Finite Group Databasebeta

← #6435616

Subgroups — #6435616

Order 512.

1684 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic, complemented
2#34058C211yesmaximal, characteristic
3#30003C214yesmaximal, complemented
4#30002C214yesmaximal, complemented
5#29998C212yesmaximal, complemented
6#29995C212yesmaximal, complemented
7#29995C212yesmaximal, complemented
8#29994C212yesmaximal, complemented
9#23671C212yesmaximal, complemented
10#23669C212yesmaximal, complemented
11#23614C212yesmaximal
12#23612C212yesmaximal
13#21171C211yesmaximal, characteristic, complemented
14#21169C211yesmaximal, characteristic, complemented
15#20793C211yesmaximal, characteristic, complemented
16#20646C211yesmaximal, characteristic
17C2×#1802C211yesmaximal, characteristic
18C22×C4×D4C2211yescharacteristic
19C2×#428C2212yes—
20C2×#427C2212yes—
21C2×#424C2212yes—
22C2×#424C2212yes—
23C2×#423C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C2×#418C2212yes—
26C22×#102C2212yes—
27#2859C2212yescomplemented
28#2859C2212yescomplemented
29#2647C2214yescomplemented
30#2571—21no—
31#2571—21no—
32#2552—21no—
33#2552—21no—
34#2442—22no—
35#2442—22no—
36#2319—21no—
37#2319—21no—
38#2319C2212yes—
39#2319C2212yes—
40#2317C2212yes—
41#2316C2211yescharacteristic
42#2316C2211yescharacteristic
43#2316C2212yes—
44#2312—22no—
45#2309C2211yescharacteristic
46#2309C2211yescharacteristic
47#2293—21no—
48#2293—21no—
49#2293C2212yes—
50#2293C2212yes—
51#2285C2211yescharacteristic
52C2×#384C2211yescharacteristic
53#2280—22no—
54C4×#117C2211yescharacteristic
55C4×#117C2211yescharacteristic
56#2088C2214yescomplemented
57#2088C2214yescomplemented
58#2088C2214yescomplemented
59#2088C2214yescomplemented
60#2087C2214yescomplemented
61#2087C2214yescomplemented
62#2086C2214yescomplemented
63#2086C2214yescomplemented
64#2067C2214yes—
65#2067C2214yes—
66#2067C2214yes—
67#2067C2214yes—
68#2065C2214yes—
69#2065C2214yes—
70#2065C2214yes—
71#2065C2214yes—
72#1949C2212yescomplemented
73#1949C2212yescomplemented
74#1948C2211yescharacteristic, complemented
75#1948C2212yescomplemented
76#1948C2211yescharacteristic, complemented
77#1948C2212yescomplemented
78#1947C2211yescharacteristic, complemented
79#1947C2212yescomplemented
80#1947C2211yescharacteristic, complemented
81#1947C2212yescomplemented
82#1843C2214yescomplemented
83#1842C2212yescomplemented
84#1842C2212yescomplemented
85#1804C2214yes—
86#1802C2212yes—
87#1802C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1802C2214yes—
91#1736C2212yes—
92#1736C2212yes—
93#1735C2212yes—
94C2×#340C2212yes—
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C22×M16C2312yes—
98C23×C8C2312yes—
99C2×#125—22no—
100C2×#125—22no—
101C2×C4×D4—21no—
102C2×C4×D4—21no—
103C2×C4×D4—24no—
104C2×C4×D4Q818yes—
105C2×C4×D4C2312yes—
106C2×C4×D4C2312yes—
107C2×#117—22no—
108C2×#117—22no—
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C2×#117C2312yes—
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C42C2311yescharacteristic
120#428—44no—
121#427—44no—
122C2×#107C2312yes—
123C2×#107C2314yes—
124C2×#107C2312yes—
125C2×#106C2312yes—
126C2×#106C2312yes—
127C2×#106C2314yes—
128#424—44no—
129#424—44no—
130#423—24no—
131C2×C4⋊C8—24no—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2314yes—
141#418—44no—
142C2×#102—44no—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147#400—24no—
148#400—24no—
149#398—24no—
150#398—24no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390—21no—
155#390C2312yes—
156#390C2312yes—
157#390C2312yes—
158#390C2312yes—
159#389—22no—
160#389—22no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2314yes—
174#389C2314yes—
175#389C2314yes—
176#389C2314yes—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#385—24no—
186#384—22no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#382—22no—
194#382—22no—
195#382—22no—
196#382C2312yes—
197#381—22no—
198#381—22no—
199#380—21no—
200#380—21no—
201#380—22no—
202#380—22no—
203#380—24no—
204#380—22no—
205#380—22no—
206#379—22no—
207#379—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224#376—21no—
225#376C2312yes—
226C43—21no—
227C43C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#340C2314yes—
235#340C2314yes—
236#340C2312yes—
237#340C2312yes—
238C25—21no—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4C2×Q814yes—
246C22×D4C2412yes—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—28no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—82no—
253C23×C4—82no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2411yescharacteristic
257C2×M16—24no—
258C2×M16—24no—
259C2×M16—24no—
260C22×C8—24no—
261C22×C8—24no—
262C22×C8—24no—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2414yes—
266#125—84no—
267#125—84no—
268C4×D4—28no—
269C4×D4—28no—
270C4×D4—24no—
271C4×D4—24no—
272C4×D4C2×Q818yes—
273C4×D4C2×Q818yes—
274#117—22no—
275#117—22no—
276#117—22no—
277#117—22no—
278#117—24no—
279#117—22no—
280#117—22no—
281#117—22no—
