Finite Group Databasebeta

← #6856598

Subgroups — #6856598

Order 512.

1215 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39753C211yesmaximal, characteristic
2#35208C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic
4#32340C212yesmaximal
5#32310C212yesmaximal
6#31476C212yesmaximal
7#31464C212yesmaximal
8#29875C212yesmaximal
9#29289C211yesmaximal, characteristic
10#29282C212yesmaximal
11#29228C212yesmaximal
12#29219C211yesmaximal, characteristic
13#28798C211yesmaximal, characteristic
14#28792C211yesmaximal, characteristic
15#27738C211yesmaximal, characteristic
16#27729C211yesmaximal, characteristic
17#26282C211yesmaximal, characteristic
18#26279C211yesmaximal, characteristic
19#24995C211yesmaximal, characteristic
20#24994C211yesmaximal, characteristic
21#22822C211yesmaximal, characteristic
22#22125C212yesmaximal
23C4×#479C211yesmaximal, characteristic
24C2×#479C2211yescharacteristic
25C2×#477C2212yes—
26#2868C2211yescharacteristic
27#2863—21no—
28#2859—21no—
29#2791—22no—
30#2713—22no—
31#2666—22no—
32#2660C2212yes—
33#2582C2211yescharacteristic
34#2571C2211yescharacteristic
35#2570C2212yes—
36#2567C2211yescharacteristic
37#2567C2211yescharacteristic
38#2565—21no—
39#2552C2212yes—
40#2552C2211yescharacteristic
41#2552C2211yescharacteristic
42#2550—21no—
43#2511—21no—
44#2507—21no—
45#2506—21no—
46#2506C2212yes—
47#2462—22no—
48#2453C2212yes—
49#2451—21no—
50#2355C2211yescharacteristic
51Q8×C4⋊C4C2211yescharacteristic
52#2331C2212yes—
53#2330C2212yes—
54#2320C2212yes—
55C4×C4⋊Q8C2211yescharacteristic
56C4×#125C2211yescharacteristic
57#2295C2211yescharacteristic
58C42×Q8C2211yescharacteristic
59#2101C2212yes—
60#2101C2212yes—
61#2100C2212yes—
62#2100C2212yes—
63#2099C2212yes—
64#2099C2212yes—
65#2099C2212yes—
66#2099C2212yes—
67#2088C2212yes—
68#2086C2212yes—
69#2085C2212yes—
70#2085C2212yes—
71#2083C2212yes—
72#2083C2212yes—
73#2074C2212yes—
74#2074C2212yes—
75#2072C2212yes—
76#2072C2212yes—
77#2059C2212yes—
78#2059C2212yes—
79#1992C2212yes—
80#1991C2212yes—
81#1990C2211yescharacteristic
82#1990C2211yescharacteristic
83#1989C2211yescharacteristic
84#1989C2212yes—
85#1988C2212yes—
86#1988C2212yes—
87#1987C2212yes—
88#1987C2211yescharacteristic
89#1987C2211yescharacteristic
90#1984C2211yescharacteristic
91#1984C2211yescharacteristic
92#1949C2211yescharacteristic
93#1949C2212yes—
94#1948C2211yescharacteristic
95#1947C2212yes—
96#1947C2211yescharacteristic
97#1947C2211yescharacteristic
98#1947C2211yescharacteristic
99#1943C2212yes—
100#1942C2212yes—
101#1935C2212yes—
102#1933C2212yes—
103#1928C2212yes—
104#1927C2212yes—
105#1921C2212yes—
106#1921C2212yes—
107#1920C2211yescharacteristic
108#1920C2212yes—
109#1919C2212yes—
110#1918C2211yescharacteristic
111#1918C2212yes—
112#1918C2212yes—
113#1917C2212yes—
114#1916C2212yes—
115#1865C2211yescharacteristic
116#1863C2211yescharacteristic
117#1842C2211yescharacteristic
118#1836C2212yes—
119#1834C2211yescharacteristic
120#1834C2211yescharacteristic
121#1834C2212yes—
122#1832C2212yes—
123#1805C2211yescharacteristic
124#1803C2211yescharacteristic
125#1803C2211yescharacteristic
126#1803C2212yes—
127C4×#107C2211yescharacteristic
128C4×#106C2211yescharacteristic
