Finite Group Databasebeta

← #7567252

Subgroups — #7567252

Order 512.

1657 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59821C211yesmaximal, characteristic, complemented
2#36180C211yesmaximal, characteristic, complemented
3#32257C214yesmaximal, complemented
4#29218C211yesmaximal, characteristic, complemented
5#28716C211yesmaximal, characteristic, complemented
6#28645C212yesmaximal, complemented
7#27822C212yesmaximal, complemented
8#27571C214yesmaximal, complemented
9#26282C212yesmaximal, complemented
10#26025C212yesmaximal, complemented
11#25930C212yesmaximal
12#24811C212yesmaximal, complemented
13#23153C211yesmaximal, characteristic, complemented
14#22758C212yesmaximal, complemented
15#22522C212yesmaximal, complemented
16C4×#474C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#556—21no—
19C2×#554C2212yes—
20C2×#486C2212yes—
21C2×#474C2211yescharacteristic
22C2×#459C2211yescharacteristic
23C2×#446C2212yes—
24C2×C4×D8C2212yes—
25#2868C2211yescharacteristic
26#2857—21no—
27#2727—22no—
28#2567C2212yes—
29#2552C2212yes—
30#2552C2212yes—
31#2548C2212yes—
32#2542C2212yes—
33#2538—21no—
34#2512C2214yes—
35#2502—22no—
36#2448—22no—
37#2440—22no—
38#2440—22no—
39#2415—22no—
40#2354C2212yes—
41Q8×C4⋊C4C2212yes—
42D4×C4⋊C4C2212yes—
43D4×C4⋊C4C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C2×#384—21no—
48C42×D4C2211yescharacteristic
49#2100C2214yes—
50#2100C2214yes—
51#2083C2214yes—
52#2071C2214yes—
53#2067C2214yes—
54#2067C2214yes—
55#2025C2214yes—
56#2025C2214yes—
57#1992C2214yes—
58#1990C2212yes—
59#1990C2212yes—
60#1989C2214yes—
61#1988C2211yescharacteristic
62#1987C2212yes—
63#1987C2212yes—
64#1984C2212yes—
65#1982C2211yescharacteristic
66#1947C2211yescharacteristic
67#1947C2211yescharacteristic
68#1939C2212yes—
69#1936C2214yes—
70#1936C2214yes—
71#1925C2212yes—
72#1921C2214yes—
73#1921C2214yes—
74#1918C2212yes—
75#1917C2214yes—
76#1879C2214yes—
77#1879C2214yes—
78#1879C2212yes—
79#1862C2212yes—
80#1834C2212yes—
81#1802C2212yes—
82#1802C2212yes—
83#1802C2211yescharacteristic
84#1802C2214yes—
85#1802C2211yescharacteristic
86C4×#107C2212yes—
87#1770C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90#1762C2214yes—
91C4×#102C2212yes—
92C4×#102C2212yes—
93C22×D8C2312yes—
94#556—48no—
95#554—28no—
96#554—44no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8D411yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#126—24no—
104C2×#125C2314yes—
105C2×#124—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8C2312yes—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4C2312yes—
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2314yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—22no—
124C22×C42—21no—
125#486—44no—
126#474—24no—
127#459—24no—
128#446—44no—
129C4×D8—28no—
130C2×#107C2312yes—
131C2×#107C2312yes—
132C2×#107C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2314yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#97C2314yes—
147C2×C4×C8C2312yes—
148C2×C4×C8C2312yes—
149#399—24no—
150#398—24no—
151#398—24no—
152#397—24no—
153#396—24no—
154#395—24no—
155#394—24no—
156#391—22no—
157#391—21no—
158#391D412yes—
159#390—22no—
160#390—21no—
161#390—21no—
162#390C2312yes—
163#390C2314yes—
164#390C2314yes—
165#389—22no—
166#389—22no—
167#389—24no—
168#389—22no—
169#389—21no—
170#389D412yes—
171#389D412yes—
172#389C2312yes—
173#389C2314yes—
174#389C2314yes—
175#388—24no—
176#388—24no—
177#385—24no—
178#385—24no—
179#384—22no—
180#384—22no—
181#384—22no—
182#384—21no—
183#384—21no—
184#384D411yescharacteristic
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2314yes—
189#384C2312yes—
190#380—22no—
191#380—22no—
192#380—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2311yescharacteristic
196C4×C4⋊C4C2314yes—
197C4×C4⋊C4C2311yescharacteristic
198C4×C4⋊C4C2311yescharacteristic
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C22⋊C4—21no—
202C4×C22⋊C4—21no—
203C4×C22⋊C4—21no—
204C43C2311yescharacteristic
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2312yes—
209#340C2312yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#340C2312yes—
217#336C2312yes—
218#336C2314yes—
219#336C2312yes—
220#336C2314yes—
221C22×Q8—21no—
222C22×Q8—22no—
223C22×Q8C2×D412yes—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4C2412yes—
227C22×D4C2411yescharacteristic
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—22no—
232C23×C4—24no—
233C2×D8—24no—
234C2×D8—24no—
235C2×D8—24no—
236C22×C8C2412yes—
237C22×C8C2412yes—
238C22×C8C2412yes—
239C22×C8C2412yes—
240C4⋊Q8—24no—
241C4⋊Q8—24no—
242C4⋊Q8—22no—
243C4⋊Q8—24no—
244C4⋊Q8—22no—
245C4⋊Q8—82no—
246C4⋊Q8—82no—
247#127—22no—
248#127—22no—
249#126—416no—
250#125—28no—
251#125—44no—
252#124—48no—
253C22⋊Q8—416no—
254C22⋊Q8—48no—
255C22⋊Q8—48no—
256C22⋊Q8—84no—
257C4×Q8—24no—
258C4×Q8—24no—
259C4×Q8—48no—
