Finite Group Databasebeta

← #6852166

Subgroups — #6852166

Order 512.

1609 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C22⋊Q8C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic
3#33148C211yesmaximal, characteristic
4#32695C214yesmaximal, complemented
5#32595C214yesmaximal, complemented
6#29966C212yesmaximal
7#29302C212yesmaximal
8#29244C212yesmaximal
9#28682C212yesmaximal, complemented
10#26280C212yesmaximal, complemented
11#25841C212yesmaximal, complemented
12#25813C211yesmaximal, characteristic
13#25480C212yesmaximal, complemented
14#24302C211yesmaximal, characteristic
15#22510C211yesmaximal, characteristic
16#22084C212yesmaximal
17C4×#483C211yesmaximal, characteristic
18C2×Q82C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#500C2212yes—
22C2×#483C2211yescharacteristic
23C2×#477C2212yes—
24C2×#446C2212yes—
25#2857C2212yes—
26#2857C2211yescharacteristic
27#2582C2212yes—
28#2582C2211yescharacteristic
29#2567—21no—
30#2567—21no—
31#2567—22no—
32#2567C2211yescharacteristic
33#2567C2212yes—
34#2567C2214yes—
35#2567C2212yes—
36#2567C2212yes—
37#2451—21no—
38#2451—22no—
39#2440—22no—
40#2440—21no—
41C2×#395—22no—
42Q8×C4⋊C4—21no—
43Q8×C4⋊C4C2211yescharacteristic
44Q8×C22⋊C4—21no—
45C4×C22⋊Q8C2212yes—
46C4×C22⋊Q8C2211yescharacteristic
47C2×#391—21no—
48C2×#389C2212yes—
49C2×#389C2212yes—
50C2×#384—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2099C2214yes—
53#2086C2214yes—
54#2086C2214yes—
55#2086C2214yes—
56#2084C2214yes—
57#2066C2214yes—
58#2057C2214yes—
59#2057C2214yes—
60#2057C2214yes—
61#2057C2212yes—
62#2024C2212yes—
63#1990C2212yes—
64#1987C2212yes—
65#1987C2212yes—
66#1985C2214yes—
67#1984C2212yes—
68#1948C2212yes—
69#1944C2214yes—
70#1944C2214yes—
71#1944C2214yes—
72#1935C2214yes—
73#1927C2212yes—
74#1920C2214yes—
75#1920C2214yes—
76#1920C2212yes—
77#1920C2214yes—
78#1872C2211yescharacteristic
79#1872C2212yes—
80#1863C2214yes—
81#1863C2214yes—
82#1853C2212yes—
83#1853C2211yescharacteristic
84#1852C2212yes—
85#1852C2212yes—
86#1852C2212yes—
87#1823C2211yescharacteristic
88#1823C2212yes—
89#1797C2212yes—
90#1797C2211yescharacteristic
91C4×#106C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8D411yescharacteristic
97Q82—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8C2311yescharacteristic
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8—44no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D412yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2312yes—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4D412yes—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#500—44no—
133#483Q818yescomplemented
134#477—44no—
135#446—44no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×#106C2312yes—
141C2×#106C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×C22⋊C8C2311yescharacteristic
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2311yescharacteristic
150C2×#97C2314yes—
151C2×C4×C8C2312yes—
152#398—22no—
153#398—22no—
154#398—22no—
155#395—24no—
156#395—24no—
157#395—24no—
158#395—24no—
159#395—22no—
160#395—22no—
161#395—22no—
162#391—22no—
163#391—24no—
164#391—22no—
165#391—21no—
166#391—22no—
167#391—21no—
168#391—22no—
169#391—21no—
170#391—22no—
171#391C2312yes—
172#391C2311yescharacteristic
173#389—21no—
174#389—22no—
175#389—21no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—21no—
182#389—22no—
183#389C2311yescharacteristic
184#389C2314yes—
185#389C2314yes—
186#389C2314yes—
187#389C2314yes—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#387—21no—
193#387—22no—
194#385—21no—
195#385—22no—
196#384—22no—
197#384—22no—
198#384—24no—
199#384—21no—
200#384—22no—
201#384C2312yes—
202#384C2311yescharacteristic
203#380—22no—
204#380—21no—
205#380C2311yescharacteristic
206#380C2312yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—22no—
216C4×C22⋊C4—21no—
217C2×#95—21no—
218C2×#95—22no—
219C2×#95—22no—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#336C2314yes—
232#336C2312yes—
233#336C2312yes—
234#336C2314yes—
235C22×Q8—24no—
236C22×Q8—24no—
237C22×Q8—22no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—42no—
242C22×Q8—44no—
243C22×Q8—82no—
244C22×Q8—82no—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2×D412yes—
