Finite Group Databasebeta

← #357998

Subgroups — #357998

Order 512.

2118 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×#2317C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3C2×#1736C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5C2×#1735C211yesmaximal, characteristic
6C22×#340C211yesmaximal, characteristic
7C2×#1729C211yesmaximal, characteristic
8#11208C214yesmaximal, complemented
9#11207C214yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#2317—24no—
18#2317—24no—
19#2317—24no—
20#2317C418yescomplemented
21C2×#389—21no—
22C2×#389—21no—
23C2×#389C2214yes—
24C2×#385—22no—
25C2×#385—22no—
26C2×#384—21no—
27C2×#384C2212yes—
28C2×#382—21no—
29C2×#382C2212yes—
30#1736—24no—
31#1736—24no—
32#1735—24no—
33C2×#340—24no—
34C2×#340C2214yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2214yes—
39C2×#340C2212yes—
40#1729—24no—
41C2×#336C2214yes—
42C24×C4—21no—
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C22×M16—22no—
46C22×M16—22no—
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—22no—
51C22×M16Q814yes—
52C23×C8—22no—
53C23×C8—22no—
54C23×C8—22no—
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8Q814yes—
59C23×C8C2312yes—
60C23×C8C2312yes—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117C2×C414yes—
74C2×#117C2×C414yes—
75C2×#117D414yes—
76C2×#117D414yes—
77C2×#117D414yes—
78C2×#117Q814yes—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4C2×C414yes—
95C22×C4⋊C4D414yes—
96C22×C4⋊C4D411yescharacteristic
97C22×C4⋊C4D411yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4—21no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4D412yes—
106C22×C42—21no—
107C22×C42—21no—
108C22×C42C2312yes—
109#389—28no—
110#389—28no—
111#389—28no—
112#389C2×C4116yescomplemented
113#385—216no—
114#385—216no—
115#384—28no—
116#384—24no—
117#384—24no—
118#382—28no—
119#382—24no—
120#382—24no—
121C2×#95—24no—
122C2×#95—22no—
123C2×#95—22no—
124#340—28no—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—28no—
130#336—28no—
131C25C2×D411yescharacteristic
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—82no—
144C23×C4—82no—
145C23×C4—82no—
146C23×C4—82no—
147C23×C4C22×C412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×Q812yes—
156C23×C4C2×Q812yes—
157C23×C4C2×Q812yes—
158C23×C4#4112yes—
159C23×C4#4112yes—
160C23×C4#4112yes—
161C23×C4#4112yes—
162C23×C4#4112yes—
163C23×C4#4112yes—
164C23×C4C2411yescharacteristic
165C2×M16—28no—
166C2×M16—28no—
167C2×M16—28no—
168C2×M16—44no—
169C2×M16—44no—
170C2×M16—44no—
171C2×M16—44no—
172C22×C8—28no—
173C22×C8—28no—
174C22×C8—28no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—44no—
188C22×C8—44no—
189C22×C8—44no—
190C22×C8—44no—
191C22×C8C2×Q814yes—
192C22×C8C2×Q814yes—
193#117—28no—
194#117—28no—
195#117—28no—
196#117—28no—
197#117—28no—
198#117—28no—
199#117—28no—
200#117—28no—
201#117—28no—
202#117—44no—
203#117—44no—
204#117C4⋊C4116yescomplemented
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—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—216no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—84no—
250C2×C4⋊C4—84no—
251C2×C4⋊C4—84no—
252C2×C4⋊C4—84no—
253C2×C4⋊C4C22⋊C414yes—
254C2×C4⋊C4C22⋊C414yes—
255C2×C4⋊C4C22×C414yes—
256C2×C4⋊C4C22×C414yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×Q814yes—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4C22⋊C418yes—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42C22×C414yes—
282C2×C42C2×D414yes—
283#95—28no—
284#95—28no—
285#95—216no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24C2×C22⋊C412yes—
291C24#12012yes—
292C24#12112yes—
293C24C22⋊Q812yes—
294C24#12312yes—
295C24#12412yes—
296C24C4⋊Q812yes—
297C24C22×D411yescharacteristic
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—48no—
334C22×C4—48no—
335C22×C4—88no—
336C22×C4—84no—
337C22×C4—84no—
338C22×C4—88no—
339C22×C4—88no—
340C22×C4—84no—
341C22×C4—84no—
342C22×C4—88no—
343C22×C4C2×C22⋊C414yes—
344C22×C4C2×C22⋊C414yes—
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C4⋊C414yes—
347C22×C4#11714yes—
348C22×C4C4×D414yes—
349C22×C4C4×D414yes—
350C22×C4C4×Q814yes—
351C22×C4C4×Q814yes—
352C22×C4#12012yes—
353C22×C4#12114yes—
354C22×C4#12114yes—
355C22×C4#12114yes—
356C22×C4#12114yes—
357C22×C4#12114yes—
358C22×C4#12112yes—
359C22×C4C22⋊Q814yes—
360C22×C4C22⋊Q814yes—
361C22×C4C22⋊Q814yes—
362C22×C4C22⋊Q814yes—
363C22×C4C22⋊Q814yes—
364C22×C4C22⋊Q814yes—
365C22×C4C22⋊Q814yes—
366C22×C4C22⋊Q812yes—
367C22×C4#12314yes—
368C22×C4#12314yes—
369C22×C4#12312yes—
370C22×C4#12412yes—
371C22×C4#12514yes—
