Finite Group Databasebeta

← #225188

Subgroups — #225188

Order 512.

757 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#22640C211yesmaximal, characteristic, complemented
2#22402C211yesmaximal, characteristic
3#21405C211yesmaximal, characteristic
4#9556C211yesmaximal, characteristic, complemented
5#9554C211yesmaximal, characteristic, complemented
6#9545C211yesmaximal, characteristic, complemented
7#9542C211yesmaximal, characteristic, complemented
8#8849C211yesmaximal, characteristic, complemented
9#8843C211yesmaximal, characteristic, complemented
10#8711C211yesmaximal, characteristic, complemented
11#8597C211yesmaximal, characteristic, complemented
12#8313C211yesmaximal, characteristic
13#8208C211yesmaximal, characteristic, complemented
14C8×#102C211yesmaximal, characteristic, complemented
15#8146C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#408—21no—
20C2×C4×M16C2211yescharacteristic
21#2283C2211yescharacteristic
22#1993C411yescharacteristic
23#1993C411yescharacteristic
24#1961C411yescharacteristic
25#1961C411yescharacteristic
26#1954C2211yescharacteristic
27#1954C2211yescharacteristic
28#1939C2211yescharacteristic
29#1939C2211yescharacteristic
30#1924—21no—
31#1923—21no—
32#1895C411yescharacteristic
33#1895C411yescharacteristic
34#1895C411yescharacteristic
35#1895C411yescharacteristic
36#1870—21no—
37#1869—21no—
38#1840—21no—
39#1797C2211yescharacteristic
40#1797C2211yescharacteristic
41#1780—21no—
42#1770C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44#1768—21no—
45C4×#102C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×#322C2211yescharacteristic
58C2×C82C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×#41—21no—
68C22×M16—21no—
69C22×M16C2×C411yescharacteristic
70C22×M16C2×C411yescharacteristic
71C23×C8—21no—
72#532—22no—
73#532—22no—
74#532—41no—
75#532—41no—
76C2×C4⋊Q8C2×C411yescharacteristic
77C2×#127C2×C411yescharacteristic
78C2×#124—21no—
79C2×#121—21no—
80C2×#121C2×C411yescharacteristic
81C2×#121C2×C411yescharacteristic
82C2×C4×D4—21no—
83C2×C4×D4C2311yescharacteristic
84C2×C4×D4C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#436—22no—
87#436—41no—
88C8×D4—22no—
89C8×D4—41no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#102C2×C411yescharacteristic
97C2×#102C2×C411yescharacteristic
98C2×#102C2×C411yescharacteristic
99C2×#102C2×C411yescharacteristic
100C2×#102C2×C411yescharacteristic
101C2×#102C2×C411yescharacteristic
102C2×#102C2×C411yescharacteristic
103C2×#102C2×C411yescharacteristic
104#408—44no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8D411yescharacteristic
115C2×C22⋊C8D411yescharacteristic
116C4×M16—41no—
117C4×M16—41no—
118C2×#97—21no—
119C2×#97C2×C411yescharacteristic
120C2×#97C2×C411yescharacteristic
121C2×C4×C8—21no—
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128#383—21no—
129#382C2×C411yescharacteristic
130#382C2×C411yescharacteristic
131#381—21no—
132#380C2×C411yescharacteristic
133#380C2×C411yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136C4×C22⋊C4—21no—
137#376—21no—
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336D411yescharacteristic
154#336D411yescharacteristic
155#335—41no—
156#334—41no—
157D4⋊C8—22no—
158D4⋊C8—22no—
159D4⋊C8—41no—
160D4⋊C8—41no—
161#322—41no—
162C82—41no—
163C2×#41—42no—
164C2×#41—42no—
165C2×#41—42no—
166C2×#41—42no—
167C2×#41—81no—
168C2×#41—81no—
169C22×Q8—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4C22×C411yescharacteristic
175C22×D4C22×C411yescharacteristic
176C23×C4—21no—
177C23×C4—21no—
178C23×C4C22×C411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4#4111yescharacteristic
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—41no—
185C2×M16—41no—
186C2×M16—41no—
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—81no—
199C22×C8C22×C411yescharacteristic
200C22×C8C22×C411yescharacteristic
201C22×C8C22×C411yescharacteristic
202C22×C8C22×C411yescharacteristic
203C22×C8C2×D411yescharacteristic
204C22×C8C2×D411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C22×C8C2×D411yescharacteristic
207C4⋊Q8—22no—
208C4⋊Q8—41no—
209#127—22no—
210#127—41no—
211#124—42no—
212#124—42no—
213#121—22no—
214#121—22no—
215#121—42no—
216#121—42no—
217#121—41no—
218#121—41no—
219C4×D4—21no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4—21no—
225C4×D4—42no—
226C4×D4—42no—
227C4×D4#4112yes—
228C4×D4#4112yes—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42C22×C411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42#4111yescharacteristic