247C22×Q8C2×D411yescharacteristic
248C22×Q8C2411yescharacteristic
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2411yescharacteristic
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—
267C4⋊Q8—48no—
268C4⋊Q8—44no—
269C22⋊Q8—22no—
270C22⋊Q8—82no—
271C22⋊Q8—84no—
272C22⋊Q8C2×Q814yes—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8—42no—
279C4×Q8—42no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×Q8—44no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—84no—
313C2×C4⋊C4—84no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×Q814yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—44no—
341C2×C42—44no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350#107—44no—
351#107—44no—
352#107—44no—
353#106—24no—
354#106—28no—
355#106C2×Q818yescomplemented
356C4⋊C8—24no—
357C4⋊C8—44no—
358#103—24no—
359#103—44no—
360#103C2×Q818yescomplemented
361C22⋊C8—22no—
362C22⋊C8—24no—
363C22⋊C8C2×Q814yes—
364#97—44no—
365C4×C8—24no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—22no—
373#95—21no—
374#95—22no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—22no—
385C24C22×D411yescharacteristic
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—42no—
397C2×Q8—416no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—82no—
403C2×Q8—82no—
404C2×Q8—82no—
405C2×Q8—82no—
406C2×Q8—81no—
407C2×Q8—82no—
408C2×Q8—81no—
409C2×Q8—82no—
410C2×Q8—816no—
411C2×Q8#13714yes—
412C2×Q8C22×Q812yes—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—42no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4#13714yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×Q812yes—
455C22×C42−1+412yes—
456C22×C4C2511yescharacteristic
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8C22×Q814yes—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—82no—
495C4⋊C4—81no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—81no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4C22×Q814yes—
506C4⋊C4C22×Q812yes—
507C4⋊C42−1+414yes—
508C22⋊C4—24no—
509C22⋊C4—44no—
510C22⋊C4—42no—
511C42—22no—
512C42—22no—
513C42—24no—
514C42—22no—
515C42—24no—
516C42—42no—
517C42—42no—
518C42—44no—
519C42—42no—
520C42—44no—
521C42—44no—
522C42—44no—
523C42—42no—
524C42H(5,𝔽2)14yes—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23D4×Q812yes—
530C23C2×#13712yes—
531C23#57612yes—
532C23C23×D411yescharacteristic
533Q8—28no—
534Q8—24no—
535Q8—48no—
536Q8—44no—
537Q8—42no—
538Q8—88no—
539Q8—88no—
540Q8—162no—
541Q8—164no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—48no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—48no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—84no—
597C2×C4—88no—
598C2×C4#53412yes—
599C2×C4#53412yes—
600C2×C4D4×Q812yes—
601C2×C4C2×#13712yes—
602C2×C4C2×#13714yes—
603C2×C4C2×#13712yes—
604C2×C4C2×#13712yes—
605C2×C4C23×Q811yescharacteristic
606C2×C4C2×H(5,𝔽2)12yes—
607C2×C4C2×2−1+412yes—
608C2×C4C2×2−1+411yescharacteristic
609C8—24no—
610C8—42no—
611C8—44no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22#338514yes—
619C22C2×#53411yescharacteristic
620C22C2×D4×Q811yescharacteristic
621C22#348611yescharacteristic
622C22C22×#13711yescharacteristic
623C22C22×#13711yescharacteristic
624C22C2×#57611yescharacteristic
625C22#358511yescharacteristic
626C4—22no—
627C4—24no—
628C4—22no—
629C4—22no—
630C4—42no—
631C4—41no—
632C4—42no—
633C4—42no—
634C4—44no—
635C4—42no—
636C4—44no—
637C4—42no—
638C4—84no—
639C4—82no—
640C4—81no—
641C4—82no—
642C4—81no—
643C4#319514yes—
644C4#350312yes—
645C4#355212yes—
646C2—24no—
647C2#3959311yescharacteristic
648C2C2×#319511yescharacteristic
649C2#6075611yescharacteristic
650C2C2×#338511yescharacteristic
651C2Q8×#13711yescharacteristic
652C2#6182611yescharacteristic
653C2#6184411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32695C2semidirect
5. #32595C2semidirect
9. #28682C2semidirect
10. #26280C2semidirect
11. #25841C2semidirect
13. #25480C2semidirect
133. #483Q8semidirect
355. #106C2×Q8semidirect
360. #103C2×Q8semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊Q82
C2×#25672
#331482
#326952
#325952
#299662
#293022
#292442
#286822
#262802
#258412
#258132
#254802
#243022
#225102
#220842
C4×#4832