Finite Group Databasebeta

← #6621207

Subgroups — #6621207

Order 512.

1372 conjugacy classes of subgroups.

The subgroup lattice has 776 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39753C211yesmaximal, characteristic
2#39619C211yesmaximal, characteristic
3#33293C211yesmaximal, characteristic
4#33292C212yesmaximal
5#29229C212yesmaximal
6#29222C211yesmaximal, characteristic
7#28729C212yesmaximal
8#28717C211yesmaximal, characteristic
9#27742C212yesmaximal
10#27741C212yesmaximal
11#27738C211yesmaximal, characteristic
12#27737C211yesmaximal, characteristic
13#27737C212yesmaximal
14#27734C212yesmaximal
15#27730C211yesmaximal, characteristic
16#27729C211yesmaximal, characteristic
17#22525C211yesmaximal, characteristic
18#22524C211yesmaximal, characteristic
19#22521C211yesmaximal, characteristic
20Q8×#103C211yesmaximal, characteristic
21C4×#491C211yesmaximal, characteristic
22#21645C212yesmaximal
23#20521C211yesmaximal, characteristic
24C2×Q82C2211yescharacteristic
25C2×#557C2211yescharacteristic
26C2×#491C2211yescharacteristic
27C2×#487C2212yes—
28C2×#446C2211yescharacteristic
29C2×C8×Q8C2211yescharacteristic
30#2868C2211yescharacteristic
31#2862C2211yescharacteristic
32#2791—22no—
33#2713—22no—
34#2713—22no—
35#2713—22no—
36#2660C2212yes—
37#2660C2212yes—
38#2571C2211yescharacteristic
39#2570C2212yes—
40#2568C2211yescharacteristic
41#2568C2212yes—
42#2567C2211yescharacteristic
43#2565—21no—
44#2565C2212yes—
45#2552C2211yescharacteristic
46#2552C2212yes—
47#2550—21no—
48#2506—21no—
49#2451—21no—
50#2451—21no—
51#2451—21no—
52#2446—21no—
53#2446—21no—
54#2355C2211yescharacteristic
55Q8×C4⋊C4C2211yescharacteristic
56#2331C2212yes—
57#2327C2212yes—
58C4×C4⋊Q8C2211yescharacteristic
59C4×#125C2211yescharacteristic
60#2295C2211yescharacteristic
61#2101C2212yes—
62#2101C2212yes—
63#2101C2212yes—
64#2101C2212yes—
65#2100C2212yes—
66#2099C2212yes—
67#2074C2212yes—
68#2074C2212yes—
69#2072C2212yes—
70#2072C2212yes—
71#2072C2212yes—
72#2072C2212yes—
73#1993C2212yes—
74#1993C2212yes—
75#1993C2212yes—
76#1992C2211yescharacteristic
77#1992C2212yes—
78#1991C2212yes—
79#1991C2212yes—
80#1990C2212yes—
81#1990C2212yes—
82#1990C2211yescharacteristic
83#1990C2211yescharacteristic
84#1990C2211yescharacteristic
85#1990C2211yescharacteristic
86#1989C2212yes—
87#1989C2212yes—
88#1988C2211yescharacteristic
89#1988C2211yescharacteristic
90#1988C2212yes—
91#1987C2211yescharacteristic
92#1987C2211yescharacteristic
93#1987C2212yes—
94#1960C2211yescharacteristic
95#1935C2211yescharacteristic
96#1935C2212yes—
97#1935C2211yescharacteristic
98#1935C2212yes—
99#1935C2211yescharacteristic
100#1935C2211yescharacteristic
101#1933C2212yes—
102#1933C2212yes—
103#1836C2212yes—
104#1836C2212yes—
105#1836C2212yes—
106#1836C2212yes—
107#1836C2211yescharacteristic
108#1835C2212yes—
109#1835C2212yes—
110#1834C2212yes—
111#1805C2212yes—
112#1805C2212yes—
113#1803C2211yescharacteristic
114#1803C2211yescharacteristic
115#1803C2211yescharacteristic
