Finite Group Databasebeta

← #6498074

Subgroups — #6498074

Order 512.

1305 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic
2#35284C211yesmaximal, characteristic
3#33316C211yesmaximal, characteristic, complemented
4#33294C211yesmaximal, characteristic, complemented
5#33291C211yesmaximal, characteristic, complemented
6#33289C211yesmaximal, characteristic, complemented
7#29885C214yesmaximal
8#29188C214yesmaximal, complemented
9#29179C211yesmaximal, characteristic, complemented
10#29178C211yesmaximal, characteristic, complemented
11#29177C211yesmaximal, characteristic, complemented
12#29176C211yesmaximal, characteristic, complemented
13#27888C214yesmaximal
14#27878C211yesmaximal, characteristic
15#27877C211yesmaximal, characteristic
16#27876C211yesmaximal, characteristic
17#27875C211yesmaximal, characteristic
18#22255C214yesmaximal, complemented
19C4×#433C211yesmaximal, characteristic
20C2×#501C2214yes—
21C2×#433C2212yes—
22#2868C2211yescharacteristic
23#2868C2211yescharacteristic
24#2859C2211yescharacteristic
25#2859C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2857C2211yescharacteristic
29#2857C2211yescharacteristic
30#2647C2212yes—
31#2629C2212yes—
32#2629C2212yes—
33#2628C2212yes—
34#2570—21no—
35#2570—21no—
36#2550—21no—
37#2550—21no—
38#2507—21no—
39#2507—21no—
40#2497—21no—
41#2497—21no—
42#2446—21no—
43#2446—21no—
44#2441—21no—
45#2441—21no—
46#2439—21no—
47#2439—21no—
48#2429—21no—
49#2429—21no—
50#2331C2212yes—
51#2330C2212yes—
52#2330C2212yes—
53#2314C2212yes—
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56#2285C2211yescharacteristic
57#2285C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2088C2214yes—
60#2088C2214yes—
61#2087C2214yes—
62#2086C2214yes—
63#2077C2214yes—
64#2077C2214yes—
65#2076C2214yes—
66#2075C2214yes—
67#1992C2214yes—
68#1991C2214yes—
69#1991C2214yes—
70#1990C2214yes—
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1990C2211yescharacteristic
79#1921C2214yes—
80#1921C2214yes—
81#1920C2214yes—
82#1920C2214yes—
83#1919C2214yes—
84#1919C2214yes—
85#1918C2214yes—
86#1918C2214yes—
87#1847C2214yes—
88#1845C2211yescharacteristic
89#1845C2211yescharacteristic
90#1844C2211yescharacteristic
91#1844C2211yescharacteristic
92#1836C2214yes—
93#1834C2214yes—
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1768C2212yes—
99C4×C4⋊C8C2211yescharacteristic
100C4×C4⋊C8C2211yescharacteristic
101#1750C2212yes—
102C4×C22⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104#1746C2212yes—
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2312yes—
114C2×#125—21no—
115C2×#125—21no—
116C2×#125C2312yes—
117C2×#125C2312yes—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×#117—21no—
123C2×#117—21no—
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#501—216no—
128#433—28no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×#97C2312yes—
138C2×#97C2312yes—
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2312yes—
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2312yes—
145#400—22no—
146#400—22no—
147#400—22no—
148#399—22no—
149#398—22no—
150#398—22no—
151#398—22no—
152#398—22no—
153#398—22no—
154#398—22no—
155#396—22no—
156#396—22no—
157#396—22no—
158#396—22no—
159#395—22no—
160#393—22no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—22no—
166#391—22no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#388—22no—
196#388—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#384—21no—
206#384—21no—
207#384—21no—
208#384—21no—
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#384C2312yes—
212#384C2312yes—
213#384C2312yes—
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#382—21no—
219#382—21no—
220#382—21no—
221#382—21no—
222#382C2312yes—
223#382C2312yes—
224#380—21no—
225#380—21no—
226#380—21no—
227#380—21no—
228#380C2312yes—
229#380C2312yes—
230#380C2312yes—
231#380C2312yes—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4D411yescharacteristic
237C4×C4⋊C4D411yescharacteristic
238C4×C4⋊C4D411yescharacteristic
