Finite Group Databasebeta

← #6989175

Subgroups — #6989175

Order 512.

1631 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#36274C211yesmaximal, characteristic
2#35279C211yesmaximal, characteristic
3#32744C212yesmaximal, complemented
4#31776C214yesmaximal, complemented
5#31669C214yesmaximal, complemented
6#30358C212yesmaximal, complemented
7#29953C212yesmaximal, complemented
8#29923C211yesmaximal, characteristic, complemented
9#29105C212yesmaximal, complemented
10#28638C212yesmaximal, complemented
11#26302C212yesmaximal, complemented
12#25902C211yesmaximal, characteristic, complemented
13#25888C212yesmaximal, complemented
14#25471C212yesmaximal, complemented
15#24302C211yesmaximal, characteristic, complemented
16#23757C211yesmaximal, characteristic, complemented
17#22530C211yesmaximal, characteristic
18C2×D4×Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#487C2212yes—
22C2×#477C2212yes—
23C2×C8×Q8C2212yes—
24#2857C2212yes—
25#2857C2212yes—
26#2857C2211yescharacteristic
27#2855C2212yes—
28#2628—22no—
29#2582C2211yescharacteristic
30#2568—21no—
31#2568C2211yescharacteristic
32#2567C2212yes—
33#2563C2212yes—
34#2542—21no—
35#2487C2214yes—
36#2451—21no—
37#2450—22no—
38#2439—21no—
39#2438—22no—
40#2417C2212yes—
41#2409—22no—
42#2409C2212yes—
43#2349—21no—
44D4×C4⋊C4C2211yescharacteristic
45#2339—21no—
46#2317—21no—
47#2311—21no—
48C4×C22⋊Q8C2211yescharacteristic
49C2×#391C2212yes—
50#2285C2211yescharacteristic
51#2100C2211yescharacteristic
52#2095C2211yescharacteristic
53#2095C2212yes—
54#2087C2214yes—
55#2083C2211yescharacteristic
56#2076C2212yes—
57#2076C2214yes—
58#2076C2214yes—
59#2072C2211yescharacteristic
60#2072C2212yes—
61#2068C2214yes—
62#2068C2214yes—
63#2068C2214yes—
64#2057C2212yes—
65#2036C2214yes—
66#2036C2214yes—
67#2036C2214yes—
68#2036C2212yes—
69#2024C2212yes—
70#2024C2212yes—
71#2024C2212yes—
72#1990C2212yes—
73#1987C2212yes—
74#1981C2212yes—
75#1954C2212yes—
76#1954C2214yes—
77#1947C2212yes—
78#1942C2214yes—
79#1933C2214yes—
80#1933C2214yes—
81#1933C2214yes—
82#1872C2214yes—
83#1872C2212yes—
84#1852C2214yes—
85#1852C2212yes—
86#1824C2214yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1787C2214yes—
90#1787C2214yes—
91C4×#107C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8—21no—
97C23×Q8C2312yes—
98D4×Q8—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8C2311yescharacteristic
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2312yes—
112C4×#41—44no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D412yes—
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2312yes—
122C2×#117—21no—
123C2×#117D411yescharacteristic
124C2×#117D412yes—
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#487—44no—
131#477—44no—
132C8×Q8—44no—
133C2×#107C2314yes—
134C2×#107C2312yes—
135C2×#107C2312yes—
136C2×#107C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C4×C8C2312yes—
151C2×C4×C8C2314yes—
152#400—24no—
153#400—22no—
154#399—24no—
155#398—22no—
156#398—22no—
157#396—22no—
158#395—24no—
159#395—22no—
160#393—24no—
161#393—22no—
162#391—22no—
163#391—21no—
164#391—21no—
165#391—22no—
166#391—22no—
167#391—22no—
168#391—22no—
169#391C2312yes—
170#391C2312yes—
171#391C2312yes—
172#391C2312yes—
173#391C2311yescharacteristic
174#390—22no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389—22no—
179#389—21no—
180#389—21no—
181#389C2314yes—
182#389C2314yes—
183#389C2311yescharacteristic
184#389C2312yes—
185#387—22no—
186#387—22no—
187#387—21no—
188#386—24no—
189#386—22no—
190#386—22no—
191#385—22no—
192#385—21no—
193#385—24no—
194#385—22no—
195#385—22no—
196#384—22no—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#382—22no—
201#382—21no—
202#382—21no—
203#381—22no—
204#380—22no—
205#380—22no—
206#380—21no—
207#380—21no—
208#380—21no—
209#380—22no—
210#380C2314yes—
211#380C2314yes—
212#380C2312yes—
213#380C2311yescharacteristic
214#380C2312yes—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C2×#95—22no—
221C2×#95—22no—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234C2×#41—42no—
235C2×#41—42no—
236C2×#41Q1618yes—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—24no—
241C22×Q8—44no—
242C22×Q8—42no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8—82no—
246C22×Q8—82no—
247C22×Q8C2×D411yescharacteristic
