Finite Group Databasebeta

← #6627452

Subgroups — #6627452

Order 512.

1659 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2C4×D4×Q8C211yesmaximal, characteristic
3#33201C212yesmaximal
4#33197C211yesmaximal, characteristic
5#29290C212yesmaximal, complemented
6#29286C211yesmaximal, characteristic, complemented
7#29224C212yesmaximal, complemented
8#29218C211yesmaximal, characteristic, complemented
9#26521C214yesmaximal, complemented
10#26513C212yesmaximal, complemented
11#26119C214yesmaximal
12#26111C212yesmaximal
13#22743C212yesmaximal, complemented
14#22722C211yesmaximal, characteristic, complemented
15#22523C212yesmaximal
16#21614C212yesmaximal
17C4×#474C211yesmaximal, characteristic
18C2×D4×Q8C2211yescharacteristic
19C22×C4×Q8—21no—
20C2×#476C2212yes—
21C2×#474C2211yescharacteristic
22C2×#446C2212yes—
23#2868C2211yescharacteristic
24#2862C2212yes—
25#2582—21no—
26#2571C2212yes—
27#2570C2214yes—
28#2568—21no—
29#2511—22no—
30#2446—22no—
31#2365—21no—
32#2365C2211yescharacteristic
33#2365C2212yes—
34#2360—21no—
35#2360—22no—
36#2355—21no—
37#2355C2212yes—
38Q8×C4⋊C4C2211yescharacteristic
39D4×C4⋊C4C2212yes—
40D4×C4⋊C4C2211yescharacteristic
41Q8×C22⋊C4—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2212yes—
46C4×C22⋊Q8—21no—
47C4×C22⋊Q8—22no—
48C42×Q8C2211yescharacteristic
49C42×D4C2211yescharacteristic
50C42×D4C2212yes—
51#2101C2214yes—
52#2101C2214yes—
53#2100C2214yes—
54#2087C2214yes—
55#2087C2214yes—
56#2086C2214yes—
57#1993C2214yes—
58#1990C2212yes—
59#1988C2211yescharacteristic
60#1988C2212yes—
61#1988C2212yes—
62#1987C2212yes—
63#1987C2212yes—
64#1987C2212yes—
65#1987C2214yes—
66#1949C2214yes—
67#1949C2214yes—
68#1948C2212yes—
69#1948C2212yes—
70#1947C2214yes—
71#1947C2212yes—
72#1947C2212yes—
73#1947C2211yescharacteristic
74#1940C2214yes—
75#1939C2212yes—
76#1919C2214yes—
77#1918C2212yes—
78#1862C2212yes—
79#1862C2214yes—
80#1842C2211yescharacteristic
81#1842C2212yes—
82#1836C2214yes—
83#1834C2212yes—
84#1804C2214yes—
85#1804C2214yes—
86#1802C2212yes—
87#1802C2211yescharacteristic
88#1802C2212yes—
89#1776C2214yes—
90C4×#107C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1762C2214yes—
94C4×#102C2212yes—
95C23×Q8—21no—
96D4×Q8—24no—
97D4×Q8Q818yescomplemented
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—22no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—22no—
109C2×C4×Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8—22no—
113C2×C4×Q8—48no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2312yes—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C2×C4×D4C2311yescharacteristic
122C22×C4⋊C4—21no—
123C22×C4⋊C4—22no—
124C22×C42—22no—
125C22×C42—21no—
126#476—28no—
127#474—24no—
128#446—28no—
129C2×#107C2312yes—
130C2×#107C2314yes—
131C2×#107C2312yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#97C2314yes—
144C2×C4×C8C2312yes—
145#400—24no—
146#400—24no—
147#398—24no—
148#391—22no—
149#391—21no—
150#391—22no—
151#391—22no—
152#391—21no—
153#391—22no—
154#391—21no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391—22no—
159#389—22no—
160#389—21no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389C2312yes—
167#389C2314yes—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#387—24no—
172#387—24no—
173#387—24no—
174#387—22no—
175#387—22no—
176#387—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#384—21no—
181#384—21no—
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#384C2312yes—
185#384C2312yes—
186#384C2312yes—
187#382—22no—
188#382C2314yes—
189#382C2314yes—
190#380—21no—
191#380—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4D411yescharacteristic
198C4×C4⋊C4D411yescharacteristic
199C4×C4⋊C4D412yes—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2314yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—21no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—21no—
217C43C2312yes—
218C43C2311yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234C22×Q8—28no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—21no—
239C22×Q8—44no—
240C22×Q8—44no—
241C22×Q8C2×D411yescharacteristic
242C22×D4C2411yescharacteristic
243C23×C4—22no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—21no—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C4⋊Q8—22no—
252C4⋊Q8—24no—
253C4⋊Q8—22no—
254C4⋊Q8—24no—
255C4⋊Q8—22no—
256C4⋊Q8—24no—
257C4⋊Q8—44no—
258C4⋊Q8—82no—
259C4⋊Q8C2×Q818yes—
