Finite Group Databasebeta

← #6987037

Subgroups — #6987037

Order 512.

1655 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36445C211yesmaximal, characteristic
2#35316C211yesmaximal, characteristic
3#32774C212yesmaximal, complemented
4#32039C214yesmaximal, complemented
5#32016C214yesmaximal, complemented
6#30199C212yesmaximal, complemented
7#30175C212yesmaximal, complemented
8#29923C211yesmaximal, characteristic, complemented
9#29166C212yesmaximal, complemented
10#28606C212yesmaximal, complemented
11#26276C212yesmaximal, complemented
12#25902C211yesmaximal, characteristic, complemented
13#25647C212yesmaximal, complemented
14#25348C212yesmaximal, complemented
15#24301C211yesmaximal, characteristic, complemented
16#23757C211yesmaximal, characteristic, complemented
17#22473C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C2×#536—21no—
20C22×C4⋊Q8C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#487C2212yes—
23C2×#475C2212yes—
24C2×#436C2212yes—
25#2857C2212yes—
26#2857C2212yes—
27#2857C2212yes—
28#2857C2211yescharacteristic
29#2621—22no—
30#2547—21no—
31#2547C2212yes—
32#2546—21no—
33#2546C2212yes—
34#2542C2212yes—
35#2542C2211yescharacteristic
36#2538C2211yescharacteristic
37#2538C2212yes—
38#2486—22no—
39#2486C2214yes—
40#2454—22no—
41#2454—21no—
42#2441—21no—
43#2441—22no—
44Q8×C4⋊C4—21no—
45Q8×C4⋊C4C2211yescharacteristic
46#2341—21no—
47#2310—21no—
48C4×C22⋊Q8C2211yescharacteristic
49C4×C22⋊Q8C2212yes—
50#2285C2211yescharacteristic
51#2100C2211yescharacteristic
52#2099C2214yes—
53#2095C2211yescharacteristic
54#2095C2212yes—
55#2083C2214yes—
56#2083C2211yescharacteristic
57#2077C2214yes—
58#2076C2212yes—
59#2072C2211yescharacteristic
60#2072C2212yes—
61#2058C2212yes—
62#2043C2214yes—
63#2043C2214yes—
64#2036C2214yes—
65#2036C2214yes—
66#2036C2214yes—
67#2036C2212yes—
68#2036C2214yes—
69#2024C2212yes—
70#2024C2212yes—
71#2024C2212yes—
72#1990C2212yes—
73#1988C2212yes—
74#1984C2212yes—
75#1981C2212yes—
76#1947C2212yes—
77#1918C2214yes—
78#1918C2214yes—
79#1894C2214yes—
80#1894C2214yes—
81#1894C2214yes—
82#1872C2214yes—
83#1872C2212yes—
84#1870C2214yes—
85#1865C2214yes—
86#1865C2214yes—
87#1853C2212yes—
88#1817C2212yes—
89#1809C2214yes—
90#1797C2211yescharacteristic
91C4×#107C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1728C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8—21no—
97C22×M16C2314yes—
98#557—44no—
99#536—48no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×#125—21no—
109C2×#125—22no—
110C2×#124—22no—
111C2×#123—24no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8C2311yescharacteristic
115C2×C22⋊Q8C2312yes—
116C2×C22⋊Q8C2312yes—
117C2×C22⋊Q8C2312yes—
118C2×C22⋊Q8C2312yes—
119C4×#41—44no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8D411yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2311yescharacteristic
126C2×C4×D4D412yes—
127C2×C4×D4D411yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2312yes—
130C2×#117—21no—
131C2×#117D411yescharacteristic
132C2×#117D412yes—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137#487—44no—
138#475—44no—
139#436—44no—
140C2×#107C2312yes—
141C2×#107C2314yes—
142C2×#107C2312yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2312yes—
155C2×C4×C8C2312yes—
156#399—22no—
157#399—22no—
158#399—22no—
159#398—24no—
160#398—24no—
161#397—24no—
162#397—24no—
163#396—22no—
164#396—22no—
165#396—22no—
166#391—22no—
167#391—22no—
168#391—22no—
169#391—22no—
170#391C2311yescharacteristic
171#391C2312yes—
172#390—21no—
173#390—22no—
174#390—21no—
175#390—22no—
176#389—21no—
177#389—22no—
178#389—21no—
179#389—22no—
180#389—21no—
181#389—22no—
182#389—21no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389C2311yescharacteristic
187#389C2314yes—
188#389C2314yes—
189#387—21no—
190#387—22no—
191#385—21no—
192#385—22no—
193#384—21no—
194#384—22no—
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#382—21no—
200#381—24no—
201#380—22no—
202#380—21no—
203#380—21no—
204#380—22no—
205#380—22no—
206#380—22no—
207#380—21no—
208#380C2312yes—
209#380C2314yes—
210#380C2311yescharacteristic
211#380C2312yes—
212#380C2314yes—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2314yes—