239C4×C4⋊C4D411yescharacteristic
240C4×C4⋊C4C2312yes—
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2312yes—
244C4×C22⋊C4D411yescharacteristic
245C4×C22⋊C4D411yescharacteristic
246C4×C22⋊C4D411yescharacteristic
247C4×C22⋊C4D411yescharacteristic
248#376C2311yescharacteristic
249#376C2311yescharacteristic
250C43C2311yescharacteristic
251C43C2311yescharacteristic
252#340C2314yes—
253#340C2314yes—
254#340C2314yes—
255#340C2314yes—
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#336C2312yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268C22×Q8—21no—
269C22×Q8—21no—
270C22×Q8—22no—
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2411yescharacteristic
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C4⋊Q8—28no—
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—82no—
284C4⋊Q8—82no—
285C4⋊Q8—82no—
286C4⋊Q8—82no—
287#125—28no—
288#125—28no—
289#125—82no—
290#125—82no—
291C22⋊Q8—82no—
292C22⋊Q8—82no—
293C4×Q8—82no—
294C4×Q8—82no—
295#117—24no—
296#117—24no—
297#117—82no—
298#117—82no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4C2×D411yescharacteristic
349C2×C22⋊C4C2×D411yescharacteristic
350C2×C22⋊C4C2×D411yescharacteristic
351C2×C22⋊C4C2×D411yescharacteristic
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374#107—28no—
375#107—28no—
376#106—28no—
377#106—28no—
378C4⋊C8—24no—
379C4⋊C8—24no—
380C22⋊C8—24no—
381C22⋊C8—24no—
382#97—24no—
383#97#4118yes—
384C4×C8—24no—
385C4×C8—22no—
386C4×C8—22no—
387C4×C8—22no—
388C4×C8—22no—
389C4×C8—24no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95C2×D411yescharacteristic
411#95C2×D411yescharacteristic
412#95C2×D411yescharacteristic
413#95C2×D411yescharacteristic
414#95C2×D411yescharacteristic
415#95C2×D411yescharacteristic
416#95C2×D411yescharacteristic
417#95C2×D411yescharacteristic
418C24C22×D411yescharacteristic
419C2×Q8—44no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—82no—
427C2×Q8—82no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2511yescharacteristic
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8C2×#4114yes—
471C2×C8C2×#4114yes—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—21no—
539C42—21no—
540C42—22no—
541C42—82no—
542C42—82no—
543C42—82no—
544C42—82no—
545C42C2×#4114yes—
546C422−1+412yes—
547C422−1+412yes—
548C422−1+414yes—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23C23×D411yescharacteristic
557Q8—84no—
558Q8—84no—
559Q8—164no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4#53212yes—
618C2×C4#53212yes—
619C2×C4#53612yes—
620C2×C4#53612yes—
621C2×C4#53612yes—
622C2×C4#53612yes—
623C2×C4#57612yes—
624C2×C4#57712yes—
625C2×C4#57712yes—
626C2×C4#57812yes—
627C2×C4C22×#4112yes—
628C2×C4C2×2−1+411yescharacteristic
629C2×C4C2×2−1+411yescharacteristic
630C2×C4C2×2−1+412yes—
631C8—24no—
632C8—24no—
633C8—42no—
634C8—42no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22C2×#53211yescharacteristic
643C22C2×#53611yescharacteristic
644C22C2×#53611yescharacteristic
645C22C2×#57611yescharacteristic
646C22C2×#57711yescharacteristic
647C22C2×#57711yescharacteristic
648C22C2×#57811yescharacteristic
649C4—22no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4#324114yes—
669C4#324214yes—
670C4#350912yes—
671C4#350912yes—
672C2—81no—
673C2#3532211yescharacteristic
674C2#6072211yescharacteristic
675C2#6072311yescharacteristic
676C2C2×#324111yescharacteristic
677C2C2×#324211yescharacteristic
678C2#6088011yescharacteristic
679C2#6088111yescharacteristic
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. #33316C2semidirect
4. #33294C2semidirect
5. #33291C2semidirect
6. #33289C2semidirect
8. #29188C2semidirect
9. #29179C2semidirect
10. #29178C2semidirect
11. #29177C2semidirect
12. #29176C2semidirect
18. #22255C2semidirect

Maximal subgroups

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

subgroupindex
#352942
#352842
#333162
#332942
#332912
#332892
#298852
#291882
#291792
#291782
#291772
#291762
#278882
#278782
#278772
#278762
#278752
#222552
C4×#4332