Finite Group Databasebeta

← #7189433

Subgroups — #7189433

Order 512.

1262 conjugacy classes of subgroups.

The subgroup lattice has 761 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59668C211yesmaximal, characteristic
2#59445C211yesmaximal, characteristic
3#32478C212yesmaximal
4#32316C212yesmaximal
5#32232C212yesmaximal
6#31968C212yesmaximal
7#29290C211yesmaximal, characteristic
8#28284C211yesmaximal, characteristic
9#27815C211yesmaximal, characteristic
10#27729C211yesmaximal, characteristic
11#27364C212yesmaximal
12#26694C212yesmaximal
13#26512C212yesmaximal
14#26112C211yesmaximal, characteristic
15#26026C212yesmaximal
16#26026C211yesmaximal, characteristic
17#25455C211yesmaximal, characteristic
18#25455C211yesmaximal, characteristic
19#23816C211yesmaximal, characteristic
20#23063C211yesmaximal, characteristic
21#22965C211yesmaximal, characteristic
22#22299C211yesmaximal, characteristic
23#22248C211yesmaximal, characteristic
24C2×#500C2211yescharacteristic
25C2×#477C2211yescharacteristic
26C2×#477C2211yescharacteristic
27#2806—22no—
28#2774—22no—
29#2584—21no—
30#2570—21no—
31#2567C2211yescharacteristic
32#2567C2211yescharacteristic
33#2552C2211yescharacteristic
34#2552C2212yes—
35#2551C2211yescharacteristic
36#2551C2212yes—
37#2550—21no—
38#2546—21no—
39#2511C2212yes—
40#2508—21no—
41#2507C2212yes—
42#2507C2212yes—
43#2506—21no—
44#2462—22no—
45#2451—22no—
46#2446C2212yes—
47#2370—21no—
48#2370—21no—
49#2369C2211yescharacteristic
50#2369C2212yes—
51#2369C2211yescharacteristic
52#2355C2211yescharacteristic
53#2355C2212yes—
54Q8×C4⋊C4C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×#125C2211yescharacteristic
60#2100C2212yes—
61#2100C2211yescharacteristic
62#2100C2211yescharacteristic
63#2100C2212yes—
64#2100C2212yes—
65#2087C2212yes—
66#2087C2212yes—
67#2087C2212yes—
68#2083C2212yes—
69#2083C2212yes—
70#2072C2211yescharacteristic
71#2072C2211yescharacteristic
72#2072C2212yes—
73#2068C2212yes—
74#2068C2212yes—
75#2068C2212yes—
76#2068C2212yes—
77#2068C2212yes—
78#2043C2211yescharacteristic
79#2043C2211yescharacteristic
80#2024C2211yescharacteristic
81#2024C2211yescharacteristic
82#2024C2212yes—
83#2024C2212yes—
84#1990C2211yescharacteristic
85#1990C2211yescharacteristic
86#1990C2212yes—
87#1987C2211yescharacteristic
88#1987C2211yescharacteristic
89#1987C2212yes—
90#1987C2212yes—
91#1981C2212yes—
92#1947C2211yescharacteristic
93#1947C2212yes—
94#1947C2211yescharacteristic
95#1947C2212yes—
96#1947C2212yes—
97#1947C2212yes—
98#1942C2212yes—
99#1939C2211yescharacteristic
100#1939C2211yescharacteristic
101#1939C2212yes—
102#1939C2212yes—
103#1933C2212yes—
104#1933C2212yes—
105#1933C2212yes—
106#1928C2212yes—
107#1928C2212yes—
108#1928C2212yes—
109#1921C2212yes—
110#1921C2212yes—
111#1921C2212yes—
112#1921C2212yes—
