Finite Group Databasebeta

← #7541666

Subgroups — #7541666

Order 512.

1266 conjugacy classes of subgroups.

The subgroup lattice has 758 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59875C211yesmaximal, characteristic
2#36285C211yesmaximal, characteristic
3#32449C212yesmaximal
4#32434C212yesmaximal
5#29212C211yesmaximal, characteristic
6#29144C211yesmaximal, characteristic
7#27742C212yesmaximal
8#27151C212yesmaximal
9#27138C212yesmaximal
10#26322C211yesmaximal, characteristic
11#26317C211yesmaximal, characteristic
12#26101C212yesmaximal
13#25302C211yesmaximal, characteristic
14#25294C211yesmaximal, characteristic
15#24997C211yesmaximal, characteristic
16#24995C211yesmaximal, characteristic
17#24699C212yesmaximal
18#23150C211yesmaximal, characteristic
19#22818C212yesmaximal
20#22743C211yesmaximal, characteristic
21#22741C211yesmaximal, characteristic
22C4×#477C211yesmaximal, characteristic
23#21506C211yesmaximal, characteristic
24C2×#477C2211yescharacteristic
25C2×#462C2211yescharacteristic
26C2×#446C2211yescharacteristic
27C2×C8×Q8C2211yescharacteristic
28C2×#441C2212yes—
29#2863—21no—
30#2859C2211yescharacteristic
31#2810—22no—
32#2606—22no—
33#2590—22no—
34#2582C2211yescharacteristic
35#2582C2211yescharacteristic
36#2570C2212yes—
37#2568—21no—
38#2568C2211yescharacteristic
39#2568C2211yescharacteristic
40#2568C2212yes—
41#2565C2212yes—
42#2514—21no—
43#2514—21no—
44#2514—22no—
45#2514C2212yes—
46#2514C2212yes—
47#2507—21no—
48#2446—21no—
49#2370C2212yes—
50#2356C2212yes—
51#2355—21no—
52#2355C2212yes—
53#2355C2211yescharacteristic
54#2355C2211yescharacteristic
55#2355C2211yescharacteristic
56Q8×C4⋊C4C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60#2295—21no—
61C42×Q8C2211yescharacteristic
62#2101C2212yes—
63#2101C2212yes—
64#2097C2212yes—
65#2095C2212yes—
66#2074C2212yes—
67#2074C2212yes—
68#2074C2212yes—
69#2072C2212yes—
70#2068C2212yes—
71#2066C2212yes—
72#2059C2212yes—
73#2059C2212yes—
74#2056C2212yes—
75#2056C2212yes—
76#2026C2212yes—
77#2026C2212yes—
78#1993C2212yes—
79#1987C2211yescharacteristic
80#1984C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1971C2212yes—
84#1969C2212yes—
85#1956C2211yescharacteristic
86#1956C2211yescharacteristic
87#1949C2212yes—
88#1949C2212yes—
89#1949C2212yes—
90#1948C2211yescharacteristic
91#1948C2211yescharacteristic
92#1948C2211yescharacteristic
93#1947C2211yescharacteristic
94#1947C2211yescharacteristic
95#1947C2211yescharacteristic
96#1947C2212yes—
97#1940C2212yes—
98#1935C2212yes—
99#1933C2212yes—
100#1928C2212yes—
101#1928C2212yes—
102#1928C2212yes—
103#1928C2212yes—
104#1927C2212yes—
105#1927C2212yes—
106#1927C2212yes—
107#1921C2211yescharacteristic
108#1921C2211yescharacteristic
109#1919C2212yes—
110#1918C2212yes—
111#1917C2212yes—
112#1916C2212yes—
113#1878C2212yes—
114#1863C2211yescharacteristic
115#1863C2211yescharacteristic
116#1847C2212yes—
117#1836C2212yes—
118#1835C2212yes—
119#1834C2211yescharacteristic
120#1834C2211yescharacteristic
121#1805C2212yes—
122#1805C2212yes—
123#1803C2211yescharacteristic
124#1803C2211yescharacteristic
125#1776C2212yes—
126C4×#107C2211yescharacteristic
127C4×#107C2211yescharacteristic
128C4×#106C2212yes—
129#1771C2212yes—
130#1767C2211yescharacteristic
131C4×C4⋊C8C2211yescharacteristic
132#1763C2212yes—
133C4×#103C2211yescharacteristic
134C4×#103C2211yescharacteristic
135C22×Q16C2312yes—
136C2×C4⋊Q8D411yescharacteristic
137C2×C4⋊Q8C2311yescharacteristic
138C2×C4⋊Q8C2311yescharacteristic
