Finite Group Databasebeta

← #10693111

Subgroups — #10693111

Order 768.

660 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 103 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×#2312C211yesmaximal, characteristic
2#82114C211yesmaximal, characteristic
3#71542C211yesmaximal, characteristic
4#69495C212yesmaximal
5C3⋊#1304C212yesmaximal
6#9500—31nomaximal, Sylow
7C22×C3×C22⋊C4C2211yescharacteristic
8C22×#217C2211yescharacteristic
9C3×#388—24no—
10C3×#380—21no—
11C3×#379C2212yes—
12C3×C4×C22⋊C4—21no—
13#5247C2212yes—
14#5247C2211yescharacteristic
15#5211—21no—
16#5210—21no—
17#4952—22no—
18#4946C412yescomplemented
19#4943C412yescomplemented
20#4527—24no—
21#4526—22no—
22#2483—31no—
23#2312D311yescharacteristic
24#1819—32no—
25#1819—31no—
26#1304—32no—
27C25×C3C2311yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C23×Dic3C2×C411yescharacteristic
31C23×Dic3C2×C411yescharacteristic
32C2×C3×C4⋊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⋊C4C2×C412yes—
38C2×C3×C22⋊C4D412yes—
39C2×C3×C22⋊C4D414yes—
40C2×C3×C42—22no—
41C2×#217—21no—
42C2×#217—21no—
43C2×#217—21no—
44C2×#217—22no—
45C2×#217—22no—
46C2×#217—22no—
47C2×#217—22no—
48C2×#217—21no—
49C2×#217—22no—
50C2×#217C2×C412yes—
51C2×Dic3⋊C4—22no—
52C3×#95—22no—
53C3×#95—22no—
54C3⋊#95—22no—
55C3⋊#95—22no—
56C3⋊#95—22no—
57C22×C22⋊C4—31no—
58C22×C22⋊C4C2×D311yescharacteristic
59#397—62no—
60#388—24no—
61#385—32no—
62#385—61no—
63#381—32no—
64#381—61no—
65#380—21no—
66#379—31no—
67#379—32no—
68#379C2×D312yes—
69C4×C22⋊C4—21no—
70#342—62no—
71#342—64no—
72C24×C3—42no—
73C24×C3—42no—
74C24×C3—81no—
75C24×C3C22×C411yescharacteristic
76C24×C3C2×D411yescharacteristic
77C24×C3C2×D411yescharacteristic
78C24×C3C2×D412yes—
79C24×C3#4112yes—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—24no—
85C22×C3×C4—42no—
86C22×C3×C4—81no—
87C22×C3×C4C22⋊C412yes—
88C22×C3×C4C22⋊C412yes—
89C22×Dic3—21no—
90C22×Dic3—22no—
91C22×Dic3—21no—
92C22×Dic3—22no—
93C22×Dic3—22no—
94C22×Dic3—42no—
95C22×Dic3—42no—
96C22×Dic3—81no—
97C3×C4⋊C4—82no—
98C3×C22⋊C4—24no—
99C3×C22⋊C4—24no—
100C3×C22⋊C4—44no—
101C3×C22⋊C4—44no—
102C3×C22⋊C4—82no—
103C3×C42—82no—
104#217—42no—
105#217—41no—
106#217—42no—
107#217—41no—
108#217—42no—
109#217—44no—
110#217—44no—
111#217—81no—
112#217—82no—
113Dic3⋊C4—82no—
114C25C22×D311yescharacteristic
115C23×C4—31no—
116C23×C4—31no—
117C23×C4#6711yescharacteristic
118C23×C4#6711yescharacteristic
119C2×C4⋊C4—22no—
120C2×C4⋊C4—62no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—32no—
126C2×C22⋊C4—61no—
127C2×C22⋊C4—62no—
128C2×C22⋊C4—62no—
129C2×C22⋊C4—61no—
130C2×C22⋊C4—61no—
131C2×C22⋊C4—62no—
132C2×C22⋊C4—62no—
133C2×C22⋊C4—61no—
134C2×C22⋊C4—62no—
135C2×C22⋊C4C4×D312yes—
136C2×C22⋊C4D1212yes—
137C2×C22⋊C4#6714yes—
138C2×C42—22no—
139#95—22no—
140#95—22no—
141#95—62no—
142#95—62no—
143#95—62no—
144C23×C3—22no—
145C23×C3—22no—
146C23×C3—22no—
147C23×C3—22no—
148C23×C3—24no—
149C23×C3—41no—
150C23×C3—41no—
151C23×C3—41no—
152C23×C3—41no—
153C23×C3—42no—
154C23×C3—42no—
155C23×C3—41no—
156C23×C3—42no—
157C23×C3—42no—
158C23×C3—42no—
159C23×C3—88no—
160C23×C3C2×C22⋊C411yescharacteristic
161C23×C3C4×D412yes—
162C23×C3#12011yescharacteristic
163C23×C3#12112yes—
164C23×C3#12311yescharacteristic
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4—44no—
172C2×C3×C4—44no—