113#1921C2212yes—
114#1921C2212yes—
115#1921C2212yes—
116#1916C2212yes—
117#1916C2212yes—
118#1863C2212yes—
119#1863C2212yes—
120#1842C2211yescharacteristic
121#1834C2211yescharacteristic
122#1834C2212yes—
123#1803C2211yescharacteristic
124#1803C2212yes—
125#1803C2211yescharacteristic
126C4×#107C2212yes—
127C8×C4⋊C4C2211yescharacteristic
128C8×C4⋊C4C2211yescharacteristic
129C4×#103C2211yescharacteristic
130C4×#103C2211yescharacteristic
131C4×#103C2211yescharacteristic
132C4×#103C2211yescharacteristic
133C4×#103C2212yes—
134#1744C2211yescharacteristic
135C2×C4⋊Q8—22no—
136C2×C4⋊Q8—21no—
137C2×C4⋊Q8C2311yescharacteristic
138C2×C4⋊Q8C2311yescharacteristic
139C2×C4⋊Q8C2311yescharacteristic
140C2×C4⋊Q8C2312yes—
141C2×C4⋊Q8C2311yescharacteristic
142C2×#125—21no—
143C2×#125—22no—
144C2×#125D411yescharacteristic
145C2×C4×Q8—21no—
146C2×C4×Q8—21no—
147C2×C4×Q8—21no—
148C2×C4×Q8C2311yescharacteristic
149C2×C4×Q8C2311yescharacteristic
150C2×C4×Q8C2312yes—
151#500—24no—
152#477—24no—
153#477—24no—
154C2×#107C2312yes—
155C2×#107C2312yes—
156C2×#107C2312yes—
157C2×#107C2312yes—
158C2×#107C2312yes—
159C2×#107C2312yes—
160C2×C4⋊C8C2311yescharacteristic
161C2×C4⋊C8C2311yescharacteristic
162C2×C4⋊C8C2312yes—
163C2×C4⋊C8C2312yes—
164C2×C4⋊C8C2312yes—
165C2×C4⋊C8C2312yes—
166C2×#103C2312yes—
167C2×#103C2312yes—
168C2×#103C2311yescharacteristic
169C2×#103C2311yescharacteristic
170C2×#103C2311yescharacteristic
171C2×#103C2311yescharacteristic
172C2×#103C2311yescharacteristic
173C2×#103C2312yes—
174C2×#103C2312yes—
175C2×#103C2311yescharacteristic
176C2×#103C2311yescharacteristic
177C2×#103C2311yescharacteristic
178C2×C4×C8C2311yescharacteristic
179C2×C4×C8C2311yescharacteristic
180C2×C4×C8C2312yes—
181C2×C4×C8C2312yes—
182#400—22no—
183#400—22no—
184#400—22no—
185#398—22no—
186#398—22no—
187#398—22no—
188#398—22no—
189#398—22no—
190#398—22no—
191#398—22no—
192#398—22no—
193#395—22no—
194#395—22no—
195#395—22no—
196#395—22no—
197#395—22no—
198#391—21no—
199#391—21no—
200#391—21no—
201#391—22no—
202#391—22no—
203#391—22no—
204#391—22no—
205#391C2311yescharacteristic
206#391C2311yescharacteristic
207#391C2312yes—
208#391C2312yes—
209#391C2312yes—
210#389—21no—
211#389—21no—
212#389—21no—
213#389—21no—
214#389—21no—
215#389—22no—
216#389—21no—
217#389—21no—
218#389—22no—
219#389—21no—
220#389—22no—
221#389—21no—
222#389—21no—
223#389—21no—
224#389—21no—
225#389—21no—
226#389—22no—
227#389—22no—
228#389—22no—
229#389D411yescharacteristic
230#389D411yescharacteristic
231#389C2312yes—
232#389C2311yescharacteristic
233#389C2311yescharacteristic
234#389C2312yes—
235#389C2312yes—
236#389C2312yes—
237#389C2312yes—
238#389C2312yes—
239#389C2312yes—
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#387D412yes—
250#387D412yes—
251#384—21no—
