Finite Group Databasebeta

← #6591301

Subgroups — #6591301

Order 512.

1336 conjugacy classes of subgroups.

The subgroup lattice has 769 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2#36286C211yesmaximal, characteristic
3#33293C211yesmaximal, characteristic
4#33202C211yesmaximal, characteristic
5#33198C211yesmaximal, characteristic
6#28198C212yesmaximal
7#27742C212yesmaximal
8#27738C211yesmaximal, characteristic
9#27730C211yesmaximal, characteristic
10#26518C212yesmaximal
11#26511C211yesmaximal, characteristic
12#26509C211yesmaximal, characteristic
13#25008C212yesmaximal
14#25005C212yesmaximal
15#24699C212yesmaximal
16#24683C211yesmaximal, characteristic
17#24682C211yesmaximal, characteristic
18#23073C212yesmaximal
19#23068C211yesmaximal, characteristic
20#23063C211yesmaximal, characteristic
21#22744C212yesmaximal
22#21713C211yesmaximal, characteristic
23C4×#441C211yesmaximal, characteristic
24C2×#477C2211yescharacteristic
25C2×#477C2212yes—
26C2×#475C2211yescharacteristic
27C2×#446C2212yes—
28C2×#441C2211yescharacteristic
29C2×#441C2211yescharacteristic
30C2×C4×Q16C2211yescharacteristic
31#2868C2211yescharacteristic
32#2863—21no—
33#2862—21no—
34#2862C2211yescharacteristic
35#2605—22no—
36#2588—22no—
37#2582C2212yes—
38#2581C2212yes—
39#2571C2211yescharacteristic
40#2571C2211yescharacteristic
41#2570—21no—
42#2570—21no—
43#2570C2212yes—
44#2568C2211yescharacteristic
45#2568C2212yes—
46#2567C2211yescharacteristic
47#2511—22no—
48#2511—21no—
49#2451—21no—
50#2446—22no—
51#2369C2211yescharacteristic
52#2365C2211yescharacteristic
53#2365C2212yes—
54#2356C2212yes—
55#2355C2211yescharacteristic
56#2355C2211yescharacteristic
57#2355C2212yes—
58#2355C2212yes—
59C4×C4⋊Q8—21no—
60C4×C4⋊Q8C2211yescharacteristic
61#2295—21no—
62C42×Q8C2211yescharacteristic
63C42×Q8C2211yescharacteristic
64#2101C2212yes—
65#2101C2212yes—
66#2101C2212yes—
67#2099C2212yes—
68#2088C2212yes—
69#2088C2212yes—
70#2074C2212yes—
71#2074C2212yes—
72#2074C2212yes—
73#2072C2212yes—
74#2056C2212yes—
75#2056C2212yes—
76#1993C2212yes—
77#1993C2212yes—
78#1990C2211yescharacteristic
79#1990C2211yescharacteristic
80#1990C2211yescharacteristic
81#1990C2211yescharacteristic
82#1988C2211yescharacteristic
83#1987C2211yescharacteristic
84#1987C2212yes—
85#1985C2212yes—
86#1949C2212yes—
87#1949C2212yes—
88#1949C2212yes—
89#1949C2212yes—
90#1949C2212yes—
91#1947C2211yescharacteristic
92#1947C2211yescharacteristic
93#1947C2212yes—
94#1947C2212yes—
95#1940C2212yes—
96#1939C2211yescharacteristic
97#1939C2211yescharacteristic
98#1935C2212yes—
99#1935C2212yes—
100#1933C2212yes—
101#1933C2212yes—
102#1928C2212yes—
103#1928C2212yes—
104#1928C2212yes—
105#1927C2212yes—
106#1921C2212yes—
107#1921C2212yes—
108#1920C2212yes—
109#1918C2212yes—
110#1842C2211yescharacteristic
111#1836C2212yes—
112#1836C2212yes—
113#1836C2212yes—
114#1835C2212yes—
