Finite Group Databasebeta

← #393514

Subgroups — #393514

Order 512.

1736 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#405C211yesmaximal, characteristic
2C2×#2625C211yesmaximal, characteristic
3C2×#1894C211yesmaximal, characteristic
4C2×#1892C211yesmaximal, characteristic
5#14343C211yesmaximal, characteristic
6#10928C212yesmaximal, complemented
7#10926C212yesmaximal, complemented
8#10881C212yesmaximal, complemented
9#10880C212yesmaximal, complemented
10#10722C211yesmaximal, characteristic
11#10721C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C2×#405—24no—
20C2×#405—24no—
21C2×#405—24no—
22C22×#97C2212yes—
23C22×C4×C8C2212yes—
24#2625—24no—
25#2625C418yescomplemented
26C2×#400—21no—
27C2×#398—21no—
28C2×#397—22no—
29C2×#391C2212yes—
30C2×#389C2212yes—
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34#1894—24no—
35#1892—24no—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2212yes—
43#1445—21no—
44#1445—21no—
45#1440—22no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×Q8C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—24no—
53C22×M16—24no—
54C22×M16D414yes—
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—24no—
58C23×C8—24no—
59C23×C8D414yes—
60C23×C8C2311yescharacteristic
61C23×C8C2312yes—
62C23×C8C2311yescharacteristic
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8C2×C414yes—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117C2×C414yes—
74C2×#117D414yes—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4D412yes—
89C22×C42C2312yes—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97#405—232no—
98C2×#97—24no—
99C2×#97—24no—
100C2×#97—24no—
101C2×C4×C8—24no—
102C2×C4×C8—24no—
103C2×C4×C8—24no—
104#400—28no—
105#398—28no—
106#397—216no—
107#391—24no—
108#391C2×C418yes—
109#389—24no—
110#389C2×C418yes—
111#380—22no—
112#380—22no—
113#380—22no—
114#380—22no—
115#380—22no—
116#380C2×C414yes—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127#340—22no—
128#340—22no—
129#340—24no—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—24no—
134C25C2×D411yescharacteristic
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8C22×C412yes—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4C22×C412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4#4112yes—
163C23×C4#4112yes—
164C23×C4C2411yescharacteristic
165C2×M16—216no—
166C2×M16—28no—
167C2×M16—28no—
168C2×M16—28no—
169C2×M16—216no—
170C22×C8—216no—
171C22×C8—216no—
172C22×C8—28no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—28no—
184C22×C8—28no—
185C22×C8C2×D412yes—
186C22×C8C2×D412yes—
187C22×C8C2×D414yes—
188C22×C8#4114yes—
189C22×C8#4114yes—
190C22×C8#4114yes—
191C22×C8#4114yes—
192C22⋊Q8—28no—
193C22⋊Q8—28no—
194C22⋊Q8—44no—
195C22⋊Q8—44no—
196#117—28no—
197#117—28no—
198#117—28no—
199#117—28no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4—82no—
234C2×C4⋊C4C22⋊C414yes—
235C2×C4⋊C4C22⋊C414yes—
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C22×C414yes—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—81no—
247C2×C22⋊C4—81no—
248C2×C22⋊C4—81no—
249C2×C22⋊C4—82no—
250C2×C22⋊C4—81no—
251C2×C22⋊C4C22⋊C418yes—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42C22×C414yes—
257C2×C42C2×D414yes—
258#103—44no—
259#103—44no—
260#103—48no—
261#97—216no—
262C4×C8—216no—
263#95—24no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—42no—
268#95—42no—
269#95—82no—
270#95—82no—
271#95—82no—
272#95—82no—
273C24—22no—
274C24—22no—
275C24—21no—
276C24—21no—
277C24—22no—
278C24—21no—
279C24—22no—
280C24—21no—
281C24C2×C22⋊C412yes—
282C24#12112yes—
283C24#12112yes—
284C24C22×D411yescharacteristic
