Finite Group Databasebeta

← #7545513

Subgroups — #7545513

Order 512.

1655 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59823C211yesmaximal, characteristic, complemented
2#36134C211yesmaximal, characteristic, complemented
3#32544C214yesmaximal, complemented
4#29219C211yesmaximal, characteristic, complemented
5#28716C211yesmaximal, characteristic, complemented
6#27727C212yesmaximal, complemented
7#27727C212yesmaximal, complemented
8#26506C212yesmaximal, complemented
9#26106C212yesmaximal, complemented
10#25318C212yesmaximal
11#24990C214yesmaximal, complemented
12#24676C212yesmaximal, complemented
13#23247C211yesmaximal, characteristic, complemented
14#23216C212yesmaximal, complemented
15#22766C212yesmaximal, complemented
16C4×#474C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#554—21no—
19C2×#533—21no—
20C2×#486C2212yes—
21C2×#474C2211yescharacteristic
22C2×#459C2211yescharacteristic
23C2×C4×D8C2212yes—
24#2868C2211yescharacteristic
25#2857—21no—
26#2732—22no—
27#2552C2212yes—
28#2552C2212yes—
29#2542C2212yes—
30#2542C2212yes—
31#2538—21no—
32#2538C2212yes—
33#2538C2214yes—
34#2454—22no—
35#2441—22no—
36#2429—22no—
37#2423—22no—
38#2417—22no—
39#2366C2212yes—
40#2365C2212yes—
41#2354C2212yes—
42Q8×C4⋊C4C2212yes—
43D4×C4⋊C4C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C42×D4C2211yescharacteristic
48#2099C2214yes—
49#2099C2214yes—
50#2099C2214yes—
51#2086C2214yes—
52#2071C2214yes—
53#2071C2214yes—
54#2071C2214yes—
55#2054C2214yes—
56#1990C2212yes—
57#1990C2212yes—
58#1988C2212yes—
59#1988C2212yes—
60#1988C2211yescharacteristic
61#1984C2214yes—
62#1982C2211yescharacteristic
63#1947C2214yes—
64#1947C2211yescharacteristic
65#1947C2211yescharacteristic
66#1939C2212yes—
67#1939C2212yes—
68#1934C2214yes—
69#1934C2214yes—
70#1925C2214yes—
71#1925C2212yes—
72#1918C2212yes—
73#1918C2214yes—
74#1879C2212yes—
75#1879C2214yes—
76#1862C2212yes—
77#1862C2214yes—
78#1834C2212yes—
79#1834C2214yes—
80#1802C2212yes—
81#1802C2212yes—
82#1802C2211yescharacteristic
83#1802C2211yescharacteristic
84#1802C2214yes—
85C4×#107C2212yes—
86C8×C4⋊C4C2212yes—
87C8×C4⋊C4C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90C4×#102C2212yes—
91C4×#102C2212yes—
92C4×#102C2214yes—
93C22×D8C2312yes—
94#554—48no—
95#533—48no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8D411yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×#127C2311yescharacteristic
102C2×#125—22no—
103C2×#125—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2314yes—
121C2×C4×D4C2311yescharacteristic
122C2×#117—22no—
123C22×C4⋊C4—22no—
124C22×C42—21no—
125#486—28no—
126#474—24no—
127#459—24no—
128C4×D8—28no—
129C2×#107C2312yes—
130C2×#107C2312yes—
131C2×#107C2312yes—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2314yes—
137C2×#102C2312yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2314yes—
148#399—24no—
149#399—24no—
150#398—24no—
151#398—24no—
152#396—24no—
153#396—24no—
154#396—24no—
155#391—22no—
156#391—21no—
157#391D412yes—
158#390—22no—
159#390—21no—
160#390—21no—
161#390C2312yes—
162#390C2314yes—
163#390C2314yes—
164#389—22no—
165#389—22no—
166#389—24no—
167#389—22no—
168#389—21no—
169#389D412yes—
170#389D412yes—
171#389C2312yes—
172#389C2314yes—
173#389C2314yes—
174#388—24no—
175#387—24no—
176#386—24no—
177#385—24no—
178#384—21no—
