Finite Group Databasebeta

← #392920

Subgroups — #392920

Order 512.

1786 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×M16C211yesmaximal, characteristic
2C2×#1943C211yesmaximal, characteristic, complemented
3C2×#1941C211yesmaximal, characteristic, complemented
4C2×#1809C211yesmaximal, characteristic
5C2×#1809C211yesmaximal, characteristic
6#14697C212yesmaximal
7#10881C212yesmaximal, complemented
8#10880C212yesmaximal, complemented
9#10863C211yesmaximal, characteristic, complemented
10#10862C211yesmaximal, characteristic, complemented
11#10722C212yesmaximal
12C23×M16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C2×C4×M16—24no—
20C2×C4×M16—24no—
21C2×C4×M16—24no—
22C22×#97C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#385—22no—
29C2×#380C2212yes—
30C2×#380C2212yes—
31#1943—24no—
32#1941—24no—
33#1809—24no—
34#1809C418yescomplemented
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41#1436—22no—
42#1436—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C22⋊#95—22no—
46C24×C4D412yes—
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—22no—
57C22×M16—22no—
58C22×M16C2×C414yes—
59C22×M16D414yes—
60C22×M16Q814yes—
61C22×M16Q814yes—
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C42—22no—
75C22×C42—22no—
76C22×C42—22no—
77C22×C42—22no—
78C22×C42—22no—
79C22×C42—22no—
80C22×C42D412yes—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#107—22no—
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#107C2×C414yescomplemented
89C2×#106—22no—
90C2×#106—22no—
91C2×#106—22no—
92C2×#106—22no—
93C2×#106—22no—
94C2×#106C2×C414yescomplemented
95C4×M16—232no—
96C2×#97—22no—
97C2×#97—22no—
98C2×#97—22no—
99C2×#97—22no—
100C2×#97—22no—
101C2×#97D414yes—
102C2×C4×C8—22no—
103C2×C4×C8—22no—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8D414yes—
108#389—24no—
109#389—24no—
110#389—24no—
111#389—24no—
112#385—48no—
113#380—24no—
114#380—24no—
115#380—24no—
116#380C2×C418yes—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340C2×C418yescomplemented
131C25C2×D411yescharacteristic
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4C22×C412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×Q812yes—
157C23×C4C2×Q812yes—
158C23×C4#4112yes—
159C23×C4#4112yes—
160C23×C4C2411yescharacteristic
161C2×M16—28no—
162C2×M16—28no—
163C2×M16—28no—
164C2×M16—28no—
165C2×M16—28no—
166C2×M16—28no—
167C2×M16—28no—
168C2×M16—28no—
169C2×M16—28no—
170C2×M16—28no—
171C2×M16—28no—
172C2×M16—28no—
173C2×M16—28no—
174C2×M16C4⋊C4116yescomplemented
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8C22×C412yes—
188C22×C8C22×C412yes—
189C22×C8C2×D412yes—
190C22×C8C2×D412yes—
191C22×C8C2×D412yes—
192C22×C8C2×D412yes—
193C22×C8C2×D412yes—
194C22×C8C2×D412yes—
195C22×C8C2×Q812yes—
196C22×C8C2×Q812yes—
197C22×C8C2×Q812yes—
198C22×C8C2×Q812yes—
199C22×C8#4112yes—
200C22×C8#4112yes—
201C22×C8#4112yes—
202C22×C8#4112yes—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4C22×C414yes—
232C2×C4⋊C4C22×C414yes—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C42—28no—
240C2×C42—28no—
241C2×C42—28no—
242C2×C42—28no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42#4112yes—
257C2×C42#4112yes—
258#107—28no—
259#107—44no—
260#106—28no—
261#106—44no—
262#97—28no—
263#97—28no—
264C4×C8—28no—
265C4×C8—28no—
266#95—28no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24C4×D412yes—
283C24#12112yes—
284C24C4⋊Q812yes—
285C24C22×D411yescharacteristic
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—216no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—48no—
328C22×C4—48no—
329C22×C4—48no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4—48no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4C2×C4⋊C414yes—
360C22×C4C4×D412yes—
361C22×C4C4×Q814yes—
362C22×C4#12114yes—
363C22×C4#12112yes—
364C22×C4#12114yes—
365C22×C4C22⋊Q814yes—
366C22×C4#12514yes—
367C22×C4C4⋊Q812yes—
368C22×C4AGL(1,ℤ/8)14yes—
369C22×C4#13714yes—
370C22×C4C23×C411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×Q811yescharacteristic
