Finite Group Databasebeta

← #7545555

Subgroups — #7545555

Order 512.

1541 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59823C211yesmaximal, characteristic, complemented
2#36194C211yesmaximal, characteristic, complemented
3#32559C214yesmaximal, complemented
4#29218C211yesmaximal, characteristic, complemented
5#28716C211yesmaximal, characteristic, complemented
6#27736C212yesmaximal, complemented
7#27735C212yesmaximal, complemented
8#26512C212yesmaximal, complemented
9#26111C212yesmaximal, complemented
10#25321C212yesmaximal
11#25007C214yesmaximal, complemented
12#24676C212yesmaximal, complemented
13#23250C211yesmaximal, characteristic, complemented
14#23223C212yesmaximal, complemented
15#22758C212yesmaximal, complemented
16C4×#474C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#554—21no—
19C2×#486C2212yes—
20C2×#474C2211yescharacteristic
21C2×#459C2211yescharacteristic
22C2×C4×D8C2212yes—
23#2868C2211yescharacteristic
24#2857—21no—
25#2732—22no—
26#2571C2212yes—
27#2563C2212yes—
28#2552C2212yes—
29#2542C2212yes—
30#2538—21no—
31#2538C2212yes—
32#2538C2214yes—
33#2497—22no—
34#2492—22no—
35#2454—22no—
36#2439—22no—
37#2417—22no—
38#2369C2212yes—
39#2366C2212yes—
40#2355C2212yes—
41#2354C2212yes—
42D4×C4⋊C4C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46#2285—21no—
47C42×D4C2211yescharacteristic
48#2101C2214yes—
49#2100C2214yes—
50#2100C2214yes—
51#2087C2214yes—
52#2073C2214yes—
53#2071C2214yes—
54#2071C2214yes—
55#2054C2214yes—
56#1990C2212yes—
57#1990C2212yes—
58#1988C2211yescharacteristic
59#1987C2212yes—
60#1987C2212yes—
61#1985C2214yes—
62#1982C2211yescharacteristic
63#1947C2214yes—
64#1947C2211yescharacteristic
65#1947C2211yescharacteristic
66#1939C2212yes—
67#1939C2212yes—
68#1936C2214yes—
69#1936C2214yes—
70#1925C2214yes—
71#1925C2212yes—
72#1921C2214yes—
73#1918C2212yes—
74#1879C2212yes—
75#1879C2214yes—
76#1862C2212yes—
77#1862C2214yes—
78#1836C2214yes—
79#1834C2212yes—
80#1804C2214yes—
81#1802C2212yes—
82#1802C2212yes—
83#1802C2211yescharacteristic
84#1802C2211yescharacteristic
85C4×#107C2212yes—
86#1770C2212yes—
87#1770C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90#1762C2214yes—
91C4×#102C2212yes—
92C4×#102C2212yes—
93C22×D8C2312yes—
94#554—48no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8D411yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×#127C2311yescharacteristic
101C2×#125—22no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×C4×Q8—21no—
107C2×C4×Q8C2312yes—
108C2×C4×D4—22no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4C2312yes—
112C2×C4×D4C2311yescharacteristic
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2314yes—
115C2×C4×D4C2311yescharacteristic
116C2×#117—22no—
117C22×C4⋊C4—22no—
118C22×C42—21no—
119#486—28no—
120#474—24no—
121#459—24no—
122C4×D8—28no—
123C2×#107C2312yes—
124C2×#107C2312yes—
125C2×#107C2312yes—
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2314yes—
131C2×#102C2312yes—
132C2×#102C2314yes—
133C2×#102C2312yes—
134C2×#102C2314yes—
135C2×#102C2312yes—
136C2×#102C2312yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×#97C2314yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2312yes—
142#400—24no—
143#399—24no—
144#398—24no—
145#396—24no—
146#396—24no—
147#396—24no—
148#393—24no—
149#391—22no—
150#391—21no—
151#391—24no—
152#391D412yes—
153#390—22no—
154#390—21no—
155#390—21no—
156#390C2312yes—
157#390C2314yes—
