Finite Group Databasebeta

← #6495162

Subgroups — #6495162

Order 512.

1610 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35330C211yesmaximal, characteristic, complemented
2#35298C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic, complemented
4#33293C211yesmaximal, characteristic, complemented
5#33237C211yesmaximal, characteristic, complemented
6#33233C211yesmaximal, characteristic, complemented
7#30215C211yesmaximal, characteristic, complemented
8#30214C211yesmaximal, characteristic, complemented
9#30213C211yesmaximal, characteristic, complemented
10#30212C211yesmaximal, characteristic, complemented
11#29151C214yesmaximal, complemented
12#28730C214yesmaximal, complemented
13#27984C214yesmaximal, complemented
14#25916C214yesmaximal, complemented
15#21870C211yesmaximal, characteristic, complemented
16#21869C211yesmaximal, characteristic
17#21866C211yesmaximal, characteristic, complemented
18#21865C211yesmaximal, characteristic
19#20467C211yesmaximal, characteristic, complemented
20C2×#544C2212yes—
21C2×#533C2211yescharacteristic
22C2×#532C2211yescharacteristic
23C2×#489C2214yes—
24C2×#488C2214yes—
25C2×#432C2211yescharacteristic
26C2×#432C2211yescharacteristic
27#2868C2211yescharacteristic
28#2862C2211yescharacteristic
29#2862C2211yescharacteristic
30#2862C2211yescharacteristic
31#2859C2211yescharacteristic
32#2859C2211yescharacteristic
33#2858C2211yescharacteristic
34#2858C2211yescharacteristic
35#2660C2212yes—
36#2660C2212yes—
37#2620C2212yes—
38#2620C2212yes—
39#2566—22no—
40#2550—22no—
41#2550—22no—
42#2549—22no—
43#2549—22no—
44#2545—22no—
45#2500—22no—
46#2432—22no—
47#2326C2212yes—
48#2326C2212yes—
49#2317C2212yes—
50#2285C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C4×#117C2211yescharacteristic
53#2074C2214yes—
54#2073C2214yes—
55#2072C2214yes—
56#2071C2214yes—
57#2026C2214yes—
58#2025C2214yes—
59#2024C2214yes—
60#2023C2214yes—
61#1993C2211yescharacteristic
62#1993C2211yescharacteristic
63#1993C2211yescharacteristic
64#1993C2211yescharacteristic
65#1993C2211yescharacteristic
66#1993C2211yescharacteristic
67#1993C2211yescharacteristic
68#1993C2211yescharacteristic
69#1992C2214yes—
70#1990C2214yes—
71#1961C2211yescharacteristic
72#1961C2211yescharacteristic
73#1961C2211yescharacteristic
74#1961C2211yescharacteristic
75#1960C2211yescharacteristic
76#1960C2211yescharacteristic
77#1960C2211yescharacteristic
78#1960C2211yescharacteristic
79#1959C2214yes—
80#1958C2214yes—
81#1875C2214yes—
82#1874C2214yes—
83#1873C2214yes—
84#1872C2214yes—
85#1836C2214yes—
86#1834C2214yes—
87#1805C2214yes—
88#1804C2214yes—
89#1803C2214yes—
90#1802C2214yes—
91#1768C2212yes—
92#1768C2212yes—
93#1750C2212yes—
94#1750C2212yes—
95#1745C2211yescharacteristic
96#1745C2211yescharacteristic
97C4×#97C2211yescharacteristic
98C4×#97C2211yescharacteristic
99C22×#41—21no—
100#544—48no—
101#533—24no—
102#533—24no—
103#532—24no—
104#532—24no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2312yes—
112C2×#127C2311yescharacteristic
113C2×#125—22no—
114C2×#125—21no—
115C2×#125C2312yes—
116C2×#125C2312yes—
117C2×#124—21no—
118C2×#124—22no—
119C2×#124C2312yes—
120C2×#124C2312yes—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8—22no—
124C2×C22⋊Q8C2312yes—
125C2×#121—21no—
126C2×#121C2312yes—
127C2×C4×Q8—21no—
128C2×C4×Q8C2312yes—
129C2×C4×D4—21no—
130C2×C4×D4C2312yes—
131C2×#117—21no—
132C2×#117C2312yes—
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C42C2311yescharacteristic
137#489—48no—
138#488—48no—
139#432—24no—
140#432—24no—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153C2×#97C2312yes—
154C2×#97C2312yes—
155C2×#97C2311yescharacteristic
156C2×#97C2311yescharacteristic
157C2×#97C2311yescharacteristic
158C2×#97C2311yescharacteristic
159C2×C4×C8C2312yes—
160C2×C4×C8C2312yes—
161#398—22no—
162#398—22no—
163#398—22no—
164#398—22no—
165#396—24no—
166#396—24no—
167#395—22no—
168#395—22no—
169#395—22no—
170#395—22no—
171#394—24no—
172#394—24no—
173#391—22no—
174#391—22no—
175#391—22no—
176#391—22no—
177#391—22no—
178#391C2312yes—
179#391C2312yes—
180#390—22no—
181#390—22no—
182#390—22no—
183#390C2312yes—
184#390C2312yes—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389—22no—
190#389—22no—
191#388—24no—
192#388—24no—
193#385—22no—
194#385—22no—
195#385—24no—
196#385—24no—
197#384—22no—
198#384C2312yes—
199#384C2312yes—
200#384C2312yes—
201#384C2312yes—
202#384C2312yes—
203#382—22no—
204#382—21no—
205#382—21no—
206#382C2312yes—
207#382C2312yes—
208#382C2312yes—
209#382C2311yescharacteristic
210#382C2311yescharacteristic
