2×D411yescharacteristic
87C22×C3×C4#4111yescharacteristic
88C2×#67—416no—
89C2×#67—416no—
90C2×#67—88no—
91C22×Dic3—24no—
92C2×D12—24no—
93C2×D12—28no—
94C2×D12—42no—
95C2×D12—44no—
96C2×D12—48no—
97C2×D12—44no—
98C2×D12—41no—
99C2×D12—42no—
100C2×D12—41no—
101C3×D8—82no—
102C3×M16—42no—
103C2×C3×C8—22no—
104C3×C4⋊C4—22no—
105C3×C4⋊C4—22no—
106C3×C22⋊C4—42no—
107C3×C22⋊C4—42no—
108#217—28no—
109(C2×D3)⋊C4—416no—
110C4×Dic3—44no—
111#208—42no—
112C2×C3⋊C8—41no—
113C25—32no—
114C22×D4—22no—
115C22×D4—68no—
116C22×D4—62no—
117C22×D4—61no—
118C22×D4—62no—
119C22×D4—61no—
120C22×D4—124no—
121C22×D4#6712yescomplemented
122C23×C4C22×D311yescharacteristic
123C2×D8—22no—
124C2×M16—61no—
125C2×M16#6712yescomplemented
126#127—128no—
127#121—22no—
128#121—22no—
129#121—22no—
130#121—22no—
131#120—1216no—
132C2×C4⋊C4#6712yescomplemented
133C2×C22⋊C4—22no—
134C2×C22⋊C4—62no—
135C2×C22⋊C4—68no—
136C2×C42—62no—
137#106—22no—
138#102—24no—
139#100—124no—
140C22⋊C8—22no—
141C22⋊C8—62no—
142#95—34no—
143C23×C3—24no—
144C23×C3—41no—
145C23×C3—44no—
146C23×C3—44no—
147C23×C3—42no—
148C23×C3—44no—
149C23×C3#12012yes—
150C23×C3#12111yescharacteristic
151C22×D3—28no—
152C22×D3—24no—
153C22×D3—48no—
154C22×D3—44no—
155C22×D3—42no—
156C22×D3—42no—
157C22×D3—48no—
158C22×D3—816no—
159C22×D3—84no—
160C3×D4—44no—
161C3×D4—82no—
162C3×D4—84no—
163C3×D4—84no—
164C2×C3×C4—22no—
165C2×C3×C4—24no—
166C2×C3×C4—42no—
167C2×C3×C4—41no—
168C2×C3×C4—44no—
169C2×C3×C4—42no—
170C2×C3×C4#12011yescharacteristic
171C2×C3×C4#12112yes—
172C2×C3×C4#12711yescharacteristic
173#67—832no—
174C2×Dic3—28no—
175C2×Dic3—44no—
176C2×Dic3—44no—
177D12—44no—
178D12—48no—
179D12—44no—
180C3×C8—42no—
181C3⋊C8—81no—
182C24—22no—
183C24—64no—
184C24—62no—
185C24—128no—
186C24—124no—
187C24D3×D411yescharacteristic
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—44no—
191C2×D4—48no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—64no—
195C2×D4—68no—
196C2×D4—124no—
197C2×D4—122no—
198C2×D4—128no—
199C2×D4—124no—
200C2×D4—1216no—
201C2×D4—1216no—
202C2×D4—122no—
203C2×D4—121no—
204C2×D4—121no—
205C2×D4—248no—
206C22×C4—22no—
207C22×C4—42no—
208C22×C4—64no—
209C22×C4D3×D412yes—
210C22×C4#23711yescharacteristic
211C22×C4C2×#6712yes—
212C22×C4C2×#6711yescharacteristic
213D8—82no—
214M16—42no—
215M16—122no—
216C2×C8—22no—
217C2×C8—121no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—68no—
223C22⋊C4—1216no—
224C42—124no—
225C22×C3—22no—
226C22×C3—22no—
227C22×C3—22no—
228C22×C3—21no—
229C22×C3—44no—
230C22×C3—88no—
231C22×C3—82no—
232C22×C3—82no—
233C22×C3—82no—
234C22×C3#39211yescharacteristic
235C22×C3#45314yes—
236C2×D3—416no—
237C2×D3—44no—
238C2×D3—88no—
239C2×D3—84no—
240C2×D3—84no—
241C2×D3—84no—
242C3×C4—21no—
243C3×C4—22no—
244C3×C4—82no—
245C3×C4B(4,𝔽2)12yes—
246Dic3—48no—
247C23—24no—
248C23—41no—
249C23—44no—
250C23—44no—
251C23—44no—
252C23—42no—
253C23—68no—
254C23—64no—
255C23—128no—
256C23—124no—
257C23—122no—
258C23—128no—
259C23—122no—
260C23—244no—
261C23—2416no—
262C23#93712yes—
263C23#93811yescharacteristic
264D4—44no—
265D4—82no—
266D4—84no—
267D4—84no—
268D4—124no—
269D4—128no—
270D4—124no—
271D4—2432no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—68no—
279C2×C4—124no—
280C2×C4—124no—
281C2×C4#93912yes—
282C2×C4#94011yescharacteristic
283C2×C4#95311yescharacteristic
284C8—42no—
285C8—241no—
286C2×C3—24no—
287C2×C3—41no—
288C2×C3—162no—
289C2×C3#200212yes—
290C2×C3#202111yescharacteristic
291D3—84no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—21no—
296C22—44no—
297C22—88no—
298C22—82no—
299C22—82no—
300C22—82no—
301C22—1216no—
302C22—124no—
303C22—248no—
304C22—244no—
305C22—244no—
306C22—244no—
307C22#519814yes—
308C22#522311yescharacteristic
309C4—21no—
310C4—22no—
311C4—82no—
312C4—128no—
313C4#524412yes—
314C3#1032611yescharacteristic, complemented, Sylow
315C2—24no—
316C2—41no—
317C2—162no—
318C2—244no—
319C2#7185111yescharacteristic
320C2#7209412yes—
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3192C2
socleC22×C3#392
hypercenter#3192D3
hypocenterC3#10326
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. #82653C2semidirect
3. #71673C2semidirect
4. #69988C2semidirect
5. #67683C2semidirect
21. #3192D3semidirect
28. C22×C3×D4D4semidirect
37. C2×C3×M16D4semidirect
42. C2×C3×C4⋊C4D4semidirect
53. C3⋊#95D4semidirect
121. C22×D4#67semidirect
125. C2×M16#67semidirect
132. C2×C4⋊C4#67semidirect
314. C3#10326semidirect

Maximal subgroups

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

subgroupindex
C3×#31922
#826532
#716732
#699882
#676832
#103263