Finite Group Databasebeta

← #6569122

Subgroups — #6569122

Order 512.

1554 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic, complemented
2#36300C211yesmaximal, characteristic, complemented
3#30336C212yesmaximal, complemented
4#30203C211yesmaximal, characteristic, complemented
5#29558C212yesmaximal, complemented
6#29521C212yesmaximal, complemented
7#29213C211yesmaximal, characteristic, complemented
8#29088C214yesmaximal, complemented
9#28381C211yesmaximal, characteristic
10#28017C214yesmaximal, complemented
11#25477C212yesmaximal, complemented
12#25300C212yesmaximal, complemented
13#23819C211yesmaximal, characteristic, complemented
14#23775C212yesmaximal, complemented
15#23093C212yesmaximal, complemented
16#23061C212yesmaximal, complemented
17#20573C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×#476C2212yes—
20C2×#459C2212yes—
21C2×#446C2212yes—
22#2868C2211yescharacteristic
23#2860C2212yes—
24#2858C2212yes—
25#2858C2211yescharacteristic
26#2567C2211yescharacteristic
27#2548C2211yescharacteristic
28#2547—21no—
29#2545—22no—
30#2538—21no—
31#2538C2212yes—
32#2521—22no—
33#2513C2212yes—
34#2512—21no—
35#2512—22no—
36#2512C2214yes—
37#2474—22no—
38#2454C2212yes—
39#2448—21no—
40#2369C2211yescharacteristic
41#2367—21no—
42#2366C2211yescharacteristic
43#2354—21no—
44#2354—21no—
45#2354C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×#127C2211yescharacteristic
48C4×#125C2211yescharacteristic
49#2293C2212yes—
50C42×Q8C2211yescharacteristic
51#2100C2211yescharacteristic
52#2100C2211yescharacteristic
53#2094C2212yes—
54#2088C2214yes—
55#2073C2212yes—
56#2071C2214yes—
57#2067C2214yes—
58#2065C2214yes—
59#2058C2212yes—
60#2057C2212yes—
61#2054C2214yes—
62#2041C2211yescharacteristic
63#2041C2211yescharacteristic
64#2025C2212yes—
65#2023C2212yes—
66#2023C2212yes—
67#2011C2212yes—
68#1992C2212yes—
69#1990C2212yes—
70#1972C2214yes—
71#1972C2214yes—
72#1970C2214yes—
73#1957C2212yes—
74#1948C2212yes—
75#1947C2212yes—
76#1943C2214yes—
77#1936C2214yes—
78#1936C2214yes—
79#1934C2214yes—
80#1926C2214yes—
81#1925C2214yes—
82#1920C2214yes—
83#1879C2214yes—
84#1847C2212yes—
85C4×#106C2214yes—
86#1771C2214yes—
87C8×C4⋊C4C2212yes—
88#1767C2211yescharacteristic
89C4×C4⋊C8C2212yes—
90C4×#102C2212yes—
91C4×#102C2212yes—
92C4×#102C2214yes—
93#1745C2212yes—
94#1744C2211yescharacteristic
95C22×D8C2314yes—
96#556—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×#127C2311yescharacteristic
102C2×#127C2311yescharacteristic
103C2×#126—24no—
104C2×#126—22no—
105C2×#125C2311yescharacteristic
106C2×#124—21no—
107C2×#124—22no—
108C2×#124—21no—
109C2×#124—21no—
110C2×#124—22no—
111C2×C22⋊Q8—22no—
112C2×C4×Q8—22no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4C2312yes—
120C2×C4×D4C2312yes—
121C2×#117—22no—
122C2×#117—22no—
123#476—28no—
124#459—28no—
125#446—28no—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2312yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2312yes—
134C2×#102C2312yes—
135C2×#102C2314yes—
136C2×#102C2312yes—
137C2×#102C2314yes—
138C2×#102C2312yes—
139C2×#97C2314yes—
140C2×#97C2312yes—
141C2×C4×C8C2312yes—
142C2×C4×C8C2314yes—
143#399—22no—
144#397—22no—
145#397—22no—
146#397—24no—
147#396—24no—
148#396—24no—
149#396—22no—
150#395—22no—
151#394—22no—
152#391—21no—
153#391—22no—
154#391—22no—
155#390—21no—
156#390—21no—
157#390—21no—
158#390—21no—
159#390—21no—
160#390—22no—
161#390—21no—
162#390—22no—
163#390C2312yes—
164#390C2312yes—
165#390C2314yes—
166#390C2312yes—
167#390C2312yes—
168#390C2312yes—
169#390C2314yes—
170#390C2312yes—
171#389—21no—
172#389—22no—
173#389C2311yescharacteristic
174#389C2311yescharacteristic
175#389C2314yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—24no—
186#388—22no—
187#388—24no—
188#387—21no—
189#387—21no—
190#386—22no—
191#385—22no—
192#384—21no—
193#384—21no—
194#384—21no—
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2312yes—
198#382C2311yescharacteristic
199#380—22no—
200#380—22no—
201#380—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—22no—
217C43C2311yescharacteristic
218C43C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#336C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2314yes—
232C22×Q8—22no—
233C22×Q8—21no—
234C22×Q8C2×D411yescharacteristic
235C22×D4—21no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4C2412yes—
240C22×D4C2412yes—
241C23×C4—22no—
242C23×C4—22no—
243C2×D8—28no—
