Finite Group Databasebeta

← #6616256

Subgroups — #6616256

Order 512.

1679 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36133C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic
3#33108C211yesmaximal, characteristic
4#33100C212yesmaximal
5#27021C214yesmaximal, complemented
6#26711C214yesmaximal, complemented
7#26442C212yesmaximal, complemented
8#25902C211yesmaximal, characteristic
9#25278C212yesmaximal, complemented
10#24609C212yesmaximal
11#24302C211yesmaximal, characteristic
12#24063C212yesmaximal
13#23221C212yesmaximal, complemented
14#22971C212yesmaximal, complemented
15#22603C211yesmaximal, characteristic
16C4×#483C211yesmaximal, characteristic
17#21394C212yesmaximal
18C2×#533C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C2×#487C2212yes—
21C2×#483C2211yescharacteristic
22C2×#483C2212yes—
23C2×#477C2212yes—
24C2×#432C2212yes—
25#2857C2211yescharacteristic
26#2855C2212yes—
27#2551C2211yescharacteristic
28#2546C2211yescharacteristic
29#2506—21no—
30#2462—22no—
31#2451C2212yes—
32#2446C2212yes—
33#2440—21no—
34#2438—22no—
35#2369—21no—
36#2369—22no—
37#2369—21no—
38#2369C2212yes—
39#2369C2214yes—
40#2369C2212yes—
41#2365—21no—
42#2365C2211yescharacteristic
43Q8×C4⋊C4C2211yescharacteristic
44#2344—21no—
45C4×C22⋊Q8C2211yescharacteristic
46C2×#391—22no—
47C2×#389—21no—
48C2×#389C2212yes—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2088C2214yes—
54#2068C2214yes—
55#2058C2214yes—
56#2057C2212yes—
57#2043C2214yes—
58#2024C2214yes—
59#2024C2212yes—
60#1990C2212yes—
61#1988C2214yes—
62#1987C2212yes—
63#1981C2212yes—
64#1981C2214yes—
65#1948C2212yes—
66#1943C2214yes—
67#1939C2212yes—
68#1935C2214yes—
69#1928C2214yes—
70#1927C2212yes—
71#1927C2214yes—
72#1927C2214yes—
73#1920C2212yes—
74#1919C2214yes—
75#1918C2214yes—
76#1917C2214yes—
77#1872C2212yes—
78#1872C2212yes—
79#1872C2211yescharacteristic
80#1872C2212yes—
81#1872C2212yes—
82#1865C2214yes—
83#1865C2214yes—
84#1853C2211yescharacteristic
85#1852C2212yes—
86#1823C2211yescharacteristic
87#1817C2212yes—
88#1797C2211yescharacteristic
89C4×#106C2212yes—
90#1770C2212yes—
91#1763C2214yes—
92#1763C2214yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96#533—28no—
97#533—44no—
98C2×C4⋊Q8—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×#125—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2311yescharacteristic
111C2×C4×Q8—22no—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×#117—22no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C42—21no—
124C22×C42—22no—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127#487—28no—
128#483—24no—
129#483—44no—
130#477—28no—
131#432—44no—
132C2×#107C2314yes—
133C2×#106C2312yes—
134C2×#106C2312yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2311yescharacteristic
147C2×C22⋊C8C2311yescharacteristic
148C2×#97C2314yes—
149C2×C4×C8C2312yes—
150#400—22no—
151#398—22no—
152#398—22no—
153#398—22no—
154#395—22no—
155#395—22no—
156#391—22no—
157#391—24no—
158#391—24no—
159#391—24no—
160#391—24no—
161#391C2312yes—
162#391C2312yes—
163#391C2312yes—
164#391C2311yescharacteristic
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389—24no—
171#389—22no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389C2311yescharacteristic
178#389C2314yes—
179#389C2312yes—
180#389C2312yes—
181#389C2314yes—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—21no—
187#387—22no—
188#387—22no—
189#385—22no—
190#385—21no—
191#384—21no—
192#384C2311yescharacteristic
193#381—22no—
194#380—21no—
195#380—21no—
196#380C2311yescharacteristic
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—24no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D412yes—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C2×#95—22no—
220C2×#95—22no—
221C2×#95—21no—
222C2×#95D412yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—28no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—82no—
241C22×Q8—82no—
242C22×Q8C2412yes—
243C22×Q8C2411yescharacteristic
