Finite Group Databasebeta

← #10804130

Subgroups — #10804130

Order 768.

1570 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#123C211yesmaximal, characteristic
2#84027C212yesmaximal, complemented
3#72130C212yesmaximal
4#71705C218yesmaximal, complemented
5#71542C212yesmaximal, complemented
6#21747—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C24×C3×C4C2211yescharacteristic
9C2×C3×#123—24no—
10C2×C3×#123—24no—
11C2×C3×#123C414yescomplemented
12C2×C3×#123C2214yes—
13C22×C3×C4⋊C4C412yescomplemented
14C22×C3×C22⋊C4C412yescomplemented
15C22×C3×C22⋊C4C2211yescharacteristic
16C2×#926C2214yescomplemented
17#5247C2214yes—
18C2×#834C2218yes—
19#5212—24no—
20#5211C2214yescomplemented
21C4×#217—24no—
22#4526C2218yescomplemented
23C22×#123D311yescharacteristic
24#2280—32no—
25#1899—38no—
26#1819—32no—
27#1781—32no—
28C25×C3C2×C411yescharacteristic
29C25×C3C2311yescharacteristic
30C22×C3×D4—24no—
31C22×C3×D4—44no—
32C22×C3×D4C2×C412yes—
33C22×C3×D4C2312yes—
34C23×C3×C4—44no—
35C23×C3×C4C2×C411yescharacteristic
36C23×C3×C4C2×C412yes—
37C23×C3×C4C2×C412yes—
38C23×C3×C4C2×C412yes—
39C23×C3×C4D414yes—
40C23×C3×C4D414yes—
41C23×C3×C4C2312yes—
42C3×#123—24no—
43C3×#123—28no—
44C3×#123—24no—
45C3×#123—28no—
46C3×#123—24no—
47C3×#123—24no—
48C2×C3×C4⋊C4—24no—
49C2×C3×C4⋊C4—22no—
50C2×C3×C4⋊C4—22no—
51C2×C3×C4⋊C4C2×C418yes—
52C2×C3×C22⋊C4—24no—
53C2×C3×C22⋊C4—22no—
54C2×C3×C22⋊C4—22no—
55C2×C3×C22⋊C4—24no—
56C2×C3×C22⋊C4C2×C418yes—
57C2×C3×C22⋊C4C2×C412yes—
58C2×C3×C22⋊C4C2312yes—
59C2×#217—28no—
60C2×#217C2318yes—
61#926—48no—
62C2×C4⋊Dic3—24no—
63C2×C4×Dic3—24no—
64#834—48no—
65C3⋊#95—28no—
66C23×D4C2×D311yescharacteristic
67C24×C4C2×D311yescharacteristic
68C2×#123—24no—
69C2×#123—24no—
70C2×#123Dic314yescomplemented
71C2×#123C2×D314yes—
72C2×#117—34no—
73C22×C4⋊C4Dic312yescomplemented
74C22×C22⋊C4Dic312yescomplemented
75C22×C22⋊C4C2×D311yescharacteristic
76C2×#99—38no—
77#381—34no—
78#380—64no—
79#379—34no—
80C4×C22⋊C4—64no—
81#342—38no—
82C24×C3—24no—
83C24×C3—22no—
84C24×C3—44no—
85C24×C3—42no—
86C24×C3—44no—
87C24×C3C22×C411yescharacteristic
88C24×C3C22×C412yes—
89C24×C3C22×C411yescharacteristic
90C24×C3C2×D414yes—
91C24×C3C2411yescharacteristic
92C2×C3×D4—24no—
93C2×C3×D4—24no—
94C2×C3×D4—28no—
95C2×C3×D4—44no—
96C2×C3×D4—416no—
97C22×C3×C4—28no—
98C22×C3×C4—22no—
99C22×C3×C4—22no—
100C22×C3×C4—22no—
101C22×C3×C4—44no—
102C22×C3×C4—44no—
103C22×C3×C4—48no—
104C22×C3×C4—48no—
105C22×C3×C4—42no—
106C22×C3×C4—44no—
107C22×C3×C4C22⋊C412yes—
108C22×C3×C4C22⋊C414yes—
109C22×C3×C4C22⋊C412yes—
110C22×C3×C4C22⋊C412yes—
