Finite Group Databasebeta

← #6851930

Subgroups — #6851930

Order 512.

1683 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36564C211yesmaximal, characteristic
2C2×#2551C211yesmaximal, characteristic
3#33148C211yesmaximal, characteristic
4#32132C214yesmaximal, complemented
5#30949C214yesmaximal, complemented
6#29968C212yesmaximal
7#28809C212yesmaximal, complemented
8#27948C211yesmaximal, characteristic
9#27815C212yesmaximal, complemented
10#27680C211yesmaximal, characteristic
11#26442C212yesmaximal, complemented
12#26042C212yesmaximal, complemented
13#25811C212yesmaximal
14#24301C212yesmaximal
15#22625C211yesmaximal, characteristic
16#22078C212yesmaximal
17C4×#483C211yesmaximal, characteristic
18C22×C4⋊Q8—21no—
19C22×C4⋊Q8C2212yes—
20C2×#498C2214yes—
21C2×#483C2211yescharacteristic
22C2×#475C2214yes—
23#2857C2212yes—
24#2857C2211yescharacteristic
25#2586—22no—
26#2567C2211yescharacteristic
27#2551—22no—
28#2551—21no—
29#2551—21no—
30#2551C2212yes—
31#2551C2214yes—
32#2551C2212yes—
33#2551C2211yescharacteristic
34#2549—22no—
35#2507—21no—
36#2507C2212yes—
37#2451C2212yes—
38#2440—21no—
39C2×#395—22no—
40#2369—21no—
41#2369C2211yescharacteristic
42Q8×C4⋊C4C2211yescharacteristic
43#2345—21no—
44C4×C22⋊Q8C2211yescharacteristic
45C4×C22⋊Q8C2212yes—
46C2×#389—21no—
47C2×#389C2212yes—
48C2×#384C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51#2100C2214yes—
52#2099C2212yes—
53#2096C2212yes—
54#2088C2214yes—
55#2088C2214yes—
56#2084C2214yes—
57#2058C2214yes—
58#2058C2214yes—
59#2058C2214yes—
60#2058C2214yes—
61#2024C2214yes—
62#1990C2212yes—
63#1990C2214yes—
64#1988C2214yes—
65#1987C2212yes—
66#1948C2212yes—
67#1948C2212yes—
68#1941C2214yes—
69#1939C2212yes—
70#1933C2214yes—
71#1927C2212yes—
72#1920C2212yes—
73#1920C2214yes—
74#1919C2214yes—
75#1916C2214yes—
76#1872C2212yes—
77#1872C2211yescharacteristic
78#1865C2214yes—
79#1853C2212yes—
80#1853C2212yes—
81#1853C2212yes—
82#1853C2211yescharacteristic
83#1853C2212yes—
84#1844C2212yes—
85#1823C2211yescharacteristic
86#1823C2212yes—
87#1803C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90C4×#106C2212yes—
91C8×C4⋊C4C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C23×Q8—22no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—48no—
103C2×C4⋊Q8—48no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2314yes—
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2312yes—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2312yes—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C42—21no—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125#498—48no—
126#483Q818yescomplemented
127#475—48no—
128C2×#106C2314yes—
129C2×#106C2312yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×#106C2312yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×C22⋊C8C2311yescharacteristic
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2311yescharacteristic
142C2×C4×C8C2312yes—
143C2×C4×C8C2314yes—
144#398—22no—
145#398—22no—
146#398—22no—
147#395—22no—
148#395—22no—
149#395—22no—
150#395—24no—
151#395—24no—
152#395—24no—
153#395—24no—
154#391—21no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391C2312yes—
159#391C2311yescharacteristic
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—22no—
165#389—24no—
166#389—22no—
167#389—22no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2314yes—
177#389C2312yes—
178#389C2314yes—
179#389C2311yescharacteristic
180#387—22no—
181#387—22no—
182#387—21no—
183#387—22no—
184#385—22no—
185#385—21no—
186#384—21no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2312yes—
191#384C2312yes—
192#384C2311yescharacteristic
193#380—21no—
194#380—22no—
195#380C2312yes—
196#380C2311yescharacteristic
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C22⋊C4—21no—
214C4×C22⋊C4—22no—
215C2×#95—21no—
216C2×#95—22no—
217C2×#95D412yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#336C2314yes—
230#336C2312yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—24no—
234C22×Q8—24no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8—48no—
238C22×Q8—48no—
239C22×Q8—84no—
240C22×Q8C2411yescharacteristic
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—22no—
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D412yes—
248C23×C4C2411yescharacteristic
249C22×C8C2414yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C4⋊Q8—28no—
253C4⋊Q8—28no—
254C4⋊Q8—24no—
255C4⋊Q8—44no—
256C4⋊Q8—44no—
257C4⋊Q8—44no—
258C4⋊Q8—48no—
259C4⋊Q8—48no—
260C4⋊Q8—416no—
