Finite Group Databasebeta

← #229646

Subgroups — #229646

Order 512.

831 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23174C211yesmaximal, characteristic, complemented
2D4×C4⋊C8C211yesmaximal, characteristic, complemented
3C4×#459C211yesmaximal, characteristic
4C2×#1586C211yesmaximal, characteristic, complemented
5C2×#1580C211yesmaximal, characteristic, complemented
6#9644C211yesmaximal, characteristic, complemented
7#9605C211yesmaximal, characteristic, complemented
8#9560C211yesmaximal, characteristic, complemented
9#9556C211yesmaximal, characteristic, complemented
10#8947C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8849C211yesmaximal, characteristic, complemented
13#8845C211yesmaximal, characteristic, complemented
14#8265C211yesmaximal, characteristic, complemented
15#8174C211yesmaximal, characteristic, complemented
16C2×#459C411yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×#435—21no—
21C2×C8×D4C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23C2×C8×D4C2211yescharacteristic
24C22×C4⋊C8—21no—
25C4×#127C2211yescharacteristic
26C42×D4C2211yescharacteristic
27#1982C411yescharacteristic
28#1955—21no—
29#1954C2211yescharacteristic
30#1954C2211yescharacteristic
31#1947C411yescharacteristic
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1925C411yescharacteristic
35#1925C411yescharacteristic
36#1879C411yescharacteristic
37#1879C411yescharacteristic
38#1870—21no—
39#1869—21no—
40#1850—21no—
41#1850—21no—
42#1842C2211yescharacteristic
43#1802C411yescharacteristic
44#1797—21no—
45#1767C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×#102C2211yescharacteristic
50#1744C2211yescharacteristic
51#1586—24no—
52#1580—24no—
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×D8C2×C411yescharacteristic
70C22×D8C2×C411yescharacteristic
71C22×M16—21no—
72C23×C8—21no—
73C23×C8—21no—
74C23×C8—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—41no—
91#435—44no—
92C8×D4—22no—
93C8×D4—22no—
94C8×D4—22no—
95C8×D4—41no—
96C8×D4—41no—
97C8×D4—41no—
98C2×#107C2×C411yescharacteristic
99C2×#107C2×C411yescharacteristic
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—44no—
105C2×C4⋊C8C2×C411yescharacteristic
106C2×C4⋊C8C2×C411yescharacteristic
107C2×C4⋊C8D411yescharacteristic
108C2×C4⋊C8D411yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×#97—21no—
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136#390—21no—
137#390—21no—
138#390C2×C411yescharacteristic
139#390C2×C411yescharacteristic
140#384C2×C411yescharacteristic
141C4×C4⋊C4C2×C411yescharacteristic
142C4×C4⋊C4C2×C411yescharacteristic
143C4×C4⋊C4C2311yescharacteristic
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C43C2311yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#340C2×C411yescharacteristic
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336C2×C411yescharacteristic
158#336C2×C411yescharacteristic
159#336D411yescharacteristic
160#336D411yescharacteristic
161#334—22no—
162#334—22no—
163D4⋊C8—22no—
164D4⋊C8—22no—
165D4⋊C8—22no—
166D4⋊C8—41no—
167D4⋊C8—41no—
168D4⋊C8D414yescomplemented
169#322—22no—
170#322—22no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4C22×C411yescharacteristic
174C22×D4C22×C411yescharacteristic
175C22×D4C22×C411yescharacteristic
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C2×D8—22no—
182C2×D8—22no—
183C2×D8—22no—
184C2×D8—22no—
185C2×D8—22no—
186C2×D8M1614yescomplemented
187C2×M16—42no—
188C2×M16—42no—
189C2×M16—81no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—42no—
197C22×C8—42no—
198C22×C8—42no—
199C22×C8—42no—
200C22×C8—42no—
201C22×C8—42no—
202C22×C8—81no—
203C22×C8—81no—
204C22×C8—81no—
205C22×C8C22×C411yescharacteristic
206C22×C8C22×C411yescharacteristic
207C22×C8C22×C411yescharacteristic
208C22×C8C22×C411yescharacteristic
209C22×C8C2×D411yescharacteristic
210C22×C8C2×D411yescharacteristic
211C22×C8C2×D411yescharacteristic
212C22×C8C2×D411yescharacteristic
213#127—22no—
214#127—41no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—22no—
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—21no—
228C4×D4—41no—
229C4×D4—42no—
230C4×D4—42no—
231C4×D4—81no—
232C4×D4—81no—
233C4×D4C2×D412yes—
234C4×D4#4112yes—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42#4111yescharacteristic
262C2×C42C2411yescharacteristic
263#107—22no—
264#107—22no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—41no—
276C4⋊C8—42no—
277C4⋊C8—42no—
278C4⋊C8—42no—
279C4⋊C8—81no—