173C2×C3×C4—44no—
174C2×C3×C4—81no—
175C2×C3×C4—81no—
176C2×Dic3—42no—
177C2×Dic3—42no—
178C2×Dic3—42no—
179C2×Dic3—41no—
180C2×Dic3—41no—
181C2×Dic3—42no—
182C2×Dic3—42no—
183C2×Dic3—82no—
184C2×Dic3—82no—
185C2×Dic3—82no—
186C2×Dic3—82no—
187C2×Dic3—82no—
188C24—42no—
189C24—42no—
190C24—81no—
191C24C2×C4×D311yescharacteristic
192C24C2×D1211yescharacteristic
193C24#23512yes—
194C24C2×#6711yescharacteristic
195C24C2×#6712yes—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—24no—
201C22×C4—42no—
202C22×C4—61no—
203C22×C4—62no—
204C22×C4—61no—
205C22×C4—62no—
206C22×C4—62no—
207C22×C4—81no—
208C22×C4—122no—
209C22×C4—122no—
210C22×C4—241no—
211C22×C4(C2×D3)⋊C412yes—
212C22×C4(C2×D3)⋊C412yes—
213C4⋊C4—82no—
214C4⋊C4—242no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—44no—
218C22⋊C4—44no—
219C22⋊C4—82no—
220C22⋊C4—122no—
221C22⋊C4—121no—
222C22⋊C4—122no—
223C22⋊C4—121no—
224C22⋊C4—122no—
225C22⋊C4—124no—
226C22⋊C4—124no—
227C22⋊C4—241no—
228C22⋊C4—242no—
229C42—82no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—21no—
234C22×C3—22no—
235C22×C3—21no—
236C22×C3—22no—
237C22×C3—22no—
238C22×C3—22no—
239C22×C3—21no—
240C22×C3—22no—
241C22×C3—22no—
242C22×C3—21no—
243C22×C3—44no—
244C22×C3—44no—
245C22×C3—82no—
246C22×C3—84no—
247C22×C3—84no—
248C22×C3#38611yescharacteristic
249C22×C3#41012yes—
250C22×C3#45812yes—
251C3×C4—42no—
252C3×C4—42no—
253C3×C4—84no—
254Dic3—82no—
255Dic3—81no—
256Dic3—81no—
257Dic3—162no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—24no—
262C23—22no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—42no—
267C23—41no—
268C23—42no—
269C23—41no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—88no—
274C23C2×(C2×D3)⋊C411yescharacteristic
275C23C4×#6712yes—
276C23#92911yescharacteristic
277C23#93012yes—
278C23#95311yescharacteristic
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—122no—
291C2×C4—122no—
292C2×C4—122no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—122no—
296C2×C4—122no—
297C2×C4—242no—
298C2×C4—242no—
299C2×C4—242no—
300C2×C4—242no—
301C2×C4—242no—
302C2×C3—22no—
303C2×C3—22no—
304C2×C3—42no—
305C2×C3—41no—
306C2×C3—42no—
307C2×C3#189011yescharacteristic
308C2×C3#189911yescharacteristic
309C2×C3#191311yescharacteristic
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—21no—
317C22—21no—
318C22—22no—
319C22—22no—
320C22—21no—
321C22—22no—
322C22—22no—
323C22—44no—
324C22—44no—
325C22—84no—
326C22—82no—
327C22—84no—
328C22#500612yes—
329C22#521311yescharacteristic
330C22#524312yes—
331C4—42no—
332C4—42no—
333C4—84no—
334C4—242no—
335C4—241no—
336C4—241no—
337C4—482no—
338C3#950011yescharacteristic, complemented, Sylow
339C2—22no—
340C2—22no—
341C2—42no—
342C2—42no—
343C2—41no—
344C2#7159511yescharacteristic
345C2#7165511yescharacteristic
346C2#7165611yescharacteristic
subgroupG/H
centerC22#5213
commutatorC24×C3C22×C4
FrattiniC25C22×D3
FittingC3×#2312C2
socleC22×C3#386
prime radical, cosocleC25×C3C23
hypercenter#2312D3
hypocenterC3#9500
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #2312

Complements

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

NH ≅ G/Ntype
18. #4946C4semidirect
19. #4943C4semidirect
338. C3#9500semidirect

Maximal subgroups

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

subgroupindex
C3×#23122
#821142
#715422
#694952
C3⋊#13042
#95003