Finite Group Databasebeta

← #6714906

Subgroups — #6714906

Order 512.

1248 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39181C211yesmaximal, characteristic
2#36598C211yesmaximal, characteristic
3#33073C211yesmaximal, characteristic
4#32566C212yesmaximal
5#32557C212yesmaximal
6#32319C212yesmaximal
7#31481C212yesmaximal
8#30230C211yesmaximal, characteristic
9#29866C211yesmaximal, characteristic
10#27823C211yesmaximal, characteristic
11#27730C211yesmaximal, characteristic
12#27428C212yesmaximal
13#27380C212yesmaximal
14#26509C211yesmaximal, characteristic
15#26444C211yesmaximal, characteristic
16#26294C212yesmaximal
17#25001C212yesmaximal
18#24818C211yesmaximal, characteristic
19#24683C211yesmaximal, characteristic
20#23068C211yesmaximal, characteristic
21#22964C211yesmaximal, characteristic
22#22309C211yesmaximal, characteristic
23#21116C211yesmaximal, characteristic
24C2×#475C2211yescharacteristic
25C2×#475C2211yescharacteristic
26C2×C4×Q16C2211yescharacteristic
27#2868C2211yescharacteristic
28#2862—21no—
29#2859C2211yescharacteristic
30#2774—22no—
31#2713—22no—
32#2647—21no—
33#2582C2212yes—
34#2581C2212yes—
35#2571C2211yescharacteristic
36#2571C2212yes—
37#2570—22no—
38#2570—21no—
39#2568C2211yescharacteristic
40#2568C2211yescharacteristic
41#2551—21no—
42#2551C2211yescharacteristic
43#2546C2212yes—
44#2511—22no—
45#2509C2212yes—
46#2451C2212yes—
47#2446—21no—
48#2446—21no—
49#2365C2211yescharacteristic
50#2365C2211yescharacteristic
51#2355C2211yescharacteristic
52#2355C2212yes—
53#2355C2212yes—
54Q8×C4⋊C4C2211yescharacteristic
55#2331—21no—
56C4×C4⋊Q8—21no—
57C4×C4⋊Q8C2211yescharacteristic
58#2295C2211yescharacteristic
59C42×Q8C2211yescharacteristic
60#2101C2212yes—
61#2101C2212yes—
62#2101C2212yes—
63#2100C2212yes—
64#2100C2212yes—
65#2100C2212yes—
66#2100C2212yes—
67#2088C2212yes—
68#2088C2212yes—
69#2085C2212yes—
70#2084C2212yes—
71#2084C2212yes—
72#2074C2212yes—
73#2074C2212yes—
74#2072C2212yes—
75#2072C2212yes—
76#2066C2212yes—
77#2058C2212yes—
78#2056C2212yes—
79#2024C2212yes—
80#1993C2212yes—
81#1990C2211yescharacteristic
82#1990C2211yescharacteristic
83#1990C2211yescharacteristic
84#1990C2211yescharacteristic
85#1988C2211yescharacteristic
86#1988C2211yescharacteristic
87#1987C2212yes—
88#1985C2212yes—
89#1984C2212yes—
90#1949C2212yes—
91#1949C2212yes—
92#1948C2211yescharacteristic
93#1947C2211yescharacteristic
94#1941C2212yes—
95#1939C2211yescharacteristic
96#1939C2211yescharacteristic
97#1935C2212yes—
98#1935C2212yes—
99#1933C2212yes—
100#1928C2212yes—
101#1927C2212yes—
102#1927C2211yescharacteristic
103#1927C2211yescharacteristic
104#1921C2212yes—
105#1920C2212yes—
106#1920C2212yes—
107#1920C2212yes—
108#1920C2212yes—
109#1918C2212yes—
110#1918C2211yescharacteristic
