Finite Group Databasebeta

← #198455

Subgroups — #198455

Order 512.

1253 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3195C211yesmaximal, characteristic
2#34380C211yesmaximal, characteristic
3#21825C211yesmaximal, characteristic
4#21762C211yesmaximal, characteristic
5C2×#1520C211yesmaximal, characteristic
6#9258C214yesmaximal, complemented
7#9256C214yesmaximal, complemented
8#7983C211yesmaximal, characteristic
9#7925C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C22×C22⋊Q8C2211yescharacteristic
12#3195—22no—
13#3195—22no—
14#3195—22no—
15#3195—22no—
16C2×#483C412yescomplemented
17C2×#483C2212yes—
18C2×#470C414yescomplemented
19C2×#451C412yescomplemented
20C2×#451C2212yes—
21C2×#407C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23#2378—22no—
24#2360—22no—
25#2344C2212yes—
26#2342—22no—
27Q8×C22⋊C4C2212yes—
28#2329—22no—
29#2323—22no—
30#2316—21no—
31#2313—21no—
32#2280C2211yescharacteristic
33#2280C2211yescharacteristic
34C2×#380C2211yescharacteristic
35#1996—21no—
36#1996—21no—
37#1996C2214yes—
38#1892—22no—
39#1873—22no—
40#1823C2212yes—
41#1819—22no—
42#1787C2212yes—
43#1520—22no—
44#1520C2214yes—
45C2×#328—22no—
46#1289C2214yes—
47#1282C2214yes—
48C23×Q8C2×C411yescharacteristic
49C23×Q8C2311yescharacteristic
50C24×C4C2311yescharacteristic
51C22×Q16C2×C414yescomplemented
52C22×M16C2×C412yes—
53C23×C8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8C2×C412yes—
71C2×C22⋊Q8C2×C412yes—
72C2×C22⋊Q8C2×C412yes—
73C2×C22⋊Q8D412yes—
74C2×C22⋊Q8D412yes—
75C2×C22⋊Q8D412yes—
76C2×C22⋊Q8D412yes—
77C2×C22⋊Q8C2312yes—
78C2×C4×Q8—22no—
79C2×#117—21no—
80C2×#117—22no—
81C2×#117—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90#483—24no—
91#483—24no—
92#483—24no—
93#483—24no—
94#470—28no—
95#470—28no—
96#451—24no—
97#451—24no—
98#451—24no—
99#451—24no—
100C2×#106C2×C414yescomplemented
101C2×#103—24no—
102C2×#103C2×C414yescomplemented
103C2×#103C2×C414yescomplemented
104#407—21no—
105#407—21no—
106#407—21no—
107#407—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8C2×C412yes—
114C2×C22⋊C8C2312yes—
115C2×C22⋊C8C2312yes—
116#391—22no—
117#391—24no—
118#389—24no—
119#389—24no—
120#389—22no—
121#388—24no—
122#388—24no—
123#387—24no—
124#387—24no—
125#387—24no—
126#385—24no—
127#381—21no—
128#381—22no—
129#381—21no—
130#380—21no—
131#380—21no—
132#380—21no—
133#380—22no—
134#380—21no—
135#380D411yescharacteristic
136#380D411yescharacteristic
137#380D412yes—
138#380C2312yes—
139#380C2312yes—
140#379—21no—
141#379D412yes—
142C4×C4⋊C4—22no—
143C4×C22⋊C4—24no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—22no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4C2312yes—
149C2×#95D412yes—
150#342—24no—
151#342—24no—
152#340—24no—
153#328—28no—
154C25C2×D411yescharacteristic
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—82no—
164C22×Q8—82no—
165C22×Q8C22×C412yes—
166C22×Q8C22×C412yes—
167C22×Q8C2×D412yes—
168C22×Q8C2×D412yes—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4—82no—
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4#4111yescharacteristic
185C23×C4#4111yescharacteristic
186C23×C4#4111yescharacteristic
187C23×C4#4111yescharacteristic
188C23×C4C2411yescharacteristic
189C2×Q16—28no—
190C2×Q16—84no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—82no—
194C22×C8—22no—
195C22×C8—41no—
196C22×C8—41no—
197C22×C8—82no—
198C22×C8C22×C414yes—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203C22⋊Q8—22no—
204C22⋊Q8—22no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218C22⋊Q8—22no—
219C22⋊Q8—41no—
220C22⋊Q8—41no—
221C22⋊Q8—41no—
222C22⋊Q8—41no—
223C22⋊Q8—41no—
224C22⋊Q8—41no—
225C22⋊Q8—41no—
226C22⋊Q8—41no—
227C22⋊Q8C22⋊C414yes—
228C22⋊Q8C22⋊C414yes—
229C22⋊Q8C22⋊C414yes—
230C22⋊Q8C22⋊C414yes—
231C4×Q8—84no—
232#117—82no—
233#117—84no—
234#117—82no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—82no—
263C2×C4⋊C4C22×C412yes—
264C2×C4⋊C4C22×C412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—82no—
287C2×C22⋊C4C2×D412yes—
288C2×C22⋊C4C2×D412yes—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—41no—
293#106—28no—
294#103—28no—
295#103—28no—
296#103—84no—
297C22⋊C8—22no—
298C22⋊C8—22no—
299C22⋊C8—22no—
