Finite Group Databasebeta

← #193707

Subgroups — #193707

Order 512.

1372 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3105C211yesmaximal, characteristic
2C2×#2966C211yesmaximal, characteristic, complemented
3C2×#1954C211yesmaximal, characteristic, complemented
4C2×#1483C211yesmaximal, characteristic, complemented
5C2×#1490C211yesmaximal, characteristic, complemented
6#8952C212yesmaximal, complemented
7#8951C212yesmaximal, complemented
8#8948C212yesmaximal, complemented
9#8947C212yesmaximal, complemented
10#7879C211yesmaximal, characteristic, complemented
11#7848C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14#3105—24no—
15#3105—24no—
16C2×#484C411yescharacteristic
17C2×#481C411yescharacteristic
18C2×#478C412yes—
19C2×#463C412yes—
20C2×#450C411yescharacteristic
21C2×#447C411yescharacteristic
22#2966—24no—
23#2966—24no—
24C2×#446C2211yescharacteristic, complemented
25C2×C8×Q8C2211yescharacteristic, complemented
26C2×C4×SD16C2212yes—
27C2×C4×D8C2212yes—
28C2×#436C2211yescharacteristic, complemented
29C2×#435—21no—
30C2×C8×D4—21no—
31C2×C8×D4C2211yescharacteristic, complemented
32C2×#433—21no—
33C2×#432C2211yescharacteristic
34C2×#431C2211yescharacteristic
35C22×C4⋊C8C2211yescharacteristic
36C22×C22⋊C8—21no—
37C22×C22⋊C8—21no—
38C2×#405—21no—
39C2×C4×M16C2211yescharacteristic
40C22×C4×C8C2211yescharacteristic
41#1954—21no—
42#1954—21no—
43#1954C2212yes—
44#1954C2212yes—
45#1490—24no—
46#1483—24no—
47C2×Q8⋊C8C2212yescomplemented
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×C412yes—
58C22×D8C2×C412yes—
59C22×M16—21no—
60C22×M16—21no—
61C23×C8—21no—
62C23×C8—21no—
63C23×C8D411yescharacteristic
64C23×C8D411yescharacteristic
65#529—22no—
66#529—22no—
67#529D414yes—
68#529D414yes—
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—22no—
81C2×C4×D4—21no—
82C2×C4×D4—22no—
83C2×C4×D4C2×C412yes—
84C2×C4×D4C2312yes—
85C2×C4×D4C2311yescharacteristic
86C2×#117—21no—
87C22×C4⋊C4C2×C411yescharacteristic
88C22×C22⋊C4—21no—
89C22×C42C2311yescharacteristic
90#484—24no—
91#481—24no—
92#478—28no—
93#463—28no—
94#450—24no—
95#447—24no—
96#446—22no—
97#446—22no—
98C8×Q8—22no—
99C8×Q8—22no—
100C4×SD16—24no—
101C4×SD16—24no—
102C4×D8—24no—
103C4×D8—24no—
104#436—22no—
105#436—22no—
106#435—28no—
107C8×D4—28no—
108C8×D4—22no—
109C8×D4—22no—
110#433—28no—
111#432—22no—
112#432D414yes—
113#431—22no—
114#431—22no—
115C2×#107C2×C412yes—
116C2×#106C2×C412yes—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—22no—
120C2×C4⋊C8—22no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8C2×C412yes—
123C2×C4⋊C8C2312yes—
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×#103C2×C412yes—
127C2×#102C2×C412yes—
128C2×#102C2×C412yes—
129C2×#102C2×C412yes—
130C2×C22⋊C8—21no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—22no—
134C2×C22⋊C8—22no—
135C2×C22⋊C8—22no—
136C2×C22⋊C8—22no—
137C2×C22⋊C8—44no—
138C2×C22⋊C8—44no—
139C2×C22⋊C8C2×C411yescharacteristic
140C2×C22⋊C8C2×C411yescharacteristic
141#405—28no—
142C4×M16—22no—
143C4×M16—22no—
144C2×#97—21no—
145C2×#97—21no—
146C2×#97—21no—
147C2×C4×C8—21no—
148C2×C4×C8—22no—
149C2×C4×C8—22no—
150C2×C4×C8D411yescharacteristic
151C2×C4×C8D411yescharacteristic
152C2×C4×C8C2312yes—
153C2×C4×C8C2312yes—
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156Q8⋊C8—24no—
157D4⋊C8—24no—
158D4⋊C8—24no—
159D4⋊C8—24no—
160C25—21no—
