Finite Group Databasebeta

← #208267

Subgroups — #208267

Order 512.

878 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35288C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic, complemented
3#20431C211yesmaximal, characteristic, complemented
4#9906C211yesmaximal, characteristic, complemented
5#9901C211yesmaximal, characteristic, complemented
6#9710C211yesmaximal, characteristic, complemented
7#9698C211yesmaximal, characteristic, complemented
8#9141C211yesmaximal, characteristic
9#9137C211yesmaximal, characteristic, complemented
10#9133C211yesmaximal, characteristic
11#9129C211yesmaximal, characteristic, complemented
12#8236C211yesmaximal, characteristic, complemented
13#8234C211yesmaximal, characteristic, complemented
14#8233C211yesmaximal, characteristic, complemented
15#8231C211yesmaximal, characteristic, complemented
16C2×#423C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C2×#405—21no—
19#2859C2211yescharacteristic
20#2858C2211yescharacteristic
21#2857C2211yescharacteristic
22#2856C2211yescharacteristic
23#2644—21no—
24#2624—21no—
25#2513C412yes—
26#2490C412yes—
27#2454C412yes—
28#2448—21no—
29#2447—21no—
30#2427—21no—
31#2426—21no—
32#2420C412yes—
33#2324—21no—
34#2319—21no—
35C4×C4⋊Q8C2211yescharacteristic
36C4×#127C2211yescharacteristic
37C4×#117C2211yescharacteristic
38#1924—21no—
39#1924—21no—
40#1917—21no—
41#1917—21no—
42#1842C2211yescharacteristic
43#1831C2211yescharacteristic
44#1797C2211yescharacteristic
45#1797C2211yescharacteristic
46#1768—21no—
47#1768—21no—
48#1767C2211yescharacteristic
49#1767C2211yescharacteristic
50#1767C2211yescharacteristic
51#1767C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1729—21no—
55C2×#332C2211yescharacteristic
56C2×#332C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×#331C2211yescharacteristic
61C2×#331C2211yescharacteristic
62C2×#331C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71C22×M16—21no—
72C23×C8—21no—
73C2×C4⋊Q8C2311yescharacteristic
74C2×#127C2311yescharacteristic
75C2×#125C2×C412yes—
76C2×#124—21no—
77C2×#124—21no—
78C2×C22⋊Q8—21no—
79C2×#121—21no—
80C2×#121—21no—
81C2×#121C2×C412yes—
82C2×C4×Q8—21no—
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4C2×C412yes—
86C2×#117D411yescharacteristic
87C2×#117D411yescharacteristic
88C22×C42C2311yescharacteristic
89#423—22no—
90#423—22no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107#405—44no—
108C2×#97—21no—
109C2×#97—21no—
110C2×#97—21no—
111C2×#97—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114#399—22no—
115#397—22no—
116#397—22no—
117#397—22no—
118#397—22no—
119#396—22no—
120#394—22no—
121#394—22no—
122#391—21no—
123#391—21no—
124#390—22no—
125#390—21no—
126#390—21no—
127#390—21no—
128#390—21no—
129#390—21no—
130#390—21no—
131#390C2×C412yes—
132#390C2×C412yes—
133#390C2×C412yes—
134#390C2×C412yes—
135#389—21no—
136#389—21no—
137#389—21no—
138#389—21no—
139#389—21no—
140#389—21no—
141#389C2×C412yes—
142#389C2×C412yes—
143#388—22no—
144#388—22no—
145#388—22no—
146#386—22no—
147#385—22no—
148#384C2311yescharacteristic
149#384C2311yescharacteristic
150#382C2×C412yes—
151#381—22no—
152#380—21no—
153#380—22no—
154#380C2×C412yes—
155C4×C4⋊C4—21no—
156C4×C4⋊C4C2×C412yes—
157C4×C4⋊C4C2×C411yescharacteristic
158C4×C4⋊C4C2×C411yescharacteristic
159C4×C4⋊C4D411yescharacteristic
160C4×C4⋊C4D411yescharacteristic
161C4×C22⋊C4C2×C411yescharacteristic
162C4×C22⋊C4C2×C411yescharacteristic
163C4×C22⋊C4D411yescharacteristic
164C4×C22⋊C4D411yescharacteristic
165#376D411yescharacteristic
166#376D411yescharacteristic
167C43C2311yescharacteristic
168C43C2311yescharacteristic
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#332—22no—
180#332—22no—
181#332—22no—
182#332—22no—
183#331—22no—
184#331—22no—
185#331—22no—
186#331—22no—
187C22×Q8—21no—
188C22×D4—22no—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4C22×C412yes—
192C23×C4—22no—
193C23×C4C22×C411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C2×M16—24no—
197C2×M16—42no—
198C2×M16—42no—
199C22×C8—24no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—42no—
209C22×C8—42no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214#127—21no—
215#127—21no—
216#127—21no—
217#127—21no—
218#125—82no—
219#124—81no—
220#124—81no—
221#124—81no—
222#124—81no—
223C22⋊Q8—81no—
224C22⋊Q8—81no—
225#121—42no—
226#121—42no—
227#121—81no—
228#121—81no—
229#121—82no—
230C4×Q8—81no—
231C4×Q8—81no—
232C4×D4—42no—
233C4×D4—42no—
234C4×D4—81no—
235C4×D4—81no—
236C4×D4—82no—
237#117—22no—
238#117—22no—
239#117—41no—
240#117—41no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4C22⋊C411yescharacteristic
