Finite Group Databasebeta

← #556801

Subgroups — #556801

Order 512.

3642 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#533C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic
3C22×#383C211yesmaximal, characteristic
4#14663C212yesmaximal, complemented
5#14663C212yesmaximal, complemented
6#14346C214yesmaximal, complemented
7#14289C214yesmaximal, complemented
8#14281C214yesmaximal
9#14159C214yesmaximal
10#13961C214yesmaximal, complemented
11#13958C214yesmaximal
12C2×#533—28no—
13C2×#533—28no—
14C2×#533C4116yescomplemented
15C22×C4⋊Q8C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#125C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C22⋊Q8C2212yes—
20C22×C4×Q8C2212yes—
21C22×C4×Q8C2212yes—
22C22×#117C2212yes—
23C23×C42C2211yescharacteristic
24#2283—28no—
25C2×#383—28no—
26C2×#383C2212yes—
27C2×#383C2212yes—
28C2×#383C2212yes—
29C2×#383C2212yes—
30C2×#383C2212yes—
31C2×#382C2212yes—
32C2×#381C2212yes—
33C2×#380C2212yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C22⋊C4C2212yes—
36C2×#376C2212yes—
37C22×#95C2212yes—
38C22×#95C2212yes—
39C22×#95C2212yes—
40#1446C2218yes—
41#1446C2218yes—
42#1445C2218yes—
43#1445C2218yes—
44#1445C2218yes—
45#1445C2218yes—
46#1441C22116yes—
47#1436C22116yes—
48#1434C22116yes—
49C22⋊#95C22116yes—
50C23×Q8C2312yes—
51C23×Q8C2312yes—
52C24×C4C2311yescharacteristic
53C24×C4C2314yes—
54C24×C4C2311yescharacteristic
55C24×C4C2311yescharacteristic
56#533—2128no—
57C2×C4⋊Q8—24no—
58C2×C4⋊Q8—24no—
59C2×C4⋊Q8—24no—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8C2×C418yescomplemented
62C2×C4⋊Q8C2×C418yescomplemented
63C2×#125—28no—
64C2×#125—28no—
65C2×#125C2×C4116yescomplemented
66C2×C22⋊Q8—28no—
67C2×C22⋊Q8—28no—
68C2×C22⋊Q8—28no—
69C2×C22⋊Q8—28no—
70C2×C22⋊Q8C2×C4116yes—
71C2×C22⋊Q8C2×C4116yes—
72C2×C4×Q8—28no—
73C2×C4×Q8—28no—
74C2×C4×Q8—28no—
75C2×C4×Q8—28no—
76C2×C4×Q8C2×C4116yescomplemented
77C2×C4×Q8C2×C4116yescomplemented
78C2×#117—28no—
79C2×#117—28no—
80C2×#117C2×C4116yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C42C2×C418yes—
91C22×C42D418yes—
92C22×C42D418yes—
93C22×C42C2314yes—
94C22×C42C2314yes—
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99#383—28no—
100#383—28no—
101#383—28no—
102#383—28no—
103#383—28no—
104#382—28no—
105#381—28no—
106#380—28no—
107C4×C4⋊C4—28no—
108C4×C22⋊C4—28no—
109#376—28no—
110C2×#95—28no—
111C2×#95—28no—
112C2×#95—28no—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C23132yes—
124C2×#95C23132yes—
125C25C2411yescharacteristic
126C22×Q8—24no—
127C22×Q8—24no—
128C22×Q8—24no—
129C22×Q8—24no—
130C22×Q8—48no—
131C22×Q8—48no—
132C22×Q8C22×C418yes—
133C22×Q8C22×C418yes—
134C23×C4—28no—
135C23×C4—28no—
136C23×C4—216no—
137C23×C4—44no—
138C23×C4—44no—
139C23×C4—48no—
140C23×C4—48no—
141C23×C4C22×C414yes—
142C23×C4C22×C414yes—
143C23×C4C22×C414yes—
144C23×C4C2×D414yes—
145C23×C4C2×D414yes—
146C23×C4C2×D414yes—
147C23×C4C2×D414yes—
148C23×C4C2×D414yes—
149C23×C4C2×D414yes—
150C23×C4C2418yes—
151C23×C4C2418yes—
152C23×C4C2414yes—
153C23×C4C2414yes—
154C23×C4C2411yescharacteristic
155C23×C4C2411yescharacteristic
156C23×C4C2411yescharacteristic
157C23×C4C2411yescharacteristic
158C23×C4C2411yescharacteristic
159C23×C4C2411yescharacteristic
160C4⋊Q8—232no—
161C4⋊Q8—232no—
162#125—264no—
163C22⋊Q8—264no—
164C22⋊Q8—264no—
165C4×Q8—264no—
166C4×Q8—264no—
167#117—264no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4C22×C418yes—
183C2×C4⋊C4C22×C418yes—
184C2×C4⋊C4C22×C4116yes—
185C2×C4⋊C4C22×C4116yes—
186C2×C4⋊C4C22×C418yes—
187C2×C4⋊C4C22×C418yes—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—216no—
192C2×C22⋊C4C22×C4116yes—
193C2×C42—28no—
194C2×C42—28no—
195C2×C42—28no—
196C2×C42—48no—
197C2×C42C22⋊C4164yes—
198C2×C42C22×C4116yes—
199C2×C42C22×C414yes—
