Finite Group Databasebeta

← #6556925

Subgroups — #6556925

Order 512.

1346 conjugacy classes of subgroups.

The subgroup lattice has 782 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39753C211yesmaximal, characteristic
2#39170C211yesmaximal, characteristic
3#30210C212yesmaximal
4#30205C211yesmaximal, characteristic
5#29359C212yesmaximal
6#29349C211yesmaximal, characteristic
7#29153C212yesmaximal
8#29141C211yesmaximal, characteristic
9#27762C212yesmaximal
10#27757C211yesmaximal, characteristic
11#27749C211yesmaximal, characteristic
12#27749C212yesmaximal
13#27741C212yesmaximal
14#27737C211yesmaximal, characteristic
15#27732C212yesmaximal
16#27730C211yesmaximal, characteristic
17#23829C212yesmaximal
18#23820C211yesmaximal, characteristic
19#22547C211yesmaximal, characteristic
20#22542C211yesmaximal, characteristic
21#22524C211yesmaximal, characteristic
22#22521C211yesmaximal, characteristic
23#20571C211yesmaximal, characteristic
24C2×#557C2211yescharacteristic
25C2×#557C2211yescharacteristic
26C2×#446C2211yescharacteristic
27C2×C8×Q8C2211yescharacteristic
28#2868C2211yescharacteristic
29#2862C2211yescharacteristic
30#2791—22no—
31#2719—22no—
32#2719—22no—
33#2713—22no—
34#2660C2212yes—
35#2642C2212yes—
36#2571C2211yescharacteristic
37#2570C2212yes—
38#2568C2211yescharacteristic
39#2565—21no—
40#2565C2212yes—
41#2552C2211yescharacteristic
42#2552C2212yes—
43#2550—21no—
44#2546C2211yescharacteristic
45#2546C2212yes—
46#2506—21no—
47#2453—21no—
48#2453—21no—
49#2451—21no—
50#2446—21no—
51#2446—21no—
52#2355C2211yescharacteristic
53Q8×C4⋊C4C2211yescharacteristic
54#2331C2212yes—
55#2327C2212yes—
56C4×C4⋊Q8C2211yescharacteristic
57C4×#125C2211yescharacteristic
58#2295C2211yescharacteristic
59#2101C2212yes—
60#2101C2212yes—
61#2099C2211yescharacteristic
62#2099C2211yescharacteristic
63#2074C2212yes—
64#2074C2212yes—
65#2074C2211yescharacteristic
66#2074C2211yescharacteristic
67#2074C2212yes—
68#2072C2212yes—
69#2072C2212yes—
70#2072C2212yes—
71#2072C2212yes—
72#2043C2211yescharacteristic
73#2043C2211yescharacteristic
74#2043C2212yes—
75#2043C2212yes—
76#2026C2211yescharacteristic
77#2026C2212yes—
78#2026C2211yescharacteristic
79#2024C2212yes—
80#2024C2212yes—
81#2024C2212yes—
82#2024C2212yes—
83#1992C2212yes—
84#1991C2212yes—
85#1990C2211yescharacteristic
86#1990C2211yescharacteristic
87#1989C2212yes—
88#1989C2212yes—
89#1988C2211yescharacteristic
90#1988C2211yescharacteristic
91#1981C2211yescharacteristic
92#1981C2212yes—
93#1981C2212yes—
94#1981C2211yescharacteristic
95#1960C2212yes—
96#1960C2212yes—
97#1956C2212yes—
98#1956C2211yescharacteristic
99#1956C2211yescharacteristic
100#1956C2212yes—
101#1956C2212yes—
102#1935C2212yes—
103#1935C2212yes—
104#1933C2212yes—
105#1933C2212yes—
106#1933C2212yes—
107#1933C2212yes—
108#1836C2212yes—
109#1835C2212yes—
110#1834C2212yes—
111#1805C2212yes—
