Finite Group Databasebeta

← #6526553

Subgroups — #6526553

Order 512.

1560 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35289C211yesmaximal, characteristic
2#34978C211yesmaximal, characteristic, complemented
3#29356C212yesmaximal, complemented
4#29346C212yesmaximal, complemented
5#29224C212yesmaximal, complemented
6#29222C212yesmaximal, complemented
7#27821C212yesmaximal, complemented
8#27814C212yesmaximal, complemented
9#27735C212yesmaximal, complemented
10#27728C212yesmaximal, complemented
11#27725C214yesmaximal, complemented
12#27703C214yesmaximal, complemented
13#21893C211yesmaximal, characteristic
14#21833C211yesmaximal, characteristic, complemented
15#21390C211yesmaximal, characteristic
16#21313C211yesmaximal, characteristic, complemented
17#20550C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22#2862C2212yes—
23#2858C2212yes—
24#2640—22no—
25#2612—22no—
26#2571C2211yescharacteristic
27#2563C2212yes—
28#2563C2212yes—
29#2552C2211yescharacteristic
30#2552C2211yescharacteristic
31#2551C2211yescharacteristic
32#2547—21no—
33#2547—21no—
34#2504C2214yes—
35#2441—21no—
36#2441C2212yes—
37#2439—21no—
38#2439C2212yes—
39#2436—22no—
40#2396—22no—
41#2311—21no—
42#2311—21no—
43C4×C4⋊Q8—21no—
44C4×C4⋊Q8C2212yes—
45C4×#125—21no—
46C4×#125C2212yes—
47#2285C2211yescharacteristic
48C4×#117C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2101C2214yes—
53#2101C2214yes—
54#2073C2214yes—
55#2071C2214yes—
56#2025C2214yes—
57#2023C2214yes—
58#1993C2214yes—
59#1991C2214yes—
60#1990C2212yes—
61#1990C2212yes—
62#1988C2212yes—
63#1988C2212yes—
64#1988C2212yes—
65#1988C2212yes—
66#1987C2211yescharacteristic
67#1987C2212yes—
68#1987C2211yescharacteristic
69#1987C2212yes—
70#1987C2212yes—
71#1982C2212yes—
72#1982C2211yescharacteristic
73#1982C2211yescharacteristic
74#1936C2214yes—
75#1936C2214yes—
76#1934C2214yes—
77#1934C2214yes—
78#1895C2214yes—
79#1875C2212yes—
80#1875C2212yes—
81#1836C2214yes—
82#1835C2214yes—
83#1823C2214yes—
84#1802C2212yes—
85#1802C2212yes—
86#1770C2212yes—
87C8×C4⋊C4C2212yes—
88#1767C2212yes—
89C4×C4⋊C8C2212yes—
90C4×#102C2212yes—
91C4×#102C2212yes—
92#1753C2214yes—
93#1736C2212yes—
94#1736C2212yes—
95C22×#41—21no—
96C23×Q8—21no—
97C22×M16C2312yes—
98#532—24no—
99#532—24no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—24no—
104C2×C4⋊Q8—24no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×#127C2311yescharacteristic
110C2×#125—21no—
111C2×#125C2312yes—
112C2×#124—21no—
113C2×#124—22no—
114C2×#124—22no—
115C2×#121—21no—
116C2×#121C2312yes—
117C2×C4×Q8—22no—
118C2×C4×D4—21no—
119C2×C4×D4C2312yes—
120C2×#117—21no—
121C2×#117D411yescharacteristic
122C2×#117D411yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2312yes—
127C22×C42C2311yescharacteristic
128#431—24no—
129#423—24no—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×#102C2312yes—
135C2×#102C2312yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×C22⋊C8C2312yes—
141C2×#97C2314yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2314yes—
144#400—22no—
145#399—22no—
146#398—22no—
147#398—22no—
148#397—24no—
149#397—24no—
150#396—22no—
151#396—22no—
152#395—22no—
153#393—22no—
154#393—24no—
155#393—24no—
156#391—22no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#390—22no—
163#390—22no—
164#390C2314yes—
165#390C2314yes—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#386—24no—
181#386—24no—
182#384—21no—
183#384C2312yes—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#382—21no—
187#382—21no—
188#382C2312yes—
189#382C2314yes—
190#382C2314yes—
191#380—21no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380C2312yes—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—21no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4C2312yes—
214#376—21no—
215#376D412yes—
216C43—21no—
217C43C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2314yes—
234C2×#41—44no—
235C2×#41—44no—
236C2×#41—44no—
237C22×Q8—22no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—44no—
