Finite Group Databasebeta

← #188842

Subgroups — #188842

Order 512.

1227 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×M16C211yesmaximal, characteristic
2#20667C212yesmaximal, complemented
3#17362C211yesmaximal, characteristic
4C2×#1465C211yesmaximal, characteristic
5#8613C214yesmaximal, complemented
6#8612C212yesmaximal, complemented
7#8611C212yesmaximal, complemented
8#8009C212yesmaximal, complemented
9C23×M16C411yescharacteristic
10C23×M16C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12D4×M16—22no—
13D4×M16—24no—
14D4×M16—22no—
15C2×#436C412yes—
16C2×#435C412yes—
17C2×#435C2212yes—
18C2×C8×D4C2212yes—
19C2×#430—22no—
20C2×#423C411yescharacteristic
21C2×#407C411yescharacteristic
22C2×#407C2212yes—
23C2×#407C2212yes—
24C2×#407C2211yescharacteristic
25C2×C4×M16C411yescharacteristic
26#1811—24no—
27#1810C2214yescomplemented
28#1810C2214yescomplemented
29#1738—22no—
30#1465—21no—
31#1465—21no—
32#1465C2211yescharacteristic
33#1465C2212yes—
34#1465C2211yescharacteristic
35#1465C2212yes—
36#1465C2212yes—
37#1463—21no—
38#1462—21no—
39#1459—21no—
40#1459—21no—
41C2×#324—21no—
42C2×#323—21no—
43#1293C2214yescomplemented
44#1293C2214yescomplemented
45C23×D4Q811yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16—21no—
49C22×M16—22no—
50C22×M16—21no—
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—44no—
54C22×M16—44no—
55C22×M16C2×C411yescharacteristic
56C22×M16C2×C411yescharacteristic
57C22×M16C2×C411yescharacteristic
58C22×M16C2×C412yes—
59C22×M16C2×C411yescharacteristic
60C22×M16C2×C412yes—
61C22×M16D412yes—
62C22×M16D414yes—
63C22×M16D412yes—
64C23×C8C2×C412yes—
65C23×C8C2312yes—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4Q811yescharacteristic
75C2×C4×D4Q811yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C22⋊C4D411yescharacteristic
80C22×C22⋊C4Q811yescharacteristic
81C22×C42C2×C411yescharacteristic
82#436—24no—
83#436—24no—
84#435—24no—
85#435—24no—
86#435—24no—
87#435—24no—
88C8×D4—24no—
89C8×D4—24no—
90#430—48no—
91C2×#108—24no—
92C2×#108—24no—
93#423—22no—
94#423—22no—
95C2×C4⋊C8C2×C412yes—
96C2×C4⋊C8C2×C412yes—
97#407—22no—
98#407—22no—
99#407—22no—
100#407—22no—
101#407—22no—
102#407—22no—
103#407—22no—
104#407—22no—
105#407—22no—
106#407—22no—
107#407C2312yes—
108#407C2314yes—
109#407C2312yes—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8C2×C412yes—
115C2×C22⋊C8C2×C412yes—
116C2×C22⋊C8C2×C412yes—
117C2×C22⋊C8C2312yes—
118C4×M16—21no—
119C4×M16—22no—
120C4×M16—21no—
121C2×#97C2×C412yes—
122C2×C4×C8C2×C412yes—
123#341—24no—
124#341—24no—
125#324—22no—
126#324—22no—
127#324—21no—
128#324—21no—
129#324—22no—
130#323—22no—
131#323—22no—
132#323—21no—
133#323—21no—
134#323—22no—
135C25C4⋊C411yescharacteristic
136C25C2×Q811yescharacteristic
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4C4⋊C411yescharacteristic
142C22×D4C4⋊C411yescharacteristic
143C22×D4C2×Q811yescharacteristic
144C22×D4C2×Q811yescharacteristic
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4C4⋊C411yescharacteristic
154C23×C4C4⋊C411yescharacteristic
155C23×C4C4⋊C411yescharacteristic
156C23×C4C4⋊C411yescharacteristic
157C23×C4C4⋊C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×Q811yescharacteristic
167C23×C4#4112yes—
168C23×C4#4112yes—
169C23×C4C2411yescharacteristic
170C2×M16—24no—
171C2×M16—28no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—24no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—44no—
182C2×M16—48no—
183C2×M16—44no—
184C2×M16—42no—
185C2×M16—44no—
186C2×M16—44no—
187C2×M16—48no—
188C2×M16—44no—
189C2×M16—42no—
190C2×M16—42no—
191C2×M16—42no—
192C2×M16—42no—
193C2×M16—44no—
194C2×M16—44no—
195C2×M16—44no—
196C2×M16—44no—
197C2×M16C22⋊C411yescharacteristic
198C2×M16C22⋊C412yes—
199C2×M16C22⋊C412yes—
200C2×M16C22⋊C414yes—
201C2×M16C22⋊C412yes—
202C2×M16C22⋊C414yes—
203C2×M16C22⋊C411yescharacteristic
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—44no—
207C22×C8—44no—
208C22×C8—42no—
209C22×C8—42no—
210C22×C8C22×C412yes—
211C22×C8C22×C412yes—
212C22×C8C22×C412yes—
213C22×C8C22×C412yes—
214C22×C8C2×D414yes—
215C22×C8C2×D414yes—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4C4⋊C411yescharacteristic
235C2×C4⋊C4C4⋊C411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4C4⋊C411yescharacteristic
243C2×C22⋊C4C4⋊C411yescharacteristic
244C2×C22⋊C4C4⋊C411yescharacteristic
245C2×C22⋊C4C4⋊C411yescharacteristic
246C2×C22⋊C4C2×D411yescharacteristic
247C2×C22⋊C4C2×D411yescharacteristic
248C2×C22⋊C4C2×Q811yescharacteristic
249C2×C22⋊C4C2×Q811yescharacteristic
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42C4⋊C411yescharacteristic
