Finite Group Databasebeta

← #10742713

Subgroups — #10742713

Order 768.

2354 conjugacy classes of subgroups; only the 315 normal subgroups are shown.

normal subgroups only

This group has 2354 conjugacy classes of subgroups — too many to draw. Only its 315 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
1C22×#935C212yesmaximal
2C2×#4950C214yesmaximal, complemented
3C3×#1438C211yesmaximal, characteristic
4#71549C214yesmaximal, complemented
5#69420C214yesmaximal, complemented
6#15949—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C24×Dic3C2212yes—
9C22×C3×C22⋊C4C2214yes—
10C22×#217C2212yes—
11C22×#217C2214yes—
12C2×#935—28no—
13C2×#935—216no—
14C22×C4⋊Dic3C2214yes—
15C22×Dic3⋊C4C2214yes—
16C2×C3×#95C2212yes—
17C2×C3⋊#95C2214yes—
18C2×C3⋊#95C2218yes—
19#4950—216no—
20C22×#123—32no—
21C2×#385—34no—
22#1444—34no—
23#1440—34no—
24#1438D311yescharacteristic
25C25×C3C2312yes—
26C22×C3×D4—28no—
27C22×C3×D4—42no—
28C22×C3×D4D414yes—
29C23×C3×C4C2311yescharacteristic
30C23×C3×C4C2314yes—
31C23×Dic3—24no—
32C23×Dic3—28no—
33C23×Dic3—28no—
34C23×Dic3—44no—
35C23×Dic3C2314yes—
36C23×Dic3C2314yes—
37C2×C3×C22⋊C4—28no—
38C2×C3×C22⋊C4—216no—
39C2×#217—24no—
40C2×#217—28no—
41C2×#217—28no—
42C2×#217—28no—
43C2×#217D4116yes—
44#935—264no—
45C2×C4⋊Dic3—216no—
46C2×C4⋊Dic3—44no—
47C2×Dic3⋊C4—28no—
48C2×Dic3⋊C4—28no—
49C2×Dic3⋊C4—28no—
50C3×#95—42no—
51C3⋊#95—28no—
52C3⋊#95—216no—
53C23×D4C2×D311yescharacteristic
54C24×C4—32no—
55C2×#123—616no—
56C2×#123—68no—
57C22×C4⋊C4—34no—
58C22×C4⋊C4—34no—
59C22×C22⋊C4—34no—
60C22×C22⋊C4—32no—
61C22×C22⋊C4C2×D314yes—
62#385—616no—
63C2×#95—34no—
64C2×#95—38no—
65C2×#95C2×D312yes—
66C24×C3—24no—
67C24×C3—28no—
68C24×C3—24no—
69C24×C3—28no—
70C24×C3C2×D414yes—
71C24×C3C2×D418yes—
72C24×C3C2411yescharacteristic
73C2×C3×D4—216no—
74C2×C3×D4—44no—
75C2×C3×D4—416no—
76C22×C3×C4—28no—
77C22×C3×C4—24no—
78C22×C3×C4—28no—
79C22×C3×C4—24no—
80C22×C3×C4—22no—
81C22×C3×C4—24no—
82C22×C3×C4C2×D412yes—
83C22×C3×C4#4118yes—
84C22×Dic3—216no—
85C22×Dic3—28no—
86C22×Dic3—24no—
87C22×Dic3—24no—
88C22×Dic3—24no—
89C22×Dic3—24no—
90C22×Dic3—28no—
91C22×Dic3—28no—
92C22×Dic3—28no—
93C22×Dic3—432no—
94C22×Dic3—48no—
95C22×Dic3—48no—
96C22×Dic3C2×D418yes—
97C22×Dic3C2×D418yes—
98C3×C22⋊C4—232no—
99#217—216no—
100#217—232no—
101C4⋊Dic3—416no—
102Dic3⋊C4—232no—
103C25C22×D312yes—
104C22×D4—28no—
105C22×D4—42no—
106C22×D4D1214yes—
107C23×C4—34no—
108C23×C4—34no—
109C23×C4—68no—
110C23×C4—64no—
111C23×C4—68no—
112C23×C4—124no—
113C23×C4C22×D311yescharacteristic
114C23×C4C22×D314yes—
115#123—664no—
116C2×C4⋊C4—68no—
117C2×C4⋊C4—68no—
118C2×C4⋊C4—68no—
119C2×C4⋊C4—616no—
120C2×C4⋊C4—124no—
121C2×C22⋊C4—28no—
122C2×C22⋊C4—216no—
123C2×C22⋊C4—316no—
124C2×C22⋊C4—68no—
125C2×C22⋊C4—64no—
126C2×C22⋊C4—68no—
127C2×C22⋊C4—68no—
128#95—42no—
129#95—616no—
130#95—68no—
131C23×C3—24no—
132C23×C3—28no—
133C23×C3—24no—
134C23×C3—28no—
135C23×C3—216no—
136C23×C3—216no—
137C23×C3—216no—
138C23×C3—44no—
139C23×C3—44no—
140C23×C3—48no—
141C23×C3—48no—
142C23×C3—44no—
143C23×C3—44no—
144C23×C3#12018yes—
145C23×C3C22×D411yescharacteristic
