Finite Group Databasebeta

← #6852332

Subgroups — #6852332

Order 512.

1553 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36564C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic
3#32957C211yesmaximal, characteristic
4#32369C214yesmaximal, complemented
5#31545C214yesmaximal, complemented
6#29951C212yesmaximal
7#28542C212yesmaximal, complemented
8#27956C212yesmaximal
9#27823C212yesmaximal, complemented
10#27686C212yesmaximal
11#25811C211yesmaximal, characteristic
12#25454C212yesmaximal, complemented
13#24818C212yesmaximal, complemented
14#24301C211yesmaximal, characteristic
15#22629C211yesmaximal, characteristic
16#22050C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#498C2212yes—
21C2×#477C2212yes—
22C2×#475C2212yes—
23C2×#451C2211yescharacteristic
24C2×C4×Q16C2212yes—
25#2857C2212yes—
26#2857C2211yescharacteristic
27#2586—22no—
28#2568—22no—
29#2568—21no—
30#2568—21no—
31#2568C2214yes—
32#2568C2212yes—
33#2568C2212yes—
34#2567C2211yescharacteristic
35#2551C2211yescharacteristic
36#2549—22no—
37#2507—21no—
38#2507C2212yes—
39#2451C2212yes—
40#2440—21no—
41C2×#400—22no—
42#2369—21no—
43#2369C2211yescharacteristic
44Q8×C4⋊C4C2211yescharacteristic
45#2345—21no—
46C4×C22⋊Q8C2212yes—
47C4×C22⋊Q8C2211yescharacteristic
48C2×#391—21no—
49C2×#391—21no—
50C2×#391C2212yes—
51C2×#382C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2101C2214yes—
54#2097C2214yes—
55#2096C2214yes—
56#2085C2214yes—
57#2085C2214yes—
58#2066C2214yes—
59#2066C2214yes—
60#2058C2214yes—
61#2058C2212yes—
62#2024C2212yes—
63#2024C2214yes—
64#1993C2214yes—
65#1990C2212yes—
66#1988C2212yes—
67#1971C2214yes—
68#1949C2214yes—
69#1939C2212yes—
70#1935C2214yes—
71#1927C2214yes—
72#1917C2214yes—
73#1878C2214yes—
74#1878C2212yes—
75#1872C2212yes—
76#1872C2211yescharacteristic
77#1863C2214yes—
78#1863C2212yes—
79#1853C2212yes—
80#1853C2212yes—
81#1853C2212yes—
82#1852C2212yes—
83#1852C2211yescharacteristic
84#1847C2214yes—
85#1805C2214yes—
86#1803C2212yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1787C2212yes—
90#1787C2211yescharacteristic
91#1770C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8C2311yescharacteristic
97C22×Q16C2312yes—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8C2311yescharacteristic
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—44no—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C42—21no—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125#498—44no—
126#477—44no—
127#475—44no—
128#451Q818yescomplemented
129C4×Q16—44no—
130C2×#107C2314yes—
131C2×#107C2312yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2311yescharacteristic
146C2×C22⋊C8C2311yescharacteristic
147C2×#97C2314yes—
148C2×C4×C8C2312yes—
149#400—24no—
150#400—24no—
151#400—24no—
152#400—24no—
153#398—22no—
154#398—22no—
155#398—22no—
156#395—22no—
157#395—22no—
158#395—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391—24no—
163#391—22no—
164#391—24no—
165#391—22no—
166#391—22no—
167#391—21no—
168#391—22no—
169#391—22no—
170#391—22no—
171#391C2312yes—
172#391C2314yes—
173#391C2314yes—
174#391C2311yescharacteristic
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389—21no—
181#389—22no—
182#389—22no—
183#389C2312yes—
184#389C2311yescharacteristic
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#387—22no—
189#387—22no—
190#387—21no—
191#387—22no—
192#385—22no—
193#385—21no—
194#384—21no—
195#384C2312yes—
196#384C2311yescharacteristic
197#382—22no—
198#382—22no—
199#382C2314yes—
200#382C2314yes—
201#380—21no—
202#380—22no—
203#380C2312yes—
204#380C2311yescharacteristic
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
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—21no—
218C4×C22⋊C4—22no—
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—22no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—42no—
242C22×Q8—42no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8C2×Q814yes—