251C2×C4⋊C4C22⋊C411yescharacteristic
252C2×C4⋊C4C22⋊C411yescharacteristic
253C2×C4⋊C4C22⋊C411yescharacteristic
254C2×C4⋊C4C22×C412yes—
255C2×C4⋊C4C22×C412yes—
256C2×C4⋊C4C22×C412yes—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4C22⋊C411yescharacteristic
269C2×C22⋊C4C22⋊C411yescharacteristic
270C2×C22⋊C4C22⋊C411yescharacteristic
271C2×C22⋊C4C22⋊C411yescharacteristic
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—22no—
277C2×C42C22×C411yescharacteristic
278C2×C42C22×C411yescharacteristic
279C2×C42C22×C411yescharacteristic
280C2×C42C22×C411yescharacteristic
281C2×C42C22×C411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2411yescharacteristic
293C4⋊C8—24no—
294C4⋊C8—24no—
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—21no—
306C4⋊C8—21no—
307C4⋊C8—81no—
308C4⋊C8—81no—
309C4⋊C8#4112yes—
310C4⋊C8#4112yes—
311C4⋊C8#4112yes—
312C4⋊C8#4112yes—
313C22⋊C8—81no—
314C22⋊C8—81no—
315#97—42no—
316#97—42no—
317#97—81no—
318#97—81no—
319C4×C8—42no—
320C4×C8—42no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95C22⋊C411yescharacteristic
328#95C22⋊C411yescharacteristic
329#95C22⋊C411yescharacteristic
330#95C22⋊C411yescharacteristic
331#95C22⋊C411yescharacteristic
332#95C22⋊C411yescharacteristic
333#95C22⋊C411yescharacteristic
334#95C22⋊C411yescharacteristic
335C24—22no—
336C24C2×C22⋊C411yescharacteristic
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—44no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—82no—
361C2×D4—82no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—44no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4AGL(1,ℤ/8)12yes—
394C22×C4#13712yes—
395C22×C4C23×C411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398M16—48no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—48no—
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—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C4⋊C4—82no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—82no—
453C22⋊C4—82no—
454C22⋊C4—82no—
455C22⋊C4—82no—
456C22⋊C4—82no—
457C22⋊C4—82no—
458C22⋊C4—82no—
459C22⋊C4—82no—
460C22⋊C4—82no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—41no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—82no—
488C42C2×#4111yescharacteristic
489C42C2×#4111yescharacteristic
490C42C2×#4111yescharacteristic
491C42C2×#4111yescharacteristic
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23#41712yes—
506C23C22×C22⋊C411yescharacteristic
507Q8—81no—
508Q8—81no—
509Q8—81no—
510Q8—81no—
511D4—82no—
512D4—82no—
513D4—81no—
514D4—81no—
515D4—81no—
516D4—81no—
517D4—81no—
518D4—81no—
519D4—81no—
520D4—81no—
521D4—164no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—164no—
576C2×C4#41712yes—
577C2×C4C4×#4111yescharacteristic
578C2×C4C4×#4111yescharacteristic
579C2×C4#52511yescharacteristic
580C2×C4#52511yescharacteristic
581C2×C4#53211yescharacteristic
582C2×C4#53211yescharacteristic
583C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
584C2×C4C2×#13711yescharacteristic
585C2×C4#57511yescharacteristic
586C2×C4#57511yescharacteristic
587C2×C4#57611yescharacteristic
588C2×C4#57611yescharacteristic
589C2×C4#57711yescharacteristic
590C2×C4#57711yescharacteristic
591C8—42no—
592C8—42no—
593C8—42no—
594C8—42no—
595C8—81no—
596C8—81no—
597C8—81no—
598C8—81no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22#229711yescharacteristic
614C22#288811yescharacteristic
615C22#288911yescharacteristic
616C22C2×#41711yescharacteristic
617C22#289611yescharacteristic
618C22#289711yescharacteristic
619C22#289711yescharacteristic
620C4—22no—
621C4—21no—
622C4—21no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4#311411yescharacteristic
637C4#311411yescharacteristic
638C4#311711yescharacteristic
639C4#311711yescharacteristic
640C4#317411yescharacteristic
641C4#317611yescharacteristic
642C4#317911yescharacteristic
643C4#317911yescharacteristic
644C2—42no—
645C2—162no—
646C2#1776211yescharacteristic
647C2#1776511yescharacteristic
648C2#1776611yescharacteristic
649C2#1776911yescharacteristic
650C2#2139111yescharacteristic
651C2#2141411yescharacteristic
652C2#2142711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21926C2semidirect
3. #20431C2semidirect
4. #9906C2semidirect
5. #9901C2semidirect
6. #9710C2semidirect
7. #9698C2semidirect
9. #9137C2semidirect
11. #9129C2semidirect
12. #8236C2semidirect
13. #8234C2semidirect
14. #8233C2semidirect
15. #8231C2semidirect

Maximal subgroups

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

subgroupindex
#352882
#219262
#204312
#99062
#99012
#97102
#96982
#91412
#91372
#91332
#91292
#82362
#82342
#82332
#82312