Finite Group Databasebeta

← #6555340

Subgroups — #6555340

Order 512.

1753 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic, complemented
2#34316C211yesmaximal, characteristic
3#29980C211yesmaximal, characteristic, complemented
4#29955C212yesmaximal, complemented
5#29255C211yesmaximal, characteristic, complemented
6#29165C212yesmaximal, complemented
7#28512C211yesmaximal, characteristic, complemented
8#27680C212yesmaximal, complemented
9#26532C212yesmaximal, complemented
10#26353C214yesmaximal, complemented
11#26090C212yesmaximal, complemented
12#25967C214yesmaximal, complemented
13#23774C211yesmaximal, characteristic, complemented
14#23734C212yesmaximal, complemented
15#22700C211yesmaximal, characteristic, complemented
16#22663C212yesmaximal, complemented
17#22431C212yesmaximal
18C2×#554—21no—
19C2×#547C2211yescharacteristic
20C2×#475C2214yes—
21C2×#436C2212yes—
22C2×#428C2214yes—
23C2×#417C2214yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2567—21no—
27#2552—21no—
28#2552C2212yes—
29#2502—22no—
30#2440—22no—
31#2440C2214yes—
32#2374—22no—
33#2365C2211yescharacteristic
34#2363—21no—
35#2362C2212yes—
36#2354C2211yescharacteristic
37#2354C2211yescharacteristic
38D4×C4⋊C4—21no—
39D4×C4⋊C4C2212yes—
40#2345—21no—
41#2341C2212yes—
42#2335—22no—
43#2324—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C22⋊Q8C2212yes—
47#2297C2212yes—
48#2293C2211yescharacteristic
49C2×#384C2212yes—
50#2274—21no—
51C42×D4C2211yescharacteristic
52#2099C2212yes—
53#2099C2211yescharacteristic
54#2096C2212yes—
55#2096C2212yes—
56#2096C2212yes—
57#2095C2214yes—
58#2094C2214yes—
59#2093C2212yes—
60#2093C2211yescharacteristic
61#2086C2212yes—
62#2068C2214yes—
63#2067C2212yes—
64#2065C2214yes—
65#2057C2212yes—
66#2057C2211yescharacteristic
67#2024C2212yes—
68#2023C2211yescharacteristic
69#2023C2212yes—
70#1981C2212yes—
71#1956C2212yes—
72#1948C2214yes—
73#1948C2212yes—
74#1947C2212yes—
75#1939C2212yes—
76#1920C2214yes—
77#1894C2214yes—
78#1872C2214yes—
79#1865C2214yes—
80#1863C2214yes—
81#1854C2214yes—
82#1853C2214yes—
83#1851C2214yes—
84#1844C2212yes—
85#1831C2211yescharacteristic
86#1809C2214yes—
87#1803C2214yes—
88#1802C2212yes—
89#1797C2212yes—
90#1788C2214yes—
91C4×#107C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94#1728C2212yes—
95C22×#41C2312yes—
96C22×M16C2314yes—
97#554—48no—
98#547—24no—
99#547Q818yescomplemented
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#125—22no—
106C2×#123—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8C2312yes—
110C2×#121—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D411yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×#117—21no—
120C2×#117—21no—
121C2×#117C2312yes—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4C2314yes—
126C22×C42C2312yes—
127#475—48no—
128#436—28no—
129#428—48no—
130C2×#107C2312yes—
131C2×#107C2312yes—
132C2×#107C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139#417—48no—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×C22⋊C8C2314yes—
148C2×C4×C8C2312yes—
149#398—24no—
150#398—24no—
151#395—24no—
152#391—21no—
153#391—22no—
154#391C2312yes—
155#391C2312yes—
156#390—21no—
157#390—22no—
158#390—22no—
159#390C2312yes—
160#390C2311yescharacteristic
161#390C2311yescharacteristic
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389C2312yes—
170#389C2311yescharacteristic
171#389C2311yescharacteristic
172#389C2314yes—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—21no—
177#388—21no—
178#388—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#386—22no—
183#386—22no—
184#385—22no—
185#385—24no—
186#385—24no—
187#384—22no—
188#384—21no—
189#384—21no—
190#384—22no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2314yes—
195#384C2312yes—
196#380—22no—
197#380—21no—
198#380—22no—
199#380C2312yes—
200#380C2314yes—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4C2312yes—
222C43C2311yescharacteristic
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2314yes—
235C2×#41—24no—
236C2×#41—44no—
237C2×#41—44no—
238C2×#41C2×Q818yes—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8C2412yes—
242C22×D4—22no—
243C22×D4C2×D411yescharacteristic
244C22×D4C2411yescharacteristic
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—24no—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2412yes—
251C2×M16—28no—
252C2×M16—28no—
