Finite Group Databasebeta

← #10698079

Subgroups — #10698079

Order 768.

1174 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 1174 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
1#84049C212yesmaximal, complemented
2#78641C211yesmaximal, characteristic, complemented
3C3×#1783C211yesmaximal, characteristic, complemented
4#69977C212yesmaximal, complemented
5#67683C211yesmaximal, characteristic, complemented
6#10326—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C2×#953—24no—
9C2×#940—22no—
10C2×#930C2212yes—
11C2×C3×#100—22no—
12C3×#407C2211yescharacteristic
13C3×#407C2211yescharacteristic
14#5223—24no—
15#5221—22no—
16C4×#217C2212yes—
17#5135—22no—
18#4735—22no—
19#4732—22no—
20#4470—24no—
21#3192—31no—
22#2433—32no—
23#1783D311yescharacteristic, complemented
24#1507—32no—
25#1297—31no—
26C25×C3D412yes—
27C22×C3×D4—22no—
28C22×C3×D4—21no—
29C22×C3×D4—22no—
30C22×C3×D4—21no—
31C22×C3×D4—44no—
32C23×C3×C4C2311yescharacteristic
33C22×#67—22no—
34C22×#67—28no—
35C22×D12D412yescomplemented
36C2×C3×M16—21no—
37C2×C3×M16D412yescomplemented
38#953—416no—
39C2×#217—22no—
40C2×#217—28no—
41#940—48no—
42#930—22no—
43#930—22no—
44#930—22no—
45#930—22no—
46C2×(C2×D3)⋊C4—22no—
47C2×C4⋊Dic3D412yescomplemented
48C2×C4×Dic3—22no—
49C2×D24—22no—
50C3×#100—44no—
51C3×C22⋊C8—22no—
52C3×C22⋊C8—22no—
53C3⋊#95D414yescomplemented
54#821—24no—
55#817—22no—
56C23×D4C2×D311yescharacteristic
57C2×#127—62no—
58C2×#121—32no—
59C2×#120—64no—
60#481—62no—
61#469—62no—
62#447—62no—
63C2×#100—22no—
64#407C2×D311yescharacteristic
65#407C2×D311yescharacteristic
66#392—64no—
67#390—62no—
68C4×C22⋊C4—32no—
69#327—64no—
70C24×C3—24no—
71C24×C3—22no—
72C24×C3—44no—
73C24×C3—48no—
74C24×C3C2×D411yescharacteristic
75C23×D3—22no—
76C2×C3×D4—24no—
77C2×C3×D4—28no—
78C2×C3×D4—44no—
79C2×C3×D4—42no—
80C2×C3×D4—48no—
81C2×C3×D4—44no—
82C2×C3×D4—41no—
83C2×C3×D4—42no—
84C2×C3×D4—41no—
85C22×C3×C4—42no—
86C22×C3×C4C2×D412yes—
87C22×C3×C4C2×D412yes—
88C22×C3×C4C2×D411yescharacteristic
89C22×C3×C4#4111yescharacteristic
90C2×#67—416no—
91C2×#67—416no—
92C2×#67—48no—
93C2×#67—42no—
94C2×#67—42no—
95C2×#67—88no—
96C22×Dic3—24no—
97C22×Dic3—22no—
98C2×D12—22no—
99C2×D12—22no—
100C2×D12—44no—
101C3×M16—42no—
102C3×M16—42no—
103C2×C3×C8—22no—
104C2×C3×C8—41no—
105#217—28no—
106#217—416no—
107(C2×D3)⋊C4—42no—
108(C2×D3)⋊C4—42no—
109C4⋊Dic3—22no—
110C4⋊Dic3—22no—
111C4×Dic3—44no—
112D24—82no—
113C25D1212yes—
114C22×D4—21no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—21no—
118C22×D4—32no—
119C22×D4—44no—
120C22×D4—62no—
121C22×D4—68no—
122C23×C4C22×D311yescharacteristic
123C2×D8—62no—
124C2×M16—21no—
125C2×M16#6712yescomplemented
126#127—128no—
127#121—62no—
128#121—62no—
129#121—62no—
130#121—62no—
131#120—1216no—
132C2×C4⋊C4—32no—
133C2×C22⋊C4—62no—
134C2×C22⋊C4—68no—
135C2×C22⋊C4—62no—
136C2×C42—62no—
137#106—62no—
138#102—64no—
139#100—44no—
140C22⋊C8—22no—
141C22⋊C8—22no—
142#95—34no—
143C23×C3—24no—
144C23×C3—28no—
145C23×C3—24no—
146C23×C3—41no—
147C23×C3—48no—
148C23×C3—44no—
149C23×C3—42no—
150C23×C3—42no—
151C23×C3—48no—
152C23×C3—84no—
153C23×C3—816no—
154C23×C3#12012yes—
155C23×C3#12111yescharacteristic
156C22×D3—44no—
157C22×D3—44no—
158C22×D3—42no—
159C22×D3—44no—
160C3×D4—44no—
