Finite Group Databasebeta

← #6464142

Subgroups — #6464142

Order 512.

1674 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35320C211yesmaximal, characteristic
2C4×#544C211yesmaximal, characteristic
3#29361C214yesmaximal, complemented
4#29359C214yesmaximal, complemented
5#29357C212yesmaximal, complemented
6#29355C212yesmaximal, complemented
7#29349C212yesmaximal, complemented
8#29347C212yesmaximal, complemented
9#23633C212yesmaximal
10#23457C212yesmaximal
11#21312C211yesmaximal, characteristic
12#21311C211yesmaximal, characteristic
13#21259C211yesmaximal, characteristic
14C2×#1981C211yesmaximal, characteristic
15#21090C212yesmaximal
16#21088C212yesmaximal
17#20431C211yesmaximal, characteristic
18C2×#544C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C2×#419C2212yes—
24C2×#419C2212yes—
25C22×#103C2212yes—
26C22×#103C2212yes—
27C2×#405C2212yes—
28#2862C2212yes—
29#2862C2212yes—
30#2642C2214yes—
31#2568—21no—
32#2568—21no—
33#2568—21no—
34#2568—21no—
35#2503—22no—
36#2503—22no—
37#2331—21no—
38#2331C2212yes—
39#2330—21no—
40#2330C2212yes—
41#2329—22no—
42#2317C2211yescharacteristic
43#2317C2211yescharacteristic
44#2316C2211yescharacteristic
45#2310C2212yes—
46#2310C2212yes—
47C4×C4⋊Q8—21no—
48C4×C4⋊Q8C2212yes—
49C4×#125—21no—
50C4×#125C2212yes—
51C4×C22⋊Q8—22no—
52C4×#117C2211yescharacteristic
53C4×#117C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55#2074C2214yes—
56#2074C2214yes—
57#2072C2214yes—
58#2072C2214yes—
59#2026C2214yes—
60#2026C2214yes—
61#2024C2214yes—
62#2024C2214yes—
63#1989C2212yes—
64#1989C2212yes—
65#1989C2214yes—
66#1988C2211yescharacteristic
67#1988C2211yescharacteristic
68#1988C2212yes—
69#1987C2211yescharacteristic
70#1987C2211yescharacteristic
71#1987C2212yes—
72#1981C2212yes—
73#1981C2212yes—
74#1981C2214yes—
75#1981C2212yes—
76#1981C2212yes—
77#1981C2212yes—
78#1981C2212yes—
79#1935C2214yes—
80#1935C2214yes—
81#1935C2214yes—
82#1935C2214yes—
83#1933C2214yes—
84#1933C2214yes—
85#1933C2214yes—
86#1933C2214yes—
87#1768C2214yes—
88#1767C2212yes—
89C4×C4⋊C8C2212yes—
90#1736C2212yes—
91#1736C2212yes—
92#1735C2212yes—
93#1735C2212yes—
94#1729C2212yes—
95C23×Q8C2312yes—
96C22×M16C2312yes—
97C23×C8C2312yes—
98#544—24no—
99#544—24no—
100#533—24no—
101#533—24no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—24no—
106C2×C4⋊Q8—24no—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×#125—21no—
113C2×#125C2312yes—
114C2×#125C2312yes—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—22no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—22no—
121C2×#117C2312yes—
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132#423—24no—
133C2×C4⋊C8—24no—
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139#419—28no—
140#419—44no—
141C2×#103—28no—
142C2×#103—44no—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151#405—44no—
152C2×#97C2314yes—
153C2×C4×C8C2314yes—
154#400—24no—
155#400—24no—
156#400—24no—
157#400—24no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391—22no—
164#391—22no—
165#391—22no—
166#391—22no—
167#391—22no—
168#391—22no—
169#391C2314yes—
170#391C2314yes—
171#391C2314yes—
172#391C2314yes—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#385—22no—
191#385—22no—
192#384—21no—
193#384C2312yes—
194#382—21no—
195#382—22no—
196#382—22no—
197#382—22no—
198#382—22no—
199#382C2312yes—
200#381—24no—
201#381—24no—
202#380—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4D412yes—
222#376—21no—
223#376C2312yes—
224C2×#95—21no—
225C43—21no—
226C43C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#340C2314yes—
236#340C2314yes—
237#340C2312yes—
238#340C2312yes—
239#336C2314yes—
240#336C2314yes—
241C22×Q8—22no—
242C22×Q8—21no—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8—24no—
246C22×Q8—24no—
247C22×Q8—24no—
248C22×Q8—24no—
249C22×Q8—24no—
250C22×Q8—24no—
251C22×Q8C2414yes—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2411yescharacteristic
258C2×M16—24no—
259C2×M16—24no—
260C2×M16—24no—
261C22×C8—24no—
262C22×C8—24no—
263C22×C8—24no—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2414yes—
267C4⋊Q8—28no—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—22no—
271C4⋊Q8—22no—
272C4⋊Q8—22no—
273C4⋊Q8—22no—
274C4⋊Q8—24no—