179#384—21no—
180#384D411yescharacteristic
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2312yes—
184#384C2314yes—
185#384C2312yes—
186#384C2314yes—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—24no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2311yescharacteristic
194C4×C4⋊C4C2314yes—
195C4×C4⋊C4C2311yescharacteristic
196C4×C4⋊C4C2311yescharacteristic
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C22⋊C4—21no—
200C4×C22⋊C4—21no—
201C4×C22⋊C4—21no—
202C43C2311yescharacteristic
203#340C2314yes—
204#340C2312yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2312yes—
209#340C2312yes—
210#340C2312yes—
211#340C2314yes—
212#340C2312yes—
213#340C2314yes—
214#340C2312yes—
215#336C2312yes—
216#336C2314yes—
217#336C2314yes—
218#336C2312yes—
219C22×Q8—21no—
220C22×Q8—22no—
221C22×Q8C2×D412yes—
222C22×D4—21no—
223C22×D4—21no—
224C22×D4C2412yes—
225C22×D4C2411yescharacteristic
226C23×C4—21no—
227C23×C4—22no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—24no—
231C2×D8—24no—
232C2×D8—24no—
233C2×D8—42no—
234C22×C8C2412yes—
235C22×C8C2412yes—
236C22×C8C2412yes—
237C22×C8C2412yes—
238C4⋊Q8—22no—
239C4⋊Q8—24no—
240C4⋊Q8—22no—
241C4⋊Q8—24no—
242C4⋊Q8—44no—
243C4⋊Q8—82no—
244C4⋊Q8D818yescomplemented
245#127—22no—
246#127#4114yes—
247#125—48no—
248#125—48no—
249C22⋊Q8—48no—
250C22⋊Q8—416no—
251C22⋊Q8—48no—
252C22⋊Q8—48no—
253C22⋊Q8—84no—
254C22⋊Q8—84no—
255C4×Q8—42no—
256C4×Q8—42no—
257C4×Q8—48no—
258C4×Q8—48no—
259C4×Q8—44no—
260C4×D4—24no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—24no—
264C4×D4—22no—
265C4×D4—41no—
266C4×D4—41no—
267C4×D4—44no—
268C4×D4—44no—
269C4×D4—48no—
270C4×D4—44no—
271C4×D4—82no—
272C4×D4#4114yes—
273#117—48no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—48no—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D414yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C22⋊C4—21no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—44no—
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2411yescharacteristic
327C2×C42C2414yes—
328C2×C42C2411yescharacteristic
329C2×C42C2411yescharacteristic
330#107—24no—
331#107—24no—
332#107—42no—
333C4⋊C8—22no—
334C4⋊C8—22no—
335C4⋊C8—22no—
336C4⋊C8—22no—
337C4⋊C8—44no—
338#102—24no—
339#102—24no—
340#102—24no—
341#102—28no—
342#102—28no—
343#102—24no—
344#102—42no—
345#102—42no—
346C4×C8—24no—
347C4×C8—24no—
348C4×C8—44no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—22no—
353#95—24no—
354#95—22no—
355#95—24no—
356#95—24no—
357#95C2×D414yes—
358#95C2×D414yes—
359C24—21no—
360C24—22no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—44no—
365C2×Q8—42no—
366C2×Q8—44no—
367C2×Q8—42no—
368C2×Q8—44no—
369C2×Q8C2×D818yes—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—22no—
376C2×D4—21no—
377C2×D4—22no—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—81no—
385C2×D4—81no—
386C2×D4C2×#4114yes—
387C2×D4C2×#4112yes—
388C22×C4—21no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—42no—
407C22×C4—48no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4C22×D414yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D414yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2511yescharacteristic
419D8—48no—
420C2×C8—22no—
421C2×C8—22no—
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—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—81no—