129#1770C2211yescharacteristic
130C8×C4⋊C4C2211yescharacteristic
131#1768C2212yes—
132C4×C4⋊C8C2211yescharacteristic
133C4×#103C2211yescharacteristic
134C4×#103C2211yescharacteristic
135C2×C4⋊Q8—21no—
136C2×C4⋊Q8—21no—
137C2×C4⋊Q8C2312yes—
138C2×C4⋊Q8C2311yescharacteristic
139C2×#125—21no—
140C2×#125—22no—
141C2×#125—22no—
142C2×#125C2311yescharacteristic
143C2×#125C2312yes—
144C2×C4×Q8—21no—
145C2×C4×Q8—21no—
146C2×C4×Q8C2311yescharacteristic
147C2×C4×Q8C2311yescharacteristic
148C2×C4×Q8C2311yescharacteristic
149C2×C4×Q8C2311yescharacteristic
150#479—24no—
151#477—28no—
152C2×#107C2312yes—
153C2×#107C2312yes—
154C2×#107C2311yescharacteristic
155C2×#107C2311yescharacteristic
156C2×#107C2312yes—
157C2×#106C2311yescharacteristic
158C2×#106C2311yescharacteristic
159C2×#106C2312yes—
160C2×C4⋊C8C2312yes—
161C2×C4⋊C8C2311yescharacteristic
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2311yescharacteristic
164C2×C4⋊C8C2311yescharacteristic
165C2×C4⋊C8C2312yes—
166C2×#103C2312yes—
167C2×#103C2312yes—
168C2×#103C2311yescharacteristic
169C2×#103C2311yescharacteristic
170C2×#103C2311yescharacteristic
171C2×#103C2311yescharacteristic
172C2×#97C2312yes—
173C2×C4×C8C2311yescharacteristic
174C2×C4×C8C2311yescharacteristic
175C2×C4×C8C2312yes—
176#400—22no—
177#400—22no—
178#400—22no—
179#400—22no—
180#398—22no—
181#398—22no—
182#398—22no—
183#398—22no—
184#398—22no—
185#398—22no—
186#398—22no—
187#398—22no—
188#398—22no—
189#398—22no—
190#395—22no—
191#395—22no—
192#395—22no—
193#395—22no—
194#391—21no—
195#391—21no—
196#391—21no—
197#391—21no—
198#391—22no—
199#391—22no—
200#391D411yescharacteristic
201#391D411yescharacteristic
202#391C2312yes—
203#391C2312yes—
204#389—21no—
205#389—21no—
206#389—21no—
207#389—21no—
208#389—22no—
209#389—22no—
210#389—21no—
211#389—21no—
212#389—21no—
213#389—22no—
214#389—21no—
215#389—22no—
216#389—22no—
217#389D411yescharacteristic
218#389D411yescharacteristic
219#389C2311yescharacteristic
220#389C2311yescharacteristic
221#389C2312yes—
222#389C2312yes—
223#389C2312yes—
224#389C2312yes—
225#389C2312yes—
226#389C2312yes—
227#389C2312yes—
228#387—22no—
229#387—21no—
230#387—21no—
231#387—22no—
232#387—22no—
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#387D412yes—
242#387D412yes—
243#384—21no—
244#384—21no—
245#384—21no—
246#384—21no—
247#384C2311yescharacteristic
248#384C2312yes—
249#384C2312yes—
250#384C2311yescharacteristic
251#384C2311yescharacteristic
252#384C2312yes—
253#384C2312yes—
254#383—22no—
255#382—21no—
256#382—22no—
257#382C2311yescharacteristic
258#382C2312yes—
259#382C2312yes—
260C4×C4⋊C4—21no—
261C4×C4⋊C4—21no—
262C4×C4⋊C4—21no—
263C4×C4⋊C4—21no—
264C4×C4⋊C4—21no—
265C4×C4⋊C4—21no—
266C4×C4⋊C4—21no—
267C4×C4⋊C4C2311yescharacteristic
268C4×C4⋊C4C2311yescharacteristic
269C4×C4⋊C4C2312yes—
270C4×C4⋊C4C2311yescharacteristic
271C4×C4⋊C4C2312yes—
272C4×C4⋊C4C2311yescharacteristic
273C4×C4⋊C4C2312yes—
274C4×C4⋊C4C2311yescharacteristic
275C4×C4⋊C4C2311yescharacteristic
276C4×C4⋊C4C2312yes—
277#376C2312yes—
278C43—21no—
279C43C2311yescharacteristic
280#340C2312yes—
281#340C2312yes—
282#340C2312yes—
283#340C2312yes—
284#340C2312yes—
285#340C2312yes—
286#340C2311yescharacteristic
