Finite Group Databasebeta

← #6611555

Subgroups — #6611555

Order 512.

1641 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36185C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic, complemented
3#33293C211yesmaximal, characteristic, complemented
4#33051C212yesmaximal, complemented
5#29224C211yesmaximal, characteristic, complemented
6#28726C211yesmaximal, characteristic, complemented
7#27968C212yesmaximal, complemented
8#27824C212yesmaximal, complemented
9#27822C212yesmaximal, complemented
10#27718C214yesmaximal, complemented
11#27716C214yesmaximal, complemented
12#25909C212yesmaximal, complemented
13#22490C212yesmaximal, complemented
14#22489C212yesmaximal
15C4×#489C211yesmaximal, characteristic, complemented
16#21321C212yesmaximal, complemented
17D4×#97C211yesmaximal, characteristic, complemented
18C2×#556C2211yescharacteristic
19C2×#555C2211yescharacteristic
20C2×#533C2212yes—
21C2×#489C2211yescharacteristic
22C2×#489C2212yes—
23C2×C8×D4C2212yes—
24C22×#97C2212yes—
25#2868C2211yescharacteristic
26#2862C2211yescharacteristic
27#2857C2212yes—
28#2850C2212yes—
29#2567C2211yescharacteristic
30#2552C2212yes—
31#2542—21no—
32#2542C2211yescharacteristic
33#2538—21no—
34#2521—22no—
35#2521—22no—
36#2500C2212yes—
37#2499C2212yes—
38#2441C2214yes—
39#2429—22no—
40#2427—21no—
41#2427—21no—
42#2423—22no—
43#2365—21no—
44#2362—21no—
45#2362—21no—
46#2354C2212yes—
47Q8×C4⋊C4—21no—
48C4×C4⋊Q8C2211yescharacteristic
49C4×#124C2211yescharacteristic
50#2285C2212yes—
51C42×D4C2211yescharacteristic
52#2101C2214yes—
53#2100C2214yes—
54#2099C2214yes—
55#2072C2212yes—
56#2071C2212yes—
57#2026C2214yes—
58#2025C2214yes—
59#1993C2212yes—
60#1993C2212yes—
61#1993C2212yes—
62#1993C2212yes—
63#1993C2211yescharacteristic
64#1992C2214yes—
65#1992C2214yes—
66#1990C2212yes—
67#1990C2212yes—
68#1988C2212yes—
69#1987C2212yes—
70#1959C2211yescharacteristic
71#1936C2214yes—
72#1936C2212yes—
73#1935C2214yes—
74#1935C2212yes—
75#1893C2214yes—
76#1892C2214yes—
77#1875C2214yes—
78#1873C2214yes—
79#1836C2214yes—
80#1836C2212yes—
81#1836C2211yescharacteristic
82#1825C2214yes—
83#1825C2214yes—
84#1803C2212yes—
85#1802C2212yes—
86#1770C2212yes—
87#1767C2211yescharacteristic
88C4×#103C2212yes—
89C4×#102C2212yes—
90#1752C2214yes—
91#1749C2212yes—
92#1735C2214yes—
93#1735C2214yes—
94C4×#97C2211yescharacteristic
95C23×C8C2314yes—
96#556—28no—
97#555—28no—
98#533—28no—
99#533—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×#126—22no—
107C2×#125C2314yes—
108C2×#124—21no—
109C2×#124—22no—
110C2×#124C2311yescharacteristic
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8C2312yes—
117C2×#121—21no—
118C2×#121C2312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—22no—
122C2×C4×Q8—22no—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—21no—
129C2×#117C2312yes—
130C2×#117C2312yes—
131C2×#117C2314yes—
132C22×C42C2312yes—
133#489—24no—
134#489—44no—
135C8×D4—28no—
136C2×C4⋊C8C2312yes—
137C2×#103C2312yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#102C2312yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×C22⋊C8C2314yes—
144C2×#97—44no—
145C2×#97C2312yes—
146C2×#97C2312yes—
147C2×#97C2311yescharacteristic
148C2×#97C2311yescharacteristic
149C2×#97C2314yes—
150C2×C4×C8C2312yes—
151#399—24no—
152#399—22no—
153#398—24no—
154#397—22no—
155#397—22no—
156#396—24no—
157#395—22no—
158#394—22no—
159#394—22no—
160#391—21no—
161#391—22no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391C2312yes—
167#391C2311yescharacteristic
168#390—22no—
169#390C2311yescharacteristic
170#390C2312yes—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—21no—
175#389—22no—
176#389—24no—
177#389C2312yes—
178#389C2314yes—
179#388—24no—
180#388—22no—
181#388—22no—
182#388—21no—
183#388—21no—
184#388—22no—
185#388—22no—
186#387—22no—
187#387—22no—
188#387—24no—
189#386—22no—
190#386—22no—
191#384—22no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2312yes—
197#384C2312yes—
198#384C2312yes—
199#382C2312yes—
200#382C2312yes—
201#382C2311yescharacteristic
202#380—22no—
203#380—22no—
204#380—22no—
205#380C2314yes—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219C43C2311yescharacteristic
