Finite Group Databasebeta

← #7623513

Subgroups — #7623513

Order 512.

1450 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3169C211yesmaximal, characteristic, complemented
2C2×#3168C211yesmaximal, characteristic, complemented
3#33813C214yesmaximal, complemented
4#33804C212yesmaximal, complemented
5#33800C212yesmaximal, complemented
6#33783C212yesmaximal, complemented
7#33779C212yesmaximal, complemented
8#33753C212yesmaximal, complemented
9#33752C212yesmaximal, complemented
10#33746C214yesmaximal, complemented
11#33674C214yesmaximal, complemented
12#33672C212yesmaximal, complemented
13#33669C212yesmaximal, complemented
14C2×#2163C211yesmaximal, characteristic, complemented
15C2×#532—21no—
16C2×#529—21no—
17#3169—21no—
18#3169—21no—
19#3169—21no—
20#3169—21no—
21#3169C2212yes—
22#3169C2212yes—
23#3169C2212yescomplemented
24#3169C2212yescomplemented
25#3168—21no—
26#3168—21no—
27#3168—21no—
28#3168—21no—
29#3168C2212yes—
30#3168C2212yes—
31#3168C2212yescomplemented
32#3168C2212yescomplemented
33C2×#500C2211yescharacteristic, complemented
34C2×#499C2211yescharacteristic, complemented
35C2×#495C2211yescharacteristic, complemented
36C2×#493C2211yescharacteristic, complemented
37C2×#488—21no—
38C2×#486—21no—
39C2×#467C2211yescharacteristic
40C2×#466—21no—
41C2×#466C2211yescharacteristic
42C2×#466C2212yes—
43C2×#465—21no—
44C2×C4×Q16C2211yescharacteristic, complemented
45C2×C4×SD16—21no—
46C2×C4×D8—21no—
47C2×C4×D8C2211yescharacteristic, complemented
48C2×C4×D8C2212yescomplemented
49C22×C4×C8C2211yescharacteristic
50#2255C2214yescomplemented
51#2250C2214yescomplemented
52#2244C2214yescomplemented
53#2243C2214yescomplemented
54#2235C2214yescomplemented
55#2233C2214yescomplemented
56#2232C2214yes—
57#2232C2214yescomplemented
58#2228C2218yescomplemented
59#2227C2214yescomplemented
60#2226C2214yescomplemented
61#2217C2218yescomplemented
62#2215C2218yescomplemented
63#2210C2218yescomplemented
64#2207C2214yescomplemented
65#2206C2214yescomplemented
66#2187C2214yescomplemented
67#2185C2214yescomplemented
68#2184C2214yes—
69#2184C2214yescomplemented
70#2177C2218yescomplemented
71#2175C2218yescomplemented
72#2163C2212yescomplemented
73#2163C2212yescomplemented
74#2163C2212yescomplemented
75#2163C2212yescomplemented
76C2×C4⋊C16C2212yescomplemented
77C2×C22⋊C16C2212yes—
78C2×#346C2212yescomplemented
79C22×#41—41no—
80C22×Q16C2311yescharacteristic
81C22×SD16—21no—
82C22×D8—21no—
83C22×D8C2312yes—
84C22×D8C2311yescharacteristic
85C23×C8D411yescharacteristic
86C23×C8C2311yescharacteristic
87#532—22no—
88#532—22no—
89#532—22no—
90#532—22no—
91#532—41no—
92#532—41no—
93#532—41no—
94#532—41no—
95C2×C4⋊Q8—21no—
96C2×#127—21no—
97#529—22no—
98#529—22no—
99#529—22no—
100#529—22no—
101#529—41no—
102#529—41no—
103#529—41no—
104#529—41no—
105C2×#126—41no—
106C2×#125—21no—
107C2×#124—21no—
108C2×#124—41no—
109C2×#123—41no—
110C2×C22⋊Q8—21no—
111C2×#121—21no—
112C2×#121—22no—
113C2×#121—41no—
114C2×C4×Q8—21no—
115C2×C4×D4—21no—
116C2×C4×D4—22no—
117C2×C4×D4—41no—
118C2×C4×D4—41no—
119C2×#117—41no—
120C22×C42D411yescharacteristic
121C2×SD32C2318yes—
122C2×D16C2318yes—
123C22×C16C2314yes—
124#500—21no—
125#500—21no—
126#500C2312yes—
127#500C2312yes—
128#499—21no—
129#499—21no—
130#499C2312yes—
131#499C2312yes—
132#495—21no—
133#495—21no—
134#495C2312yes—
135#495C2312yes—
136#493—21no—
137#493—21no—
138#493C2312yes—
139#493C2312yes—
140#488—22no—
141#488—22no—
142#488—22no—
143#488—22no—
144#486—22no—
145#486—22no—
