Finite Group Databasebeta

← #10694033

Subgroups — #10694033

Order 768.

640 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4928C211yesmaximal, characteristic
2C3×#1749C211yesmaximal, characteristic
3#71530C211yesmaximal, characteristic
4C2×#4469C214yesmaximal
5#9654—31nomaximal, Sylow
6C22×C3×C42C2211yescharacteristic
7C22×C4⋊Dic3C411yescharacteristic
8C22×C4⋊Dic3C411yescharacteristic
9C22×Dic3⋊C4C2212yes—
10C2×C3×C22⋊C8C2212yes—
11C2×C3×#97—21no—
12C2×C3⋊#95C412yes—
13C2×#830C2212yes—
14#4928—21no—
15#4928—21no—
16#4928—21no—
17#4928—21no—
18C2×C4⋊(C3⋊C8)—21no—
19C3×#336—21no—
20#4524—21no—
21#4469—28no—
22C2×#389—31no—
23#1797—31no—
24#1749D311yescharacteristic
25C2×#324—34no—
26C23×C3×C4C2×C411yescharacteristic
27C23×C3×C4C2×C411yescharacteristic
28C23×C3×C4C2311yescharacteristic
29C23×Dic3C2×C412yes—
30C23×Dic3C2×C412yes—
31C22×C3×C8—22no—
32C2×C3×C42—42no—
33C2×C3×C42D411yescharacteristic
34C2×C3×C42D411yescharacteristic
35C2×C3×C42D411yescharacteristic
36C2×C3×C42D411yescharacteristic
37C2×C4⋊Dic3—21no—
38C2×C4⋊Dic3—21no—
39C2×C4⋊Dic3—24no—
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3—42no—
43C2×Dic3⋊C4—22no—
44C2×Dic3⋊C4—22no—
45C2×Dic3⋊C4—24no—
46C2×Dic3⋊C4—24no—
47C22×C3⋊C8—22no—
48C3×C22⋊C8—22no—
49C3×C22⋊C8D414yes—
50C3×#97—42no—
51C3⋊#95—22no—
52C3⋊#95—22no—
53#830—24no—
54C4⋊(C3⋊C8)—42no—
55C22×C4⋊C4—31no—
56C22×C4⋊C4—32no—
57C22×C4⋊C4—31no—
58C22×C42C2×D311yescharacteristic
59C2×C4⋊C8—61no—
60C2×C22⋊C8—32no—
61C2×C22⋊C8C2×D312yes—
62C2×#97—21no—
63#389—61no—
64#389—61no—
65#389—61no—
66#389—61no—
67C2×#95—32no—
68#336—21no—
69#336—61no—
70#324—68no—
71C24×C3C22×C411yescharacteristic
72C22×C3×C4—22no—
73C22×C3×C4—22no—
74C22×C3×C4—22no—
75C22×C3×C4—22no—
76C22×C3×C4—22no—
77C22×C3×C4—41no—
78C22×C3×C4—41no—
79C22×C3×C4—41no—
80C22×C3×C4C22⋊C411yescharacteristic
81C22×C3×C4C22⋊C411yescharacteristic
82C22×C3×C4C22⋊C411yescharacteristic
83C22×C3×C4C22⋊C411yescharacteristic
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4C2×D412yes—
86C22×C3×C4C2×D411yescharacteristic
87C22×C3×C4#4112yes—
88C22×Dic3—22no—
89C22×Dic3—24no—
90C22×Dic3—22no—
91C22×Dic3—24no—
92C22×Dic3—22no—
93C22×Dic3—24no—
94C22×Dic3—42no—
95C22×Dic3—42no—
96C22×Dic3M1614yes—
97C2×C3×C8—24no—
98C2×C3×C8—42no—
99C2×C3×C8—42no—
100C3×C42—22no—
101C3×C42—22no—
102C3×C42—41no—
103C3×C42—41no—
104C4⋊Dic3—24no—
105C4⋊Dic3—24no—
106C4⋊Dic3—42no—
107C4⋊Dic3—42no—
108Dic3⋊C4—44no—
109Dic3⋊C4—42no—
110Dic3⋊C4—42no—
111C2×C3⋊C8—42no—
112C2×C3⋊C8—42no—
113C2×C3⋊C8—42no—
114C23×C4—32no—
115C23×C4—32no—
116C23×C4C4×D311yescharacteristic
117C23×C4C4×D311yescharacteristic
118C23×C4C22×D311yescharacteristic
119C22×C8—22no—
120C22×C8—62no—
121C2×C4⋊C4—64no—
122C2×C4⋊C4—61no—
123C2×C4⋊C4—61no—
124C2×C4⋊C4—61no—
125C2×C4⋊C4—61no—
126C2×C4⋊C4—64no—
127C2×C4⋊C4—62no—
128C2×C4⋊C4—64no—
129C2×C4⋊C4—62no—
130C2×C4⋊C4—122no—
131C2×C42—42no—
132C2×C42D1211yescharacteristic
133C2×C42D1211yescharacteristic
134C2×C42#6711yescharacteristic
135C2×C42#6711yescharacteristic
136C4⋊C8—122no—
137C22⋊C8—22no—
138C22⋊C8—64no—
139C22⋊C8D1214yes—
140#97—42no—
141#95—62no—
142#95—62no—
143C23×C3—21no—
144C23×C3—21no—
145C23×C3—21no—
146C23×C3—21no—
147C23×C3C2×C22⋊C411yescharacteristic
148C23×C3C4×D412yes—
149C23×C3C2×M1612yes—
150C23×C3#13112yes—
151C2×C3×C4—28no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—44no—
168C2×C3×C4#9912yes—
169C2×C3×C4#10112yes—
170C2×C3×C4#10112yes—