112#1805C2212yes—
113#1803C2211yescharacteristic
114#1803C2212yes—
115#1803C2211yescharacteristic
116#1803C2211yescharacteristic
117#1803C2212yes—
118#1803C2211yescharacteristic
119#1771C2212yes—
120#1770C2211yescharacteristic
121C8×C4⋊C4C2211yescharacteristic
122C8×C4⋊C4C2212yes—
123#1768C2212yes—
124C4×C4⋊C8C2211yescharacteristic
125#1763C2212yes—
126#1763C2212yes—
127C4×#103C2211yescharacteristic
128C4×#103C2211yescharacteristic
129C4×#103C2211yescharacteristic
130C4×#103C2211yescharacteristic
131C4×#103C2212yes—
132C4×#103C2212yes—
133#1746C2212yes—
134#1744C2211yescharacteristic
135#557—24no—
136#557—24no—
137#557—24no—
138#557—24no—
139C2×C4⋊Q8—21no—
140C2×C4⋊Q8—21no—
141C2×C4⋊Q8—21no—
142C2×C4⋊Q8C2311yescharacteristic
143C2×C4⋊Q8C2312yes—
144C2×C4⋊Q8C2311yescharacteristic
145C2×C4⋊Q8C2312yes—
146C2×C4⋊Q8C2311yescharacteristic
147C2×C4⋊Q8C2312yes—
148C2×#125—21no—
149C2×#125—22no—
150C2×#125—22no—
151C2×#125—21no—
152C2×#125—22no—
153C2×#125C2312yes—
154C2×#125C2311yescharacteristic
155C2×C4×Q8—21no—
156C2×C4×Q8—21no—
157C2×C4×Q8—21no—
158C2×C4×Q8C2311yescharacteristic
159C2×C4×Q8C2311yescharacteristic
160C2×C4×Q8C2312yes—
161C2×C4×Q8C2312yes—
162C2×C4×Q8C2311yescharacteristic
163#446—24no—
164C8×Q8—24no—
165C2×C4⋊C8C2311yescharacteristic
166C2×C4⋊C8C2311yescharacteristic
167C2×C4⋊C8C2312yes—
168C2×C4⋊C8C2312yes—
169C2×#103C2311yescharacteristic
170C2×#103C2311yescharacteristic
171C2×#103C2311yescharacteristic
172C2×#103C2312yes—
173C2×#103C2312yes—
174C2×#103C2312yes—
175C2×#103C2312yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#103C2311yescharacteristic
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2311yescharacteristic
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2311yescharacteristic
185C2×#103C2311yescharacteristic
186C2×#103C2312yes—
187C2×#103C2312yes—
188C2×#103C2311yescharacteristic
189C2×#97C2312yes—
190C2×#97C2312yes—
191C2×C4×C8C2311yescharacteristic
192C2×C4×C8C2312yes—
193C2×C4×C8C2311yescharacteristic
194C2×C4×C8C2312yes—
195C2×C4×C8C2312yes—
196#400—22no—
197#400—22no—
198#400—22no—
199#400—22no—
200#400—22no—
201#400—22no—
202#400—22no—
203#398—22no—
204#398—22no—
205#398—22no—
206#398—22no—
207#398—22no—
208#398—22no—
209#398—22no—
210#398—22no—
211#398—22no—
212#398—22no—
213#398—22no—
214#395—22no—
215#395—22no—
216#391—21no—
217#391—21no—
218#391—21no—
219#391—21no—
220#391—21no—
221#391—22no—
222#391—22no—
223#391D411yescharacteristic
224#391D411yescharacteristic
225#391C2311yescharacteristic
226#391C2311yescharacteristic
227#391C2312yes—
228#391C2312yes—
229#391C2312yes—
230#391C2312yes—
231#389—21no—
232#389—22no—
233#389—21no—
234#389—21no—
235#389—21no—
236#389—21no—
237#389—22no—
238#389—22no—
239#389—22no—
240#389—21no—
241#389—21no—
242#389D411yescharacteristic
243#389D411yescharacteristic
244#389C2311yescharacteristic
