Finite Group Databasebeta

← #204342

Subgroups — #204342

Order 512.

973 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3172C211yesmaximal, characteristic
2#21848C211yesmaximal, characteristic
3#21819C211yesmaximal, characteristic, complemented
4C2×#1565C211yesmaximal, characteristic, complemented
5C2×#1482C211yesmaximal, characteristic
6C2×#1481C211yesmaximal, characteristic, complemented
7C2×#1453C211yesmaximal, characteristic, complemented
8#9666C211yesmaximal, characteristic, complemented
9#9664C211yesmaximal, characteristic, complemented
10#9662C211yesmaximal, characteristic, complemented
11#9661C211yesmaximal, characteristic, complemented
12#9660C211yesmaximal, characteristic, complemented
13#9658C211yesmaximal, characteristic, complemented
14#9656C211yesmaximal, characteristic, complemented
15#9655C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#529C2211yescharacteristic
18#3172—24no—
19#3172—24no—
20C2×#501C411yescharacteristic
21C2×#497C411yescharacteristic
22C2×#491C411yescharacteristic
23C2×#489C411yescharacteristic
24C2×#470C411yescharacteristic
25C2×#470C411yescharacteristic
26C2×#468C411yescharacteristic
27C2×#468C411yescharacteristic
28C2×C4×Q16C2211yescharacteristic
29C2×C4×Q16C2211yescharacteristic
30C2×C4×SD16C2211yescharacteristic
31C2×C4×SD16C2211yescharacteristic
32C2×#431C2211yescharacteristic
33C2×#431C2211yescharacteristic
34C2×#431C2211yescharacteristic
35C22×C4⋊C8C2211yescharacteristic
36C2×C4×M16C2211yescharacteristic
37#1984—22no—
38#1984C2211yescharacteristic
39#1984C2211yescharacteristic
40#1984C2211yescharacteristic
41#1984C2211yescharacteristic
42#1984C2211yescharacteristic
43#1984C2211yescharacteristic
44#1954C2211yescharacteristic
45#1954C2211yescharacteristic
46#1869—22no—
47#1818—22no—
48#1798—22no—
49#1565—24no—
50C2×#335C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×#334C2211yescharacteristic
54#1482—24no—
55#1481—24no—
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×Q8⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64#1453—24no—
65C2×#322C2211yescharacteristic
66C2×#322C2211yescharacteristic
67C2×#322C2211yescharacteristic
68C2×C82C2211yescharacteristic
69C22×Q16C2×C411yescharacteristic
70C22×Q16C2×C411yescharacteristic
71C22×SD16C2×C411yescharacteristic
72C22×SD16C2×C411yescharacteristic
73C22×M16C2×C411yescharacteristic
74C22×M16C2×C411yescharacteristic
75C23×C8—21no—
76#533—22no—
77#533—22no—
78#533—22no—
79#533—22no—
80C2×C4⋊Q8C2×C411yescharacteristic
81C2×C4⋊Q8C2×C411yescharacteristic
82#529—22no—
83#529—22no—
84#529—22no—
85#529—22no—
86C2×#126—21no—
87C2×#125—21no—
88C2×#125C2×C411yescharacteristic
89C2×#124C2×C411yescharacteristic
90C2×#123—21no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8C2×C411yescharacteristic
93C2×C22⋊Q8C2×C411yescharacteristic
94C2×C22⋊Q8C2×C411yescharacteristic
95C2×#121C2×C411yescharacteristic
96C2×C4×Q8—21no—
97C2×C4×Q8C2311yescharacteristic
98C2×C4×Q8C2311yescharacteristic
99C2×C4×Q8C2311yescharacteristic
100C2×C4×D4—21no—
101C2×C4×D4C2311yescharacteristic
102C2×#117—21no—
103C2×#117—21no—
104C22×C42C2311yescharacteristic
105#501—24no—
106#497—24no—
107#491—24no—
108#489—24no—
109#470—24no—
110#470—24no—
111#468—24no—
112#468—24no—
113C4×Q16—22no—
114C4×Q16—22no—
115C4×Q16—22no—
116C4×Q16—22no—
117C4×SD16—22no—
118C4×SD16—22no—
119C4×SD16—22no—
120C4×SD16—22no—
121#431—22no—
122#431—22no—
123#431—22no—
124#431—22no—
125#431—22no—
126#431D414yes—
127C2×#107C2×C411yescharacteristic
128C2×#107C2×C411yescharacteristic