375C22×C4C22×Q811yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384M16—232no—
385M16—232no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8C2×C4⋊C414yes—
405C2×C8C2×C4⋊C414yes—
406C2×C8C4×D414yes—
407C2×C8C4×D414yes—
408C2×C8#12014yes—
409C2×C8C22⋊Q814yes—
410C2×C8C22⋊Q814yes—
411C2×C8C22⋊Q814yes—
412C2×C8C22⋊Q814yes—
413C2×C8#12314yes—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—88no—
431C4⋊C4—88no—
432C22⋊C4—416no—
433C22⋊C4—88no—
434C42—216no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23—42no—
456C23—42no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23#38914yes—
466C23C2×C4×D411yescharacteristic
467C23C2×#12111yescharacteristic
468C23C2×#12111yescharacteristic
469C23C2×#12111yescharacteristic
470C23C2×C4⋊Q811yescharacteristic
471C23#53211yescharacteristic
472C23#53211yescharacteristic
473C23C2×AGL(1,ℤ/8)12yes—
474C23C2×#13712yes—
475C23#57512yes—
476C23#57512yes—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—432no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4#38914yes—
534C2×C4#46818yes—
535C2×C4C22×C4⋊C411yescharacteristic
536C2×C4C2×C4×D411yescharacteristic
537C2×C4C2×C4×D411yescharacteristic
538C2×C4C2×C4×Q811yescharacteristic
539C2×C4C4×#4111yescharacteristic
540C2×C4C4×#4111yescharacteristic
541C2×C4C2×#12011yescharacteristic
542C2×C4C2×C22⋊Q811yescharacteristic
543C2×C4C2×C22⋊Q811yescharacteristic
544C2×C4C2×C22⋊Q811yescharacteristic
545C2×C4C2×C22⋊Q811yescharacteristic
546C2×C4C2×C22⋊Q811yescharacteristic
547C2×C4C2×#12311yescharacteristic
548C2×C4#52511yescharacteristic
549C2×C4#52511yescharacteristic
550C2×C4#52511yescharacteristic
551C2×C4#52511yescharacteristic
552C2×C4#52511yescharacteristic
553C2×C4#52511yescharacteristic
554C2×C4C2×#12511yescharacteristic
555C2×C4#52911yescharacteristic
556C2×C4#52911yescharacteristic
557C2×C4#52911yescharacteristic
558C2×C4#52911yescharacteristic
559C2×C4#53311yescharacteristic
560C2×C4#53311yescharacteristic
561C2×C4#53311yescharacteristic
562C2×C4#53311yescharacteristic
563C8—24no—
564C8—24no—
565C8—24no—
566C8#38518yescomplemented
567C22—24no—
568C22—28no—
569C22—44no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—44no—
575C22—42no—
576C22—42no—
577C22—44no—
578C22C2×#38911yescharacteristic
579C22C4×#12111yescharacteristic
580C22C4×#12111yescharacteristic
581C22#284711yescharacteristic
582C22#285411yescharacteristic
583C22#285711yescharacteristic
584C22#285711yescharacteristic
585C22C4×AGL(1,ℤ/8)12yes—
586C22C4×#13712yes—
587C22C2×#46812yes—
588C22#305412yes—
589C22#305512yes—
590C22#305512yes—
591C22#305612yes—
592C22#305612yes—
593C22#316512yes—
594C22#316612yes—
595C22#317112yes—
596C22#317212yes—
597C22#318311yescharacteristic
598C22#318311yescharacteristic
599C22#318411yescharacteristic
600C22#318511yescharacteristic
601C4—24no—
602C4—28no—
603C4—44no—
604C4—84no—
605C4—84no—
606C4—162no—
607C4—162no—
608C4C2×#38511yescharacteristic
609C4#229611yescharacteristic
610C4C4×C22⋊Q811yescharacteristic
611C4C4×C22⋊Q811yescharacteristic
612C4#284811yescharacteristic
613C4#285111yescharacteristic
614C4#285111yescharacteristic
615C4#285511yescharacteristic
616C2—44no—
617C2#1469711yescharacteristic
618C2#2115912yes—
619C2C4×#46812yes—
620C2#2153811yescharacteristic
621C2#2153911yescharacteristic
622C2#3293512yes—
623C2#3314311yescharacteristic
624C2#3314411yescharacteristic
625C2#3321011yescharacteristic
626C2#3321111yescharacteristic
627C2#3321812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1943C2semidirect
3. C2×#1941C2semidirect
7. #10881C2semidirect
8. #10880C2semidirect
9. #10863C2semidirect
10. #10862C2semidirect
34. #1809C4semidirect
88. C2×#107C2×C4semidirect
94. C2×#106C2×C4semidirect
130. #340C2×C4semidirect
174. C2×M16C4⋊C4semidirect
566. C8#385semidirect

Maximal subgroups

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

subgroupindex
C22×C4×M162
C2×#19432
C2×#19412
C2×#18092
#146972
#108812
#108802
#108632
#108622
#107222