242C22×Q8—82no—
243C22×Q8—82no—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4C2412yes—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C2×M16—24no—
254C2×M16—42no—
255C2×M16—42no—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—42no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270#127—22no—
271#127—22no—
272#125—24no—
273#125—42no—
274#125—44no—
275#124—44no—
276#124—84no—
277#124—84no—
278#121—24no—
279#121—24no—
280#121—44no—
281C4×Q8—84no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—44no—
285#117—22no—
286#117—22no—
287#117—42no—
288#117—42no—
289#117—41no—
290#117—41no—
291#117—82no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42#4114yes—
349C2×C42#4114yes—
350C2×C42C2412yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C4⋊C8—24no—
355C4⋊C8—24no—
356C4⋊C8—22no—
357C4⋊C8—22no—
358#102—24no—
359#102—24no—
360#102—42no—
361#102—42no—
362#102—44no—
363#102—44no—
364C22⋊C8—24no—
365#97—44no—
366C4×C8—24no—
367C4×C8—44no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24—22no—
389C24C22×D411yescharacteristic
390#41—816no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—82no—
401C2×Q8—82no—
402C2×Q8—82no—
403C2×Q8—816no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4#12414yes—
442C22×C4#12414yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2511yescharacteristic
455M16—84no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C42−1+412yes—
501C4⋊C42−1+412yes—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C22⋊C4—82no—
513C42—24no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—42no—
520C42—42no—
521C42—42no—
522C42—42no—
523C42—42no—
524C42—42no—
525C42—44no—
526C42—42no—
527C42—42no—
528C42#12414yes—
529C42#12414yes—
530C42C2×#4112yes—
531C42C2×#4112yes—
532C42C2×#4112yes—
533C42C2×#4112yes—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23C2×#12412yes—
548C23C23×D411yescharacteristic
549Q8—84no—
550Q8—88no—
551Q8—88no—
552Q8—88no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—84no—
558D4—88no—
559D4—84no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—88no—
609C2×C4—88no—
610C2×C4C2×#12412yes—
611C2×C4C2×#12412yes—
612C2×C4C2×#12412yes—
613C2×C4C2×#12412yes—
614C2×C4C2×#12412yes—
615C2×C4#57512yes—
616C2×C4#57512yes—
617C2×C4#57612yes—
618C2×C4#57712yes—
619C2×C4C22×#4111yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C2×2−1+411yescharacteristic
622C2×C4C2×2−1+411yescharacteristic
623C2×C4#58511yescharacteristic
624C2×C4#58511yescharacteristic
625C8—42no—
626C8—42no—
627C8—44no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22C22×#12411yescharacteristic
643C22#348911yescharacteristic
644C22#348911yescharacteristic
645C22C2×#57511yescharacteristic
646C22C2×#57511yescharacteristic
647C22C2×#57611yescharacteristic
648C22C2×#57711yescharacteristic
649C4—22no—
650C4—22no—
651C4—22no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—81no—
659C4—82no—
660C4—82no—
661C4—81no—
662C4—84no—
663C4—82no—
664C4#313614yes—
665C4#314014yes—
666C4#350711yescharacteristic
667C4#350711yescharacteristic
668C4#350911yescharacteristic
669C4#351011yescharacteristic
670C2—42no—
671C2—162no—
672C2#3562811yescharacteristic
673C2C2×#313611yescharacteristic
674C2C2×#314011yescharacteristic
675C2#6213411yescharacteristic
676C2#6213611yescharacteristic
677C2#6214311yescharacteristic
678C2#6214511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34978C2semidirect
3. #29356C2semidirect
4. #29346C2semidirect
5. #29224C2semidirect
6. #29222C2semidirect
7. #27821C2semidirect
8. #27814C2semidirect
9. #27735C2semidirect
10. #27728C2semidirect
11. #27725C2semidirect
12. #27703C2semidirect
14. #21833C2semidirect
16. #21313C2semidirect
17. #20550C2semidirect

Maximal subgroups

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

subgroupindex
#352892
#349782
#293562
#293462
#292242
#292222
#278212
#278142
#277352
#277282
#277252
#277032
#218932
#218332
#213902
#213132
#205502