115#1805C2212yes—
116#1805C2212yes—
117#1803C2212yes—
118#1803C2211yescharacteristic
119#1803C2211yescharacteristic
120#1776C2211yescharacteristic
121#1776C2211yescharacteristic
122#1776C2212yes—
123C4×#107C2212yes—
124C4×#106C2211yescharacteristic
125#1771C2212yes—
126#1770C2211yescharacteristic
127C8×C4⋊C4C2211yescharacteristic
128#1763C2211yescharacteristic
129#1763C2211yescharacteristic
130#1763C2211yescharacteristic
131#1763C2211yescharacteristic
132C4×#103C2211yescharacteristic
133C4×#103C2211yescharacteristic
134C4×#97C2211yescharacteristic
135C22×Q16C2311yescharacteristic
136C2×C4⋊Q8—21no—
137C2×C4⋊Q8—21no—
138C2×C4⋊Q8D411yescharacteristic
139C2×C4⋊Q8C2311yescharacteristic
140C2×C4⋊Q8C2311yescharacteristic
141C2×C4⋊Q8C2312yes—
142C2×C4×Q8—21no—
143C2×C4×Q8—22no—
144C2×C4×Q8C2311yescharacteristic
145C2×C4×Q8C2311yescharacteristic
146C2×C4×Q8C2311yescharacteristic
147C2×C4×Q8C2311yescharacteristic
148C2×C4×Q8C2311yescharacteristic
149C2×C4×Q8C2311yescharacteristic
150C2×C4×Q8C2312yes—
151#477—24no—
152#477—28no—
153#475—24no—
154#446—28no—
155#441—24no—
156#441—24no—
157C4×Q16—24no—
158C2×#107C2312yes—
159C2×#107C2312yes—
160C2×#107C2312yes—
161C2×#107C2311yescharacteristic
162C2×#106C2312yes—
163C2×#106C2311yescharacteristic
164C2×#106C2311yescharacteristic
165C2×C4⋊C8C2311yescharacteristic
166C2×C4⋊C8C2311yescharacteristic
167C2×C4⋊C8C2312yes—
168C2×C4⋊C8C2312yes—
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2312yes—
171C2×#103C2312yes—
172C2×#103C2312yes—
173C2×#103C2312yes—
174C2×#103C2312yes—
175C2×#103C2311yescharacteristic
176C2×#103C2311yescharacteristic
177C2×#103C2311yescharacteristic
178C2×#103C2311yescharacteristic
179C2×#103C2311yescharacteristic
180C2×#103C2311yescharacteristic
181C2×#97C2311yescharacteristic
182C2×#97C2312yes—
183C2×#97C2311yescharacteristic
184C2×#97C2311yescharacteristic
185C2×#97C2311yescharacteristic
186C2×C4×C8C2312yes—
187C2×C4×C8C2311yescharacteristic
188C2×C4×C8C2311yescharacteristic
189#400—22no—
190#400—22no—
191#400—22no—
192#400—22no—
193#400—22no—
194#400—22no—
195#400—22no—
196#398—22no—
197#398—22no—
198#398—22no—
199#395—22no—
200#395—22no—
201#391—22no—
202#391—22no—
203#391—22no—
204#391—22no—
205#391—22no—
206#391—21no—
207#391—21no—
208#391—22no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391—21no—
214#391—22no—
215#391D412yes—
216#391C2312yes—
217#391C2312yes—
218#391C2312yes—
219#391C2312yes—
220#389—21no—
221#389—21no—
222#389—22no—
223#389—21no—
224#389—21no—
225#389—21no—
226#389D411yescharacteristic
227#389D411yescharacteristic
228#389D412yes—
229#389C2312yes—
230#389C2312yes—
231#389C2312yes—
232#389C2312yes—
233#387—22no—
234#387—22no—
235#387—22no—
236#387—22no—
237#387—22no—
238#387—22no—
239#387—22no—
240#387—22no—
241#387—22no—
242#387—22no—
243#387—22no—
244#387—22no—
245#387—22no—
246#387—22no—
