Finite Group Databasebeta

← #6583862

Subgroups — #6583862

Order 512.

1888 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3372C211yesmaximal, characteristic
2C2×#3359C211yesmaximal, characteristic
3#36560C211yesmaximal, characteristic, complemented
4#36276C211yesmaximal, characteristic
5#33236C211yesmaximal, characteristic, complemented
6#33202C211yesmaximal, characteristic, complemented
7#33162C211yesmaximal, characteristic
8#28646C211yesmaximal, characteristic, complemented
9#28638C211yesmaximal, characteristic, complemented
10#27782C214yesmaximal, complemented
11#26473C214yesmaximal, complemented
12#25934C211yesmaximal, characteristic
13#25929C211yesmaximal, characteristic
14#25651C211yesmaximal, characteristic, complemented
15#25647C211yesmaximal, characteristic, complemented
16#24748C214yesmaximal
17#23012C214yesmaximal
18#21714C211yesmaximal, characteristic, complemented
19C4×#440C211yesmaximal, characteristic
20C2×#557C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×D4×Q8C2211yescharacteristic
24#3372—28no—
25#3359—28no—
26C2×#491C2212yes—
27C2×#487C2212yes—
28C2×#487C2211yescharacteristic
29C2×#487C2211yescharacteristic
30C2×#487C2211yescharacteristic
31C2×#487C2211yescharacteristic
32C2×#479C2212yes—
33C2×#477C2211yescharacteristic
34C2×#475C2211yescharacteristic
35C2×#475C2212yes—
36C2×#461C2211yescharacteristic
37C2×#461C2211yescharacteristic
38C2×#461C2212yes—
39C2×#461C2212yes—
40C2×#461C2214yes—
41C2×#446C2211yescharacteristic
42C2×C8×Q8C2211yescharacteristic
43C2×#440C2212yes—
44C2×#440C2212yes—
45C2×C4×SD16C2212yes—
46C2×C4×SD16C2211yescharacteristic
47#2868C2211yescharacteristic
48#2862C2211yescharacteristic
49#2859—21no—
50#2857—21no—
51#2568C2211yescharacteristic
52#2568C2211yescharacteristic
53#2567C2211yescharacteristic
54#2565—21no—
55#2565—21no—
56#2550C2214yes—
57#2549—24no—
58#2546C2211yescharacteristic
59#2546C2214yes—
60#2506—24no—
61#2502—21no—
62#2434—21no—
63#2365C2214yes—
64#2349C2214yes—
65C4×C4⋊Q8—21no—
66C4×C4⋊Q8C2211yescharacteristic
67C2×#391—21no—
68C42×Q8C2211yescharacteristic
69C42×D4C2211yescharacteristic
70#2068C2214yes—
71#2068C2214yes—
72#2043C2214yes—
73#2043C2214yes—
74#2026C2214yes—
75#2024C2214yes—
76#1988C2211yescharacteristic
77#1987C2211yescharacteristic
78#1984C2211yescharacteristic
79#1984C2211yescharacteristic
80#1981C2211yescharacteristic
81#1981C2211yescharacteristic
82#1981C2214yes—
83#1960C2214yes—
84#1960C2214yes—
85#1949C2214yes—
86#1940C2214yes—
87#1935C2214yes—
88#1933C2214yes—
89#1877C2214yes—
90#1877C2214yes—
91#1865C2214yes—
92#1863C2214yes—
93#1842C2211yescharacteristic
94#1805C2214yes—
95#1776C2212yes—
96C4×#107C2211yescharacteristic
97#1771C2214yes—
98#1763C2212yes—
99#1762C2212yes—
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102C4×#97C2211yescharacteristic
103C23×Q8—21no—
104C22×SD16C2312yes—
105C22×SD16C2312yes—
106C22×SD16C2311yescharacteristic
107#557—24no—
108#557—24no—
109#557—42no—
110#557—42no—
111#554—24no—
112#554—42no—
113D4×Q8—24no—
114D4×Q8—42no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8D411yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2314yes—
120C2×C4⋊Q8C2312yes—
121C2×C4⋊Q8C2312yes—
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2312yes—
125C2×#125—21no—
126C2×#125—21no—
127C2×#125—22no—
128C2×#125—22no—
129C2×#125C2312yes—
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8—22no—
133C2×C22⋊Q8—22no—
134C2×C4×Q8—21no—
135C2×C4×Q8—21no—
136C2×C4×Q8C2312yes—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2311yescharacteristic
140C2×C4×Q8C2311yescharacteristic
141C2×C4×Q8C2311yescharacteristic
142C2×C4×D4C2312yes—
143C2×C4×D4C2312yes—
144C2×C4×D4C2311yescharacteristic
145C2×C4×D4C2312yes—
146C22×C4⋊C4—21no—
147C22×C42—21no—
148#491—28no—
149#487—28no—
150#487—24no—
151#487—24no—
152#487—42no—
