Finite Group Databasebeta

← #6631345

Subgroups — #6631345

Order 512.

1576 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35307C211yesmaximal, characteristic
2#35280C211yesmaximal, characteristic
3#32757C214yesmaximal, complemented
4#32755C214yesmaximal, complemented
5#32745C212yesmaximal, complemented
6#32744C212yesmaximal, complemented
7#32741C212yesmaximal, complemented
8#32740C212yesmaximal, complemented
9#23669C212yesmaximal, complemented
10#23658C211yesmaximal, characteristic, complemented
11#23643C212yesmaximal
12C2×#2072C211yesmaximal, characteristic, complemented
13#23615C212yesmaximal, complemented
14#23551C211yesmaximal, characteristic, complemented
15#23487C212yesmaximal
16#23455C211yesmaximal, characteristic, complemented
17#20630C211yesmaximal, characteristic
18C2×#536C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#428C2212yes—
22C2×#427C2212yes—
23C2×#424C2212yes—
24C2×#419C2212yes—
25C22×#103C2212yes—
26C2×#408C2212yes—
27#2857C2212yes—
28#2857C2212yes—
29#2855C2212yes—
30#2855C2212yes—
31#2629C2214yes—
32#2625C2212yes—
33#2625C2212yes—
34#2582—21no—
35#2582—21no—
36#2538—21no—
37#2538—21no—
38#2447—21no—
39#2447—21no—
40#2446—21no—
41#2446—21no—
42#2419—22no—
43#2419—22no—
44#2417—22no—
45#2417—22no—
46#2316C2212yes—
47#2316C2211yescharacteristic
48#2311C2211yescharacteristic
49#2311C2212yes—
50#2310C2211yescharacteristic
51#2296C2211yescharacteristic
52C2×#391C2211yescharacteristic
53C2×#384C2211yescharacteristic
54#2088C2214yes—
55#2088C2214yes—
56#2087C2214yes—
57#2086C2214yes—
58#2084C2212yes—
59#2084C2212yes—
60#2083C2212yes—
61#2083C2212yes—
62#2077C2214yes—
63#2077C2214yes—
64#2076C2214yes—
65#2075C2214yes—
66#2072C2212yes—
67#2072C2212yes—
68#2072C2214yes—
69#2068C2214yes—
70#2068C2214yes—
71#2066C2214yes—
72#2066C2214yes—
73#2059C2212yes—
74#2059C2212yes—
75#2058C2211yescharacteristic
76#2058C2211yescharacteristic
77#2057C2211yescharacteristic
78#2057C2211yescharacteristic
79#2036C2214yes—
80#2036C2214yes—
81#2035C2214yes—
82#2035C2214yes—
83#2026C2214yes—
84#2024C2212yes—
85#2024C2212yes—
86#1798C2214yes—
87#1797C2212yes—
88#1797C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#536—44no—
100#533—44no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#125C2312yes—
105C2×#124—21no—
106C2×#123—22no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2312yes—
113C4×#41—44no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D411yescharacteristic
118C2×C4×D4D412yes—
119C2×C4×D4D412yes—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2312yes—
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2312yes—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#428—44no—
134#427—44no—
135C2×#107C2312yes—
136C2×#107C2312yes—
137C2×#107C2314yes—
138C2×#106C2312yes—
139C2×#106C2312yes—
140C2×#106C2314yes—
141#424—44no—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2314yes—
144#419—44no—
145C2×#103—44no—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150#408—44no—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2314yes—
153#400—22no—
154#400—22no—
155#398—24no—
156#398—24no—
157#398—22no—
158#398—22no—
159#397—22no—
160#397—22no—
161#397—22no—
162#397—22no—
163#396—24no—
164#396—24no—
165#391—22no—
166#391—21no—
167#391—21no—
168#391—21no—
169#391—22no—
170#391—22no—
171#391—21no—
172#391—21no—
173#391C2312yes—
174#391C2312yes—
175#391C2312yes—
176#391C2312yes—
177#390—21no—
178#390—22no—
179#389—22no—
180#389—22no—
181#389—21no—
182#389—21no—
183#389—22no—
184#389—22no—
185#389C2311yescharacteristic
186#389C2311yescharacteristic
187#389C2311yescharacteristic
188#389C2311yescharacteristic
189#389C2314yes—
190#389C2314yes—
191#389C2312yes—
192#389C2312yes—
193#387—22no—
194#387—22no—
195#387—22no—
196#387—22no—
197#386—24no—
198#386—22no—
199#385—22no—
200#385—22no—
201#385—21no—
202#385—21no—
203#384—21no—
204#384—21no—
205#384C2312yes—
206#384C2312yes—
207#381—22no—
208#381—22no—
209#381—22no—
210#380—22no—
211#380—22no—
212#380—24no—
213#380—22no—
214#380C2312yes—
215#380C2314yes—
216#380C2314yes—
217#380C2312yes—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C2×#95—21no—
221C2×#95—21no—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237C2×#41—24no—