146#486—22no—
147#486—22no—
148#467—21no—
149#467—21no—
150#467C2312yes—
151#467C2312yes—
152#466—21no—
153#466—21no—
154#466—22no—
155#466—22no—
156#466—22no—
157#466—22no—
158#466—22no—
159#466—22no—
160#466C2312yes—
161#466C2312yes—
162#466C2314yes—
163#466C2314yes—
164#465—22no—
165#465—22no—
166#465—22no—
167#465—22no—
168C4×Q16—21no—
169C4×Q16—21no—
170C4×Q16C2312yes—
171C4×Q16C2312yes—
172C4×SD16—22no—
173C4×SD16—22no—
174C4×SD16—22no—
175C4×SD16—22no—
176C4×D8—21no—
177C4×D8—21no—
178C4×D8—22no—
179C4×D8—22no—
180C4×D8—22no—
181C4×D8—22no—
182C4×D8—22no—
183C4×D8—22no—
184C4×D8C2312yes—
185C4×D8C2312yes—
186C4×D8C2314yes—
187C4×D8C2314yes—
188C2×#107—21no—
189C2×#107C2311yescharacteristic
190C2×#107C2311yescharacteristic
191C2×#107C2312yes—
192C2×#106—21no—
193C2×#103—21no—
194C2×#103—21no—
195C2×#102—22no—
196C2×#102—22no—
197C2×#102—21no—
198C2×#102—21no—
199C2×C4×C8D411yescharacteristic
200C2×C4×C8D411yescharacteristic
201C2×C4×C8D411yescharacteristic
202C2×C4×C8D411yescharacteristic
203C2×C4×C8D411yescharacteristic
204C2×C4×C8D411yescharacteristic
205C2×C4×C8C2311yescharacteristic
206C2×C4×C8C2311yescharacteristic
207C2×C4×C8C2311yescharacteristic
208C2×C4×C8C2311yescharacteristic
209C2×C4×C8C2311yescharacteristic
210C2×C4×C8C2311yescharacteristic
211#367C2318yes—
212#366C2318yes—
213C4⋊C16C2314yes—
214C4⋊C16C2314yes—
215#358C2318yes—
216#357C2318yes—
217#357C2318yes—
218#357C2318yes—
219C22⋊C16C2314yes—
220C22⋊C16C2314yes—
221#346C2314yes—
222#346C2314yes—
223C2×#41—42no—
224C2×#41—42no—
225C2×#41—42no—
226C2×#41—42no—
227C2×#41—88no—
228C22×Q8—21no—
229C22×Q8—41no—
230C22×D4—21no—
231C22×D4—22no—
232C22×D4—41no—
233C22×D4—42no—
234C22×D4—41no—
235C22×D4—41no—
236C23×C4—41no—
237C23×C4—42no—
238C23×C4D811yescharacteristic
239C23×C4D811yescharacteristic
240C23×C4C2×D411yescharacteristic
241C2×Q16—21no—
242C2×Q16—42no—
243C2×Q16C2412yes—
244C2×SD16—22no—
245C2×SD16—22no—
246C2×SD16—44no—
247C2×D8—22no—
248C2×D8—22no—
249C2×D8—22no—
250C2×D8—21no—
251C2×D8—44no—
252C2×D8—42no—
253C2×D8—44no—
254C2×D8C2414yes—
255C2×D8C2412yes—
256C22×C8—42no—
257C22×C8—42no—
258C22×C8C2×D411yescharacteristic
259C22×C8C2×D411yescharacteristic
260C22×C8C2×D411yescharacteristic
261C22×C8C2×D411yescharacteristic
262C22×C8C2×D411yescharacteristic
263C22×C8C2×D411yescharacteristic
264C22×C8C2×D411yescharacteristic
265C22×C8C2×D411yescharacteristic
266C22×C8C2×D411yescharacteristic
267C22×C8C2411yescharacteristic
268C22×C8C2411yescharacteristic
269C22×C8C2411yescharacteristic
270C4⋊Q8—22no—
271C4⋊Q8—22no—
272C4⋊Q8—41no—
273C4⋊Q8—41no—
274#127—22no—
275#127—22no—
276#127—41no—
277#127—41no—
278#126—42no—
279#126—42no—
280#126—42no—
281#126—42no—
282#125—22no—
283#125—22no—
284#125—41no—
285#125—41no—
286#124—22no—
287#124—22no—
288#124—41no—
289#124—41no—
290#124—42no—
291#124—42no—
292#124—42no—
293#124—42no—
294#123—42no—
295#123—42no—
296#123—42no—
297#123—42no—
298C22⋊Q8—22no—
299C22⋊Q8—22no—
300C22⋊Q8—41no—
301C22⋊Q8—41no—
302#121—22no—
303#121—22no—
304#121—24no—
305#121—24no—
306#121—41no—
307#121—41no—
308#121—42no—
309#121—42no—
310#121—42no—
311#121—42no—
312#121—42no—
313#121—42no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—41no—
317C4×Q8—41no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—24no—
321C4×D4—24no—
322C4×D4—42no—
323C4×D4—42no—
324C4×D4—42no—
325C4×D4—42no—
326C4×D4—41no—
327C4×D4—41no—
328C4×D4—42no—
329C4×D4—42no—
330C4×D4—42no—
331C4×D4—42no—
332C4×D4—42no—