161C3×D4—48no—
162C3×D4—44no—
163C2×C3×C4—24no—
164C2×C3×C4—22no—
165C2×C3×C4—42no—
166C2×C3×C4—41no—
167C2×C3×C4#12011yescharacteristic
168C2×C3×C4#12112yes—
169C2×C3×C4#12711yescharacteristic
170#67—832no—
171#67—84no—
172#67—84no—
173C2×Dic3—28no—
174C2×Dic3—44no—
175C2×Dic3—44no—
176C2×Dic3—44no—
177C2×Dic3—42no—
178D12—44no—
179D12—82no—
180C3×C8—42no—
181C3×C8—81no—
182C24—24no—
183C24—22no—
184C24—44no—
185C24—48no—
186C24—62no—
187C24C2×D1211yescharacteristic
188C2×D4—24no—
189C2×D4—28no—
190C2×D4—44no—
191C2×D4—42no—
192C2×D4—44no—
193C2×D4—48no—
194C2×D4—42no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—62no—
198C2×D4—62no—
199C2×D4—124no—
200C2×D4—1216no—
201C2×D4—128no—
202C2×D4—1216no—
203C2×D4—122no—
204C2×D4—122no—
205C2×D4—248no—
206C22×C4—42no—
207C22×C4—64no—
208C22×C4—62no—
209C22×C4#23511yescharacteristic
210C22×C4D3×D412yes—
211C22×C4D3×D412yes—
212C22×C4C2×#6711yescharacteristic
213D8—242no—
214M16—42no—
215M16—42no—
216C2×C8—22no—
217C2×C8—41no—
218C4⋊C4—62no—
219C4⋊C4—62no—
220C22⋊C4—68no—
221C22⋊C4—1216no—
222C22⋊C4—122no—
223C22⋊C4—122no—
224C42—124no—
225C22×C3—22no—
226C22×C3—22no—
227C22×C3—22no—
228C22×C3—21no—
229C22×C3—44no—
230C22×C3—416no—
231C22×C3—44no—
232C22×C3—88no—
233C22×C3—84no—
234C22×C3—84no—
235C22×C3—84no—
236C22×C3#39211yescharacteristic
237C22×C3#45314yes—
238C2×D3—88no—
239C2×D3—82no—
240C2×D3—82no—
241C2×D3—82no—
242C3×C4—21no—
243C3×C4—22no—
244C3×C4B(4,𝔽2)12yes—
245Dic3—48no—
246Dic3—82no—
247C23—24no—
248C23—28no—
249C23—24no—
250C23—41no—
251C23—48no—
252C23—44no—
253C23—42no—
254C23—42no—
255C23—48no—
256C23—84no—
257C23—816no—
258C23—124no—
259C23—124no—
260C23—122no—
261C23—124no—
262C23#88212yes—
263C23#93011yescharacteristic
264D4—44no—
265D4—48no—
266D4—44no—
267D4—124no—
268D4—242no—
269D4—2432no—
270D4—244no—
271D4—244no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—68no—
277C2×C4—124no—
278C2×C4—124no—
279C2×C4—124no—
280C2×C4—122no—
281C2×C4#88412yes—
282C2×C4#93711yescharacteristic
283C2×C4#94011yescharacteristic
284C8—42no—
285C8—81no—
286C2×C3—24no—
287C2×C3—41no—
288C2×C3—84no—
289C2×C3#200212yes—
290C2×C3#202111yescharacteristic
291D3—162no—
292C22—22no—
293C22—22no—
294C22—21no—
295C22—22no—
296C22—44no—
297C22—416no—
298C22—44no—
299C22—88no—
300C22—84no—
301C22—84no—
302C22—84no—
303C22—248no—
304C22—242no—
305C22—242no—
306C22—242no—
307C22#474714yes—
308C22#496011yescharacteristic
309C4—21no—
310C4—22no—
311C4—128no—
312C4—242no—
313C4#503212yes—
314C3#1032611yescharacteristic, complemented, Sylow
315C2—24no—
316C2—41no—
317C2—84no—
318C2—482no—
319C2#6950011yescharacteristic
320C2#7008212yes—
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1783C2
socleC22×C3#392
hypercenter#1783D3
hypocenterC3#10326
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. #84049C2semidirect
2. #78641C2semidirect
3. C3×#1783C2semidirect
4. #69977C2semidirect
5. #67683C2semidirect
23. #1783D3semidirect
35. C22×D12D4semidirect
37. C2×C3×M16D4semidirect
47. C2×C4⋊Dic3D4semidirect
53. C3⋊#95D4semidirect
125. C2×M16#67semidirect
314. C3#10326semidirect

Maximal subgroups

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

subgroupindex
#840492
#786412
C3×#17832
#699772
#676832
#103263