Finite Group Databasebeta

← #6435222

Subgroups — #6435222

Order 512.

1646 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic, complemented
2#34253C211yesmaximal, characteristic
3#29993C214yesmaximal, complemented
4#29991C214yesmaximal, complemented
5#29989C212yesmaximal, complemented
6#29987C212yesmaximal, complemented
7#29984C212yesmaximal, complemented
8#29981C212yesmaximal, complemented
9#23747C211yesmaximal, characteristic, complemented
10C2×#2100C211yesmaximal, characteristic, complemented
11#23734C211yesmaximal, characteristic
12#23730C211yesmaximal, characteristic
13#23557C212yesmaximal, complemented
14#23458C212yesmaximal
15#21175C212yesmaximal, complemented
16#20754C211yesmaximal, characteristic, complemented
17#20652C212yesmaximal
18C2×#544C2212yes—
19C2×#520C2211yescharacteristic
20C2×#428C2212yes—
21C2×#427C2212yes—
22C2×#424C2212yes—
23C2×#417C2212yes—
24C2×#416C2212yes—
25#2868C2212yes—
26#2859C2212yes—
27#2660C2214yes—
28#2552—21no—
29#2552—21no—
30#2551—21no—
31#2551—21no—
32#2440—22no—
33#2440—22no—
34#2353—22no—
35#2352—22no—
36#2347—21no—
37#2346—21no—
38#2345—21no—
39#2344—21no—
40#2320C2212yes—
41#2319C2212yes—
42#2317C2212yes—
43#2316C2211yescharacteristic
44#2311C2211yescharacteristic
45#2297C2211yescharacteristic
46#2295C2212yes—
47#2293C2212yes—
48C2×#389C2211yescharacteristic
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51#2274C2211yescharacteristic
52C4×#117C2211yescharacteristic
53#2101C2214yes—
54#2100C2212yes—
55#2100C2212yes—
56#2100C2212yes—
57#2100C2214yes—
58#2099C2212yes—
59#2097C2212yes—
60#2096C2211yescharacteristic
61#2096C2212yes—
62#2095C2211yescharacteristic
63#2095C2212yes—
64#2094C2211yescharacteristic
65#2094C2212yes—
66#2093C2211yescharacteristic
67#2093C2212yes—
68#2092C2212yes—
69#2059C2214yes—
70#2059C2214yes—
71#2058C2214yes—
72#2057C2214yes—
73#2026C2214yes—
74#2025C2214yes—
75#2024C2214yes—
76#2023C2214yes—
77#1949C2214yes—
78#1949C2214yes—
79#1948C2214yes—
80#1947C2214yes—
81#1832C2214yes—
82#1831C2212yes—
83#1831C2212yes—
84#1805C2214yes—
85#1804C2214yes—
86#1803C2214yes—
87#1802C2214yes—
88#1736C2212yes—
89#1736C2212yes—
90C2×#340C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C22×M16C2312yes—
97C23×C8C2312yes—
98#544—44no—
99#544—44no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8C2314yes—
104C2×#125—22no—
105C2×#125C2314yes—
106C2×C22⋊Q8—24no—
107#520—24no—
108#520Q818yescomplemented
109C2×C4×Q8C2312yes—
110C2×C4×D4—22no—
111C2×C4×D4C2312yes—
112C2×#117—21no—
113C2×#117—22no—
114C2×#117C2311yescharacteristic
115C2×#117C2312yes—
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#428—44no—
127#427—44no—
128C2×#107C2312yes—
129C2×#107C2314yes—
130C2×#107C2312yes—
131C2×#106C2312yes—
132C2×#106C2314yes—
133C2×#106C2312yes—
134#424—28no—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2314yes—
137#417—44no—
138#416—44no—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145#398—24no—
146#398—24no—
147#395—24no—
148#395—24no—
149#391C2312yes—
150#391C2311yescharacteristic
151#391C2311yescharacteristic
152#390C2312yes—
153#390C2311yescharacteristic
154#390C2311yescharacteristic
155#389—21no—
156#389—21no—
157#389—22no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#386—21no—
178#386—21no—
179#386—22no—
180#385—24no—
181#384—22no—
182#384—22no—
183#384—21no—
184#384—21no—
185#384C2314yes—
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#384C2311yescharacteristic
190#383—22no—
191#382—22no—
192#382C2314yes—
193#382C2312yes—
194#381—22no—
195#380—21no—
196#380—21no—
197#380—21no—
198#380—21no—
199#380—22no—
200#380—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—21no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4D412yes—
218#376—22no—
219#376C2312yes—
220C2×#95—21no—
221C43C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#336C2314yes—
231#336C2314yes—
232#336C2311yescharacteristic
233#336C2312yes—
234#336C2311yescharacteristic
235#336C2312yes—
236#336C2312yes—
237C2×#41—22no—
238C2×#41—24no—
239C2×#41—24no—
240C2×#41C2×Q814yes—
241C22×Q8—24no—
242C22×Q8C2411yescharacteristic
243C22×D4—21no—
244C22×D4C2411yescharacteristic