111#1918C2211yescharacteristic
112#1917C2212yes—
113#1916C2212yes—
114#1878C2211yescharacteristic
115#1878C2212yes—
116#1878C2211yescharacteristic
117#1865C2212yes—
118#1865C2212yes—
119#1865C2211yescharacteristic
120#1865C2211yescharacteristic
121#1863C2212yes—
122#1835C2212yes—
123#1831C2211yescharacteristic
124#1805C2212yes—
125#1803C2211yescharacteristic
126#1803C2211yescharacteristic
127#1776C2212yes—
128C4×#106C2211yescharacteristic
129#1770C2211yescharacteristic
130C8×C4⋊C4C2211yescharacteristic
131C4×#103C2211yescharacteristic
132C4×#103C2212yes—
133C4×#103C2211yescharacteristic
134#1744C2211yescharacteristic
135C22×Q16C2311yescharacteristic
136C2×C4⋊Q8—22no—
137C2×C4⋊Q8—21no—
138C2×C4⋊Q8—21no—
139C2×C4⋊Q8—21no—
140C2×C4⋊Q8D411yescharacteristic
141C2×C4⋊Q8C2311yescharacteristic
142C2×C4⋊Q8C2311yescharacteristic
143C2×C4⋊Q8C2312yes—
144C2×#125—21no—
145C2×C4×Q8—21no—
146C2×C4×Q8—21no—
147C2×C4×Q8C2311yescharacteristic
148C2×C4×Q8C2311yescharacteristic
149C2×C4×Q8C2311yescharacteristic
150C2×C4×Q8C2312yes—
151C2×C4×Q8C2311yescharacteristic
152C2×C4×Q8C2312yes—
153#475—24no—
154#475—24no—
155C4×Q16—24no—
156C2×#107C2311yescharacteristic
157C2×#107C2312yes—
158C2×#106C2312yes—
159C2×#106C2312yes—
160C2×#106C2311yescharacteristic
161C2×#106C2311yescharacteristic
162C2×#106C2312yes—
163C2×C4⋊C8C2311yescharacteristic
164C2×C4⋊C8C2311yescharacteristic
165C2×C4⋊C8C2312yes—
166C2×C4⋊C8C2312yes—
167C2×#103C2312yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#103C2311yescharacteristic
171C2×#103C2311yescharacteristic
172C2×#103C2311yescharacteristic
173C2×#103C2311yescharacteristic
174C2×#103C2312yes—
175C2×#103C2311yescharacteristic
176C2×#103C2311yescharacteristic
177C2×#97C2312yes—
178C2×C4×C8C2311yescharacteristic
179C2×C4×C8C2311yescharacteristic
180C2×C4×C8C2312yes—
181#400—22no—
182#400—22no—
183#400—22no—
184#400—22no—
185#400—22no—
186#400—22no—
187#400—22no—
188#400—22no—
189#398—22no—
190#398—22no—
191#398—22no—
192#398—22no—
193#398—22no—
194#395—22no—
195#395—22no—
196#395—22no—
197#391—22no—
198#391—22no—
199#391—21no—
200#391—21no—
201#391—21no—
202#391—21no—
203#391—22no—
204#391—21no—
205#391—21no—
206#391—22no—
207#391D412yes—
208#391C2312yes—
209#391C2312yes—
210#391C2312yes—
211#391C2311yescharacteristic
212#391C2311yescharacteristic
213#389—22no—
214#389—22no—
215#389—21no—
216#389—21no—
217#389—22no—
218#389—21no—
219#389—21no—
220#389—22no—
221#389D412yes—
222#389D411yescharacteristic
223#389D411yescharacteristic
224#389C2312yes—
225#389C2312yes—
226#389C2312yes—
227#389C2312yes—
228#389C2312yes—
229#389C2312yes—
230#389C2311yescharacteristic
231#389C2311yescharacteristic
232#389C2312yes—
233#387—22no—
234#387—22no—
235#387—22no—
236#387—22no—
237#387—22no—
238#387—21no—
239#387—21no—
240#387—22no—
