Finite Group Databasebeta

← #193604

Subgroups — #193604

Order 512.

1319 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3105C211yesmaximal, characteristic
2C2×#2974C211yesmaximal, characteristic, complemented
3#21848C211yesmaximal, characteristic, complemented
4C2×#1494C211yesmaximal, characteristic, complemented
5C22×D4⋊C8C211yesmaximal, characteristic, complemented
6#8951C212yesmaximal, complemented
7#8949C212yesmaximal, complemented
8#8947C212yesmaximal, complemented
9#8945C212yesmaximal, complemented
10#7879C211yesmaximal, characteristic, complemented
11#7849C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14#3105—24no—
15#3105—24no—
16C2×#484C411yescharacteristic
17C2×#481C411yescharacteristic
18C2×#478C411yescharacteristic
19C2×#478C411yescharacteristic
20C2×#463C411yescharacteristic
21C2×#463C411yescharacteristic
22C2×#450C411yescharacteristic
23C2×#447C411yescharacteristic
24#2974—24no—
25#2974—24no—
26C2×C4×SD16C2211yescharacteristic
27C2×C4×SD16C2211yescharacteristic
28C2×C4×D8C2211yescharacteristic
29C2×C4×D8C2211yescharacteristic
30C2×#436C2212yes—
31C2×#435—22no—
32C2×C8×D4C2212yes—
33C2×#432C2211yescharacteristic
34C2×#431C2211yescharacteristic
35C2×#431C2211yescharacteristic
36C22×C4⋊C8C2211yescharacteristic
37C2×#407—21no—
38C22×C22⋊C8—21no—
39C22×C4×C8C2211yescharacteristic
40#1984C2212yescomplemented
41#1954C2212yescomplemented
42#1869—21no—
43#1869—21no—
44#1798—22no—
45#1494—24no—
46C2×Q8⋊C8C2212yescomplemented
47C2×D4⋊C8—24no—
48C2×D4⋊C8C2212yes—
49C2×D4⋊C8C2212yes—
50C2×D4⋊C8C2212yescomplemented
51#1277C2212yescomplemented
52#1277C2212yescomplemented
53#1273C2212yescomplemented
54#1273C2212yescomplemented
55C23×D4C2×C411yescharacteristic
56C24×C4—21no—
57C22×SD16C2×C411yescharacteristic
58C22×SD16C2×C411yescharacteristic
59C22×D8C2×C411yescharacteristic
60C22×D8C2×C411yescharacteristic
61C22×M16—22no—
62C23×C8—21no—
63C23×C8—22no—
64C23×C8—21no—
65#529—22no—
66#529—22no—
67#529—22no—
68#529—22no—
69C2×#126—21no—
70C2×#125C2×C411yescharacteristic
71C2×#124C2×C411yescharacteristic
72C2×#123—21no—
73C2×C22⋊Q8C2×C411yescharacteristic
74C2×#121C2×C411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4—21no—
81C2×C4×D4C2×C411yescharacteristic
82C2×C4×D4C2×C411yescharacteristic
83C2×C4×D4D414yes—
84C2×C4×D4D414yes—
85C2×C4×D4C2311yescharacteristic
86C2×C4×D4C2311yescharacteristic
87C2×C4×D4C2311yescharacteristic
88C2×#117—21no—
89C22×C4⋊C4C2×C411yescharacteristic
90C22×C22⋊C4—21no—
91C22×C42C2311yescharacteristic
92#484—24no—
93#481—24no—
94#478—24no—
95#478—24no—
96#463—24no—
97#463—24no—
98#450—24no—
99#447—24no—
100C4×SD16—22no—
101C4×SD16—22no—
102C4×SD16—22no—
103C4×SD16—22no—
104C4×D8—22no—
105C4×D8—22no—
106C4×D8—22no—
107C4×D8—22no—
108#436—24no—
109#436—24no—
110#435—216no—
111C8×D4—24no—
112C8×D4—24no—
113#432—22no—
114#432D414yes—
115#431—22no—
116#431—22no—
117#431—22no—
118#431—22no—
119C2×#107C2×C411yescharacteristic
120C2×#107C2×C411yescharacteristic
121C2×#106C2×C411yescharacteristic
122C2×#106C2×C411yescharacteristic
123C2×C4⋊C8—22no—
124C2×C4⋊C8—22no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8C2×C411yescharacteristic
127C2×C4⋊C8C2×C411yescharacteristic
128C2×C4⋊C8D412yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131C2×#103C2×C411yescharacteristic
132C2×#103C2×C411yescharacteristic
133C2×#102C2×C411yescharacteristic
134C2×#102C2×C411yescharacteristic
135C2×#102C2×C411yescharacteristic
136C2×#102C2×C411yescharacteristic
137C2×#102C2×C411yescharacteristic