245#389C2311yescharacteristic
246#389C2312yes—
247#389C2312yes—
248#387—22no—
249#387—21no—
250#387—21no—
251#387—22no—
252#387—22no—
253#387—22no—
254#387—22no—
255#387—22no—
256#387—22no—
257#387—22no—
258#387—22no—
259#387—22no—
260#387—22no—
261#387D412yes—
262#387D412yes—
263#384—21no—
264#384—21no—
265#384C2311yescharacteristic
266#384C2312yes—
267#384C2311yescharacteristic
268#384C2312yes—
269#384C2311yescharacteristic
270#384C2312yes—
271#382—21no—
272#382—21no—
273#382—21no—
274#382C2311yescharacteristic
275#382C2312yes—
276#382C2312yes—
277#382C2312yes—
278C4×C4⋊C4—21no—
279C4×C4⋊C4—21no—
280C4×C4⋊C4—21no—
281C4×C4⋊C4—21no—
282C4×C4⋊C4—21no—
283C4×C4⋊C4C2311yescharacteristic
284C4×C4⋊C4C2311yescharacteristic
285C4×C4⋊C4C2312yes—
286C4×C4⋊C4C2312yes—
287C4×C4⋊C4C2312yes—
288C4×C4⋊C4C2311yescharacteristic
289#376C2312yes—
290C43C2311yescharacteristic
291#340C2312yes—
292#340C2311yescharacteristic
293#340C2312yes—
294#340C2312yes—
295#340C2312yes—
296#340C2312yes—
297#340C2312yes—
298#340C2311yescharacteristic
299#340C2312yes—
300#340C2312yes—
301#340C2311yescharacteristic
302#340C2312yes—
303#340C2312yes—
304#340C2311yescharacteristic
305#340C2311yescharacteristic
306#340C2311yescharacteristic
307#340C2311yescharacteristic
308#340C2311yescharacteristic
309#340C2312yes—
310#340C2312yes—
311#336C2311yescharacteristic
312#336C2311yescharacteristic
313#336C2312yes—
314#336C2312yes—
315#336C2311yescharacteristic
316#336C2311yescharacteristic
317#336C2312yes—
318C22×Q8—21no—
319C22×Q8—22no—
320C22×Q8C2×D411yescharacteristic
321C22×Q8C2411yescharacteristic
322C22×Q8C2412yes—
323C22×Q8C2411yescharacteristic
324C22×C8C2411yescharacteristic
325C22×C8C2411yescharacteristic
326C22×C8C2411yescharacteristic
327C22×C8C2412yes—
328C22×C8C2411yescharacteristic
329C22×C8C2412yes—
330C4⋊Q8—22no—
331C4⋊Q8—24no—
332C4⋊Q8—22no—
333C4⋊Q8—22no—
334C4⋊Q8—24no—
335C4⋊Q8—22no—
336C4⋊Q8—22no—
337C4⋊Q8—24no—
338C4⋊Q8—42no—
339C4⋊Q8—42no—
340C4⋊Q8—41no—
341C4⋊Q8—42no—
342C4⋊Q8—44no—
343C4⋊Q8—82no—
344C4⋊Q8—82no—
345#125—24no—
346#125—24no—
347#125—41no—
348#125—41no—
349#125—44no—
350#125—48no—
351#125—48no—
352#125—82no—
353#125—84no—
354C4×Q8—24no—
355C4×Q8—24no—
356C4×Q8—22no—
357C4×Q8—22no—
358C4×Q8—24no—
359C4×Q8—24no—
360C4×Q8—22no—
361C4×Q8—22no—
362C4×Q8—44no—
363C4×Q8—44no—
364C4×Q8—82no—
365C4×Q8#4114yes—
366C4×Q8#4114yes—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4C2×D411yescharacteristic
402C2×C4⋊C4C2×D412yes—
403C2×C4⋊C4C2×D412yes—
404C2×C4⋊C4C2×D411yescharacteristic
405C2×C4⋊C4C2×D411yescharacteristic
406C2×C4⋊C4C2×D412yes—
407C2×C4⋊C4C2411yescharacteristic
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—21no—
417C2×C42—22no—
418C2×C42—22no—
419C2×C42—21no—
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×C42C2411yescharacteristic
430C2×C42C2412yes—
431C2×C42C2412yes—
432C2×C42C2411yescharacteristic