280C4⋊C8C2×D412yes—
281#102—22no—
282#102—22no—
283#102—22no—
284#102—22no—
285#102—22no—
286#102—22no—
287#102—22no—
288#102—22no—
289C22⋊C8—42no—
290C22⋊C8—42no—
291C22⋊C8—42no—
292C22⋊C8—42no—
293C22⋊C8—42no—
294C22⋊C8—42no—
295C22⋊C8—81no—
296C22⋊C8—81no—
297#97—42no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8C2×D412yes—
310#95—21no—
311#95—21no—
312#95—21no—
313C24—21no—
314C24—21no—
315C24—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—
337C2×D4C2×M1612yes—
338C2×D4C2×M1612yes—
339C2×D4#13112yes—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—81no—
369C22×C4C4×D411yescharacteristic
370C22×C4C4×D411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4C23×C411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C2×#4111yescharacteristic
376D8—28no—
377D8—44no—
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—22no—
389C2×C8—22no—
390C2×C8—21no—
391C2×C8—21no—
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—21no—
403C2×C8—21no—
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×C8—41no—
419C2×C8—41no—
420C2×C8—84no—
421C2×C8—84no—
422C2×C8—84no—
423C2×C8C2×M1612yes—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—42no—
433C4⋊C4—81no—
434C4⋊C4C4×D412yes—
435C4⋊C4#13112yes—
436C22⋊C4—24no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—42no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—81no—
447C22⋊C4—81no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—42no—
459C42—42no—
460C42AGL(1,ℤ/8)12yes—
461C42AGL(1,ℤ/8)12yes—
462C42C22×D411yescharacteristic
463C42C2×#4111yescharacteristic
464C42H(5,𝔽2)11yescharacteristic
465C422−1+411yescharacteristic
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
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—
484C23—41no—
485C23—41no—
486C23—41no—
487C23C2×C4×D411yescharacteristic
488D4—22no—
489D4—22no—
490D4—22no—
491D4—22no—
492D4—22no—
493D4—41no—
494D4—41no—
495D4—41no—
496D4—41no—
497D4—41no—
498D4—41no—
499D4—84no—
500D4—84no—
501D4#43514yescomplemented
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—48no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—84no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4#44212yes—
544C2×C4#44212yes—
545C2×C4C2×C4×D411yescharacteristic
546C2×C4#52011yescharacteristic
547C2×C4D4211yescharacteristic
548C2×C4#54711yescharacteristic
549C2×C4#54811yescharacteristic
550C2×C4#55311yescharacteristic
551C2×C4C22×M1611yescharacteristic
552C2×C4C2×#13111yescharacteristic
553C2×C4C2×#13111yescharacteristic
554C2×C4#56811yescharacteristic
555C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
556C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
557C2×C4#57611yescharacteristic
558C2×C4#57711yescharacteristic
559C8—22no—
560C8—42no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—41no—
565C8—41no—
566C8—42no—
567C8—41no—
568C8—41no—
569C8—81no—
570C8—81no—
571C8—81no—
572C8—81no—
573C22—24no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22#236111yescharacteristic
595C22#292911yescharacteristic
596C22D4×M1611yescharacteristic
597C22C2×#44211yescharacteristic
598C22#294611yescharacteristic
599C22#295511yescharacteristic
600C22#295711yescharacteristic
601C4—22no—
602C4—21no—
603C4—21no—
604C4—22no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—42no—
609C4—81no—
610C4—81no—
611C4—81no—
612C4C2×#43511yescharacteristic
613C4#293511yescharacteristic
614C4#299011yescharacteristic
615C4#299211yescharacteristic
616C4#327211yescharacteristic
617C4#327411yescharacteristic
618C4#336611yescharacteristic
619C4#337311yescharacteristic
620C2—44no—
621C2—82no—
622C2—161no—
623C2#1790711yescharacteristic
624C2#1792011yescharacteristic
625C2#1804411yescharacteristic
626C2#1805011yescharacteristic
627C2#2298711yescharacteristic
628C2#2300111yescharacteristic
629C2#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. #23174C2semidirect
2. D4×C4⋊C8C2semidirect
4. C2×#1586C2semidirect
5. C2×#1580C2semidirect
6. #9644C2semidirect
7. #9605C2semidirect
8. #9560C2semidirect
9. #9556C2semidirect
10. #8947C2semidirect
11. #8947C2semidirect
12. #8849C2semidirect
13. #8845C2semidirect
14. #8265C2semidirect
15. #8174C2semidirect
168. D4⋊C8D4semidirect
186. C2×D8M16semidirect
501. D4#435semidirect

Maximal subgroups

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

subgroupindex
#231742
D4×C4⋊C82
C4×#4592
C2×#15862
C2×#15802
#96442
#96052
#95602
#95562
#89472
#88492
#88452
#82652
#81742