241#387—22no—
242#387—22no—
243#387—22no—
244#387—22no—
245#387—22no—
246#384—21no—
247#384—21no—
248#384—21no—
249#384D411yescharacteristic
250#384C2311yescharacteristic
251#384C2311yescharacteristic
252#384C2312yes—
253#384C2312yes—
254#382—21no—
255#382—22no—
256#382—22no—
257#382—21no—
258#382—21no—
259#382C2312yes—
260#382C2312yes—
261#382C2312yes—
262C4×C4⋊C4—22no—
263C4×C4⋊C4—22no—
264C4×C4⋊C4—21no—
265C4×C4⋊C4—21no—
266C4×C4⋊C4—21no—
267C4×C4⋊C4—21no—
268C4×C4⋊C4—21no—
269C4×C4⋊C4—21no—
270C4×C4⋊C4C2311yescharacteristic
271C4×C4⋊C4C2311yescharacteristic
272C4×C4⋊C4C2312yes—
273C4×C4⋊C4C2312yes—
274C4×C4⋊C4C2311yescharacteristic
275C4×C4⋊C4C2312yes—
276C4×C4⋊C4C2311yescharacteristic
277C4×C4⋊C4C2311yescharacteristic
278C4×C4⋊C4C2311yescharacteristic
279C4×C4⋊C4C2312yes—
280#376—21no—
281C43—21no—
282C43C2311yescharacteristic
283#340C2312yes—
284#340C2312yes—
285#340C2312yes—
286#340C2312yes—
287#340C2312yes—
288#340C2312yes—
289#340C2312yes—
290#340C2312yes—
291#340C2312yes—
292#340C2312yes—
293#340C2312yes—
294#340C2311yescharacteristic
295#340C2312yes—
296#340C2311yescharacteristic
297#340C2312yes—
298#340C2312yes—
299#340C2312yes—
300#340C2311yescharacteristic
301#340C2312yes—
302#340C2311yescharacteristic
303#340C2311yescharacteristic
304#340C2311yescharacteristic
305#336C2312yes—
306#336C2312yes—
307#336C2312yes—
308#336C2312yes—
309#336C2312yes—
310#336C2311yescharacteristic
311#336C2311yescharacteristic
312#336C2311yescharacteristic
313#336C2311yescharacteristic
314#336C2312yes—
315C22×Q8—21no—
316C22×Q8—21no—
317C22×Q8—22no—
318C22×Q8C2×D412yes—
319C22×Q8C2411yescharacteristic
320C22×Q8C2411yescharacteristic
321C2×Q16—22no—
322C2×Q16—22no—
323C2×Q16—41no—
324C22×C8C2411yescharacteristic
325C22×C8C2411yescharacteristic
326C22×C8C2411yescharacteristic
327C22×C8C2411yescharacteristic
328C22×C8C2412yes—
329C22×C8C2412yes—
330C4⋊Q8—22no—
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333C4⋊Q8—22no—
334C4⋊Q8—42no—
335C4⋊Q8—42no—
336C4⋊Q8—42no—
337C4⋊Q8—44no—
338C4⋊Q8—44no—
339C4⋊Q8—82no—
340C4⋊Q8—84no—
341#125—82no—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—42no—
348C4×Q8—42no—
349C4×Q8—41no—
350C4×Q8—41no—
351C4×Q8—42no—
352C4×Q8—42no—
353C4×Q8—82no—
354C4×Q8—82no—
355C4×Q8#4114yes—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4C2×D411yescharacteristic
382C2×C4⋊C4C2×D411yescharacteristic
383C2×C4⋊C4C2×D411yescharacteristic
384C2×C4⋊C4C2×D411yescharacteristic
385C2×C4⋊C4C2×D412yes—
386C2×C4⋊C4C2×D412yes—
387C2×C4⋊C4C2×D412yes—
388C2×C4⋊C4C2×D412yes—
389C2×C4⋊C4C2×D412yes—
390C2×C4⋊C4C2412yes—
391C2×C4⋊C4C2411yescharacteristic
392C2×C4⋊C4C2411yescharacteristic
393C2×C4⋊C4C2411yescharacteristic
394C2×C4⋊C4C2412yes—
395C2×C4⋊C4C2412yes—
396C2×C4⋊C4C2411yescharacteristic