138C2×#102C2×C411yescharacteristic
139#407—28no—
140C2×C22⋊C8—28no—
141C2×C22⋊C8—21no—
142C2×C22⋊C8—22no—
143C2×C22⋊C8—22no—
144C2×C22⋊C8—22no—
145C2×C22⋊C8—22no—
146C2×C22⋊C8C2×C411yescharacteristic
147C2×C22⋊C8C2×C411yescharacteristic
148C2×C22⋊C8D411yescharacteristic
149C2×C22⋊C8D411yescharacteristic
150C2×#97—21no—
151C2×C4×C8—22no—
152C2×C4×C8—22no—
153C2×C4×C8—21no—
154C2×C4×C8C2312yes—
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2312yes—
158#336—22no—
159#336—22no—
160Q8⋊C8—24no—
161D4⋊C8—24no—
162D4⋊C8—24no—
163D4⋊C8—24no—
164C25—21no—
165C22×Q8C22×C411yescharacteristic
166C22×D4—24no—
167C22×D4—21no—
168C22×D4—22no—
169C22×D4—21no—
170C22×D4—42no—
171C22×D4C22⋊C414yes—
172C22×D4C22×C411yescharacteristic
173C22×D4C22×C411yescharacteristic
174C22×D4C22×C411yescharacteristic
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—21no—
183C23×C4—82no—
184C23×C4—82no—
185C23×C4C22×C411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4#4111yescharacteristic
188C2×SD16—22no—
189C2×SD16—22no—
190C2×SD16—24no—
191C2×SD16—42no—
192C2×D8—22no—
193C2×D8—22no—
194C2×D8—24no—
195C2×D8—42no—
196C2×M16—28no—
197C2×M16—82no—
198C2×M16—82no—
199C22×C8—24no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—28no—
203C22×C8—22no—
204C22×C8—42no—
205C22×C8—42no—
206C22×C8—42no—
207C22×C8—82no—
208C22×C8—82no—
209C22×C8—82no—
210C22×C8C22×C411yescharacteristic
211C22×C8C22×C411yescharacteristic
212C22×C8C22×C411yescharacteristic
213C22×C8C22×C411yescharacteristic
214C22×C8C2×D412yes—
215C22×C8C2×D412yes—
216#126—42no—
217#126—42no—
218#125—22no—
219#125—22no—
220#124—22no—
221#124—22no—
222#123—42no—
223#123—42no—
224C22⋊Q8—22no—
225C22⋊Q8—22no—
226#121—22no—
227#121—22no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—24no—
235C4×D4—24no—
236C4×D4—24no—
237C4×D4—24no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—42no—
247C4×D4—42no—
248C4×D4C22⋊C414yes—
249C4×D4C22⋊C414yes—
250C4×D4C2×D412yes—
251C4×D4C2×D412yes—
252C4×D4C2×D412yes—
253C4×D4C2×D412yes—
254#117—42no—
255#117—42no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4C22⋊C414yes—
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C42—21no—
279C2×C42C22×C411yescharacteristic
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D411yescharacteristic
284C2×C42#4112yes—
285C2×C42#4111yescharacteristic
286C2×C42C2411yescharacteristic
287#107—22no—
288#107—22no—
289#106—22no—
290#106—22no—
291C4⋊C8—22no—
292C4⋊C8—22no—
293C4⋊C8—24no—
294C4⋊C8—22no—
295C4⋊C8—22no—
296C4⋊C8—22no—
297C4⋊C8—42no—
298C4⋊C8C2×D414yes—
299#103—22no—
300#103—22no—
301#102—22no—
302#102—22no—
303#102—22no—
304#102—22no—
305#102—22no—
306#102—22no—
307C22⋊C8—28no—
308C22⋊C8—28no—
309C22⋊C8—22no—
310C22⋊C8—22no—
311C22⋊C8—22no—
312C22⋊C8—24no—
313C22⋊C8—22no—
314C22⋊C8—82no—
315C22⋊C8—82no—
316#97—24no—
317C4×C8—22no—
318C4×C8—22no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—22no—
324C4×C8—22no—
325C4×C8—42no—
326C24—21no—
327C24—21no—
328C24—24no—
329C24—21no—
330C24—44no—
331C24—42no—
332C24—42no—
333C24—82no—
334C24C4×D411yescharacteristic
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×D4—28no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—24no—
350C2×D4—44no—
351C2×D4—48no—
352C2×D4—44no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4C2×C22⋊C412yes—
357C2×D4C2×C22⋊C412yes—
358C2×D4#13114yes—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—88no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4C2×C22⋊C412yes—
392C22×C4C4×D412yes—
393C22×C4C4×D411yescharacteristic
