Finite Group Databasebeta

← #6986982

Subgroups — #6986982

Order 512.

1601 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36445C211yesmaximal, characteristic
2#35279C211yesmaximal, characteristic
3#32774C212yesmaximal, complemented
4#32029C214yesmaximal, complemented
5#32000C214yesmaximal, complemented
6#30187C212yesmaximal, complemented
7#30169C212yesmaximal, complemented
8#29923C211yesmaximal, characteristic, complemented
9#29160C212yesmaximal, complemented
10#28598C212yesmaximal, complemented
11#26282C212yesmaximal, complemented
12#25902C211yesmaximal, characteristic, complemented
13#25651C212yesmaximal, complemented
14#25340C212yesmaximal, complemented
15#24302C211yesmaximal, characteristic, complemented
16#23757C211yesmaximal, characteristic, complemented
17#22469C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#487C2212yes—
22C2×#477C2212yes—
23C2×C8×D4C2212yes—
24#2857C2212yes—
25#2857C2212yes—
26#2857C2212yes—
27#2857C2211yescharacteristic
28#2628—22no—
29#2568—21no—
30#2567C2212yes—
31#2563C2212yes—
32#2542—21no—
33#2542C2212yes—
34#2542C2211yescharacteristic
35#2538C2212yes—
36#2538C2211yescharacteristic
37#2487C2214yes—
38#2454—22no—
39#2454—21no—
40#2441—21no—
41#2441—22no—
42#2409—22no—
43Q8×C4⋊C4—21no—
44Q8×C4⋊C4C2211yescharacteristic
45#2341—21no—
46#2317—21no—
47#2311—21no—
48C4×C22⋊Q8C2211yescharacteristic
49C4×C22⋊Q8C2212yes—
50#2285C2211yescharacteristic
51#2100C2214yes—
52#2100C2211yescharacteristic
53#2095C2211yescharacteristic
54#2095C2212yes—
55#2083C2214yes—
56#2083C2211yescharacteristic
57#2076C2212yes—
58#2076C2214yes—
59#2072C2211yescharacteristic
60#2072C2212yes—
61#2057C2212yes—
62#2043C2214yes—
63#2043C2214yes—
64#2036C2214yes—
65#2036C2214yes—
66#2036C2214yes—
67#2036C2212yes—
68#2036C2214yes—
69#2024C2212yes—
70#2024C2212yes—
71#2024C2212yes—
72#1990C2212yes—
73#1987C2212yes—
74#1984C2212yes—
75#1981C2212yes—
76#1947C2212yes—
77#1921C2214yes—
78#1921C2214yes—
79#1892C2214yes—
80#1892C2214yes—
81#1892C2214yes—
82#1872C2214yes—
83#1872C2212yes—
84#1869C2214yes—
85#1863C2214yes—
86#1863C2214yes—
87#1852C2212yes—
88#1817C2212yes—
89#1808C2214yes—
90#1797C2211yescharacteristic
91C4×#107C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C2×#336C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8—21no—
97C23×C8C2314yes—
98#557—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×#125—21no—
107C2×#125—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2311yescharacteristic
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2312yes—
115C4×#41—44no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4D412yes—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2312yes—
126C2×#117—21no—
127C2×#117D411yescharacteristic
128C2×#117D412yes—
129C22×C4⋊C4—21no—
130C22×C4⋊C4C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#487—44no—
134#477—44no—
135C8×D4—44no—
136C2×#107C2312yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#107C2312yes—
140C2×C4⋊C8C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2312yes—
151C2×C4×C8C2312yes—
152#400—24no—
153#399—24no—
154#399—22no—
155#399—22no—
156#399—22no—
157#396—22no—
158#396—22no—
159#396—22no—
160#395—24no—
161#393—24no—
162#391—22no—
163#391—22no—
164#391—22no—
165#391C2311yescharacteristic
166#391C2312yes—
167#390—22no—
168#390—21no—
169#390—22no—
170#390—21no—
171#390—22no—
172#389—21no—
173#389—22no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—21no—
179#389—22no—
180#389—21no—
181#389—22no—
182#389C2314yes—
183#389C2314yes—
184#389C2311yescharacteristic
185#387—21no—
186#387—22no—
187#386—24no—
188#385—21no—
189#385—22no—
190#385—24no—
191#384—22no—
192#384—21no—
193#384—22no—
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#382—22no—
199#382—21no—
200#380—22no—
201#380—22no—
202#380—21no—
203#380—22no—
204#380—21no—
205#380—22no—
206#380—21no—
207#380C2314yes—
208#380C2314yes—
209#380C2312yes—
210#380C2311yescharacteristic
211#380C2312yes—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2314yes—
234C2×#41—42no—
235C2×#41—42no—
236C2×#41Q1618yes—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—44no—
240C22×Q8—82no—
241C22×Q8—82no—
242C22×Q8C2×D411yescharacteristic
243C22×Q8C2411yescharacteristic
244C22×Q8C2412yes—
