Finite Group Databasebeta

← #6583646

Subgroups — #6583646

Order 512.

1854 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
1C2×#3365C211yesmaximal, characteristic
2C2×#3363C211yesmaximal, characteristic, complemented
3#36538C211yesmaximal, characteristic
4#36474C211yesmaximal, characteristic, complemented
5#33227C211yesmaximal, characteristic
6#33202C211yesmaximal, characteristic, complemented
7#33165C211yesmaximal, characteristic
8#28638C211yesmaximal, characteristic, complemented
9#28637C211yesmaximal, characteristic
10#27772C214yesmaximal, complemented
11#26468C214yesmaximal, complemented
12#25932C211yesmaximal, characteristic
13#25931C211yesmaximal, characteristic, complemented
14#25648C211yesmaximal, characteristic
15#25647C211yesmaximal, characteristic, complemented
16#24736C214yesmaximal, complemented
17#23000C214yesmaximal, complemented
18C8×C4⋊Q8C211yesmaximal, characteristic, complemented
19C42×Q16C211yesmaximal, characteristic, complemented
20C2×#559C2211yescharacteristic
21C2×Q82C2211yescharacteristic
22C2×#557C2211yescharacteristic
23C2×#557C2211yescharacteristic
24#3365—28no—
25#3363—28no—
26C2×#488C2212yes—
27C2×#487C2212yes—
28C2×#487C2211yescharacteristic
29C2×#487C2211yescharacteristic
30C2×#486C2211yescharacteristic
31C2×#486C2211yescharacteristic
32C2×#477C2211yescharacteristic
33C2×#477C2212yes—
34C2×#475C2211yescharacteristic
35C2×#478C2212yes—
36C2×#464C2212yes—
37C2×#464C2214yes—
38C2×#462C2211yescharacteristic
39C2×#462C2212yes—
40C2×#460C2211yescharacteristic
41C2×C8×Q8C2211yescharacteristic
42C2×C8×Q8C2211yescharacteristic
43C2×C4×Q16C2212yes—
44C2×C4×Q16C2212yes—
45C2×C4×Q16C2211yescharacteristic
46C2×C4×SD16C2212yes—
47#2862—21no—
48#2862C2211yescharacteristic
49#2861—21no—
50#2858C2211yescharacteristic
51#2568C2211yescharacteristic
52#2567—21no—
53#2567C2211yescharacteristic
54#2566—21no—
55#2558—24no—
56#2550—24no—
57#2550C2214yes—
58#2548C2211yescharacteristic
59#2546—21no—
60#2546C2211yescharacteristic
61#2545—21no—
62#2545C2214yes—
63#2367C2214yes—
64#2351C2214yes—
65C4×C4⋊Q8—21no—
66C4×C4⋊Q8C2211yescharacteristic
67C4×#124—21no—
68C42×Q8C2211yescharacteristic
69C42×Q8C2211yescharacteristic
70#2068C2214yes—
71#2067C2214yes—
72#2043C2214yes—
73#2042C2214yes—
74#2026C2214yes—
75#2023C2214yes—
76#1988C2211yescharacteristic
77#1987C2211yescharacteristic
78#1984C2211yescharacteristic
79#1984C2211yescharacteristic
80#1983C2214yes—
81#1982C2211yescharacteristic
82#1981C2211yescharacteristic
83#1958C2214yes—
84#1958C2214yes—
85#1949C2214yes—
86#1939C2214yes—
87#1934C2214yes—
88#1933C2214yes—
89#1880C2214yes—
90#1878C2214yes—
91#1865C2214yes—
92#1862C2214yes—
93#1842C2211yescharacteristic
94#1805C2214yes—
95C4×#107C2212yes—
96C4×#107C2211yescharacteristic
97C8×C4⋊C4C2214yes—
98C4×#103C2211yescharacteristic
99C4×#103C2211yescharacteristic
100C4×#103C2212yes—
101C4×#103C2212yes—
102C42×C8C2211yescharacteristic
103C22×Q16C2312yes—
104C22×Q16C2311yescharacteristic
105C22×SD16C2312yes—
106#559—24no—
107#559—42no—
108Q82—24no—
109Q82—42no—
110#557—24no—
111#557—24no—
112#557—42no—
113#557—42no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—22no—
118C2×C4⋊Q8D411yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2312yes—
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2312yes—
124C2×#127C2311yescharacteristic
125C2×#125—21no—
126C2×#125—22no—
127C2×#125—22no—
128C2×#125C2312yes—
129C2×#124—21no—
130C2×#124—21no—
131C2×#124C2314yes—
132C2×#124C2312yes—
133C2×#123—22no—
134C2×C4×Q8—21no—
135C2×C4×Q8—21no—
136C2×C4×Q8—21no—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2312yes—
140C2×C4×Q8C2312yes—
141C2×C4×Q8C2311yescharacteristic
142C2×C4×Q8C2311yescharacteristic
143C2×C4×Q8C2312yes—
