Finite Group Databasebeta

← #556758

Subgroups — #556758

Order 512.

6138 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#525C211yesmaximal, characteristic
2C22×#380C212yesmaximal, complemented
3#14658C214yesmaximal, complemented
4#14343C212yesmaximal, complemented
5#14337C212yesmaximal, complemented
6#14281C214yesmaximal, complemented
7#14271C214yesmaximal, complemented
8#14134C214yesmaximal, complemented
9#13872C218yesmaximal, complemented
10C23×#41C2211yescharacteristic
11C25×C4C2211yescharacteristic
12C2×#525—216no—
13C2×#525C4116yescomplemented
14C22×#123C2212yes—
15C22×C22⋊Q8C2212yes—
16C22×#121C2212yes—
17C22×#120C2212yes—
18C22×C4×D4C2214yes—
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2214yes—
21C23×C22⋊C4C2214yes—
22C2×#386C22116yes—
23C2×#380—216no—
24C2×#380C2214yes—
25C2×#380C2214yes—
26C2×#380C2214yes—
27C2×#380C2214yes—
28C2×#380C2218yescomplemented
29C22×#95C2214yes—
30#1446C2214yescomplemented
31#1445C2218yescomplemented
32#1441C22116yescomplemented
33#1440C2218yes—
34#1439C22116yescomplemented
35#1438C2218yes—
36#1438C2214yescomplemented
37C4⋊#95C2218yescomplemented
38C22⋊#95C22116yes—
39C26C2311yescharacteristic
40C22×#41—28no—
41C22×#41—48no—
42C22×#41C2×C418yescomplemented
43C23×Q8C2311yescharacteristic
44C23×D4C2314yes—
45C23×D4C2312yes—
46C23×D4C2311yescharacteristic
47C24×C4—48no—
48C24×C4C2×C418yes—
49C24×C4D4116yes—
50C24×C4C2312yes—
51C24×C4C2311yescharacteristic
52C24×C4C2312yes—
53C24×C4C2318yes—
54C24×C4C2312yes—
55C24×C4C2311yescharacteristic
56#525—2128no—
57C2×#123—216no—
58C2×#123C2×C4116yescomplemented
59C2×C22⋊Q8—28no—
60C2×C22⋊Q8—28no—
61C2×C22⋊Q8C2×C4116yescomplemented
62C2×#121—28no—
63C2×#121—28no—
64C2×#121C2×C4116yescomplemented
65C2×#120—216no—
66C2×#120C2×C4116yes—
67C2×C4×D4—216no—
68C2×C4×D4—216no—
69C2×C4×D4C2×C4132yescomplemented
70C2×#117—28no—
71C2×#117C2×C418yescomplemented
72C22×C4⋊C4—216no—
73C22×C4⋊C4—416no—
74C22×C4⋊C4C2318yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2318yes—
78C22×C22⋊C4—216no—
79C22×C22⋊C4—416no—
80C22×C22⋊C4C2318yes—
81C22×C22⋊C4C2312yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2318yes—
85C22×C22⋊C4C23116yes—
86C22×C42C2312yes—
87#386—432no—
88#380—216no—
89#380—216no—
90#380—216no—
91#380—216no—
92#380—416no—
93C2×#95—216no—
94C2×#95C2318yes—
95C2×#95C2318yes—
96C2×#95C23116yes—
97C2×#95C23132yes—
98C25—28no—
99C25—28no—
100C25—44no—
101C25C22×C414yes—
102C25C2×D418yes—
103C25C2412yes—
104C25C2411yescharacteristic
105C25C2412yes—
106C2×#41—232no—
107C2×#41—432no—
108C2×#41—464no—
109C22×Q8—24no—
110C22×Q8—44no—
111C22×Q8C22×C414yes—
112C22×D4—28no—
113C22×D4—28no—
114C22×D4—28no—
115C22×D4—216no—
116C22×D4—216no—
117C22×D4—24no—
118C22×D4—44no—
119C22×D4—48no—
120C22×D4C22×C4116yes—
121C22×D4C22×C418yes—
122C22×D4C22×C414yes—
123C23×C4—216no—
124C23×C4—28no—
125C23×C4—28no—