161C22×Q8C22×C411yescharacteristic
162C22×D4—21no—
163C22×D4—24no—
164C22×D4—22no—
165C22×D4—21no—
166C22×D4—22no—
167C22×D4—42no—
168C22×D4C22×C411yescharacteristic
169C22×D4C22×C412yes—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4C22×C411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4#4111yescharacteristic
183C2×SD16—24no—
184C2×SD16—24no—
185C2×SD16—42no—
186C2×D8—24no—
187C2×D8—24no—
188C2×D8—42no—
189C2×M16—24no—
190C2×M16—24no—
191C2×M16—42no—
192C2×M16—42no—
193C2×M16—42no—
194C2×M16—42no—
195C22×C8—24no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—41no—
205C22×C8—41no—
206C22×C8—42no—
207C22×C8—42no—
208C22×C8—42no—
209C22×C8—42no—
210C22×C8—42no—
211C22×C8C22×C412yes—
212C22×C8C22×C412yes—
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C22×C8C2×D411yescharacteristic
216C22×C8C2×D411yescharacteristic
217#126—24no—
218#126—24no—
219#125—22no—
220#125C22⋊C414yes—
221#124—22no—
222#124C22⋊C414yes—
223#123—24no—
224#123—24no—
225C22⋊Q8—22no—
226C22⋊Q8C22⋊C414yes—
227#121—22no—
228#121C22⋊C414yes—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8C2×D412yes—
232C4×Q8C2×D412yes—
233C4×D4—24no—
234C4×D4—24no—
235C4×D4—24no—
236C4×D4—24no—
237C4×D4—21no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—21no—
242C4×D4—22no—
243C4×D4—42no—
244C4×D4—42no—
245C4×D4—42no—
246C4×D4—42no—
247C4×D4C2×D412yes—
248C4×D4C2×D412yes—
249#117—24no—
250#117—24no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4C22×C412yes—
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C42—21no—
271C2×C42C22×C411yescharacteristic
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D411yescharacteristic
276C2×C42#4112yes—
277C2×C42#4111yescharacteristic
278C2×C42C2411yescharacteristic
279#107—24no—
280#106—24no—
281C4⋊C8—24no—
282C4⋊C8—24no—
283C4⋊C8—24no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—22no—
287C4⋊C8—22no—
288C4⋊C8—42no—
289C4⋊C8C2×D412yes—
290C4⋊C8C2×D412yes—
291#103—24no—
292#102—24no—
293#102—24no—
294#102—24no—
295C22⋊C8—24no—
296C22⋊C8—24no—
297C22⋊C8—22no—
298C22⋊C8—22no—
299C22⋊C8—24no—
300C22⋊C8—44no—
301C22⋊C8—44no—
302C22⋊C8—44no—
303C22⋊C8—44no—
304#97—24no—
305#97—24no—
306#97—42no—
307C4×C8—24no—
308C4×C8—22no—
309C4×C8—22no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—22no—
315C4×C8—22no—
316C4×C8—22no—
317C4×C8—22no—
318C24—24no—
319C24—21no—
320C24—22no—
321C24—42no—
322C24—42no—
323C24—42no—
324C24—42no—
325C24—42no—
326C24—42no—
327C24C4×D411yescharacteristic
328C2×Q8—21no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8C2×C22⋊C412yes—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—21no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4C2×C22⋊C412yes—
349C2×D4#13114yes—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—84no—
381C22×C4—84no—
382C22×C4—84no—
383C22×C4—84no—
384C22×C4C2×C22⋊C412yes—
385C22×C4C4×D412yes—
386C22×C4C4×D411yescharacteristic
387C22×C4C4×D411yescharacteristic
388C22×C4C4×D411yescharacteristic
389C22×C4#12012yes—
390C22×C4#12112yes—
391C22×C4#12112yes—
392C22×C4#12312yes—
393C22×C4C23×C411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C2×#4111yescharacteristic
396SD16—48no—
397D8—48no—
398M16—48no—
399M16—48no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—24no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—44no—
420C2×C8—44no—