238C2×#41—22no—
239C2×#41—24no—
240C2×#41—22no—
241C22×Q8—21no—
242C22×Q8—21no—
243C22×Q8—21no—
244C22×Q8—24no—
245C22×Q8—22no—
246C22×Q8—24no—
247C22×Q8—22no—
248C22×Q8C2×D411yescharacteristic
249C22×Q8C2412yes—
250C22×D4C2×D412yes—
251C22×D4C2×D411yescharacteristic
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2×D412yes—
259C23×C4C2412yes—
260C23×C4C2411yescharacteristic
261C2×M16—24no—
262C2×M16—24no—
263C2×M16—24no—
264C22×C8—24no—
265C22×C8—24no—
266C22×C8—24no—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2414yes—
270C4⋊Q8—42no—
271C4⋊Q8—42no—
272C4⋊Q8—44no—
273#125—42no—
274#125—42no—
275#124—44no—
276#123—48no—
277C22⋊Q8—44no—
278C22⋊Q8—44no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281C22⋊Q8—84no—
282C22⋊Q8—84no—
283C4×Q8—42no—
284C4×Q8—42no—
285C4×Q8—44no—
286C4×Q8—44no—
287C4×D4—42no—
288C4×D4—42no—
289C4×D4—44no—
290C4×D4—44no—
291#117—24no—
292#117—24no—
293#117—24no—
294#117—24no—
295#117—22no—
296#117—22no—
297#117—44no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—82no—
322C2×C4⋊C4—82no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C42—24no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—44no—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359#107—42no—
360#107—42no—
361#107—44no—
362#106—42no—
363#106—42no—
364#106—44no—
365C4⋊C8—44no—
366C4⋊C8—44no—
367#103—44no—
368#103—44no—
369#103—44no—
370#103—44no—
371C22⋊C8—44no—
372C22⋊C8—44no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393C24C22×D412yes—
394C24C22×D411yescharacteristic
395#41—48no—
396#41—48no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—48no—
414C2×Q8—48no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4H(5,𝔽2)14yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C42−1+414yes—
466C22×C42−1+412yes—
467C22×C4C2511yescharacteristic
468M16—48no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—48no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—81no—
509C4⋊C4—82no—
510C4⋊C4—81no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C22⋊C4—24no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C42—22no—
529C42—22no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—42no—
534C42—42no—
535C42—44no—
536C42—42no—
537C42—42no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23#53412yes—
550C23#53612yes—
551C23#53712yes—
552C23C23×D411yescharacteristic
553Q8—44no—
554Q8—44no—
555Q8—44no—
556Q8—84no—
557Q8—84no—
558Q8—84no—
559D4—48no—
560D4—44no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—162no—
615C2×C4—164no—
616C2×C4#53412yes—
617C2×C4#53612yes—
618C2×C4#53712yes—
619C2×C4#56218yes—
620C2×C4#56318yes—
621C2×C4#57712yes—
622C2×C4#57812yes—
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4C2×H(5,𝔽2)12yes—
625C2×C4C2×2−1+412yes—
626C2×C4C2×2−1+411yescharacteristic
627C8—42no—
628C8—42no—
629C8—44no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22#281414yes—
644C22C2×#53411yescharacteristic
645C22C2×#53611yescharacteristic
646C22C2×#53711yescharacteristic
647C22C2×#57711yescharacteristic
648C22C2×#57811yescharacteristic
649C22#358511yescharacteristic
650C22#358611yescharacteristic
651C4—22no—
652C4—22no—
653C4—42no—
654C4—41no—
655C4—42no—
656C4—41no—
657C4—42no—
658C4—82no—
659C4—81no—
660C4—82no—
661C4—81no—
662C4—82no—
663C4—82no—
664C4—81no—
665C4—84no—
666C4—81no—
667C4#281414yes—
668C4C2×#56212yes—
669C4C2×#56312yes—
670C2—24no—
671C2—82no—
672C2C2×#281411yescharacteristic
673C2#6076311yescharacteristic
674C2#6076411yescharacteristic
675C2#6087811yescharacteristic
676C2#6088511yescharacteristic
677C2#6092411yescharacteristic
678C2#6092511yescharacteristic
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. #32757C2semidirect
4. #32755C2semidirect
5. #32745C2semidirect
6. #32744C2semidirect
7. #32741C2semidirect
8. #32740C2semidirect
9. #23669C2semidirect
10. #23658C2semidirect
12. C2×#2072C2semidirect
13. #23615C2semidirect
14. #23551C2semidirect
16. #23455C2semidirect

Maximal subgroups

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

subgroupindex
#353072
#352802
#327572
#327552
#327452
#327442
#327412
#327402
#236692
#236582
#236432
C2×#20722
#236152
#235512
#234872
#234552
#206302