Finite Group Databasebeta

← #7567522

Subgroups — #7567522

Order 512.

1725 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59823C211yesmaximal, characteristic, complemented
2#36180C211yesmaximal, characteristic, complemented
3#32593C214yesmaximal, complemented
4#29206C211yesmaximal, characteristic, complemented
5#29139C211yesmaximal, characteristic, complemented
6#28798C212yesmaximal, complemented
7#27822C212yesmaximal, complemented
8#26467C212yesmaximal, complemented
9#26110C212yesmaximal, complemented
10#26045C212yesmaximal, complemented
11#24811C212yesmaximal, complemented
12#24347C214yesmaximal, complemented
13#23250C211yesmaximal, characteristic, complemented
14#23001C212yesmaximal
15#22670C212yesmaximal, complemented
16C4×#474C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#554—21no—
19C2×#554C2212yes—
20C2×#478C2214yes—
21C2×#474C2211yescharacteristic
22C2×#459C2211yescharacteristic
23C2×#442C2214yes—
24C2×C4×D8C2212yes—
25#2868C2211yescharacteristic
26#2857—21no—
27#2732—22no—
28#2567C2212yes—
29#2552C2212yes—
30#2552C2212yes—
31#2552C2212yes—
32#2542C2212yes—
33#2538—21no—
34#2538C2214yes—
35#2502—22no—
36#2454—22no—
37#2440—22no—
38#2440—22no—
39#2417—22no—
40#2366C2212yes—
41D4×C4⋊C4C2212yes—
42D4×C4⋊C4C2212yes—
43D4×C4⋊C4C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C2×#384—21no—
48C42×D4C2211yescharacteristic
49#2100C2214yes—
50#2088C2214yes—
51#2083C2214yes—
52#2067C2214yes—
53#2059C2214yes—
54#2058C2214yes—
55#2025C2214yes—
56#2023C2214yes—
57#1992C2214yes—
58#1990C2212yes—
59#1988C2211yescharacteristic
60#1985C2214yes—
61#1982C2211yescharacteristic
62#1957C2212yes—
63#1948C2212yes—
64#1948C2212yes—
65#1947C2211yescharacteristic
66#1947C2211yescharacteristic
67#1943C2214yes—
68#1939C2212yes—
69#1939C2212yes—
70#1936C2214yes—
71#1925C2212yes—
72#1918C2212yes—
73#1879C2212yes—
74#1879C2214yes—
75#1864C2214yes—
76#1864C2214yes—
77#1864C2214yes—
78#1862C2212yes—
79#1844C2212yes—
80#1834C2212yes—
81#1802C2214yes—
82#1802C2211yescharacteristic
83#1802C2212yes—
84#1802C2212yes—
85#1802C2211yescharacteristic
86C4×#107C2212yes—
87#1770C2212yes—
88#1770C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90C4×C4⋊C8C2211yescharacteristic
91C4×#102C2212yes—
92C4×#102C2212yes—
93C22×D8C2312yes—
94#554—44no—
95#554—44no—
96#554—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8D411yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#125—22no—
104C2×#125C2314yes—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—22no—
110C2×C4×Q8—21no—
111C2×C4×Q8C2312yes—
112C2×C4×D4—22no—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4C2312yes—
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2314yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2314yes—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—22no—
125C22×C42—21no—
126#478—48no—
127#474—24no—
128#459Q818yescomplemented
129#442—48no—
130C4×D8—28no—
131C2×#107C2312yes—
132C2×#107C2312yes—
133C2×#107C2312yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×#97C2314yes—
150C2×C4×C8C2312yes—
151C2×C4×C8C2312yes—
152#399—24no—
153#398—24no—
154#398—24no—
155#398—24no—
156#396—24no—
157#396—24no—
158#395—24no—
159#391—22no—
160#391—21no—
161#391D412yes—
162#390—22no—
163#390—21no—
164#390—21no—
165#390C2312yes—
166#390C2314yes—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—21no—
171#389D412yes—
172#389D412yes—
173#389C2312yes—
174#389C2314yes—
175#389C2314yes—
176#389C2314yes—
177#386—24no—
178#385—24no—
179#385—24no—
180#385—24no—
181#384—22no—
182#384—22no—
183#384—22no—
184#384—21no—
185#384—21no—
186#384D411yescharacteristic
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2314yes—
191#384C2312yes—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—24no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2311yescharacteristic
199C4×C4⋊C4C2311yescharacteristic
200C4×C4⋊C4C2311yescharacteristic
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C22⋊C4—21no—
204C4×C22⋊C4—21no—
205C4×C22⋊C4—21no—
206C43C2311yescharacteristic
207#340C2314yes—
208#340C2314yes—
209#340C2312yes—
210#340C2314yes—
211#340C2312yes—
212#340C2314yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#340C2312yes—
217#336C2312yes—
218#336C2314yes—
219#336C2314yes—
220#336C2312yes—
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—
233C23×C4—24no—
234C2×D8—24no—
235C2×D8—24no—
236C2×D8C2×Q818yes—
237C22×C8C2412yes—
238C22×C8C2412yes—
239C22×C8C2412yes—
240C22×C8C2412yes—
241C4⋊Q8—24no—
242C4⋊Q8—22no—
243C4⋊Q8—22no—
244C4⋊Q8—42no—
245C4⋊Q8—42no—
246C4⋊Q8—82no—
247C4⋊Q8—82no—
248#127—22no—
249#127C2×Q814yes—
250#125—28no—
251#125—44no—
252#125—48no—
253C22⋊Q8—416no—
254C22⋊Q8—48no—
255C22⋊Q8—48no—
256C22⋊Q8—416no—