397C2×C4⋊C4C2412yes—
398C2×C4⋊C4C2411yescharacteristic
399C2×C4⋊C4C2411yescharacteristic
400C2×C42—21no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—22no—
409C2×C42—22no—
410C2×C42—22no—
411C2×C42—22no—
412C2×C42C2×D411yescharacteristic
413C2×C42C2×D411yescharacteristic
414C2×C42C2×D411yescharacteristic
415C2×C42C2×D411yescharacteristic
416C2×C42C2411yescharacteristic
417C2×C42C2412yes—
418C2×C42C2412yes—
419C2×C42C2411yescharacteristic
420C2×C42C2411yescharacteristic
421#107—22no—
422#107—24no—
423#106—24no—
424#106—24no—
425#106—22no—
426#106—41no—
427#106—42no—
428C4⋊C8—22no—
429C4⋊C8—22no—
430C4⋊C8—42no—
431C4⋊C8—42no—
432#103—22no—
433#103—24no—
434#103—24no—
435#103—24no—
436#103—22no—
437#103—22no—
438#103—22no—
439#103—41no—
440#103—42no—
441#103—41no—
442#97—42no—
443C4×C8—22no—
444C4×C8—22no—
445C4×C8—42no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—22no—
454#95—22no—
455#95—21no—
456#95—21no—
457#95—22no—
458#95—22no—
459#95—22no—
460#95—22no—
461#95—22no—
462#95—22no—
463#95—22no—
464#95—21no—
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#95C2×D412yes—
479#95C2×D412yes—
480#95C2×D412yes—
481#95C2×D412yes—
482C2×Q8—24no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—42no—
494C2×Q8—42no—
495C2×Q8—44no—
496C2×Q8—42no—
497C2×Q8—42no—
498C2×Q8—42no—
499C2×Q8—42no—
500C2×Q8—42no—
501C2×Q8—42no—
502C2×Q8—82no—
503C2×Q8—82no—
504C2×Q8C2×#4112yes—
505C2×Q82−1+412yes—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—22no—
513C22×C4—21no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—21no—
517C22×C4—22no—
518C22×C4C22×D412yes—
519C22×C4C22×D412yes—
520C22×C4C22×D412yes—
521C22×C4C22×D412yes—
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D411yescharacteristic
524C22×C4C22×D411yescharacteristic
525C22×C4C22×D411yescharacteristic
526C22×C4C22×D411yescharacteristic
527C22×C4C22×D411yescharacteristic
528C22×C4C2511yescharacteristic
529Q16—44no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—22no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—22no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—22no—
544C2×C8—21no—
545C2×C8—22no—
546C2×C8—21no—
547C2×C8—21no—
548C4⋊C4—24no—
549C4⋊C4—24no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—24no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—22no—
562C4⋊C4—21no—
563C4⋊C4—22no—
564C4⋊C4—21no—
565C4⋊C4—22no—
566C4⋊C4—22no—
567C4⋊C4—21no—
568C4⋊C4—22no—
569C4⋊C4—21no—
570C4⋊C4—21no—
571C4⋊C4—22no—
572C4⋊C4—21no—
573C4⋊C4—42no—
574C4⋊C4—42no—
575C4⋊C4—42no—
576C4⋊C4—42no—
577C4⋊C4—42no—
578C4⋊C4—44no—
579C4⋊C4—44no—
580C4⋊C4—44no—
581C4⋊C4—42no—
582C4⋊C4—42no—
583C4⋊C4—42no—
584C4⋊C4—42no—
585C4⋊C4—42no—
586C4⋊C4—44no—
587C4⋊C4—42no—
588C4⋊C4—42no—
589C4⋊C4—82no—
590C4⋊C4—82no—
591C4⋊C4—82no—