252#384C2311yescharacteristic
253#384C2311yescharacteristic
254#384C2311yescharacteristic
255#384C2311yescharacteristic
256#384C2312yes—
257#384C2312yes—
258#384C2312yes—
259#382—21no—
260#382—22no—
261#382D411yescharacteristic
262C4×C4⋊C4—21no—
263C4×C4⋊C4—21no—
264C4×C4⋊C4—22no—
265C4×C4⋊C4—21no—
266C4×C4⋊C4—22no—
267C4×C4⋊C4—21no—
268C4×C4⋊C4—21no—
269C4×C4⋊C4—21no—
270C4×C4⋊C4—21no—
271C4×C4⋊C4—21no—
272C4×C4⋊C4—21no—
273C4×C4⋊C4C2312yes—
274C4×C4⋊C4C2312yes—
275C4×C4⋊C4C2311yescharacteristic
276C4×C4⋊C4C2311yescharacteristic
277C4×C4⋊C4C2312yes—
278C4×C4⋊C4C2312yes—
279C4×C4⋊C4C2312yes—
280C4×C4⋊C4C2311yescharacteristic
281C4×C4⋊C4C2311yescharacteristic
282C4×C4⋊C4C2312yes—
283C4×C4⋊C4C2312yes—
284C43C2311yescharacteristic
285#340C2312yes—
286#340C2312yes—
287#340C2312yes—
288#340C2312yes—
289#340C2312yes—
290#340C2312yes—
291#340C2311yescharacteristic
292#340C2311yescharacteristic
293#340C2311yescharacteristic
294#340C2311yescharacteristic
295#340C2312yes—
296#340C2311yescharacteristic
297#340C2312yes—
298#340C2312yes—
299#340C2312yes—
300#340C2311yescharacteristic
301#340C2312yes—
302#340C2312yes—
303#340C2311yescharacteristic
304#340C2311yescharacteristic
305#340C2312yes—
306#340C2312yes—
307#336C2312yes—
308#336C2311yescharacteristic
309#336C2311yescharacteristic
310#336C2312yes—
311#336C2312yes—
312#336C2311yescharacteristic
313#336C2312yes—
314#336C2311yescharacteristic
315C22×Q8—21no—
316C22×Q8—21no—
317C22×Q8—21no—
318C22×Q8—22no—
319C22×Q8C2411yescharacteristic
320C22×Q8C2411yescharacteristic
321C22×C8C2411yescharacteristic
322C22×C8C2411yescharacteristic
323C22×C8C2411yescharacteristic
324C22×C8C2411yescharacteristic
325C22×C8C2412yes—
326C22×C8C2412yes—
327C4⋊Q8—22no—
328C4⋊Q8—24no—
329C4⋊Q8—22no—
330C4⋊Q8—22no—
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333C4⋊Q8—42no—
334C4⋊Q8—41no—
335C4⋊Q8—41no—
336C4⋊Q8—82no—
337C4⋊Q8—84no—
338C4⋊Q8#4114yes—
339#125—82no—
340#125—84no—
341#125Q1618yes—
342C4×Q8—24no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—42no—
348C4×Q8—82no—
349C4×Q8—82no—
350C4×Q8—82no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
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—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
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—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4C2×D411yescharacteristic
384C2×C4⋊C4C2×D411yescharacteristic
385C2×C4⋊C4C2×D412yes—
386C2×C4⋊C4C2×D412yes—
387C2×C4⋊C4C2×D411yescharacteristic
388C2×C4⋊C4C2×D411yescharacteristic
389C2×C4⋊C4C2×D412yes—
390C2×C4⋊C4C2×D412yes—
391C2×C4⋊C4C2411yescharacteristic
392C2×C4⋊C4C2412yes—
393C2×C4⋊C4C2411yescharacteristic
394C2×C4⋊C4C2411yescharacteristic
395C2×C4⋊C4C2412yes—
396C2×C4⋊C4C2412yes—
397C2×C4⋊C4C2411yescharacteristic