461C4⋊C4—81no—
462C4⋊C4—84no—
463C4⋊C4—84no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4C2×D818yes—
467C4⋊C4C2×D818yes—
468C4⋊C4C2×#4112yes—
469C22⋊C4—24no—
470C22⋊C4—42no—
471C22⋊C4—44no—
472C22⋊C4—42no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—44no—
476C22⋊C4—84no—
477C22⋊C4—82no—
478C22⋊C4—84no—
479C22⋊C4—84no—
480C42—22no—
481C42—22no—
482C42—24no—
483C42—24no—
484C42—21no—
485C42—21no—
486C42—41no—
487C42—42no—
488C42—42no—
489C42—42no—
490C42—48no—
491C42—44no—
492C42C2×D814yes—
493C42C2×#4112yes—
494C42C2×#4112yes—
495C422−1+412yes—
496C422−1+412yes—
497C23—22no—
498C23—24no—
499C23—41no—
500C23—42no—
501C23—41no—
502C23—42no—
503C23—41no—
504C23—42no—
505C23—41no—
506C23—42no—
507C23—41no—
508C23—42no—
509C23—41no—
510C23—42no—
511C23C23×D411yescharacteristic
512Q8—48no—
513Q8—84no—
514Q8—88no—
515D4—22no—
516D4—24no—
517D4—22no—
518D4—24no—
519D4—48no—
520D4—42no—
521D4—41no—
522D4—41no—
523D4—42no—
524D4—84no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—21no—
530C2×C4—24no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—24no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—84no—
562C2×C4—88no—
563C2×C4—84no—
564C2×C4—168no—
565C2×C4#54812yes—
566C2×C4#54812yes—
567C2×C4#54812yes—
568C2×C4#54812yes—
569C2×C4C22×D812yes—
570C2×C4C22×D818yes—
571C2×C4C22×D812yes—
572C2×C4C22×D812yes—
573C2×C4#57712yes—
574C2×C4#57712yes—
575C2×C4C22×#4111yescharacteristic
576C2×C4C22×#4111yescharacteristic
577C2×C4C2×2−1+411yescharacteristic
578C2×C4C2×2−1+411yescharacteristic
579C2×C4#58511yescharacteristic
580C2×C4#58511yescharacteristic
581C8—42no—
582C8—42no—
583C8—42no—
584C8—42no—
585C22—42no—
586C22—44no—
587C22—44no—
588C22—42no—
589C22—42no—
590C22—44no—
591C22—81no—
592C22—82no—
593C22—82no—
594C22—81no—
595C22—81no—
596C22—82no—
597C22—81no—
598C22—82no—
599C22C2×#54811yescharacteristic
600C22C2×#54811yescharacteristic
601C22#348511yescharacteristic
602C22C23×D811yescharacteristic
603C22C2×#57711yescharacteristic
604C22C2×#57711yescharacteristic
605C22#358611yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—28no—
610C4—41no—
611C4—44no—
612C4—42no—
613C4—82no—
614C4—81no—
615C4—82no—
616C4—84no—
617C4—81no—
618C4—82no—
619C4#335814yes—
620C4#335814yes—
621C4#347511yescharacteristic
622C4#350711yescharacteristic
623C4#351511yescharacteristic
624C4#355511yescharacteristic
625C2—82no—
626C2—84no—
627C2#5907911yescharacteristic
628C2C2×#335811yescharacteristic
629C2C2×#335811yescharacteristic
630C2#6134011yescharacteristic
631C2#6134611yescharacteristic
632C2#6169411yescharacteristic
633C2#6179211yescharacteristic
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. #59823C2semidirect
2. #36134C2semidirect
3. #32544C2semidirect
4. #29219C2semidirect
5. #28716C2semidirect
6. #27727C2semidirect
7. #27727C2semidirect
8. #26506C2semidirect
9. #26106C2semidirect
11. #24990C2semidirect
12. #24676C2semidirect
13. #23247C2semidirect
14. #23216C2semidirect
15. #22766C2semidirect
16. C4×#474C2semidirect
244. C4⋊Q8D8semidirect

Maximal subgroups

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

subgroupindex
#598232
#361342
#325442
#292192
#287162
#277272
#265062
#261062
#253182
#249902
#246762
#232472
#232162
#227662
C4×#4742
C4×#4592