Finite Group Databasebeta

← #6986208

Subgroups — #6986208

Order 512.

1563 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36456C211yesmaximal, characteristic
2#35280C211yesmaximal, characteristic
3#32786C212yesmaximal, complemented
4#31398C214yesmaximal, complemented
5#31334C214yesmaximal, complemented
6#30199C212yesmaximal, complemented
7#30116C212yesmaximal, complemented
8#29691C211yesmaximal, characteristic, complemented
9#29165C212yesmaximal, complemented
10#28151C212yesmaximal, complemented
11#27960C211yesmaximal, characteristic, complemented
12#27680C211yesmaximal, characteristic, complemented
13#25349C212yesmaximal, complemented
14#24994C212yesmaximal, complemented
15#24333C212yesmaximal, complemented
16#23571C211yesmaximal, characteristic, complemented
17#22473C211yesmaximal, characteristic
18C2×Q82C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#477C2212yes—
22C2×C4×Q16C2212yes—
23C2×#436C2212yes—
24#2857C2212yes—
25#2857C2212yes—
26#2857C2211yescharacteristic
27#2857C2212yes—
28#2629—22no—
29#2582—21no—
30#2582C2212yes—
31#2563C2211yescharacteristic
32#2563C2212yes—
33#2538—21no—
34#2538C2212yes—
35#2538C2212yes—
36#2538C2211yescharacteristic
37#2452—22no—
38#2452—21no—
39#2439—21no—
40#2439—22no—
41#2417—22no—
42#2417C2214yes—
43#2351—21no—
44Q8×C4⋊C4C2211yescharacteristic
45#2341—21no—
46#2316—21no—
47#2311—21no—
48C4×C22⋊Q8C2211yescharacteristic
49C4×C22⋊Q8C2212yes—
50C2×#384C2211yescharacteristic
51#2099C2212yes—
52#2099C2212yes—
53#2099C2214yes—
54#2099C2212yes—
55#2096C2212yes—
56#2075C2214yes—
57#2075C2214yes—
58#2072C2214yes—
59#2057C2214yes—
60#2057C2211yescharacteristic
61#2057C2212yes—
62#2043C2211yescharacteristic
63#2043C2212yes—
64#2036C2212yes—
65#2036C2214yes—
66#2035C2214yes—
67#2024C2214yes—
68#2024C2211yescharacteristic
69#2016C2211yescharacteristic
70#2001C2212yes—
71#2001C2214yes—
72#1990C2212yes—
73#1984C2212yes—
74#1948C2212yes—
75#1927C2214yes—
76#1920C2214yes—
77#1894C2214yes—
78#1894C2214yes—
79#1894C2214yes—
80#1878C2214yes—
81#1872C2212yes—
82#1870C2214yes—
83#1863C2214yes—
84#1853C2212yes—
85#1834C2212yes—
86#1823C2214yes—
87#1817C2212yes—
88#1809C2214yes—
89#1803C2212yes—
90#1803C2212yes—
91#1797C2211yescharacteristic
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1728C2212yes—
95C22×#41D411yescharacteristic
96C22×Q16C2312yes—
97C22×M16C2314yes—
98Q82—44no—
99#533—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×#125—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C2311yescharacteristic
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2314yes—
114C2×C22⋊Q8C2312yes—
115C4×#41—44no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4D412yes—
125C2×C4×D4D411yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2312yes—
128C2×#117—21no—
129C2×#117D411yescharacteristic
130C2×#117D412yes—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4C2312yes—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#477—44no—
137C4×Q16—44no—
138#436—44no—
139C2×#107C2314yes—
140C2×#107C2312yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2312yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2312yes—
153C2×C4×C8C2312yes—
154#398—24no—
155#398—24no—
156#396—24no—
157#396—24no—
158#396—22no—
159#396—22no—
160#396—22no—
161#393—22no—
162#393—22no—
163#393—22no—
164#391—22no—
165#391—22no—
166#391—21no—
167#391—22no—
168#391C2314yes—
169#391C2311yescharacteristic
170#390—21no—
171#390—22no—
172#390—22no—
173#390—21no—
174#390—22no—
175#389—22no—
176#389—22no—
177#389—21no—
178#389—22no—
179#389—21no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—21no—
185#389—22no—
186#389C2314yes—
187#389C2311yescharacteristic
188#389C2312yes—
189#387—21no—
190#387—22no—
191#386—24no—
192#385—21no—
193#385—22no—
194#384—21no—
195#384—21no—
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2312yes—
199#384C2312yes—
200#382—21no—
201#382—22no—
202#380—22no—
203#380—22no—
204#380—22no—
205#380—21no—
206#380—22no—
207#380—24no—
208#380C2312yes—
209#380C2311yescharacteristic
210#380C2312yes—
211#380C2312yes—
212#380C2312yes—
213#380C2314yes—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2314yes—
233#336C2311yescharacteristic
234C2×#41—24no—
235C2×#41—24no—
236C2×#41—42no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8C2×D411yescharacteristic
241C22×Q8C2412yes—
242C22×Q8C2411yescharacteristic
243C22×D4C2×D411yescharacteristic
244C22×D4C2×D412yes—