251C2×C42C2411yescharacteristic
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—42no—
261C4⋊C8—42no—
262#102—22no—
263#102—22no—
264#102—22no—
265#102—22no—
266#102—22no—
267#102—22no—
268#102—41no—
269#102—41no—
270C22⋊C8—22no—
271C22⋊C8—41no—
272C22⋊C8—42no—
273C22⋊C8—42no—
274C22⋊C8—42no—
275C22⋊C8—42no—
276C22⋊C8—42no—
277C22⋊C8—42no—
278C22⋊C8—81no—
279C22⋊C8—81no—
280C22⋊C8—81no—
281#97—41no—
282#97—41no—
283#97—81no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—81no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305C24—21no—
306C24—21no—
307C24C4×D411yescharacteristic
308#41—84no—
309#41—84no—
310#41—84no—
311#41—84no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—81no—
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—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—81no—
335C2×D4—81no—
336C2×D4—81no—
337C2×D4—81no—
338C2×D4#11712yes—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4C4×D411yescharacteristic
374C22×C4C4×D411yescharacteristic
375C22×C4C4×D411yescharacteristic
376C22×C4C23×C411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C2×#4111yescharacteristic
379M16—44no—
380M16—44no—
381M16—84no—
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—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—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×C8—41no—
419C2×C8—41no—
420C2×C8—84no—
421C2×C8#13112yes—
422C2×C8#13112yes—
423C2×C8#13112yes—
424C2×C8AGL(1,ℤ/8)12yes—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—42no—
437C4⋊C4#11712yes—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—81no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42C2×#4111yescharacteristic
457C42C2×#4111yescharacteristic
458C42H(5,𝔽2)11yescharacteristic
459C42H(5,𝔽2)11yescharacteristic
460C23—22no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
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—
481C23C2×C4×D411yescharacteristic
482Q8—84no—
483D4—22no—
484D4—22no—
485D4—41no—
486D4—41no—
487D4—41no—
488D4—41no—
489D4—41no—
490D4—41no—
491D4—84no—
492D4—84no—
493D4—84no—
494D4—84no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
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—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4C2×#11711yescharacteristic
543C2×C4#51811yescharacteristic
544C2×C4#54611yescharacteristic
545C2×C4#54611yescharacteristic
546C2×C4#55111yescharacteristic
547C2×C4#55111yescharacteristic
548C2×C4C2×#13111yescharacteristic
549C2×C4C2×#13111yescharacteristic
550C2×C4C2×#13111yescharacteristic
551C2×C4#56811yescharacteristic
552C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
553C2×C4#57511yescharacteristic
554C2×C4#57611yescharacteristic
555C2×C4#57711yescharacteristic
556C8—22no—
557C8—22no—
558C8—22no—
559C8—22no—
560C8—42no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—41no—
565C8—81no—
566C8—81no—
567C8—81no—
568C8—81no—
569C8—81no—
570C22—41no—
571C22—42no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—42no—
576C22—41no—
577C22—42no—
578C22—41no—
579C22—41no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22#234211yescharacteristic
592C22#293111yescharacteristic
593C22#293511yescharacteristic
594C22C4×AGL(1,ℤ/8)11yescharacteristic
595C22#294611yescharacteristic
596C22#295511yescharacteristic
597C22#295611yescharacteristic
598C4—21no—
599C4—21no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—81no—
605C4—81no—
606C4—81no—
607C4—81no—
608C4—81no—
609C4#292211yescharacteristic
610C4#292511yescharacteristic
611C4#297611yescharacteristic
612C4#297711yescharacteristic
613C4#331111yescharacteristic
614C4#331811yescharacteristic
615C4#332211yescharacteristic
616C4#332411yescharacteristic
617C2—81no—
618C2—82no—
619C2—161no—
620C2#1796411yescharacteristic
621C2#1796811yescharacteristic
622C2#1807811yescharacteristic
623C2#1808011yescharacteristic
624C2#2256011yescharacteristic
625C2#2258011yescharacteristic
626C2#2258811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22640C2semidirect
4. #9556C2semidirect
5. #9554C2semidirect
6. #9545C2semidirect
7. #9542C2semidirect
8. #8849C2semidirect
9. #8843C2semidirect
10. #8711C2semidirect
11. #8597C2semidirect
13. #8208C2semidirect
14. C8×#102C2semidirect
15. #8146C2semidirect

Maximal subgroups

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

subgroupindex
#226402
#224022
#214052
#95562
#95542
#95452
#95422
#88492
#88432
#87112
#85972
#83132
#82082
C8×#1022
#81462