Finite Group Databasebeta

← #10693017

Subgroups — #10693017

Order 768.

614 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2482C211yesmaximal, characteristic
2#82093C211yesmaximal, characteristic
3#71590C212yesmaximal, complemented
4#71539C211yesmaximal, characteristic
5#67628C212yesmaximal, complemented
6#9481—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C22×C3×C22⋊C4C2211yescharacteristic
9C3×#399C412yescomplemented
10C3×#397C412yescomplemented
11C3×#388—22no—
12C3×#385—21no—
13C3×#381C2211yescharacteristic
14C3×#379—21no—
15#5247—21no—
16#5212—21no—
17C4×#217C2212yes—
18#5207C2212yes—
19#4951—24no—
20#4532—24no—
21#4525—22no—
22#2482D311yescharacteristic
23#2312—31no—
24#1826—32no—
25#1462—31no—
26#1284—32no—
27C25×C3D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C3×#117—22no—
31C3×#117—22no—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4C2×C411yescharacteristic
34C2×C3×C4⋊C4C2×C411yescharacteristic
35C2×C3×C22⋊C4—22no—
36C2×C3×C22⋊C4—22no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4—21no—
39C2×C3×C22⋊C4—21no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4C2×C412yes—
42C2×C3×C22⋊C4D412yes—
43C2×C3×C42—21no—
44C2×#217—22no—
45C2×#217—22no—
46C2×#217—22no—
47C2×C4⋊Dic3—22no—
48C2×C4×Dic3D414yes—
49C2×#208—22no—
50C3×#95—22no—
51C3×#95—21no—
52C3×#95C2×C412yes—
53C3⋊#95—22no—
54C3⋊#95—22no—
55#830—22no—
56#830—22no—
57C2×#117C2×D311yescharacteristic
58C22×C22⋊C4C2×D311yescharacteristic
59#407—32no—
60#399Dic312yescomplemented
61#397Dic312yescomplemented
62#388—22no—
63#388—64no—
64#385—21no—
65#381C2×D311yescharacteristic
66#380—61no—
67#379—21no—
68#379—61no—
69C4×C22⋊C4—32no—
70#339—64no—
71#323—62no—
72C24×C3—21no—
73C24×C3—21no—
74C24×C3—42no—
75C24×C3—42no—
76C24×C3—81no—
77C24×C3C22⋊C412yes—
78C24×C3C2×D411yescharacteristic
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4—22no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—42no—
85C22×C3×C4—81no—
86C22×C3×C4C22⋊C412yes—
87C22×C3×C4C22×C411yescharacteristic
88C22×C3×C4C2×D412yes—
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4#4112yes—
91C22×Dic3—24no—
92C3×C4⋊C4—22no—
93C3×C4⋊C4—22no—
94C3×C4⋊C4—82no—
95C3×C22⋊C4—24no—
96C3×C22⋊C4—24no—
97C3×C22⋊C4—42no—
98C3×C22⋊C4—44no—
99C3×C22⋊C4—41no—
100C3×C22⋊C4—41no—
101C3×C22⋊C4—82no—
102C3×C42—42no—
103#217—44no—
104#217—82no—
105#217—82no—
106C4⋊Dic3—82no—
107C4×Dic3—44no—
108#208—44no—
109C2×C3⋊C8—42no—
110C25#6711yescharacteristic
111C23×C4#6711yescharacteristic
112C23×C4C22×D311yescharacteristic
113C2×M16—62no—
114#117—22no—
115#117—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—62no—
118C2×C4⋊C4C2×Dic311yescharacteristic
119C2×C4⋊C4C2×Dic311yescharacteristic
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—62no—
127C2×C22⋊C4—62no—
128C2×C22⋊C4—62no—
129C2×C22⋊C4C2×Dic312yes—
130C2×C22⋊C4#6712yes—
131C2×C42—21no—
132C2×C42—34no—
133C22⋊C8—62no—
134C22⋊C8—62no—
135#95—22no—
136#95—21no—
137#95—62no—
138#95—62no—
139#95C2×Dic312yes—
140C23×C3—22no—
141C23×C3—22no—
142C23×C3—22no—
143C23×C3—22no—
144C23×C3—22no—
145C23×C3—41no—
146C23×C3—42no—
147C23×C3—42no—
148C23×C3—42no—
149C23×C3—42no—
150C23×C3—42no—
151C23×C3—42no—
