Finite Group Databasebeta

← #4462532

Subgroups — #4462532

Order 512.

2806 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2626C211yesmaximal, characteristic
2C22×#399C211yesmaximal, characteristic
3#17168C214yesmaximal, complemented
4#17110C218yesmaximal, complemented
5#17106C214yesmaximal, complemented
6#15092C212yesmaximal, complemented
7#15026C212yesmaximal, complemented
8#14822C212yesmaximal, complemented
9#14727C214yesmaximal
10#14720C212yesmaximal, complemented
11#14341C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C23×C4⋊C4C2212yes—
14C23×C4⋊C4C2211yescharacteristic
15C23×C22⋊C4C2212yes—
16C23×C22⋊C4C2211yescharacteristic
17#2626—28no—
18C2×#400C2214yescomplemented
19C2×#399—28no—
20C2×#399C2212yes—
21C2×#399C2214yes—
22C2×#399C2214yes—
23C2×#397C2218yes—
24C2×#396C2214yes—
25C2×#395C2214yescomplemented
26C2×#393C2212yes—
27C2×#385C2212yes—
28C2×#385C2214yes—
29C2×#380C2211yescharacteristic
30C2×#379C2212yes—
31C2×#379C2212yes—
32C22×#95C2211yescharacteristic
33#1444C2218yescomplemented
34#1444C2218yescomplemented
35#1443C2218yescomplemented
36#1442C2218yescomplemented
37#1442C2218yescomplemented
38#1441C2214yescomplemented
39#1441C2218yescomplemented
40#1441C2214yescomplemented
41#1440C2214yes—
42#1440C2214yes—
43#1440C2218yes—
44#1440C2218yes—
45#1439C2218yes—
46C22⋊#95C2212yes—
47C22⋊#95C2212yes—
48C22⋊#95C2214yes—
49C22⋊#95C2214yes—
50C22⋊#95C2214yes—
51C26C2311yescharacteristic
52C23×Q8C2311yescharacteristic
53C24×C4C2312yes—
54C24×C4C2312yes—
55C24×C4C2311yescharacteristic
56C24×C4C2311yescharacteristic
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8—44no—
59C22×C4⋊C4—28no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—48no—
62C22×C4⋊C4—44no—
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2318yes—
71C22×C22⋊C4—28no—
72C22×C22⋊C4—48no—
73C22×C22⋊C4—44no—
74C22×C22⋊C4D418yes—
75C22×C22⋊C4C2312yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2312yes—
78C22×C22⋊C4C2318yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2312yes—
84C22×C22⋊C4C2312yes—
85#400—48no—
86#399—28no—
87#399—216no—
88#399—216no—
89#397—416no—
90#396—216no—
91#395—48no—
92#393—28no—
93#385—28no—
94#385—48no—
95#380—24no—
96#379—28no—
97#379—44no—
98C2×#95—24no—
99C2×#95C2314yes—
100C2×#95C2312yes—
101C2×#95C2318yes—
102C2×#95C2318yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2318yes—
106C2×#95C2318yes—
107C2×#95C2314yes—
108C2×#95C2318yes—
109C2×#95C2318yes—
110C2×#95C2318yes—
111C2×#95C2318yes—
112C25—24no—
113C25—24no—
114C25—24no—
115C25—44no—
116C25—44no—
117C25C2×D414yes—
118C25C2412yes—
119C25C2411yescharacteristic
120C22×Q8—22no—
121C22×Q8—42no—
122C22×Q8—44no—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—28no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—28no—
131C23×C4—44no—
132C23×C4—44no—
133C23×C4—44no—
134C23×C4—44no—
135C23×C4—44no—
136C23×C4C2×D414yes—
137C23×C4C2×D414yes—
138C23×C4C2412yes—
139C23×C4C2412yes—
140C23×C4C2414yes—
141C23×C4C2418yes—
142C23×C4C2414yes—
143C23×C4C2411yescharacteristic
144C23×C4C2411yescharacteristic
145C23×C4C2412yes—
146C23×C4C2414yes—
147C22⋊Q8—416no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—432no—
164C2×C4⋊C4—416no—
165C2×C4⋊C4—48no—
166C2×C4⋊C4—48no—
167C2×C4⋊C4—48no—
168C2×C4⋊C4—48no—
169C2×C4⋊C4—44no—
170C2×C22⋊C4—216no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—216no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—416no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—48no—
187C2×C22⋊C4—432no—
188C2×C22⋊C4—48no—
189C2×C22⋊C4—48no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—48no—
193C2×C22⋊C4—48no—
194C2×C22⋊C4C2×D418yes—
195C2×C22⋊C4C2×D418yes—
196#95—28no—
197#95—24no—
198#95—28no—
199#95—216no—
200#95—216no—
201#95—48no—
202#95—48no—
203#95—48no—
204#95—48no—