129C2×#106C2×C411yescharacteristic
130C2×#106C2×C411yescharacteristic
131C2×C4⋊C8—21no—
132C2×C4⋊C8—22no—
133C2×C4⋊C8—22no—
134C2×C4⋊C8—21no—
135C2×C4⋊C8D412yes—
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2311yescharacteristic
140C2×#103C2×C411yescharacteristic
141C2×#103C2×C411yescharacteristic
142C2×#103C2×C411yescharacteristic
143C2×#103C2×C411yescharacteristic
144C2×#103C2×C411yescharacteristic
145C2×#103C2×C411yescharacteristic
146C2×#102C2×C411yescharacteristic
147C2×#102C2×C411yescharacteristic
148C2×C22⋊C8—21no—
149C2×C22⋊C8—21no—
150C2×C22⋊C8—22no—
151C2×C22⋊C8—22no—
152C2×C22⋊C8D412yes—
153C4×M16D414yes—
154C4×M16D414yes—
155C2×#97C2×C411yescharacteristic
156C2×#97C2×C411yescharacteristic
157C2×C4×C8—21no—
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164#336—22no—
165#336—22no—
166#336—22no—
167#336—22no—
168#336—22no—
169#336—22no—
170#335—22no—
171#335—22no—
172#334—22no—
173#334—22no—
174Q8⋊C8—22no—
175Q8⋊C8—22no—
176Q8⋊C8—22no—
177Q8⋊C8—22no—
178Q8⋊C8—22no—
179Q8⋊C8—22no—
180D4⋊C8—22no—
181D4⋊C8—22no—
182#322—22no—
183#322—22no—
184#322—22no—
185C82—22no—
186C22×Q8—21no—
187C22×Q8C22×C411yescharacteristic
188C22×Q8C22×C411yescharacteristic
189C22×Q8C22×C411yescharacteristic
190C22×D4—21no—
191C22×D4C22×C411yescharacteristic
192C23×C4—21no—
193C23×C4C22×C411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4#4111yescharacteristic
196C2×Q16—22no—
197C2×Q16—22no—
198C2×Q16—22no—
199C2×Q16—22no—
200C2×Q16—41no—
201C2×Q16—41no—
202C2×SD16—22no—
203C2×SD16—22no—
204C2×SD16—22no—
205C2×SD16—22no—
206C2×SD16—41no—
207C2×SD16—41no—
208C2×M16—22no—
209C2×M16—22no—
210C2×M16—41no—
211C2×M16—41no—
212C2×M16C22⋊C414yes—
213C2×M16C22⋊C414yes—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—42no—
221C22×C8—82no—
222C22×C8C22×C411yescharacteristic
223C22×C8C22×C411yescharacteristic
224C22×C8C22×C411yescharacteristic
225C22×C8C22×C411yescharacteristic
226C22×C8C2×D412yes—
227C22×C8C2×D412yes—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232#126—42no—
233#126—42no—
234#125—22no—
235#125—22no—
236#125—42no—
237#125—42no—
238#124—22no—
239#124—22no—
240#123—42no—
241#123—42no—
242C22⋊Q8—22no—
243C22⋊Q8—22no—
244C22⋊Q8—22no—
245C22⋊Q8—22no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8—42no—
249C22⋊Q8—42no—
250#121—22no—
251#121—22no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×Q8—42no—
265C4×Q8—42no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—42no—
271C4×D4—42no—
272#117—42no—
273#117—42no—
274#117—42no—
275#117—42no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C4⋊C4C22×C411yescharacteristic
289C2×C4⋊C4C22×C411yescharacteristic
290C2×C4⋊C4C22×C411yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42C22×C411yescharacteristic
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D411yescharacteristic
306C2×C42#4112yes—
307C2×C42#4111yescharacteristic
308C2×C42C2411yescharacteristic
309#107—22no—
310#107—22no—
311#106—22no—
312#106—22no—
313C4⋊C8—24no—
314C4⋊C8—24no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—42no—
324#103—22no—
325#103—22no—
326#103—22no—
327#103—22no—
328#103—22no—
329#103—22no—
330#102—22no—
331#102—22no—
332C22⋊C8—24no—
333C22⋊C8—24no—
334C22⋊C8—42no—
335C22⋊C8—82no—
336C22⋊C8—82no—
337#97C22⋊C414yes—
338#97C22⋊C414yes—
339C4×C8—21no—
340C4×C8—21no—
341C4×C8—21no—
342C4×C8—21no—
343C4×C8—21no—
344C4×C8—21no—
345C4×C8—42no—
346C4×C8C2×D412yes—
347C4×C8C2×D412yes—
348C4×C8C2×D412yes—
349C4×C8C2×D412yes—
350C4×C8C2×D412yes—
351C4×C8C2×D412yes—
352C24—21no—
