Finite Group Databasebeta

← #6557792

Subgroups — #6557792

Order 512.

1314 conjugacy classes of subgroups.

The subgroup lattice has 766 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39619C211yesmaximal, characteristic
2#39182C211yesmaximal, characteristic
3#30232C212yesmaximal
4#30229C211yesmaximal, characteristic
5#29229C212yesmaximal
6#29222C211yesmaximal, characteristic
7#29135C212yesmaximal
8#29123C211yesmaximal, characteristic
9#27828C212yesmaximal
10#27823C211yesmaximal, characteristic
11#27818C212yesmaximal
12#27815C211yesmaximal, characteristic
13#27745C212yesmaximal
14#27738C211yesmaximal, characteristic
15#27730C211yesmaximal, characteristic
16#27730C212yesmaximal
17#23829C212yesmaximal
18#23820C211yesmaximal, characteristic
19#22525C211yesmaximal, characteristic
20#22524C211yesmaximal, characteristic
21#22521C211yesmaximal, characteristic
22Q8×#103C211yesmaximal, characteristic
23#20571C211yesmaximal, characteristic
24C2×Q82C2211yescharacteristic
25C2×#557C2211yescharacteristic
26C2×#446C2211yescharacteristic
27C2×C8×Q8C2211yescharacteristic
28#2862C2211yescharacteristic
29#2859C2211yescharacteristic
30#2805—22no—
31#2755—22no—
32#2713—22no—
33#2713—22no—
34#2660C2212yes—
35#2647C2212yes—
36#2571C2211yescharacteristic
37#2571C2212yes—
38#2568—21no—
39#2568C2211yescharacteristic
40#2568C2212yes—
41#2567C2211yescharacteristic
42#2565C2212yes—
43#2551C2211yescharacteristic
44#2550—21no—
45#2550C2212yes—
46#2508—21no—
47#2453—21no—
48#2451—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#2101C2212yes—
62#2101C2212yes—
63#2101C2212yes—
64#2101C2212yes—
65#2101C2212yes—
66#2101C2212yes—
67#2100C2212yes—
68#2100C2212yes—
69#2099C2211yescharacteristic
70#2099C2211yescharacteristic
71#2099C2212yes—
72#2074C2211yescharacteristic
73#2074C2211yescharacteristic
74#2074C2212yes—
75#2074C2212yes—
76#2043C2211yescharacteristic
77#2043C2211yescharacteristic
78#2043C2212yes—
79#2026C2211yescharacteristic
80#2026C2211yescharacteristic
81#2024C2212yes—
82#2024C2212yes—
83#1993C2212yes—
84#1993C2212yes—
85#1992C2212yes—
86#1991C2212yes—
87#1991C2212yes—
88#1990C2211yescharacteristic
89#1990C2211yescharacteristic
90#1990C2211yescharacteristic
91#1990C2211yescharacteristic
92#1990C2212yes—
93#1990C2212yes—
94#1989C2212yes—
95#1989C2212yes—
96#1988C2211yescharacteristic
97#1988C2211yescharacteristic
98#1988C2212yes—
99#1987C2211yescharacteristic
100#1987C2211yescharacteristic
101#1987C2212yes—
102#1935C2212yes—
103#1935C2212yes—
104#1933C2212yes—
105#1933C2212yes—
106#1933C2212yes—
107#1933C2212yes—
108#1836C2212yes—
109#1835C2212yes—
110#1835C2212yes—
111#1805C2212yes—
112#1805C2212yes—
113#1803C2211yescharacteristic
114#1803C2212yes—
115#1803C2211yescharacteristic
116#1803C2212yes—
117#1803C2211yescharacteristic
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
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⋊Q8C2311yescharacteristic
144C2×C4⋊Q8C2311yescharacteristic
145C2×C4⋊Q8C2311yescharacteristic
146C2×C4⋊Q8C2312yes—
147C2×C4⋊Q8C2312yes—
148C2×#125—22no—
149C2×#125—21no—
150C2×#125C2312yes—
151C2×#125C2311yescharacteristic
152C2×C4×Q8—21no—
153C2×C4×Q8—21no—
154C2×C4×Q8—21no—
155C2×C4×Q8C2311yescharacteristic
156C2×C4×Q8C2311yescharacteristic
157C2×C4×Q8C2312yes—
158C2×C4×Q8C2312yes—
159C2×C4×Q8C2311yescharacteristic
160#446—24no—
161C8×Q8—24no—