333C4×D4—42no—
334#117—42no—
335#117—42no—
336#117—42no—
337#117—42no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—41no—
342C2×C4⋊C4—41no—
343C2×C4⋊C4—41no—
344C2×C4⋊C4—41no—
345C2×C4⋊C4—41no—
346C2×C22⋊C4—41no—
347C2×C22⋊C4—41no—
348C2×C22⋊C4—41no—
349C2×C22⋊C4—42no—
350C2×C22⋊C4—41no—
351C2×C22⋊C4—42no—
352C2×C22⋊C4—41no—
353C2×C42—41no—
354C2×C42D811yescharacteristic
355C2×C42D811yescharacteristic
356C2×C42D811yescharacteristic
357C2×C42D811yescharacteristic
358C2×C42D811yescharacteristic
359C2×C42D811yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366SD32—48no—
367D16—48no—
368C2×C16—44no—
369C2×C16C2418yes—
370#107—22no—
371#107—22no—
372#107—22no—
373#107—21no—
374#107—21no—
375#107C2412yes—
376#107C2412yes—
377#107C2414yes—
378#106—22no—
379#106—22no—
380#103—22no—
381#103—22no—
382#103—22no—
383#103—22no—
384#102—24no—
385#102—24no—
386#102—22no—
387#102—22no—
388#102—24no—
389#102—24no—
390#102—22no—
391#102—22no—
392C4×C8C2×D411yescharacteristic
393C4×C8C2×D411yescharacteristic
394C4×C8C2×D411yescharacteristic
395C4×C8C2×D411yescharacteristic
396C4×C8C2×D411yescharacteristic
397C4×C8C2×D411yescharacteristic
398C4×C8C2×D411yescharacteristic
399C4×C8C2×D411yescharacteristic
400C4×C8C2×D411yescharacteristic
401C4×C8C2×D411yescharacteristic
402C4×C8C2×D411yescharacteristic
403C4×C8C2×D411yescharacteristic
404C4×C8C2411yescharacteristic
405C4×C8C2411yescharacteristic
406C4×C8C2411yescharacteristic
407C4×C8C2411yescharacteristic
408C24—41no—
409C24—42no—
410C24C2×D811yescharacteristic
411#41—88no—
412#41—88no—
413#41—88no—
414#41—88no—
415C2×Q8—22no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—41no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—84no—
422C2×D4—22no—
423C2×D4—24no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—44no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—41no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—84no—
439C2×D4—84no—
440C2×D4—84no—
441C2×D4—88no—
442C22×C4—24no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—84no—
455C22×C4—88no—
456C22×C4C2×D811yescharacteristic
457C22×C4C2×D811yescharacteristic
458C22×C4C2×D811yescharacteristic
459C22×C4C2×D811yescharacteristic
460C22×C4C2×D811yescharacteristic
461C22×C4C2×D811yescharacteristic
462C22×C4C2×D811yescharacteristic
463C22×C4C2×D811yescharacteristic
464C22×C4C2×D811yescharacteristic
465C22×C4C2×D811yescharacteristic
466C22×C4C2×D811yescharacteristic
467C22×C4C2×D811yescharacteristic
468C22×C4C2×D811yescharacteristic
469C22×C4C2×D811yescharacteristic
470C22×C4C2×D811yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474Q16—24no—
475Q16—42no—
476SD16—44no—
477SD16—44no—
478D8—24no—
479D8—28no—
480D8—44no—
481D8—42no—
482D8—44no—
483D8—44no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8C22×D411yescharacteristic
497C2×C8C22×D411yescharacteristic
498C2×C8C22×D411yescharacteristic
499C2×C8C22×D411yescharacteristic
500C2×C8C22×D411yescharacteristic
501C2×C8C22×D411yescharacteristic
502C2×C8C22×D411yescharacteristic
503C2×C8C2511yescharacteristic
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—42no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C42—42no—
535C42—42no—
536C42C2×D811yescharacteristic
537C42C2×D811yescharacteristic
538C42C2×D811yescharacteristic
539C42C2×D811yescharacteristic
540C42C2×D811yescharacteristic
541C42C2×D811yescharacteristic
542C42C2×D811yescharacteristic
543C42C2×D811yescharacteristic
544C42C2×D811yescharacteristic
