Finite Group Databasebeta

← #6435668

Subgroups — #6435668

Order 512.

1574 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35257C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#29947C214yesmaximal, complemented
4#29946C214yesmaximal, complemented
5#29943C212yesmaximal, complemented
6#29943C212yesmaximal, complemented
7#29938C212yesmaximal, complemented
8#29936C212yesmaximal, complemented
9#23745C211yesmaximal, characteristic, complemented
10C2×#2099C211yesmaximal, characteristic, complemented
11#23657C211yesmaximal, characteristic, complemented
12#23656C211yesmaximal, characteristic, complemented
13#23643C212yesmaximal
14#23641C212yesmaximal
15#21315C212yesmaximal, complemented
16#21175C212yesmaximal, complemented
17#20630C211yesmaximal, characteristic
18C2×#544C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#428C2212yes—
21C2×#427C2212yes—
22C2×#424C2212yes—
23C2×#408C2212yes—
24#2857C2212yes—
25#2851C2212yes—
26#2629C2212yes—
27#2628C2212yes—
28#2454—21no—
29#2453—21no—
30#2452—21no—
31#2451—21no—
32#2450—22no—
33#2449—22no—
34#2355—21no—
35#2355—21no—
36#2354—21no—
37#2354—21no—
38#2352—22no—
39#2352—22no—
40#2317C2211yescharacteristic
41#2314C2214yes—
42#2311C2211yescharacteristic
43#2311C2212yes—
44#2311C2211yescharacteristic
45#2296C2211yescharacteristic
46C2×#389C2211yescharacteristic
47#2285C2212yes—
48#2285C2212yes—
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51#2274C2212yes—
52#2100C2212yes—
53#2100C2212yes—
54#2099C2212yes—
55#2099C2212yes—
56#2099C2214yes—
57#2099C2212yes—
58#2099C2212yes—
59#2085C2212yes—
60#2085C2212yes—
61#2085C2214yes—
62#2084C2211yescharacteristic
63#2084C2211yescharacteristic
64#2084C2212yes—
65#2083C2211yescharacteristic
66#2083C2211yescharacteristic
67#2083C2212yes—
68#2077C2214yes—
69#2077C2214yes—
70#2077C2214yes—
71#2077C2214yes—
72#2076C2214yes—
73#2076C2214yes—
74#2075C2214yes—
75#2075C2214yes—
76#1989C2214yes—
77#1989C2214yes—
78#1988C2214yes—
79#1987C2214yes—
80#1949C2214yes—
81#1949C2214yes—
82#1948C2214yes—
83#1947C2214yes—
84#1798C2214yes—
85#1797C2212yes—
86#1797C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C22×M16C2312yes—
97C23×C8C2312yes—
98#544—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×#125—21no—
102C2×#125C2312yes—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105C4×#41—28no—
106C2×C4×Q8D411yescharacteristic
107C2×C4×Q8D411yescharacteristic
108C2×C4×D4D412yes—
109C2×C4×D4D412yes—
110C2×C4×D4D411yescharacteristic
111C2×C4×D4D411yescharacteristic
112C2×#117—22no—
113C2×#117—22no—
114C2×#117C2311yescharacteristic
115C2×#117C2312yes—
116C2×#117C2311yescharacteristic
117C2×#117C2312yes—
118C2×#117C2311yescharacteristic
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127#428—44no—
128#427—44no—
129C2×#107C2312yes—
130C2×#107C2314yes—
131C2×#107C2312yes—
132C2×#106C2312yes—
133C2×#106C2312yes—
134C2×#106C2314yes—
135#424—28no—
136C2×C4⋊C8C2314yes—
137C2×C4⋊C8C2314yes—
138#408—44no—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141#400—22no—
142#399—22no—
143#398—22no—
144#398—22no—
145#396—22no—
146#396—22no—
147#395—22no—
148#393—22no—
149#391—22no—
150#391—21no—
151#391—21no—
152#391—22no—
153#390—22no—
154#390—21no—
155#390—21no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389C2311yescharacteristic
162#389C2312yes—
163#389C2311yescharacteristic
164#389C2312yes—
165#389C2311yescharacteristic
166#389C2311yescharacteristic
167#388—24no—
168#388—24no—
169#387—24no—
170#387—24no—
171#387—22no—
172#387—22no—
173#386—24no—
174#386—22no—
175#386—22no—
176#385—22no—
177#385—22no—
178#385—21no—
179#385—21no—
180#385—22no—
181#384—21no—
182#384—21no—
183#384C2312yes—
184#384C2312yes—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2312yes—
188#382—22no—
189#382—22no—
190#382C2312yes—
191#382C2312yes—
192#380—22no—
193#380—21no—
194#380—21no—
195#380—22no—
196#380—22no—
197#380C2314yes—
198#380C2314yes—
199#380C2312yes—
200#380C2312yes—
201#380C2312yes—
202#380C2312yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2314yes—
213C4×C22⋊C4—24no—
214C2×#95—21no—
215#340C2312yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2312yes—
234C2×#41—24no—
235C2×#41—22no—
236C2×#41—24no—
237C2×#41—22no—