247C22×Q8C2411yescharacteristic
248C22×Q8C2412yes—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2411yescharacteristic
257C2×Q16—42no—
258C2×Q16—42no—
259C2×Q16C2×Q818yescomplemented
260C22×C8C2412yes—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C22⋊Q8—22no—
272C22⋊Q8—82no—
273C22⋊Q8—84no—
274C22⋊Q8C2×Q814yes—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—42no—
279C4×Q8—42no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×Q8—44no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—44no—
333C2×C42—44no—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2412yes—
340C2×C42C2412yes—
341C2×C42C2411yescharacteristic
342#107—42no—
343#107—44no—
344#106—44no—
345#106—44no—
346#106—44no—
347C4⋊C8—24no—
348C4⋊C8—44no—
349#103—44no—
350#103—42no—
351#103—42no—
352#103—42no—
353#103—44no—
354#103—44no—
355#103C2×Q818yescomplemented
356C22⋊C8—24no—
357C22⋊C8—22no—
358C22⋊C8C2×Q814yes—
359#97—44no—
360C4×C8—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—21no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—21no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24C22×D411yescharacteristic
387C2×Q8—24no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—48no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—48no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—81no—
409C2×Q8—82no—
410C2×Q8—82no—
411C2×Q8—81no—
412C2×Q8—816no—
413C2×Q8C22×Q814yes—
414C2×Q8C22×Q812yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4#13714yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×Q812yes—
457C22×C42−1+412yes—
458C22×C4C2511yescharacteristic
459Q16—48no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8C22×Q814yes—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—81no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4C22×Q812yes—
505C22⋊C4—24no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—24no—
512C42—24no—
513C42—42no—
514C42—42no—
515C42—44no—
516C42—44no—
517C42—42no—
518C42—42no—
519C42—82no—
520C42—82no—
521C42H(5,𝔽2)14yes—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23D4×Q812yes—
527C23C2×#13712yes—
528C23#57712yes—
529C23C23×D411yescharacteristic
530Q8—24no—
531Q8—24no—
532Q8—24no—
533Q8—48no—
534Q8—42no—
535Q8—42no—
536Q8—42no—
537Q8—88no—
538Q8—88no—
539Q8—162no—
540Q8—164no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—88no—
598C2×C4#53512yes—
599C2×C4#53512yes—
600C2×C4D4×Q812yes—
601C2×C4C2×#13712yes—
602C2×C4C2×#13712yes—
603C2×C4#57512yes—
604C2×C4C23×Q811yescharacteristic
605C2×C4C2×H(5,𝔽2)12yes—
606C2×C4C2×2−1+411yescharacteristic
607C2×C4#58512yes—
608C8—24no—
609C8—42no—
610C8—44no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22#338714yes—
618C22C2×#53511yescharacteristic
619C22C2×D4×Q811yescharacteristic
620C22#348511yescharacteristic
621C22C22×#13711yescharacteristic
622C22C2×#57511yescharacteristic
623C22C2×#57711yescharacteristic
624C22#358611yescharacteristic
625C4—22no—
626C4—22no—
627C4—22no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—41no—
633C4—44no—
634C4—42no—
635C4—42no—
636C4—82no—
637C4—81no—
638C4—84no—
639C4—81no—
640C4—82no—
641C4—82no—
642C4#321514yes—
643C4#350112yes—
644C4#354112yes—
645C2—24no—
646C2#3959411yescharacteristic
647C2C2×#321511yescharacteristic
648C2#6081311yescharacteristic
649C2C2×#338711yescharacteristic
650C2#6136911yescharacteristic
651C2#6172311yescharacteristic
652C2#6173911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
4. #32369C2semidirect
5. #31545C2semidirect
7. #28542C2semidirect
9. #27823C2semidirect
12. #25454C2semidirect
13. #24818C2semidirect
128. #451Q8semidirect
259. C2×Q16C2×Q8semidirect
355. #103C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365642
C2×#25682
#329572
#323692
#315452
#299512
#285422
#279562
#278232
#276862
#258112
#254542
#248182
#243012
#226292
#220502
C4×#4512