282#117—22no—
283#117—24no—
284#117—84no—
285#117—84no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×Q814yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—28no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×Q814yes—
352C2×C42#4114yes—
353C2×C42C2412yes—
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—42no—
358#107—42no—
359#107—44no—
360#106—42no—
361#106—42no—
362#106—44no—
363C4⋊C8—22no—
364C4⋊C8—22no—
365C4⋊C8—24no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368C4⋊C8—44no—
369C4⋊C8—44no—
370C4⋊C8—44no—
371C4⋊C8—44no—
372#102—44no—
373#102—44no—
374#102—44no—
375#102—44no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—22no—
395C24—24no—
396C24—42no—
397C24—42no—
398C24—81no—
399C24—81no—
400C24—81no—
401C24—81no—
402C24C22×D411yescharacteristic
403C2×D4—24no—
404C2×D4—28no—
405C2×D4—28no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4C22×Q814yes—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—816no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4C22⋊Q814yes—
442C22×C4C22⋊Q814yes—
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×C4C22×Q812yes—
453C22×C4C2×#4112yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C42−1+412yes—
457C22×C42−1+412yes—
458C22×C4C2511yescharacteristic
459M16—48no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—48no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4C22×Q814yes—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—28no—
511C22⋊C4—28no—
512C22⋊C4—42no—
513C22⋊C4—44no—
514C22⋊C4—42no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C42—24no—
524C42—24no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—42no—
530C42—42no—
531C42—42no—
532C42—44no—
533C42—42no—
534C42—42no—
535C42—42no—
536C42—81no—
537C42—81no—
538C42C22⋊Q814yes—
539C42C22⋊Q814yes—
540C42C22×Q812yes—
541C42C22×Q812yes—
542C42C2×#4112yes—
543C42C2×#4112yes—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—24no—
549C23—42no—
550C23—42no—
551C23—48no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—84no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23C2×C22⋊Q812yes—
563C23#53712yes—
564C23#53712yes—
565C23C23×D411yescharacteristic
566D4—24no—
567D4—24no—
568D4—48no—
569D4—42no—
570D4—42no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—28no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—416no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—88no—
617C2×C4—84no—
618C2×C4—164no—
619C2×C4C2×C22⋊Q812yes—
620C2×C4C2×C22⋊Q812yes—
621C2×C4C2×C22⋊Q812yes—
622C2×C4C2×C22⋊Q812yes—
623C2×C4C2×C22⋊Q812yes—
624C2×C4#53712yes—
625C2×C4#53712yes—
626C2×C4#54014yes—
627C2×C4#54114yes—
628C2×C4#54414yes—
629C2×C4#54414yes—
630C2×C4#57612yes—
631C2×C4#57712yes—
632C2×C4C23×Q811yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×2−1+411yescharacteristic
637C2×C4C2×2−1+411yescharacteristic
638C8—42no—
639C8—42no—
640C8—44no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—816no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22#262914yes—
656C22C22×C22⋊Q811yescharacteristic
657C22C2×#53711yescharacteristic
658C22C2×#53711yescharacteristic
659C22C2×#57611yescharacteristic
660C22C2×#57711yescharacteristic
661C22#358511yescharacteristic
662C22#358611yescharacteristic
663C4—22no—
664C4—22no—
665C4—22no—
666C4—42no—
667C4—42no—
668C4—44no—
669C4—42no—
670C4—42no—
671C4—81no—
672C4—82no—
673C4—81no—
674C4—82no—
675C4—81no—
676C4—84no—
677C4—81no—
678C4#262914yes—
679C4#308114yes—
680C4C2×#54011yescharacteristic
681C4C2×#54111yescharacteristic
682C4C2×#54411yescharacteristic
683C4C2×#54411yescharacteristic
684C2—24no—
685C2—84no—
686C2C2×#262911yescharacteristic
687C2C2×#308111yescharacteristic
688C2#6049311yescharacteristic
689C2#6092211yescharacteristic
690C2#6092311yescharacteristic
691C2#6092811yescharacteristic
692C2#6092911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
1. #35286C2semidirect
3. #30003C2semidirect
4. #30002C2semidirect
5. #29998C2semidirect
6. #29995C2semidirect
7. #29995C2semidirect
8. #29994C2semidirect
9. #23671C2semidirect
10. #23669C2semidirect
13. #21171C2semidirect
14. #21169C2semidirect
15. #20793C2semidirect
27. #2859C22semidirect
28. #2859C22semidirect
29. #2647C22semidirect
56. #2088C22semidirect
57. #2088C22semidirect
58. #2088C22semidirect
59. #2088C22semidirect
60. #2087C22semidirect
61. #2087C22semidirect
62. #2086C22semidirect
63. #2086C22semidirect
72. #1949C22semidirect
73. #1949C22semidirect
74. #1948C22semidirect
75. #1948C22semidirect
76. #1948C22semidirect
77. #1948C22semidirect
78. #1947C22semidirect
79. #1947C22semidirect
80. #1947C22semidirect
81. #1947C22semidirect
82. #1843C22semidirect
83. #1842C22semidirect
84. #1842C22semidirect

Maximal subgroups

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

subgroupindex
#352862
#340582
#300032
#300022
#299982
#299952
#299942
#236712
#236692
#236142
#236122
#211712
#211692
#207932
#206462
C2×#18022