116#1803C2211yescharacteristic
117#1803C2212yes—
118#1803C2212yes—
119#1771C2212yes—
120#1770C2212yes—
121#1770C2211yescharacteristic
122C8×C4⋊C4C2211yescharacteristic
123#1768C2212yes—
124C4×C4⋊C8C2211yescharacteristic
125#1763C2212yes—
126#1763C2212yes—
127C4×#103C2211yescharacteristic
128C4×#103C2211yescharacteristic
129C4×#103C2212yes—
130C4×#103C2212yes—
131C4×#103C2211yescharacteristic
132C4×#103C2211yescharacteristic
133#1726C2212yes—
134C4×#97C2211yescharacteristic
135Q82—24no—
136Q82—24no—
137#557—24no—
138#557—24no—
139C2×C4⋊Q8—21no—
140C2×C4⋊Q8—21no—
141C2×C4⋊Q8—22no—
142C2×C4⋊Q8—21no—
143C2×C4⋊Q8—21no—
144C2×C4⋊Q8C2311yescharacteristic
145C2×C4⋊Q8C2311yescharacteristic
146C2×C4⋊Q8C2311yescharacteristic
147C2×C4⋊Q8C2312yes—
148C2×C4⋊Q8C2312yes—
149C2×#125—22no—
150C2×#125—21no—
151C2×#125—22no—
152C2×#125C2311yescharacteristic
153C2×#125C2312yes—
154C2×C4×Q8—21no—
155C2×C4×Q8—21no—
156C2×C4×Q8—21no—
157C2×C4×Q8C2311yescharacteristic
158C2×C4×Q8C2311yescharacteristic
159C2×C4×Q8C2312yes—
160C2×C4×Q8C2312yes—
161C2×C4×Q8C2311yescharacteristic
162#491—24no—
163#487—28no—
164#446—24no—
165C8×Q8—24no—
166C2×C4⋊C8C2311yescharacteristic
167C2×C4⋊C8C2311yescharacteristic
168C2×C4⋊C8C2312yes—
169C2×C4⋊C8C2312yes—
170C2×#103C2311yescharacteristic
171C2×#103C2311yescharacteristic
172C2×#103C2312yes—
173C2×#103C2312yes—
174C2×#103C2312yes—
175C2×#103C2312yes—
176C2×#103C2311yescharacteristic
177C2×#103C2311yescharacteristic
178C2×#103C2311yescharacteristic
179C2×#103C2311yescharacteristic
180C2×#103C2311yescharacteristic
181C2×#103C2311yescharacteristic
182C2×#97C2311yescharacteristic
183C2×#97C2311yescharacteristic
184C2×#97C2312yes—
185C2×#97C2311yescharacteristic
186C2×#97C2312yes—
187C2×#97C2311yescharacteristic
188C2×C4×C8C2312yes—
189C2×C4×C8C2312yes—
190C2×C4×C8C2311yescharacteristic
191C2×C4×C8C2312yes—
192C2×C4×C8C2311yescharacteristic
193#400—22no—
194#400—22no—
195#400—22no—
196#400—22no—
197#400—22no—
198#398—22no—
199#398—22no—
200#398—22no—
201#398—22no—
202#398—22no—
203#398—22no—
204#398—22no—
205#398—22no—
206#398—22no—
207#398—22no—
208#395—22no—
209#395—22no—
210#395—22no—
211#395—22no—
212#395—22no—
213#391—21no—
214#391—21no—
215#391—21no—
216#391—22no—
217#391—22no—
218#391—21no—
219#391—21no—
220#391—21no—
221#391—21no—
222#391—22no—
223#391—22no—
224#391D411yescharacteristic
225#391D411yescharacteristic
226#391C2311yescharacteristic
227#391C2311yescharacteristic
228#391C2312yes—
229#391C2312yes—
230#389—21no—
231#389—21no—
232#389—21no—
233#389—22no—
234#389—21no—
235#389—21no—
236#389—22no—
237#389—21no—
238#389—21no—
239#389D411yescharacteristic
240#389D411yescharacteristic
241#389C2311yescharacteristic
242#389C2311yescharacteristic
243#389C2312yes—
244#389C2312yes—
245#387—22no—
246#387—21no—
247#387—21no—
248#387—22no—
249#387—22no—
250#387—22no—
251#387—22no—
252#387—22no—
253#387—22no—
254#387—22no—
255#387—22no—
256#387—22no—
257#387—22no—
258#387D412yes—
259#387D412yes—
260#384—21no—
