Finite Group Databasebeta

← #6557589

Subgroups — #6557589

Order 512.

1708 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36559C211yesmaximal, characteristic, complemented
2#36271C211yesmaximal, characteristic, complemented
3#30384C212yesmaximal, complemented
4#30203C211yesmaximal, characteristic, complemented
5#29578C212yesmaximal, complemented
6#29499C212yesmaximal, complemented
7#29095C214yesmaximal, complemented
8#28687C211yesmaximal, characteristic, complemented
9#28369C211yesmaximal, characteristic
10#26490C214yesmaximal, complemented
11#25478C212yesmaximal, complemented
12#25300C212yesmaximal, complemented
13#23815C211yesmaximal, characteristic, complemented
14#23682C212yesmaximal, complemented
15#23001C212yesmaximal, complemented
16#22968C212yesmaximal, complemented
17#20570C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×#554—21no—
20C2×#498C2212yes—
21C2×#474C2212yes—
22C2×#459C2212yes—
23C2×#446C2212yes—
24C2×#442C2214yes—
25#2868C2211yescharacteristic
26#2860C2212yes—
27#2859C2212yes—
28#2858C2211yescharacteristic
29#2601—22no—
30#2592—21no—
31#2566—22no—
32#2563—21no—
33#2560—22no—
34#2548C2211yescharacteristic
35#2546C2211yescharacteristic
36#2545—21no—
37#2538—21no—
38#2538C2212yes—
39#2512C2212yes—
40#2452—22no—
41#2452C2212yes—
42#2452C2214yes—
43#2367—21no—
44#2366C2211yescharacteristic
45#2365C2211yescharacteristic
46D4×C4⋊C4—21no—
47D4×C4⋊C4C2212yes—
48C4×C4⋊Q8C2211yescharacteristic
49C4×#127C2211yescharacteristic
50C4×#127C2212yes—
51#2293C2212yes—
52C42×Q8C2211yescharacteristic
53#2099C2211yescharacteristic
54#2099C2211yescharacteristic
55#2093C2212yes—
56#2087C2214yes—
57#2084C2212yes—
58#2084C2212yes—
59#2071C2212yes—
60#2071C2212yes—
61#2071C2212yes—
62#2065C2214yes—
63#2065C2214yes—
64#2054C2214yes—
65#2041C2211yescharacteristic
66#2041C2211yescharacteristic
67#2023C2212yes—
68#2023C2214yes—
69#2011C2212yes—
70#1990C2212yes—
71#1990C2212yes—
72#1972C2214yes—
73#1970C2214yes—
74#1957C2212yes—
75#1948C2212yes—
76#1948C2212yes—
77#1944C2214yes—
78#1943C2214yes—
79#1936C2214yes—
80#1936C2214yes—
81#1936C2214yes—
82#1921C2214yes—
83#1879C2214yes—
84#1864C2214yes—
85#1864C2214yes—
86#1771C2214yes—
87#1770C2212yes—
88#1767C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90#1762C2214yes—
91C4×#102C2212yes—
92C4×#102C2212yes—
93#1745C2212yes—
94#1744C2211yescharacteristic
95C22×D8C2314yes—
96#556—48no—
97#554—48no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#127C2311yescharacteristic
105C2×#127C2311yescharacteristic
106C2×#126—22no—
107C2×#126—22no—
108C2×#125—22no—
109C2×#124—21no—
110C2×#124—21no—
111C2×#124—21no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—24no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×C4×D4—22no—
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×C4×D4C2314yes—
128C2×#117—22no—
129C22×C4⋊C4—22no—
130#498—44no—
131#474—28no—
132#459—28no—
133#446—28no—
134#442—48no—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2312yes—
140C2×#102C2312yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#97C2314yes—
149C2×#97C2314yes—
150C2×C4×C8C2312yes—
151C2×C4×C8C2312yes—
152#398—22no—
153#397—22no—
154#396—24no—
155#396—24no—
156#396—22no—
157#396—22no—
158#394—22no—
159#393—22no—
160#393—24no—
161#391—21no—
162#391—21no—
163#391—21no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390—22no—
168#390—21no—
169#390—21no—
170#390—22no—
171#390—21no—
172#390C2314yes—
173#390C2312yes—
174#390C2312yes—
175#390C2312yes—
176#390C2312yes—
177#390C2312yes—
178#390C2312yes—
179#389—21no—
180#389—22no—
181#389—22no—
182#389C2311yescharacteristic
183#389C2311yescharacteristic
184#389C2312yes—
185#389C2314yes—
186#389C2312yes—
187#389C2314yes—
188#389C2312yes—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#387—22no—
196#386—22no—
197#385—24no—
198#385—24no—
199#385—22no—
200#384—21no—
201#384—21no—
202#384C2312yes—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#384C2312yes—
206#382—22no—
207#380—22no—
208#380—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D412yes—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D412yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C43C2311yescharacteristic
221C43C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2312yes—