126C23×C4—216no—
127C23×C4—216no—
128C23×C4—216no—
129C23×C4—216no—
130C23×C4—432no—
131C23×C4—464no—
132C23×C4—44no—
133C23×C4—48no—
134C23×C4—416no—
135C23×C4—416no—
136C23×C4C22⋊C4164yescomplemented
137C23×C4C22×C418yes—
138C23×C4C22×C414yes—
139C23×C4C22×C418yes—
140C23×C4C22×C418yes—
141C23×C4C22×C414yes—
142C23×C4C2×D418yes—
143C23×C4C2×D418yes—
144C23×C4C2×D418yes—
145C23×C4C2×D418yes—
146C23×C4C2×D418yes—
147C23×C4C2×D418yes—
148C23×C4C24116yes—
149C23×C4C2414yes—
150C23×C4C2412yes—
151C23×C4C2412yes—
152C23×C4C2411yescharacteristic
153C23×C4C2411yescharacteristic
154#123—264no—
155C22⋊Q8—264no—
156#121—264no—
157#120—264no—
158C4×D4—2128no—
159#117—232no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—216no—
165C2×C4⋊C4—216no—
166C2×C4⋊C4—232no—
167C2×C4⋊C4—416no—
168C2×C4⋊C4—464no—
169C2×C4⋊C4C22×C418yes—
170C2×C4⋊C4C22×C4116yes—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—216no—
177C2×C22⋊C4—216no—
178C2×C22⋊C4—232no—
179C2×C22⋊C4—416no—
180C2×C22⋊C4—464no—
181C2×C22⋊C4—416no—
182C2×C22⋊C4—432no—
183C2×C22⋊C4C22×C418yes—
184C2×C22⋊C4C22×C4116yes—
185C2×C22⋊C4C22×C4116yes—
186C2×C42—28no—
187C2×C42C22×C418yes—
188#95—216no—
189#95—216no—
190#95—416no—
191#95—432no—
192C24—28no—
193C24—24no—
194C24—24no—
195C24—232no—
196C24—216no—
197C24—216no—
198C24—232no—
199C24—24no—
200C24—24no—
201C24—24no—
202C24—24no—
203C24—28no—
204C24—28no—
205C24—28no—
206C24—48no—
207C24—416no—
208C24—432no—
209C24—416no—
210C24—416no—
211C24C2×C22⋊C4116yes—
212C24C23×C414yes—
213C24C23×C412yes—
214C24C23×C414yes—
215C24C22×D414yes—
216C24C22×D414yes—
217C24C22×D414yes—
218C24H(5,𝔽2)18yes—
219C242−1+418yes—
220C24C2511yescharacteristic
221#41—4128no—
222C2×Q8—28no—
223C2×Q8—48no—
224C2×Q8—416no—
225C2×D4—232no—
226C2×D4—216no—
227C2×D4—264no—
228C2×D4—28no—
229C2×D4—48no—
230C2×D4—416no—
231C2×D4—416no—
232C2×D4—432no—
233C2×D4—416no—
234C2×D4—416no—
235C2×D4—432no—
236C22×C4—28no—
237C22×C4—24no—
238C22×C4—232no—
239C22×C4—216no—
240C22×C4—232no—
241C22×C4—232no—
242C22×C4—232no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—28no—
247C22×C4—28no—
248C22×C4—28no—
249C22×C4—216no—
250C22×C4—232no—
251C22×C4—216no—
252C22×C4—216no—
253C22×C4—232no—
254C22×C4—232no—
255C22×C4—216no—
256C22×C4—4128no—
257C22×C4—48no—
258C22×C4—416no—
259C22×C4—416no—
260C22×C4—416no—
261C22×C4—416no—
262C22×C4—432no—
263C22×C4—432no—
264C22×C4—432no—
265C22×C4—432no—
266C22×C4—832no—
267C22×C4—832no—
268C22×C4C2×C22⋊C4116yes—
269C22×C4C2×C22⋊C4132yes—
270C22×C4C2×C22⋊C4132yes—
271C22×C4C2×C22⋊C4116yes—
272C22×C4C23×C418yes—
273C22×C4C23×C414yes—
274C22×C4C23×C414yes—
275C22×C4C23×C412yes—
276C22×C4C23×C412yes—
277C22×C4C22×D414yes—
278C22×C4C22×D414yes—
279C22×C4C22×D414yes—
280C22×C4C22×D414yes—
281C22×C4H(5,𝔽2)18yes—
282C4⋊C4—216no—
283C4⋊C4—232no—
284C4⋊C4—432no—
285C4⋊C4—432no—