545C42C2×D811yescharacteristic
546C42C2×D811yescharacteristic
547C42C2×D811yescharacteristic
548C42C22×D411yescharacteristic
549C42C22×D411yescharacteristic
550C42C22×D411yescharacteristic
551C42C22×D411yescharacteristic
552C16—28no—
553C23—41no—
554C23—41no—
555C23—42no—
556C23—44no—
557C23—82no—
558C23—82no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23—82no—
563C23#50914yes—
564C23C22×D811yescharacteristic
565C23C22×D811yescharacteristic
566C23C22×D811yescharacteristic
567Q8—42no—
568Q8—42no—
569Q8—82no—
570Q8—84no—
571Q8—84no—
572D4—42no—
573D4—44no—
574D4—44no—
575D4—42no—
576D4—84no—
577D4—82no—
578D4—84no—
579D4—84no—
580D4—84no—
581D4—84no—
582D4—84no—
583D4—88no—
584D4—88no—
585D4—84no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—84no—
612C2×C4—88no—
613C2×C4—88no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—84no—
617C2×C4—82no—
618C2×C4—84no—
619C2×C4C22×D811yescharacteristic
620C2×C4C22×D811yescharacteristic
621C2×C4C22×D811yescharacteristic
622C2×C4C22×D811yescharacteristic
623C2×C4C22×D811yescharacteristic
624C2×C4C22×D811yescharacteristic
625C2×C4C22×D811yescharacteristic
626C2×C4C22×D811yescharacteristic
627C2×C4C22×D811yescharacteristic
628C2×C4C22×D811yescharacteristic
629C2×C4C22×D811yescharacteristic
630C2×C4C23×D411yescharacteristic
631C8—21no—
632C8—21no—
633C8—21no—
634C8—21no—
635C8—21no—
636C8—21no—
637C8#57512yes—
638C8#58512yes—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—44no—
646C22—84no—
647C22—82no—
648C22—84no—
649C22—82no—
650C22—82no—
651C22—84no—
652C22—168no—
653C22—164no—
654C22C2×#50912yes—
655C22C2×#50912yes—
656C22C2×#50912yes—
657C22C23×D811yescharacteristic
658C4—21no—
659C4—21no—
660C4—21no—
661C4—21no—
662C4—21no—
663C4—21no—
664C4—21no—
665C4—21no—
666C4—21no—
667C4—21no—
668C4—82no—
669C4—84no—
670C4—82no—
671C4—82no—
672C4#341412yes—
673C4#344911yescharacteristic
674C4C2×#57511yescharacteristic
675C2—22no—
676C2—22no—
677C2—22no—
678C2—164no—
679C2—162no—
680C2#6094011yescharacteristic
681C2C22×#50911yescharacteristic
682C2C2×#341411yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3169C2semidirect
2. C2×#3168C2semidirect
3. #33813C2semidirect
4. #33804C2semidirect
5. #33800C2semidirect
6. #33783C2semidirect
7. #33779C2semidirect
8. #33753C2semidirect
9. #33752C2semidirect
10. #33746C2semidirect
11. #33674C2semidirect
12. #33672C2semidirect
13. #33669C2semidirect
14. C2×#2163C2semidirect
23. #3169C22semidirect
24. #3169C22semidirect
31. #3168C22semidirect
32. #3168C22semidirect
33. C2×#500C22semidirect
34. C2×#499C22semidirect
35. C2×#495C22semidirect
36. C2×#493C22semidirect
44. C2×C4×Q16C22semidirect
47. C2×C4×D8C22semidirect
48. C2×C4×D8C22semidirect
50. #2255C22semidirect
51. #2250C22semidirect
52. #2244C22semidirect
53. #2243C22semidirect
54. #2235C22semidirect
55. #2233C22semidirect
57. #2232C22semidirect
58. #2228C22semidirect
59. #2227C22semidirect
60. #2226C22semidirect
61. #2217C22semidirect
62. #2215C22semidirect
63. #2210C22semidirect
64. #2207C22semidirect
65. #2206C22semidirect
66. #2187C22semidirect
67. #2185C22semidirect
69. #2184C22semidirect
70. #2177C22semidirect
71. #2175C22semidirect
72. #2163C22semidirect
73. #2163C22semidirect
74. #2163C22semidirect
75. #2163C22semidirect
76. C2×C4⋊C16C22semidirect
78. C2×#346C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31692
C2×#31682
#338132
#338042
#338002
#337832
#337792
#337532
#337522
#337462
#336742
#336722
#336692
C2×#21632