Finite Group Databasebeta

← #6852260

Subgroups — #6852260

Order 512.

1595 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36551C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic, complemented
3#32957C211yesmaximal, characteristic, complemented
4#32355C214yesmaximal, complemented
5#31531C214yesmaximal, complemented
6#29957C212yesmaximal, complemented
7#28529C212yesmaximal, complemented
8#27952C212yesmaximal, complemented
9#27816C212yesmaximal, complemented
10#27684C212yesmaximal
11#25811C211yesmaximal, characteristic, complemented
12#25454C212yesmaximal, complemented
13#24816C212yesmaximal, complemented
14#24298C211yesmaximal, characteristic
15#22625C211yesmaximal, characteristic, complemented
16#22041C212yesmaximal
17C4×#448C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#498C2212yes—
21C2×#477C2212yes—
22C2×#474C2212yes—
23C2×#448C2211yescharacteristic
24C2×C4×SD16C2212yes—
25#2857C2211yescharacteristic
26#2851C2212yes—
27#2589—22no—
28#2568—22no—
29#2568—21no—
30#2568—21no—
31#2568C2214yes—
32#2568C2212yes—
33#2568C2212yes—
34#2563C2211yescharacteristic
35#2552C2211yescharacteristic
36#2509C2212yes—
37#2497—21no—
38#2487—21no—
39#2454C2212yes—
40#2450—22no—
41C2×#400—22no—
42#2369C2211yescharacteristic
43D4×C4⋊C4—21no—
44D4×C4⋊C4C2211yescharacteristic
45#2345—21no—
46C4×#121C2212yes—
47C4×#121C2211yescharacteristic
48C2×#391—21no—
49C2×#391—21no—
50C2×#391C2212yes—
51C2×#382C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2101C2214yes—
54#2097C2214yes—
55#2092C2214yes—
56#2085C2214yes—
57#2085C2214yes—
58#2068C2214yes—
59#2066C2214yes—
60#2058C2214yes—
61#2057C2212yes—
62#2026C2214yes—
63#2023C2212yes—
64#1991C2214yes—
65#1990C2212yes—
66#1988C2212yes—
67#1968C2214yes—
68#1949C2214yes—
69#1939C2212yes—
70#1933C2214yes—
71#1927C2214yes—
72#1916C2214yes—
73#1880C2212yes—
74#1877C2214yes—
75#1874C2212yes—
76#1874C2211yescharacteristic
77#1865C2212yes—
78#1863C2214yes—
79#1853C2212yes—
80#1853C2212yes—
81#1853C2211yescharacteristic
82#1852C2212yes—
83#1852C2212yes—
84#1846C2214yes—
85#1804C2214yes—
86#1803C2212yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1787C2211yescharacteristic
90#1787C2212yes—
91C8×C4⋊C4C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8C2311yescharacteristic
97C22×SD16C2312yes—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—44no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×#125C2312yes—
105C2×C22⋊Q8—21no—
106C2×#121C2311yescharacteristic
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—44no—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2312yes—
112C2×C4×D4—21no—
113C2×C4×D4—22no—
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2312yes—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126#498—44no—
127#477—44no—
128#474—44no—
129#448Q818yescomplemented
130C4×SD16—44no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×#106C2312yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2311yescharacteristic
147C2×C22⋊C8C2311yescharacteristic
148C2×C4×C8C2312yes—
149C2×C4×C8C2314yes—
150#400—22no—
151#400—24no—
152#400—24no—
153#400—24no—
154#400—24no—
155#400—22no—
156#399—22no—
157#398—22no—
158#396—22no—
159#393—22no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391—22no—
164#391—24no—
165#391—24no—
166#391—22no—
167#391—22no—
168#391—21no—
169#391—22no—
170#391C2314yes—
171#391C2314yes—
172#390C2312yes—
173#390C2311yescharacteristic
174#389—22no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389C2311yescharacteristic
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#388—21no—
184#388—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#386—21no—
189#386—22no—
190#385—22no—
191#385—22no—
192#385—22no—
193#385—22no—
194#384—21no—
195#384C2311yescharacteristic
196#382—22no—
197#382—22no—
198#382C2312yes—
199#382C2314yes—
200#382C2314yes—
201#380—21no—
202#380—21no—
203#380—21no—
204#380—21no—
205#380—22no—
206#380C2311yescharacteristic
207#380C2312yes—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—21no—
219C2×#95—22no—
220C2×#95—22no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8C2×Q814yes—
246C22×Q8C2412yes—
247C22×D4—21no—