248C22×Q8C2412yes—
249C22×Q8C2411yescharacteristic
250C22×D4C2×D411yescharacteristic
251C22×D4C2×D412yes—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2×D412yes—
259C23×C4C2411yescharacteristic
260C23×C4C2412yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C22⋊Q8—22no—
272C22⋊Q8—48no—
273C22⋊Q8—41no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C22⋊Q8—84no—
277C22⋊Q8—84no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—42no—
281C4×D4—24no—
282C4×D4—42no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—44no—
286C4×D4—44no—
287#117—44no—
288#117—42no—
289#117—44no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—44no—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—42no—
348#107—42no—
349#107—44no—
350#107—44no—
351C4⋊C8—24no—
352C4⋊C8—44no—
353#103—24no—
354#103—44no—
355#103—44no—
356#103—42no—
357#103—44no—
358#103—44no—
359#103—44no—
360C22⋊C8—44no—
361C22⋊C8—42no—
362C22⋊C8—42no—
363C22⋊C8—44no—
364C4×C8—24no—
365C4×C8—44no—
366#95—21no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387C24C22×D412yes—
388C24C22×D411yescharacteristic
389#41—416no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—82no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8—88no—
409C2×Q8—88no—
410C2×Q8—816no—
411C2×Q8C2×Q1614yes—
412C2×Q82−1+412yes—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4C2×Q1614yes—
418C2×D4C2×Q1618yes—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4C2×Q1614yes—
456C22×C4C2×Q1618yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C2511yescharacteristic
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—81no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C22⋊C4—24no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—81no—
518C22⋊C4—82no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C42—22no—
522C42—22no—
523C42—24no—
524C42—22no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—44no—
529C42—42no—
530C42H(5,𝔽2)14yes—
531C23—22no—
532C23—21no—
533C23—22no—
534C23—21no—
535C23—21no—
536C23—22no—
537C23—21no—
538C23—22no—
539C23—21no—
540C23—22no—
541C23—22no—
542C23—21no—
543C23C22×Q1612yes—
544C23C22×Q1614yes—
545C23C23×D411yescharacteristic
546Q8—28no—
547Q8—24no—
548Q8—42no—
549Q8—44no—
550Q8—44no—
551Q8—88no—
552Q8—88no—
553Q8—88no—
554D4—28no—
555D4—48no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—48no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—84no—
608C2×C4—88no—
609C2×C4—88no—
610C2×C4—164no—
611C2×C4#53412yes—
612C2×C4#53412yes—
613C2×C4C22×Q1612yes—
614C2×C4C22×Q1614yes—
615C2×C4C22×Q1612yes—
616C2×C4#57512yes—
617C2×C4#57712yes—
618C2×C4C2×H(5,𝔽2)12yes—
619C2×C4C2×2−1+411yescharacteristic
620C2×C4#58512yes—
621C2×C4#58511yescharacteristic
622C8—42no—
623C8—42no—
624C8—44no—
625C22—22no—
626C22—24no—
627C22—22no—
628C22—24no—
629C22—24no—
630C22—22no—
631C22—41no—
632C22—42no—
633C22—42no—
634C22—41no—
635C22—42no—
636C22—41no—
637C22—41no—
638C22—42no—
639C22C2×#53411yescharacteristic
640C22#348911yescharacteristic
641C22#350011yescharacteristic
642C22C23×Q1611yescharacteristic
643C22C2×#57511yescharacteristic
644C22C2×#57711yescharacteristic
645C22#358511yescharacteristic
646C4—22no—
647C4—24no—
648C4—22no—
649C4—22no—
650C4—42no—
651C4—41no—
652C4—42no—
653C4—42no—
654C4—81no—
655C4—84no—
656C4—82no—
657C4—82no—
658C4—81no—
659C4—82no—
660C4—81no—
661C4—82no—
662C4#318914yes—
663C4#356112yes—
664C4#356212yes—
665C2—42no—
666C2—44no—
667C2#4294311yescharacteristic
668C2C2×#318911yescharacteristic
669C2#6075711yescharacteristic
670C2#6210011yescharacteristic
671C2#6214511yescharacteristic
672C2#6253011yescharacteristic
673C2#6259211yescharacteristic
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. #32744C2semidirect
4. #31776C2semidirect
5. #31669C2semidirect
6. #30358C2semidirect
7. #29953C2semidirect
8. #29923C2semidirect
9. #29105C2semidirect
10. #28638C2semidirect
11. #26302C2semidirect
12. #25902C2semidirect
13. #25888C2semidirect
14. #25471C2semidirect
15. #24302C2semidirect
16. #23757C2semidirect

Maximal subgroups

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

subgroupindex
#362742
#352792
#327442
#317762
#316692
#303582
#299532
#299232
#291052
#286382
#263022
#259022
#258882
#254712
#243022
#237572
#225302