261C4⋊Q8—48no—
262C4⋊Q8—48no—
263C4⋊Q8—44no—
264C22⋊Q8—22no—
265C22⋊Q8—84no—
266C22⋊Q8—82no—
267C22⋊Q8C2×Q814yes—
268C4×Q8—24no—
269C4×Q8—44no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4—48no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4SD1618yes—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×Q814yes—
311C2×C4⋊C4C2414yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C42—24no—
323C2×C42—21no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—21no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—44no—
330C2×C42—44no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2412yes—
336C2×C42C2412yes—
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339#106—28no—
340#106—24no—
341#106—44no—
342#106—44no—
343#106—44no—
344#106C2×Q818yescomplemented
345C4⋊C8—24no—
346C4⋊C8—44no—
347#103—24no—
348#103—44no—
349#103C2×Q818yescomplemented
350C22⋊C8—22no—
351C22⋊C8—24no—
352C22⋊C8C2×Q814yes—
353C4×C8—24no—
354C4×C8—44no—
355#95—22no—
356#95—22no—
357#95—24no—
358#95—24no—
359#95—21no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—21no—
367#95—22no—
368#95—21no—
369#95—22no—
370#95—21no—
371#95—22no—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376C24C22×D411yescharacteristic
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—42no—
383C2×Q8—48no—
384C2×Q8—48no—
385C2×Q8—44no—
386C2×Q8—84no—
387C2×Q8—84no—
388C2×Q8—81no—
389C2×Q8—82no—
390C2×Q8—82no—
391C2×Q8—81no—
392C2×Q8—832no—
393C2×Q8C22×Q812yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—41no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4C2×SD1614yes—
425C22×C4C2×SD1618yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×Q812yes—
436C22×C42−1+412yes—
437C22×C4C2511yescharacteristic
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—24no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8C22×Q814yes—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—82no—
473C4⋊C4—81no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—81no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4C2×SD1618yes—
484C4⋊C4C2×SD1614yes—
485C4⋊C4C2×SD1614yes—
486C4⋊C4#13714yes—
487C4⋊C4C22×Q814yes—
488C4⋊C4C22×Q812yes—
489C4⋊C4H(5,𝔽2)14yes—
490C22⋊C4—24no—
491C22⋊C4—42no—
492C22⋊C4—44no—
493C42—22no—
494C42—22no—
495C42—24no—
496C42—24no—
497C42—22no—
498C42—42no—
499C42—42no—
500C42—42no—
501C42—44no—
502C42—44no—
503C42—42no—
504C42—82no—
505C42—82no—
506C422−1+414yes—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23D4×Q812yes—
512C23C22×SD1612yes—
513C23#57712yes—
514C23C23×D411yescharacteristic
515Q8—24no—
516Q8—42no—
517Q8—816no—
518Q8—816no—
519Q8—164no—
520Q8—162no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—48no—
547C2×C4—48no—
548C2×C4—44no—
549C2×C4—48no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—48no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—88no—
575C2×C4#53612yes—
576C2×C4#53612yes—
577C2×C4D4×Q812yes—
578C2×C4C22×SD1612yes—
579C2×C4C22×SD1614yes—
580C2×C4C22×SD1614yes—
581C2×C4C22×SD1612yes—
582C2×C4C2×#13714yes—
583C2×C4C2×#13712yes—
584C2×C4C23×Q811yescharacteristic
585C2×C4C2×H(5,𝔽2)12yes—
586C2×C4C2×2−1+412yes—
587C2×C4C2×2−1+411yescharacteristic
588C8—24no—
589C8—44no—
590C8—42no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22Q8×SD1614yes—
598C22C2×#53611yescharacteristic
599C22C2×D4×Q811yescharacteristic
600C22#348411yescharacteristic
601C22C23×SD1611yescharacteristic
602C22C22×#13711yescharacteristic
603C22C2×#57711yescharacteristic
604C22#358511yescharacteristic
605C4—24no—
606C4—24no—
607C4—24no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—44no—
612C4—42no—
613C4—42no—
614C4—44no—
615C4—42no—
616C4—41no—
617C4—84no—
618C4—82no—
619C4—81no—
620C4—81no—
621C4—82no—
622C4#322914yes—
623C4#350312yes—
624C4#355212yes—
625C2—24no—
626C2#3959211yescharacteristic
627C2C2×#322911yescharacteristic
628C2#6087311yescharacteristic
629C2C2×Q8×SD1611yescharacteristic
630C2Q8×#13711yescharacteristic
631C2#6156011yescharacteristic
632C2#6159311yescharacteristic
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. #32132C2semidirect
5. #30949C2semidirect
7. #28809C2semidirect
9. #27815C2semidirect
11. #26442C2semidirect
12. #26042C2semidirect
126. #483Q8semidirect
344. #106C2×Q8semidirect
349. #103C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365642
C2×#25512
#331482
#321322
#309492
#299682
#288092
#279482
#278152
#276802
#264422
#260422
#258112
#243012
#226252
#220782
C4×#4832