433C4⋊C8—24no—
434C4⋊C8—22no—
435C4⋊C8—22no—
436C4⋊C8—24no—
437#103—22no—
438#103—22no—
439#103—41no—
440#103—42no—
441#103—42no—
442#103—41no—
443#103—41no—
444#103—41no—
445#103—42no—
446#103—42no—
447#103—41no—
448#103—41no—
449#103—42no—
450#103—42no—
451#103—42no—
452#103—42no—
453#103—42no—
454#103—42no—
455#103—42no—
456#103—42no—
457#97—24no—
458#97—42no—
459C4×C8—24no—
460C4×C8—22no—
461C4×C8—22no—
462C4×C8—42no—
463C4×C8—42no—
464#95—21no—
465#95—21no—
466#95—22no—
467#95—21no—
468#95—21no—
469#95—21no—
470#95—21no—
471#95—21no—
472#95—21no—
473#95—21no—
474#95—21no—
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#95—22no—
489#95—22no—
490#95—22no—
491#95—22no—
492#95C2×D412yes—
493#95C2×D412yes—
494#95C2×D412yes—
495#95C2×D412yes—
496#95C2×D412yes—
497#95C2×D412yes—
498#95C2×D412yes—
499C2×Q8—22no—
500C2×Q8—22no—
501C2×Q8—21no—
502C2×Q8—21no—
503C2×Q8—21no—
504C2×Q8—22no—
505C2×Q8—21no—
506C2×Q8—21no—
507C2×Q8—22no—
508C2×Q8—21no—
509C2×Q8—22no—
510C2×Q8—21no—
511C2×Q8—21no—
512C2×Q8—22no—
513C2×Q8—22no—
514C2×Q8—21no—
515C2×Q8—21no—
516C2×Q8—22no—
517C2×Q8—21no—
518C2×Q8—44no—
519C2×Q8—42no—
520C2×Q8—42no—
521C2×Q8—42no—
522C2×Q8—82no—
523C2×Q8—82no—
524C2×Q8#12414yes—
525C2×Q82−1+412yes—
526C22×C4—21no—
527C22×C4—21no—
528C22×C4—21no—
529C22×C4—21no—
530C22×C4—22no—
531C22×C4—21no—
532C22×C4—21no—
533C22×C4—21no—
534C22×C4—22no—
535C22×C4—22no—
536C22×C4—22no—
537C22×C4—21no—
538C22×C4C22×D411yescharacteristic
539C22×C4C22×D411yescharacteristic
540C22×C4C22×D412yes—
541C22×C4C22×D412yes—
542C22×C4C22×D412yes—
543C22×C4C22×D412yes—
544C22×C4C22×D411yescharacteristic
545C22×C4C22×D411yescharacteristic
546C22×C4C22×D411yescharacteristic
547C22×C4C22×D411yescharacteristic
548C22×C4C2511yescharacteristic
549C2×C8—21no—
550C2×C8—21no—
551C2×C8—22no—
552C2×C8—21no—
553C2×C8—21no—
554C2×C8—22no—
555C2×C8—21no—
556C2×C8—21no—
557C2×C8—22no—
558C2×C8—21no—
559C2×C8—22no—
560C2×C8—21no—
561C2×C8—21no—
562C2×C8—21no—
563C2×C8—21no—
564C2×C8—22no—
565C2×C8—21no—
566C2×C8—22no—
567C4⋊C4—24no—
568C4⋊C4—22no—
569C4⋊C4—24no—
570C4⋊C4—22no—
571C4⋊C4—21no—
572C4⋊C4—21no—
573C4⋊C4—21no—
574C4⋊C4—21no—
575C4⋊C4—22no—
576C4⋊C4—22no—
577C4⋊C4—21no—
578C4⋊C4—21no—
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—22no—
588C4⋊C4—22no—
589C4⋊C4—42no—
590C4⋊C4—42no—
591C4⋊C4—44no—
592C4⋊C4—44no—
593C4⋊C4—42no—
594C4⋊C4—42no—
595C4⋊C4—44no—
596C4⋊C4—42no—
597C4⋊C4—44no—
598C4⋊C4—42no—
599C4⋊C4—44no—
600C4⋊C4—44no—
601C4⋊C4—44no—
602C4⋊C4—42no—
603C4⋊C4—44no—
604C4⋊C4—42no—
605C4⋊C4—44no—
606C4⋊C4—42no—
607C4⋊C4—42no—
608C4⋊C4—42no—
609C4⋊C4—42no—
610C4⋊C4—44no—
611C4⋊C4—42no—
612C4⋊C4—42no—
613C4⋊C4—42no—
614C4⋊C4—82no—
615C4⋊C4—82no—
616C4⋊C4—82no—
617C4⋊C4—82no—
618C4⋊C4—82no—
619C4⋊C4—82no—
620C4⋊C4—81no—
621C4⋊C4—82no—
622C4⋊C4—81no—
623C4⋊C4#12414yes—
624C4⋊C4#12418yes—
625C42—22no—