287#340C2311yescharacteristic
288#340C2312yes—
289#340C2312yes—
290#340C2312yes—
291#340C2312yes—
292#340C2312yes—
293#340C2312yes—
294#340C2312yes—
295#340C2312yes—
296#340C2311yescharacteristic
297#340C2312yes—
298#340C2312yes—
299#340C2312yes—
300#340C2311yescharacteristic
301#340C2312yes—
302#340C2311yescharacteristic
303#340C2311yescharacteristic
304#340C2311yescharacteristic
305#340C2311yescharacteristic
306#336C2312yes—
307#336C2312yes—
308#336C2312yes—
309#336C2312yes—
310#336C2311yescharacteristic
311#336C2311yescharacteristic
312#336C2312yes—
313#336C2312yes—
314C22×Q8—21no—
315C22×Q8—22no—
316C22×Q8C2×D411yescharacteristic
317C22×Q8C2411yescharacteristic
318C22×C8C2411yescharacteristic
319C22×C8C2411yescharacteristic
320C22×C8C2411yescharacteristic
321C22×C8C2411yescharacteristic
322C22×C8C2412yes—
323C22×C8C2412yes—
324C4⋊Q8—24no—
325C4⋊Q8—22no—
326C4⋊Q8—22no—
327C4⋊Q8—24no—
328C4⋊Q8—82no—
329C4⋊Q8—82no—
330#125—24no—
331#125—22no—
332#125—22no—
333#125—42no—
334#125—84no—
335#125—82no—
336#125—84no—
337C4×Q8—22no—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×Q8—41no—
344C4×Q8—82no—
345C4×Q8—82no—
346C4×Q8#4114yes—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4C2×D411yescharacteristic
380C2×C4⋊C4C2×D412yes—
381C2×C4⋊C4C2×D412yes—
382C2×C4⋊C4C2×D411yescharacteristic
383C2×C4⋊C4C2×D411yescharacteristic
384C2×C4⋊C4C2×D412yes—
385C2×C4⋊C4C2411yescharacteristic
386C2×C4⋊C4C2411yescharacteristic
387C2×C4⋊C4C2411yescharacteristic
388C2×C4⋊C4C2412yes—
389C2×C4⋊C4C2412yes—
390C2×C4⋊C4C2412yes—
391C2×C4⋊C4C2412yes—
392C2×C4⋊C4C2411yescharacteristic
393C2×C4⋊C4C2411yescharacteristic
394C2×C4⋊C4C2411yescharacteristic
395C2×C4⋊C4C2411yescharacteristic
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—21no—
399C2×C42—21no—
400C2×C42—21no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—22no—
404C2×C42—22no—
405C2×C42—22no—
406C2×C42—21no—
407C2×C42C2×D411yescharacteristic
408C2×C42C2×D411yescharacteristic
409C2×C42C2×D411yescharacteristic
410C2×C42C2×D411yescharacteristic
411C2×C42C2412yes—
412C2×C42C2411yescharacteristic
413C2×C42C2412yes—
414C2×C42C2411yescharacteristic
415C2×C42C2411yescharacteristic
416#107—24no—
417#107—24no—
418#107—22no—
419#107—22no—
420#107—42no—
421#106—22no—
422#106—22no—
423#106—42no—
424C4⋊C8—24no—
425C4⋊C8—22no—
426C4⋊C8—22no—
427C4⋊C8—22no—
428C4⋊C8—22no—
429C4⋊C8—42no—
430#103—24no—
431#103—24no—
432#103—22no—
433#103—22no—
434#103—41no—
435#103—41no—
436#97—42no—
437C4×C8—22no—
438C4×C8—22no—
439C4×C8—42no—
440#95—21no—
441#95—21no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—22no—
446#95—21no—
447#95—21no—
448#95—22no—
449#95—22no—
450#95—21no—
451#95—21no—
452#95—22no—
453#95—22no—
454#95—22no—
455#95—22no—
456#95—22no—
457#95—22no—
458#95—22no—
459#95—22no—
460#95—22no—
461#95—22no—
462#95—22no—
463#95—22no—
464#95—22no—
465#95—22no—
466#95—22no—
467#95C2×D412yes—
468#95C2×D412yes—
469#95C2×D412yes—
470#95C2×D412yes—
471#95C2×D412yes—
472#95C2×D412yes—
473#95C2×D412yes—
474C2×Q8—22no—
475C2×Q8—22no—
476C2×Q8—22no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—42no—
483C2×Q8—44no—
484C2×Q8—81no—
485C2×Q8—81no—
486C2×Q8—82no—