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2311yescharacteristic
164C2×C4⋊C8C2312yes—
165C2×C4⋊C8C2312yes—
166C2×#103C2312yes—
167C2×#103C2311yescharacteristic
168C2×#103C2312yes—
169C2×#103C2311yescharacteristic
170C2×#103C2311yescharacteristic
171C2×#103C2311yescharacteristic
172C2×#103C2311yescharacteristic
173C2×#103C2311yescharacteristic
174C2×#103C2312yes—
175C2×#103C2312yes—
176C2×#103C2311yescharacteristic
177C2×#103C2311yescharacteristic
178C2×#97C2312yes—
179C2×#97C2312yes—
180C2×C4×C8C2311yescharacteristic
181C2×C4×C8C2312yes—
182C2×C4×C8C2311yescharacteristic
183C2×C4×C8C2312yes—
184C2×C4×C8C2312yes—
185#400—22no—
186#400—22no—
187#400—22no—
188#400—22no—
189#400—22no—
190#400—22no—
191#400—22no—
192#400—22no—
193#398—22no—
194#398—22no—
195#398—22no—
196#398—22no—
197#398—22no—
198#398—22no—
199#398—22no—
200#395—22no—
201#395—22no—
202#395—22no—
203#395—22no—
204#395—22no—
205#391—21no—
206#391—21no—
207#391—21no—
208#391—22no—
209#391—22no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391—21no—
214#391—22no—
215#391—22no—
216#391—22no—
217#391D411yescharacteristic
218#391D411yescharacteristic
219#391C2311yescharacteristic
220#391C2311yescharacteristic
221#391C2312yes—
222#391C2312yes—
223#389—21no—
224#389—21no—
225#389—21no—
226#389—22no—
227#389—22no—
228#389—21no—
229#389—21no—
230#389—21no—
231#389—21no—
232#389—21no—
233#389—21no—
234#389—22no—
235#389D411yescharacteristic
236#389D411yescharacteristic
237#389C2311yescharacteristic
238#389C2311yescharacteristic
239#389C2312yes—
240#389C2312yes—
241#389C2312yes—
242#389C2312yes—
243#387—22no—
244#387—21no—
245#387—21no—
246#387—22no—
247#387—22no—
248#387—22no—
249#387—22no—
250#387—22no—
251#387—22no—
252#387—22no—
253#387—22no—
254#387D412yes—
255#387D412yes—
256#384C2311yescharacteristic
257#384C2312yes—
258#384C2311yescharacteristic
259#384C2312yes—
260#384C2311yescharacteristic
261#382—21no—
262#382—22no—
263#382—21no—
264#382C2311yescharacteristic
265#382C2312yes—
266#382C2312yes—
267#382C2312yes—
268#382C2312yes—
269#382C2312yes—
270C4×C4⋊C4—21no—
271C4×C4⋊C4—21no—
272C4×C4⋊C4—21no—
273C4×C4⋊C4—21no—
274C4×C4⋊C4—21no—
275C4×C4⋊C4C2311yescharacteristic
276C4×C4⋊C4C2311yescharacteristic
277C4×C4⋊C4C2312yes—
278C4×C4⋊C4C2312yes—
279C4×C4⋊C4C2312yes—
280C4×C4⋊C4C2311yescharacteristic
281#376C2312yes—
282C43C2311yescharacteristic
283#340C2312yes—
284#340C2312yes—
285#340C2312yes—
286#340C2312yes—
287#340C2312yes—
288#340C2312yes—
289#340C2312yes—
290#340C2312yes—
291#340C2312yes—
292#340C2312yes—
293#340C2312yes—
294#340C2312yes—
295#340C2311yescharacteristic
296#340C2312yes—
297#340C2312yes—
298#340C2311yescharacteristic
299#340C2311yescharacteristic
300#340C2311yescharacteristic
301#340C2312yes—
302#340C2312yes—
303#340C2311yescharacteristic
304#340C2311yescharacteristic
305#340C2311yescharacteristic
306#340C2311yescharacteristic
307#340C2312yes—
308#340C2312yes—
309#340C2312yes—
310#340C2312yes—
311#336C2312yes—
312#336C2312yes—
313#336C2311yescharacteristic
314#336C2311yescharacteristic
315#336C2311yescharacteristic
316#336C2311yescharacteristic
317#336C2312yes—
318C22×Q8—21no—
319C22×Q8—22no—
320C22×Q8—21no—
321C22×Q8—22no—
322C22×Q8C2×D411yescharacteristic
323C22×Q8C2411yescharacteristic
324C22×Q8C2411yescharacteristic
325C22×C8C2411yescharacteristic
326C22×C8C2411yescharacteristic
327C22×C8C2412yes—
328C22×C8C2411yescharacteristic