257C22⋊Q8—84no—
258C4×Q8—42no—
259C4×Q8—42no—
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—44no—
271C4×D4—48no—
272C4×D4—44no—
273C4×D4—82no—
274C4×D4C2×Q814yes—
275C4×D4#4114yes—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—48no—
295C2×C4⋊C4—48no—
296C2×C4⋊C4—84no—
297C2×C4⋊C4—84no—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C42—22no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—44no—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2414yes—
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334#107—24no—
335#107—24no—
336#107—24no—
337#106—44no—
338#106—44no—
339C4⋊C8—22no—
340C4⋊C8—22no—
341C4⋊C8—22no—
342C4⋊C8—44no—
343C4⋊C8C2×Q814yes—
344#102—24no—
345#102—24no—
346#102—24no—
347#102—24no—
348#102—24no—
349#102—44no—
350#102—44no—
351#102C2×Q818yes—
352#97—44no—
353C4×C8—24no—
354C4×C8—24no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—22no—
359#95—22no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95C2×D414yes—
364#95C2×D414yes—
365C24—21no—
366C24—22no—
367C2×Q8—24no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—44no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—82no—
375C2×Q8—82no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—81no—
389C2×D4—81no—
390C2×D4C22×Q814yes—
391C2×D4C22×Q812yes—
392C2×D4C2×#4112yes—
393C2×D4H(5,𝔽2)12yes—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—42no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—84no—
416C22×C4—84no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D414yes—
421C22×C4C22×D414yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C2511yescharacteristic
428D8—216no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8C22×Q814yes—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—81no—
470C4⋊C4—84no—
471C4⋊C4—84no—
472C4⋊C4—81no—
473C4⋊C4—84no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4C22×Q812yes—
477C4⋊C4C2×#4112yes—
478C4⋊C4H(5,𝔽2)12yes—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—84no—
487C22⋊C4—82no—
488C22⋊C4—84no—
489C42—22no—
490C42—22no—
491C42—24no—
492C42—24no—
493C42—21no—
494C42—21no—
495C42—41no—
496C42—42no—
497C42—42no—
498C42—42no—
499C42—44no—
500C42AGL(1,ℤ/8)14yes—
501C42C22×Q812yes—
502C42C2×#4112yes—
503C422−1+412yes—
504C422−1+412yes—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23C23×D411yescharacteristic
520Q8—84no—
521Q8—84no—
522Q8—164no—
523D4—24no—
524D4—22no—
525D4—24no—
526D4—22no—
527D4—22no—
528D4—48no—
529D4—42no—
530D4—42no—
531D4—84no—
532D4#55414yescomplemented
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—21no—
548C2×C4—24no—
549C2×C4—21no—
550C2×C4—24no—
551C2×C4—48no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—88no—
573C2×C4—168no—
574C2×C4—168no—
575C2×C4#54812yes—
576C2×C4#54812yes—
577C2×C4D4×Q812yes—
578C2×C4D4×Q812yes—
579C2×C4C2×AGL(1,ℤ/8)12yes—
580C2×C4C2×AGL(1,ℤ/8)12yes—
581C2×C4C2×AGL(1,ℤ/8)12yes—
582C2×C4#57612yes—
583C2×C4#57712yes—
584C2×C4C23×Q811yescharacteristic
585C2×C4C22×#4111yescharacteristic
586C2×C4C2×H(5,𝔽2)11yescharacteristic
587C2×C4C2×H(5,𝔽2)11yescharacteristic
588C2×C4C2×2−1+411yescharacteristic
589C2×C4C2×2−1+411yescharacteristic
590C8—24no—
591C8—42no—
592C8—42no—
593C8—42no—
594C22—24no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22C2×#54811yescharacteristic
609C22C2×D4×Q811yescharacteristic
610C22#348411yescharacteristic
611C22C22×AGL(1,ℤ/8)11yescharacteristic
612C22C2×#57611yescharacteristic
613C22C2×#57711yescharacteristic
614C22#358611yescharacteristic
615C4—22no—
616C4—22no—
617C4—22no—
618C4—42no—
619C4—44no—
620C4—42no—
621C4—42no—
622C4—44no—
623C4—41no—
624C4—82no—
625C4—82no—
626C4—82no—
627C4—84no—
628C4#336614yes—
629C4#338414yes—
630C4C2×#55411yescharacteristic
631C4#350311yescharacteristic
632C4#351111yescharacteristic
633C4#354411yescharacteristic
634C2—44no—
635C2—84no—
636C2#5913611yescharacteristic
637C2C2×#336611yescharacteristic
638C2#6134611yescharacteristic
639C2C2×#338411yescharacteristic
640C2#6138111yescharacteristic
641C2#6158511yescharacteristic
642C2#6166811yescharacteristic
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. #36180C2semidirect
3. #32593C2semidirect
4. #29206C2semidirect
5. #29139C2semidirect
6. #28798C2semidirect
7. #27822C2semidirect
8. #26467C2semidirect
9. #26110C2semidirect
10. #26045C2semidirect
11. #24811C2semidirect
12. #24347C2semidirect
13. #23250C2semidirect
15. #22670C2semidirect
16. C4×#474C2semidirect
128. #459Q8semidirect
532. D4#554semidirect

Maximal subgroups

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

subgroupindex
#598232
#361802
#325932
#292062
#291392
#287982
#278222
#264672
#261102
#260452
#248112
#243472
#232502
#230012
#226702
C4×#4742
C4×#4592