300C22⋊C8—22no—
301C22⋊C8—22no—
302C22⋊C8—22no—
303C22⋊C8—41no—
304C22⋊C8—41no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—22no—
310#95—21no—
311#95—21no—
312#95—22no—
313#95—22no—
314#95—41no—
315#95C2×D414yes—
316C24—22no—
317C24—22no—
318C24—42no—
319C24—81no—
320C24C4×D411yescharacteristic
321C24C4×D411yescharacteristic
322C24#12011yescharacteristic
323C24#12011yescharacteristic
324C24#12111yescharacteristic
325C24#12111yescharacteristic
326C24C22×D411yescharacteristic
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—44no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—82no—
339C2×Q8—82no—
340C2×Q8—82no—
341C2×Q8—82no—
342C2×Q8—82no—
343C2×Q8—82no—
344C2×Q8—82no—
345C2×Q8—82no—
346C2×Q8C2×C22⋊C414yes—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—81no—
374C22×C4—81no—
375C22×C4—81no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C4×D411yescharacteristic
390C22×C4C4×D411yescharacteristic
391C22×C4C4×D411yescharacteristic
392C22×C4C4×D411yescharacteristic
393C22×C4C4×D411yescharacteristic
394C22×C4C4×D411yescharacteristic
395C22×C4#12011yescharacteristic
396C22×C4#12011yescharacteristic
397C22×C4#12111yescharacteristic
398C22×C4#12111yescharacteristic
399C22×C4#12111yescharacteristic
400C22×C4#12111yescharacteristic
401C22×C4#12111yescharacteristic
402C22×C4#12111yescharacteristic
403C22×C4#12311yescharacteristic
404C22×C4#12311yescharacteristic
405C22×C4#12311yescharacteristic
406C22×C4#12311yescharacteristic
407C22×C4C23×C411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413Q16—88no—
414M16—82no—
415M16—82no—
416C2×C8—24no—
417C2×C8—44no—
418C2×C8—44no—
419C2×C8—42no—
420C2×C8—82no—
421C2×C8—82no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—82no—
440C4⋊C4—81no—
441C4⋊C4—81no—
442C4⋊C4—84no—
443C4⋊C4—84no—
444C4⋊C4C2×C22⋊C414yes—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—44no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—44no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—82no—
460C22⋊C4—82no—
461C22⋊C4—82no—
462C22⋊C4—82no—
463C42—44no—
464C42—84no—
465C42—84no—
466C42—82no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—21no—
472C23—21no—
473C23—22no—
474C23—22no—
475C23—21no—
476C23—21no—
477C23—22no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—84no—
490C23—84no—
491C23#38611yescharacteristic
492C23#38611yescharacteristic
493C23#38611yescharacteristic
494C23#38611yescharacteristic
495C23C2×C4×D411yescharacteristic
496C23C2×#12011yescharacteristic
497C23C2×#12111yescharacteristic
498C23#57712yes—
499C23#57812yes—
500Q8—28no—
501Q8—44no—
502Q8—44no—
503Q8—82no—
504Q8—82no—
505Q8—82no—
506Q8—82no—
507Q8—82no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
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—41no—
537C2×C4—41no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—164no—
566C2×C4#38611yescharacteristic
567C2×C4#38611yescharacteristic
568C2×C4#38611yescharacteristic
569C2×C4#38611yescharacteristic
570C2×C4C22×C22⋊C411yescharacteristic
571C2×C4C2×C4×D411yescharacteristic
572C2×C4C2×#12111yescharacteristic
573C2×C4C2×#12311yescharacteristic
574C8—44no—
575C8—82no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—44no—
598C22—44no—
599C22—81no—
600C22—81no—
601C22—82no—
602C22—82no—
603C22#190512yes—
604C22C2×#38611yescharacteristic
605C22#294712yes—
606C22#300911yescharacteristic
607C22#301111yescharacteristic
608C22#304612yes—
609C4—21no—
610C4—21no—
611C4—21no—
612C4—21no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—44no—
617C4—44no—
618C4—82no—
619C4—82no—
620C4—84no—
621C4—82no—
622C4—82no—
623C4—162no—
624C4#288312yes—
625C2—21no—
626C2—21no—
627C2—22no—
628C2—22no—
629C2—22no—
630C2—41no—
631C2—41no—
632C2—41no—
633C2#2093111yescharacteristic
634C2#2098411yescharacteristic
635C2C2×#190511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9258C2semidirect
7. #9256C2semidirect
16. C2×#483C4semidirect
18. C2×#470C4semidirect
19. C2×#451C4semidirect
51. C22×Q16C2×C4semidirect
100. C2×#106C2×C4semidirect
102. C2×#103C2×C4semidirect
103. C2×#103C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31952
#343802
#218252
#217622
C2×#15202
#92582
#92562
#79832
#79252