626C42—22no—
627C42—22no—
628C42—22no—
629C42—21no—
630C42—22no—
631C42—21no—
632C42—22no—
633C42—21no—
634C42—21no—
635C42—44no—
636C42—44no—
637C42—42no—
638C42—42no—
639C42—42no—
640C42—42no—
641C42—82no—
642C42—82no—
643C42C2×#4114yes—
644C42C2×#4112yes—
645C42C2×#4112yes—
646C42C2×#4114yes—
647C23C23×D411yescharacteristic
648Q8—28no—
649Q8—22no—
650Q8—22no—
651Q8—41no—
652Q8—41no—
653Q8—42no—
654Q8—41no—
655Q8—41no—
656Q8—42no—
657Q8—42no—
658Q8—41no—
659Q8—41no—
660Q8—42no—
661Q8—84no—
662Q8—164no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—21no—
670C2×C4—21no—
671C2×C4—22no—
672C2×C4—22no—
673C2×C4—21no—
674C2×C4—21no—
675C2×C4—22no—
676C2×C4—22no—
677C2×C4—22no—
678C2×C4—21no—
679C2×C4—21no—
680C2×C4—22no—
681C2×C4—22no—
682C2×C4—21no—
683C2×C4—21no—
684C2×C4—22no—
685C2×C4—22no—
686C2×C4—41no—
687C2×C4—41no—
688C2×C4—41no—
689C2×C4—41no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—41no—
693C2×C4—41no—
694C2×C4—41no—
695C2×C4—42no—
696C2×C4—42no—
697C2×C4—41no—
698C2×C4—42no—
699C2×C4—41no—
700C2×C4—41no—
701C2×C4—41no—
702C2×C4—42no—
703C2×C4—41no—
704C2×C4—41no—
705C2×C4—41no—
706C2×C4—42no—
707C2×C4—42no—
708C2×C4—41no—
709C2×C4—42no—
710C2×C4—42no—
711C2×C4—41no—
712C2×C4—41no—
713C2×C4—42no—
714C2×C4—41no—
715C2×C4—41no—
716C2×C4—41no—
717C2×C4—41no—
718C2×C4—41no—
719C2×C4—41no—
720C2×C4—42no—
721C2×C4C2×#12414yes—
722C2×C4C2×#12414yes—
723C2×C4C2×#12412yes—
724C2×C4C2×#12412yes—
725C2×C4#57712yes—
726C2×C4#57712yes—
727C2×C4#57812yes—
728C2×C4#57812yes—
729C2×C4C22×#4111yescharacteristic
730C2×C4C22×#4111yescharacteristic
731C2×C4C2×2−1+411yescharacteristic
732C2×C4#58511yescharacteristic
733C2×C4#58511yescharacteristic
734C2×C4#58511yescharacteristic
735C8—41no—
736C8—41no—
737C8—42no—
738C8—41no—
739C8—41no—
740C8—42no—
741C22C22×#12411yescharacteristic
742C22#348711yescharacteristic
743C22#348911yescharacteristic
744C22C2×#57711yescharacteristic
745C22C2×#57711yescharacteristic
746C22C2×#57811yescharacteristic
747C22C2×#57811yescharacteristic
748C4—24no—
749C4—22no—
750C4—24no—
751C4—22no—
752C4—22no—
753C4—42no—
754C4—41no—
755C4—41no—
756C4—42no—
757C4—42no—
758C4—82no—
759C4—81no—
760C4—82no—
761C4—81no—
762C4—81no—
763C4—81no—
764C4—82no—
765C4—82no—
766C4—81no—
767C4—81no—
768C4—81no—
769C4#313714yes—
770C4#351611yescharacteristic
771C4#351611yescharacteristic
772C4#352411yescharacteristic
773C4#352411yescharacteristic
774C2#3585411yescharacteristic
775C2C2×#313711yescharacteristic
776C2#6062311yescharacteristic
777C2#6193311yescharacteristic
778C2#6193611yescharacteristic
779C2#6220311yescharacteristic
780C2#6220411yescharacteristic
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
#391702
#302102
#302052
#293592
#293492
#291532
#291412
#277622
#277572
#277492
#277412
#277372
#277322
#277302
#238292
#238202
#225472
#225422
#225242
#225212
#205712