144C2×C4×Q8C2311yescharacteristic
145C2×C4×Q8C2311yescharacteristic
146C2×C4×D4—21no—
147C2×C4×D4C2312yes—
148#488—28no—
149#487—28no—
150#487—24no—
151#487—42no—
152#486—24no—
153#486—42no—
154#478—28no—
155#477—24no—
156#477—28no—
157#475—24no—
158#464—28no—
159#464—216no—
160#462—28no—
161#462—24no—
162#460—24no—
163C8×Q8—24no—
164C8×Q8—24no—
165C4×Q16—28no—
166C4×Q16—28no—
167C4×Q16D418yes—
168C4×SD16—28no—
169C2×#107C2312yes—
170C2×#107C2312yes—
171C2×#107C2311yescharacteristic
172C2×#106C2312yes—
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2311yescharacteristic
175C2×C4⋊C8C2312yes—
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2314yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2314yes—
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#103C2312yes—
186C2×#103C2314yes—
187C2×#103C2311yescharacteristic
188C2×#103C2311yescharacteristic
189C2×#102C2312yes—
190C2×#102C2312yes—
191C2×#102C2314yes—
192C2×#102C2312yes—
193C2×#102C2314yes—
194C2×C4×C8C2311yescharacteristic
195C2×C4×C8C2312yes—
196C2×C4×C8C2312yes—
197C2×C4×C8C2312yes—
198C2×C4×C8C2312yes—
199C2×C4×C8C2311yescharacteristic
200#400—22no—
201#398—24no—
202#398—22no—
203#396—24no—
204#396—24no—
205#395—24no—
206#395—22no—
207#394—22no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391—22no—
214#391—24no—
215#391D411yescharacteristic
216#391D411yescharacteristic
217#391C2314yes—
218#390—21no—
219#390—21no—
220#390—21no—
221#390—22no—
222#390C2314yes—
223#389—24no—
224#389—24no—
225#389—24no—
226#389D414yes—
227#389C2314yes—
228#389C2314yes—
229#388—24no—
230#388—24no—
231#388—24no—
232#388—24no—
233#385—24no—
234#384—21no—
235#384D411yescharacteristic
236#382—21no—
237#382—24no—
238#382C2314yes—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—22no—
242C4×C4⋊C4—22no—
243C4×C4⋊C4—24no—
244C4×C4⋊C4C2311yescharacteristic
245C4×C4⋊C4C2311yescharacteristic
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2314yes—
249C4×C22⋊C4—24no—
250C43—21no—
251C43—21no—
252C43C2311yescharacteristic
253#340C2314yes—
254#340C2314yes—
255#340C2312yes—
256#340C2312yes—
257#340C2312yes—
258#336C2314yes—
259#336C2314yes—
260C22×Q8—21no—
261C22×Q8—22no—
262C22×Q8—21no—
263C22×Q8C2×D411yescharacteristic
264C22×Q8C2×D411yescharacteristic
265C22×Q8C2412yes—
266C22×Q8C2411yescharacteristic
267C22×Q8C2411yescharacteristic
268C22×D4—21no—
269C22×D4—21no—
270C22×D4C2412yes—
271C23×C4—22no—
272C2×Q16—24no—
273C2×Q16—24no—
274C2×Q16—28no—
275C2×Q16C2×D414yes—
276C2×SD16—24no—
277C2×SD16—28no—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2411yescharacteristic
282C22×C8C2411yescharacteristic
283C4⋊Q8—24no—
284C4⋊Q8—24no—
285C4⋊Q8—24no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—42no—
293C4⋊Q8—44no—
294C4⋊Q8—42no—
295C4⋊Q8—44no—
296C4⋊Q8—48no—
297C4⋊Q8—44no—
298#127—22no—
299#127—22no—
300#125—24no—
301#125—42no—
302#125—44no—
303#125—48no—
304#125—48no—
305#124—24no—
306#124—28no—
307#124—28no—
308#124—42no—
309#124—44no—
310#124—44no—
311#123—48no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—24no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—24no—
322C4×Q8—24no—
323C4×Q8—24no—
324C4×Q8—42no—
325C4×Q8—42no—
326C4×Q8—42no—
327C4×Q8—44no—
328C4×Q8—44no—
329C4×Q8—44no—
330C4×Q8C2×D414yes—
331C4×Q8C2×D414yes—
332C4×Q8#4118yes—
333C4×D4—24no—
334C4×D4—42no—
335C4×D4—44no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2×D414yes—
359C2×C4⋊C4C2×D414yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2411yescharacteristic
364C2×C4⋊C4C2411yescharacteristic
365C2×C4⋊C4C2412yes—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D412yes—
387C2×C42C2412yes—
388C2×C42C2412yes—
389C2×C42C2411yescharacteristic
390C2×C42C2412yes—
391#107—24no—