248C22×D4C2411yescharacteristic
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2411yescharacteristic
259C2×SD16—42no—
260C2×SD16—42no—
261C2×SD16C2×Q818yescomplemented
262C22×C8C2412yes—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271#125—44no—
272C22⋊Q8—82no—
273#121—22no—
274#121C2×Q814yes—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—42no—
278C4×Q8—42no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×D4—24no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—84no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C42—24no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—44no—
336C2×C42—44no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345#107—44no—
346#107—44no—
347#106—42no—
348#106—44no—
349#106—44no—
350C4⋊C8—24no—
351C4⋊C8—44no—
352#103—44no—
353#103—42no—
354#103—42no—
355#103—44no—
356#103C2×Q818yescomplemented
357#102—42no—
358#102—44no—
359C22⋊C8—24no—
360C22⋊C8—22no—
361C22⋊C8C2×Q814yes—
362C4×C8—24no—
363C4×C8—44no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—21no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24—21no—
388C24C22×D411yescharacteristic
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—48no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—48no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—81no—
407C2×Q8—81no—
408C2×Q8—816no—
409C2×Q8C22×Q814yes—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4C22×Q812yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4#13514yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×Q812yes—
464C22×C42−1+412yes—
465C22×C4C2511yescharacteristic
466SD16—48no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8C22×Q814yes—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4—81no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4C22×Q812yes—
508C22⋊C4—24no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—81no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—24no—
521C42—24no—
522C42—42no—
523C42—42no—
524C42—44no—
525C42—44no—
526C42—42no—
527C42—82no—
528C42H(5,𝔽2)14yes—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23D4×Q812yes—
541C23C2×#13512yes—
542C23#57712yes—
543C23C23×D411yescharacteristic
544Q8—24no—
545Q8—24no—
546Q8—48no—
547Q8—42no—
548Q8—42no—
549Q8—88no—
550Q8—88no—
551Q8—162no—
552D4—24no—
553D4—48no—
554D4—42no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—24no—
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—21no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—88no—
609C2×C4—84no—
610C2×C4—164no—
611C2×C4#53512yes—
612C2×C4#53512yes—
613C2×C4D4×Q812yes—
614C2×C4C2×#13512yes—
615C2×C4C2×#13512yes—
616C2×C4#57512yes—
617C2×C4C23×Q811yescharacteristic
618C2×C4C2×H(5,𝔽2)12yes—
619C2×C4C2×2−1+411yescharacteristic
620C2×C4#58512yes—
621C8—24no—
622C8—42no—
623C8—44no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22#338114yes—
638C22C2×#53511yescharacteristic
639C22C2×D4×Q811yescharacteristic
640C22#348511yescharacteristic
641C22C22×#13511yescharacteristic
642C22C2×#57511yescharacteristic
643C22C2×#57711yescharacteristic
644C22#358611yescharacteristic
645C4—22no—
646C4—22no—
647C4—22no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—41no—
653C4—44no—
654C4—42no—
655C4—81no—
656C4—82no—
657C4—81no—
658C4—84no—
659C4—82no—
660C4—82no—
661C4#321114yes—
662C4#350112yes—
663C4#354112yes—
664C2—24no—
665C2—82no—
666C2#3959411yescharacteristic
667C2C2×#321111yescharacteristic
668C2#6081311yescharacteristic
669C2C2×#338111yescharacteristic
670C2#6136911yescharacteristic
671C2#6170111yescharacteristic
672C2#6173911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2568C2semidirect
3. #32957C2semidirect
4. #32355C2semidirect
5. #31531C2semidirect
6. #29957C2semidirect
7. #28529C2semidirect
8. #27952C2semidirect
9. #27816C2semidirect
11. #25811C2semidirect
12. #25454C2semidirect
13. #24816C2semidirect
15. #22625C2semidirect
129. #448Q8semidirect
261. C2×SD16C2×Q8semidirect
356. #103C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365512
C2×#25682
#329572
#323552
#315312
#299572
#285292
#279522
#278162
#276842
#258112
#254542
#248162
#242982
#226252
#220412
C4×#4482