247#387—22no—
248#384—21no—
249#384—21no—
250#384D411yescharacteristic
251#384C2312yes—
252#384C2311yescharacteristic
253#384C2311yescharacteristic
254#384C2312yes—
255#383—22no—
256#382—21no—
257#382—21no—
258#382—21no—
259#382—21no—
260#382C2312yes—
261#382C2312yes—
262#382C2312yes—
263#382C2312yes—
264#382C2312yes—
265C4×C4⋊C4—21no—
266C4×C4⋊C4—21no—
267C4×C4⋊C4—22no—
268C4×C4⋊C4—22no—
269C4×C4⋊C4—21no—
270C4×C4⋊C4—21no—
271C4×C4⋊C4—21no—
272C4×C4⋊C4—21no—
273C4×C4⋊C4—21no—
274C4×C4⋊C4—21no—
275C4×C4⋊C4—21no—
276C4×C4⋊C4C2311yescharacteristic
277C4×C4⋊C4C2311yescharacteristic
278C4×C4⋊C4C2312yes—
279C4×C4⋊C4C2312yes—
280C4×C4⋊C4C2311yescharacteristic
281C4×C4⋊C4C2311yescharacteristic
282C4×C4⋊C4C2312yes—
283C4×C4⋊C4C2312yes—
284C4×C4⋊C4C2311yescharacteristic
285C4×C4⋊C4C2311yescharacteristic
286C4×C4⋊C4C2311yescharacteristic
287C4×C4⋊C4C2312yes—
288C4×C4⋊C4C2311yescharacteristic
289#376—22no—
290C43—21no—
291C43—21no—
292C43C2311yescharacteristic
293#340C2312yes—
294#340C2312yes—
295#340C2312yes—
296#340C2312yes—
297#340C2312yes—
298#340C2312yes—
299#340C2312yes—
300#340C2312yes—
301#340C2312yes—
302#340C2312yes—
303#340C2312yes—
304#340C2312yes—
305#340C2312yes—
306#340C2312yes—
307#340C2311yescharacteristic
308#340C2312yes—
309#340C2312yes—
310#340C2311yescharacteristic
311#340C2311yescharacteristic
312#340C2311yescharacteristic
313#340C2311yescharacteristic
314#340C2311yescharacteristic
315#336C2312yes—
316#336C2312yes—
317#336C2312yes—
318#336C2312yes—
319C22×Q8—22no—
320C22×Q8—21no—
321C22×Q8—21no—
322C22×Q8C2×D412yes—
323C22×Q8C2411yescharacteristic
324C22×Q8C2411yescharacteristic
325C2×Q16—22no—
326C2×Q16—22no—
327C2×Q16—22no—
328C22×C8C2411yescharacteristic
329C22×C8C2411yescharacteristic
330C22×C8C2412yes—
331C22×C8C2412yes—
332C22×C8C2411yescharacteristic
333C22×C8C2411yescharacteristic
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—82no—
343C4×Q8—24no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×Q8—24no—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8—22no—
352C4×Q8—22no—
353C4×Q8—22no—
354C4×Q8—22no—
355C4×Q8—44no—
356C4×Q8—84no—
357C4×Q8#4114yes—
358C4×Q8#4114yes—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4C2×D411yescharacteristic
381C2×C4⋊C4C2×D412yes—
382C2×C4⋊C4C2×D412yes—
383C2×C4⋊C4C2×D412yes—
384C2×C4⋊C4C2×D411yescharacteristic
385C2×C4⋊C4C2×D411yescharacteristic
386C2×C4⋊C4C2×D411yescharacteristic
387C2×C4⋊C4C2×D412yes—
388C2×C4⋊C4C2×D412yes—
389C2×C4⋊C4C2411yescharacteristic
390C2×C4⋊C4C2411yescharacteristic
391C2×C4⋊C4C2411yescharacteristic
392C2×C4⋊C4C2411yescharacteristic
393C2×C4⋊C4C2412yes—
394C2×C4⋊C4C2412yes—
395C2×C4⋊C4C2412yes—
396C2×C4⋊C4C2412yes—
397C2×C4⋊C4C2411yescharacteristic
398C2×C4⋊C4C2411yescharacteristic
399C2×C42—21no—
400C2×C42—21no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—22no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—22no—