260C4⋊Q8C2×Q814yes—
261C22⋊Q8—216no—
262C22⋊Q8—28no—
263C4×Q8—24no—
264C4×Q8—42no—
265C4×Q8—48no—
266C4×Q8—44no—
267C4×Q8—48no—
268C4×Q8—44no—
269C4×Q8—44no—
270C4×Q8—44no—
271C4×Q8—84no—
272C4×Q8C2×Q814yes—
273C4×Q8#4114yes—
274C4×D4—22no—
275C4×D4—24no—
276C4×D4—22no—
277C4×D4—24no—
278C4×D4—22no—
279C4×D4—41no—
280C4×D4—42no—
281C4×D4C2×Q814yes—
282C4×D4C2×Q818yes—
283C4×D4#4114yes—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—48no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C42—22no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—48no—
333C2×C42—44no—
334C2×C42C2×D412yes—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—24no—
348#107—44no—
349#107—42no—
350#106—28no—
351C4⋊C8—24no—
352C4⋊C8—28no—
353C4⋊C8—22no—
354C4⋊C8—24no—
355C4⋊C8—22no—
356C4⋊C8—42no—
357C4⋊C8—44no—
358#102—24no—
359#102—28no—
360#102—24no—
361#97—28no—
362C4×C8—24no—
363#95—22no—
364#95—21no—
365#95—22no—
366#95—21no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—21no—
384C2×Q8—28no—
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—432no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8C22⋊Q814yes—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4C22×Q812yes—
404C2×D4C2×#4112yes—
405C2×D42−1+412yes—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—81no—
428C22×C4—82no—
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C2511yescharacteristic
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—24no—
449C4⋊C4—24no—
450C4⋊C4—28no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—82no—
481C4⋊C4—81no—
482C4⋊C4—84no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4C22⋊Q818yes—
486C4⋊C4C22⋊Q814yes—
487C4⋊C4C22×Q812yes—
488C4⋊C4C22×Q814yes—
489C4⋊C4C2×#4112yes—
490C22⋊C4—24no—
491C22⋊C4—28no—
492C22⋊C4—24no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C42—24no—
496C42—22no—
497C42—24no—
498C42—22no—
499C42—21no—
500C42—22no—
501C42—21no—
502C42—22no—
503C42—42no—
504C42—42no—
505C42—41no—
506C42—44no—
507C42—44no—
508C42—42no—
509C42—42no—
510C42—44no—
511C42—42no—
512C42—44no—
513C42—84no—
514C42AGL(1,ℤ/8)14yes—
515C42C22×Q814yes—
516C42C22×Q812yes—
517C42C2×#4112yes—
518C42C2×#4114yes—
519C42C2×#4112yes—
520C422−1+412yes—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23C23×D411yescharacteristic
529Q8—28no—
530Q8—416no—
531Q8—48no—
532Q8—88no—
533Q8—84no—
534D4—22no—
535D4—22no—
536D4—22no—
537D4#55714yes—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—416no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—84no—
585C2×C4—88no—
586C2×C4C2×C22⋊Q812yes—
587C2×C4C2×C22⋊Q814yes—
588C2×C4C2×C22⋊Q814yes—
589C2×C4C2×C22⋊Q812yes—
590C2×C4#54812yes—
591C2×C4#54812yes—
592C2×C4C2×AGL(1,ℤ/8)12yes—
593C2×C4C2×AGL(1,ℤ/8)12yes—
594C2×C4C2×AGL(1,ℤ/8)12yes—
595C2×C4#57512yes—
596C2×C4#57812yes—
597C2×C4C23×Q811yescharacteristic
598C2×C4C22×#4111yescharacteristic
599C2×C4C22×#4111yescharacteristic
600C2×C4C2×2−1+411yescharacteristic
601C2×C4C2×2−1+411yescharacteristic
602C2×C4#58511yescharacteristic
603C8—42no—
604C8—44no—
605C8—42no—
606C22—24no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22C22×C22⋊Q811yescharacteristic
614C22C2×#54811yescharacteristic
615C22#348911yescharacteristic
616C22C22×AGL(1,ℤ/8)11yescharacteristic
617C22C2×#57511yescharacteristic
618C22C2×#57811yescharacteristic
619C22#358511yescharacteristic
620C4—24no—
621C4—24no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—44no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—41no—
630C4—42no—
631C4—82no—
632C4—84no—
633C4—82no—
634C4—81no—
635C4—82no—
636C4#307714yes—
637C4#336714yes—
638C4C2×#55711yescharacteristic
639C4#348011yescharacteristic
640C4#350211yescharacteristic
641C4#352311yescharacteristic
642C2—44no—
643C2#3657711yescharacteristic
644C2C2×#307711yescharacteristic
645C2#6049811yescharacteristic
646C2C2×#336711yescharacteristic
647C2#6132411yescharacteristic
648C2#6211011yescharacteristic
649C2#6213311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36598C2semidirect
5. #29290C2semidirect
6. #29286C2semidirect
7. #29224C2semidirect
8. #29218C2semidirect
9. #26521C2semidirect
10. #26513C2semidirect
13. #22743C2semidirect
14. #22722C2semidirect
97. D4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#365982
C4×D4×Q82
#332012
#331972
#292902
#292862
#292242
#292182
#265212
#265132
#261192
#261112
#227432
#227222
#225232
#216142
C4×#4742