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—21no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C2×M16—24no—
254C2×M16—24no—
255C2×M16—24no—
256C22×C8—24no—
257C22×C8—24no—
258C22×C8—24no—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C4⋊Q8—44no—
263C4⋊Q8—44no—
264C4⋊Q8—48no—
265C4⋊Q8—84no—
266C4⋊Q8—84no—
267#125—44no—
268#125—44no—
269#125—48no—
270C22⋊Q8—416no—
271C4×Q8—24no—
272C4×Q8C2×Q818yes—
273C4×D4—216no—
274C4×D4—24no—
275C4×D4C2×Q818yes—
276#117—24no—
277#117—28no—
278#117—24no—
279#117—24no—
280#117—24no—
281#117—84no—
282#117C2×Q814yes—
283#117#4114yes—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—82no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×Q814yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2412yes—
352#107—42no—
353#107—42no—
354#107—44no—
355#106—42no—
356#106—42no—
357#106—44no—
358C4⋊C8—28no—
359C4⋊C8—28no—
360#103—42no—
361#103—42no—
362#103—44no—
363#102—42no—
364#102—42no—
365#102—44no—
366#95—22no—
367#95—22no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—21no—
387C24C22×D411yescharacteristic
388#41—216no—
389#41—48no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—48no—
394C2×Q8—84no—
395C2×Q8—84no—
396C2×Q8C22×Q812yes—
397C2×D4—24no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4C22×Q812yes—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×Q812yes—
443C22×C4C2×#4112yes—
444C22×C4H(5,𝔽2)14yes—
445C22×C42−1+414yes—
446C22×C4C2511yescharacteristic
447M16—48no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—48no—
458C4⋊C4—24no—
459C4⋊C4—28no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4C22⋊Q814yes—
492C4⋊C4C22⋊Q814yes—
493C4⋊C4C22×Q814yes—
494C22⋊C4—24no—
495C22⋊C4—28no—
496C22⋊C4—42no—
497C22⋊C4—44no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—82no—
501C22⋊C4—81no—
502C22⋊C4—81no—
503C22⋊C4—82no—
504C22⋊C4C22⋊Q818yes—
505C42—28no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—24no—
510C42—24no—
511C42—22no—
512C42—22no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—44no—
517C42—82no—
518C42—82no—
519C42C22×Q814yes—
520C42C2×#4114yes—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C2×C22⋊Q812yes—
533C23#53512yes—
534C23#53612yes—
535C23C23×D411yescharacteristic
536Q8—24no—
537Q8—42no—
538Q8—168no—
539D4—24no—
540D4—48no—
541D4—42no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—48no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—88no—
590C2×C4—84no—
591C2×C4—164no—
592C2×C4—162no—
593C2×C4C2×C22⋊Q814yes—
594C2×C4C2×C22⋊Q814yes—
595C2×C4C2×C22⋊Q812yes—
596C2×C4#53512yes—
597C2×C4#53612yes—
598C2×C4#54218yes—
599C2×C4#54418yes—
600C2×C4#57712yes—
601C2×C4#57812yes—
602C2×C4C23×Q811yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C2×C4C2×H(5,𝔽2)12yes—
605C2×C4C2×2−1+412yes—
606C8—42no—
607C8—42no—
608C8—44no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22#262814yes—
623C22C22×C22⋊Q811yescharacteristic
624C22C2×#53511yescharacteristic
625C22C2×#53611yescharacteristic
626C22C2×#57711yescharacteristic
627C22C2×#57811yescharacteristic
628C22#358511yescharacteristic
629C22#358511yescharacteristic
630C4—24no—
631C4—24no—
632C4—22no—
633C4—22no—
634C4—42no—
635C4—44no—
636C4—42no—
637C4—42no—
638C4—41no—
639C4—41no—
640C4—82no—
641C4—81no—
642C4—81no—
643C4—84no—
644C4—82no—
645C4—82no—
646C4#262814yes—
647C4C2×#54212yes—
648C4C2×#54412yes—
649C2—24no—
650C2—82no—
651C2C2×#262811yescharacteristic
652C2#6049511yescharacteristic
653C2#6049811yescharacteristic
654C2#6082411yescharacteristic
655C2#6083011yescharacteristic
656C2#6087311yescharacteristic
657C2#6087411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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
1. #35275C2semidirect
3. #29993C2semidirect
4. #29991C2semidirect
5. #29989C2semidirect
6. #29987C2semidirect
7. #29984C2semidirect
8. #29981C2semidirect
9. #23747C2semidirect
10. C2×#2100C2semidirect
13. #23557C2semidirect
15. #21175C2semidirect
16. #20754C2semidirect
108. #520Q8semidirect

Maximal subgroups

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

subgroupindex
#352752
#342532
#299932
#299912
#299892
#299872
#299842
#299812
#237472
C2×#21002
#237342
#237302
#235572
#234582
#211752
#207542
#206522