275C4⋊Q8—24no—
276C4⋊Q8—24no—
277C4⋊Q8—24no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280#125—28no—
281#125—24no—
282#125—24no—
283#125—24no—
284#125—24no—
285C22⋊Q8—216no—
286C22⋊Q8—44no—
287C22⋊Q8—44no—
288C4×Q8—44no—
289C4×Q8—44no—
290C4×Q8—84no—
291#117—22no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—24no—
297#117—24no—
298#117—22no—
299#117—22no—
300#117#4114yes—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4#4114yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—82no—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42#4114yes—
360C2×C42#4114yes—
361C2×C42C2412yes—
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365C4⋊C8—24no—
366C4⋊C8—22no—
367C4⋊C8—22no—
368C4⋊C8—24no—
369C4⋊C8—24no—
370#103—28no—
371#103—28no—
372#103—28no—
373#103—28no—
374#103—44no—
375#103—44no—
376#103—44no—
377#103—44no—
378#97—44no—
379C4×C8—44no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404C24C22×D411yescharacteristic
405C2×Q8—28no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—44no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—48no—
422C2×Q8—48no—
423C2×Q8—48no—
424C2×Q8—48no—
425C2×Q8—82no—
426C2×Q8—82no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—81no—
446C22×C4—81no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4#12414yes—
450C22×C4#12414yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C42−1+412yes—
466C22×C4C2511yescharacteristic
467M16—48no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—48no—
478C4⋊C4—28no—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4C2×#4114yes—
515C4⋊C4C2×#4112yes—
516C4⋊C4C2×#4112yes—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C42—24no—
523C42—24no—
524C42—24no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—42no—
530C42—42no—
531C42—44no—
532C42—42no—
533C42—44no—
534C42—42no—
535C42—42no—
536C42—81no—
537C42—81no—
538C42—82no—
539C42#12414yes—
540C42#12414yes—
541C42C2×#4112yes—
542C42C2×#4112yes—
543C42C2×#4112yes—
544C42C2×#4112yes—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23#52512yes—
550C23C2×#12412yes—
551C23#53712yes—
552C23C23×D411yescharacteristic
553Q8—44no—
554Q8—44no—
555Q8—44no—
556Q8—44no—
557Q8—84no—
558Q8—88no—
559Q8—88no—
560Q8—84no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
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—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—88no—
611C2×C4#52512yes—
612C2×C4C2×#12412yes—
613C2×C4C2×#12412yes—
614C2×C4C2×#12412yes—
615C2×C4C2×#12412yes—
616C2×C4C2×#12412yes—
617C2×C4#53712yes—
618C2×C4#55614yes—
619C2×C4#55614yes—
620C2×C4#55914yes—
621C2×C4#55914yes—
622C2×C4#57712yes—
623C2×C4#57812yes—
624C2×C4C22×#4111yescharacteristic
625C2×C4C22×#4111yescharacteristic
626C2×C4C22×#4111yescharacteristic
627C2×C4C22×#4111yescharacteristic
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C2×C4C2×2−1+411yescharacteristic
630C8—42no—
631C8—42no—
632C8—44no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22#259214yes—
640C22C2×#52511yescharacteristic
641C22C22×#12411yescharacteristic
642C22C2×#53711yescharacteristic
643C22C2×#57711yescharacteristic
644C22C2×#57811yescharacteristic
645C22#358611yescharacteristic
646C22#358611yescharacteristic
647C4—22no—
648C4—22no—
649C4—22no—
650C4—42no—
651C4—44no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—84no—
658C4—82no—
659C4—82no—
660C4—81no—
661C4—82no—
662C4—81no—
663C4#259214yes—
664C4#313714yes—
665C4C2×#55611yescharacteristic
666C4C2×#55611yescharacteristic
667C4C2×#55911yescharacteristic
668C4C2×#55911yescharacteristic
669C2—24no—
670C2C2×#259211yescharacteristic
671C2#6059111yescharacteristic
672C2#6059411yescharacteristic
673C2C2×#313711yescharacteristic
674C2#6062011yescharacteristic
675C2#6092311yescharacteristic
676C2#6092511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
3. #29361C2semidirect
4. #29359C2semidirect
5. #29357C2semidirect
6. #29355C2semidirect
7. #29349C2semidirect
8. #29347C2semidirect

Maximal subgroups

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

subgroupindex
#353202
C4×#5442
#293612
#293592
#293572
#293552
#293492
#293472
#236332
#234572
#213122
#213112
#212592
C2×#19812
#210902
#210882
#204312