411C2×C42—22no—
412C2×C42—21no—
413C2×C42—21no—
414C2×C42—22no—
415C2×C42C2×D411yescharacteristic
416C2×C42C2×D411yescharacteristic
417C2×C42C2×D411yescharacteristic
418C2×C42C2×D411yescharacteristic
419C2×C42C2412yes—
420C2×C42C2412yes—
421C2×C42C2411yescharacteristic
422C2×C42C2411yescharacteristic
423C2×C42C2411yescharacteristic
424#107—22no—
425#107—24no—
426#107—24no—
427#107—24no—
428#106—22no—
429#106—22no—
430#106—24no—
431C4⋊C8—24no—
432C4⋊C8—24no—
433C4⋊C8—24no—
434C4⋊C8—22no—
435C4⋊C8—22no—
436C4⋊C8—24no—
437#103—22no—
438#103—22no—
439#103—24no—
440#103—24no—
441#103—24no—
442#103—24no—
443#103—22no—
444#103—22no—
445#103—22no—
446#103—22no—
447#97—24no—
448#97—22no—
449#97—22no—
450#97—22no—
451#97#4114yes—
452C4×C8—24no—
453C4×C8—22no—
454C4×C8—22no—
455#95—22no—
456#95—22no—
457#95—22no—
458#95—21no—
459#95—21no—
460#95—21no—
461#95—21no—
462#95—21no—
463#95—21no—
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#95—22no—
481#95—22no—
482#95—22no—
483#95—22no—
484#95—22no—
485#95—22no—
486#95—22no—
487#95C2×D412yes—
488#95C2×D412yes—
489#95C2×D412yes—
490#95C2×D412yes—
491C2×Q8—24no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—21no—
499C2×Q8—21no—
500C2×Q8—21no—
501C2×Q8—21no—
502C2×Q8—42no—
503C2×Q8—42no—
504C2×Q8—44no—
505C2×Q8—44no—
506C2×Q8—44no—
507C2×Q8—42no—
508C2×Q8—42no—
509C2×Q8—42no—
510C2×Q8—42no—
511C2×Q8—81no—
512C2×Q8—81no—
513C2×Q8C2×#4112yes—
514C2×Q8C2×#4112yes—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—22no—
521C22×C4—22no—
522C22×C4—21no—
523C22×C4—21no—
524C22×C4—21no—
525C22×C4—22no—
526C22×C4—22no—
527C22×C4C22×D412yes—
528C22×C4C22×D412yes—
529C22×C4C22×D412yes—
530C22×C4C22×D412yes—
531C22×C4C22×D411yescharacteristic
532C22×C4C22×D411yescharacteristic
533C22×C4C22×D411yescharacteristic
534C22×C4C22×D411yescharacteristic
535C22×C4C22×D411yescharacteristic
536C22×C4C22×D411yescharacteristic
537C22×C4C2511yescharacteristic
538Q16—28no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—22no—
545C2×C8—21no—
546C2×C8—22no—
547C2×C8—22no—
548C2×C8—22no—
549C2×C8—22no—
550C2×C8—22no—
551C2×C8—21no—
552C2×C8—21no—
553C2×C8—21no—
554C2×C8—21no—
555C2×C8C2×#4112yes—
556C2×C8C2×#4112yes—
557C4⋊C4—24no—
558C4⋊C4—24no—
559C4⋊C4—24no—
560C4⋊C4—24no—
561C4⋊C4—24no—
562C4⋊C4—22no—
563C4⋊C4—22no—
564C4⋊C4—22no—
565C4⋊C4—22no—
566C4⋊C4—21no—
567C4⋊C4—21no—
568C4⋊C4—21no—
569C4⋊C4—21no—
570C4⋊C4—21no—
571C4⋊C4—22no—
572C4⋊C4—21no—
573C4⋊C4—22no—
574C4⋊C4—22no—
575C4⋊C4—22no—
576C4⋊C4—21no—
577C4⋊C4—21no—
578C4⋊C4—21no—
579C4⋊C4—21no—
580C4⋊C4—22no—
581C4⋊C4—22no—
582C4⋊C4—22no—
583C4⋊C4—22no—
584C4⋊C4—42no—
585C4⋊C4—42no—
586C4⋊C4—44no—
587C4⋊C4—44no—
588C4⋊C4—42no—
589C4⋊C4—42no—
590C4⋊C4—42no—
591C4⋊C4—42no—
592C4⋊C4—44no—
593C4⋊C4—42no—
594C4⋊C4—42no—
595C4⋊C4—42no—
596C4⋊C4—42no—
597C4⋊C4—82no—
