Finite Group Databasebeta

← #10804103

Subgroups — #10804103

Order 768.

1802 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D4C211yesmaximal, characteristic
2C2×#5207C212yesmaximal, complemented
3#72126C212yesmaximal, complemented
4#71611C218yesmaximal, complemented
5#71531C212yesmaximal, complemented
6#21742—31nomaximal, Sylow
7C23×C3×D4C411yescharacteristic
8C24×C3×C4C2211yescharacteristic
9C24×C3×C4C2211yescharacteristic
10C2×C3×C4×D4—28no—
11C2×C3×C4×D4C414yes—
12C2×C3×C4×D4C2214yes—
13C22×C3×C4⋊C4C411yescharacteristic
14C22×C3×C22⋊C4C411yescharacteristic
15C22×C3×C22⋊C4C411yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×#208C2214yescomplemented
18#5207—28no—
19C2×#830C2214yes—
20C2×#830C2214yescomplemented
21C2×#830C2218yescomplemented
22#4524C2218yescomplemented
23C22×C4×D4D311yescharacteristic
24C2×#407—32no—
25#1870—38no—
26#1817—32no—
27#1779—32no—
28C25×C3C2×C411yescharacteristic
29C25×C3C2×C411yescharacteristic
30C22×C3×D4—24no—
31C22×C3×D4—24no—
32C22×C3×D4—24no—
33C22×C3×D4C2×C414yes—
34C23×C3×C4—24no—
35C23×C3×C4—44no—
36C23×C3×C4—44no—
37C23×C3×C4C2×C411yescharacteristic
38C23×C3×C4C2×C411yescharacteristic
39C23×C3×C4C2×C412yes—
40C23×C3×C4C2×C411yescharacteristic
41C23×C3×C4C2×C411yescharacteristic
42C23×C3×C4C2×C412yes—
43C23×C3×C4D418yes—
44C23×C3×C4C2312yes—
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2312yes—
47C3×C4×D4—216no—
48C3×C4×D4—28no—
49C3×C4×D4—28no—
50C2×C3×C4⋊C4—24no—
51C2×C3×C4⋊C4C2×C414yes—
52C2×C3×C22⋊C4—24no—
53C2×C3×C22⋊C4—24no—
54C2×C3×C22⋊C4C2×C414yes—
55C2×C3×C22⋊C4C2×C414yes—
56C2×C3×C42—24no—
57C2×C3×C42C2×C412yes—
58C2×C3×C42C2312yes—
59C2×#208—28no—
60C2×#208—48no—
61C22×C3⋊C8C2318yes—
62#830—28no—
63#830—28no—
64#830—48no—
65C23×D4Dic311yescharacteristic
66C24×C4C2×D311yescharacteristic
67C24×C4C2×D311yescharacteristic
68C22×M16—34no—
69C2×C4×D4—28no—
70C2×C4×D4Dic314yes—
71C2×C4×D4C2×D314yes—
72C22×C4⋊C4Dic311yescharacteristic
73C22×C22⋊C4Dic311yescharacteristic
74C22×C22⋊C4Dic311yescharacteristic
75C22×C42C2×D311yescharacteristic
76#407—68no—
77C2×C22⋊C8—34no—
78C2×C22⋊C8—34no—
79C2×C22⋊C8—38no—
80#336—38no—
81C24×C3—22no—
82C24×C3—22no—
83C24×C3—22no—
84C24×C3—22no—
85C24×C3—42no—
86C24×C3—42no—
87C24×C3—42no—
88C24×C3—42no—
89C24×C3C22⋊C414yes—
90C24×C3M1614yescomplemented
91C24×C3C22×C412yes—
92C24×C3C22×C412yes—
93C24×C3C22×C411yescharacteristic
94C2×C3×D4—232no—
95C2×C3×D4—28no—
96C2×C3×D4—28no—
97C2×C3×D4—28no—
98C22×C3×C4—22no—
99C22×C3×C4—22no—
100C22×C3×C4—28no—
101C22×C3×C4—24no—
102C22×C3×C4—22no—
103C22×C3×C4—22no—
104C22×C3×C4—24no—
105C22×C3×C4—24no—
106C22×C3×C4—28no—
107C22×C3×C4—416no—
108C22×C3×C4—44no—
109C22×C3×C4—44no—
110C22×C3×C4—42no—
111C22×C3×C4—42no—
112C22×C3×C4—42no—
113C22×C3×C4—416no—
114C22×C3×C4C22⋊C414yes—
115C22×C3×C4C22⋊C418yes—
116C22×C3×C4M1614yescomplemented
117C22×C3×C4C22×C412yes—
118C22×C3×C4C22×C412yes—
119C22×C3×C4C22×C411yescharacteristic
