Finite Group Databasebeta

← #6571390

Subgroups — #6571390

Order 512.

1527 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36302C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#30061C212yesmaximal, complemented
4#29921C211yesmaximal, characteristic
5#29388C212yesmaximal, complemented
6#29352C212yesmaximal, complemented
7#27410C214yesmaximal, complemented
8#27348C214yesmaximal, complemented
9#26441C212yesmaximal, complemented
10#25903C211yesmaximal, characteristic
11#25277C212yesmaximal, complemented
12#24302C211yesmaximal, characteristic, complemented
13#23780C212yesmaximal, complemented
14#23757C211yesmaximal, characteristic, complemented
15#23223C212yesmaximal, complemented
16#23069C212yesmaximal, complemented
17#22643C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#486C2212yes—
20C2×#477C2212yes—
21#2861C2212yes—
22#2856C2211yescharacteristic
23#2552C2211yescharacteristic
24#2548C2211yescharacteristic
25#2531—22no—
26#2513—21no—
27#2500—21no—
28#2490C2212yes—
29#2476—22no—
30#2427C2212yes—
31#2369C2212yes—
32#2369C2212yes—
33#2366C2211yescharacteristic
34#2365—21no—
35#2365—21no—
36#2363—21no—
37#2355C2211yescharacteristic
38#2354—21no—
39#2345C2214yes—
40Q8×C22⋊C4—22no—
41#2331—22no—
42#2316—21no—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×C22⋊Q8C2211yescharacteristic
46#2293C2212yes—
47#2285C2211yescharacteristic
48#2274—21no—
49C4×#117C2211yescharacteristic
50#2100C2211yescharacteristic
51#2095C2211yescharacteristic
52#2094C2212yes—
53#2088C2214yes—
54#2083C2211yescharacteristic
55#2076C2212yes—
56#2072C2211yescharacteristic
57#2071C2212yes—
58#2067C2214yes—
59#2059C2214yes—
60#2057C2212yes—
61#2042C2214yes—
62#2036C2212yes—
63#2034C2214yes—
64#2024C2212yes—
65#2023C2212yes—
66#2023C2212yes—
67#1990C2212yes—
68#1989C2214yes—
69#1987C2212yes—
70#1983C2214yes—
71#1982C2212yes—
72#1948C2212yes—
73#1939C2212yes—
74#1936C2214yes—
75#1935C2214yes—
76#1927C2214yes—
77#1925C2214yes—
78#1925C2214yes—
79#1924C2214yes—
80#1918C2214yes—
81#1918C2214yes—
82#1874C2214yes—
83#1872C2212yes—
84#1862C2214yes—
85#1852C2212yes—
86#1839C2212yes—
87#1797C2211yescharacteristic
88#1776C2214yes—
89C4×#107C2212yes—
90#1770C2212yes—
91#1767C2212yes—
92#1762C2214yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×#127C2311yescharacteristic
102C2×#125C2312yes—
103C2×#124C2312yes—
104C2×C22⋊Q8C2311yescharacteristic
105C2×#121—21no—
106C2×#121—21no—
107C2×#121C2312yes—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×D4—21no—
112C2×C4×D4C2312yes—
113C2×#117—21no—
114C2×#117—22no—
115C2×#117D411yescharacteristic
116C2×#117D411yescharacteristic
117C22×C4⋊C4—21no—
118C22×C4⋊C4C2312yes—
119C22×C42C2311yescharacteristic
120#486—28no—
121#477—28no—
122C2×#107C2314yes—
123C2×#107C2312yes—
124C2×#107C2312yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2314yes—
128C2×#103C2314yes—
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×#97C2314yes—
138C2×C4×C8C2312yes—
139#398—22no—
140#397—22no—
141#397—22no—
142#394—22no—
143#394—22no—
144#394—22no—
145#391—22no—
146#391—22no—
147#391—24no—
148#391—24no—
149#391C2311yescharacteristic
150#391C2312yes—
151#390—21no—
152#390—21no—
153#390—21no—
154#390—22no—
155#390—21no—
156#390—21no—
157#390C2312yes—
158#390C2312yes—
159#390C2312yes—
160#389—22no—
161#389—22no—
162#389—21no—
163#389—22no—
164#389—22no—
165#389C2314yes—
166#389C2314yes—
167#389C2311yescharacteristic
168#389C2312yes—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#387—21no—
177#387—24no—
178#387—24no—
179#387—22no—
180#386—22no—
181#386—22no—
182#385—21no—
183#384—21no—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#382—21no—
188#382—21no—
189#380—21no—
190#380—24no—
191#380—22no—
192#380C2311yescharacteristic
193#380C2312yes—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4C2314yes—
204C4×C4⋊C4C2314yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—21no—
211C4×C22⋊C4—24no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4C2312yes—
215#376D412yes—
216C43C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#336C2312yes—
227#336C2314yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2314yes—
231C22×Q8—28no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—82no—
235C22×Q8—82no—
236C22×Q8C2411yescharacteristic
237C22×D4—22no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2412yes—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C22×C8C2412yes—
247C22×C8C2414yes—
248C22×C8C2412yes—
249C4⋊Q8—28no—
250C4⋊Q8—28no—
251C4⋊Q8—28no—
252C4⋊Q8—24no—
253C4⋊Q8#4118yes—
254#127—24no—
255#125—44no—
256#124—44no—
257C22⋊Q8—22no—
258C22⋊Q8—41no—