286C22⋊C4—216no—
287C22⋊C4—232no—
288C22⋊C4—232no—
289C22⋊C4—432no—
290C22⋊C4—432no—
291C22⋊C4—832no—
292C42—216no—
293C23—28no—
294C23—28no—
295C23—28no—
296C23—28no—
297C23—24no—
298C23—24no—
299C23—28no—
300C23—216no—
301C23—216no—
302C23—216no—
303C23—216no—
304C23—264no—
305C23—24no—
306C23—28no—
307C23—28no—
308C23—24no—
309C23—24no—
310C23—28no—
311C23—24no—
312C23—28no—
313C23—216no—
314C23—416no—
315C23—464no—
316C23—464no—
317C23—464no—
318C23—48no—
319C23—44no—
320C23—44no—
321C23—48no—
322C23—44no—
323C23—44no—
324C23—48no—
325C23—416no—
326C23—48no—
327C23—48no—
328C23C22×C22⋊C414yes—
329C23C22×C22⋊C418yes—
330C23#520116yes—
331C23#53518yes—
332C23#53618yes—
333C23#537116yes—
334C23C24×C411yescharacteristic
335C23C23×D412yes—
336C23C2×H(5,𝔽2)14yes—
337C23C2×H(5,𝔽2)14yes—
338C23C2×2−1+414yes—
339Q8—416no—
340D4—416no—
341D4—432no—
342D4—464no—
343C2×C4—28no—
344C2×C4—24no—
345C2×C4—28no—
346C2×C4—28no—
347C2×C4—28no—
348C2×C4—264no—
349C2×C4—232no—
350C2×C4—28no—
351C2×C4—24no—
352C2×C4—28no—
353C2×C4—216no—
354C2×C4—416no—
355C2×C4—432no—
356C2×C4—48no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—48no—
360C2×C4—432no—
361C2×C4—416no—
362C2×C4—416no—
363C2×C4—416no—
364C2×C4—432no—
365C2×C4—432no—
366C2×C4—864no—
367C2×C4C22×C22⋊C418yes—
368C2×C4C22×C22⋊C414yes—
369C2×C4C22×C22⋊C414yes—
370C2×C4#51818yes—
371C2×C4#534116yes—
372C22—232no—
373C22—216no—
374C22—216no—
375C22—216no—
376C22—216no—
377C22—28no—
378C22—28no—
379C22—24no—
380C22—28no—
381C22—464no—
382C22—48no—
383C22—48no—
384C22—48no—
385C22—416no—
386C22—44no—
387C22—48no—
388C22—88no—
389C22—88no—
390C22#2311116yes—
391C22C23×C22⋊C411yescharacteristic
392C22C2×#51812yes—
393C22C2×#52014yes—
394C22C2×#53414yes—
395C22C2×#53512yes—
396C22C2×#53612yes—
397C22C2×#53714yes—
398C22#353618yes—
399C22#353714yes—
400C22#353814yes—
401C4—28no—
402C4—24no—
403C4—44no—
404C4—48no—
405C4—816no—
406C4#230818yes—
407C2—216no—
408C2—44no—
409C2—84no—
410C2C2×#230811yescharacteristic
411C2C2×#231112yes—
412C2#3436014yes—
413C2#3743914yes—
414C2#3744214yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#525C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#380C2semidirect
3. #14658C2semidirect
4. #14343C2semidirect
5. #14337C2semidirect
6. #14281C2semidirect
7. #14271C2semidirect
8. #14134C2semidirect
9. #13872C2semidirect
13. C2×#525C4semidirect
28. C2×#380C22semidirect
30. #1446C22semidirect
31. #1445C22semidirect
32. #1441C22semidirect
34. #1439C22semidirect
36. #1438C22semidirect
37. C4⋊#95C22semidirect
42. C22×#41C2×C4semidirect
58. C2×#123C2×C4semidirect
61. C2×C22⋊Q8C2×C4semidirect
64. C2×#121C2×C4semidirect
69. C2×C4×D4C2×C4semidirect
71. C2×#117C2×C4semidirect
136. C23×C4C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5252
C22×#3802
#146582
#143432
#143372
#142812
#142712
#141342
#138722