Finite Group Databasebeta

← #6848234

Subgroups — #6848234

Order 512.

1646 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36176C211yesmaximal, characteristic
2C2×#2551C211yesmaximal, characteristic, complemented
3#33148C211yesmaximal, characteristic, complemented
4#32246C214yesmaximal, complemented
5#32152C214yesmaximal, complemented
6#29638C212yesmaximal
7#28797C212yesmaximal, complemented
8#27960C211yesmaximal, characteristic, complemented
9#27821C212yesmaximal, complemented
10#27644C211yesmaximal, characteristic
11#27616C212yesmaximal, complemented
12#26446C212yesmaximal, complemented
13#26024C212yesmaximal, complemented
14#24608C212yesmaximal
15#22603C211yesmaximal, characteristic, complemented
16#21840C212yesmaximal, complemented
17C4×#480C211yesmaximal, characteristic
18C22×C4⋊Q8—21no—
19C22×C4⋊Q8C2212yes—
20C2×#480C2211yescharacteristic
21C2×#480C2212yes—
22C2×#432C2212yes—
23#2857C2211yescharacteristic
24#2854C2212yes—
25#2571C2211yescharacteristic
26#2563C2211yescharacteristic
27#2551—21no—
28#2551—21no—
29#2551—22no—
30#2551C2212yes—
31#2551C2212yes—
32#2551C2214yes—
33#2532—22no—
34#2513C2212yes—
35#2508C2212yes—
36#2495—21no—
37#2488—22no—
38#2486—21no—
39C2×#395—22no—
40#2369C2211yescharacteristic
41#2354C2211yescharacteristic
42#2352—21no—
43#2346—21no—
44C4×#121C2211yescharacteristic
45C2×#389—21no—
46C2×#389C2212yes—
47C2×#384C2211yescharacteristic
48#2283C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51#2101C2214yes—
52#2099C2214yes—
53#2087C2214yes—
54#2085C2214yes—
55#2083C2212yes—
56#2071C2212yes—
57#2065C2214yes—
58#2057C2214yes—
59#2057C2214yes—
60#2023C2214yes—
61#2023C2214yes—
62#1993C2214yes—
63#1993C2214yes—
64#1990C2212yes—
65#1987C2214yes—
66#1987C2212yes—
67#1987C2214yes—
68#1948C2212yes—
69#1947C2212yes—
70#1944C2214yes—
71#1939C2212yes—
72#1936C2214yes—
73#1925C2212yes—
74#1921C2214yes—
75#1920C2214yes—
76#1920C2214yes—
77#1918C2212yes—
78#1917C2214yes—
79#1874C2211yescharacteristic
80#1853C2212yes—
81#1852C2211yescharacteristic
82#1834C2212yes—
83#1823C2212yes—
84#1823C2212yes—
85#1823C2211yescharacteristic
86#1823C2212yes—
87#1823C2212yes—
88#1817C2212yes—
89#1802C2212yes—
90#1797C2211yescharacteristic
91C4×#107C2212yes—
92#1770C2212yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—22no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8—48no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2314yes—
106C2×C4⋊Q8C2314yes—
107C2×#123—22no—
108C2×C22⋊Q8—21no—
109C2×#121C2311yescharacteristic
110C2×C4×D4—21no—
111C2×C4×D4C2311yescharacteristic
112C2×#117—21no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C42—21no—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125#480—24no—
126#480—44no—
127#432—44no—
128C2×#107C2312yes—
129C2×#107C2312yes—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2314yes—
134C2×#102C2312yes—
135C2×#102C2312yes—
136C2×#102C2314yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2311yescharacteristic
139C2×C22⋊C8C2311yescharacteristic
140C2×#97C2314yes—
141C2×C4×C8C2312yes—
142#400—22no—
143#400—22no—
144#398—22no—
145#397—22no—
146#397—22no—
147#395—24no—
148#395—24no—
149#395—24no—
150#395—24no—
151#393—22no—
152#391—21no—
153#391—22no—
154#390C2312yes—
155#390C2312yes—
156#390C2311yescharacteristic
157#389—21no—
158#389—21no—
159#389—22no—
160#389—24no—
161#389—22no—
162#389—21no—
163#389—22no—
164#389—22no—
165#389C2312yes—
166#389C2312yes—
167#389C2314yes—
168#389C2311yescharacteristic
169#389C2314yes—
170#389C2312yes—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—21no—
175#388—22no—
176#388—22no—
177#387—22no—
178#386—22no—
179#386—21no—
180#384—21no—
181#384—21no—
182#384C2312yes—
183#384C2311yescharacteristic
184#384C2312yes—
185#383—22no—
186#382—21no—
187#382—22no—
188#382C2312yes—
189#382C2312yes—
190#381—22no—
191#380—21no—
192#380—21no—
193#380—21no—
194#380—21no—
195#380C2312yes—
196#380C2311yescharacteristic
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D412yes—
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—21no—
213#376—22no—
214C2×#95—21no—
215C2×#95—22no—
216C2×#95D412yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—24no—
235C22×Q8—24no—
236C22×Q8—21no—
237C22×Q8—48no—
238C22×Q8—48no—