259#121—44no—
260#121—44no—
261#121—82no—
262C4×Q8—24no—
263C4×Q8—84no—
264C4×Q8—84no—
265C4×D4—24no—
266C4×D4—42no—
267C4×D4—44no—
268#117—24no—
269#117—42no—
270#117—44no—
271#117—84no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2411yescharacteristic
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2411yescharacteristic
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C22⋊C4—22no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42C2×D412yes—
325C2×C42C2×D412yes—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2412yes—
331C2×C42C2412yes—
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334#107—28no—
335#107—24no—
336#107—42no—
337#106—44no—
338C4⋊C8—24no—
339C4⋊C8—44no—
340#103—28no—
341#103—42no—
342#103—42no—
343#102—28no—
344#102—28no—
345#102—44no—
346#102—44no—
347C22⋊C8—42no—
348C22⋊C8—42no—
349#97—44no—
350C4×C8—24no—
351#95—21no—
352#95—24no—
353#95—24no—
354#95—22no—
355#95—21no—
356#95—22no—
357#95—22no—
358#95—21no—
359#95—21no—
360#95—22no—
361#95—24no—
362#95—24no—
363#95—22no—
364#95—21no—
365#95—22no—
366#95C2×D412yes—
367#95C2×D412yes—
368#95C2×D412yes—
369#95C2×D412yes—
370C24—22no—
371C24C22×D411yescharacteristic
372C2×Q8—28no—
373C2×Q8—28no—
374C2×Q8—22no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—82no—
381C2×Q8—82no—
382C2×Q8—816no—
383C2×D4—24no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—24no—
387C2×D4—44no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—81no—
393C2×D4—81no—
394C2×D4—82no—
395C2×D4—82no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—28no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—41no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4#13714yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C2511yescharacteristic
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—24no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—24no—
439C4⋊C4—28no—
440C4⋊C4—28no—
441C4⋊C4—28no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—82no—
464C4⋊C4—84no—
465C4⋊C4—84no—
466C4⋊C4—81no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4C2×#4118yes—
470C4⋊C42−1+412yes—
471C22⋊C4—24no—
472C22⋊C4—42no—
473C22⋊C4—44no—
474C22⋊C4—42no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—82no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C22⋊C4—81no—
486C42—22no—
487C42—22no—
488C42—24no—
489C42—24no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—24no—
494C42—42no—
495C42—42no—
496C42—42no—
497C42—44no—
498C42—42no—
499C42—82no—
500C42—84no—
501C42—84no—
502C42C2×#4114yes—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23C2×#13712yes—
517C23C23×D411yescharacteristic
518Q8—416no—
519Q8—42no—
520Q8—42no—
521Q8—88no—
522D4—44no—
523D4—44no—
524D4—84no—
525D4—162no—
526D4—164no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—28no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—28no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—41no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—88no—
577C2×C4—88no—
578C2×C4#52512yes—
579C2×C4#52512yes—
580C2×C4C2×#13712yes—
581C2×C4C2×#13712yes—
582C2×C4#57512yes—
583C2×C4#57612yes—
584C2×C4C22×#4112yes—
585C2×C4C2×2−1+411yescharacteristic
586C2×C4#58512yes—
587C2×C4#58511yescharacteristic
588C8—42no—
589C8—42no—
590C8—44no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22C2×#52511yescharacteristic
606C22#348911yescharacteristic
607C22#350011yescharacteristic
608C22C22×#13711yescharacteristic
609C22C2×#57511yescharacteristic
610C22C2×#57611yescharacteristic
611C22#358511yescharacteristic
612C4—22no—
613C4—22no—
614C4—22no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—48no—
619C4—42no—
620C4—42no—
621C4—41no—
622C4—82no—
623C4—82no—
624C4—81no—
625C4—81no—
626C4—82no—
627C4—81no—
628C4#311714yes—
629C4#351812yes—
630C4#352412yes—
631C2—42no—
632C2—162no—
633C2#3607711yescharacteristic
634C2C2×#311711yescharacteristic
635C2#6057811yescharacteristic
636C2#6211211yescharacteristic
637C2#6214311yescharacteristic
638C2#6253711yescharacteristic
639C2#6258611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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. #34216C2semidirect
3. #30061C2semidirect
5. #29388C2semidirect
6. #29352C2semidirect
7. #27410C2semidirect
8. #27348C2semidirect
9. #26441C2semidirect
11. #25277C2semidirect
12. #24302C2semidirect
13. #23780C2semidirect
14. #23757C2semidirect
15. #23223C2semidirect
16. #23069C2semidirect
17. #22643C2semidirect

Maximal subgroups

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

subgroupindex
#363022
#342162
#300612
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