398C2×C4⋊C4C2411yescharacteristic
399C2×C4⋊C4C2412yes—
400C2×C4⋊C4C2411yescharacteristic
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—22no—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42—22no—
409C2×C42—22no—
410C2×C42—22no—
411C2×C42C2×D411yescharacteristic
412C2×C42C2×D411yescharacteristic
413C2×C42C2×D411yescharacteristic
414C2×C42C2×D411yescharacteristic
415C2×C42C2412yes—
416C2×C42C2411yescharacteristic
417C2×C42C2411yescharacteristic
418C2×C42C2412yes—
419C2×C42C2411yescharacteristic
420#107—24no—
421#107—24no—
422#107—24no—
423#107—24no—
424#107—42no—
425#107—42no—
426C4⋊C8—22no—
427C4⋊C8—22no—
428C4⋊C8—42no—
429C4⋊C8—42no—
430C4⋊C8—42no—
431C4⋊C8—42no—
432#103—22no—
433#103—22no—
434#103—24no—
435#103—24no—
436#103—22no—
437#103—22no—
438#103—41no—
439#103—42no—
440#103—41no—
441#103—42no—
442#103—41no—
443#103—41no—
444C4×C8—22no—
445C4×C8—22no—
446C4×C8—42no—
447C4×C8—42no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—22no—
453#95—21no—
454#95—22no—
455#95—22no—
456#95—22no—
457#95—22no—
458#95—22no—
459#95—22no—
460#95—22no—
461#95—21no—
462#95—22no—
463#95—22no—
464#95—22no—
465#95—22no—
466#95—21no—
467#95—22no—
468#95—21no—
469#95—22no—
470#95—22no—
471#95—22no—
472#95—22no—
473#95—22no—
474#95C2×D412yes—
475#95C2×D412yes—
476#95C2×D412yes—
477#95C2×D412yes—
478#95C2×D412yes—
479#95C2×D412yes—
480C2×Q8—24no—
481C2×Q8—21no—
482C2×Q8—21no—
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—42no—
493C2×Q8—44no—
494C2×Q8—42no—
495C2×Q8—42no—
496C2×Q8—81no—
497C2×Q8—81no—
498C2×Q8—81no—
499C2×Q8—81no—
500C2×Q8—81no—
501C2×Q8—82no—
502C2×Q8—82no—
503C2×Q82−1+412yes—
504C22×C4—21no—
505C22×C4—22no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—22no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—22no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—22no—
515C22×C4—21no—
516C22×C4C22×D412yes—
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D412yes—
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D412yes—
524C22×C4C22×D412yes—
525C22×C4C22×D411yescharacteristic
526C22×C4C2511yescharacteristic
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—22no—
544C2×C8—22no—
545C4⋊C4—24no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—22no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—21no—
560C4⋊C4—22no—
561C4⋊C4—21no—
562C4⋊C4—22no—
563C4⋊C4—21no—
564C4⋊C4—21no—
565C4⋊C4—22no—
566C4⋊C4—42no—
567C4⋊C4—42no—
568C4⋊C4—44no—
569C4⋊C4—42no—
570C4⋊C4—44no—
571C4⋊C4—42no—
572C4⋊C4—42no—
573C4⋊C4—44no—
574C4⋊C4—44no—
575C4⋊C4—42no—
576C4⋊C4—42no—
577C4⋊C4—42no—
578C4⋊C4—42no—
579C4⋊C4—42no—
580C4⋊C4—42no—
581C4⋊C4—81no—
582C4⋊C4—81no—
583C4⋊C4—82no—
584C4⋊C4—82no—
585C4⋊C4—82no—
586C4⋊C4—81no—
587C4⋊C4—81no—
588C4⋊C4—81no—
589C4⋊C4—81no—
590C4⋊C4—81no—
591C4⋊C4—82no—
592C4⋊C4—82no—