239C22×Q8—84no—
240C22×D4—21no—
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—21no—
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C4⋊Q8—28no—
255C4⋊Q8—28no—
256C4⋊Q8—24no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8—48no—
260C4⋊Q8—48no—
261C4⋊Q8—416no—
262C4⋊Q8—48no—
263C4⋊Q8—48no—
264#123—84no—
265C22⋊Q8—82no—
266#121—22no—
267#121—22no—
268C4×D4—24no—
269C4×D4—44no—
270#117—44no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—48no—
296C2×C4⋊C4—48no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—82no—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4#4114yes—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—44no—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2412yes—
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#107—24no—
342#107—24no—
343#107—44no—
344#106—28no—
345C4⋊C8—24no—
346C4⋊C8—44no—
347#102—24no—
348#102—24no—
349#102—44no—
350C22⋊C8—22no—
351C22⋊C8—22no—
352C22⋊C8—24no—
353#97—44no—
354C4×C8—24no—
355#95—22no—
356#95—22no—
357#95—24no—
358#95—24no—
359#95—21no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376C24—21no—
377C24C22×D411yescharacteristic
378C2×Q8—48no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—42no—
383C2×Q8—84no—
384C2×Q8—84no—
385C2×Q8—81no—
386C2×Q8—81no—
387C2×Q8—832no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—42no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—42no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4AGL(1,ℤ/8)14yes—
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C2×#4112yes—
439C22×C4H(5,𝔽2)12yes—
440C22×C4C2511yescharacteristic
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—24no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—21no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—81no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4#13714yes—
484C4⋊C4C2×#4114yes—
485C4⋊C42−1+414yes—
486C22⋊C4—42no—
487C22⋊C4—44no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—44no—
492C22⋊C4—81no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C42—22no—
497C42—22no—
498C42—24no—
499C42—24no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—44no—
506C42—44no—
507C42—42no—
508C42—44no—
509C42—82no—
510C422−1+414yes—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23#54612yes—
523C23C2×AGL(1,ℤ/8)12yes—
524C23#57612yes—
525C23C23×D411yescharacteristic
526Q8—816no—
527Q8—816no—
528Q8—162no—
529D4—42no—
530D4—42no—
531D4—84no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—44no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—88no—
585C2×C4—84no—
586C2×C4#53612yes—
587C2×C4#53612yes—
588C2×C4#54612yes—
589C2×C4C2×AGL(1,ℤ/8)12yes—
590C2×C4C2×AGL(1,ℤ/8)12yes—
591C2×C4C2×#13714yes—
592C2×C4C2×#13712yes—
593C2×C4C22×#4111yescharacteristic
594C2×C4C2×H(5,𝔽2)11yescharacteristic
595C2×C4C2×2−1+412yes—
596C2×C4C2×2−1+412yes—
597C8—42no—
598C8—42no—
599C8—44no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22#332514yes—
614C22C2×#53611yescharacteristic
615C22C2×#54611yescharacteristic
616C22#348611yescharacteristic
617C22C22×AGL(1,ℤ/8)11yescharacteristic
618C22C22×#13711yescharacteristic
619C22C2×#57611yescharacteristic
620C22#358611yescharacteristic
621C4—22no—
622C4—22no—
623C4—22no—
624C4—24no—
625C4—44no—
626C4—41no—
627C4—42no—
628C4—42no—
629C4—44no—
630C4—42no—
631C4—81no—
632C4—82no—
633C4—81no—
634C4—82no—
635C4—84no—
636C4—81no—
637C4#323814yes—
638C4#350912yes—
639C4#354412yes—
640C2—24no—
641C2—161no—
642C2#3946511yescharacteristic
643C2C2×#323811yescharacteristic
644C2#6087111yescharacteristic
645C2C2×#332511yescharacteristic
646C2#6115711yescharacteristic
647C2#6182711yescharacteristic
648C2#6184411yescharacteristic
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×#2551C2semidirect
3. #33148C2semidirect
4. #32246C2semidirect
5. #32152C2semidirect
7. #28797C2semidirect
8. #27960C2semidirect
9. #27821C2semidirect
11. #27616C2semidirect
12. #26446C2semidirect
13. #26024C2semidirect
15. #22603C2semidirect
16. #21840C2semidirect

Maximal subgroups

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

subgroupindex
#361762
C2×#25512
#331482
#322462
#321522
#296382
#287972
#279602
#278212
#276442
#276162
#264462
#260242
#246082
#226032
#218402
C4×#4802