Finite Group Databasebeta

← #480095

Subgroups — #480095

Order 512.

1211 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#535C211yesmaximal, characteristic
2#17349C211yesmaximal, characteristic, complemented
3#17349C211yesmaximal, characteristic, complemented
4#11473C214yesmaximal, complemented
5#11471C214yesmaximal, complemented
6#11454C212yesmaximal, complemented
7#11427C212yesmaximal, complemented
8C2×#535C411yescharacteristic
9C2×C4×#41C2211yescharacteristic
10C22×C4×D4C411yescharacteristic
11C2×#431—21no—
12C2×#431—21no—
13#2328—21no—
14#2326—21no—
15#2324—21no—
16#2324—21no—
17#2321C412yes—
18#2321C2212yes—
19#2317C2211yescharacteristic
20#2311C411yescharacteristic
21#2309C2211yescharacteristic
22C4×#127—21no—
23C4×#124—21no—
24C4×#121—21no—
25C4×#121—21no—
26C4×#120C412yes—
27C4×#120C2212yes—
28C4×#117C411yescharacteristic
29#1461—21no—
30#1461—21no—
31#1460C2212yes—
32#1460C2212yes—
33#1459—21no—
34#1459—21no—
35#1459C2212yes—
36#1459C2212yes—
37#1458—21no—
38#1458—21no—
39#1458C2212yes—
40#1458C2212yes—
41#1326C2218yes—
42#1316C2218yes—
43C22×#41C2×C411yescharacteristic
44C23×D4C2×C411yescharacteristic
45C24×C4C2×C412yes—
46#535—24no—
47#535C818yescomplemented
48C2×#127—21no—
49C2×#124—21no—
50C2×#121—21no—
51C2×#121—21no—
52C2×#120C2×C412yes—
53C2×#120C2×C412yes—
54C4×#41—41no—
55C4×#41—41no—
56C4×#41—41no—
57C4×#41—41no—
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—41no—
67C2×C4×D4—41no—
68C2×C4×D4—41no—
69C2×C4×D4—41no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×#117C2×C411yescharacteristic
77C2×#117C2×C411yescharacteristic
78C2×#117D411yescharacteristic
79C2×#117D411yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C22⋊C4C2×C412yes—
83C22×C42C2×C411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86#431—22no—
87#431—22no—
88#431—22no—
89#431—22no—
90#431—41no—
91#431—41no—
92#431—41no—
93#431—41no—
94C2×C4⋊C8—41no—
95C2×C4⋊C8—41no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C4×C8—41no—
99C2×C4×C8—41no—
100#391—21no—
101#390—21no—
102#390—21no—
103#390—21no—
104#390—22no—
105#390—22no—
106#390—21no—
107#390—21no—
108#389—21no—
109#389—21no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#386C2×C412yes—
119#386C2×C412yes—
120#385—22no—
121#385—22no—
122#385C2×C412yes—
123#385C2312yes—
124#384—21no—
125#382—21no—
126#380—21no—
127#380—21no—
128#379C2×C412yes—
129#379C2312yes—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4C2×C412yes—
135C4×C22⋊C4C2×C412yes—
136#376—21no—
137C43—21no—
138#324—22no—
139#324—22no—
140#324—22no—
141#324—22no—
142#323—22no—
143#323—22no—
144#323C2314yes—
145#323C2314yes—
146C25C22×C412yes—
147C2×#41—41no—
148C2×#41—41no—
149C2×#41—42no—
150C2×#41C2×C814yes—
151C2×#41M1614yes—
152C22×Q8C22⋊C411yescharacteristic
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—22no—
158C22×D4—41no—
159C22×D4—41no—
160C22×D4—42no—
161C22×D4C22⋊C411yescharacteristic
162C22×D4C2×C814yescomplemented
163C22×D4C22×C411yescharacteristic
164C22×D4C22×C411yescharacteristic
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—84no—
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C412yes—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C22×C8—42no—
181C22×C8—42no—
182#127—28no—
183#124—28no—
184#121—28no—
185#121—28no—
186#120—24no—
187#120—24no—
188#120C2×C818yescomplemented
189#120C2×C818yescomplemented
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×Q8—41no—
194C4×D4—41no—
195C4×D4—41no—
196C4×D4—41no—
197C4×D4—41no—
198C4×D4—41no—
199C4×D4—41no—
200C4×D4—41no—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—41no—
205C4×D4—41no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—41no—
213C4×D4—41no—
214C4×D4—41no—
215C4×D4—41no—
216C4×D4—84no—
217C4×D4—84no—
218#117—22no—
219#117—41no—
220#117—41no—
221#117—41no—
222#117—41no—
223#117—41no—
224#117—41no—
225#117—42no—
226#117C2×C814yescomplemented
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4C22⋊C411yescharacteristic
254C2×C22⋊C4C22⋊C411yescharacteristic
255C2×C22⋊C4C22×C412yes—
256C2×C22⋊C4C22×C412yes—
257C2×C22⋊C4C22×C411yescharacteristic
258C2×C22⋊C4C2×D411yescharacteristic
259C2×C22⋊C4C2×D411yescharacteristic
260C2×C22⋊C4C2411yescharacteristic
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42C22⋊C411yescharacteristic
278C2×C42C22⋊C411yescharacteristic