394C22×C4C4×D411yescharacteristic
395C22×C4C4×D411yescharacteristic
396C22×C4#12012yes—
397C22×C4#12112yes—
398C22×C4#12112yes—
399C22×C4#12312yes—
400C22×C4C23×C411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C2×#4111yescharacteristic
403SD16—44no—
404SD16—44no—
405D8—44no—
406D8—44no—
407M16—88no—
408C2×C8—24no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—24no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—48no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—88no—
433C2×C8—84no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—24no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4C2×C22⋊C412yes—
447C4⋊C4C2×C22⋊C412yes—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—42no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C42—41no—
461C42—41no—
462C42C2×C22⋊C412yes—
463C42C4×D412yes—
464C42#12012yes—
465C42#12112yes—
466C42#12112yes—
467C42#12312yes—
468C42C22×D411yescharacteristic
469C42C22×D411yescharacteristic
470C42C22×D411yescharacteristic
471C42C2×#4111yescharacteristic
472C23—22no—
473C23—22no—
474C23—24no—
475C23—22no—
476C23—22no—
477C23—48no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—48no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—84no—
492C23—88no—
493C23—88no—
494C23#38612yes—
495C23#44312yes—
496C23#44412yes—
497C23C2×C4×D411yescharacteristic
498Q8—41no—
499Q8—41no—
500Q8—42no—
501D4—22no—
502D4—22no—
503D4—22no—
504D4—24no—
505D4—48no—
506D4—41no—
507D4—41no—
508D4—42no—
509D4—42no—
510D4—84no—
511D4#40814yes—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4—82no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4#38612yes—
550C2×C4#38612yes—
551C2×C4#38612yes—
552C2×C4C22×C22⋊C411yescharacteristic
553C2×C4C2×C4×D411yescharacteristic
554C2×C4C2×#12011yescharacteristic
555C2×C4C2×#12111yescharacteristic
556C2×C4C2×#12111yescharacteristic
557C2×C4C2×#12311yescharacteristic
558C2×C4C2×#13112yes—
559C2×C4#56812yes—
560C2×C4#57612yes—
561C2×C4#57712yes—
562C8—44no—
563C8—41no—
564C8—41no—
565C8—41no—
566C8—41no—
567C8—42no—
568C8—42no—
569C8—44no—
570C8—82no—
571C22—24no—
572C22—24no—
573C22—22no—
574C22—22no—
575C22—24no—
576C22—44no—
577C22—44no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—816no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22#187614yes—
592C22C2×#38611yescharacteristic
593C22#293312yes—
594C22#294612yes—
595C22C2×#44311yescharacteristic
596C22C2×#44411yescharacteristic
597C4—22no—
598C4—21no—
599C4—21no—
600C4—22no—
601C4—22no—
602C4—42no—
603C4—42no—
604C4—81no—
605C4—81no—
606C4—81no—
607C4C2×#40811yescharacteristic
608C4#287911yescharacteristic
609C4#293312yes—
610C4#301212yes—
611C4#304111yescharacteristic
612C4#304211yescharacteristic
613C2—24no—
614C2—44no—
615C2—82no—
616C2—161no—
617C2#1801412yes—
618C2#1802012yes—
619C2#2089711yescharacteristic
620C2C2×#187611yescharacteristic
621C2#2092911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#447C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2974C2semidirect
3. #21848C2semidirect
4. C2×#1494C2semidirect
5. C22×D4⋊C8C2semidirect
6. #8951C2semidirect
7. #8949C2semidirect
8. #8947C2semidirect
9. #8945C2semidirect
10. #7879C2semidirect
11. #7849C2semidirect
40. #1984C22semidirect
41. #1954C22semidirect
46. C2×Q8⋊C8C22semidirect
50. C2×D4⋊C8C22semidirect
51. #1277C22semidirect
52. #1277C22semidirect
53. #1273C22semidirect
54. #1273C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31052
C2×#29742
#218482
C2×#14942
C22×D4⋊C82
#89512
#89492
#89472
#89452
#78792
#78492