111C22×C3×C4C22⋊C414yes—
112C22×C3×C4C22⋊C412yes—
113C22×C3×C4C22×C412yes—
114C22×C3×C4C22×C414yes—
115C22×C3×C4C22×C414yes—
116C22×C3×C4C2×D414yes—
117C22×C3×C4C2×D412yes—
118C22×C3×C4C2×D412yes—
119C22×Dic3—28no—
120C3×C4⋊C4—28no—
121C3×C4⋊C4—28no—
122C3×C22⋊C4—24no—
123C3×C22⋊C4—24no—
124C3×C22⋊C4—28no—
125C3×C22⋊C4—28no—
126#217—216no—
127#217—416no—
128C4⋊Dic3—48no—
129C4×Dic3—48no—
130C25C2×Dic311yescharacteristic
131C25C22×D311yescharacteristic
132C22×D4—24no—
133C22×D4—44no—
134C22×D4C2×Dic312yes—
135C22×D4C22×D312yes—
136C23×C4—44no—
137C23×C4C2×Dic312yes—
138C23×C4C2×Dic311yescharacteristic
139C23×C4C2×Dic312yes—
140C23×C4C2×Dic312yes—
141C23×C4#6714yes—
142C23×C4#6714yes—
143C23×C4C22×D312yes—
144#123—24no—
145#123—28no—
146#123—24no—
147#123—28no—
148#123—24no—
149#123—24no—
150#117—128no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—64no—
155C2×C4⋊C4C2×Dic318yes—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—38no—
161C2×C22⋊C4—68no—
162C2×C22⋊C4C2×Dic318yes—
163C2×C22⋊C4C2×Dic312yes—
164C2×C22⋊C4C22×D312yes—
165C2×C42—64no—
166#99—128no—
167#95—68no—
168C23×C3—22no—
169C23×C3—24no—
170C23×C3—22no—
171C23×C3—28no—
172C23×C3—28no—
173C23×C3—22no—
174C23×C3—24no—
175C23×C3—42no—
176C23×C3—44no—
177C23×C3—42no—
178C23×C3—416no—
179C23×C3—44no—
180C23×C3—44no—
181C23×C3—44no—
182C23×C3—44no—
183C23×C3—88no—
184C23×C3C2×C22⋊C412yes—
185C23×C3C2×C22⋊C412yes—
186C23×C3C23×C411yescharacteristic
187C23×C3C22×D412yes—
188C23×C3H(5,𝔽2)14yes—
189C3×D4—48no—
190C3×D4—48no—
191C2×C3×C4—28no—
192C2×C3×C4—24no—
193C2×C3×C4—24no—
194C2×C3×C4—28no—
195C2×C3×C4—24no—
196C2×C3×C4—22no—
197C2×C3×C4—24no—
198C2×C3×C4—42no—
199C2×C3×C4—44no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—44no—
203C2×C3×C4—42no—
204C2×C3×C4—44no—
205C2×C3×C4—44no—
206C2×C3×C4—44no—
207C2×C3×C4—44no—
208C2×C3×C4C2×C22⋊C414yes—
209C2×C3×C4C2×C22⋊C414yes—
210C2×Dic3—416no—
211C2×Dic3—48no—
212C24—24no—
213C24—22no—
214C24—44no—
215C24—42no—
216C24—44no—
217C24C22×Dic311yescharacteristic
218C24C22×Dic312yes—
219C24C22×Dic311yescharacteristic
220C24C2×#6714yes—
221C24C23×D311yescharacteristic
222C2×D4—24no—
223C2×D4—28no—
224C2×D4—24no—
225C2×D4—44no—
226C2×D4—416no—
227C22×C4—28no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—48no—
234C22×C4—48no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—68no—
238C22×C4#21712yes—
239C22×C4#21714yes—
240C22×C4#21712yes—
241C22×C4#21712yes—
242C22×C4#21714yes—
243C22×C4#21712yes—
244C22×C4C22×Dic312yes—
245C22×C4C22×Dic314yes—
246C22×C4C22×Dic314yes—
247C22×C4C2×#6714yes—
248C22×C4C2×#6712yes—
