Finite Group Databasebeta

← #10694034

Subgroups — #10694034

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×C3×#389C211yesmaximal, characteristic
2C2×#4528C214yesmaximal
3#71530C211yesmaximal, characteristic
4#71529C211yesmaximal, characteristic
5#9654—31nomaximal, Sylow
6C22×C3×C4⋊C4C411yescharacteristic
7C22×C3×C4⋊C4C411yescharacteristic
8C22×C3×C4⋊C4C2212yes—
9C22×C3×C42C2211yescharacteristic
10C3×#389—21no—
11C3×#389—21no—
12C3×#389—21no—
13C3×#389—21no—
14C2×C3×#95C412yes—
15C2×#830C2212yes—
16C2×#830C2212yes—
17C2×C4⋊(C3⋊C8)—21no—
18C2×#803—21no—
19#4528—28no—
20#4524—21no—
21#4524—21no—
22C2×#389D311yescharacteristic
23#1797—31no—
24#1749—31no—
25C2×#324—34no—
26C23×C3×C4C2×C411yescharacteristic
27C23×C3×C4C2×C412yes—
28C23×C3×C4C2×C411yescharacteristic
29C23×C3×C4C2×C412yes—
30C23×C3×C4C2311yescharacteristic
31C2×C3×C4⋊C4—21no—
32C2×C3×C4⋊C4—21no—
33C2×C3×C4⋊C4—24no—
34C2×C3×C4⋊C4—21no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4—24no—
39C2×C3×C4⋊C4—24no—
40C2×C3×C4⋊C4—42no—
41C2×C3×C42—42no—
42C2×C3×C42D411yescharacteristic
43C2×C3×C42D411yescharacteristic
44C2×C3×C42D411yescharacteristic
45C2×C3×C42D411yescharacteristic
46C22×C3⋊C8—22no—
47C22×C3⋊C8—22no—
48C3×#95—22no—
49C3×#95—22no—
50#830—22no—
51#830—24no—
52#830D414yes—
53C4⋊(C3⋊C8)—42no—
54#803—42no—
55C22×C4⋊C4Dic311yescharacteristic
56C22×C4⋊C4Dic311yescharacteristic
57C22×C4⋊C4C2×D312yes—
58C22×C42C2×D311yescharacteristic
59C2×C4⋊C8—61no—
60C2×C22⋊C8—32no—
61C2×C22⋊C8—32no—
62C2×#97—61no—
63#389—21no—
64#389—21no—
65#389—21no—
66#389—21no—
67C2×#95Dic312yes—
68#336—61no—
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—24no—
77C22×C3×C4—22no—
78C22×C3×C4—22no—
79C22×C3×C4—24no—
80C22×C3×C4—22no—
81C22×C3×C4—22no—
82C22×C3×C4—24no—
83C22×C3×C4—41no—
84C22×C3×C4—41no—
85C22×C3×C4—41no—
86C22×C3×C4—42no—
87C22×C3×C4—42no—
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4M1614yes—
93C22×C3×C4C2×D412yes—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4#4112yes—
97C3×C4⋊C4—24no—
98C3×C4⋊C4—24no—
99C3×C4⋊C4—42no—
100C3×C4⋊C4—42no—
101C3×C4⋊C4—44no—
102C3×C4⋊C4—42no—
103C3×C4⋊C4—42no—
104C3×C42—22no—
105C3×C42—22no—
106C3×C42—41no—
107C3×C42—41no—
108C2×C3⋊C8—24no—
109C2×C3⋊C8—42no—
110C2×C3⋊C8—42no—
111C2×C3⋊C8—42no—
112C2×C3⋊C8—42no—
113C2×C3⋊C8—42no—
114C23×C4C2×Dic312yes—
115C23×C4C2×Dic311yescharacteristic
116C23×C4C2×Dic311yescharacteristic
117C23×C4C2×Dic312yes—
118C23×C4C22×D311yescharacteristic
119C22×C8—62no—
120C22×C8—62no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—42no—
131C2×C42—42no—
132C2×C42#6711yescharacteristic
133C2×C42#6711yescharacteristic
134C2×C42#6711yescharacteristic
135C2×C42#6711yescharacteristic
136C4⋊C8—122no—
137C22⋊C8—34no—
138C22⋊C8—62no—
139C22⋊C8—64no—
140#97—122no—
141#95—22no—
142#95—22no—
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—24no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—28no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—44no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—44no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—42no—
176C2×C3×C4—84no—
177C2×C3×C4#9912yes—
178C2×C3×C4#10112yes—
179C2×C3×C4#10112yes—
180C2×C3×C4#12112yes—
181C2×C3×C4#12411yescharacteristic
182C2×C3×C4#12711yescharacteristic
183C3⋊C8—44no—
184C3⋊C8—82no—
185C24C22×Dic311yescharacteristic
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4#20814yes—
203C22×C4#21711yescharacteristic
204C22×C4#21711yescharacteristic
205C22×C4#21711yescharacteristic
206C22×C4#21711yescharacteristic
207C22×C4D3×D412yes—
208C22×C4#23712yes—
209C22×C4C2×#6711yescharacteristic
210C22×C4C2×#6711yescharacteristic
211C2×C8—64no—
212C2×C8—122no—
213C2×C8—122no—
214C2×C8—122no—
215C2×C8—122no—
216C2×C8—122no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—44no—
223C4⋊C4—42no—
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—22no—
243C3×C4—21no—
244C3×C4—21no—
245C3×C4—22no—
246C3×C4—22no—
247C3×C4—22no—
248C3×C4—44no—
249C3×C4—41no—
250C3×C4—82no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23C2×#20812yes—
256C23D4×Dic312yes—
257C23#94812yes—
258C23C2×#21711yescharacteristic
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—24no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—28no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—84no—
285C2×C4#83412yes—
286C2×C4#83612yes—
287C2×C4#83612yes—
288C2×C4#93611yescharacteristic
289C2×C4#93812yes—
290C2×C4#94011yescharacteristic
291C8—124no—
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#501114yes—
308C22#520711yescharacteristic
309C22C2×#83411yescharacteristic
310C22#522111yescharacteristic
311C22C2×#83611yescharacteristic
312C22C2×#83611yescharacteristic
313C22#523411yescharacteristic
314C22#524011yescharacteristic
315C4—22no—
316C4—21no—
317C4—21no—
318C4—22no—
319C4—22no—
320C4—22no—
321C4—41no—
322C4—44no—
323C4—82no—
324C3#965411yescharacteristic, complemented, Sylow
325C2—24no—
326C2#7155211yescharacteristic
327C2#7161511yescharacteristic
328C2#7162511yescharacteristic
329C2#7168611yescharacteristic
330C2#7169111yescharacteristic
331C2#7173711yescharacteristic
332C2#7178911yescharacteristic
subgroupG/H
centerC23C2×#217
commutator, prime radicalC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#389C2
socleC23×C3C2×C22⋊C4
cosocleC23×C3×C4C23
hypercenterC2×#389D3
hypocenterC3#9654
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#389

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×C3×#3892
C2×#45282
#715302
#715292
#96543