Finite Group Databasebeta

← #6484968

Subgroups — #6484968

Order 512.

1474 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35332C211yesmaximal, characteristic
2#35284C211yesmaximal, characteristic, complemented
3#33057C212yesmaximal, complemented
4#33051C212yesmaximal, complemented
5#30191C214yesmaximal, complemented
6#29361C212yesmaximal, complemented
7#29360C212yesmaximal, complemented
8#29223C212yesmaximal, complemented
9#29146C212yesmaximal, complemented
10#29137C214yesmaximal, complemented
11#28754C212yesmaximal, complemented
12#28726C212yesmaximal, complemented
13#21898C212yesmaximal, complemented
14#21897C212yesmaximal
15#20427C211yesmaximal, characteristic, complemented
16C2×#541C2211yescharacteristic
17C2×#540C2211yescharacteristic
18C2×#489C2212yes—
19C2×#489C2212yes—
20C2×#424C2212yes—
21C2×C4×M16C2211yescharacteristic
22C22×#97C2211yescharacteristic
23#2868C2211yescharacteristic
24#2859C2211yescharacteristic
25#2859C2211yescharacteristic
26#2858C2211yescharacteristic
27#2857C2212yes—
28#2850C2212yes—
29#2647C2212yes—
30#2642C2211yescharacteristic
31#2641C2211yescharacteristic
32#2629C2212yes—
33#2620C2212yes—
34#2559—22no—
35#2559—22no—
36#2558—22no—
37#2558—22no—
38#2507—22no—
39#2497—22no—
40#2446—22no—
41#2429—22no—
42#2330C2212yes—
43#2326C2212yes—
44#2314C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×#124C2212yes—
47C4×#117C2211yescharacteristic
48#2100C2214yes—
49#2099C2214yes—
50#2072C2214yes—
51#2071C2214yes—
52#2042C2214yes—
53#2042C2214yes—
54#2026C2214yes—
55#2025C2214yes—
56#1993C2212yes—
57#1993C2212yes—
58#1989C2214yes—
59#1989C2214yes—
60#1982C2214yes—
61#1981C2214yes—
62#1959C2212yes—
63#1959C2212yes—
64#1936C2214yes—
65#1935C2214yes—
66#1934C2214yes—
67#1933C2214yes—
68#1895C2214yes—
69#1894C2214yes—
70#1893C2214yes—
71#1892C2214yes—
72#1836C2212yes—
73#1836C2212yes—
74#1805C2214yes—
75#1804C2214yes—
76#1803C2214yes—
77#1802C2214yes—
78#1768C2212yes—
79#1768C2212yes—
80#1767C2212yes—
81#1767C2212yes—
82#1729C2212yes—
83C22×M16C2312yes—
84C23×C8C2312yes—
85#541—28no—
86#540—28no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2312yes—
90C2×#127C2311yescharacteristic
91C2×#126—22no—
92C2×#125—22no—
93C2×#125C2311yescharacteristic
94C2×#125C2311yescharacteristic
95C2×#125C2312yes—
96C2×#124C2312yes—
97C2×#124C2312yes—
98C2×#123—22no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8C2312yes—
101C2×#121—22no—
102C2×#121C2312yes—
103C2×C4×Q8—22no—
104C2×C4×Q8C2312yes—
105C2×C4×D4C2312yes—
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C22×C42C2311yescharacteristic
109#489—44no—
110#489—44no—
111#424—28no—
112C2×C4⋊C8C2314yes—
113C2×C4⋊C8C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C4×M16—42no—
123C2×#97—42no—
124C2×#97C2312yes—
125C2×#97C2312yes—
126C2×#97C2312yes—
127C2×C4×C8C2312yes—
128#400—22no—
129#400—22no—
130#400—24no—
131#398—24no—
132#398—24no—
133#396—22no—
134#396—22no—
135#396—24no—
136#394—22no—
137#394—22no—
138#394—22no—
139#394—22no—
140#391—22no—
141#391—22no—
142#391C2312yes—
143#391C2312yes—
144#390—22no—
145#390—22no—
146#390C2312yes—
147#390C2312yes—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—24no—
155#389C2314yes—
156#389C2314yes—
157#388—24no—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#388—22no—
163#388—22no—
164#387—24no—
165#387—24no—
166#386—22no—
167#386—22no—
168#385—22no—
169#385—22no—
170#384—22no—
171#384C2312yes—
172#384C2312yes—
173#384C2312yes—
174#384C2312yes—
175#382C2312yes—
176#382C2312yes—
177#380—22no—
178#380C2314yes—
179C4×C4⋊C4—22no—
180C4×C4⋊C4D412yes—
181C4×C4⋊C4D412yes—
182C4×C4⋊C4C2314yes—
183C4×C22⋊C4—22no—
184C4×C22⋊C4—22no—
185C4×C22⋊C4D412yes—
186C4×C22⋊C4D412yes—
187#376C2311yescharacteristic
188#376C2311yescharacteristic
189C43C2311yescharacteristic
190C43C2311yescharacteristic
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C22×Q8—22no—
202C22×Q8C2412yes—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4C2412yes—
206C23×C4—22no—
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2411yescharacteristic
210C2×M16—24no—
211C2×M16—42no—
212C2×M16—42no—
213C22×C8—24no—
