Finite Group Databasebeta

← #6815448

Subgroups — #6815448

Order 512.

1778 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3374C211yesmaximal, characteristic, complemented
2C2×#3328C211yesmaximal, characteristic
3#39744C211yesmaximal, characteristic
4#39616C211yesmaximal, characteristic
5#33225C211yesmaximal, characteristic, complemented
6#31119C214yesmaximal, complemented
7#30990C214yesmaximal, complemented
8#29866C211yesmaximal, characteristic, complemented
9#29799C211yesmaximal, characteristic, complemented
10#28646C211yesmaximal, characteristic, complemented
11#28606C211yesmaximal, characteristic
12#27283C214yesmaximal, complemented
13#26988C214yesmaximal, complemented
14#25931C211yesmaximal, characteristic, complemented
15#25884C211yesmaximal, characteristic
16#25647C211yesmaximal, characteristic, complemented
17#25570C211yesmaximal, characteristic
18#21631C211yesmaximal, characteristic
19#21276C211yesmaximal, characteristic, complemented
20C2×#557C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#556C2211yescharacteristic
23C2×D4×Q8C2211yescharacteristic
24#3374—28no—
25#3328—28no—
26C2×#491C2212yes—
27C2×#487C2211yescharacteristic
28C2×#487C2211yescharacteristic
29C2×#487C2211yescharacteristic
30C2×#487C2211yescharacteristic
31C2×#485C2212yes—
32C2×#479C2212yes—
33C2×#475C2211yescharacteristic
34C2×#475C2211yescharacteristic
35C2×#470C2212yes—
36C2×#462C2212yes—
37C2×#462C2211yescharacteristic
38C2×#462C2211yescharacteristic
39C2×#451C2212yes—
40C2×#446C2211yescharacteristic
41C2×#441C2212yes—
42C2×C4×Q16C2211yescharacteristic
43C2×#436C2211yescharacteristic
44C2×#419C2212yes—
45#2868C2211yescharacteristic
46#2868C2211yescharacteristic
47#2721—24no—
48#2720—24no—
49#2660—21no—
50#2658—21no—
51#2568C2211yescharacteristic
52#2567—21no—
53#2566—21no—
54#2547C2211yescharacteristic
55#2546—21no—
56#2546C2211yescharacteristic
57#2545—21no—
58#2542C2211yescharacteristic
59#2441C2214yes—
60#2434C2214yes—
61#2429C2214yes—
62#2423C2214yes—
63#2327—21no—
64#2326—21no—
65C4×#124C2211yescharacteristic
66#2295C2211yescharacteristic
67#2295C2211yescharacteristic
68#2084C2214yes—
69#2072C2214yes—
70#2068C2214yes—
71#2043C2214yes—
72#2043C2214yes—
73#2036C2214yes—
74#2035C2214yes—
75#2026C2214yes—
76#2016C2214yes—
77#2001C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1984C2211yescharacteristic
81#1981C2211yescharacteristic
82#1981C2211yescharacteristic
83#1969C2212yes—
84#1956C2214yes—
85#1956C2214yes—
86#1954C2211yescharacteristic
87#1942C2212yes—
88#1935C2214yes—
89#1933C2212yes—
90#1933C2212yes—
91#1923C2214yes—
92#1894C2214yes—
93#1878C2214yes—
94#1873C2214yes—
95#1865C2214yes—
96#1831C2211yescharacteristic
97#1787C2214yes—
98C4×#107C2211yescharacteristic
99C4×#103C2211yescharacteristic
100C4×#103C2211yescharacteristic
101#1753C2214yes—
102C42×C8C2211yescharacteristic
103C23×Q8C2312yes—
104C22×Q16C2312yes—
105C22×Q16C2312yes—
106C22×Q16C2311yescharacteristic
107C22×M16C2312yes—
108#557—24no—
109#557—24no—
110#557—42no—
111#557—42no—
112#556—24no—
113#556—42no—
114D4×Q8—24no—
115D4×Q8—42no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—21no—
119C2×C4⋊Q8—21no—
120C2×C4⋊Q8—21no—
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2312yes—
124C2×C4⋊Q8C2314yes—
125C2×C4⋊Q8C2311yescharacteristic
126C2×C4⋊Q8C2311yescharacteristic
127C2×#126—22no—
128C2×#125—21no—
129C2×#125—22no—
130C2×#125—22no—
131C2×#125C2312yes—
132C2×#124—21no—
133C2×#124—21no—
134C2×#124D411yescharacteristic
135C2×C22⋊Q8—22no—
136C2×C22⋊Q8C2312yes—
137C2×C22⋊Q8C2312yes—
138C2×C22⋊Q8C2314yes—
139C2×C4×Q8—21no—
140C2×C4×Q8—21no—
141C2×C4×Q8—21no—
142C2×C4×Q8C2312yes—