487C2×Q8—82no—
488C2×Q8C2×#4112yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—21no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—21no—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C2511yescharacteristic
512C2×C8—21no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—22no—
528C2×C8—21no—
529C2×C8—22no—
530C4⋊C4—22no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—22no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—42no—
561C4⋊C4—42no—
562C4⋊C4—42no—
563C4⋊C4—42no—
564C4⋊C4—42no—
565C4⋊C4—44no—
566C4⋊C4—44no—
567C4⋊C4—42no—
568C4⋊C4—44no—
569C4⋊C4—42no—
570C4⋊C4—42no—
571C4⋊C4—42no—
572C4⋊C4—82no—
573C4⋊C4—82no—
574C4⋊C4—81no—
575C4⋊C4—82no—
576C4⋊C4—82no—
577C4⋊C4—82no—
578C4⋊C4—82no—
579C4⋊C4—82no—
580C4⋊C4—82no—
581C4⋊C4—82no—
582C4⋊C4—82no—
583C4⋊C4—82no—
584C4⋊C4—82no—
585C4⋊C4—81no—
586C4⋊C4—81no—
587C4⋊C4—82no—
588C4⋊C4C2×#4112yes—
589C4⋊C42−1+412yes—
590C42—24no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—21no—
595C42—22no—
596C42—22no—
597C42—22no—
598C42—22no—
599C42—21no—
600C42—41no—
601C42—42no—
602C42—42no—
603C42—42no—
604C42—42no—
605C42—42no—
606C42—42no—
607C42—81no—
608C42—82no—
609C42—82no—
610C42—82no—
611C42C2×#4112yes—
612C422−1+412yes—
613C422−1+412yes—
614C422−1+412yes—
615C23C23×D411yescharacteristic
616Q8—22no—
617Q8—22no—
618Q8—44no—
619Q8—41no—
620Q8—41no—
621Q8—84no—
622Q8—164no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—21no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—21no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—41no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—41no—
674C2×C4—42no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—42no—
682C2×C4#53612yes—
683C2×C4#53612yes—
684C2×C4#54812yes—
685C2×C4#54812yes—
686C2×C4#57612yes—
687C2×C4#57712yes—
688C2×C4#57812yes—
689C2×C4C22×#4111yescharacteristic
690C2×C4C2×2−1+411yescharacteristic
691C2×C4C2×2−1+411yescharacteristic
692C2×C4C2×2−1+411yescharacteristic
693C2×C4C2×2−1+411yescharacteristic
694C2×C4#58511yescharacteristic
695C8—42no—
696C8—42no—
697C8—41no—
698C8—41no—
699C8—41no—
700C8—41no—
701C22C2×#53611yescharacteristic
702C22C2×#54811yescharacteristic
703C22#348911yescharacteristic
704C22C2×#57611yescharacteristic
705C22C2×#57711yescharacteristic
706C22C2×#57811yescharacteristic
707C22#358611yescharacteristic
708C4—22no—
709C4—22no—
710C4—22no—
711C4—22no—
712C4—42no—
713C4—42no—
714C4—42no—
715C4—42no—
716C4—41no—
717C4—42no—
718C4—42no—
719C4—42no—
720C4—81no—
721C4—82no—
722C4—82no—
723C4—81no—
724C4—81no—
725C4—82no—
726C4—81no—
727C4—82no—
728C4—81no—
729C4#324214yes—
730C4#350711yescharacteristic
731C4#350911yescharacteristic
732C4#355511yescharacteristic
733C4#355611yescharacteristic
734C2#3972411yescharacteristic
735C2C2×#324211yescharacteristic
736C2#6088811yescharacteristic
737C2#6132911yescharacteristic
738C2#6134211yescharacteristic
739C2#6218211yescharacteristic
740C2#6220611yescharacteristic
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
#397532
#352082
#331982
#323402
#323102
#314762
#314642
#298752
#292892
#292822
#292282
#292192
#287982
#287922
#277382
#277292
#262822
#262792
#249952
#249942
#228222
#221252
C4×#4792