329C22×C8C2411yescharacteristic
330C22×C8C2412yes—
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333C4⋊Q8—24no—
334C4⋊Q8—22no—
335C4⋊Q8—22no—
336C4⋊Q8—24no—
337C4⋊Q8—22no—
338C4⋊Q8—22no—
339C4⋊Q8—24no—
340C4⋊Q8—42no—
341C4⋊Q8—44no—
342C4⋊Q8—44no—
343C4⋊Q8—48no—
344C4⋊Q8—82no—
345#125—22no—
346#125—24no—
347#125—24no—
348#125—41no—
349#125—48no—
350#125—82no—
351C4×Q8—24no—
352C4×Q8—24no—
353C4×Q8—22no—
354C4×Q8—22no—
355C4×Q8—24no—
356C4×Q8—22no—
357C4×Q8—22no—
358C4×Q8—24no—
359C4×Q8—44no—
360C4×Q8—44no—
361C4×Q8—82no—
362C4×Q8#4114yes—
363C4×Q8#4114yes—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4C2×D411yescharacteristic
395C2×C4⋊C4C2×D411yescharacteristic
396C2×C4⋊C4C2×D412yes—
397C2×C4⋊C4C2×D412yes—
398C2×C4⋊C4C2×D411yescharacteristic
399C2×C4⋊C4C2×D412yes—
400C2×C4⋊C4C2411yescharacteristic
401C2×C4⋊C4C2412yes—
402C2×C4⋊C4C2411yescharacteristic
403C2×C4⋊C4C2412yes—
404C2×C4⋊C4C2411yescharacteristic
405C2×C4⋊C4C2412yes—
406C2×C4⋊C4C2411yescharacteristic
407C2×C4⋊C4C2411yescharacteristic
408C2×C4⋊C4C2412yes—
409C2×C4⋊C4C2411yescharacteristic
410C2×C42—22no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—22no—
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—22no—
417C2×C42—22no—
418C2×C42C2×D411yescharacteristic
419C2×C42C2×D411yescharacteristic
420C2×C42C2×D411yescharacteristic
421C2×C42C2×D411yescharacteristic
422C2×C42C2411yescharacteristic
423C2×C42C2411yescharacteristic
424C2×C42C2412yes—
425C2×C42C2412yes—
426C2×C42C2411yescharacteristic
427C4⋊C8—24no—
428C4⋊C8—22no—
429C4⋊C8—22no—
430C4⋊C8—24no—
431#103—24no—
432#103—22no—
433#103—24no—
434#103—22no—
435#103—22no—
436#103—22no—
437#103—22no—
438#103—22no—
439#103—41no—
440#103—41no—
441#103—42no—
442#103—42no—
443#97—24no—
444#97—42no—
445C4×C8—24no—
446C4×C8—22no—
447C4×C8—22no—
448C4×C8—42no—
449C4×C8—42no—
450#95—21no—
451#95—21no—
452#95—22no—
453#95—21no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—21no—
461#95—22no—
462#95—22no—
463#95—22no—
464#95—22no—
465#95—22no—
466#95—22no—
467#95—22no—
468#95—22no—
469#95—22no—
470#95—22no—
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#95C2×D412yes—
481#95C2×D412yes—
482#95C2×D412yes—
483#95C2×D412yes—
484#95C2×D412yes—
485#95C2×D412yes—
486#95C2×D412yes—
487C2×Q8—22no—
488C2×Q8—22no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—21no—
499C2×Q8—44no—
500C2×Q8—42no—
501C2×Q8—44no—
502C2×Q8—42no—
503C2×Q8—42no—
504C2×Q8—42no—
505C2×Q8—82no—
506C2×Q8—81no—
507C2×Q8—81no—
508C2×Q8—82no—
509C2×Q8—82no—
510C2×Q8—82no—
511C2×Q8#12414yes—
512C2×Q82−1+412yes—
513C2×Q82−1+412yes—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—22no—
520C22×C4—21no—
521C22×C4—22no—
522C22×C4—22no—
523C22×C4—21no—
524C22×C4—22no—
525C22×C4—21no—
526C22×C4C22×D411yescharacteristic
527C22×C4C22×D411yescharacteristic
528C22×C4C22×D412yes—
529C22×C4C22×D412yes—
530C22×C4C22×D411yescharacteristic
531C22×C4C22×D411yescharacteristic
532C22×C4C22×D412yes—
533C22×C4C22×D412yes—
534C22×C4C22×D411yescharacteristic
535C22×C4C22×D411yescharacteristic
536C22×C4C2511yescharacteristic
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—22no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—22no—
547C2×C8—21no—
548C2×C8—21no—
549C2×C8—21no—
550C2×C8—21no—
551C2×C8—22no—
552C2×C8—21no—
553C2×C8—22no—
554C2×C8—21no—
555C4⋊C4—24no—
556C4⋊C4—24no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—22no—