353C24C4×D411yescharacteristic
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—44no—
373C2×Q8—42no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—44no—
381C2×D4—42no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4C2×C22⋊C412yes—
401C22×C4C4×D412yes—
402C22×C4C4×D411yescharacteristic
403C22×C4C4×D411yescharacteristic
404C22×C4C4×D411yescharacteristic
405C22×C4#12112yes—
406C22×C4#12112yes—
407C22×C4#12412yes—
408C22×C4#12712yes—
409C22×C4C23×C411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C2×#4111yescharacteristic
412Q16—44no—
413Q16—44no—
414SD16—44no—
415SD16—44no—
416M16—28no—
417M16—44no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—84no—
443C2×C8C2×C22⋊C412yes—
444C2×C8C2×C22⋊C412yes—
445C2×C8C2×C22⋊C412yes—
446C2×C8C2×C22⋊C412yes—
447C2×C8#13112yes—
448C2×C8#13112yes—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C42—41no—
480C42—41no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42C2×C22⋊C412yes—
486C42C4×D412yes—
487C42#12112yes—
488C42#12112yes—
489C42#12412yes—
490C42#12712yes—
491C42C22×D411yescharacteristic
492C42C22×D411yescharacteristic
493C42C22×D411yescharacteristic
494C42C2×#4111yescharacteristic
495C23—21no—
496C23—21no—
497C23—22no—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—42no—
503C23—41no—
504C23#39012yes—
505C23C2×C4×D411yescharacteristic
506Q8—41no—
507Q8—41no—
508Q8—41no—
509Q8—41no—
510Q8—41no—
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—41no—
515Q8—41no—
516Q8—41no—
517Q8—41no—
518Q8—84no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—41no—
523D4—84no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—82no—
558C2×C4#39012yes—
559C2×C4#39012yes—
560C2×C4#39012yes—
561C2×C4C22×C22⋊C411yescharacteristic
562C2×C4C2×C4×D411yescharacteristic
563C2×C4C2×#12111yescharacteristic
564C2×C4C2×#12111yescharacteristic
565C2×C4C2×#12411yescharacteristic
566C2×C4C2×#12711yescharacteristic
567C2×C4C2×#13111yescharacteristic
568C2×C4C2×#13111yescharacteristic
569C2×C4#56811yescharacteristic
570C2×C4#56811yescharacteristic
571C2×C4#57712yes—
572C2×C4#57812yes—
573C8—22no—
574C8—22no—
575C8—22no—
576C8—42no—
577C8—42no—
578C8—42no—
579C8—42no—
580C8—81no—
581C8—81no—
582C8—81no—
583C8—81no—
584C8#40814yes—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—42no—
589C22—42no—
590C22—82no—
591C22—82no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C2×#39011yescharacteristic
596C22#293511yescharacteristic
597C22#293511yescharacteristic
598C22#294712yes—
599C22#295512yes—
600C4—22no—
601C4—22no—
602C4—21no—
603C4—21no—
604C4—22no—
605C4—81no—
606C4—81no—
607C4—81no—
608C4—81no—
609C4—81no—
610C4—81no—
611C4—81no—
612C4C2×#40811yescharacteristic
613C4#287911yescharacteristic
614C4#293511yescharacteristic
615C4#293511yescharacteristic
616C4#306312yes—
617C4#315411yescharacteristic
618C4#315611yescharacteristic
619C2—42no—
620C2—161no—
621C2#1808112yes—
622C2#1808312yes—
623C2#2118911yescharacteristic
624C2#2123411yescharacteristic
625C2#2124211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
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
3. #21819C2semidirect
4. C2×#1565C2semidirect
6. C2×#1481C2semidirect
7. C2×#1453C2semidirect
8. #9666C2semidirect
9. #9664C2semidirect
10. #9662C2semidirect
11. #9661C2semidirect
12. #9660C2semidirect
13. #9658C2semidirect
14. #9656C2semidirect
15. #9655C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31722
#218482
#218192
C2×#15652
C2×#14822
C2×#14812
C2×#14532
#96662
#96642
#96622
#96612
#96602
#96582
#96562
#96552