Finite Group Databasebeta

← #229711

Subgroups — #229711

Order 512.

783 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23176C211yesmaximal, characteristic, complemented
2#22700C211yesmaximal, characteristic, complemented
3C4×#459C211yesmaximal, characteristic
4C2×#1594C211yesmaximal, characteristic, complemented
5C2×#1589C211yesmaximal, characteristic, complemented
6#9642C211yesmaximal, characteristic, complemented
7#9603C211yesmaximal, characteristic, complemented
8#9556C211yesmaximal, characteristic, complemented
9#9554C211yesmaximal, characteristic, complemented
10#8951C211yesmaximal, characteristic, complemented
11#8951C211yesmaximal, characteristic, complemented
12#8849C211yesmaximal, characteristic, complemented
13#8847C211yesmaximal, characteristic, complemented
14#8264C211yesmaximal, characteristic, complemented
15#8170C211yesmaximal, characteristic, complemented
16C2×#459C411yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×#436—21no—
20C2×#436C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×#436C2211yescharacteristic
23C2×#436C2211yescharacteristic
24C4×#127C2211yescharacteristic
25C42×D4C2211yescharacteristic
26#1982C411yescharacteristic
27#1954—21no—
28#1954C2211yescharacteristic
29#1954C2211yescharacteristic
30#1947C411yescharacteristic
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1925C411yescharacteristic
34#1925C411yescharacteristic
35#1879C411yescharacteristic
36#1879C411yescharacteristic
37#1870—21no—
38#1870—21no—
39#1851—21no—
40#1851—21no—
41#1831C2211yescharacteristic
42#1802C411yescharacteristic
43#1797—21no—
44C4×C4⋊C8C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1744C2211yescharacteristic
50#1728—21no—
51#1594—24no—
52#1589—24no—
53C2×#335C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×D8C2×C411yescharacteristic
70C22×D8C2×C411yescharacteristic
71C22×M16—21no—
72C22×M16—21no—
73C22×M16—21no—
74C22×M16—21no—
75C2×#127C2×C411yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4C2×C411yescharacteristic
80C2×C4×D4C2311yescharacteristic
81C2×C4×D4C2311yescharacteristic
82C2×C4×D4C2311yescharacteristic
83C22×C42—21no—
84#459—24no—
85C4×D8—22no—
86C4×D8—22no—
87C4×D8—22no—
88C4×D8—22no—
89#436—22no—
90#436—22no—
91#436—22no—
92#436—22no—
93#436—41no—
94#436—41no—
95#436—41no—
96#436—41no—
97#436—44no—
98C2×#107C2×C411yescharacteristic
99C2×#107C2×C411yescharacteristic
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8C2×C411yescharacteristic
103C2×C4⋊C8C2×C411yescharacteristic
104C2×C4⋊C8D411yescharacteristic
105C2×C4⋊C8D411yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C4×C8—21no—
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133#390—21no—
134#390—21no—
135#390C2×C411yescharacteristic
136#390C2×C411yescharacteristic
137#384C2×C411yescharacteristic
138C4×C4⋊C4C2×C411yescharacteristic
139C4×C4⋊C4C2×C411yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C43C2311yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336C2×C411yescharacteristic
157#336C2×C411yescharacteristic
158#336D411yescharacteristic
159#336D411yescharacteristic
160#335—22no—
161#334—22no—
162D4⋊C8—22no—
163D4⋊C8—22no—
164D4⋊C8—22no—
165D4⋊C8—41no—
166D4⋊C8—41no—
167D4⋊C8D414yescomplemented
168#322—22no—
169C82—22no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4C22×C411yescharacteristic
173C22×D4C22×C411yescharacteristic
174C22×D4C22×C411yescharacteristic
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C2×D8—22no—
181C2×D8—22no—
182C2×D8—22no—
183C2×D8—22no—
184C2×D8—41no—
185C2×D8—41no—
186C2×M16—42no—
187C2×M16—42no—
188C2×M16—42no—
189C2×M16—42no—
190C2×M16—42no—
191C2×M16—42no—
192C2×M16—42no—
193C2×M16—42no—
194C2×M16—81no—
195C2×M16—81no—
196C2×M16—81no—
197C2×M16—81no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8C22×C411yescharacteristic
205C22×C8C22×C411yescharacteristic
206C22×C8C22×C411yescharacteristic
207C22×C8C22×C411yescharacteristic
208C22×C8C2×D411yescharacteristic
209C22×C8C2×D411yescharacteristic
210C22×C8C2×D411yescharacteristic
211C22×C8C2×D411yescharacteristic
212#127—22no—
213#127—22no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4—21no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4—41no—
228C4×D4—42no—
229C4×D4—42no—
230C4×D4—81no—
231C4×D4—81no—
232C4×D4C2×D412yes—
233C4×D4#4112yes—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—81no—
254C2×C42—81no—
