Finite Group Databasebeta

← #6434774

Subgroups — #6434774

Order 512.

1608 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35320C211yesmaximal, characteristic
2#34272C211yesmaximal, characteristic
3#30073C214yesmaximal, complemented
4#30070C214yesmaximal, complemented
5#30067C212yesmaximal, complemented
6#30067C212yesmaximal, complemented
7#30063C212yesmaximal, complemented
8#30060C212yesmaximal, complemented
9#23736C212yesmaximal
10#23554C211yesmaximal, characteristic
11#23551C211yesmaximal, characteristic
12#23507C211yesmaximal, characteristic
13C2×#2043C211yesmaximal, characteristic
14#23457C212yesmaximal
15#21066C212yesmaximal
16#21064C212yesmaximal
17#20774C211yesmaximal, characteristic
18C2×#544C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C2×#428C2212yes—
22C2×#427C2212yes—
23C2×#419C2212yes—
24C2×#419C2212yes—
25C22×#103C2212yes—
26C22×#103C2212yes—
27#2862C2212yes—
28#2862C2212yes—
29#2642C2214yes—
30#2568—21no—
31#2568—21no—
32#2568—21no—
33#2568—21no—
34#2503—22no—
35#2503—22no—
36#2370—21no—
37#2370—21no—
38#2369—21no—
39#2368—22no—
40#2365—21no—
41#2360—22no—
42#2331C2212yes—
43#2330C2212yes—
44#2317C2211yescharacteristic
45#2310C2212yes—
46#2310C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×#125C2212yes—
49#2296C2211yescharacteristic
50C2×#389C2211yescharacteristic
51#2274C2211yescharacteristic
52#2274C2211yescharacteristic
53C4×#117C2211yescharacteristic
54#2097C2214yes—
55#2097C2214yes—
56#2096C2214yes—
57#2095C2214yes—
58#2059C2214yes—
59#2059C2212yes—
60#2059C2212yes—
61#2058C2212yes—
62#2058C2211yescharacteristic
63#2058C2211yescharacteristic
64#2057C2212yes—
65#2057C2211yescharacteristic
66#2057C2211yescharacteristic
67#2043C2212yes—
68#2043C2212yes—
69#2043C2212yes—
70#2043C2212yes—
71#2043C2212yes—
72#2043C2214yes—
73#2043C2212yes—
74#2026C2214yes—
75#2026C2214yes—
76#2024C2214yes—
77#2024C2214yes—
78#1928C2214yes—
79#1928C2214yes—
80#1928C2214yes—
81#1928C2214yes—
82#1927C2214yes—
83#1927C2214yes—
84#1927C2214yes—
85#1927C2214yes—
86#1840C2214yes—
87#1839C2212yes—
88#1839C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91#1729C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C23×Q8C2312yes—
96C22×M16C2312yes—
97C23×C8C2312yes—
98#544—24no—
99#544—24no—
100#533—44no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×#125—21no—
111C2×#125C2312yes—
112C2×#125C2312yes—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—22no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—22no—
119C2×#117C2312yes—
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#428—44no—
131#427—44no—
132C2×#107C2312yes—
133C2×#107C2314yes—
134C2×#107C2312yes—
135C2×#106C2312yes—
136C2×#106C2314yes—
137C2×#106C2312yes—
138#419—28no—
139#419—44no—
140C2×#103—28no—
141C2×#103—44no—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150#400—24no—
151#400—24no—
152#400—24no—
153#400—24no—
154#391—22no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391—22no—
164#391—22no—
165#391C2314yes—
166#391C2314yes—
167#391C2314yes—
168#391C2314yes—
169#389—21no—
170#389—21no—
171#389—22no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389C2311yescharacteristic
177#389C2312yes—
178#389C2311yescharacteristic
179#389C2311yescharacteristic
180#389C2312yes—
181#389C2311yescharacteristic
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#385—21no—
189#385—21no—
190#385—22no—
191#384C2312yes—
192#382—22no—
193#382—22no—
194#382—22no—
195#382—22no—
196#382C2312yes—
197#381—24no—
198#381—24no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4D412yes—
220#376C2312yes—
221C2×#95—21no—
222C43C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2314yes—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—24no—