261#384—21no—
262#384C2311yescharacteristic
263#384C2312yes—
264#384C2312yes—
265#384C2311yescharacteristic
266#384C2312yes—
267#384C2311yescharacteristic
268#384C2312yes—
269#382—21no—
270#382C2311yescharacteristic
271#382C2312yes—
272#382C2312yes—
273#382C2312yes—
274#382C2312yes—
275#382C2312yes—
276C4×C4⋊C4—21no—
277C4×C4⋊C4—21no—
278C4×C4⋊C4—21no—
279C4×C4⋊C4—21no—
280C4×C4⋊C4—21no—
281C4×C4⋊C4C2311yescharacteristic
282C4×C4⋊C4C2311yescharacteristic
283C4×C4⋊C4C2312yes—
284C4×C4⋊C4C2312yes—
285C4×C4⋊C4C2312yes—
286C4×C4⋊C4C2311yescharacteristic
287#376C2312yes—
288C43C2311yescharacteristic
289#340C2312yes—
290#340C2312yes—
291#340C2312yes—
292#340C2312yes—
293#340C2312yes—
294#340C2312yes—
295#340C2312yes—
296#340C2312yes—
297#340C2312yes—
298#340C2312yes—
299#340C2312yes—
300#340C2312yes—
301#340C2311yescharacteristic
302#340C2312yes—
303#340C2311yescharacteristic
304#340C2312yes—
305#340C2312yes—
306#340C2312yes—
307#340C2312yes—
308#340C2312yes—
309#340C2311yescharacteristic
310#340C2312yes—
311#340C2312yes—
312#340C2311yescharacteristic
313#340C2311yescharacteristic
314#340C2311yescharacteristic
315#340C2311yescharacteristic
316#340C2311yescharacteristic
317#336C2312yes—
318#336C2312yes—
319#336C2312yes—
320C22×Q8—21no—
321C22×Q8—22no—
322C22×Q8—21no—
323C22×Q8—22no—
324C22×Q8C2×D411yescharacteristic
325C22×Q8C2411yescharacteristic
326C22×Q8C2411yescharacteristic
327C22×C8C2412yes—
328C22×C8C2412yes—
329C22×C8C2411yescharacteristic
330C22×C8C2411yescharacteristic
331C22×C8C2411yescharacteristic
332C22×C8C2411yescharacteristic
333C4⋊Q8—22no—
334C4⋊Q8—22no—
335C4⋊Q8—24no—
336C4⋊Q8—22no—
337C4⋊Q8—22no—
338C4⋊Q8—24no—
339C4⋊Q8—24no—
340C4⋊Q8—22no—
341C4⋊Q8—24no—
342C4⋊Q8—44no—
343C4⋊Q8—44no—
344C4⋊Q8—48no—
345C4⋊Q8—82no—
346C4⋊Q8—82no—
347C4⋊Q8#4114yes—
348#125—24no—
349#125—24no—
350#125—22no—
351#125—22no—
352#125—48no—
353#125—82no—
354#125—84no—
355C4×Q8—24no—
356C4×Q8—24no—
357C4×Q8—22no—
358C4×Q8—22no—
359C4×Q8—24no—
360C4×Q8—22no—
361C4×Q8—22no—
362C4×Q8—24no—
363C4×Q8—44no—
364C4×Q8—44no—
365C4×Q8—82no—
366C4×Q8#4114yes—
367C4×Q8#4114yes—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4C2×D411yescharacteristic
401C2×C4⋊C4C2×D411yescharacteristic
402C2×C4⋊C4C2×D412yes—
403C2×C4⋊C4C2×D412yes—
404C2×C4⋊C4C2×D411yescharacteristic
405C2×C4⋊C4C2×D412yes—
406C2×C4⋊C4C2411yescharacteristic
407C2×C4⋊C4C2412yes—
408C2×C4⋊C4C2411yescharacteristic
409C2×C4⋊C4C2412yes—
410C2×C4⋊C4C2411yescharacteristic
411C2×C4⋊C4C2411yescharacteristic
412C2×C4⋊C4C2412yes—
413C2×C4⋊C4C2412yes—
414C2×C4⋊C4C2411yescharacteristic
415C2×C4⋊C4C2411yescharacteristic
416C2×C42—22no—
417C2×C42—21no—
418C2×C42—21no—
419C2×C42—22no—
420C2×C42—21no—
421C2×C42—21no—
422C2×C42—22no—
423C2×C42—22no—
424C2×C42C2×D411yescharacteristic
425C2×C42C2×D411yescharacteristic
426C2×C42C2×D411yescharacteristic
427C2×C42C2×D411yescharacteristic
428C2×C42C2411yescharacteristic
429C2×C42C2412yes—
430C2×C42C2412yes—
431C2×C42C2411yescharacteristic