143C2×C4×Q8C2312yes—
144C2×C4×Q8C2311yescharacteristic
145C2×C4×Q8C2311yescharacteristic
146C2×C4×Q8C2311yescharacteristic
147C2×C4×D4—21no—
148C2×C4×D4C2311yescharacteristic
149C2×#117C2312yes—
150#491—28no—
151#487—24no—
152#487—24no—
153#487—42no—
154#487—42no—
155#485—28no—
156#479—28no—
157#475—24no—
158#475—24no—
159#470—28no—
160#462—28no—
161#462—24no—
162#462—24no—
163#451—28no—
164#446—24no—
165#441—28no—
166C4×Q16D418yes—
167#436—24no—
168C2×#107C2312yes—
169C2×#107C2311yescharacteristic
170C2×#106C2312yes—
171C2×#106C2312yes—
172C2×C4⋊C8C2311yescharacteristic
173C2×C4⋊C8C2311yescharacteristic
174C2×C4⋊C8C2312yes—
175#419—28no—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#103C2314yes—
187C2×#103C2314yes—
188C2×#103C2312yes—
189C2×#103C2312yes—
190C2×#103C2311yescharacteristic
191C2×#103C2311yescharacteristic
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2314yes—
194C2×#97C2312yes—
195C2×C4×C8C2311yescharacteristic
196C2×C4×C8C2312yes—
197C2×C4×C8C2312yes—
198C2×C4×C8C2311yescharacteristic
199#400—22no—
200#399—24no—
201#399—24no—
202#399—22no—
203#398—22no—
204#398—24no—
205#398—24no—
206#397—22no—
207#396—24no—
208#396—24no—
209#395—24no—
210#395—24no—
211#391—21no—
212#391—21no—
213#391—21no—
214#391D411yescharacteristic
215#391C2314yes—
216#391C2312yes—
217#391C2312yes—
218#390—21no—
219#390D411yescharacteristic
220#389—24no—
221#389—24no—
222#389C2314yes—
223#389C2312yes—
224#389C2312yes—
225#388—24no—
226#388—24no—
227#388D414yes—
228#387—22no—
229#387—22no—
230#387—24no—
231#387—24no—
232#385—24no—
233#385—24no—
234#384—21no—
235#384—21no—
236#384—21no—
237#384—22no—
238#384—22no—
239#384C2314yes—
240#382—21no—
241#382D411yescharacteristic
242#380C2314yes—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4C2311yescharacteristic
246C4×C4⋊C4C2311yescharacteristic
247C4×C22⋊C4C2314yes—
248#376—21no—
249#376—21no—
250C43C2311yescharacteristic
251#340C2314yes—
252#340C2314yes—
253#340C2312yes—
254#340C2312yes—
255#340C2312yes—
256#336C2314yes—
257#336C2314yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—22no—
261C22×Q8—24no—
262C22×Q8—42no—
263C22×Q8—44no—
264C22×Q8—44no—
265C22×Q8C2×D411yescharacteristic
266C22×Q8C2412yes—
267C22×Q8C2412yes—
268C22×Q8C2411yescharacteristic
269C22×Q8C2411yescharacteristic
270C22×D4C2×D411yescharacteristic
271C23×C4—22no—
272C23×C4C2412yes—
273C2×Q16—24no—
274C2×Q16—24no—
275C2×Q16—42no—
276C2×Q16—44no—
277C2×Q16—44no—
278C2×Q16C2×D414yes—
279C2×M16—24no—
280C2×M16—44no—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2411yescharacteristic
285C22×C8C2411yescharacteristic
286C4⋊Q8—24no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—44no—
296C4⋊Q8—42no—
297C4⋊Q8—44no—
298C4⋊Q8—44no—
299C4⋊Q8—44no—
300C4⋊Q8—44no—
301C4⋊Q8—82no—
302C4⋊Q8—82no—
303#126—48no—
304#125—24no—
305#125—42no—
306#125—44no—
307#125—48no—
308#125—48no—
309#124—24no—
310#124—44no—
311#124—82no—
312C22⋊Q8—24no—
313C22⋊Q8—24no—
314C22⋊Q8—48no—
315C22⋊Q8—42no—
316C22⋊Q8—42no—
317C22⋊Q8—44no—
318C22⋊Q8—44no—
319C4×Q8—22no—
320C4×Q8—22no—
321C4×Q8—22no—
322C4×Q8—22no—
323C4×Q8—24no—
324C4×Q8—24no—
325C4×Q8—42no—
326C4×Q8—42no—
327C4×Q8—44no—
328C4×Q8—44no—
329C4×Q8—44no—
330C4×Q8C2×D414yes—
331C4×Q8C2×D414yes—
332C4×D4—22no—
333C4×D4—22no—
334C4×D4—44no—
335#117—24no—
336#117—42no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—24no—
360C2×C4⋊C4—24no—
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2×D412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2411yescharacteristic
369C2×C4⋊C4C2411yescharacteristic
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4C2×D412yes—
376C2×C22⋊C4C2×D414yes—