139C2×#125C2312yes—
140C2×C4×Q8—21no—
141C2×C4×Q8—21no—
142C2×C4×Q8C2312yes—
143C2×C4×Q8C2311yescharacteristic
144C2×C4×Q8C2311yescharacteristic
145C2×C4×Q8C2311yescharacteristic
146C2×C4×Q8C2311yescharacteristic
147C2×C4×Q8C2311yescharacteristic
148#477—24no—
149#462—24no—
150#446—24no—
151C8×Q8—24no—
152#441—28no—
153C2×#107C2311yescharacteristic
154C2×#107C2311yescharacteristic
155C2×#107C2312yes—
156C2×#107C2312yes—
157C2×#107C2311yescharacteristic
158C2×#107C2311yescharacteristic
159C2×#106C2312yes—
160C2×#106C2312yes—
161C2×#106C2312yes—
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2312yes—
164C2×C4⋊C8C2312yes—
165C2×C4⋊C8C2311yescharacteristic
166C2×C4⋊C8C2311yescharacteristic
167C2×C4⋊C8C2311yescharacteristic
168C2×#103C2311yescharacteristic
169C2×#103C2311yescharacteristic
170C2×#103C2312yes—
171C2×#103C2312yes—
172C2×#103C2312yes—
173C2×#103C2312yes—
174C2×#103C2312yes—
175C2×#103C2311yescharacteristic
176C2×#103C2311yescharacteristic
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#97C2312yes—
181C2×#97C2312yes—
182C2×C4×C8C2311yescharacteristic
183C2×C4×C8C2311yescharacteristic
184C2×C4×C8C2312yes—
185#401—22no—
186#401—22no—
187#400—22no—
188#400—22no—
189#400—22no—
190#400—22no—
191#400—22no—
192#400—22no—
193#400—22no—
194#398—22no—
195#398—22no—
196#398—22no—
197#398—22no—
198#398—22no—
199#391—21no—
200#391—21no—
201#391—21no—
202#391—21no—
203#391—21no—
204#391—21no—
205#391—21no—
206#391—22no—
207#391—22no—
208#391—21no—
209#391D411yescharacteristic
210#391D411yescharacteristic
211#391C2312yes—
212#391C2312yes—
213#391C2312yes—
214#391C2312yes—
215#391C2312yes—
216#391C2312yes—
217#391C2312yes—
218#389—21no—
219#389—21no—
220#389—22no—
221#389—21no—
222#389—22no—
223#389—21no—
224#389—21no—
225#389—21no—
226#389—21no—
227#389—21no—
228#389—21no—
229#389—21no—
230#389D412yes—
231#389D412yes—
232#389C2311yescharacteristic
233#389C2311yescharacteristic
234#389C2312yes—
235#389C2312yes—
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#387—22no—
249#387—22no—
250#387—22no—
251#384D411yescharacteristic
252#384C2311yescharacteristic
253#384C2311yescharacteristic
254#384C2312yes—
255#384C2311yescharacteristic
256#384C2311yescharacteristic
257#383—22no—
258#383—22no—
259#383—22no—
260#382—22no—
261#382—21no—
262#382—21no—
263#382C2312yes—
264#382C2312yes—
265#382C2312yes—
266#382C2312yes—
267C4×C4⋊C4—22no—
268C4×C4⋊C4—21no—
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⋊C4—21no—
277C4×C4⋊C4—22no—
278C4×C4⋊C4C2311yescharacteristic
279C4×C4⋊C4C2311yescharacteristic
280C4×C4⋊C4C2311yescharacteristic
281C4×C4⋊C4C2311yescharacteristic
282C4×C4⋊C4C2312yes—
283C4×C4⋊C4C2312yes—
284C4×C4⋊C4C2312yes—
285C4×C4⋊C4C2312yes—
286C4×C4⋊C4C2311yescharacteristic
287C4×C4⋊C4C2312yes—
288C4×C4⋊C4C2312yes—
289#376—22no—
290C43—21no—
291C43C2311yescharacteristic
292#340C2312yes—
293#340C2312yes—
294#340C2312yes—
295#340C2311yescharacteristic
296#340C2311yescharacteristic
297#340C2312yes—
298#340C2312yes—
299#340C2312yes—
300#340C2312yes—
301#340C2312yes—
302#340C2312yes—
303#340C2311yescharacteristic
304#340C2311yescharacteristic
305#340C2311yescharacteristic
306#340C2311yescharacteristic
307#340C2312yes—
308#340C2311yescharacteristic
309#340C2311yescharacteristic
310#336C2312yes—
311#336C2312yes—
312#336C2312yes—
313#336C2312yes—
314#336C2312yes—
315#336C2311yescharacteristic
316#336C2311yescharacteristic
317C22×Q8—21no—
318C22×Q8C2×D411yescharacteristic
319C22×Q8C2×D411yescharacteristic