432C2×C42C2411yescharacteristic
433C4⋊C8—24no—
434C4⋊C8—22no—
435C4⋊C8—22no—
436C4⋊C8—24no—
437#103—24no—
438#103—24no—
439#103—24no—
440#103—24no—
441#103—22no—
442#103—22no—
443#103—22no—
444#103—22no—
445#103—22no—
446#103—22no—
447#103—22no—
448#103—22no—
449#97—22no—
450#97—22no—
451#97—24no—
452#97—24no—
453#97—22no—
454#97—22no—
455C4×C8—24no—
456C4×C8—24no—
457C4×C8—22no—
458C4×C8—22no—
459C4×C8—24no—
460#95—21no—
461#95—21no—
462#95—22no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95—21no—
470#95—21no—
471#95—22no—
472#95—22no—
473#95—22no—
474#95—22no—
475#95—22no—
476#95—22no—
477#95—22no—
478#95—22no—
479#95—22no—
480#95—22no—
481#95—22no—
482#95—22no—
483#95—22no—
484#95—22no—
485#95—22no—
486#95—22no—
487#95—22no—
488#95C2×D412yes—
489#95C2×D412yes—
490#95C2×D412yes—
491#95C2×D412yes—
492#95C2×D412yes—
493#95C2×D412yes—
494#95C2×D412yes—
495C2×Q8—22no—
496C2×Q8—22no—
497C2×Q8—21no—
498C2×Q8—21no—
499C2×Q8—21no—
500C2×Q8—21no—
501C2×Q8—21no—
502C2×Q8—21no—
503C2×Q8—21no—
504C2×Q8—21no—
505C2×Q8—21no—
506C2×Q8—21no—
507C2×Q8—42no—
508C2×Q8—44no—
509C2×Q8—42no—
510C2×Q8—44no—
511C2×Q8—42no—
512C2×Q8—42no—
513C2×Q8—82no—
514C2×Q8—82no—
515C2×Q8—81no—
516C2×Q8—82no—
517C2×Q8—81no—
518C2×Q8—82no—
519C2×Q8#12414yes—
520C2×Q8C2×#4112yes—
521C2×Q8C2×#4112yes—
522C22×C4—21no—
523C22×C4—21no—
524C22×C4—21no—
525C22×C4—21no—
526C22×C4—21no—
527C22×C4—22no—
528C22×C4—21no—
529C22×C4—22no—
530C22×C4—22no—
531C22×C4—22no—
532C22×C4—21no—
533C22×C4—21no—
534C22×C4C22×D412yes—
535C22×C4C22×D412yes—
536C22×C4C22×D411yescharacteristic
537C22×C4C22×D411yescharacteristic
538C22×C4C22×D412yes—
539C22×C4C22×D412yes—
540C22×C4C22×D411yescharacteristic
541C22×C4C22×D411yescharacteristic
542C22×C4C22×D411yescharacteristic
543C22×C4C22×D411yescharacteristic
544C22×C4C2511yescharacteristic
545C2×C8—21no—
546C2×C8—21no—
547C2×C8—21no—
548C2×C8—21no—
549C2×C8—22no—
550C2×C8—22no—
551C2×C8—22no—
552C2×C8—21no—
553C2×C8—21no—
554C2×C8—21no—
555C2×C8—21no—
556C2×C8—22no—
557C2×C8—22no—
558C2×C8—22no—
559C2×C8—21no—
560C2×C8—21no—
561C2×C82−1+412yes—
562C2×C82−1+412yes—
563C4⋊C4—24no—
564C4⋊C4—24no—
565C4⋊C4—22no—
566C4⋊C4—22no—
567C4⋊C4—21no—
568C4⋊C4—21no—
569C4⋊C4—22no—
570C4⋊C4—22no—
571C4⋊C4—21no—
572C4⋊C4—21no—
573C4⋊C4—22no—
574C4⋊C4—22no—
575C4⋊C4—21no—
576C4⋊C4—22no—
577C4⋊C4—21no—
578C4⋊C4—22no—
579C4⋊C4—21no—
580C4⋊C4—21no—
581C4⋊C4—21no—
582C4⋊C4—21no—
583C4⋊C4—21no—
584C4⋊C4—21no—
585C4⋊C4—22no—
586C4⋊C4—22no—
587C4⋊C4—42no—
588C4⋊C4—42no—
589C4⋊C4—42no—
590C4⋊C4—42no—
591C4⋊C4—44no—
592C4⋊C4—42no—
593C4⋊C4—44no—
594C4⋊C4—44no—
595C4⋊C4—42no—
596C4⋊C4—44no—
597C4⋊C4—44no—
598C4⋊C4—42no—
599C4⋊C4—44no—
600C4⋊C4—42no—
601C4⋊C4—42no—
602C4⋊C4—42no—
603C4⋊C4—44no—
604C4⋊C4—42no—
605C4⋊C4—42no—
606C4⋊C4—81no—
607C4⋊C4—81no—
608C4⋊C4—82no—
609C4⋊C4—82no—
610C4⋊C4—82no—
611C4⋊C4—82no—
612C4⋊C4—82no—
613C4⋊C4—82no—
614C4⋊C4—82no—
615C4⋊C4—82no—
616C4⋊C4—81no—
617C4⋊C4—81no—
618C4⋊C4—82no—
619C4⋊C4#12414yes—
620C4⋊C4#12418yes—