200C2×C42C22×C414yes—
201C2×C42C22×C414yes—
202C2×C42C22×C414yes—
203C2×C42C2×D414yes—
204C2×C42C2×D414yes—
205C2×C42C2×D414yes—
206C2×C42C2×D414yes—
207C2×C42C2×D414yes—
208C2×C42C2×D414yes—
209C2×C42C2×D414yes—
210C2×C42C2×D414yes—
211#95—28no—
212#95—28no—
213#95—28no—
214#95—28no—
215#95—28no—
216#95—28no—
217#95—28no—
218#95—28no—
219#95—28no—
220#95—28no—
221#95—432no—
222#95—432no—
223C24—24no—
224C24—24no—
225C24C23×C412yes—
226C24C22×D412yes—
227C24C22×D412yes—
228C242−1+418yes—
229C24C2511yescharacteristic
230C2×Q8—216no—
231C2×Q8—216no—
232C2×Q8—416no—
233C2×Q8—416no—
234C2×Q8—416no—
235C2×Q8—416no—
236C2×Q8—416no—
237C2×Q8—416no—
238C22×C4—216no—
239C22×C4—216no—
240C22×C4—216no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—216no—
252C22×C4—216no—
253C22×C4—28no—
254C22×C4—28no—
255C22×C4—216no—
256C22×C4—216no—
257C22×C4—216no—
258C22×C4—216no—
259C22×C4—216no—
260C22×C4—216no—
261C22×C4—48no—
262C22×C4—48no—
263C22×C4—48no—
264C22×C4—48no—
265C22×C4—48no—
266C22×C4—48no—
267C22×C4—416no—
268C22×C4—416no—
269C22×C4—416no—
270C22×C4—832no—
271C22×C4C2×C22⋊C4116yes—
272C22×C4C2×C22⋊C4116yes—
273C22×C4C2×C22⋊C4116yes—
274C22×C4C23×C418yes—
275C22×C4C23×C418yes—
276C22×C4C23×C412yes—
277C22×C4C23×C412yes—
278C22×C4C23×C412yes—
279C22×C4C23×C412yes—
280C22×C4C23×C412yes—
281C22×C4C23×C412yes—
282C22×C4C22×D412yes—
283C22×C4C22×D412yes—
284C22×C4C22×D412yes—
285C22×C4C22×D412yes—
286C22×C4C22×D412yes—
287C22×C4C22×D412yes—
288C22×C4C22×D412yes—
289C22×C4C22×D412yes—
290C22×C4C22×D412yes—
291C22×C4C22×D412yes—
292C22×C4C22×D412yes—
293C22×C4C22×D412yes—
294C4⋊C4—216no—
295C4⋊C4—216no—
296C4⋊C4—232no—
297C4⋊C4—232no—
298C4⋊C4—216no—
299C4⋊C4—216no—
300C4⋊C4—416no—
301C22⋊C4—232no—
302C22⋊C4—416no—
303C42—232no—
304C42—416no—
305C42C2×C22⋊C4116yes—
306C42C2×C22⋊C4116yes—
307C42C2×C22⋊C4116yes—
308C42C2×C22⋊C4116yes—
309C23—24no—
310C23—24no—
311C23—24no—
312C23—24no—
313C23—24no—
314C23—24no—
315C23—232no—
316C23C22×C22⋊C414yes—
317C23#51918yes—
318C23#53618yes—
319C23#53618yes—
320C23C24×C411yescharacteristic
321C23C23×D411yescharacteristic
322C23C23×D411yescharacteristic
323C23C2×2−1+414yes—
324C23#58514yes—
325C23#58514yes—
326Q8—432no—
327Q8—432no—
328C2×C4—232no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—28no—
348C2×C4—28no—
349C2×C4—216no—
350C2×C4—216no—
351C2×C4—416no—
352C2×C4—416no—
353C2×C4—48no—
354C2×C4—48no—
355C2×C4—48no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—48no—
359C2×C4—48no—
360C2×C4—48no—
361C2×C4—48no—
362C2×C4—48no—
363C2×C4—432no—
364C2×C4—432no—
365C2×C4—832no—
366C2×C4C22×C22⋊C414yes—
367C2×C4C22×C22⋊C414yes—
368C2×C4C22×C22⋊C414yes—
369C2×C4C22×C22⋊C414yes—
370C2×C4C22×C22⋊C414yes—
371C2×C4C22×C22⋊C414yes—
372C22—216no—
373C22—216no—
374C22—216no—
375C22—24no—
376C22—24no—
377C22#231018yes—
378C22C23×C22⋊C411yescharacteristic
379C22C2×#51912yes—
380C22#342812yes—
381C22#342812yes—
382C22C2×#53612yes—
383C22C2×#53612yes—
384C22#344912yes—
385C22#344912yes—
386C22#344912yes—
387C22#344912yes—
388C22#3564116yes—
389C4—24no—
390C4—24no—
391C4—24no—
392C4—24no—
393C4—24no—
394C4—24no—
395C4—48no—
396C4—48no—
397C4—88no—
398C4—88no—
399C2—28no—
400C2C2×#231011yescharacteristic
401C2#3404411yescharacteristic
402C2#3404411yescharacteristic
403C2#3438914yes—
404C2#3754814yes—
405C2#3755714yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #14663C2semidirect
5. #14663C2semidirect
6. #14346C2semidirect
7. #14289C2semidirect
10. #13961C2semidirect
14. C2×#533C4semidirect
61. C2×C4⋊Q8C2×C4semidirect
62. C2×C4⋊Q8C2×C4semidirect
65. C2×#125C2×C4semidirect
76. C2×C4×Q8C2×C4semidirect
77. C2×C4×Q8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5332
C2×#22832
C22×#3832
#146632
#143462
#142892
#142812
#141592
#139612
#139582