Finite Group Databasebeta

← #392814

Subgroups — #392814

Order 512.

1688 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#405C211yesmaximal, characteristic
2C2×#1933C212yesmaximal, complemented
3C2×#1894C211yesmaximal, characteristic
4C2×#1892C211yesmaximal, characteristic
5#14264C211yesmaximal, characteristic
6#14258C211yesmaximal, characteristic
7#10928C212yesmaximal, complemented
8#10881C212yesmaximal, complemented
9#10820C212yesmaximal, complemented
10#10722C211yesmaximal, characteristic
11#10721C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C2×#405—28no—
20C2×#405—24no—
21C22×#97C2212yes—
22C22×C4×C8C2212yes—
23C2×#391C2212yes—
24C2×#389C2212yes—
25C2×#380C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28#1933—28no—
29#1894C418yescomplemented
30#1892—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38#1444—21no—
39#1442—21no—
40#1441—21no—
41#1441—21no—
42#1440—22no—
43#1439—22no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C23×Q8C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—24no—
51C22×M16—24no—
52C22×M16C2×C414yes—
53C23×C8—22no—
54C23×C8—22no—
55C23×C8—24no—
56C23×C8—24no—
57C23×C8D414yes—
58C23×C8C2312yes—
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8C2×C414yes—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—24no—
70C2×#117—24no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4D412yes—
85C22×C42C2312yes—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103C2×C418yes—
93#405—232no—
94C2×#97—24no—
95C2×#97—24no—
96C2×#97—24no—
97C2×C4×C8—24no—
98C2×C4×C8—24no—
99C2×C4×C8D418yes—
100#391—24no—
101#391—24no—
102#389—24no—
103#389—24no—
104#380—22no—
105#380—22no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380C2×C414yes—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120#340—22no—
121#340—22no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#340C2×C418yes—
127C25C2×D411yescharacteristic
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8C22×C412yes—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4C22×C412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4#4112yes—
154C23×C4#4112yes—
155C23×C4#4112yes—
156C23×C4#4112yes—
157C23×C4C2411yescharacteristic
158C2×M16—216no—
159C2×M16—28no—
160C2×M16—28no—
161C2×M16—28no—
162C2×M16—216no—
163C22×C8—216no—
164C22×C8—216no—
165C22×C8—28no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—28no—
177C22×C8—28no—
178C22×C8C22×C414yes—
179C22×C8C2×D414yes—
180C22×C8C2×D414yes—
181C22×C8C2×D412yes—
182C22×C8C2×D412yes—
183C22×C8#4114yes—
184C22×C8#4114yes—
185C22⋊Q8—28no—
186C22⋊Q8—28no—
187C22⋊Q8—44no—
188C22⋊Q8—44no—
189#117—216no—
190#117—28no—
191#117—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C414yes—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—81no—
238C2×C22⋊C4—81no—
239C2×C22⋊C4—81no—
240C2×C22⋊C4—81no—
241C2×C22⋊C4—82no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42C2×D414yes—
247C2×C42#4114yes—
248#103—216no—
249#103—44no—
250#103—44no—
251#97—216no—
252C4×C8—216no—
253#95—24no—
254#95—42no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—42no—
259#95—82no—
260#95—82no—
261#95—82no—
262#95—82no—
263C24—22no—
264C24—22no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—22no—
269C24—22no—
270C24—21no—
271C24C4×D412yes—
272C24#12112yes—
273C24#12412yes—
274C24C22×D411yescharacteristic
275C2×Q8—22no—
276C2×Q8—24no—
277C2×Q8—22no—
278C2×Q8—24no—
279C2×Q8—41no—
280C2×Q8—42no—
281C2×Q8—41no—
282C2×Q8—42no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—42no—
286C2×Q8—41no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
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—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4#11714yes—