146C23×C3C22×D414yes—
147C23×C3C2×#4114yes—
148C23×C3H(5,𝔽2)12yes—
149C23×C32−1+414yes—
150C3×D4—416no—
151C2×C3×C4—24no—
152C2×C3×C4—216no—
153C2×C3×C4—22no—
154C2×C3×C4—28no—
155C2×C3×C4—44no—
156C2×C3×C4—44no—
157C2×C3×C4—48no—
158C2×C3×C4—44no—
159C2×Dic3—28no—
160C2×Dic3—28no—
161C2×Dic3—216no—
162C2×Dic3—216no—
163C2×Dic3—432no—
164C2×Dic3—44no—
165C2×Dic3—44no—
166C2×Dic3—44no—
167C2×Dic3—44no—
168C2×Dic3—48no—
169C2×Dic3—48no—
170C24—24no—
171C24—28no—
172C24—24no—
173C24—28no—
174C24C2×D1214yes—
175C24D3×D418yes—
176C24C23×D311yescharacteristic
177C2×D4—216no—
178C2×D4—44no—
179C2×D4—416no—
180C22×C4—28no—
181C22×C4—24no—
182C22×C4—28no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—38no—
187C22×C4—38no—
188C22×C4—616no—
189C22×C4—64no—
190C22×C4—64no—
191C22×C4—68no—
192C22×C4—64no—
193C22×C4—64no—
194C22×C4—68no—
195C22×C4—68no—
196C22×C4—68no—
197C22×C4—1232no—
198C22×C4—128no—
199C22×C4—128no—
200C22×C4C2×D1212yes—
201C22×C4#23718yes—
202C4⋊C4—632no—
203C4⋊C4—1216no—
204C22⋊C4—232no—
205C22⋊C4—616no—
206C22⋊C4—632no—
207C22×C3—28no—
208C22×C3—216no—
209C22×C3—48no—
210C22×C3—416no—
211C22×C3—416no—
212C22×C3—48no—
213C22×C3C2×#12012yes—
214C22×C3#52514yes—
215C22×C3#53411yescharacteristic
216C22×C3#53614yes—
217C22×C3#53914yes—
218C22×C3D4214yes—
219C22×C3#54714yes—
220C22×C3#54718yes—
221C22×C3#55912yes—
222C22×C3#56312yes—
223C3×C4—42no—
224C3×C4—48no—
225Dic3—48no—
226Dic3—48no—
227C23—28no—
228C23—24no—
229C23—24no—
230C23—216no—
231C23—28no—
232C23—216no—
233C23—216no—
234C23—44no—
235C23—44no—
236C23—44no—
237C23—44no—
238C23—48no—
239C23—48no—
240C23#88218yes—
241C23C22×D1211yescharacteristic
242C23C2×D3×D414yes—
243C23C2×#23714yes—
244C23#100412yes—
245C23#101014yes—
246D4—416no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—216no—
250C2×C4—28no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—48no—
255C2×C4—68no—
256C2×C4—68no—
257C2×C4—616no—
258C2×C4—616no—
259C2×C4—1232no—
260C2×C4—124no—
261C2×C4—124no—
262C2×C4—124no—
263C2×C4—128no—
264C2×C4—124no—
265C2×C4—128no—
266C2×C3—48no—
267C2×C3#240311yescharacteristic
268C2×C3#240614yes—
269C2×C3#245514yes—
270C2×C3#268014yes—
271C2×C3#272412yes—
272C22—28no—
273C22—216no—
274C22—48no—
275C22—416no—
276C22—48no—
277C22—416no—
278C22C2×#88212yes—
279C22#549311yescharacteristic
280C22#555514yes—
281C22#558614yes—
282C22#559014yes—
283C22#560214yes—
284C22#561118yes—
285C22#564814yes—
286C22#567212yes—
287C22#571312yes—
288C4—42no—
289C4—48no—
290C4—128no—
291C4—128no—
292C3#1594911yescharacteristic, complemented, Sylow
293C2—48no—
294C2#7385211yescharacteristic
295C2#7468714yes—
296C2#7639812yes—
297C2#7667414yes—
298C2#7700414yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1438C2
prime radicalC23×C3×D4C22
hypercenter#1438D3
hypocenterC3#15949
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
2. C2×#4950C2semidirect
4. #71549C2semidirect
5. #69420C2semidirect
292. C3#15949semidirect

Maximal subgroups

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

subgroupindex
C22×#9352
C2×#49502
C3×#14382
#715492
#694202
#159493