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#336C2312yes—
235#336C2314yes—
236C22×Q8—21no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8C2411yescharacteristic
241C22×D4—22no—
242C22×D4C2×D411yescharacteristic
243C22×D4C2411yescharacteristic
244C23×C4—21no—
245C23×C4C2×D412yes—
246C23×C4C2×D412yes—
247C23×C4C2412yes—
248C22×C8—28no—
249C22×C8—28no—
250C22×C8—44no—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—24no—
259C4⋊Q8—44no—
260C4⋊Q8—82no—
261C4⋊Q8#4114yes—
262#126—48no—
263#125—28no—
264#125—44no—
265#124—22no—
266#124—22no—
267#124—44no—
268#124—48no—
269C22⋊Q8—28no—
270C22⋊Q8—48no—
271C22⋊Q8—44no—
272C22⋊Q8—44no—
273C22⋊Q8—48no—
274C22⋊Q8—84no—
275#121—28no—
276#121—44no—
277C4×Q8—24no—
278C4×Q8—48no—
279C4×Q8—48no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×D4—24no—
283C4×D4—42no—
284C4×D4—42no—
285C4×D4#4118yes—
286#117—28no—
287#117—28no—
288#117—28no—
289#117—44no—
290#117—44no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—22no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C4⋊C8—24no—
355#103—24no—
356#103—24no—
357#103—44no—
358#102—24no—
359#102—24no—
360#102—44no—
361C22⋊C8—28no—
362#97—24no—
363#97—22no—
364#97—22no—
365#97—24no—
366#97—44no—
367C4×C8—24no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—21no—
386C24C22×D412yes—
387C2×Q8—24no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—81no—
402C2×Q8—81no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—44no—
410C2×D4—82no—
411C2×D4—82no—
412C2×D4#12414yes—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C2×#4118yes—
443C22×C4C2511yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—416no—
453C2×C82−1+414yes—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—84no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4#12414yes—
486C4⋊C4C2×#4114yes—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—48no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—81no—
500C22⋊C4—81no—
501C22⋊C4—84no—
502C22⋊C4#12418yes—
503C42—24no—
504C42—24no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—41no—
511C42—41no—
512C42—48no—
513C42—44no—
514C42—44no—
515C42—42no—
516C42—44no—
517C42C2×#4114yes—
518C42C2×#4112yes—
519C42H(5,𝔽2)12yes—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23C2×#12414yes—
534C23C23×D411yescharacteristic
535Q8—48no—
536Q8—42no—
537Q8—42no—
538Q8—88no—
539Q8—84no—
540Q8—162no—
541D4—28no—
542D4—42no—
543D4—42no—
544D4—164no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4C2×#12414yes—
595C2×C4C2×#12412yes—
596C2×C4C2×#12412yes—
597C2×C4#54612yes—
598C2×C4#54612yes—
599C2×C4#57512yes—
600C2×C4#57512yes—
601C2×C4C22×#4111yescharacteristic
602C2×C4C22×#4112yes—
603C2×C4C2×H(5,𝔽2)11yescharacteristic
604C2×C4C2×2−1+412yes—
605C8—24no—
606C8—42no—
607C8—44no—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22C22×#12411yescharacteristic
623C22C2×#54611yescharacteristic
624C22#349911yescharacteristic
625C22C2×#57511yescharacteristic
626C22C2×#57511yescharacteristic
627C22#358511yescharacteristic
628C22#358611yescharacteristic
629C4—24no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—41no—
634C4—41no—
635C4—42no—
636C4—44no—
637C4—42no—
638C4—82no—
639C4—81no—
640C4—82no—
641C4—81no—
642C4—81no—
643C4—82no—
644C4—84no—
645C4#313514yes—
646C4#348212yes—
647C4#352012yes—
648C2—44no—
649C2—161no—
650C2#3630311yescharacteristic
651C2C2×#313511yescharacteristic
652C2#6062511yescharacteristic
653C2#6115911yescharacteristic
654C2#6116011yescharacteristic
655C2#6250711yescharacteristic
656C2#6250811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36134C2semidirect
3. #33293C2semidirect
4. #33051C2semidirect
5. #29224C2semidirect
6. #28726C2semidirect
7. #27968C2semidirect
8. #27824C2semidirect
9. #27822C2semidirect
10. #27718C2semidirect
11. #27716C2semidirect
12. #25909C2semidirect
13. #22490C2semidirect
15. C4×#489C2semidirect
16. #21321C2semidirect
17. D4×#97C2semidirect

Maximal subgroups

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

subgroupindex
#361852
#361342
#332932
#330512
#292242
#287262
#279682
#278242
#278222
#277182
#277162
#259092
#224902
#224892
C4×#4892
#213212
D4×#972