598C4⋊C4—82no—
599C4⋊C4—82no—
600C4⋊C4—82no—
601C4⋊C4—82no—
602C4⋊C4—82no—
603C4⋊C4—81no—
604C4⋊C4—81no—
605C4⋊C4C2×#4112yes—
606C4⋊C4C2×#4112yes—
607C42—24no—
608C42—24no—
609C42—22no—
610C42—22no—
611C42—22no—
612C42—22no—
613C42—22no—
614C42—22no—
615C42—41no—
616C42—41no—
617C42—42no—
618C42—44no—
619C42—42no—
620C42—42no—
621C42—42no—
622C42—42no—
623C42—42no—
624C42—42no—
625C42—42no—
626C42—81no—
627C42—81no—
628C42—82no—
629C42—82no—
630C42—82no—
631C42C2×#4112yes—
632C42C2×#4112yes—
633C42C2×#4112yes—
634C422−1+412yes—
635C422−1+412yes—
636C422−1+412yes—
637C23C23×D411yescharacteristic
638Q8—22no—
639Q8—22no—
640Q8—22no—
641Q8—22no—
642Q8—41no—
643Q8—41no—
644Q8—41no—
645Q8—41no—
646Q8—84no—
647Q8—88no—
648Q8—84no—
649Q8—162no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—22no—
672C2×C4—22no—
673C2×C4—22no—
674C2×C4—22no—
675C2×C4—22no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—41no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—41no—
688C2×C4—42no—
689C2×C4—41no—
690C2×C4—41no—
691C2×C4—41no—
692C2×C4—42no—
693C2×C4—41no—
694C2×C4—41no—
695C2×C4—41no—
696C2×C4—41no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4—42no—
700C2×C4—42no—
701C2×C4—41no—
702C2×C4—42no—
703C2×C4—42no—
704C2×C4—42no—
705C2×C4—41no—
706C2×C4—41no—
707C2×C4—42no—
708C2×C4#54712yes—
709C2×C4#54712yes—
710C2×C4#54812yes—
711C2×C4#54812yes—
712C2×C4#54812yes—
713C2×C4#54812yes—
714C2×C4#57512yes—
715C2×C4#57812yes—
716C2×C4C22×#4111yescharacteristic
717C2×C4C22×#4111yescharacteristic
718C2×C4C22×#4111yescharacteristic
719C2×C4C2×2−1+411yescharacteristic
720C2×C4C2×2−1+411yescharacteristic
721C2×C4C2×2−1+411yescharacteristic
722C8—22no—
723C8—22no—
724C8—42no—
725C8—42no—
726C8—41no—
727C8—41no—
728C22C2×#54711yescharacteristic
729C22C2×#54811yescharacteristic
730C22C2×#54811yescharacteristic
731C22C2×#57511yescharacteristic
732C22C2×#57811yescharacteristic
733C22#358511yescharacteristic
734C22#358611yescharacteristic
735C4—22no—
736C4—22no—
737C4—22no—
738C4—22no—
739C4—22no—
740C4—42no—
741C4—42no—
742C4—42no—
743C4—42no—
744C4—42no—
745C4—42no—
746C4—42no—
747C4—42no—
748C4—81no—
749C4—81no—
750C4—82no—
751C4—82no—
752C4—82no—
753C4—81no—
754C4—82no—
755C4—81no—
756C4#337714yes—
757C4#344411yescharacteristic
758C4#350911yescharacteristic
759C4#350911yescharacteristic
760C4#350911yescharacteristic
761C2#3615111yescharacteristic
762C2#6125711yescharacteristic
763C2#6128211yescharacteristic
764C2C2×#337711yescharacteristic
765C2#6132411yescharacteristic
766C2#6133611yescharacteristic
767C2#6134211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ 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
#365982
#362862
#332932
#332022
#331982
#281982
#277422
#277382
#277302
#265182
#265112
#265092
#250082
#250052
#246992
#246832
#246822
#230732
#230682
#230632
#227442
#217132
C4×#4412