245C23×C4—22no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2411yescharacteristic
251C23×C4C2412yes—
252C2×Q16—24no—
253C2×Q16—42no—
254C2×Q16—42no—
255C2×M16—44no—
256C2×M16—44no—
257C2×M16—44no—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—42no—
263C4⋊Q8—44no—
264C4⋊Q8—42no—
265C4⋊Q8—48no—
266C4⋊Q8—44no—
267#125—44no—
268C22⋊Q8—22no—
269C22⋊Q8—24no—
270C22⋊Q8—41no—
271C22⋊Q8—42no—
272C22⋊Q8—42no—
273C22⋊Q8—42no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277C22⋊Q8—82no—
278C22⋊Q8—84no—
279C22⋊Q8—84no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—42no—
285C4×Q8—44no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×Q8—44no—
289C4×D4—24no—
290C4×D4—42no—
291C4×D4—44no—
292C4×D4—44no—
293#117—44no—
294#117—42no—
295#117—44no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—82no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C42—24no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—44no—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—42no—
362#107—44no—
363#106—44no—
364C4⋊C8—24no—
365#103—44no—
366#103—42no—
367#103—42no—
368#103—44no—
369#103—42no—
370#103—42no—
371#103—44no—
372C22⋊C8—42no—
373C22⋊C8—42no—
374C22⋊C8—44no—
375C4×C8—24no—
376#95—21no—
377#95—22no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95—21no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—22no—
389#95—22no—
390#95—22no—
391C24C22×D412yes—
392C24C22×D411yescharacteristic
393#41—416no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—81no—
411C2×Q8—82no—
412C2×Q8—81no—
413C2×Q8—82no—
414C2×Q8#13714yes—
415C2×Q82−1+412yes—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—42no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—42no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4#13714yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C2511yescharacteristic
463Q16—48no—
464M16—88no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—84no—
510C4⋊C4—81no—
511C4⋊C4—82no—
512C4⋊C4—84no—
513C22⋊C4—24no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—81no—
520C22⋊C4—82no—
521C22⋊C4—84no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—84no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—24no—
530C42—42no—
531C42—42no—
532C42—44no—
533C42—44no—
534C42—42no—
535C42—44no—
536C42—42no—
537C42H(5,𝔽2)14yes—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23C2×#13712yes—
552C23C23×D411yescharacteristic
553Q8—28no—
554Q8—24no—
555Q8—42no—
556Q8—42no—
557Q8—42no—
558Q8—42no—
559Q8—42no—
560Q8—84no—
561Q8—162no—
562Q8—164no—
563D4—48no—
564D4—44no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—28no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—44no—
585C2×C4—48no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—84no—
617C2×C4—164no—
618C2×C4#53412yes—
619C2×C4#53412yes—
620C2×C4C2×#13714yes—
621C2×C4C2×#13712yes—
622C2×C4C2×#13712yes—
623C2×C4C2×#13712yes—
624C2×C4#57612yes—
625C2×C4C2×H(5,𝔽2)12yes—
626C2×C4C2×2−1+411yescharacteristic
627C2×C4#58512yes—
628C2×C4#58511yescharacteristic
629C8—42no—
630C8—44no—
631C8—42no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22C2×#53411yescharacteristic
647C22#348911yescharacteristic
648C22#350011yescharacteristic
649C22C22×#13711yescharacteristic
650C22C22×#13711yescharacteristic
651C22C2×#57611yescharacteristic
652C22#358611yescharacteristic
653C4—22no—
654C4—24no—
655C4—22no—
656C4—22no—
657C4—42no—
658C4—41no—
659C4—42no—
660C4—42no—
661C4—84no—
662C4—82no—
663C4—82no—
664C4—81no—
665C4—81no—
666C4—82no—
667C4—81no—
668C4—82no—
669C4#319514yes—
670C4#354112yes—
671C4#355112yes—
672C2—42no—
673C2—82no—
674C2#4290811yescharacteristic
675C2C2×#319511yescharacteristic
676C2#6076211yescharacteristic
677C2#6211311yescharacteristic
678C2#6214911yescharacteristic
679C2#6253711yescharacteristic
680C2#6258911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
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. #32786C2semidirect
4. #31398C2semidirect
5. #31334C2semidirect
6. #30199C2semidirect
7. #30116C2semidirect
8. #29691C2semidirect
9. #29165C2semidirect
10. #28151C2semidirect
11. #27960C2semidirect
12. #27680C2semidirect
13. #25349C2semidirect
14. #24994C2semidirect
15. #24333C2semidirect
16. #23571C2semidirect

Maximal subgroups

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

subgroupindex
#364562
#352802
#327862
#313982
#313342
#301992
#301162
#296912
#291652
#281512
#279602
#276802
#253492
#249942
#243332
#235712
#224732