Finite Group Databasebeta

← #6541350

Subgroups — #6541350

Order 512.

1849 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C4⋊Q8C211yesmaximal, characteristic
2#34332C211yesmaximal, characteristic
3#29983C211yesmaximal, characteristic, complemented
4#29967C212yesmaximal, complemented
5#29273C211yesmaximal, characteristic, complemented
6#29206C211yesmaximal, characteristic, complemented
7#29159C212yesmaximal, complemented
8#26442C212yesmaximal, complemented
9#26340C214yesmaximal, complemented
10#26042C212yesmaximal, complemented
11#25992C214yesmaximal, complemented
12#24301C212yesmaximal, complemented
13#23776C211yesmaximal, characteristic, complemented
14C2×#2058C212yesmaximal, complemented
15#22696C211yesmaximal, characteristic
16D4×#106C212yesmaximal
17#22487C212yesmaximal
18C2×#554C2211yescharacteristic
19C2×D4×Q8—21no—
20C22×C4⋊Q8C2212yes—
21C2×#475C2214yes—
22C2×C8×D4C2212yes—
23C22×#106C2214yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2551—21no—
27#2551C2212yes—
28#2546—21no—
29#2440—22no—
30#2440C2214yes—
31#2434—22no—
32#2370—21no—
33#2369C2211yescharacteristic
34#2360—22no—
35#2360C2212yes—
36Q8×C4⋊C4C2211yescharacteristic
37D4×C4⋊C4—21no—
38D4×C4⋊C4—21no—
39D4×C4⋊C4C2212yes—
40D4×C4⋊C4C2211yescharacteristic
41#2345C2212yes—
42#2330—21no—
43#2329—22no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C22⋊Q8C2212yes—
47#2295C2211yescharacteristic
48C2×#389C2212yes—
49#2275—21no—
50C42×D4C2211yescharacteristic
51#2099C2211yescharacteristic
52#2099C2212yes—
53#2099C2212yes—
54#2096C2212yes—
55#2096C2211yescharacteristic
56#2088C2214yes—
57#2088C2214yes—
58#2088C2212yes—
59#2066C2212yes—
60#2058C2212yes—
61#2058C2212yes—
62#2058C2212yes—
63#2058C2214yes—
64#2058C2212yes—
65#2058C2214yes—
66#2058C2211yescharacteristic
67#2024C2212yes—
68#2024C2211yescharacteristic
69#1990C2212yes—
70#1988C2214yes—
71#1987C2212yes—
72#1948C2214yes—
73#1948C2212yes—
74#1948C2212yes—
75#1939C2212yes—
76#1919C2214yes—
77#1919C2214yes—
78#1892C2214yes—
79#1872C2214yes—
80#1865C2214yes—
81#1853C2214yes—
82#1853C2214yes—
83#1850C2214yes—
84#1844C2212yes—
85#1831C2211yescharacteristic
86#1824C2214yes—
87#1807C2214yes—
88#1803C2212yes—
89#1797C2212yes—
90C4×#106C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2212yes—
93C2×#340C2214yes—
94C2×#336C2212yes—
95C23×Q8—22no—
96C23×C8C2314yes—
97#554—24no—
98#554Q818yescomplemented
99D4×Q8—48no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—48no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2314yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×#125—21no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8C2312yes—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4C2314yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C42C2312yes—
131#475—48no—
132C8×D4—28no—
133C2×#106—48no—
134C2×#106C2314yes—
135C2×#106C2312yes—
136C2×#106C2312yes—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×C22⋊C8C2314yes—
147C2×C4×C8C2312yes—
148#398—24no—
149#395—24no—
150#395—24no—
151#391—21no—
152#391—21no—
153#391—22no—
154#391—22no—
155#391C2311yescharacteristic
156#391C2311yescharacteristic
157#389—21no—
158#389—21no—
159#389—22no—
160#389—22no—
161#389—21no—
162#389—21no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389C2312yes—
169#389C2312yes—
170#389C2311yescharacteristic
171#389C2312yes—
172#389C2312yes—
173#389C2311yescharacteristic
174#389C2314yes—
175#387—21no—
176#387—22no—
177#387—21no—
178#387—22no—
179#387—22no—
180#387—22no—
181#385—22no—
182#385—22no—
183#385—22no—
184#385—24no—
185#385—24no—
186#384—21no—
187#384—21no—
188#384—21no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2312yes—
193#380—22no—
194#380—22no—
195#380C2312yes—
196#380C2314yes—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216#376—21no—
217C2×#95—22no—
218C43C2311yescharacteristic
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2312yes—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—24no—
239C22×Q8—48no—
240C22×Q8—48no—
241C22×Q8—84no—
242C22×Q8C2411yescharacteristic
243C22×D4C2×D411yescharacteristic
244C23×C4—22no—
245C23×C4—24no—
246C23×C4—22no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C22×C8—28no—
251C22×C8—28no—
252C22×C8—44no—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C4⋊Q8—24no—
257C4⋊Q8—28no—
258C4⋊Q8—22no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261C4⋊Q8—48no—
262C4⋊Q8—48no—
263C4⋊Q8—48no—
264C4⋊Q8—44no—
265C4⋊Q8—82no—
266C4⋊Q8C2×Q814yescomplemented
267#125—28no—
268C22⋊Q8—24no—
269C22⋊Q8—216no—
270C22⋊Q8—48no—
271C22⋊Q8—416no—