255C2×C42C4⋊C411yescharacteristic
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258#108—48no—
259#108—48no—
260C4⋊C8—24no—
261C4⋊C8—24no—
262C22⋊C8—24no—
263C22⋊C8—24no—
264C22⋊C8—24no—
265C22⋊C8—24no—
266C22⋊C8—24no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—24no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C22⋊C8—42no—
273C22⋊C8—42no—
274C22⋊C8—42no—
275C22⋊C8—42no—
276#97—22no—
277#97—22no—
278C4×C8—22no—
279C4×C8—22no—
280C24—21no—
281C24—21no—
282C24—22no—
283C24—42no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—42no—
288C24—81no—
289C24—81no—
290C24C2×C4⋊C411yescharacteristic
291C24C2×C4⋊C411yescharacteristic
292C24C2×C4⋊C411yescharacteristic
293C24C2×C4⋊C411yescharacteristic
294C24C22⋊Q812yes—
295C24C22⋊Q812yes—
296C24C22×Q811yescharacteristic
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—48no—
310C2×D4—48no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—81no—
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C4⋊C411yescharacteristic
360C22×C4C2×C4⋊C411yescharacteristic
361C22×C4C2×C4⋊C411yescharacteristic
362C22×C4C2×C4⋊C411yescharacteristic
363C22×C4C2×C4⋊C411yescharacteristic
364C22×C4C2×C4⋊C411yescharacteristic
365C22×C4C2×C4⋊C411yescharacteristic
366C22×C4C2×C4⋊C411yescharacteristic
367C22×C4#11711yescharacteristic
368C22×C4#11711yescharacteristic
369C22×C4#11712yes—
370C22×C4#12112yes—
371C22×C4#12112yes—
372C22×C4#12112yes—
373C22×C4#12112yes—
374C22×C4C22⋊Q812yes—
375C22×C4C22⋊Q812yes—
376C22×C4C23×C411yescharacteristic
377C22×C4C22×D412yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C2×#4112yes—
380M16—22no—
381M16—24no—
382M16—24no—
383M16—28no—
384M16—44no—
385M16—42no—
386M16—44no—
387M16—42no—
388M16—42no—
389M16—42no—
390M16—42no—
391M16—44no—
392M16—44no—
393M16—42no—
394M16—42no—
395M16—44no—
396M16—42no—
397M16—42no—
398M16—84no—
399M16—84no—
400M16#9914yes—
401C2×C8—28no—
402C2×C8—24no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—42no—
406C2×C8—44no—
407C2×C8—44no—
408C2×C8—44no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—44no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—84no—
415C2×C8—84no—
416C2×C8C2×C22⋊C412yes—
417C2×C8C2×C22⋊C414yes—
418C2×C8C2×C22⋊C412yes—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C23—22no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—24no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—41no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—81no—
471C23—81no—
472C23—84no—
473C23—84no—
474C23—84no—
475C23—84no—
476C23#38012yes—
477C23#38011yescharacteristic
478C23#38011yescharacteristic
479C23#43012yescomplemented
480C23C22×C4⋊C411yescharacteristic
481C23C2×C22⋊Q812yes—
482D4—28no—
483D4—44no—
484D4—44no—
485D4—44no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
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—81no—
535C2×C4—81no—
536C2×C4#38011yescharacteristic
537C2×C4#38012yes—
538C2×C4#38011yescharacteristic
539C2×C4C2×#9912yes—
540C2×C4#41012yes—
541C2×C4#43012yescomplemented
542C2×C4C22×C22⋊C411yescharacteristic
543C2×C4C2×#11711yescharacteristic
544C2×C4C2×#12112yes—
545C8—22no—
546C8—24no—
547C8—42no—
548C8—42no—
549C8—84no—
550C8—82no—
551C8—82no—
552C8C2×#9914yes—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—22no—
559C22—22no—
560C22—21no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—21no—
565C22—22no—
566C22—22no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—81no—
579C22—81no—
580C22—84no—
581C22—84no—
582C22—81no—
583C22—81no—
584C22—84no—
585C22—84no—
586C22#179912yes—
587C22C2×#38011yescharacteristic
588C22C2×#43011yescharacteristic
589C22C2×#43011yescharacteristic
590C4—22no—
591C4—21no—
592C4—21no—
593C4—22no—
594C4—22no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4#179912yes—
601C4C22×#9911yescharacteristic
602C4C2×#41011yescharacteristic
603C2—21no—
604C2—21no—
605C2—22no—
606C2—42no—
607C2—41no—
608C2—42no—
609C2—42no—
610C2—81no—
611C2C2×#179911yescharacteristic
612C2#2066711yescharacteristic
613C2#2067311yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4Q8
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
2. #20667C2semidirect
5. #8613C2semidirect
6. #8612C2semidirect
7. #8611C2semidirect
8. #8009C2semidirect
27. #1810C22semidirect
28. #1810C22semidirect
43. #1293C22semidirect
44. #1293C22semidirect
479. C23#430semidirect
541. C2×C4#430semidirect

Maximal subgroups

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

subgroupindex
C2×D4×M162
#206672
#173622
C2×#14652
#86132
#86122
#86112
#80092