238C22×Q8—22no—
239C22×Q8C2×D411yescharacteristic
240C22×D4C2×D412yes—
241C22×D4C2×D411yescharacteristic
242C23×C4—22no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C2×M16—24no—
252C2×M16—24no—
253C2×M16—24no—
254C22×C8—24no—
255C22×C8—24no—
256C22×C8—24no—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C4⋊Q8—28no—
261C4⋊Q8—44no—
262#125—28no—
263#125—44no—
264C22⋊Q8—48no—
265C22⋊Q8—84no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—28no—
271C4×D4—28no—
272#117—28no—
273#117—24no—
274#117—22no—
275#117—24no—
276#117—22no—
277#117—24no—
278#117—84no—
279#117—84no—
280#117#4118yes—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C42—24no—
336C2×C42—28no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349#107—42no—
350#107—42no—
351#107—44no—
352#106—42no—
353#106—42no—
354#106—44no—
355C4⋊C8—28no—
356C4⋊C8—28no—
357C22⋊C8—44no—
358C22⋊C8—44no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—24no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—24no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377C24C22×D412yes—
378C24C22×D411yescharacteristic
379#41—48no—
380#41—48no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—44no—
385C2×Q8—82no—
386C2×Q8—82no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—82no—
424C22×C4—81no—
425C22×C4—82no—
426C22×C4—81no—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C2×#4114yes—
436C22×C4H(5,𝔽2)12yes—
437C22×C42−1+414yes—
438C22×C42−1+412yes—
439C22×C4C2511yescharacteristic
440M16—48no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—24no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—48no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—28no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—82no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4#12314yes—
490C4⋊C4#12314yes—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—42no—
494C22⋊C4—44no—
495C22⋊C4—42no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C22⋊C4—84no—
499C22⋊C4—84no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4#12318yes—
503C42—22no—
504C42—22no—
505C42—24no—
506C42—24no—
507C42—22no—
508C42—22no—
509C42—42no—
510C42—44no—
511C42—42no—
512C42—84no—
513C42—84no—
514C42C2×#4118yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23C2×#12312yes—
527C23#53612yes—
528C23#53712yes—
529C23C23×D411yescharacteristic
530Q8—44no—
531Q8—164no—
532D4—48no—
533D4—44no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—48no—
552C2×C4—48no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—84no—
585C2×C4—162no—
586C2×C4—164no—
587C2×C4C2×#12314yes—
588C2×C4C2×#12314yes—
589C2×C4C2×#12312yes—
590C2×C4#53612yes—
591C2×C4#53712yes—
592C2×C4#54118yes—
593C2×C4#54218yes—
594C2×C4#57612yes—
595C2×C4#57712yes—
596C2×C4C22×#4112yes—
597C2×C4C2×H(5,𝔽2)11yescharacteristic
598C2×C4C2×2−1+411yescharacteristic
599C2×C4C2×2−1+412yes—
600C8—42no—
601C8—42no—
602C8—44no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22#262514yes—
617C22C22×#12311yescharacteristic
618C22C2×#53611yescharacteristic
619C22C2×#53711yescharacteristic
620C22C2×#57611yescharacteristic
621C22C2×#57711yescharacteristic
622C22#358611yescharacteristic
623C22#358611yescharacteristic
624C4—22no—
625C4—24no—
626C4—22no—
627C4—24no—
628C4—42no—
629C4—41no—
630C4—41no—
631C4—82no—
632C4—81no—
633C4—82no—
634C4—81no—
635C4—82no—
636C4—82no—
637C4—81no—
638C4—81no—
639C4—84no—
640C4#262514yes—
641C4C2×#54112yes—
642C4C2×#54212yes—
643C2—24no—
644C2—82no—
645C2C2×#262511yescharacteristic
646C2#6052111yescharacteristic
647C2#6052211yescharacteristic
648C2#6087711yescharacteristic
649C2#6088411yescharacteristic
650C2#6092811yescharacteristic
651C2#6093111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29947C2semidirect
4. #29946C2semidirect
5. #29943C2semidirect
6. #29943C2semidirect
7. #29938C2semidirect
8. #29936C2semidirect
9. #23745C2semidirect
10. C2×#2099C2semidirect
11. #23657C2semidirect
12. #23656C2semidirect
15. #21315C2semidirect
16. #21175C2semidirect

Maximal subgroups

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

subgroupindex
#352572
#341882
#299472
#299462
#299432
#299382
#299362
#237452
C2×#20992
#236572
#236562
#236432
#236412
#213152
#211752
#206302