243C22×Q8—24no—
244C22×Q8—24no—
245C22×Q8—24no—
246C22×Q8—24no—
247C22×Q8—24no—
248C22×Q8C2414yes—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×M16—24no—
256C2×M16—24no—
257C2×M16—24no—
258C22×C8—24no—
259C22×C8—24no—
260C22×C8—24no—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C4⋊Q8—28no—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—42no—
268C4⋊Q8—42no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273#125—28no—
274#125—24no—
275#125—24no—
276#125—42no—
277#125—42no—
278C22⋊Q8—216no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×Q8—84no—
284#117—22no—
285#117—22no—
286#117—22no—
287#117—24no—
288#117—24no—
289#117—24no—
290#117—24no—
291#117#4114yes—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4#4114yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C42—21no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2412yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—42no—
355#107—42no—
356#107—44no—
357#106—42no—
358#106—42no—
359#106—44no—
360#103—28no—
361#103—28no—
362#103—28no—
363#103—28no—
364#103—44no—
365#103—44no—
366#103—44no—
367#103—44no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—21no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24C22×D411yescharacteristic
393C2×Q8—28no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—48no—
410C2×Q8—48no—
411C2×Q8—48no—
412C2×Q8—48no—
413C2×Q8—82no—
414C2×Q8—82no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4H(5,𝔽2)14yes—
451C22×C42−1+414yes—
452C22×C4C2511yescharacteristic
453M16—48no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—48no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—82no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4C2×#4114yes—
500C4⋊C4C2×#4112yes—
501C4⋊C4C2×#4112yes—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—42no—
506C22⋊C4—44no—
507C42—24no—
508C42—22no—
509C42—22no—
510C42—24no—
511C42—24no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—24no—
517C42—42no—
518C42—42no—
519C42—42no—
520C42—44no—
521C42—82no—
522C42—82no—
523C42—82no—
524C42—82no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23#52512yes—
530C23#53512yes—
531C23#53612yes—
532C23C23×D411yescharacteristic
533Q8—44no—
534Q8—44no—
535Q8—44no—
536Q8—44no—
537Q8—84no—
538Q8—84no—
539Q8—88no—
540Q8—88no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—84no—
593C2×C4—88no—
594C2×C4—162no—
595C2×C4#52512yes—
596C2×C4#53512yes—
597C2×C4#53612yes—
598C2×C4#54218yes—
599C2×C4#54218yes—
600C2×C4#57712yes—
601C2×C4#57812yes—
602C2×C4C22×#4111yescharacteristic
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#263214yes—
616C22C2×#52511yescharacteristic
617C22C2×#53511yescharacteristic
618C22C2×#53611yescharacteristic
619C22C2×#57711yescharacteristic
620C22C2×#57811yescharacteristic
621C22#358611yescharacteristic
622C22#358611yescharacteristic
623C4—22no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—42no—
628C4—44no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—84no—
635C4—81no—
636C4—82no—
637C4—82no—
638C4—81no—
639C4—82no—
640C4#263214yes—
641C4C2×#54212yes—
642C4C2×#54212yes—
643C2—24no—
644C2C2×#263211yescharacteristic
645C2#6059111yescharacteristic
646C2#6059411yescharacteristic
647C2#6081311yescharacteristic
648C2#6081411yescharacteristic
649C2#6087811yescharacteristic
650C2#6088411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
3. #30073C2semidirect
4. #30070C2semidirect
5. #30067C2semidirect
6. #30067C2semidirect
7. #30063C2semidirect
8. #30060C2semidirect

Maximal subgroups

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

subgroupindex
#353202
#342722
#300732
#300702
#300672
#300632
#300602
#237362
#235542
#235512
#235072
C2×#20432
#234572
#210662
#210642
#207742