Finite Group Databasebeta

← #10698151

Subgroups — #10698151

Order 768.

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

normal subgroups only

This group has 738 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×#2483C211yesmaximal, characteristic
2#82114C212yesmaximal
3#71542C212yesmaximal
4#69495C211yesmaximal, characteristic
5C3⋊#1304C211yesmaximal, characteristic
6#10339—31nomaximal, Sylow
7C22×C3×C22⋊C4C2212yes—
8C22×#217C2211yescharacteristic
9C3×#397—22no—
10C3×#385—22no—
11C3×#381—22no—
12C3×#379C2211yescharacteristic
13#5247C2211yescharacteristic
14#5247C2212yes—
15#5211—22no—
16#5210—22no—
17#4952—24no—
18#4946—22no—
19#4943—22no—
20#4527—22no—
21#4526—24no—
22#2483—32no—
23#2483D311yescharacteristic
24#1819—31no—
25#1819—32no—
26#1304—31no—
27C25×C3C2311yescharacteristic
28C23×C3×C4D412yes—
29C23×C3×C4D412yes—
30C23×Dic3D412yes—
31C2×C3×C4⋊C4—22no—
32C2×C3×C22⋊C4—22no—
33C2×C3×C22⋊C4—22no—
34C2×C3×C22⋊C4—22no—
35C2×C3×C22⋊C4—22no—
36C2×C3×C22⋊C4—22no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4D412yes—
40C2×#217—22no—
41C2×#217—22no—
42C2×#217—22no—
43C2×#217—22no—
44C2×#217—21no—
45C2×#217—21no—
46C2×#217—22no—
47C2×#217—21no—
48C2×#217D414yes—
49C2×Dic3⋊C4—24no—
50C3×#95—22no—
51C3×#95—24no—
52C3⋊#95—24no—
53C3⋊#95—24no—
54C3⋊#95—24no—
55C22×C22⋊C4—31no—
56C22×C22⋊C4C2×D312yes—
57#397—22no—
58#397—64no—
59#385—22no—
60#385—62no—
61#385—62no—
62#381—22no—
63#381—62no—
64#381—62no—
65#379—32no—
66#379—31no—
67#379C2×D311yescharacteristic
68#342—64no—
69#342—62no—
70C24×C3—44no—
71C24×C3—42no—
72C24×C3C2×D411yescharacteristic
73C24×C3C2×D412yes—
74C24×C3C2×D412yes—
75C24×C3C2×D411yescharacteristic
76C24×C3#4111yescharacteristic
77C22×C3×C4—22no—
78C22×C3×C4—22no—
79C22×C3×C4—22no—
80C22×C3×C4—22no—
81C22×C3×C4—22no—
82C22×C3×C4—22no—
83C22×C3×C4—22no—
84C22×C3×C4—44no—
85C22×C3×C4—44no—
86C22×Dic3—22no—
87C22×Dic3—24no—
88C22×Dic3—22no—
89C22×Dic3—22no—
90C22×Dic3—44no—
91C22×Dic3—41no—
92C3×C4⋊C4—82no—
93C3×C22⋊C4—44no—
94C3×C22⋊C4—42no—
95C3×C22⋊C4—42no—
96C3×C22⋊C4—42no—
97C3×C22⋊C4—42no—
98C3×C22⋊C4—42no—
99#217—44no—
100#217—44no—
101#217—41no—
102#217—42no—
103#217—41no—
104#217—44no—
105#217—44no—
106#217—42no—
107Dic3⋊C4—84no—
108C25C22×D311yescharacteristic
109C23×C4—32no—
110C23×C4#6712yes—
111C23×C4#6712yes—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—64no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—34no—
122C2×C22⋊C4—62no—
123C2×C22⋊C4—62no—
124C2×C22⋊C4—61no—
125C2×C22⋊C4—61no—
126C2×C22⋊C4—62no—
127C2×C22⋊C4—62no—
128C2×C22⋊C4—62no—
129C2×C22⋊C4—61no—
130C2×C22⋊C4#6712yes—
131#95—22no—
132#95—24no—
133#95—64no—
134#95—64no—
135#95—64no—
136C23×C3—24no—
137C23×C3—24no—
138C23×C3—24no—
139C23×C3—24no—
140C23×C3—22no—
141C23×C3—41no—
142C23×C3—42no—
143C23×C3—42no—
144C23×C3—42no—
145C23×C3—42no—
146C23×C3—42no—
147C23×C3—41no—
148C23×C3—88no—
149C23×C3#12012yes—