152C23×C3—81no—
153C23×C3—81no—
154C23×C3—88no—
155C23×C3C2×C22⋊C411yescharacteristic
156C23×C3#12011yescharacteristic
157C23×C3#12311yescharacteristic
158C2×C3×C4—24no—
159C2×C3×C4—22no—
160C2×C3×C4—41no—
161C2×C3×C4—41no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—42no—
168C2×C3×C4—41no—
169C2×C3×C4—41no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—81no—
173C2×C3×C4C4×D412yes—
174C2×C3×C4#12112yes—
175C2×Dic3—44no—
176C2×Dic3—44no—
177C2×Dic3—44no—
178C3⋊C8—82no—
179C24—21no—
180C24—21no—
181C24—42no—
182C24—42no—
183C24—81no—
184C24#21712yes—
185C24C2×#6711yescharacteristic
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—42no—
192C22×C4—64no—
193C22×C4—81no—
194C22×C4#21712yes—
195C22×C4D3×D412yes—
196C22×C4#23712yes—
197C22×C4C22×Dic311yescharacteristic
198C22×C4C2×#6711yescharacteristic
199M16—124no—
200C2×C8—122no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—82no—
204C4⋊C4—242no—
205C22⋊C4—24no—
206C22⋊C4—24no—
207C22⋊C4—42no—
208C22⋊C4—44no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—82no—
212C22⋊C4—124no—
213C22⋊C4—242no—
214C22⋊C4—242no—
215C42—42no—
216C42—124no—
217C22×C3—21no—
218C22×C3—21no—
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—44no—
225C22×C3—44no—
226C22×C3—42no—
227C22×C3—42no—
228C22×C3—42no—
229C22×C3—82no—
230C22×C3—84no—
231C22×C3—84no—
232C22×C3#38611yescharacteristic
233C22×C3#42112yes—
234C22×C3#45812yes—
235C3×C4—22no—
236C3×C4—22no—
237C3×C4—42no—
238C3×C4—82no—
239C3×C4—81no—
240C3×C4—81no—
241Dic3—84no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—41no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—81no—
255C23—81no—
256C23—88no—
257C23#93511yescharacteristic
258C23#93711yescharacteristic
259C23C2×#21711yescharacteristic
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—81no—
275C2×C4—124no—
276C2×C4—124no—
277C2×C4—124no—
278C2×C4D4×Dic312yes—
279C2×C4#93912yes—
280C8—242no—
281C2×C3—22no—
282C2×C3—22no—
283C2×C3—41no—
284C2×C3—82no—
285C2×C3#189011yescharacteristic
286C2×C3#189711yescharacteristic
287C2×C3#191111yescharacteristic
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—44no—
296C22—44no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—82no—
301C22—84no—
302C22—84no—
303C22#505312yes—
304C22#522011yescharacteristic
305C22#523612yes—
306C4—22no—
307C4—22no—
308C4—42no—
309C4—82no—
310C4—81no—
311C4—81no—
312C4—244no—
313C3#948111yescharacteristic, complemented, Sylow
314C2—22no—
315C2—22no—
316C2—41no—
317C2—82no—
318C2#7170411yescharacteristic
319C2#7171211yescharacteristic
320C2#7185311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2482C2
socleC22×C3#386
prime radicalC25×C3D4
cosocleC23×C3×C4C23
hypercenter#2482D3
hypocenterC3#9481
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2482

Complements

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

NH ≅ G/Ntype
3. #71590C2semidirect
5. #67628C2semidirect
9. C3×#399C4semidirect
10. C3×#397C4semidirect
60. #399Dic3semidirect
61. #397Dic3semidirect
313. C3#9481semidirect

Maximal subgroups

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

subgroupindex
C3×#24822
#820932
#715902
#715392
#676282
#94813