260C4×Q8—44no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—28no—
264C4×D4—24no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—24no—
268C4×D4—44no—
269C4×D4—44no—
270C4×D4—82no—
271C4×D4#4114yes—
272C4×D4#4114yes—
273#117—48no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—48no—
292C2×C4⋊C4—84no—
293C2×C4⋊C4—84no—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D414yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2411yescharacteristic
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—44no—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2414yes—
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332#107—24no—
333#107—24no—
334#107—24no—
335C4⋊C8—22no—
336C4⋊C8—22no—
337C4⋊C8—22no—
338C4⋊C8—22no—
339C4⋊C8—44no—
340#102—24no—
341#102—24no—
342#102—24no—
343#102—24no—
344#102—24no—
345#102—24no—
346#102—44no—
347#102—44no—
348#97—44no—
349C4×C8—24no—
350C4×C8—24no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—22no—
355#95—24no—
356#95—22no—
357#95—24no—
358#95—24no—
359#95C2×D414yes—
360#95C2×D414yes—
361C24—21no—
362C24—22no—
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—44no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—82no—
371C2×Q8—82no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—21no—
375C2×D4—22no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—81no—
386C2×D4—81no—
387C2×D4C2×#4112yes—
388C2×D4C2×#4112yes—
389C2×D42−1+412yes—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—42no—
401C22×C4—48no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—84no—
411C22×C4—84no—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D414yes—
414C22×C4C22×D414yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C2511yescharacteristic
421D8—48no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C4⋊C4—24no—
435C4⋊C4—28no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—81no—
462C4⋊C4—84no—
463C4⋊C4—84no—
464C4⋊C4—81no—
465C4⋊C4—84no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4C2×#4112yes—
469C4⋊C4C2×#4112yes—
470C4⋊C42−1+412yes—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—84no—
480C22⋊C4—82no—
481C22⋊C4—84no—
482C42—22no—
483C42—22no—
484C42—24no—
485C42—24no—
486C42—21no—
487C42—21no—
488C42—41no—
489C42—48no—
490C42—42no—
491C42—42no—
492C42—42no—
493C42—44no—
494C42AGL(1,ℤ/8)14yes—
495C42C2×#4112yes—
496C42C2×#4112yes—
497C42H(5,𝔽2)12yes—
498C422−1+412yes—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23C23×D411yescharacteristic
514Q8—48no—
515Q8—84no—
516Q8—164no—
517D4—22no—
518D4—22no—
519D4—22no—
520D4—42no—
521D4—42no—
522D4—42no—
523D4—42no—
524D4—84no—
525D4—84no—
526D4#55614yescomplemented
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—21no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—24no—
545C2×C4—48no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—88no—
567C2×C4—168no—
568C2×C4#54612yes—
569C2×C4#54612yes—
570C2×C4#54812yes—
571C2×C4#54812yes—
572C2×C4C2×AGL(1,ℤ/8)12yes—
573C2×C4C2×AGL(1,ℤ/8)12yes—
574C2×C4C2×AGL(1,ℤ/8)12yes—
575C2×C4#57612yes—
576C2×C4#57812yes—
577C2×C4C22×#4111yescharacteristic
578C2×C4C22×#4111yescharacteristic
579C2×C4C2×H(5,𝔽2)11yescharacteristic
580C2×C4C2×2−1+411yescharacteristic
581C2×C4C2×2−1+411yescharacteristic
582C2×C4C2×2−1+411yescharacteristic
583C8—42no—
584C8—42no—
585C8—42no—
586C8—42no—
587C22—24no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22C2×#54611yescharacteristic
602C22C2×#54811yescharacteristic
603C22#348611yescharacteristic
604C22C22×AGL(1,ℤ/8)11yescharacteristic
605C22C2×#57611yescharacteristic
606C22C2×#57811yescharacteristic
607C22#358511yescharacteristic
608C4—22no—
609C4—22no—
610C4—22no—
611C4—44no—
612C4—42no—
613C4—42no—
614C4—41no—
615C4—44no—
616C4—42no—
617C4—84no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4#333014yes—
622C4#336614yes—
623C4C2×#55611yescharacteristic
624C4#350911yescharacteristic
625C4#351111yescharacteristic
626C4#355211yescharacteristic
627C2—44no—
628C2—162no—
629C2#5913411yescharacteristic
630C2C2×#333011yescharacteristic
631C2#6120511yescharacteristic
632C2C2×#336611yescharacteristic
633C2#6134811yescharacteristic
634C2#6182511yescharacteristic
635C2#6187011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59821C2semidirect
2. #36180C2semidirect
3. #32257C2semidirect
4. #29218C2semidirect
5. #28716C2semidirect
6. #28645C2semidirect
7. #27822C2semidirect
8. #27571C2semidirect
9. #26282C2semidirect
10. #26025C2semidirect
12. #24811C2semidirect
13. #23153C2semidirect
14. #22758C2semidirect
15. #22522C2semidirect
16. C4×#474C2semidirect
526. D4#556semidirect

Maximal subgroups

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

subgroupindex
#598212
#361802
#322572
#292182
#287162
#286452
#278222
#275712
#262822
#260252
#259302
#248112
#231532
#227582
#225222
C4×#4742
C4×#4592