279C2×C42C22⋊C411yescharacteristic
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22⋊C411yescharacteristic
282C2×C42C22⋊C411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C4⋊C8—42no—
288C4⋊C8—42no—
289C4⋊C8—42no—
290C4⋊C8—42no—
291C22⋊C8—24no—
292C22⋊C8—24no—
293C22⋊C8—42no—
294C22⋊C8—42no—
295C4×C8—42no—
296C4×C8—42no—
297C4×C8—42no—
298C4×C8—42no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305C24—22no—
306C24—22no—
307C24—42no—
308C24—42no—
309C24—82no—
310C24—82no—
311C24C2×C22⋊C411yescharacteristic
312C24C2×C22⋊C411yescharacteristic
313C24C22×C814yes—
314C24C23×C411yescharacteristic
315#41—44no—
316#41—44no—
317#41—44no—
318#41—44no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8C22⋊C814yes—
322C2×D4—28no—
323C2×D4—28no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—42no—
341C2×D4—82no—
342C2×D4—82no—
343C2×D4—82no—
344C2×D4—82no—
345C2×D4C22⋊C814yes—
346C2×D4C22×C812yes—
347C2×D4C22×C812yes—
348C2×D4C2×M1612yes—
349C2×D4C2×M1612yes—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
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—22no—
373C22×C4—22no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4—81no—
389C22×C4—88no—
390C22×C4—88no—
391C22×C4C22⋊C814yes—
392C22×C4C22⋊C814yes—
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C22×C812yes—
404C22×C4C2×M1612yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C2×C8—44no—
408C2×C8—44no—
409C2×C8—84no—
410C2×C8—84no—
411C4⋊C4—24no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—82no—
423C4⋊C4—82no—
424C22⋊C4—22no—
425C22⋊C4—28no—
426C22⋊C4—28no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—44no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—82no—
438C22⋊C4—82no—
439C22⋊C4—82no—
440C22⋊C4—82no—
441C22⋊C4—82no—
442C22⋊C4—82no—
443C22⋊C4C22×C814yes—
444C42—24no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—42no—
464C42—44no—
465C42—82no—
466C42#9911yescharacteristic
467C42#9911yescharacteristic
468C42#9911yescharacteristic
469C42#9911yescharacteristic
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—24no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—84no—
492C23—84no—
493C23—84no—
494C23—84no—
495C23—88no—
496C23C2×C22⋊C812yes—
497C23C2×C22⋊C812yes—
498C23C22×C22⋊C411yescharacteristic
499C23C23×C811yescharacteristic
500C23C22×M1611yescharacteristic
501Q8—44no—
502D4—44no—
503D4—42no—
504D4—42no—
505D4—42no—
506D4—42no—
507D4—48no—
508D4—44no—
509D4—44no—
510D4—88no—
511D4—88no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—84no—
562C2×C4—88no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4C2×C22⋊C812yes—
566C2×C4C2×C22⋊C812yes—
567C2×C4C2×C22⋊C812yes—
568C2×C4C2×C22⋊C812yes—
569C2×C4C2×#9911yescharacteristic
570C2×C4C2×#9911yescharacteristic
571C2×C4C2×#9911yescharacteristic
572C2×C4C2×#9911yescharacteristic
573C2×C4C2×#9911yescharacteristic
574C2×C4C2×#9911yescharacteristic
575C2×C4#41312yes—
576C8—84no—
577C8—84no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—22no—
583C22—22no—
584C22—42no—
585C22—42no—
586C22—41no—
587C22—41no—
588C22—42no—
589C22—44no—
590C22—44no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—44no—
598C22—81no—
599C22—81no—
600C22—82no—
601C22—82no—
602C22—88no—
603C22—88no—
604C22—82no—
605C22—82no—
606C22C22×C22⋊C811yescharacteristic
607C22C22×#9911yescharacteristic
608C22C2×#41311yescharacteristic
609C4—21no—
610C4—21no—
611C4—21no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—44no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—82no—
623C4—82no—
624C4#145912yes—
625C4#145912yes—
626C4#212212yes—
627C2—21no—
628C2—21no—
629C2—41no—
630C2—41no—
631C2—42no—
632C2—42no—
633C2—84no—
634C2C2×#145911yescharacteristic
635C2C2×#212211yescharacteristic
636C2#3334711yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC23C23×C8
Frattini, cosocleC2×C22⋊C4C24
socleC22C22×#99
prime radicalC2×#535C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17349C2semidirect
3. #17349C2semidirect
4. #11473C2semidirect
5. #11471C2semidirect
6. #11454C2semidirect
7. #11427C2semidirect
47. #535C8semidirect
162. C22×D4C2×C8semidirect
188. #120C2×C8semidirect
189. #120C2×C8semidirect
226. #117C2×C8semidirect

Maximal subgroups

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

subgroupindex
C4×#5352
#173492
#114732
#114712
#114542
#114272