211#382C2311yescharacteristic
212#382C2311yescharacteristic
213#380—21no—
214#380—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4D412yes—
218C4×C4⋊C4D412yes—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4D412yes—
221#376C2311yescharacteristic
222#376C2311yescharacteristic
223C2×#95—21no—
224C2×#95—21no—
225C43C2311yescharacteristic
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#336C2312yes—
236#336C2312yes—
237#336C2312yes—
238#336C2312yes—
239C2×#41—44no—
240C2×#41—84no—
241C22×Q8—21no—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8C2412yes—
246C22×D4—22no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4C2412yes—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—28no—
266C4⋊Q8—44no—
267C4⋊Q8—48no—
268C4⋊Q8—84no—
269#127—22no—
270#127—22no—
271#125—28no—
272#125—44no—
273#125—48no—
274#125—44no—
275#124—28no—
276#124—28no—
277#124—44no—
278#124—84no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—48no—
282C22⋊Q8—48no—
283C22⋊Q8—44no—
284#121—24no—
285#121—24no—
286#121—44no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—44no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—44no—
293#117—24no—
294#117—24no—
295#117—44no—
296#117—44no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42#4114yes—
346C2×C42#4114yes—
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C4⋊C8—24no—
354C4⋊C8—24no—
355#103—44no—
356#103—44no—
357#103—44no—
358#103—44no—
359#102—44no—
360#102—44no—
361#102—44no—
362#102—44no—
363C22⋊C8—24no—
364C22⋊C8—24no—
365#97—22no—
366#97—22no—
367#97—22no—
368#97—22no—
369#97—24no—
370#97—24no—
371C4×C8—24no—
372C4×C8—24no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—24no—
384#95—24no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393C24—22no—
394C24C22×D411yescharacteristic
395#41—816no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—44no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—84no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—84no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4#12414yes—
437C22×C4#12414yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2511yescharacteristic
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—84no—
492C4⋊C4—84no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—84no—
502C22⋊C4—84no—
503C42—22no—
504C42—22no—
505C42—42no—
506C42—42no—
507C42—42no—
508C42—42no—
509C42—44no—
510C42—44no—
511C42#12414yes—
512C42#12414yes—
513C42C2×#4112yes—
514C42C2×#4112yes—
515C42C2×#4112yes—
516C42C2×#4112yes—
517C42H(5,𝔽2)14yes—
518C422−1+414yes—
519C23—22no—
520C23—22no—
521C23—21no—
522C23—21no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23C2×#12412yes—
529C23C23×D411yescharacteristic
530Q8—44no—
531Q8—44no—
532Q8—84no—
533Q8—84no—
534Q8—164no—
535Q8—164no—
536D4—44no—
537D4—44no—
538D4—84no—
539D4—84no—
540D4—88no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—88no—
581C2×C4—164no—
582C2×C4C2×#12412yes—
583C2×C4C2×#12412yes—
584C2×C4C2×#12412yes—
585C2×C4C2×#12412yes—
586C2×C4C2×#12412yes—
587C2×C4#53514yes—
588C2×C4#53614yes—
589C2×C4C22×#4111yescharacteristic
590C2×C4C22×#4111yescharacteristic
591C2×C4C2×H(5,𝔽2)12yes—
592C2×C4C2×2−1+412yes—
593C8—42no—
594C8—42no—
595C8—42no—
596C8—42no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—42no—
601C22—42no—
602C22—84no—
603C22—82no—
604C22—82no—
605C22—84no—
606C22—82no—
607C22C22×#12411yescharacteristic
608C22C2×#53511yescharacteristic
609C22C2×#53611yescharacteristic
610C22#358512yes—
611C22#358612yes—
612C4—24no—
613C4—22no—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—24no—
618C4—22no—
619C4—42no—
620C4—42no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4—82no—
627C4—82no—
628C4#351112yes—
629C4#351112yes—
630C2—42no—
631C2—162no—
632C2#3528911yescharacteristic
633C2#6061911yescharacteristic
634C2#6062011yescharacteristic
635C2#6082912yes—
636C2#6088612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. #35330C2semidirect
3. #33297C2semidirect
4. #33293C2semidirect
5. #33237C2semidirect
6. #33233C2semidirect
7. #30215C2semidirect
8. #30214C2semidirect
9. #30213C2semidirect
10. #30212C2semidirect
11. #29151C2semidirect
12. #28730C2semidirect
13. #27984C2semidirect
14. #25916C2semidirect
15. #21870C2semidirect
17. #21866C2semidirect
19. #20467C2semidirect

Maximal subgroups

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

subgroupindex
#353302
#352982
#332972
#332932
#332372
#332332
#302152
#302142
#302132
#302122
#291512
#287302
#279842
#259162
#218702
#218692
#218662
#218652
#204672