593C4⋊C4—82no—
594C4⋊C4—82no—
595C4⋊C4—82no—
596C4⋊C4—82no—
597C4⋊C4—82no—
598C4⋊C4—82no—
599C4⋊C4C2×Q1618yes—
600C4⋊C4C2×Q1618yes—
601C4⋊C4C2×Q1614yes—
602C4⋊C4C2×Q1614yes—
603C4⋊C4C2×#4114yes—
604C4⋊C42−1+412yes—
605C42—22no—
606C42—22no—
607C42—21no—
608C42—21no—
609C42—22no—
610C42—22no—
611C42—22no—
612C42—21no—
613C42—21no—
614C42—41no—
615C42—41no—
616C42—44no—
617C42—42no—
618C42—42no—
619C42—42no—
620C42—42no—
621C42—82no—
622C42—82no—
623C42—82no—
624C42—82no—
625C42—82no—
626C42C2×Q1614yes—
627C42C2×#4112yes—
628C422−1+412yes—
629C23C23×D411yescharacteristic
630Q8—22no—
631Q8—22no—
632Q8—41no—
633Q8—41no—
634Q8—41no—
635Q8—41no—
636Q8—41no—
637Q8—41no—
638Q8—84no—
639Q8—84no—
640Q8—84no—
641Q8—164no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—24no—
657C2×C4—22no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—42no—
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—42no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—42no—
684C2×C4—41no—
685C2×C4—41no—
686C2×C4—41no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—41no—
690C2×C4—42no—
691C2×C4—41no—
692C2×C4—41no—
693C2×C4—41no—
694C2×C4—41no—
695C2×C4—41no—
696C2×C4—41no—
697C2×C4—41no—
698C2×C4—42no—
699C2×C4—42no—
700C2×C4#54812yes—
701C2×C4#54812yes—
702C2×C4C22×Q1612yes—
703C2×C4C22×Q1612yes—
704C2×C4C22×Q1614yes—
705C2×C4C22×Q1614yes—
706C2×C4C22×Q1612yes—
707C2×C4#57712yes—
708C2×C4C22×#4111yescharacteristic
709C2×C4C2×2−1+411yescharacteristic
710C2×C4C2×2−1+411yescharacteristic
711C2×C4C2×2−1+411yescharacteristic
712C2×C4#58511yescharacteristic
713C2×C4#58511yescharacteristic
714C8—41no—
715C8—41no—
716C8—42no—
717C8—42no—
718C8—41no—
719C8—41no—
720C22C2×#54811yescharacteristic
721C22#348911yescharacteristic
722C22#348911yescharacteristic
723C22C23×Q1611yescharacteristic
724C22C2×#57711yescharacteristic
725C22#358511yescharacteristic
726C22#358511yescharacteristic
727C4—22no—
728C4—22no—
729C4—22no—
730C4—24no—
731C4—24no—
732C4—44no—
733C4—41no—
734C4—42no—
735C4—42no—
736C4—41no—
737C4—42no—
738C4—42no—
739C4—81no—
740C4—81no—
741C4—82no—
742C4—81no—
743C4—82no—
744C4—82no—
745C4—81no—
746C4—81no—
747C4—81no—
748C4#336314yes—
749C4#352311yescharacteristic
750C4#352411yescharacteristic
751C4#355511yescharacteristic
752C4#356311yescharacteristic
753C2#4530011yescharacteristic
754C2C2×#336311yescharacteristic
755C2#6133211yescharacteristic
756C2#6210011yescharacteristic
757C2#6210011yescharacteristic
758C2#6218811yescharacteristic
759C2#6219411yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#125 ◁ 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
#596682
#594452
#324782
#323162
#322322
#319682
#292902
#282842
#278152
#277292
#273642
#266942
#265122
#261122
#260262
#254552
#238162
#230632
#229652
#222992
#222482