171C2×C3×C4#12112yes—
172C2×C3×C4#12411yescharacteristic
173C2×C3×C4#12711yescharacteristic
174C2×Dic3—24no—
175C2×Dic3—28no—
176C2×Dic3—42no—
177C2×Dic3—42no—
178C2×Dic3—44no—
179C2×Dic3—42no—
180C2×Dic3—42no—
181C2×Dic3—42no—
182C2×Dic3—84no—
183C3×C8—44no—
184C3⋊C8—82no—
185C24C2×C4×D311yescharacteristic
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—34no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—64no—
196C22×C4—62no—
197C22×C4—64no—
198C22×C4—62no—
199C22×C4—62no—
200C22×C4—64no—
201C22×C4—122no—
202C22×C4—122no—
203C22×C4(C2×D3)⋊C411yescharacteristic
204C22×C4(C2×D3)⋊C411yescharacteristic
205C22×C4(C2×D3)⋊C411yescharacteristic
206C22×C4(C2×D3)⋊C411yescharacteristic
207C22×C4C2×D1211yescharacteristic
208C22×C4C2×D1212yes—
209C22×C4#23512yes—
210C22×C4C2×#6711yescharacteristic
211C2×C8—24no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—122no—
215C2×C8—122no—
216C2×C8—122no—
217C4⋊C4—64no—
218C4⋊C4—64no—
219C4⋊C4—122no—
220C4⋊C4—122no—
221C4⋊C4—122no—
222C4⋊C4—122no—
223C4⋊C4—124no—
224C42—22no—
225C42—22no—
226C42—41no—
227C42—41no—
228C22×C3—22no—
229C22×C3—22no—
230C22×C3—22no—
231C22×C3—22no—
232C22×C3—22no—
233C22×C3—22no—
234C22×C3#39011yescharacteristic
235C22×C3#40711yescharacteristic
236C22×C3#40811yescharacteristic
237C22×C3C2×#9911yescharacteristic
238C22×C3#41011yescharacteristic
239C22×C3C2×#10111yescharacteristic
240C22×C3C2×#10111yescharacteristic
241C22×C3#43514yes—
242C3×C4—21no—
243C3×C4—22no—
244C3×C4—21no—
245C3×C4—22no—
246C3×C4—22no—
247C3×C4—22no—
248C3×C4—41no—
249Dic3—44no—
250Dic3—82no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23C4×D1212yes—
256C23D3×M1612yes—
257C23#90712yes—
258C23C2×(C2×D3)⋊C411yescharacteristic
259C2×C4—28no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—64no—
277C2×C4—68no—
278C2×C4—122no—
279C2×C4—122no—
280C2×C4—124no—
281C2×C4—122no—
282C2×C4—122no—
283C2×C4—122no—
284C2×C4—244no—
285C2×C4#80612yes—
286C2×C4#82412yes—
287C2×C4#82412yes—
288C2×C4#87411yescharacteristic
289C2×C4#87511yescharacteristic
290C2×C4#93012yes—
291C8—44no—
292C8—242no—
293C2×C3—24no—
294C2×C3#146311yescharacteristic
295C2×C3#146711yescharacteristic
296C2×C3#147011yescharacteristic
297C2×C3#147211yescharacteristic
298C2×C3#195511yescharacteristic
299C2×C3#196411yescharacteristic
300C2×C3#196611yescharacteristic
301C22—22no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22#470614yes—
308C22#493911yescharacteristic
309C22C2×#80611yescharacteristic
310C22#500611yescharacteristic
311C22#512611yescharacteristic
312C22#512911yescharacteristic
313C22C2×#82411yescharacteristic
314C22C2×#82411yescharacteristic
315C4—22no—
316C4—21no—
317C4—21no—
318C4—22no—
319C4—22no—
320C4—22no—
321C4—41no—
322C4—124no—
323C4—242no—
324C3#965411yescharacteristic, complemented, Sylow
325C2—24no—
326C2#6945611yescharacteristic
327C2#6983711yescharacteristic
328C2#6987711yescharacteristic
329C2#6997111yescharacteristic
330C2#6997211yescharacteristic
331C2#7083011yescharacteristic
332C2#7083111yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutator, prime radicalC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1749C2
socleC23×C3C2×C22⋊C4
cosocleC23×C3×C4C23
hypercenter#1749D3
hypocenterC3#9654
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
324. C3#9654semidirect

Maximal subgroups

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

subgroupindex
C2×#49282
C3×#17492
#715302
C2×#44692
#96543