377C2×C22⋊C4C2×D414yes—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D412yes—
387C2×C42C2×D411yescharacteristic
388C2×C42C2412yes—
389C2×C42C2412yes—
390C2×C42C2411yescharacteristic
391#107—42no—
392#107C2×D414yes—
393#106—24no—
394#106—24no—
395C4⋊C8—24no—
396C4⋊C8—22no—
397C4⋊C8—22no—
398#103—24no—
399#103—24no—
400#103—24no—
401#103—24no—
402#103—24no—
403#103—24no—
404#103—24no—
405#103—24no—
406#103—42no—
407#103—44no—
408#103—44no—
409#103—42no—
410#103—44no—
411#103—44no—
412#103C2×D414yes—
413#103C2×D414yes—
414C22⋊C8—24no—
415C22⋊C8—44no—
416#97—24no—
417C4×C8—22no—
418C4×C8—42no—
419C4×C8—42no—
420C4×C8C2×D414yes—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—21no—
428#95—21no—
429#95—24no—
430#95—24no—
431#95—24no—
432#95—24no—
433#95C2×D414yes—
434#95C2×D414yes—
435C24C22×D412yes—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—24no—
450C2×Q8—24no—
451C2×Q8—24no—
452C2×Q8—24no—
453C2×Q8—44no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—48no—
461C2×Q8—84no—
462C2×Q8—84no—
463C2×Q8—88no—
464C2×Q8C22×D412yes—
465C2×Q8C22×D412yes—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—44no—
484C22×C4—84no—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C2511yescharacteristic
493Q16—44no—
494Q16—48no—
495Q16—84no—
496M16—84no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8C22×D412yes—
507C2×C8C22×D412yes—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—48no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—84no—
541C4⋊C4—84no—
542C4⋊C4—84no—
543C4⋊C4—84no—
544C4⋊C4—84no—
545C4⋊C4C22×D412yes—
546C4⋊C4C22×D412yes—
547C22⋊C4—24no—
548C22⋊C4—28no—
549C22⋊C4—28no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—84no—
554C22⋊C4—84no—
555C42—24no—
556C42—24no—
557C42—22no—
558C42—24no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—44no—
564C42—44no—
565C42—44no—
566C42—42no—
567C42—42no—
568C42#13714yes—
569C42C22×D412yes—
570C42H(5,𝔽2)12yes—
571C23—22no—
572C23—24no—
573C23—22no—
574C23—22no—
575C23—24no—
576C23C23×D411yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—22no—
580Q8—22no—
581Q8—44no—
582Q8—42no—
583Q8—42no—
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—42no—
588Q8—44no—
589Q8—44no—
590Q8—88no—
591Q8—84no—
592Q8—84no—
593D4—44no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—88no—
634C2×C4D4212yes—
635C2×C4D4212yes—
636C2×C4C2×#13714yes—
637C2×C4C2×#13712yes—
638C2×C4#57712yes—
639C2×C4#57814yes—
640C2×C4C23×D411yescharacteristic
641C2×C4C2×H(5,𝔽2)11yescharacteristic
642C2×C4#58512yes—
643C2×C4#58512yes—
644C8—22no—
645C8—22no—
646C8—42no—
647C8—42no—
648C8—42no—
649C22—44no—
650C22—44no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22C2×D4211yescharacteristic
655C22#350012yes—
656C22C22×#13711yescharacteristic
657C22C2×#57711yescharacteristic
658C22C2×#57812yes—
659C4—24no—
660C4—24no—
661C4—22no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4#327814yes—
676C4#344911yescharacteristic
677C4#350011yescharacteristic
678C4#356212yes—
679C2—82no—
680C2#3935711yescharacteristic
681C2C2×#327811yescharacteristic
682C2#6109111yescharacteristic
683C2#6253612yes—
684C2#6255312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3374C2semidirect
5. #33225C2semidirect
6. #31119C2semidirect
7. #30990C2semidirect
8. #29866C2semidirect
9. #29799C2semidirect
10. #28646C2semidirect
12. #27283C2semidirect
13. #26988C2semidirect
14. #25931C2semidirect
16. #25647C2semidirect
19. #21276C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33742
C2×#33282
#397442
#396162
#332252
#311192
#309902
#298662
#297992
#286462
#286062
#272832
#269882
#259312
#258842
#256472
#255702
#216312
#212762