214C22×C8—42no—
215C22×C8—42no—
216C22×C8C2414yes—
217C22×C8C2414yes—
218C4⋊Q8—24no—
219C4⋊Q8—24no—
220C4⋊Q8—24no—
221C4⋊Q8—84no—
222#127—24no—
223#126—48no—
224#125—24no—
225#125—24no—
226#125—44no—
227#125—48no—
228#124—24no—
229#124—24no—
230#124—44no—
231#123—48no—
232C22⋊Q8—28no—
233C22⋊Q8—84no—
234#121—28no—
235#121—48no—
236C4×Q8—28no—
237C4×Q8—84no—
238C4×D4—28no—
239#117—42no—
240#117#4118yes—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2414yes—
261C2×C4⋊C4C2414yes—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4C2×D412yes—
275C2×C22⋊C4C2×D412yes—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2411yescharacteristic
287C2×C42C2411yescharacteristic
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C4⋊C8—28no—
293C4⋊C8—28no—
294#103—44no—
295#103—44no—
296#103—44no—
297#103—44no—
298#102—44no—
299#102—44no—
300#102—44no—
301#102—44no—
302#97—24no—
303#97—24no—
304#97—42no—
305C4×C8—42no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—22no—
316#95—22no—
317#95C2×D412yes—
318#95C2×D412yes—
319#95C2×D412yes—
320#95C2×D412yes—
321C24—22no—
322C24C22×D411yescharacteristic
323C2×Q8—24no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—24no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—44no—
330C2×D4—24no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—24no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—82no—
337C2×D4—82no—
338C2×D4—82no—
339C2×D4—82no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—44no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D412yes—
358C22×C4C22×D412yes—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D412yes—
361C22×C4C22×D412yes—
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C2×#4114yes—
365C22×C4C2511yescharacteristic
366M16—48no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—48no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—48no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—84no—
393C4⋊C4—84no—
394C4⋊C4—84no—
395C4⋊C4—84no—
396C4⋊C4#12418yes—
397C22⋊C4—42no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—82no—
406C22⋊C4—82no—
407C22⋊C4—82no—
408C22⋊C4—82no—
409C22⋊C4—84no—
410C22⋊C4#12418yes—
411C42—22no—
412C42—22no—
413C42—21no—
414C42—21no—
415C42—22no—
416C42—22no—
417C42—42no—
418C42—42no—
419C42—42no—
420C42—42no—
421C42—44no—
422C42—84no—
423C42C2×#4114yes—
424C42C2×#4114yes—
425C42H(5,𝔽2)12yes—
426C422−1+412yes—
427C23—22no—
428C23—22no—
429C23—21no—
430C23—21no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23C2×#12412yes—
439C23C23×D411yescharacteristic
440Q8—44no—
441Q8—44no—
442Q8—88no—
443D4—44no—
444D4—44no—
445D4—164no—
446D4—164no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—164no—
485C2×C4C2×#12414yes—
486C2×C4C2×#12414yes—
487C2×C4C2×#12412yes—
488C2×C4#53712yes—
489C2×C4#53712yes—
490C2×C4C22×#4112yes—
491C2×C4C2×H(5,𝔽2)11yescharacteristic
492C2×C4C2×2−1+411yescharacteristic
493C2×C4#58512yes—
494C8—44no—
495C8—44no—
496C22—22no—
497C22—24no—
498C22—42no—
499C22—41no—
500C22—41no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22C22×#12411yescharacteristic
509C22C2×#53711yescharacteristic
510C22#344911yescharacteristic
511C22#358512yes—
512C22#358612yes—
513C4—22no—
514C4—24no—
515C4—24no—
516C4—22no—
517C4—22no—
518C4—42no—
519C4—42no—
520C4—41no—
521C4—41no—
522C4—84no—
523C4—82no—
524C4—82no—
525C4—82no—
526C4—84no—
527C4#350612yes—
528C4#350712yes—
529C2—42no—
530C2—162no—
531C2#3518011yescharacteristic
532C2#6062512yes—
533C2#6092011yescharacteristic
534C2#6092111yescharacteristic
535C2#6102112yes—
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. #35284C2semidirect
3. #33057C2semidirect
4. #33051C2semidirect
5. #30191C2semidirect
6. #29361C2semidirect
7. #29360C2semidirect
8. #29223C2semidirect
9. #29146C2semidirect
10. #29137C2semidirect
11. #28754C2semidirect
12. #28726C2semidirect
13. #21898C2semidirect
15. #20427C2semidirect

Maximal subgroups

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

subgroupindex
#353322
#352842
#330572
#330512
#301912
#293612
#293602
#292232
#291462
#291372
#287542
#287262
#218982
#218972
#204272