158#390C2314yes—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—21no—
163#389D412yes—
164#389D412yes—
165#389C2312yes—
166#389C2314yes—
167#389C2314yes—
168#388—24no—
169#387—24no—
170#386—24no—
171#385—24no—
172#384—21no—
173#384—21no—
174#384D411yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2312yes—
178#384C2312yes—
179#384C2314yes—
180#382—22no—
181#382C2314yes—
182#380—22no—
183#380—22no—
184#380—22no—
185#380—22no—
186#380—24no—
187#380—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2311yescharacteristic
193C4×C4⋊C4C2314yes—
194C4×C4⋊C4C2311yescharacteristic
195C4×C4⋊C4C2311yescharacteristic
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C22⋊C4—21no—
199C4×C22⋊C4—21no—
200C4×C22⋊C4—21no—
201C43C2311yescharacteristic
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2312yes—
206#340C2312yes—
207#340C2312yes—
208#340C2314yes—
209#340C2314yes—
210#340C2312yes—
211#340C2312yes—
212#340C2314yes—
213#340C2312yes—
214#336C2312yes—
215#336C2314yes—
216#336C2314yes—
217#336C2312yes—
218C22×Q8—21no—
219C22×Q8—22no—
220C22×Q8C2×D412yes—
221C22×D4—21no—
222C22×D4—21no—
223C22×D4C2412yes—
224C22×D4C2411yescharacteristic
225C23×C4—21no—
226C23×C4—22no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—24no—
230C2×D8—24no—
231C2×D8—24no—
232C2×D8—42no—
233C22×C8C2412yes—
234C22×C8C2412yes—
235C22×C8C2412yes—
236C22×C8C2412yes—
237C4⋊Q8—24no—
238C4⋊Q8—22no—
239C4⋊Q8—24no—
240C4⋊Q8—22no—
241C4⋊Q8—24no—
242C4⋊Q8—82no—
243C4⋊Q8—82no—
244#127—22no—
245#127#4114yes—
246#125—48no—
247C22⋊Q8—48no—
248C22⋊Q8—416no—
249C22⋊Q8—84no—
250C22⋊Q8—84no—
251C4×Q8—42no—
252C4×Q8—42no—
253C4×Q8—44no—
254C4×D4—24no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—24no—
258C4×D4—22no—
259C4×D4—41no—
260C4×D4—41no—
261C4×D4—44no—
262C4×D4—44no—
263C4×D4—48no—
264C4×D4—44no—
265C4×D4—82no—
266C4×D4#4114yes—
267#117—84no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—48no—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2411yescharacteristic
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2411yescharacteristic
294C2×C4⋊C4C2414yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2411yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C42—22no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—24no—
313C2×C42—44no—
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2411yescharacteristic
319C2×C42C2414yes—
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322#107—24no—
323#107—24no—
324#107—42no—
325C4⋊C8—22no—
326C4⋊C8—22no—
327C4⋊C8—22no—
328C4⋊C8—22no—
329C4⋊C8—44no—
330#102—24no—
331#102—24no—
332#102—24no—
333#102—28no—
334#102—28no—
335#102—24no—
336#102—42no—
337#102—42no—
338#97—44no—
339C4×C8—24no—
340C4×C8—24no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—22no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—22no—
349#95—24no—
350#95—24no—
351#95C2×D414yes—
352#95C2×D414yes—
353C24—21no—
354C24—22no—
355C2×Q8—24no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—44no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—82no—
363C2×Q8—82no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—22no—
370C2×D4—21no—
371C2×D4—22no—
372C2×D4—21no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—81no—
379C2×D4—81no—
380C2×D4C2×#4114yes—