244C2×D8—44no—
245C2×D8—44no—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2414yes—
249C4⋊Q8—24no—
250C4⋊Q8—44no—
251C4⋊Q8—44no—
252C4⋊Q8—82no—
253#127—24no—
254#127—42no—
255#126—416no—
256#126—84no—
257#125—22no—
258#125—41no—
259#124—48no—
260#124—44no—
261#124—82no—
262#124—82no—
263#124—84no—
264C22⋊Q8—48no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—42no—
268C4×Q8—48no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—24no—
272C4×D4—42no—
273C4×D4—44no—
274C4×D4—44no—
275C4×D4—44no—
276#117—48no—
277#117—84no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2411yescharacteristic
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C42—24no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—22no—
329C2×C42C2×D412yes—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D412yes—
337C2×C42C2412yes—
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—44no—
343#106—28no—
344#106—44no—
345C4⋊C8—28no—
346C4⋊C8#4118yes—
347#102—28no—
348#102—28no—
349#102—24no—
350#102—42no—
351#102—42no—
352#102—44no—
353#102—42no—
354#102—44no—
355#97—28no—
356#97—42no—
357C4×C8—24no—
358C4×C8—44no—
359#95—21no—
360#95—22no—
361#95—22no—
362#95—24no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—24no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374C24—22no—
375C24—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—42no—
382C2×Q8—44no—
383C2×Q8—81no—
384C2×Q8—81no—
385C2×D4—22no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—24no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—81no—
406C2×D4—81no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—82no—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C2511yescharacteristic
431D8—88no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8C2×#4118yes—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4AGL(1,ℤ/8)14yes—
473C4⋊C42−1+412yes—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—48no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—48no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—82no—
486C22⋊C4—84no—
487C22⋊C4—84no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C42—22no—
493C42—24no—
494C42—22no—
495C42—24no—
496C42—22no—
497C42—24no—
498C42—22no—
499C42—24no—
500C42—22no—
501C42—21no—
502C42—22no—
503C42—21no—
504C42—22no—
505C42—42no—
506C42—48no—
507C42—42no—
508C42—42no—
509C42—82no—
510C42C2×#4114yes—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23C23×D411yescharacteristic
526Q8—44no—
527Q8—88no—
528Q8—162no—
529D4—42no—
530D4—44no—
531D4—42no—
532D4—42no—
533D4—44no—
534D4—42no—
535D4—84no—
536D4—84no—
537D4—84no—
538D4—162no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—88no—
584C2×C4—88no—
585C2×C4#53212yes—
586C2×C4#53212yes—
587C2×C4C2×AGL(1,ℤ/8)14yes—
588C2×C4C2×AGL(1,ℤ/8)12yes—
589C2×C4#57512yes—
590C2×C4#57612yes—
591C2×C4#57712yes—
592C2×C4C22×#4112yes—
593C2×C4C2×2−1+411yescharacteristic
594C2×C4#58512yes—
595C2×C4#58511yescharacteristic
596C8—28no—
597C8—42no—
598C8—42no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22C2×#53211yescharacteristic
614C22#348711yescharacteristic
615C22#348911yescharacteristic
616C22C22×AGL(1,ℤ/8)11yescharacteristic
617C22C2×#57511yescharacteristic
618C22C2×#57611yescharacteristic
619C22C2×#57711yescharacteristic
620C4—22no—
621C4—24no—
622C4—22no—
623C4—22no—
624C4—42no—
625C4—42no—
626C4—41no—
627C4—42no—
628C4—42no—
629C4—41no—
630C4—82no—
631C4—82no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—84no—
636C4#317114yes—
637C4#351812yes—
638C4#352112yes—
639C2—162no—
640C2—162no—
641C2#3604811yescharacteristic
642C2C2×#317111yescharacteristic
643C2#6072211yescharacteristic
644C2#6189911yescharacteristic
645C2#6190711yescharacteristic
646C2#6212811yescharacteristic
647C2#6213611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28381C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36594C2semidirect
2. #36300C2semidirect
3. #30336C2semidirect
4. #30203C2semidirect
5. #29558C2semidirect
6. #29521C2semidirect
7. #29213C2semidirect
8. #29088C2semidirect
10. #28017C2semidirect
11. #25477C2semidirect
12. #25300C2semidirect
13. #23819C2semidirect
14. #23775C2semidirect
15. #23093C2semidirect
16. #23061C2semidirect
17. #20573C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#363002
#303362
#302032
#295582
#295212
#292132
#290882
#283812
#280172
#254772
#253002
#238192
#237752
#230932
#230612
#205732