Finite Group Databasebeta

← #6468645

Subgroups — #6468645

Order 512.

1554 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35332C211yesmaximal, characteristic
2#34217C211yesmaximal, characteristic, complemented
3#29361C212yesmaximal, complemented
4#29360C212yesmaximal, complemented
5#29342C214yesmaximal, complemented
6#29221C212yesmaximal, complemented
7#28752C212yesmaximal, complemented
8#27976C212yesmaximal, complemented
9#25907C212yesmaximal, complemented
10#23832C214yesmaximal, complemented
11#21648C212yesmaximal, complemented
12C4×#488C212yesmaximal, complemented
13#21399C212yesmaximal
14#21398C212yesmaximal, complemented
15C4×#433C211yesmaximal, characteristic, complemented
16C2×#541C2211yescharacteristic
17C2×#540C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C2×#488C2212yes—
20C2×#488C2212yes—
21C2×#433C2212yes—
22#2859C2211yescharacteristic
23#2858C2211yescharacteristic
24#2850C2212yes—
25#2642C2211yescharacteristic
26#2641C2211yescharacteristic
27#2620C2212yes—
28#2559—24no—
29#2558—24no—
30#2370—24no—
31#2345—24no—
32#2330C2212yes—
33#2330C2212yes—
34#2326C2212yes—
35#2316C2212yes—
36C4×#125C2212yes—
37C4×#125C2212yes—
38C4×#124C2212yes—
39#2285C2211yescharacteristic
40#2274C2212yes—
41C4×#117C2211yescharacteristic
42#2100C2214yes—
43#2099C2214yes—
44#2072C2214yes—
45#2071C2214yes—
46#2042C2214yes—
47#2042C2214yes—
48#2026C2214yes—
49#2025C2214yes—
50#1991C2212yes—
51#1991C2212yes—
52#1989C2214yes—
53#1989C2214yes—
54#1982C2214yes—
55#1981C2214yes—
56#1958C2212yes—
57#1958C2212yes—
58#1936C2214yes—
59#1935C2214yes—
60#1934C2214yes—
61#1933C2214yes—
62#1875C2214yes—
63#1874C2214yes—
64#1873C2214yes—
65#1872C2214yes—
66#1835C2212yes—
67#1835C2212yes—
68#1768C2212yes—
69C4×C4⋊C8C2212yes—
70#1763C2214yes—
71#1762C2214yes—
72C4×#103C2214yes—
73C4×#102C2214yes—
74#1750C2212yes—
75C4×C22⋊C8C2212yes—
76#1746C2212yes—
77C4×#97C2211yescharacteristic
78C42×C8C2211yescharacteristic
79#541—28no—
80#540—28no—
81C2×C4⋊Q8C2311yescharacteristic
82C2×#127C2311yescharacteristic
83C2×#126—22no—
84C2×#125—28no—
85C2×#125—22no—
86C2×#125—22no—
87C2×#125C2312yes—
88C2×#125C2311yescharacteristic
89C2×#125C2311yescharacteristic
90C2×#125C2312yes—
91C2×#124C2312yes—
92C2×#124C2312yes—
93C2×#123—22no—
94C2×C22⋊Q8C2312yes—
95C2×#121—22no—
96C2×#121C2312yes—
97C2×C4×Q8C2312yes—
98C2×C4×D4C2312yes—
99C2×#117C2312yes—
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C22×C4⋊C4—22no—
103C22×C4⋊C4C2312yes—
104C22×C42C2311yescharacteristic
105#488—28no—
106#488—28no—
107#433—28no—
108C2×C4⋊C8C2314yes—
109C2×#103C2314yes—
110C2×#103C2314yes—
111C2×#103C2314yes—
112C2×#103C2314yes—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C2×#102C2314yes—
117C2×C22⋊C8C2314yes—
118C2×#97C2312yes—
119C2×#97C2312yes—
120C2×C4×C8C2312yes—
121C2×C4×C8C2312yes—
122C2×C4×C8C2312yes—
123C2×C4×C8C2312yes—
124#400—24no—
125#396—24no—
126#394—24no—
127#394—24no—
128#391C2312yes—
129#391C2312yes—
130#390—24no—
131#390C2312yes—
132#390C2312yes—
133#389—24no—
134#389—24no—
135#389—24no—
136#389—24no—
137#389C2314yes—
138#389C2314yes—
139#388—24no—
140#388—24no—
141#388—24no—
142#387—24no—
143#387—24no—
144#387—24no—
145#387—24no—
146#386—24no—
147#385—24no—
148#384C2312yes—
149#384C2312yes—
150#382C2312yes—
151#382C2312yes—
152#380—22no—
153#380—24no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—22no—
158C4×C4⋊C4D414yes—
159C4×C4⋊C4C2314yes—
160C4×C4⋊C4C2314yes—
161C4×C22⋊C4—24no—
162C4×C22⋊C4—24no—
163C4×C22⋊C4D414yes—
164#376C2311yescharacteristic
165#376C2311yescharacteristic
166C43C2311yescharacteristic
167C43C2311yescharacteristic
168#340C2314yes—
169#340C2314yes—
170#340C2314yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#336C2314yes—
177#336C2314yes—
178C22×Q8C2412yes—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4C2412yes—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4C2×D412yes—
185C23×C4C2411yescharacteristic
186C22×C8C2414yes—
187C22×C8C2414yes—
188C4⋊Q8—24no—
189#127—24no—
190#126—48no—
191#125—216no—