234#336C2314yes—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8C2×D411yescharacteristic
239C22×D4—21no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2412yes—
244C22×D4C2412yes—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—24no—
248C2×D8—28no—
249C2×D8—28no—
250C2×D8—44no—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C4⋊Q8—22no—
255C4⋊Q8—24no—
256C4⋊Q8—41no—
257C4⋊Q8—42no—
258C4⋊Q8—42no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261C4⋊Q8—82no—
262#127—24no—
263#127—24no—
264#126—48no—
265#126—48no—
266#125—48no—
267#124—44no—
268#124—44no—
269#124—44no—
270C22⋊Q8—416no—
271C22⋊Q8—48no—
272C22⋊Q8—48no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—42no—
282C4×D4—48no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—44no—
288C4×D4—84no—
289#117—48no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—48no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—28no—
356#106—28no—
357#106—44no—
358C4⋊C8—28no—
359C4⋊C8#4118yes—
360#102—24no—
361#102—24no—
362#102—28no—
363#102—28no—
364#102—24no—
365#102—42no—
366#102—44no—
367#102—44no—
368#97—28no—
369#97—44no—
370C4×C8—24no—
371C4×C8—42no—
372#95—21no—
373#95—22no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—22no—
386C24—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—81no—
397C2×Q8—82no—
398C2×Q8—82no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—81no—
419C2×D4—81no—
420C2×D42−1+414yes—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—82no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C2511yescharacteristic
448D8—416no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8C2×#4118yes—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—24no—
474C4⋊C4—42no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—81no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4H(5,𝔽2)12yes—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—48no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C42—24no—
511C42—24no—
512C42—24no—
513C42—21no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—21no—
518C42—41no—
519C42—42no—
520C42—42no—
521C42—41no—
522C42—44no—
523C42—42no—
524C42—42no—
525C42AGL(1,ℤ/8)14yes—
526C42AGL(1,ℤ/8)14yes—
527C42C2×#4114yes—
528C23—24no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23C23×D411yescharacteristic
543Q8—44no—
544Q8—84no—
545Q8—84no—
546Q8—164no—
547D4—24no—
548D4—24no—
549D4—48no—
550D4—48no—
551D4—42no—
552D4—44no—
553D4—42no—
554D4—44no—
555D4—84no—
556D4—162no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—88no—
601C2×C4—88no—
602C2×C4—168no—
603C2×C4#53212yes—
604C2×C4#53212yes—
605C2×C4C2×AGL(1,ℤ/8)12yes—
606C2×C4C2×AGL(1,ℤ/8)12yes—
607C2×C4C2×AGL(1,ℤ/8)12yes—
608C2×C4C2×AGL(1,ℤ/8)12yes—
609C2×C4C2×AGL(1,ℤ/8)12yes—
610C2×C4C2×AGL(1,ℤ/8)12yes—
611C2×C4C22×#4112yes—
612C2×C4C2×H(5,𝔽2)11yescharacteristic
613C2×C4C2×2−1+412yes—
614C2×C4#58511yescharacteristic
615C8—28no—
616C8—42no—
617C8—42no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22C2×#53211yescharacteristic
633C22#348611yescharacteristic
634C22#348811yescharacteristic
635C22C22×AGL(1,ℤ/8)11yescharacteristic
636C22C22×AGL(1,ℤ/8)11yescharacteristic
637C22#358511yescharacteristic
638C22#358611yescharacteristic
639C4—22no—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—81no—
650C4—84no—
651C4—82no—
652C4—82no—
653C4—81no—
654C4—81no—
655C4#318414yes—
656C4#351612yes—
657C4#352012yes—
658C2—84no—
659C2—162no—
660C2#3590911yescharacteristic
661C2C2×#318411yescharacteristic
662C2#6072711yescharacteristic
663C2#6182511yescharacteristic
664C2#6182711yescharacteristic
665C2#6196211yescharacteristic
666C2#6196311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28369C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36559C2semidirect
2. #36271C2semidirect
3. #30384C2semidirect
4. #30203C2semidirect
5. #29578C2semidirect
6. #29499C2semidirect
7. #29095C2semidirect
8. #28687C2semidirect
10. #26490C2semidirect
11. #25478C2semidirect
12. #25300C2semidirect
13. #23815C2semidirect
14. #23682C2semidirect
15. #23001C2semidirect
16. #22968C2semidirect
17. #20570C2semidirect

Maximal subgroups

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

subgroupindex
#365592
#362712
#303842
#302032
#295782
#294992
#290952
#286872
#283692
#264902
#254782
#253002
#238152
#236822
#230012
#229682
#205702