205#95—44no—
206#95—48no—
207#95—44no—
208#95—48no—
209C24—24no—
210C24—24no—
211C24—28no—
212C24—28no—
213C24—22no—
214C24—22no—
215C24—24no—
216C24—24no—
217C24—28no—
218C24—28no—
219C24—28no—
220C24—48no—
221C24—48no—
222C24—432no—
223C24—416no—
224C24—88no—
225C24—88no—
226C24—88no—
227C24—88no—
228C24C22×D414yes—
229C24C22×D412yes—
230C24H(5,𝔽2)14yes—
231C242−1+414yes—
232C242−1+414yes—
233C24C2511yescharacteristic
234C2×Q8—44no—
235C2×Q8—84no—
236C2×Q8—88no—
237C22×C4—216no—
238C22×C4—28no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—28no—
252C22×C4—28no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—28no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—28no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—28no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—28no—
269C22×C4—28no—
270C22×C4—28no—
271C22×C4—28no—
272C22×C4—28no—
273C22×C4—28no—
274C22×C4—28no—
275C22×C4—48no—
276C22×C4—48no—
277C22×C4—416no—
278C22×C4—416no—
279C22×C4—48no—
280C22×C4—416no—
281C22×C4—48no—
282C22×C4—48no—
283C22×C4—48no—
284C22×C4—48no—
285C22×C4—88no—
286C22×C4—816no—
287C22×C4C22×D414yes—
288C22×C4C22×D412yes—
289C22×C4C22×D412yes—
290C22×C4H(5,𝔽2)18yes—
291C4⋊C4—416no—
292C4⋊C4—416no—
293C4⋊C4—416no—
294C4⋊C4—48no—
295C4⋊C4—44no—
296C4⋊C4—84no—
297C4⋊C4—88no—
298C4⋊C4—816no—
299C22⋊C4—48no—
300C22⋊C4—48no—
301C22⋊C4—432no—
302C22⋊C4—48no—
303C22⋊C4—48no—
304C22⋊C4—416no—
305C22⋊C4—88no—
306C22⋊C4—88no—
307C22⋊C4—88no—
308C22⋊C4—88no—
309C23—24no—
310C23—24no—
311C23—28no—
312C23—28no—
313C23—28no—
314C23—216no—
315C23—28no—
316C23—24no—
317C23—24no—
318C23—24no—
319C23—28no—
320C23—28no—
321C23—432no—
322C23—416no—
323C23—44no—
324C23—44no—
325C23—44no—
326C23—44no—
327C23—48no—
328C23—48no—
329C23—416no—
330C23—44no—
331C23—44no—
332C23—88no—
333C23—816no—
334C23—88no—
335C23—864no—
336C23#53414yes—
337C23#53614yes—
338C23#53614yes—
339C23#563116yes—
340C23C23×D411yescharacteristic
341C23C2×H(5,𝔽2)12yes—
342C23C2×H(5,𝔽2)14yes—
343C23C2×2−1+412yes—
344C23C2×2−1+412yes—
345C23#58514yes—
346Q8—88no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—28no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—28no—
354C2×C4—28no—
355C2×C4—28no—
356C2×C4—416no—
357C2×C4—432no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—48no—
377C2×C4—48no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—48no—
381C2×C4—416no—
382C2×C4—48no—
383C2×C4—48no—
384C2×C4—48no—
385C2×C4—48no—
386C2×C4—48no—
387C2×C4—48no—
388C2×C4—88no—
389C2×C4—816no—
390C22—24no—
391C22—28no—
392C22—28no—
393C22—24no—
394C22—232no—
395C22—44no—
396C22—44no—
397C22—48no—
398C22—48no—
399C22—44no—
400C22—832no—
401C22—88no—
402C22—88no—
403C22#281718yes—
404C22C2×#53411yescharacteristic
405C22C2×#53611yescharacteristic
406C22C2×#53611yescharacteristic
407C22#348412yes—
408C22#348512yes—
409C22C2×#56314yes—
410C22#353314yes—
411C22#353514yes—
412C22#353514yes—
413C22#354118yes—
414C22#354314yes—
415C4—44no—
416C4—48no—
417C4—42no—
418C4—42no—
419C4—82no—
420C4—84no—
421C4—84no—
422C4—82no—
423C4—88no—
424C2—28no—
425C2—84no—
426C2C2×#281711yescharacteristic
427C2#3670512yes—
428C2#3672012yes—
429C2#3695511yescharacteristic
430C2#3723211yescharacteristic
431C2#5242618yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17168C2semidirect
4. #17110C2semidirect
5. #17106C2semidirect
6. #15092C2semidirect
7. #15026C2semidirect
8. #14822C2semidirect
10. #14720C2semidirect
18. C2×#400C22semidirect
25. C2×#395C22semidirect
33. #1444C22semidirect
34. #1444C22semidirect
35. #1443C22semidirect
36. #1442C22semidirect
37. #1442C22semidirect
38. #1441C22semidirect
39. #1441C22semidirect
40. #1441C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26262
C22×#3992
#171682
#171102
#171062
#150922
#150262
#148222
#147272
#147202
#143412