255C2×C42C22×C411yescharacteristic
256C2×C42C22×C411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42#4111yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42C2411yescharacteristic
262#107—22no—
263#107—22no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—41no—
274C4⋊C8—81no—
275C4⋊C8—81no—
276C4⋊C8M1614yescomplemented
277C4⋊C8C2×D412yes—
278#102—22no—
279#102—22no—
280#102—22no—
281#102—22no—
282#102—22no—
283#102—22no—
284#102—22no—
285#102—22no—
286C22⋊C8—42no—
287C22⋊C8—42no—
288C22⋊C8—42no—
289C22⋊C8—42no—
290C22⋊C8—42no—
291C22⋊C8—42no—
292C22⋊C8—81no—
293C22⋊C8—81no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—42no—
306C4×C8C2×D412yes—
307#95—21no—
308#95—21no—
309#95—21no—
310C24—21no—
311C24—21no—
312C24—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—81no—
335C2×D4—81no—
336C2×D4C4×D412yes—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4—81no—
366C22×C4C4×D411yescharacteristic
367C22×C4C4×D411yescharacteristic
368C22×C4C4×D411yescharacteristic
369C22×C4C4×D411yescharacteristic
370C22×C4C23×C411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C2×#4111yescharacteristic
373D8—44no—
374D8—44no—
375M16—84no—
376M16—84no—
377M16—84no—
378M16—84no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8C2×M1612yes—
419C2×C8C2×M1612yes—
420C2×C8AGL(1,ℤ/8)12yes—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—42no—
430C4⋊C4—81no—
431C4⋊C4C4×D412yes—
432C4⋊C4#13112yes—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—42no—
443C22⋊C4—81no—
444C22⋊C4—81no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—81no—
456C42—81no—
457C42C2×M1612yes—
458C42#13112yes—
459C42C22×D411yescharacteristic
460C42C2×#4111yescharacteristic
461C42H(5,𝔽2)11yescharacteristic
462C422−1+411yescharacteristic
463C23—22no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23C2×C4×D411yescharacteristic
485D4—22no—
486D4—22no—
487D4—41no—
488D4—41no—
489D4—41no—
490D4—41no—
491D4—41no—
492D4—41no—
493D4—41no—
494D4—41no—
495D4—41no—
496D4—41no—
497D4—84no—
498D4—84no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—84no—
531C2×C4—84no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4#43612yes—
541C2×C4#43612yes—
542C2×C4C2×C4×D411yescharacteristic
543C2×C4#52011yescharacteristic
544C2×C4D4211yescharacteristic
545C2×C4#54711yescharacteristic
546C2×C4#54811yescharacteristic
547C2×C4#55311yescharacteristic
548C2×C4C22×M1611yescharacteristic
549C2×C4C2×#13111yescharacteristic
550C2×C4C2×#13111yescharacteristic
551C2×C4#56811yescharacteristic
552C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
553C2×C4#57511yescharacteristic
554C2×C4#57611yescharacteristic
555C2×C4#57611yescharacteristic
556C8—22no—
557C8—22no—
558C8—22no—
559C8—42no—
560C8—41no—
561C8—41no—
562C8—41no—
563C8—42no—
564C8—41no—
565C8—41no—
566C8—81no—
567C8—81no—
568C8—81no—
569C8—81no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22#236111yescharacteristic
592C22C2×#43611yescharacteristic
593C22#293311yescharacteristic
594C22C4×AGL(1,ℤ/8)11yescharacteristic
595C22#294811yescharacteristic
596C22#295511yescharacteristic
597C22#295711yescharacteristic
598C4—22no—
599C4—22no—
600C4—21no—
601C4—21no—
602C4—41no—
603C4—41no—
604C4—42no—
605C4—42no—
606C4—81no—
607C4—81no—
608C4—81no—
609C4#293111yescharacteristic
610C4D4×M1611yescharacteristic
611C4#299111yescharacteristic
612C4#299311yescharacteristic
613C4#326411yescharacteristic
614C4#326811yescharacteristic
615C4#336611yescharacteristic
616C4#337011yescharacteristic
617C2—82no—
618C2—161no—
619C2—161no—
620C2#1789111yescharacteristic
621C2#1789511yescharacteristic
622C2#1801111yescharacteristic
623C2#1802611yescharacteristic
624C2#2299011yescharacteristic
625C2#2300811yescharacteristic
626C2#2301711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#459C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23176C2semidirect
2. #22700C2semidirect
4. C2×#1594C2semidirect
5. C2×#1589C2semidirect
6. #9642C2semidirect
7. #9603C2semidirect
8. #9556C2semidirect
9. #9554C2semidirect
10. #8951C2semidirect
11. #8951C2semidirect
12. #8849C2semidirect
13. #8847C2semidirect
14. #8264C2semidirect
15. #8170C2semidirect
167. D4⋊C8D4semidirect
276. C4⋊C8M16semidirect

Maximal subgroups

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

subgroupindex
#231762
#227002
C4×#4592
C2×#15942
C2×#15892
#96422
#96032
#95562
#95542
#89512
#88492
#88472
#82642
#81702