153#487—42no—
154#479—28no—
155#477—24no—
156#475—24no—
157#475—28no—
158#461—28no—
159#461—28no—
160#461—24no—
161#461—24no—
162#461—216no—
163#446—24no—
164C8×Q8—24no—
165#440—28no—
166#440—28no—
167C4×SD16—28no—
168C4×SD16D418yes—
169C2×#107C2312yes—
170C2×#107C2312yes—
171C2×#106C2312yes—
172C2×#106C2311yescharacteristic
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2311yescharacteristic
175C2×C4⋊C8C2311yescharacteristic
176C2×C4⋊C8C2312yes—
177C2×C4⋊C8C2314yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#103C2312yes—
187C2×#103C2312yes—
188C2×#103C2312yes—
189C2×#103C2314yes—
190C2×#103C2314yes—
191C2×#103C2311yescharacteristic
192C2×#102C2312yes—
193C2×#102C2311yescharacteristic
194C2×#97C2312yes—
195C2×#97C2312yes—
196C2×#97C2312yes—
197C2×C4×C8C2311yescharacteristic
198C2×C4×C8C2312yes—
199C2×C4×C8C2311yescharacteristic
200#400—22no—
201#400—22no—
202#398—24no—
203#398—22no—
204#398—24no—
205#398—24no—
206#395—22no—
207#395—24no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391—21no—
214#391—22no—
215#391—24no—
216#391D411yescharacteristic
217#391D411yescharacteristic
218#391C2314yes—
219#391C2314yes—
220#389—24no—
221#389—24no—
222#389—24no—
223#389—24no—
224#389D414yes—
225#389C2314yes—
226#389C2314yes—
227#387—24no—
228#387—24no—
229#385—24no—
230#385—24no—
231#384—21no—
232#384—21no—
233#384—21no—
234#384—22no—
235#384D411yescharacteristic
236#382—21no—
237#382—21no—
238#382C2314yes—
239#380—24no—
240C4×C4⋊C4—24no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—22no—
243C4×C4⋊C4C2311yescharacteristic
244C4×C4⋊C4C2311yescharacteristic
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2314yes—
248C4×C22⋊C4—21no—
249C4×C22⋊C4—22no—
250C2×#95—24no—
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×Q8—44no—
264C22×Q8—44no—
265C22×Q8—82no—
266C22×Q8C2×D411yescharacteristic
267C22×Q8C2×D411yescharacteristic
268C22×Q8C2412yes—
269C22×Q8C2412yes—
270C22×Q8C2411yescharacteristic
271C22×D4C2411yescharacteristic
272C23×C4—21no—
273C23×C4—22no—
274C23×C4—22no—
275C23×C4—22no—
276C2×SD16—24no—
277C2×SD16—24no—
278C2×SD16—24no—
279C2×SD16—28no—
280C2×SD16—28no—
281C2×SD16C2×D414yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2411yescharacteristic
286C22×C8C2411yescharacteristic
287C4⋊Q8—24no—
288C4⋊Q8—24no—
289C4⋊Q8—24no—
290C4⋊Q8—24no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—28no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—28no—
301C4⋊Q8—42no—
302C4⋊Q8—42no—
303C4⋊Q8—42no—
304C4⋊Q8—82no—
305#125—24no—
306#125—42no—
307#125—44no—
308#125—44no—
309#125—48no—
310#125—48no—
311C22⋊Q8—48no—
312C22⋊Q8—48no—
313C22⋊Q8—44no—
314C22⋊Q8—44no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—24no—
321C4×Q8—24no—
322C4×Q8—24no—
323C4×Q8—24no—
324C4×Q8—42no—
325C4×Q8—42no—
326C4×Q8—44no—
327C4×Q8—44no—
328C4×Q8C2×D414yes—
329C4×D4—24no—
330C4×D4—24no—
331C4×D4—22no—
332C4×D4—24no—
333C4×D4—42no—
334C4×D4—42no—
335C4×D4C2×D414yes—
336C4×D4#4118yes—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
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—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—44no—
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D412yes—
362C2×C4⋊C4C2×D412yes—
363C2×C4⋊C4C2×D414yes—
364C2×C4⋊C4C2×D414yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2411yescharacteristic
369C2×C4⋊C4C2411yescharacteristic
370C2×C4⋊C4C2412yes—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C42—21no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—21no—
380C2×C42—22no—
381C2×C42—21no—
382C2×C42—22no—
383C2×C42—82no—
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D412yes—
387C2×C42C2412yes—
388C2×C42C2412yes—
389C2×C42C2411yescharacteristic
390C2×C42C2412yes—
391#107—24no—
392#107—24no—
393#106—24no—
394#106C2×D414yes—
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—24no—