320C22×Q8C2412yes—
321C22×Q8C2411yescharacteristic
322C2×Q16—24no—
323C2×Q16—24no—
324C2×Q16—42no—
325C22×C8C2412yes—
326C22×C8C2411yescharacteristic
327C22×C8C2411yescharacteristic
328C22×C8C2411yescharacteristic
329C22×C8C2411yescharacteristic
330C22×C8C2412yes—
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333C4⋊Q8—22no—
334C4⋊Q8—22no—
335C4⋊Q8—42no—
336#125—42no—
337#125—42no—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×Q8—24no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×Q8—24no—
347C4×Q8—41no—
348C4×Q8—41no—
349C4×Q8—44no—
350C4×Q8—44no—
351C4×Q8#4114yes—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
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—22no—
375C2×C4⋊C4C2×D411yescharacteristic
376C2×C4⋊C4C2×D411yescharacteristic
377C2×C4⋊C4C2×D412yes—
378C2×C4⋊C4C2×D412yes—
379C2×C4⋊C4C2×D412yes—
380C2×C4⋊C4C2×D412yes—
381C2×C4⋊C4C2×D412yes—
382C2×C4⋊C4C2×D412yes—
383C2×C4⋊C4C2412yes—
384C2×C4⋊C4C2411yescharacteristic
385C2×C4⋊C4C2411yescharacteristic
386C2×C4⋊C4C2412yes—
387C2×C4⋊C4C2412yes—
388C2×C4⋊C4C2411yescharacteristic
389C2×C4⋊C4C2411yescharacteristic
390C2×C4⋊C4C2411yescharacteristic
391C2×C4⋊C4C2411yescharacteristic
392C2×C4⋊C4C2411yescharacteristic
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—22no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—21no—
399C2×C42—21no—
400C2×C42—22no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—22no—
404C2×C42—22no—
405C2×C42—22no—
406C2×C42C2×D411yescharacteristic
407C2×C42C2×D411yescharacteristic
408C2×C42C2×D411yescharacteristic
409C2×C42C2×D411yescharacteristic
410C2×C42C2411yescharacteristic
411C2×C42C2412yes—
412C2×C42C2412yes—
413C2×C42C2411yescharacteristic
414C2×C42C2411yescharacteristic
415#107—22no—
416#107—22no—
417#107—42no—
418#107—42no—
419#107—41no—
420#107—41no—
421#106—24no—
422#106—42no—
423#106—42no—
424C4⋊C8—24no—
425C4⋊C8—24no—
426C4⋊C8—22no—
427C4⋊C8—22no—
428C4⋊C8—22no—
429C4⋊C8#4114yes—
430#103—24no—
431#103—22no—
432#103—22no—
433#103—24no—
434#103—24no—
435#103—42no—
436#103—41no—
437#103—42no—
438#103—42no—
439#103—41no—
440#103—42no—
441#103—42no—
442#97—24no—
443#97—24no—
444C4×C8—24no—
445C4×C8—22no—
446C4×C8—22no—
447#95—22no—
448#95—22no—
449#95—22no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—22no—
454#95—22no—
455#95—22no—
456#95—22no—
457#95—22no—
458#95—22no—
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—21no—
468#95—21no—
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#95C2×D412yes—
483#95C2×D412yes—
484#95C2×D412yes—
485#95C2×D412yes—
486C2×Q8—22no—
487C2×Q8—22no—
488C2×Q8—22no—
489C2×Q8—22no—
490C2×Q8—22no—
491C2×Q8—22no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—22no—
495C2×Q8—22no—
496C2×Q8—21no—
497C2×Q8—22no—
498C2×Q8—22no—
499C2×Q8—21no—
500C2×Q8—21no—
501C2×Q8—22no—
502C2×Q8—22no—
503C2×Q8—42no—
504C2×Q8—42no—
505C2×Q8—42no—
506C2×Q8C2×#4112yes—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—22no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—22no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—22no—
519C22×C4C22×D412yes—
520C22×C4C22×D412yes—
521C22×C4C22×D412yes—
522C22×C4C22×D412yes—
523C22×C4C22×D411yescharacteristic
524C22×C4C22×D411yescharacteristic
525C22×C4C22×D411yescharacteristic
526C22×C4C22×D411yescharacteristic
527C22×C4C22×D411yescharacteristic
528C22×C4C22×D411yescharacteristic
529C22×C4C2511yescharacteristic
530Q16—48no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—22no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—22no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—21no—