192#125—28no—
193#125—24no—
194#125—24no—
195#125—24no—
196#125—48no—
197#125#4118yes—
198#124—24no—
199#124—24no—
200#124—44no—
201#123—48no—
202C22⋊Q8—28no—
203#121—28no—
204#121—48no—
205C4×Q8—28no—
206C4×D4—28no—
207#117—42no—
208#117—44no—
209#117#4118yes—
210C2×C4⋊C4—216no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2414yes—
227C2×C4⋊C4C2412yes—
228C2×C4⋊C4C2412yes—
229C2×C4⋊C4C2414yes—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4C2×D412yes—
239C2×C22⋊C4C2×D412yes—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42C2×D412yes—
246C2×C42C2×D414yes—
247C2×C42C2×D414yes—
248C2×C42C2411yescharacteristic
249C2×C42C2411yescharacteristic
250C2×C42C2411yescharacteristic
251C2×C42C2411yescharacteristic
252C2×C42C2411yescharacteristic
253C2×C42C2411yescharacteristic
254C4⋊C8—28no—
255#103—28no—
256#103—28no—
257#103—44no—
258#103—44no—
259#102—28no—
260#102—28no—
261#102—44no—
262#102—44no—
263C22⋊C8—28no—
264#97—24no—
265#97—42no—
266C4×C8—24no—
267C4×C8—24no—
268C4×C8—24no—
269C4×C8—42no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95C2×D414yes—
279#95C2×D414yes—
280C24—22no—
281C24C22×D411yescharacteristic
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—84no—
293C2×D4—84no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—28no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—84no—
306C22×C4—84no—
307C22×C4C22×D412yes—
308C22×C4C22×D412yes—
309C22×C4C22×D412yes—
310C22×C4C22×D412yes—
311C22×C4C22×D412yes—
312C22×C4C22×D412yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C2×#4114yes—
315C22×C4C2511yescharacteristic
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—24no—
322C4⋊C4—28no—
323C4⋊C4—216no—
324C4⋊C4—216no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—42no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—84no—
337C4⋊C4—84no—
338C4⋊C4—84no—
339C4⋊C4—84no—
340C4⋊C4#12418yes—
341C4⋊C4C2×#4118yes—
342C22⋊C4—42no—
343C22⋊C4—48no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—48no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—84no—
350C22⋊C4—84no—
351C22⋊C4#12418yes—
352C42—24no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—44no—
357C42—44no—
358C42—44no—
359C42—44no—
360C42—84no—
361C42—84no—
362C42C2×#4114yes—
363C42C2×#4114yes—
364C42C2×#4114yes—
365C23—24no—
366C23—21no—
367C23—21no—
368C23—44no—
369C23—44no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23C2×#12412yes—
374C23C23×D411yescharacteristic
375Q8—44no—
376Q8—44no—
377D4—44no—
378D4—44no—
379D4—164no—
380D4—164no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—28no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—28no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—84no—
407C2×C4—88no—
408C2×C4—164no—
409C2×C4#52514yes—
410C2×C4C2×#12414yes—
411C2×C4C2×#12414yes—
412C2×C4C2×#12412yes—
413C2×C4C22×#4112yes—
414C2×C4C22×#4112yes—
415C2×C4#58512yes—
416C8—44no—
417C8—44no—
418C22—22no—
419C22—24no—
420C22—42no—
421C22—42no—
422C22—84no—
423C22—82no—
424C22—84no—
425C22—82no—
426C22—82no—
427C22C2×#52511yescharacteristic
428C22C22×#12411yescharacteristic
429C22#344911yescharacteristic
430C22#358414yes—
431C4—22no—
432C4—24no—
433C4—24no—
434C4—24no—
435C4—42no—
436C4—42no—
437C4—42no—
438C4—48no—
439C4—84no—
440C4—82no—
441C4—82no—
442C4—82no—
443C4#348014yes—
444C2—42no—
445C2—162no—
446C2#3495511yescharacteristic
447C2#6055712yes—
448C2#6062412yes—
449C2#6100912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#540C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34217C2semidirect
3. #29361C2semidirect
4. #29360C2semidirect
5. #29342C2semidirect
6. #29221C2semidirect
7. #28752C2semidirect
8. #27976C2semidirect
9. #25907C2semidirect
10. #23832C2semidirect
11. #21648C2semidirect
12. C4×#488C2semidirect
14. #21398C2semidirect
15. C4×#433C2semidirect

Maximal subgroups

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

subgroupindex
#353322
#342172
#293612
#293602
#293422
#292212
#287522
#279762
#259072
#238322
#216482
C4×#4882
#213992
#213982
C4×#4332