245C22×D4C2×D411yescharacteristic
246C22×D4C2×D412yes—
247C23×C4—22no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D412yes—
251C23×C4C2×D412yes—
252C23×C4C2411yescharacteristic
253C23×C4C2412yes—
254C22×C8—44no—
255C22×C8—44no—
256C22×C8—44no—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267#125—48no—
268#125—44no—
269C22⋊Q8—22no—
270C22⋊Q8—24no—
271C22⋊Q8—41no—
272C22⋊Q8—42no—
273C22⋊Q8—44no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C22⋊Q8—84no—
277C22⋊Q8—84no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—42no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×D4—24no—
285C4×D4—42no—
286C4×D4—44no—
287C4×D4—44no—
288#117—44no—
289#117—42no—
290#117—44no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—44no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—24no—
357#107—42no—
358#107—44no—
359#107—44no—
360C4⋊C8—24no—
361#103—42no—
362#103—44no—
363#103—44no—
364#103—42no—
365#103—44no—
366#103—44no—
367#103—44no—
368C22⋊C8—42no—
369C22⋊C8—42no—
370C22⋊C8—44no—
371C4×C8—24no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387C24C22×D412yes—
388C24C22×D411yescharacteristic
389#41—416no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—82no—
405C2×Q8—82no—
406C2×Q8—816no—
407C2×Q8C2×Q1614yes—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4C2×Q1614yes—
413C2×D4C2×Q1618yes—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—42no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4C2×Q1614yes—
445C22×C4C2×Q1618yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2511yescharacteristic
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—88no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—81no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C4⋊C42−1+412yes—
505C22⋊C4—24no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—82no—
512C22⋊C4—81no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—24no—
522C42—42no—
523C42—42no—
524C42—44no—
525C42—44no—
526C42—42no—
527C42—44no—
528C42—42no—
529C42H(5,𝔽2)14yes—
530C23—22no—
531C23—21no—
532C23—22no—
533C23—21no—
534C23—21no—
535C23—22no—
536C23—21no—
537C23—22no—
538C23—21no—
539C23—22no—
540C23—22no—
541C23—21no—
542C23C22×Q1612yes—
543C23C22×Q1614yes—
544C23C23×D411yescharacteristic
545Q8—28no—
546Q8—42no—
547Q8—44no—
548Q8—42no—
549Q8—44no—
550Q8—88no—
551Q8—88no—
552D4—28no—
553D4—48no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—48no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—84no—
606C2×C4—88no—
607C2×C4#53412yes—
608C2×C4#53412yes—
609C2×C4C22×Q1612yes—
610C2×C4C22×Q1614yes—
611C2×C4C22×Q1612yes—
612C2×C4#57512yes—
613C2×C4#57712yes—
614C2×C4C2×H(5,𝔽2)12yes—
615C2×C4C2×2−1+411yescharacteristic
616C2×C4#58512yes—
617C2×C4#58511yescharacteristic
618C8—42no—
619C8—44no—
620C8—42no—
621C22—22no—
622C22—24no—
623C22—24no—
624C22—22no—
625C22—24no—
626C22—22no—
627C22—41no—
628C22—42no—
629C22—42no—
630C22—41no—
631C22—42no—
632C22—41no—
633C22—41no—
634C22—42no—
635C22C2×#53411yescharacteristic
636C22#348911yescharacteristic
637C22#350011yescharacteristic
638C22C23×Q1611yescharacteristic
639C22C2×#57511yescharacteristic
640C22C2×#57711yescharacteristic
641C22#358511yescharacteristic
642C4—22no—
643C4—24no—
644C4—22no—
645C4—22no—
646C4—42no—
647C4—41no—
648C4—42no—
649C4—42no—
650C4—81no—
651C4—84no—
652C4—82no—
653C4—82no—
654C4—81no—
655C4—82no—
656C4—81no—
657C4—82no—
658C4#318914yes—
659C4#355412yes—
660C4#356212yes—
661C2—42no—
662C2—44no—
663C2#4291311yescharacteristic
664C2C2×#318911yescharacteristic
665C2#6075711yescharacteristic
666C2#6210011yescharacteristic
667C2#6213411yescharacteristic
668C2#6253011yescharacteristic
669C2#6258611yescharacteristic
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. #32029C2semidirect
5. #32000C2semidirect
6. #30187C2semidirect
7. #30169C2semidirect
8. #29923C2semidirect
9. #29160C2semidirect
10. #28598C2semidirect
11. #26282C2semidirect
12. #25902C2semidirect
13. #25651C2semidirect
14. #25340C2semidirect
15. #24302C2semidirect
16. #23757C2semidirect

Maximal subgroups

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

subgroupindex
#364452
#352792
#327742
#320292
#320002
#301872
#301692
#299232
#291602
#285982
#262822
#259022
#256512
#253402
#243022
#237572
#224692