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—42no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4C2×C22⋊C414yes—
368C22×C4C2×C22⋊C414yes—
369C22×C4C2×C22⋊C412yes—
370C22×C4C4×D414yes—
371C22×C4C4×D414yes—
372C22×C4#12014yes—
373C22×C4#12112yes—
374C22×C4#12112yes—
375C22×C4#12314yes—
376C22×C4C23×C411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4112yes—
383C22×C4C2×#4112yes—
384C22×C4C2×#4112yes—
385C22×C4C2×#4112yes—
386M16—232no—
387C2×C8—232no—
388C2×C8—22no—
389C2×C8—24no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8#12314yes—
403C2×C8#12314yes—
404C2×C8#12318yes—
405C4⋊C4—28no—
406C4⋊C4—28no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—48no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—88no—
427C22⋊C4—216no—
428C22⋊C4—48no—
429C22⋊C4—84no—
430C22⋊C4—84no—
431C22⋊C4—84no—
432C42—24no—
433C42—24no—
434C42—24no—
435C42—24no—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—22no—
440C23—24no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—24no—
448C23—22no—
449C23—24no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23#38614yes—
459C23C22×C22⋊C411yescharacteristic
460C23C2×#12111yescharacteristic
461C23C2×#12111yescharacteristic
462C23#53212yes—
463C23#53212yes—
464C23#57611yescharacteristic
465C23#57711yescharacteristic
466C23#57712yes—
467C23#57711yescharacteristic
468C23#57812yes—
469C23#57811yescharacteristic
470Q8—42no—
471Q8—42no—
472Q8—44no—
473Q8—81no—
474Q8—81no—
475Q8—82no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
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—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4#38614yes—
537C2×C4C2×C4×D411yescharacteristic
538C2×C4C2×C4×D411yescharacteristic
539C2×C4C4×#4112yes—
540C2×C4C4×#4112yes—
541C2×C4C2×#12011yescharacteristic
542C2×C4C2×#12311yescharacteristic
543C2×C4C2×#12311yescharacteristic
544C2×C4C2×#12312yes—
545C2×C4C2×#12311yescharacteristic
546C2×C4#52512yes—
547C2×C4#52512yes—
548C2×C4#52511yescharacteristic
549C2×C4#52511yescharacteristic
550C2×C4#52512yes—
551C2×C4#52912yes—
552C2×C4#52912yes—
553C2×C4#52912yes—
554C2×C4#52912yes—
555C8—24no—
556C8—24no—
557C8—28no—
558C22—24no—
559C22—24no—
560C22—24no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22C2×#38611yescharacteristic
570C22#229712yes—
571C22#285411yescharacteristic
572C22#285411yescharacteristic
573C22#285412yes—
574C22#289711yescharacteristic
575C22#289812yes—
576C22#289811yescharacteristic
577C22#305711yescharacteristic
578C22#305911yescharacteristic
579C22#306011yescharacteristic
580C22#306111yescharacteristic
581C22#306211yescharacteristic
582C22#306311yescharacteristic
583C22#306312yes—
584C22#317312yes—
585C22#317412yes—
586C22#317512yes—
587C22#317712yes—
588C22#317812yes—
589C22#317812yes—
590C22#317812yes—
591C22#317912yes—
592C4—24no—
593C4—44no—
594C4—44no—
595C4—82no—
596C4—84no—
597C4—82no—
598C4—161no—
599C4—161no—
600C4—162no—
601C4C4×#12311yescharacteristic
602C4C4×#12311yescharacteristic
603C4C4×#12312yes—
604C4#284911yescharacteristic
605C4#284911yescharacteristic
606C4#285312yes—
607C2—44no—
608C2#1470611yescharacteristic
609C2#2097411yescharacteristic
610C2#2097611yescharacteristic
611C2#2141212yes—
612C2#2141312yes—
613C2#3313111yescharacteristic
614C2#3313311yescharacteristic
615C2#3313711yescharacteristic
616C2#3313911yescharacteristic
617C2#3314112yes—
618C2#3314212yes—
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
6. #10928C2semidirect
7. #10926C2semidirect
8. #10881C2semidirect
9. #10880C2semidirect
25. #2625C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4052
C2×#26252
C2×#18942
C2×#18922
#143432
#109282
#109262
#108812
#108802
#107222
#107212