408#103—24no—
409#103—28no—
410#103—28no—
411#103—44no—
412#103—44no—
413#103C2×D414yes—
414#102—24no—
415#102C2×D414yes—
416#97—24no—
417#97—24no—
418#97—24no—
419C4×C8—24no—
420C4×C8—22no—
421C4×C8C2×D414yes—
422#95—24no—
423#95—24no—
424#95—24no—
425#95—21no—
426#95—22no—
427#95—22no—
428#95—21no—
429#95—24no—
430#95—24no—
431#95—24no—
432#95—24no—
433#95—24no—
434#95C2×D414yes—
435#95C2×D414yes—
436C24—21no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—21no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—24no—
447C2×Q8—24no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—24no—
452C2×Q8—24no—
453C2×Q8—44no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—44no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—82no—
460C2×Q8—82no—
461C2×Q8—816no—
462C2×Q8#13714yes—
463C2×Q8C22×D412yes—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4C22×D412yes—
467C2×D4C2×#4114yes—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—24no—
473C22×C4—28no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—44no—
484C22×C4—82no—
485C22×C4—82no—
486C22×C4—82no—
487C22×C4—84no—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C2511yescharacteristic
497SD16—28no—
498SD16—48no—
499SD16—48no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8C22×D412yes—
510C2×C8C22×D412yes—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—28no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—44no—
543C4⋊C4—48no—
544C4⋊C4—48no—
545C4⋊C4—84no—
546C4⋊C4—84no—
547C4⋊C4C22×D412yes—
548C4⋊C4C22×D412yes—
549C4⋊C4C2×#4114yes—
550C22⋊C4—24no—
551C22⋊C4—28no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C42—24no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—42no—
562C42—44no—
563C42—44no—
564C42—44no—
565C42—44no—
566C42—42no—
567C42—42no—
568C42—82no—
569C42C22×D412yes—
570C42C2×#4114yes—
571C42H(5,𝔽2)12yes—
572C422−1+414yes—
573C23—22no—
574C23—24no—
575C23—42no—
576C23—41no—
577C23—41no—
578C23C23×D411yescharacteristic
579Q8—28no—
580Q8—22no—
581Q8—22no—
582Q8—42no—
583Q8—44no—
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—42no—
588Q8—44no—
589Q8—44no—
590Q8—88no—
591Q8—84no—
592Q8—162no—
593D4—22no—
594D4—24no—
595D4#53214yes—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—48no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—88no—
634C2×C4—88no—
635C2×C4—164no—
636C2×C4D4212yes—
637C2×C4D4212yes—
638C2×C4#54814yes—
639C2×C4#54814yes—
640C2×C4C2×#13714yes—
641C2×C4C2×#13712yes—
642C2×C4#57712yes—
643C2×C4C23×D411yescharacteristic
644C2×C4C22×#4112yes—
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4C2×2−1+412yes—
647C8—22no—
648C8—22no—
649C8—42no—
650C8—42no—
651C8—42no—
652C22—24no—
653C22—42no—
654C22—44no—
655C22—42no—
656C22—44no—
657C22C2×D4211yescharacteristic
658C22C2×#54812yes—
659C22C22×#13711yescharacteristic
660C22C2×#57711yescharacteristic
661C22#358612yes—
662C4—24no—
663C4—24no—
664C4—22no—
665C4—24no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—44no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4#326614yes—
679C4C2×#53211yescharacteristic
680C4#349911yescharacteristic
681C4#351112yes—
682C2—44no—
683C2#3612211yescharacteristic
684C2C2×#326611yescharacteristic
685C2#6108311yescharacteristic
686C2#6132212yes—
687C2#6134012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
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
3. #36560C2semidirect
5. #33236C2semidirect
6. #33202C2semidirect
8. #28646C2semidirect
9. #28638C2semidirect
10. #27782C2semidirect
11. #26473C2semidirect
14. #25651C2semidirect
15. #25647C2semidirect
18. #21714C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33722
C2×#33592
#365602
#362762
#332362
#332022
#331622
#286462
#286382
#277822
#264732
#259342
#259292
#256512
#256472
#247482
#230122
#217142
C4×#4402