392#107—24no—
393#107C2×D414yes—
394#106—24no—
395C4⋊C8—24no—
396C4⋊C8—24no—
397C4⋊C8—22no—
398C4⋊C8—22no—
399C4⋊C8—28no—
400#103—24no—
401#103—24no—
402#103—24no—
403#103—24no—
404#103—24no—
405#103—24no—
406#103—24no—
407#103—28no—
408#103—44no—
409#103C2×D414yes—
410#103C2×D414yes—
411#102—24no—
412#102—24no—
413#102—24no—
414#102—28no—
415#102—44no—
416C4×C8—24no—
417C4×C8—24no—
418C4×C8—24no—
419C4×C8—22no—
420C4×C8—24no—
421C4×C8C2×D414yes—
422#95—24no—
423#95—22no—
424#95—21no—
425#95—21no—
426#95—22no—
427#95—24no—
428#95—24no—
429#95—24no—
430#95—24no—
431#95—24no—
432#95C2×D414yes—
433#95C2×D414yes—
434C24—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—24no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—24no—
449C2×Q8—44no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—82no—
455C2×Q8—84no—
456C2×Q8#13514yes—
457C2×Q8C22×D412yes—
458C2×Q8C22×D412yes—
459C2×Q8C2×#4114yes—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—22no—
463C2×D4—24no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—82no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—44no—
479C22×C4—84no—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C2511yescharacteristic
489Q16—28no—
490Q16—48no—
491SD16—48no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8C22×D412yes—
502C2×C8C22×D412yes—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—28no—
510C4⋊C4—28no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—44no—
534C4⋊C4—48no—
535C4⋊C4—84no—
536C4⋊C4C22×D412yes—
537C4⋊C4C22×D412yes—
538C4⋊C4C2×#4114yes—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—84no—
546C42—24no—
547C42—24no—
548C42—28no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—42no—
556C42—44no—
557C42—44no—
558C42—44no—
559C42—44no—
560C42—42no—
561C42—42no—
562C42—82no—
563C42—82no—
564C42C22×D412yes—
565C42C2×#4114yes—
566C42H(5,𝔽2)12yes—
567C422−1+414yes—
568C23—44no—
569C23—42no—
570C23—44no—
571C23—42no—
572C23—42no—
573C23C23×D411yescharacteristic
574Q8—22no—
575Q8—24no—
576Q8—28no—
577Q8—22no—
578Q8—22no—
579Q8—42no—
580Q8—44no—
581Q8—42no—
582Q8—42no—
583Q8—44no—
584Q8—84no—
585Q8—162no—
586Q8—164no—
587Q8#53214yes—
588D4—42no—
589D4—42no—
590D4—44no—
591D4—84no—
592D4—162no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—164no—
629C2×C4D4212yes—
630C2×C4D4212yes—
631C2×C4#54814yes—
632C2×C4#54814yes—
633C2×C4C2×#13514yes—
634C2×C4C2×#13512yes—
635C2×C4#57812yes—
636C2×C4C23×D411yescharacteristic
637C2×C4C22×#4112yes—
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C2×C4C2×2−1+412yes—
640C8—22no—
641C8—22no—
642C8—42no—
643C8—42no—
644C8—42no—
645C22—82no—
646C22—82no—
647C22—84no—
648C22—84no—
649C22—82no—
650C22C2×D4211yescharacteristic
651C22C2×#54812yes—
652C22C22×#13511yescharacteristic
653C22C2×#57811yescharacteristic
654C22#358412yes—
655C4—24no—
656C4—24no—
657C4—24no—
658C4—24no—
659C4—22no—
660C4—42no—
661C4—42no—
662C4—44no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4#328514yes—
672C4C2×#53211yescharacteristic
673C4#349911yescharacteristic
674C4#351112yes—
675C2—162no—
676C2#3612211yescharacteristic
677C2C2×#328511yescharacteristic
678C2#6108811yescharacteristic
679C2#6130312yes—
680C2#6132812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3363C2semidirect
4. #36474C2semidirect
6. #33202C2semidirect
8. #28638C2semidirect
10. #27772C2semidirect
11. #26468C2semidirect
13. #25931C2semidirect
15. #25647C2semidirect
16. #24736C2semidirect
17. #23000C2semidirect
18. C8×C4⋊Q8C2semidirect
19. C42×Q16C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33652
C2×#33632
#365382
#364742
#332272
#332022
#331652
#286382
#286372
#277722
#264682
#259322
#259312
#256482
#256472
#247362
#230002
C8×C4⋊Q82
C42×Q162