Finite Group Databasebeta

← #6803371

Subgroups — #6803371

Order 512.

1683 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36595C211yesmaximal, characteristic
2#34951C211yesmaximal, characteristic, complemented
3#33161C211yesmaximal, characteristic
4#31918C214yesmaximal, complemented
5#31220C214yesmaximal, complemented
6#29827C212yesmaximal, complemented
7#28745C212yesmaximal, complemented
8#28441C211yesmaximal, characteristic
9#28386C212yesmaximal, complemented
10#28284C211yesmaximal, characteristic, complemented
11#28252C212yesmaximal, complemented
12#26526C212yesmaximal, complemented
13#25368C212yesmaximal, complemented
14#25303C212yesmaximal, complemented
15#22722C211yesmaximal, characteristic, complemented
16#22153C212yesmaximal, complemented
17C4×#477C211yesmaximal, characteristic, complemented
18C2×#559—21no—
19C2×#492C2212yes—
20C2×#477C2211yescharacteristic
21C2×#477C2212yes—
22C2×#460C2212yes—
23C2×#460C2212yes—
24C2×#460C2212yes—
25C2×#460C2212yes—
26C2×#460C2212yes—
27C2×#461C2214yes—
28C2×#446C2212yes—
29#2867C2211yescharacteristic
30#2858C2212yes—
31#2585—22no—
32#2582C2211yescharacteristic
33#2568C2211yescharacteristic
34#2552C2211yescharacteristic
35#2548—21no—
36#2548—21no—
37#2548C2212yes—
38#2546C2211yescharacteristic
39#2544—22no—
40#2511C2212yes—
41#2509—21no—
42#2453—21no—
43#2446C2212yes—
44#2445—22no—
45#2445C2214yes—
46#2402—22no—
47#2366—21no—
48#2366C2212yes—
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2212yes—
52C4×#127C2212yes—
53C4×#125—21no—
54#2295—21no—
55#2295C2212yes—
56C42×Q8C2211yescharacteristic
57C42×Q8C2211yescharacteristic
58#2093C2214yes—
59#2068C2214yes—
60#2067C2214yes—
61#2067C2214yes—
62#2055C2214yes—
63#2055C2214yes—
64#2055C2214yes—
65#2055C2214yes—
66#2014C2214yes—
67#2013C2214yes—
68#1987C2211yescharacteristic
69#1982C2212yes—
70#1982C2212yes—
71#1982C2212yes—
72#1981C2212yes—
73#1973C2214yes—
74#1973C2214yes—
75#1973C2214yes—
76#1957C2212yes—
77#1947C2211yescharacteristic
78#1947C2212yes—
79#1940C2214yes—
80#1939C2212yes—
81#1936C2214yes—
82#1921C2212yes—
83#1877C2214yes—
84#1863C2212yes—
85#1862C2214yes—
86#1842C2212yes—
87#1842C2211yescharacteristic
88#1803C2211yescharacteristic
89#1803C2212yes—
90#1802C2214yes—
91C4×#107C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×SD16C2314yes—
96C22×SD16C2314yes—
97C22×SD16C2314yes—
98#559—48no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127—21no—
103C2×#127—21no—
104C2×#127—21no—
105C2×#127—21no—
106C2×#127C2312yes—
107C2×#127C2312yes—
108C2×#127C2312yes—
109C2×#125—21no—
110C2×#124—22no—
111C2×#124—22no—
112C2×#124—22no—
113C2×#123—24no—
114C2×C4×Q8—21no—
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4—22no—
125C2×C4×D4C2314yes—
126#492—44no—
127#477—24no—
128#477—44no—
129#461—48no—
130#460—28no—
131#460—44no—
132#460—44no—
133#460—44no—
134#460—44no—
135#446—44no—
136C2×#107C2312yes—
137C2×#107C2314yes—
138C2×#107C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2311yescharacteristic
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#97C2314yes—
156C2×C4×C8C2312yes—
157#400—22no—
158#400—22no—
159#400—22no—
160#400—22no—
161#398—22no—
162#398—22no—
163#398—22no—
164#398—22no—
165#394—24no—
166#394—24no—
167#394—24no—
168#391—21no—
169#391—21no—
170#391—21no—
171#391C2312yes—
172#391C2312yes—
173#391C2312yes—
174#391C2312yes—
175#391C2312yes—
176#390—22no—
177#390—22no—
178#390—22no—
179#390—22no—
180#390—22no—
181#390—22no—
182#390C2314yes—
183#390C2314yes—
184#390C2314yes—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—22no—
191#389—22no—
192#389—21no—
193#389—21no—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#386—24no—
204#386—24no—
205#384C2311yescharacteristic
206#384C2312yes—
207#384C2311yescharacteristic
208#382—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D412yes—
220C4×C4⋊C4D412yes—
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2312yes—
228#376—22no—
229C43—21no—
230C43C2312yes—
231C43C2311yescharacteristic
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#340C2314yes—
236#340C2312yes—
237#336C2314yes—
238#336C2312yes—
239C22×Q8—21no—
240C22×Q8C2×D411yescharacteristic
241C22×Q8C2412yes—
242C22×Q8C2411yescharacteristic
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4C2414yes—
250C23×C4—24no—
251C2×SD16—28no—
252C2×SD16—28no—
253C2×SD16—44no—
254C2×SD16—44no—
255C2×SD16—44no—
256C2×SD16—44no—
257C2×SD16—44no—