381C2×D4C2×#4112yes—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—48no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—42no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4C22×D414yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D414yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C2511yescharacteristic
413D8—48no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—44no—
448C4⋊C4—42no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—84no—
457C4⋊C4—84no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4C2×#4112yes—
461C22⋊C4—24no—
462C22⋊C4—42no—
463C22⋊C4—48no—
464C22⋊C4—44no—
465C22⋊C4—42no—
466C22⋊C4—44no—
467C22⋊C4—84no—
468C22⋊C4—84no—
469C22⋊C4—82no—
470C22⋊C4—84no—
471C22⋊C4—84no—
472C42—22no—
473C42—22no—
474C42—24no—
475C42—24no—
476C42—21no—
477C42—21no—
478C42—41no—
479C42—42no—
480C42—42no—
481C42—42no—
482C42—44no—
483C42—84no—
484C42AGL(1,ℤ/8)14yes—
485C42C2×#4112yes—
486C42C2×#4112yes—
487C422−1+412yes—
488C422−1+412yes—
489C23—22no—
490C23—24no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—41no—
502C23—42no—
503C23C23×D411yescharacteristic
504Q8—84no—
505Q8—84no—
506Q8—164no—
507D4—22no—
508D4—24no—
509D4—22no—
510D4—24no—
511D4—48no—
512D4—42no—
513D4—41no—
514D4—41no—
515D4—42no—
516D4—84no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—21no—
530C2×C4—24no—
531C2×C4—21no—
532C2×C4—24no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—48no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—84no—
555C2×C4—88no—
556C2×C4—84no—
557C2×C4—168no—
558C2×C4#54812yes—
559C2×C4#54812yes—
560C2×C4#54812yes—
561C2×C4#54812yes—
562C2×C4C2×AGL(1,ℤ/8)12yes—
563C2×C4C2×AGL(1,ℤ/8)12yes—
564C2×C4C2×AGL(1,ℤ/8)12yes—
565C2×C4#57612yes—
566C2×C4#57812yes—
567C2×C4C22×#4111yescharacteristic
568C2×C4C22×#4111yescharacteristic
569C2×C4C2×2−1+411yescharacteristic
570C2×C4C2×2−1+411yescharacteristic
571C2×C4#58511yescharacteristic
572C2×C4#58511yescharacteristic
573C8—42no—
574C8—42no—
575C8—42no—
576C8—42no—
577C22—42no—
578C22—44no—
579C22—44no—
580C22—42no—
581C22—42no—
582C22—44no—
583C22—81no—
584C22—82no—
585C22—81no—
586C22—82no—
587C22—82no—
588C22—81no—
589C22—81no—
590C22—82no—
591C22C2×#54811yescharacteristic
592C22C2×#54811yescharacteristic
593C22#348511yescharacteristic
594C22C22×AGL(1,ℤ/8)11yescharacteristic
595C22C2×#57611yescharacteristic
596C22C2×#57811yescharacteristic
597C22#358511yescharacteristic
598C4—22no—
599C4—22no—
600C4—22no—
601C4—44no—
602C4—41no—
603C4—44no—
604C4—42no—
605C4—81no—
606C4—82no—
607C4—82no—
608C4—82no—
609C4—81no—
610C4—84no—
611C4#336614yes—
612C4#336714yes—
613C4#347511yescharacteristic
614C4#350711yescharacteristic
615C4#351511yescharacteristic
616C4#355511yescharacteristic
617C2—82no—
618C2—84no—
619C2#5907911yescharacteristic
620C2C2×#336611yescharacteristic
621C2C2×#336711yescharacteristic
622C2#6133911yescharacteristic
623C2#6134811yescharacteristic
624C2#6172011yescharacteristic
625C2#6179111yescharacteristic
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. #36194C2semidirect
3. #32559C2semidirect
4. #29218C2semidirect
5. #28716C2semidirect
6. #27736C2semidirect
7. #27735C2semidirect
8. #26512C2semidirect
9. #26111C2semidirect
11. #25007C2semidirect
12. #24676C2semidirect
13. #23250C2semidirect
14. #23223C2semidirect
15. #22758C2semidirect
16. C4×#474C2semidirect

Maximal subgroups

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

subgroupindex
#598232
#361942
#325592
#292182
#287162
#277362
#277352
#265122
#261112
#253212
#250072
#246762
#232502
#232232
#227582
C4×#4742
C4×#4592