563C4⋊C4—22no—
564C4⋊C4—21no—
565C4⋊C4—22no—
566C4⋊C4—21no—
567C4⋊C4—22no—
568C4⋊C4—21no—
569C4⋊C4—21no—
570C4⋊C4—21no—
571C4⋊C4—22no—
572C4⋊C4—22no—
573C4⋊C4—21no—
574C4⋊C4—21no—
575C4⋊C4—22no—
576C4⋊C4—21no—
577C4⋊C4—22no—
578C4⋊C4—42no—
579C4⋊C4—42no—
580C4⋊C4—42no—
581C4⋊C4—42no—
582C4⋊C4—44no—
583C4⋊C4—42no—
584C4⋊C4—44no—
585C4⋊C4—44no—
586C4⋊C4—42no—
587C4⋊C4—44no—
588C4⋊C4—44no—
589C4⋊C4—42no—
590C4⋊C4—44no—
591C4⋊C4—42no—
592C4⋊C4—42no—
593C4⋊C4—42no—
594C4⋊C4—81no—
595C4⋊C4—81no—
596C4⋊C4—82no—
597C4⋊C4—81no—
598C4⋊C4—81no—
599C4⋊C4—81no—
600C4⋊C4—81no—
601C4⋊C4—82no—
602C4⋊C4—82no—
603C4⋊C4—82no—
604C4⋊C4—82no—
605C4⋊C4—82no—
606C4⋊C4—82no—
607C4⋊C4—82no—
608C4⋊C4#12414yes—
609C4⋊C4#12418yes—
610C4⋊C42−1+412yes—
611C42—22no—
612C42—22no—
613C42—21no—
614C42—21no—
615C42—21no—
616C42—22no—
617C42—21no—
618C42—22no—
619C42—41no—
620C42—41no—
621C42—44no—
622C42—42no—
623C42—44no—
624C42—42no—
625C42—42no—
626C42—42no—
627C42—82no—
628C42—82no—
629C42C2×#4114yes—
630C42C2×#4112yes—
631C42C2×#4112yes—
632C42C2×#4114yes—
633C23C23×D411yescharacteristic
634Q8—28no—
635Q8—22no—
636Q8—22no—
637Q8—22no—
638Q8—22no—
639Q8—41no—
640Q8—41no—
641Q8—41no—
642Q8—41no—
643Q8—84no—
644Q8—162no—
645Q8—164no—
646Q8—164no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—22no—
668C2×C4—21no—
669C2×C4—21no—
670C2×C4—22no—
671C2×C4—21no—
672C2×C4—21no—
673C2×C4—22no—
674C2×C4—41no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—42no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—42no—
685C2×C4—41no—
686C2×C4—41no—
687C2×C4—41no—
688C2×C4—42no—
689C2×C4—41no—
690C2×C4—42no—
691C2×C4—41no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—41no—
695C2×C4—41no—
696C2×C4—42no—
697C2×C4—41no—
698C2×C4—41no—
699C2×C4—42no—
700C2×C4—41no—
701C2×C4—42no—
702C2×C4—42no—
703C2×C4—42no—
704C2×C4—41no—
705C2×C4—41no—
706C2×C4—42no—
707C2×C4C2×#12414yes—
708C2×C4C2×#12414yes—
709C2×C4C2×#12412yes—
710C2×C4C2×#12412yes—
711C2×C4#57712yes—
712C2×C4#57812yes—
713C2×C4C22×#4111yescharacteristic
714C2×C4C22×#4111yescharacteristic
715C2×C4C2×2−1+411yescharacteristic
716C2×C4C2×2−1+411yescharacteristic
717C2×C4C2×2−1+411yescharacteristic
718C2×C4#58511yescharacteristic
719C8—42no—
720C8—41no—
721C8—41no—
722C8—42no—
723C8—41no—
724C8—41no—
725C22C22×#12411yescharacteristic
726C22#348611yescharacteristic
727C22#348911yescharacteristic
728C22C2×#57711yescharacteristic
729C22C2×#57811yescharacteristic
730C22#358411yescharacteristic
731C22#358411yescharacteristic
732C4—24no—
733C4—22no—
734C4—24no—
735C4—41no—
736C4—42no—
737C4—42no—
738C4—41no—
739C4—42no—
740C4—42no—
741C4—42no—
742C4—41no—
743C4—41no—
744C4—82no—
745C4—81no—
746C4—81no—
747C4—82no—
748C4—81no—
749C4—82no—
750C4—81no—
751C4—81no—
752C4—82no—
753C4#313714yes—
754C4#351611yescharacteristic
755C4#351611yescharacteristic
756C4#352311yescharacteristic
757C4#352311yescharacteristic
758C2#3591011yescharacteristic
759C2C2×#313711yescharacteristic
760C2#6062411yescharacteristic
761C2#6186511yescharacteristic
762C2#6186611yescharacteristic
763C2#6221611yescharacteristic
764C2#6222011yescharacteristic
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
#396192
#391822
#302322
#302292
#292292
#292222
#291352
#291232
#278282
#278232
#278182
#278152
#277452
#277382
#277302
#238292
#238202
#225252
#225242
#225212
Q8×#1032
#205712