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—28no—
256C4⋊Q8—28no—
257C4⋊Q8—28no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—42no—
263C4⋊Q8—42no—
264C4⋊Q8—44no—
265C4⋊Q8#4118yes—
266#125—48no—
267C22⋊Q8—22no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—48no—
271C22⋊Q8#4114yes—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—42no—
276C4×Q8—42no—
277C4×Q8—48no—
278#117—48no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—24no—
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—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—84no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—44no—
337C2×C42—84no—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346#107—28no—
347#106—24no—
348#106—24no—
349#106—44no—
350C4⋊C8—24no—
351C4⋊C8—44no—
352#103—24no—
353#103—28no—
354#103—28no—
355#103—28no—
356#103—24no—
357#103—28no—
358#103—44no—
359C22⋊C8—22no—
360C22⋊C8—22no—
361C22⋊C8—24no—
362#97—44no—
363C4×C8—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—22no—
369#95—22no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24C22×D411yescharacteristic
388C2×Q8—28no—
389C2×Q8—28no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—82no—
400C2×Q8—82no—
401C2×Q8—82no—
402C2×Q8—82no—
403C2×Q8—82no—
404C2×Q8—816no—
405C2×Q8C2×#4112yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4#13714yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C2×#4112yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C4C2511yescharacteristic
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—22no—
459C4⋊C4—24no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—28no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—82no—
492C4⋊C4—81no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C4⋊C4#13714yes—
496C4⋊C4C2×#4118yes—
497C4⋊C4C2×#4112yes—
498C4⋊C42−1+414yes—
499C22⋊C4—24no—
500C22⋊C4—44no—
501C22⋊C4—42no—
502C42—22no—
503C42—22no—
504C42—24no—
505C42—24no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—44no—
512C42—44no—
513C42—42no—
514C42—44no—
515C42—44no—
516C42—42no—
517C42—84no—
518C42—84no—
519C42C2×#4114yes—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23#54612yes—
525C23C2×#13712yes—
526C23#57712yes—
527C23C23×D411yescharacteristic
528Q8—24no—
529Q8—42no—
530Q8—44no—
531Q8—44no—
532Q8—88no—
533Q8—88no—
534Q8—88no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—48no—
563C2×C4—44no—
564C2×C4—48no—
565C2×C4—416no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—84no—
591C2×C4#52512yes—
592C2×C4#52512yes—
593C2×C4#54612yes—
594C2×C4C2×#13712yes—
595C2×C4C2×#13714yes—
596C2×C4C2×#13712yes—
597C2×C4C2×#13712yes—
598C2×C4C22×#4111yescharacteristic
599C2×C4C22×#4112yes—
600C2×C4C2×H(5,𝔽2)11yescharacteristic
601C2×C4C2×2−1+412yes—
602C8—42no—
603C8—42no—
604C8—44no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22#332914yes—
612C22C2×#52511yescharacteristic
613C22C2×#54611yescharacteristic
614C22#348611yescharacteristic
615C22C22×#13711yescharacteristic
616C22C22×#13711yescharacteristic
617C22C2×#57711yescharacteristic
618C22#358511yescharacteristic
619C4—22no—
620C4—22no—
621C4—22no—
622C4—24no—
623C4—42no—
624C4—44no—
625C4—41no—
626C4—48no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—82no—
632C4—81no—
633C4—81no—
634C4—82no—
635C4—82no—
636C4#311314yes—
637C4#348212yes—
638C4#352212yes—
639C2—24no—
640C2#3642211yescharacteristic
641C2C2×#311311yescharacteristic
642C2#6057911yescharacteristic
643C2C2×#332911yescharacteristic
644C2#6116211yescharacteristic
645C2#6182611yescharacteristic
646C2#6183811yescharacteristic
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
5. #27021C2semidirect
6. #26711C2semidirect
7. #26442C2semidirect
9. #25278C2semidirect
13. #23221C2semidirect
14. #22971C2semidirect

Maximal subgroups

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

subgroupindex
#361332
C2×#23692
#331082
#331002
#270212
#267112
#264422
#259022
#252782
#246092
#243022
#240632
#232212
#229712
#226032
C4×#4832
#213942