150C23×C3#12011yescharacteristic
151C23×C3#12111yescharacteristic
152C23×C3#12112yes—
153C23×C3#12711yescharacteristic
154C2×C3×C4—44no—
155C2×C3×C4—44no—
156C2×C3×C4—44no—
157C2×C3×C4—42no—
158C2×C3×C4—42no—
159C2×C3×C4—44no—
160C2×C3×C4—42no—
161C2×C3×C4—44no—
162C2×C3×C4—82no—
163C2×C3×C4—82no—
164C2×Dic3—44no—
165C2×Dic3—44no—
166C2×Dic3—44no—
167C2×Dic3—42no—
168C2×Dic3—44no—
169C2×Dic3—44no—
170C2×Dic3—82no—
171C2×Dic3—82no—
172C2×Dic3—81no—
173C24—42no—
174C24—44no—
175C24D3×D411yescharacteristic
176C24D3×D412yes—
177C24#23711yescharacteristic
178C24C2×#6712yes—
179C24C2×#6711yescharacteristic
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—62no—
190C22×C4—64no—
191C22×C4—62no—
192C22×C4—62no—
193C22×C4—124no—
194C22×C4—121no—
195C4⋊C4—82no—
196C4⋊C4—244no—
197C22⋊C4—44no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—124no—
204C22⋊C4—124no—
205C22⋊C4—122no—
206C22⋊C4—121no—
207C22⋊C4—121no—
208C22⋊C4—124no—
209C22⋊C4—122no—
210C22⋊C4—124no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—22no—
215C22×C3—22no—
216C22×C3—21no—
217C22×C3—22no—
218C22×C3—22no—
219C22×C3—22no—
220C22×C3—21no—
221C22×C3—44no—
222C22×C3—48no—
223C22×C3—84no—
224C22×C3—82no—
225C22×C3—82no—
226C22×C3#39211yescharacteristic
227C22×C3#45812yes—
228C22×C3#45814yes—
229C3×C4—82no—
230C3×C4—82no—
231C3×C4—82no—
232Dic3—84no—
233Dic3—82no—
234Dic3—161no—
235C23—24no—
236C23—24no—
237C23—24no—
238C23—22no—
239C23—24no—
240C23—41no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—41no—
246C23—42no—
247C23—88no—
248C23#93712yes—
249C23#93811yescharacteristic
250C23#93912yes—
251C23#94011yescharacteristic
252C23#95311yescharacteristic
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—124no—
264C2×C4—124no—
265C2×C4—124no—
266C2×C4—122no—
267C2×C4—124no—
268C2×C4—124no—
269C2×C4—242no—
270C2×C4—242no—
271C2×C4—241no—
272C2×C3—24no—
273C2×C3—22no—
274C2×C3—41no—
275C2×C3—41no—
276C2×C3—42no—
277C2×C3#202212yes—
278C2×C3#202211yescharacteristic
279C22—22no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—21no—
288C22—21no—
289C22—48no—
290C22—44no—
291C22—82no—
292C22—84no—
293C22—82no—
294C22#505314yes—
295C22#522311yescharacteristic
296C22#524312yes—
297C4—82no—
298C4—82no—
299C4—82no—
300C4—244no—
301C4—242no—
302C4—481no—
303C3#1033911yescharacteristic, complemented, Sylow
304C2—22no—
305C2—24no—
306C2—41no—
307C2—42no—
308C2—41no—
309C2#7170612yes—
310C2#7184911yescharacteristic
subgroupG/H
centerC22#5223
commutator, prime radical, cosocleC25×C3C23
FrattiniC25C22×D3
FittingC3×#2483C2
socleC22×C3#392
hypercenter#2483D3
hypocenterC3#10339
LCSG ▷ C25×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #2483

Complements

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

NH ≅ G/Ntype
303. C3#10339semidirect

Maximal subgroups

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

subgroupindex
C3×#24832
#821142
#715422
#694952
C3⋊#13042
#103393