120C22×C3×C4C22×C412yes—
121C22×C3×C4C22×C412yes—
122C22×C3×C4C2×D414yes—
123C22×C3×C4C2×D414yes—
124C22×C3×C4C2×D414yes—
125C22×C3×C4C2411yescharacteristic
126C3×C4⋊C4—28no—
127C3×C22⋊C4—28no—
128C3×C22⋊C4—28no—
129C3×C42—24no—
130C3×C42—24no—
131#208—416no—
132C2×C3⋊C8—28no—
133C2×C3⋊C8—216no—
134C25C2×Dic311yescharacteristic
135C25C2×Dic311yescharacteristic
136C22×D4—24no—
137C22×D4—24no—
138C22×D4—24no—
139C22×D4C2×Dic314yes—
140C23×C4—24no—
141C23×C4—44no—
142C23×C4—44no—
143C23×C4C2×Dic311yescharacteristic
144C23×C4C2×Dic311yescharacteristic
145C23×C4C2×Dic311yescharacteristic
146C23×C4C2×Dic312yes—
147C23×C4C2×Dic311yescharacteristic
148C23×C4C2×Dic312yes—
149C23×C4#6718yes—
150C23×C4C22×D312yes—
151C23×C4C22×D312yes—
152C23×C4C22×D311yescharacteristic
153C2×M16—68no—
154C2×M16—128no—
155C22×C8—38no—
156C4×D4—216no—
157C4×D4—28no—
158C4×D4—28no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4C2×Dic314yes—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4C2×Dic314yes—
164C2×C22⋊C4C2×Dic314yes—
165C2×C42—24no—
166C2×C42C2×Dic312yes—
167C2×C42C22×D312yes—
168C22⋊C8—68no—
169C22⋊C8—68no—
170C22⋊C8—128no—
171C23×C3—22no—
172C23×C3—22no—
173C23×C3—22no—
174C23×C3—24no—
175C23×C3—24no—
176C23×C3—22no—
177C23×C3—22no—
178C23×C3—22no—
179C23×C3—24no—
180C23×C3—24no—
181C23×C3—28no—
182C23×C3—28no—
183C23×C3—44no—
184C23×C3—44no—
185C23×C3—48no—
186C23×C3—44no—
187C23×C3—44no—
188C23×C3—48no—
189C23×C3—48no—
190C23×C3—48no—
191C23×C3C2×C22⋊C414yes—
192C23×C3C2×C22⋊C412yes—
193C23×C3C2×M1614yes—
194C23×C3C2×M1612yes—
195C23×C3C2×M1612yes—
196C23×C3C23×C411yescharacteristic
197C3×D4—232no—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—24no—
201C2×C3×C4—22no—
202C2×C3×C4—28no—
203C2×C3×C4—28no—
204C2×C3×C4—28no—
205C2×C3×C4—28no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—24no—
209C2×C3×C4—24no—
210C2×C3×C4—48no—
211C2×C3×C4—44no—
212C2×C3×C4—48no—
213C2×C3×C4—44no—
214C2×C3×C4—44no—
215C2×C3×C4—48no—
216C2×C3×C4C2×C22⋊C412yes—
217C2×C3×C4C2×C22⋊C414yes—
218C2×C3×C4C2×M1614yes—
219C2×C3×C4C22×D412yes—
220C2×C3×C4H(5,𝔽2)14yes—
221C3⋊C8—48no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24—22no—
226C24—42no—
227C24—42no—
228C24—42no—
229C24—42no—
230C24#20814yescomplemented
231C24#21714yes—
232C24C22×Dic312yes—
233C24C22×Dic312yes—
234C24C22×Dic311yescharacteristic
235C2×D4—232no—
236C2×D4—28no—
237C2×D4—28no—
238C2×D4—28no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—28no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—28no—
247C22×C4—24no—
248C22×C4—416no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—416no—
255C22×C4#20814yescomplemented
256C22×C4#21714yes—
257C22×C4#21718yes—
258C22×C4C22×Dic312yes—
259C22×C4C22×Dic312yes—
260C22×C4C22×Dic312yes—
261C22×C4C22×Dic311yescharacteristic
262C22×C4C22×Dic312yes—
263C22×C4C2×#6714yes—
264C22×C4C2×#6714yes—
265C22×C4C2×#6714yes—