372C22×C4#12514yes—
373C22×C4#12514yes—
374C22×C4#12514yes—
375C22×C4#12614yes—
376C22×C4#12614yes—
377C22×C4C4⋊Q814yes—
378C22×C4C4⋊Q814yes—
379C22×C4C4⋊Q812yes—
380C22×C4C23×C411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×Q811yescharacteristic
386C22×C4C22×Q811yescharacteristic
387C22×C4C22×Q811yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394M16—416no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—416no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—28no—
414C4⋊C4—28no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—42no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—816no—
434C4⋊C4—88no—
435C4⋊C4—88no—
436C4⋊C4C2×C4⋊C418yes—
437C22⋊C4—216no—
438C22⋊C4—216no—
439C22⋊C4—48no—
440C42—24no—
441C42—24no—
442C42—24no—
443C42—24no—
444C42—216no—
445C42—48no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—24no—
461C23—24no—
462C23#38014yes—
463C23#38614yes—
464C23#39114yes—
465C23#39414yes—
466C23#39514yes—
467C23#39814yes—
468C23#40014yes—
469C23C22×C22⋊C411yescharacteristic
470C23C2×#12011yescharacteristic
471C23C2×#12111yescharacteristic
472C23C2×C22⋊Q811yescharacteristic
473C23C2×#12311yescharacteristic
474C23C2×#12411yescharacteristic
475C23C2×C4⋊Q811yescharacteristic
476C23#57612yes—
477C23#57712yes—
478C23#57712yes—
479C23#57812yes—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—48no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—816no—
509C2×C4—816no—
510C2×C4—84no—
511C2×C4—82no—
512C2×C4—84no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—84no—
518C2×C4—82no—
519C2×C4#38014yes—
520C2×C4#38614yes—
521C2×C4#38718yes—
522C2×C4#38718yes—
523C2×C4#38918yes—
524C2×C4#38918yes—
525C2×C4#39114yes—
526C2×C4#39414yes—
527C2×C4#39514yes—
528C2×C4#39618yes—
529C2×C4#39618yes—
530C2×C4#39818yes—
531C2×C4#39818yes—
532C2×C4#39814yes—
533C2×C4#40014yes—
534C2×C4C22×C4⋊C411yescharacteristic
535C2×C4C2×#11711yescharacteristic
536C2×C4C2×C4×D411yescharacteristic
537C2×C4C2×C4×D411yescharacteristic
538C2×C4C2×C4×Q811yescharacteristic
539C2×C4C2×C4×Q811yescharacteristic
540C2×C4C2×#12111yescharacteristic
541C2×C4C2×#12111yescharacteristic
542C2×C4C2×#12111yescharacteristic
543C2×C4C2×#12111yescharacteristic
544C2×C4C2×#12111yescharacteristic
545C2×C4C2×C22⋊Q811yescharacteristic
546C2×C4C2×C22⋊Q811yescharacteristic
547C2×C4C2×C22⋊Q811yescharacteristic
548C2×C4C2×C22⋊Q811yescharacteristic
549C2×C4C2×C22⋊Q811yescharacteristic
550C2×C4C2×C22⋊Q811yescharacteristic
551C2×C4C2×C22⋊Q811yescharacteristic
552C2×C4C2×#12311yescharacteristic
553C2×C4C2×#12311yescharacteristic
554C2×C4C2×#12511yescharacteristic
555C2×C4C2×#12511yescharacteristic
556C2×C4C2×#12511yescharacteristic
557C2×C4C2×#12511yescharacteristic
558C2×C4C2×#12611yescharacteristic
559C2×C4C2×#12611yescharacteristic
560C2×C4C2×C4⋊Q811yescharacteristic
561C2×C4C2×C4⋊Q811yescharacteristic
562C8—44no—
563C8—44no—
564C22—24no—
565C22—24no—
566C22—28no—
567C22—28no—
568C22—28no—
569C22—24no—
570C22—24no—
571C22—28no—
572C22—24no—
573C22—24no—
574C22#144118yes—
575C22C2×#38011yescharacteristic
576C22C2×#38611yescharacteristic
577C22C2×#39111yescharacteristic
578C22C2×#39411yescharacteristic
579C22C2×#39511yescharacteristic
580C22C2×#39811yescharacteristic
581C22C2×#40011yescharacteristic
582C22#289712yes—
583C22#289812yes—
584C22#300811yescharacteristic
585C22#300911yescharacteristic
586C22#301011yescharacteristic
587C22#301111yescharacteristic
588C22#304512yes—
589C22#304612yes—
590C22#308312yes—
591C22#308412yes—
592C22#309612yes—
593C22#309712yes—
594C22#313612yes—
595C22#313712yes—
596C22#316312yes—
597C22#316412yes—
598C4—48no—
599C4—44no—
600C4—44no—
601C4—84no—
602C4—82no—
603C4—84no—
604C4—82no—
605C4#144118yes—
606C4C2×#38711yescharacteristic
607C4C2×#38711yescharacteristic
608C4C2×#38911yescharacteristic
609C4C2×#38911yescharacteristic
610C4C2×#39611yescharacteristic
611C4C2×#39611yescharacteristic
612C4C2×#39811yescharacteristic
613C4C2×#39811yescharacteristic
614C2—28no—
615C2C2×#144111yescharacteristic
616C2#2065212yes—
617C2#2091311yescharacteristic
618C2#2091411yescharacteristic
619C2#2132612yes—
620C2#2350911yescharacteristic
621C2#2351011yescharacteristic
622C2#2355511yescharacteristic
623C2#2355611yescharacteristic
624C2#2366212yes—
625C2#2374812yes—
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
8. #11208C2semidirect
9. #11207C2semidirect
20. #2317C4semidirect
112. #389C2×C4semidirect
204. #117C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#17362
C2×#17352
C22×#3402
C2×#17292
#112082
#112072