258C2×SD16—44no—
259C2×SD16—44no—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268C4⋊Q8—42no—
269#127—24no—
270#127—24no—
271#127—24no—
272#127—24no—
273#127—42no—
274#127—42no—
275#127—44no—
276#127—44no—
277#127—82no—
278#127—82no—
279#125—44no—
280#124—48no—
281#124—48no—
282#124—84no—
283#123—416no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—22no—
289C4×Q8—24no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8—22no—
293C4×Q8—24no—
294C4×Q8—22no—
295C4×Q8—41no—
296C4×Q8—44no—
297C4×D4—48no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4—48no—
301C4×D4—84no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—24no—
361#107—42no—
362#107—44no—
363C4⋊C8—24no—
364C4⋊C8—22no—
365C4⋊C8—24no—
366C4⋊C8—44no—
367C4⋊C8—42no—
368C4⋊C8—44no—
369C4⋊C8#4114yes—
370#103—24no—
371#103—24no—
372#103—44no—
373#103—44no—
374#102—28no—
375#102—44no—
376#102—44no—
377#102—44no—
378#102—44no—
379#97—44no—
380C4×C8—24no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401C24—24no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8H(5,𝔽2)12yes—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—82no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4—82no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—48no—
453C22×C4—84no—
454C22×C4—84no—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C2511yescharacteristic
466SD16—88no—
467SD16—88no—
468SD16—88no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8C2×#4114yes—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—81no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4AGL(1,ℤ/8)14yes—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C42—24no—
522C42—22no—
523C42—24no—
524C42—24no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—21no—
529C42—22no—
530C42—22no—
531C42—21no—
532C42—42no—
533C42—41no—
534C42—42no—
535C42—42no—
536C42—42no—
537C42—44no—
538C42—44no—
539C42—42no—
540C42—42no—
541C42—81no—
542C42#13714yes—
543C42C2×#4112yes—
544C42H(5,𝔽2)12yes—
545C42H(5,𝔽2)12yes—
546C422−1+412yes—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23C23×D411yescharacteristic
555Q8—22no—
556Q8—22no—
557Q8—44no—
558Q8—41no—
559Q8—41no—
560Q8—42no—
561Q8—42no—
562Q8—42no—
563Q8—42no—
564Q8—84no—
565D4—44no—
566D4—44no—
567D4—44no—
568D4—44no—
569D4—88no—
570D4—88no—
571D4—88no—
572D4—164no—
573D4—164no—
574D4—164no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—168no—
620C2×C4#53712yes—
621C2×C4#53712yes—
622C2×C4#55012yes—
623C2×C4#55012yes—
624C2×C4C2×AGL(1,ℤ/8)14yes—
625C2×C4C2×AGL(1,ℤ/8)12yes—
626C2×C4C2×#13712yes—
627C2×C4C2×#13712yes—
628C2×C4C2×#13712yes—
629C2×C4#57712yes—
630C2×C4C22×#4111yescharacteristic
631C2×C4C2×H(5,𝔽2)11yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×2−1+411yescharacteristic
635C2×C4#58511yescharacteristic
636C8—24no—
637C8—44no—
638C8—42no—
639C22—84no—
640C22—84no—
641C22—84no—
642C22—84no—
643C22—84no—
644C22—84no—
645C22—84no—
646C22C2×#53711yescharacteristic
647C22C2×#55011yescharacteristic
648C22#348811yescharacteristic
649C22C22×AGL(1,ℤ/8)11yescharacteristic
650C22C22×#13711yescharacteristic
651C22C2×#57711yescharacteristic
652C22#358511yescharacteristic
653C4—22no—
654C4—24no—
655C4—22no—
656C4—22no—
657C4—42no—
658C4—41no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—81no—
663C4—82no—
664C4—81no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—81no—
669C4#325414yes—
670C4#340014yes—
671C4#351411yescharacteristic
672C4#352211yescharacteristic
673C4#354311yescharacteristic
674C4#355411yescharacteristic
675C2—164no—
676C2#3913711yescharacteristic
677C2C2×#325411yescharacteristic
678C2#6092411yescharacteristic
679C2C2×#340011yescharacteristic
680C2#6140211yescharacteristic
681C2#6196511yescharacteristic
682C2#6198311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2867C22
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
2. #34951C2semidirect
4. #31918C2semidirect
5. #31220C2semidirect
6. #29827C2semidirect
7. #28745C2semidirect
9. #28386C2semidirect
10. #28284C2semidirect
11. #28252C2semidirect
12. #26526C2semidirect
13. #25368C2semidirect
14. #25303C2semidirect
15. #22722C2semidirect
16. #22153C2semidirect
17. C4×#477C2semidirect

Maximal subgroups

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

subgroupindex
#365952
#349512
#331612
#319182
#312202
#298272
#287452
#284412
#283862
#282842
#282522
#265262
#253682
#253032
#227222
#221532
C4×#4772