266C22×C4C23×D311yescharacteristic
267M16—1216no—
268C2×C8—616no—
269C2×C8—68no—
270C4⋊C4—28no—
271C22⋊C4—28no—
272C22⋊C4—28no—
273C42—24no—
274C42—24no—
275C22×C3—22no—
276C22×C3—22no—
277C22×C3—24no—
278C22×C3—24no—
279C22×C3—24no—
280C22×C3—24no—
281C22×C3—24no—
282C22×C3—24no—
283C22×C3—28no—
284C22×C3—416no—
285C22×C3—416no—
286C22×C3—42no—
287C22×C3—42no—
288C22×C3—44no—
289C22×C3—44no—
290C22×C3—44no—
291C22×C3#40714yes—
292C22×C3C22×C22⋊C411yescharacteristic
293C22×C3#51812yes—
294C22×C3C22×M1611yescharacteristic
295C22×C3C22×M1611yescharacteristic
296C22×C3#56812yes—
297C3×C4—22no—
298C3×C4—24no—
299C3×C4—24no—
300C3×C4—42no—
301C3×C4—42no—
302C3×C4#53414yes—
303C23—22no—
304C23—22no—
305C23—28no—
306C23—28no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—24no—
311C23—24no—
312C23—24no—
313C23—24no—
314C23—22no—
315C23—44no—
316C23—48no—
317C23—44no—
318C23—44no—
319C23—44no—
320C23—48no—
321C23—48no—
322C23—48no—
323C23C2×#20814yes—
324C23C2×#20812yes—
325C23C2×#20812yes—
326C23C2×#21714yes—
327C23C2×#21712yes—
328C23C23×Dic311yescharacteristic
329D4—232no—
330C2×C4—28no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—28no—
336C2×C4—28no—
337C2×C4—28no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—48no—
343C2×C4—44no—
344C2×C4—48no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—48no—
348C2×C4C2×#20814yes—
349C2×C4C2×#21712yes—
350C2×C4C2×#21714yes—
351C2×C4#100414yes—
352C2×C4C22×#6712yes—
353C8—128no—
354C2×C3—24no—
355C2×C3—44no—
356C2×C3—44no—
357C2×C3#230811yescharacteristic
358C2×C3C2×#40711yescharacteristic
359C2×C3#287911yescharacteristic
360C2×C3#297312yes—
361C2×C3#297512yes—
362C22—22no—
363C22—22no—
364C22—24no—
365C22—24no—
366C22—24no—
367C22—24no—
368C22—24no—
369C22—24no—
370C22—28no—
371C22—416no—
372C22—416no—
373C22—42no—
374C22—42no—
375C22—44no—
376C22—44no—
377C22—44no—
378C22#520714yes—
379C22C22×#20811yescharacteristic
380C22C22×#20811yescharacteristic
381C22#579812yes—
382C22#581912yes—
383C22C22×#21711yescharacteristic
384C4—24no—
385C4—22no—
386C4—24no—
387C4—42no—
388C4—42no—
389C4#580414yes—
390C3#2174211yescharacteristic, complemented, Sylow
391C2—24no—
392C2—44no—
393C2—44no—
394C2#8230212yes—
395C2#8230412yes—
396C2C2×#520711yescharacteristic
397C2#8404311yescharacteristic
398C2#8407411yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC22×C3×C4×D4C2
prime radicalC23×C3×D4C4
cosocleC22×C3×C4C24
hypercenterC22×C4×D4D3
hypocenterC3#21742
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#5207C2semidirect
3. #72126C2semidirect
4. #71611C2semidirect
5. #71531C2semidirect
17. C22×#208C22semidirect
20. C2×#830C22semidirect
21. C2×#830C22semidirect
22. #4524C22semidirect
90. C24×C3M16semidirect
116. C22×C3×C4M16semidirect
230. C24#208semidirect
255. C22×C4#208semidirect
390. C3#21742semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
C2×#52072
#721262
#716112
#715312
#217423