272C22⋊Q8C2×Q818yescomplemented
273C4×Q8—22no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×Q8—82no—
277C4×Q8C2×Q814yescomplemented
278C4×D4—28no—
279C4×D4—24no—
280C4×D4—48no—
281C4×D4—44no—
282C4×D4SD1618yes—
283C4×D4C2×Q814yes—
284C4×D4#4114yes—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4—84no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×Q818yescomplemented
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C42—21no—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—44no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#106—24no—
352#106—24no—
353#106—28no—
354#106—44no—
355#106—44no—
356#106—44no—
357C4⋊C8—24no—
358C4⋊C8—24no—
359C4⋊C8—44no—
360#103—24no—
361#103—24no—
362#103—44no—
363C22⋊C8—28no—
364C4×C8—24no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—21no—
370#95—21no—
371#95—21no—
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—24no—
381#95—24no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24C22×D412yes—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—48no—
392C2×Q8—42no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—48no—
397C2×Q8—44no—
398C2×Q8—81no—
399C2×Q8—81no—
400C2×Q8—84no—
401C2×Q8—832no—
402C2×Q8C22×Q812yes—
403C2×D4C22⋊Q814yes—
404C2×D4C2×SD1614yes—
405C2×D4#13714yes—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—28no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—44no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—84no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4C2×SD1618yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×Q814yes—
442C22×C4C2×#4114yes—
443C22×C4C2511yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—416no—
453C2×C82−1+414yes—
454C4⋊C4—24no—
455C4⋊C4—28no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—28no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—81no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—81no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4C22⋊Q814yes—
491C4⋊C4C2×SD1614yes—
492C4⋊C4C22×Q812yes—
493C4⋊C4C22×Q814yes—
494C4⋊C4H(5,𝔽2)14yes—
495C22⋊C4—28no—
496C22⋊C4—24no—
497C22⋊C4—48no—
498C22⋊C4—44no—
499C22⋊C4C22⋊Q818yes—
500C22⋊C4C2×SD1618yes—
501C42—24no—
502C42—22no—
503C42—22no—
504C42—24no—
505C42—22no—
506C42—22no—
507C42—42no—
508C42—42no—
509C42—41no—
510C42—44no—
511C42—42no—
512C42—82no—
513C42—82no—
514C42—82no—
515C42C2×SD1614yes—
516C42C22×Q812yes—
517C42C2×#4112yes—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23C2×C22⋊Q814yes—
523C23C22×SD1614yes—
524C23C2×#13714yes—
525C23C23×D411yescharacteristic
526Q8—24no—
527Q8—48no—
528Q8—42no—
529Q8—84no—
530Q8—816no—
531Q8—88no—
532D4#475116yescomplemented
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—88no—
583C2×C4—816no—
584C2×C4—164no—
585C2×C4C2×C22⋊Q812yes—
586C2×C4C2×C22⋊Q814yes—
587C2×C4C2×C22⋊Q812yes—
588C2×C4C22×SD1612yes—
589C2×C4C22×SD1612yes—
590C2×C4C22×SD1614yes—
591C2×C4C22×SD1612yes—
592C2×C4C2×#13712yes—
593C2×C4#57712yes—
594C2×C4C23×Q811yescharacteristic
595C2×C4C22×#4111yescharacteristic
596C2×C4C2×H(5,𝔽2)12yes—
597C2×C4C2×2−1+412yes—
598C8—24no—
599C8—42no—
600C8—44no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22C2×#47518yes—
608C22C22×C22⋊Q811yescharacteristic
609C22#348411yescharacteristic
610C22#349911yescharacteristic
611C22C23×SD1611yescharacteristic
612C22C22×#13711yescharacteristic
613C22C2×#57711yescharacteristic
614C22#358511yescharacteristic
615C4—22no—
616C4—22no—
617C4—22no—
618C4—24no—
619C4—22no—
620C4—24no—
621C4—41no—
622C4—42no—
623C4—44no—
624C4—44no—
625C4—42no—
626C4—42no—
627C4—82no—
628C4—81no—
629C4—84no—
630C4—82no—
631C4—81no—
632C4C2×#47514yes—
633C4#350312yes—
634C4#352012yes—
635C2—28no—
636C2#3573911yescharacteristic
637C2C22×#47511yescharacteristic
638C2#6049511yescharacteristic
639C2#6156011yescharacteristic
640C2#6159311yescharacteristic
641C2#6246911yescharacteristic
642C2#6250411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29983C2semidirect
4. #29967C2semidirect
5. #29273C2semidirect
6. #29206C2semidirect
7. #29159C2semidirect
8. #26442C2semidirect
9. #26340C2semidirect
10. #26042C2semidirect
11. #25992C2semidirect
12. #24301C2semidirect
13. #23776C2semidirect
14. C2×#2058C2semidirect
98. #554Q8semidirect
266. C4⋊Q8C2×Q8semidirect
272. C22⋊Q8C2×Q8semidirect
277. C4×Q8C2×Q8semidirect
319. C2×C4⋊C4C2×Q8semidirect
532. D4#475semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#343322
#299832
#299672
#292732
#292062
#291592
#264422
#263402
#260422
#259922
#243012
#237762
C2×#20582
#226962
D4×#1062
#224872