547C2×C8—22no—
548C2×C8C2×#4114yes—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—24no—
552C4⋊C4—24no—
553C4⋊C4—24no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—22no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—22no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—21no—
564C4⋊C4—21no—
565C4⋊C4—22no—
566C4⋊C4—22no—
567C4⋊C4—21no—
568C4⋊C4—22no—
569C4⋊C4—22no—
570C4⋊C4—21no—
571C4⋊C4—21no—
572C4⋊C4—21no—
573C4⋊C4—42no—
574C4⋊C4—42no—
575C4⋊C4—42no—
576C4⋊C4—42no—
577C4⋊C4—42no—
578C4⋊C4—42no—
579C4⋊C4—42no—
580C4⋊C4—42no—
581C4⋊C4—44no—
582C4⋊C4—44no—
583C4⋊C4—44no—
584C4⋊C4—42no—
585C4⋊C4—42no—
586C4⋊C4—42no—
587C4⋊C4—44no—
588C4⋊C4—42no—
589C4⋊C4—42no—
590C4⋊C4—42no—
591C4⋊C4—42no—
592C4⋊C4—81no—
593C4⋊C4—81no—
594C4⋊C4—81no—
595C4⋊C4—81no—
596C4⋊C4—82no—
597C4⋊C4—82no—
598C4⋊C4—82no—
599C4⋊C4C2×#4112yes—
600C42—24no—
601C42—22no—
602C42—22no—
603C42—22no—
604C42—22no—
605C42—22no—
606C42—22no—
607C42—21no—
608C42—21no—
609C42—41no—
610C42—41no—
611C42—42no—
612C42—44no—
613C42—42no—
614C42—44no—
615C42—42no—
616C42—42no—
617C42—42no—
618C42—81no—
619C42—82no—
620C42—82no—
621C42—81no—
622C42—82no—
623C42C2×#4112yes—
624C42C2×#4112yes—
625C422−1+412yes—
626C422−1+412yes—
627C23C23×D411yescharacteristic
628Q8—22no—
629Q8—22no—
630Q8—44no—
631Q8—44no—
632Q8—41no—
633Q8—41no—
634Q8—42no—
635Q8—42no—
636Q8—42no—
637Q8—42no—
638Q8—84no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—21no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—21no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—42no—
675C2×C4—41no—
676C2×C4—42no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—41no—
684C2×C4—41no—
685C2×C4—41no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—41no—
689C2×C4—41no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—41no—
693C2×C4—41no—
694C2×C4—41no—
695C2×C4—41no—
696C2×C4—41no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4#54712yes—
700C2×C4#54712yes—
701C2×C4#54812yes—
702C2×C4#54812yes—
703C2×C4#57512yes—
704C2×C4#57812yes—
705C2×C4C22×#4111yescharacteristic
706C2×C4C22×#4111yescharacteristic
707C2×C4C2×2−1+411yescharacteristic
708C2×C4C2×2−1+411yescharacteristic
709C2×C4#58511yescharacteristic
710C2×C4#58511yescharacteristic
711C8—24no—
712C8—41no—
713C8—41no—
714C8—42no—
715C8—41no—
716C8—41no—
717C22C2×#54711yescharacteristic
718C22C2×#54811yescharacteristic
719C22#348711yescharacteristic
720C22C2×#57511yescharacteristic
721C22C2×#57811yescharacteristic
722C22#358411yescharacteristic
723C22#358611yescharacteristic
724C4—22no—
725C4—22no—
726C4—22no—
727C4—42no—
728C4—42no—
729C4—42no—
730C4—42no—
731C4—42no—
732C4—42no—
733C4—41no—
734C4—41no—
735C4—81no—
736C4—81no—
737C4—81no—
738C4—82no—
739C4—82no—
740C4—81no—
741C4—81no—
742C4—81no—
743C4—82no—
744C4—82no—
745C4#334914yes—
746C4#347511yescharacteristic
747C4#350711yescharacteristic
748C4#351811yescharacteristic
749C4#355511yescharacteristic
750C2#5906511yescharacteristic
751C2C2×#334911yescharacteristic
752C2#6128011yescharacteristic
753C2#6132311yescharacteristic
754C2#6132811yescharacteristic
755C2#6190211yescharacteristic
756C2#6191511yescharacteristic
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
#598752
#362852
#324492
#324342
#292122
#291442
#277422
#271512
#271382
#263222
#263172
#261012
#253022
#252942
#249972
#249952
#246992
#231502
#228182
#227432
#227412
C4×#4772
#215062