592C4⋊C4—81no—
593C4⋊C4—81no—
594C4⋊C4—82no—
595C4⋊C4—82no—
596C4⋊C4—82no—
597C4⋊C4—82no—
598C4⋊C4—82no—
599C4⋊C4—82no—
600C4⋊C4C2×#4112yes—
601C4⋊C42−1+412yes—
602C42—24no—
603C42—24no—
604C42—22no—
605C42—22no—
606C42—22no—
607C42—22no—
608C42—21no—
609C42—21no—
610C42—22no—
611C42—22no—
612C42—22no—
613C42—22no—
614C42—21no—
615C42—21no—
616C42—42no—
617C42—42no—
618C42—42no—
619C42—42no—
620C42—42no—
621C42—42no—
622C42—82no—
623C42—82no—
624C42—82no—
625C42—82no—
626C42C2×#4112yes—
627C422−1+412yes—
628C23C23×D411yescharacteristic
629Q8—22no—
630Q8—22no—
631Q8—22no—
632Q8—22no—
633Q8—41no—
634Q8—41no—
635Q8—41no—
636Q8—41no—
637Q8—84no—
638Q8—84no—
639Q8—84no—
640Q8—164no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—21no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—21no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—22no—
666C2×C4—21no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—41no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—41no—
681C2×C4—42no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—42no—
685C2×C4—41no—
686C2×C4—41no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—41no—
691C2×C4—41no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—41no—
697C2×C4—42no—
698C2×C4—41no—
699C2×C4—41no—
700C2×C4—41no—
701C2×C4#54812yes—
702C2×C4#54812yes—
703C2×C4#57512yes—
704C2×C4#57712yes—
705C2×C4#57712yes—
706C2×C4#57812yes—
707C2×C4C22×#4111yescharacteristic
708C2×C4C2×2−1+411yescharacteristic
709C2×C4C2×2−1+411yescharacteristic
710C2×C4C2×2−1+411yescharacteristic
711C2×C4#58511yescharacteristic
712C2×C4#58511yescharacteristic
713C8—41no—
714C8—42no—
715C8—42no—
716C8—41no—
717C8—41no—
718C8—41no—
719C22C2×#54811yescharacteristic
720C22#348511yescharacteristic
721C22#348611yescharacteristic
722C22C2×#57511yescharacteristic
723C22C2×#57711yescharacteristic
724C22C2×#57711yescharacteristic
725C22C2×#57811yescharacteristic
726C4—22no—
727C4—22no—
728C4—22no—
729C4—42no—
730C4—42no—
731C4—42no—
732C4—41no—
733C4—42no—
734C4—42no—
735C4—42no—
736C4—42no—
737C4—41no—
738C4—42no—
739C4—82no—
740C4—81no—
741C4—81no—
742C4—82no—
743C4—82no—
744C4—81no—
745C4—81no—
746C4—82no—
747C4#337114yes—
748C4#346111yescharacteristic
749C4#349511yescharacteristic
750C4#355511yescharacteristic
751C4#356311yescharacteristic
752C2#3775711yescharacteristic
753C2C2×#337111yescharacteristic
754C2#6133011yescharacteristic
755C2#6173911yescharacteristic
756C2#6178911yescharacteristic
757C2#6184211yescharacteristic
758C2#6187111yescharacteristic
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
#391812
#365982
#330732
#325662
#325572
#323192
#314812
#302302
#298662
#278232
#277302
#274282
#273802
#265092
#264442
#262942
#250012
#248182
#246832
#230682
#229642
#223092
#211162