621C42—22no—
622C42—22no—
623C42—22no—
624C42—21no—
625C42—21no—
626C42—22no—
627C42—41no—
628C42—41no—
629C42—44no—
630C42—42no—
631C42—44no—
632C42—42no—
633C42—42no—
634C42—42no—
635C42—82no—
636C42—82no—
637C42C2×#4114yes—
638C42C2×#4112yes—
639C42C2×#4112yes—
640C42C2×#4114yes—
641C42C2×#4112yes—
642C422−1+412yes—
643C23C23×D411yescharacteristic
644Q8—28no—
645Q8—22no—
646Q8—22no—
647Q8—22no—
648Q8—22no—
649Q8—41no—
650Q8—41no—
651Q8—41no—
652Q8—41no—
653Q8—84no—
654Q8—164no—
655Q8—162no—
656Q8—164no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—22no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—22no—
663C2×C4—21no—
664C2×C4—22no—
665C2×C4—21no—
666C2×C4—22no—
667C2×C4—21no—
668C2×C4—22no—
669C2×C4—21no—
670C2×C4—22no—
671C2×C4—22no—
672C2×C4—22no—
673C2×C4—22no—
674C2×C4—21no—
675C2×C4—22no—
676C2×C4—21no—
677C2×C4—21no—
678C2×C4—22no—
679C2×C4—21no—
680C2×C4—22no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—41no—
685C2×C4—41no—
686C2×C4—42no—
687C2×C4—41no—
688C2×C4—42no—
689C2×C4—41no—
690C2×C4—41no—
691C2×C4—42no—
692C2×C4—41no—
693C2×C4—41no—
694C2×C4—41no—
695C2×C4—41no—
696C2×C4—42no—
697C2×C4—41no—
698C2×C4—42no—
699C2×C4—42no—
700C2×C4—41no—
701C2×C4—41no—
702C2×C4—41no—
703C2×C4—42no—
704C2×C4—42no—
705C2×C4—41no—
706C2×C4—41no—
707C2×C4—41no—
708C2×C4—41no—
709C2×C4—42no—
710C2×C4—42no—
711C2×C4—42no—
712C2×C4—41no—
713C2×C4—41no—
714C2×C4—42no—
715C2×C4C2×#12414yes—
716C2×C4C2×#12414yes—
717C2×C4C2×#12412yes—
718C2×C4C2×#12412yes—
719C2×C4#54812yes—
720C2×C4#54812yes—
721C2×C4#57712yes—
722C2×C4#57812yes—
723C2×C4C22×#4111yescharacteristic
724C2×C4C22×#4111yescharacteristic
725C2×C4C22×#4111yescharacteristic
726C2×C4C2×2−1+411yescharacteristic
727C2×C4C2×2−1+411yescharacteristic
728C2×C4C2×2−1+411yescharacteristic
729C8—22no—
730C8—22no—
731C8—42no—
732C8—42no—
733C8—41no—
734C8—41no—
735C22C22×#12411yescharacteristic
736C22C2×#54811yescharacteristic
737C22#349911yescharacteristic
738C22C2×#57711yescharacteristic
739C22C2×#57811yescharacteristic
740C22#358511yescharacteristic
741C22#358611yescharacteristic
742C4—22no—
743C4—22no—
744C4—24no—
745C4—22no—
746C4—24no—
747C4—41no—
748C4—42no—
749C4—41no—
750C4—42no—
751C4—42no—
752C4—42no—
753C4—81no—
754C4—81no—
755C4—81no—
756C4—82no—
757C4—82no—
758C4—81no—
759C4—82no—
760C4—81no—
761C4—81no—
762C4—82no—
763C4#313714yes—
764C4#348211yescharacteristic
765C4#348211yescharacteristic
766C4#352311yescharacteristic
767C4#352311yescharacteristic
768C2#3644611yescharacteristic
769C2C2×#313711yescharacteristic
770C2#6062511yescharacteristic
771C2#6134111yescharacteristic
772C2#6134211yescharacteristic
773C2#6251911yescharacteristic
774C2#6252211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2295 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#397532
#396192
#332932
#332922
#292292
#292222
#287292
#287172
#277422
#277412
#277382
#277372
#277342
#277302
#277292
#225252
#225242
#225212
Q8×#1032
C4×#4912
#216452
#205212