421C2×C8—48no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—84no—
432C2×C8AGL(1,ℤ/8)12yes—
433C2×C8AGL(1,ℤ/8)12yes—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4C2×C22⋊C412yes—
445C4⋊C4C2×C22⋊C412yes—
446C22⋊C4—24no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—44no—
455C42—22no—
456C42—22no—
457C42C2×C22⋊C412yes—
458C42C4×D412yes—
459C42#12012yes—
460C42#12112yes—
461C42#12112yes—
462C42#12312yes—
463C42C22×D411yescharacteristic
464C42C22×D411yescharacteristic
465C42C22×D411yescharacteristic
466C42C2×#4111yescharacteristic
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—24no—
472C23—22no—
473C23—48no—
474C23—48no—
475C23—48no—
476C23—41no—
477C23—42no—
478C23—41no—
479C23—42no—
480C23—42no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—84no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23#38612yes—
493C23#44312yes—
494C23#44412yes—
495C23C2×C4×D411yescharacteristic
496Q8—24no—
497Q8—42no—
498D4—24no—
499D4—24no—
500D4—24no—
501D4—48no—
502D4—48no—
503D4—42no—
504D4—42no—
505D4—42no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—88no—
539C2×C4—84no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4#38612yes—
545C2×C4#38612yes—
546C2×C4#38612yes—
547C2×C4C22×C22⋊C411yescharacteristic
548C2×C4C2×C4×D411yescharacteristic
549C2×C4C2×#12011yescharacteristic
550C2×C4C2×#12111yescharacteristic
551C2×C4C2×#12111yescharacteristic
552C2×C4C2×#12311yescharacteristic
553C2×C4C2×#13112yes—
554C2×C4#56812yes—
555C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
556C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
557C2×C4#57511yescharacteristic
558C2×C4#57511yescharacteristic
559C8—22no—
560C8—22no—
561C8—44no—
562C8—42no—
563C8—42no—
564C8—41no—
565C8—41no—
566C8—42no—
567C8—42no—
568C8—82no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—44no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—44no—
580C22—44no—
581C22—816no—
582C22—81no—
583C22—82no—
584C22—81no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—81no—
589C22—81no—
590C22#187614yes—
591C22C2×#38611yescharacteristic
592C22#292911yescharacteristic
593C22#293011yescharacteristic
594C22C4×AGL(1,ℤ/8)11yescharacteristic
595C22C4×AGL(1,ℤ/8)11yescharacteristic
596C22C2×#44311yescharacteristic
597C22C2×#44411yescharacteristic
598C4—22no—
599C4—21no—
600C4—21no—
601C4—22no—
602C4—22no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—82no—
607C4#288012yes—
608C4#293312yes—
609C4#300611yescharacteristic
610C4#300611yescharacteristic
611C4#303811yescharacteristic
612C4#304011yescharacteristic
613C2—24no—
614C2—82no—
615C2—84no—
616C2#1791011yescharacteristic
617C2#1791411yescharacteristic
618C2#1793811yescharacteristic
619C2#1794211yescharacteristic
620C2#2089911yescharacteristic
621C2C2×#187611yescharacteristic
622C2#2092811yescharacteristic
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 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2966C2semidirect
3. C2×#1954C2semidirect
4. C2×#1483C2semidirect
5. C2×#1490C2semidirect
6. #8952C2semidirect
7. #8951C2semidirect
8. #8948C2semidirect
9. #8947C2semidirect
10. #7879C2semidirect
11. #7848C2semidirect
24. C2×#446C22semidirect
25. C2×C8×Q8C22semidirect
28. C2×#436C22semidirect
31. C2×C8×D4C22semidirect
47. 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×#29662
C2×#19542
C2×#14832
C2×#14902
#89522
#89512
#89482
#89472
#78792
#78482