249C22×C4C2×#6712yes—
250C4⋊C4—28no—
251C4⋊C4—28no—
252C4⋊C4—128no—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—28no—
256C22⋊C4—28no—
257C22⋊C4—616no—
258C22⋊C4—1216no—
259C42—128no—
260C22×C3—28no—
261C22×C3—22no—
262C22×C3—24no—
263C22×C3—22no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—24no—
267C22×C3—44no—
268C22×C3—416no—
269C22×C3—44no—
270C22×C3—42no—
271C22×C3—44no—
272C22×C3—84no—
273C22×C3—88no—
274C22×C3#41014yes—
275C22×C3C22×C22⋊C411yescharacteristic
276C22×C3#51812yes—
277C22×C3#53414yes—
278C3×C4—24no—
279C3×C4—24no—
280C3×C4—42no—
281C3×C4—44no—
282Dic3—88no—
283C23—24no—
284C23—22no—
285C23—28no—
286C23—22no—
287C23—28no—
288C23—22no—
289C23—24no—
290C23—42no—
291C23—44no—
292C23—42no—
293C23—416no—
294C23—44no—
295C23—44no—
296C23—44no—
297C23—44no—
298C23—88no—
299C23C2×#21712yes—
300C23C2×#21712yes—
301C23#100414yes—
302C23C23×Dic311yescharacteristic
303C23C22×#6712yes—
304D4—48no—
305D4—48no—
306C2×C4—28no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—28no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—22no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—1216no—
324C2×C4—128no—
325C2×C4C2×#21714yes—
326C2×C4C2×#21714yes—
327C2×C3—24no—
328C2×C3—22no—
329C2×C3—24no—
330C2×C3—42no—
331C2×C3—82no—
332C2×C3#230811yescharacteristic
333C2×C3C2×#41012yes—
334C22—28no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—24no—
339C22—22no—
340C22—24no—
341C22—44no—
342C22—416no—
343C22—44no—
344C22—42no—
345C22—44no—
346C22—84no—
347C22—88no—
348C22#524014yes—
349C22#579812yes—
350C22#580414yes—
351C22C22×#21711yescharacteristic
352C4—24no—
353C4—24no—
354C4—42no—
355C4—44no—
356C4—248no—
357C3#2174711yescharacteristic, complemented, Sylow
358C2—24no—
359C2—24no—
360C2—22no—
361C2—42no—
362C2—82no—
363C2#8404311yescharacteristic
364C2C2×#524012yes—
subgroupG/H
centerC22C22×#217
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC22×C3×#123C2
socleC22×C3C22×C22⋊C4
prime radicalC23×C3×D4C22
cosocleC24×C3C24
hypercenterC22×#123D3
hypocenterC3#21747
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24×C4 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
2. #84027C2semidirect
4. #71705C2semidirect
5. #71542C2semidirect
11. C2×C3×#123C4semidirect
13. C22×C3×C4⋊C4C4semidirect
14. C22×C3×C22⋊C4C4semidirect
16. C2×#926C22semidirect
20. #5211C22semidirect
22. #4526C22semidirect
70. C2×#123Dic3semidirect
73. C22×C4⋊C4Dic3semidirect
74. C22×C22⋊C4Dic3semidirect
357. C3#21747semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1232
#840272
#721302
#717052
#715422
#217473