235C2×#41—24no—
236C2×#41—24no—
237C2×#41—42no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—44no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2411yescharacteristic
245C22×Q8C2412yes—
246C22×D4C2×D411yescharacteristic
247C22×D4C2×D412yes—
248C23×C4—22no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D412yes—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D412yes—
253C23×C4C2411yescharacteristic
254C23×C4C2412yes—
255C2×M16—44no—
256C2×M16—44no—
257C2×M16—44no—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—48no—
269#125—48no—
270#125—44no—
271#124—48no—
272#123—416no—
273C22⋊Q8—22no—
274C22⋊Q8—24no—
275C22⋊Q8—41no—
276C22⋊Q8—42no—
277C22⋊Q8—48no—
278C22⋊Q8—48no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—42no—
285C4×Q8—44no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×D4—24no—
289C4×D4—42no—
290C4×D4—44no—
291C4×D4—44no—
292#117—44no—
293#117—42no—
294#117—44no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—44no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—24no—
361#107—42no—
362#107—44no—
363#106—44no—
364C4⋊C8—24no—
365#103—42no—
366#103—44no—
367#103—44no—
368#103—42no—
369#103—44no—
370#103—44no—
371#103—44no—
372C22⋊C8—42no—
373C22⋊C8—42no—
374C22⋊C8—44no—
375C4×C8—24no—
376#95—21no—
377#95—22no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391C24C22×D412yes—
392C24C22×D411yescharacteristic
393#41—416no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—816no—
411C2×Q8AGL(1,ℤ/8)14yes—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—42no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4#13714yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C2511yescharacteristic
459M16—88no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—81no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C42−1+412yes—
509C22⋊C4—24no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—81no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—24no—
526C42—42no—
527C42—42no—
528C42—44no—
529C42—42no—
530C42—44no—
531C42—44no—
532C42—42no—
533C42H(5,𝔽2)14yes—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23C2×#13712yes—
548C23C23×D411yescharacteristic
549Q8—28no—
550Q8—42no—
551Q8—44no—
552Q8—42no—
553Q8—44no—
554Q8—88no—
555Q8—88no—
556D4—48no—
557D4—44no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—28no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—44no—
578C2×C4—48no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—84no—
610C2×C4—88no—
611C2×C4#53412yes—
612C2×C4#53412yes—
613C2×C4C2×AGL(1,ℤ/8)14yes—
614C2×C4C2×AGL(1,ℤ/8)12yes—
615C2×C4C2×#13712yes—
616C2×C4C2×#13712yes—
617C2×C4#57812yes—
618C2×C4C2×H(5,𝔽2)12yes—
619C2×C4C2×2−1+411yescharacteristic
620C2×C4#58512yes—
621C2×C4#58511yescharacteristic
622C8—42no—
623C8—44no—
624C8—42no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22C2×#53411yescharacteristic
640C22#348911yescharacteristic
641C22#350011yescharacteristic
642C22C22×AGL(1,ℤ/8)11yescharacteristic
643C22C22×#13711yescharacteristic
644C22C2×#57811yescharacteristic
645C22#358611yescharacteristic
646C4—22no—
647C4—24no—
648C4—22no—
649C4—22no—
650C4—42no—
651C4—41no—
652C4—42no—
653C4—42no—
654C4—84no—
655C4—81no—
656C4—81no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4—81no—
661C4—82no—
662C4#319314yes—
663C4#355412yes—
664C4#356212yes—
665C2—42no—
666C2—82no—
667C2#4291311yescharacteristic
668C2C2×#319311yescharacteristic
669C2#6075911yescharacteristic
670C2#6211311yescharacteristic
671C2#6214811yescharacteristic
672C2#6253611yescharacteristic
673C2#6259511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32774C2semidirect
4. #32039C2semidirect
5. #32016C2semidirect
6. #30199C2semidirect
7. #30175C2semidirect
8. #29923C2semidirect
9. #29166C2semidirect
10. #28606C2semidirect
11. #26276C2semidirect
12. #25902C2semidirect
13. #25647C2semidirect
14. #25348C2semidirect
15. #24301C2semidirect
16. #23757C2semidirect

Maximal subgroups

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

subgroupindex
#364452
#353162
#327742
#320392
#320162
#301992
#301752
#299232
#291662
#286062
#262762
#259022
#256472
#253482
#243012
#237572
#224732