358C22×C4C4×D414yes—
359C22×C4C4×D412yes—
360C22×C4#12114yes—
361C22×C4#12114yes—
362C22×C4#12112yes—
363C22×C4#12314yes—
364C22×C4#12412yes—
365C22×C4#12614yes—
366C22×C4C23×C411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D412yes—
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4112yes—
373C22×C4C2×#4112yes—
374C22×C4C2×#4112yes—
375C22×C4C2×#4111yescharacteristic
376M16—232no—
377C2×C8—232no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8C4×D418yes—
393C2×C8#12014yes—
394C2×C8#12314yes—
395C4⋊C4—28no—
396C4⋊C4—28no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—41no—
402C4⋊C4—42no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—48no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—42no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—88no—
417C22⋊C4—216no—
418C22⋊C4—48no—
419C22⋊C4—84no—
420C22⋊C4—84no—
421C22⋊C4—84no—
422C42—24no—
423C42—24no—
424C42—24no—
425C42—24no—
426C23—22no—
427C23—24no—
428C23—22no—
429C23—22no—
430C23—24no—
431C23—22no—
432C23—22no—
433C23—24no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—24no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23#38814yes—
449C23C2×C4×D411yescharacteristic
450C23C2×#12112yes—
451C23C2×#12111yescharacteristic
452C23C2×#12411yescharacteristic
453C23#53212yes—
454C23#57611yescharacteristic
455C23#57711yescharacteristic
456C23#57711yescharacteristic
457C23#57712yes—
458C23#57811yescharacteristic
459C23#57812yes—
460Q8—44no—
461Q8—42no—
462Q8—42no—
463Q8—81no—
464Q8—82no—
465Q8—81no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—28no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—81no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4#38814yes—
527C2×C4C2×#11711yescharacteristic
528C2×C4C2×C4×D412yes—
529C2×C4C2×C4×D411yescharacteristic
530C2×C4C4×#4112yes—
531C2×C4C2×#12011yescharacteristic
532C2×C4C2×#12311yescharacteristic
533C2×C4C2×#12311yescharacteristic
534C2×C4#52512yes—
535C2×C4#52512yes—
536C2×C4#52512yes—
537C2×C4#52512yes—
538C2×C4#52511yescharacteristic
539C2×C4#52511yescharacteristic
540C2×C4C2×#12611yescharacteristic
541C2×C4#52912yes—
542C2×C4#52912yes—
543C2×C4#52912yes—
544C2×C4#52912yes—
545C8—24no—
546C8—24no—
547C8—28no—
548C22—24no—
549C22—24no—
550C22—24no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22C2×#38811yescharacteristic
560C22C4×#12112yes—
561C22#284711yescharacteristic
562C22#285012yes—
563C22#285411yescharacteristic
564C22#294611yescharacteristic
565C22#294712yes—
566C22#294711yescharacteristic
567C22#305712yes—
568C22#305711yescharacteristic
569C22#305912yes—
570C22#305911yescharacteristic
571C22#306012yes—
572C22#306011yescharacteristic
573C22#306112yes—
574C22#306111yescharacteristic
575C22#313611yescharacteristic
576C22#313711yescharacteristic
577C22#313712yes—
578C22#317812yes—
579C22#317812yes—
580C22#317812yes—
581C22#317912yes—
582C4—24no—
583C4—44no—
584C4—44no—
585C4—84no—
586C4—82no—
587C4—82no—
588C4—162no—
589C4—161no—
590C4—161no—
591C4#229012yes—
592C4C4×#12011yescharacteristic
593C4C4×#12311yescharacteristic
594C4#284911yescharacteristic
595C4#285212yes—
596C4#285311yescharacteristic
597C2—44no—
598C2#1469611yescharacteristic
599C2#2109011yescharacteristic
600C2#2109211yescharacteristic
601C2#2153512yes—
602C2#2153712yes—
603C2#3293211yescharacteristic
604C2#3293411yescharacteristic
605C2#3304612yes—
606C2#3304712yes—
607C2#3313111yescharacteristic
608C2#3313311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1933C2semidirect
7. #10928C2semidirect
8. #10881C2semidirect
9. #10820C2semidirect
29. #1894C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4052
C2×#19332
C2×#18942
C2×#18922
#142642
#142582
#109282
#108812
#108202
#107222
#107212