253C2×M16—44no—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—22no—
258C4⋊Q8—42no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261C4⋊Q8—82no—
262C4⋊Q8C2×Q814yes—
263#125—48no—
264#123—216no—
265C22⋊Q8—24no—
266C22⋊Q8—416no—
267C22⋊Q8—48no—
268C22⋊Q8C2×Q818yes—
269#121—28no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—22no—
276C4×D4—48no—
277C4×D4C2×Q814yes—
278C4×D4C2×Q814yes—
279C4×D4#4114yes—
280#117—44no—
281#117—44no—
282#117—44no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—48no—
309C2×C4⋊C4—84no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C42—22no—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—44no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—24no—
352#107—42no—
353#107—44no—
354#106—28no—
355#106—44no—
356#106—44no—
357C4⋊C8—24no—
358C4⋊C8—24no—
359C4⋊C8—44no—
360#103—28no—
361#103—44no—
362#103—44no—
363#103—44no—
364#102—24no—
365#102—42no—
366#102—44no—
367C22⋊C8—28no—
368C4×C8—24no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—24no—
381#95—24no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—21no—
386C24C22×D412yes—
387#41—416no—
388#41—88no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—82no—
397C2×Q8—82no—
398C2×Q8C22×Q814yes—
399C2×D4—28no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—82no—
406C2×D4—82no—
407C2×D4C22⋊Q814yes—
408C2×D4C22×Q812yes—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—28no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—48no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×Q814yes—
444C22×C4C2×#4114yes—
445C22×C4C2511yescharacteristic
446M16—416no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8H(5,𝔽2)14yes—
456C4⋊C4—24no—
457C4⋊C4—28no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—81no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4C22⋊Q814yes—
490C4⋊C4#13714yes—
491C4⋊C4C22×Q812yes—
492C22⋊C4—28no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—48no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—44no—
500C22⋊C4—42no—
501C22⋊C4—81no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4C22⋊Q818yes—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—24no—
510C42—22no—
511C42—42no—
512C42—42no—
513C42—41no—
514C42—42no—
515C42—44no—
516C42—44no—
517C42—82no—
518C42—82no—
519C42—82no—
520C42AGL(1,ℤ/8)14yes—
521C42C22×Q812yes—
522C42C2×#4112yes—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23C2×C22⋊Q814yes—
537C23C23×D411yescharacteristic
538Q8—28no—
539Q8—44no—
540Q8—84no—
541Q8—164no—
542D4—28no—
543D4—24no—
544D4—42no—
545D4—88no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—48no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—88no—
594C2×C4—168no—
595C2×C4C2×C22⋊Q812yes—
596C2×C4C2×C22⋊Q814yes—
597C2×C4C2×C22⋊Q812yes—
598C2×C4C2×AGL(1,ℤ/8)12yes—
599C2×C4C2×AGL(1,ℤ/8)12yes—
600C2×C4C2×AGL(1,ℤ/8)12yes—
601C2×C4C2×#13714yes—
602C2×C4C2×#13712yes—
603C2×C4#57712yes—
604C2×C4C23×Q811yescharacteristic
605C2×C4C22×#4111yescharacteristic
606C2×C4C2×H(5,𝔽2)12yes—
607C2×C4#58512yes—
608C8—24no—
609C8—42no—
610C8—44no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22C22×C22⋊Q811yescharacteristic
626C22#348511yescharacteristic
627C22#349811yescharacteristic
628C22C22×AGL(1,ℤ/8)11yescharacteristic
629C22C22×#13711yescharacteristic
630C22C2×#57711yescharacteristic
631C22#358611yescharacteristic
632C4—22no—
633C4—22no—
634C4—24no—
635C4—22no—
636C4—22no—
637C4—24no—
638C4—42no—
639C4—42no—
640C4—41no—
641C4—44no—
642C4—81no—
643C4—82no—
644C4—81no—
645C4—82no—
646C4—84no—
647C4—82no—
648C4#307914yes—
649C4#350112yes—
650C4#351312yes—
651C2—44no—
652C2—82no—
653C2#3583611yescharacteristic
654C2C2×#307911yescharacteristic
655C2#6049711yescharacteristic
656C2#6171711yescharacteristic
657C2#6173511yescharacteristic
658C2#6242111yescharacteristic
659C2#6245411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #36180C2semidirect
3. #29980C2semidirect
4. #29955C2semidirect
5. #29255C2semidirect
6. #29165C2semidirect
7. #28512C2semidirect
8. #27680C2semidirect
9. #26532C2semidirect
10. #26353C2semidirect
11. #26090C2semidirect
12. #25967C2semidirect
13. #23774C2semidirect
14. #23734C2semidirect
15. #22700C2semidirect
16. #22663C2semidirect
99. #547Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#361802
#343162
#299802
#299552
#292552
#291652
#285122
#276802
#265322
#263532
#260902
#259672
#237742
#237342
#227002
#226632
#224312