Finite Group Databasebeta

← #358465

Subgroups — #358465

Order 512.

2270 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3C2×#2316C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5C2×#1735C211yesmaximal, characteristic
6C22×#340C211yesmaximal, characteristic
7C2×#1729C211yesmaximal, characteristic
8#11328C214yesmaximal, complemented
9#11327C214yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#428—24no—
18C2×#428—24no—
19C2×#428—24no—
20C22×#107C2212yes—
21C22×#106C2212yes—
22#2317—24no—
23#2317—24no—
24#2316—24no—
25#2316—24no—
26C2×#389—21no—
27C2×#389—21no—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2214yes—
31C2×#385—22no—
32C2×#384—21no—
33C2×#382—21no—
34C2×#380—22no—
35#1736C418yescomplemented
36#1735—24no—
37C2×#340—24no—
38C2×#340C2212yes—
39C2×#340C2214yes—
40C2×#340C2212yes—
41C2×#340C2214yes—
42#1729—24no—
43C2×#336C2214yes—
44C24×C4—21no—
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—22no—
51C22×M16C2×C414yes—
52C22×M16D414yes—
53C22×M16Q814yes—
54C23×C8—22no—
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8Q814yes—
61C23×C8C2312yes—
62C23×C8C2312yes—
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×#117—22no—
74C2×#117—22no—
75C2×#117—22no—
76C2×#117—22no—
77C2×#117C2×C414yes—
78C2×#117D414yes—
79C2×#117D414yes—
80C2×#117Q814yes—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
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—22no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4C2×C414yes—
95C22×C4⋊C4D411yescharacteristic
96C22×C4⋊C4D411yescharacteristic
97C22×C4⋊C4D414yes—
98C22×C4⋊C4D414yes—
99C22×C4⋊C4Q814yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4D412yes—
108C22×C42—21no—
109C22×C42—21no—
110C22×C42C2312yes—
111#428—232no—
112C2×#107—24no—
113C2×#107—24no—
114C2×#107—24no—
115C2×#106—24no—
116C2×#106—24no—
117C2×#106—24no—
118#389—28no—
119#389—28no—
120#389—24no—
121#389—24no—
122#389—24no—
123#389—24no—
124#389—28no—
125#389—28no—
126#385—216no—
127#384—28no—
128#382—28no—
129#380—216no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—24no—
133#340—28no—
134#340—24no—
135#340—24no—
136#340C2×C4116yescomplemented
137#336—28no—
138C25C2×D411yescharacteristic
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—82no—
151C23×C4—82no—
152C23×C4—82no—
153C23×C4—82no—
154C23×C4C22×C412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×Q812yes—
164C23×C4C2×Q812yes—
165C23×C4C2×Q812yes—
166C23×C4C2×Q812yes—
167C23×C4#4112yes—
168C23×C4#4112yes—
169C23×C4#4112yes—
170C23×C4#4112yes—
171C23×C4C2411yescharacteristic
172C2×M16—28no—
173C2×M16—28no—
174C2×M16—28no—
175C2×M16—28no—
176C2×M16—28no—
177C2×M16—28no—
178C2×M16C4⋊C4116yescomplemented
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—28no—
184C22×C8—28no—
185C22×C8—28no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—44no—
193C22×C8—44no—
194C22×C8—44no—
195C22×C8—44no—
196C22×C8C22×C414yes—
197C22×C8C2×D414yes—
198C22×C8C2×Q814yes—
199C22×C8C2×Q814yes—
200#117—28no—
201#117—28no—
202#117—28no—
203#117—28no—
204#117—28no—
205#117—44no—
206#117—44no—
207#117—44no—
208#117—44no—
209#117—44no—
210#117—44no—
211#117C4⋊C4116yescomplemented
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
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—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—216no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—84no—
256C2×C4⋊C4—84no—
257C2×C4⋊C4—84no—
258C2×C4⋊C4—84no—
259C2×C4⋊C4C4⋊C4116yescomplemented
260C2×C4⋊C4C22×C414yes—
261C2×C4⋊C4C22×C414yes—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×Q814yes—
268C2×C4⋊C4C2×Q814yes—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—42no—
289C2×C42—42no—
290#107—216no—
291#106—216no—
292#95—28no—
293#95—28no—
294#95—216no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24C4×D412yes—
300C24#12012yes—
301C24#12012yes—
302C24#12112yes—
303C24C22⋊Q812yes—
304C24C22⋊Q812yes—
305C24C4⋊Q812yes—
306C24C22×D411yescharacteristic
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—216no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—42no—
328C22×C4—44no—
329C22×C4—42no—
330C22×C4—44no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—48no—
346C22×C4—88no—
347C22×C4—88no—
348C22×C4—84no—
349C22×C4—84no—
350C22×C4—88no—
351C22×C4—84no—
352C22×C4—84no—
353C22×C4—88no—
354C22×C4C2×C4⋊C414yes—
355C22×C4C2×C4⋊C414yes—
356C22×C4C2×C4⋊C414yes—
357C22×C4C4×D414yes—
358C22×C4C4×D414yes—
359C22×C4C4×D412yes—
360C22×C4C4×Q814yes—
361C22×C4#12014yes—
362C22×C4#12012yes—
363C22×C4#12012yes—
364C22×C4#12114yes—
365C22×C4#12114yes—
366C22×C4#12112yes—
367C22×C4C22⋊Q814yes—
368C22×C4C22⋊Q814yes—
369C22×C4C22⋊Q814yes—
370C22×C4C22⋊Q814yes—
371C22×C4C22⋊Q814yes—
372C22×C4C22⋊Q814yes—
373C22×C4C22⋊Q814yes—
374C22×C4C22⋊Q812yes—
375C22×C4C22⋊Q812yes—
376C22×C4#12314yes—
377C22×C4#12314yes—
378C22×C4#12314yes—
379C22×C4#12514yes—
380C22×C4#12514yes—
381C22×C4#12514yes—
382C22×C4#12714yes—
383C22×C4C4⋊Q814yes—
384C22×C4C4⋊Q814yes—
385C22×C4C4⋊Q814yes—
386C22×C4C4⋊Q814yes—
387C22×C4C4⋊Q814yes—
388C22×C4C4⋊Q812yes—
389C22×C4C23×C411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×Q811yescharacteristic
396C22×C4C22×Q811yescharacteristic
397C22×C4C22×Q811yescharacteristic
398C22×C4C22×Q811yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403M16—232no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—416no—
418C2×C8C2×C4⋊C418yes—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—216no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—42no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—42no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—48no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—88no—
442C4⋊C4—88no—
443C4⋊C4—816no—
444C4⋊C4C2×C4⋊C418yes—
445C4⋊C4C2×C4⋊C418yes—
446C22⋊C4—216no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C42—216no—
450C42—42no—
451C42—42no—
452C42—42no—
453C42—42no—
454C42—48no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23#38514yes—
472C23#38514yes—
473C23#38914yes—
474C23#39214yes—
475C23#39514yes—
476C23#39514yes—
477C23#39814yes—
478C23C2×C4×D411yescharacteristic
479C23C2×#12011yescharacteristic
480C23C2×#12011yescharacteristic
481C23C2×#12111yescharacteristic
482C23C2×C22⋊Q811yescharacteristic
483C23C2×C22⋊Q811yescharacteristic
484C23C2×C4⋊Q811yescharacteristic
485C23#57612yes—
486C23#57712yes—
487C23#57712yes—
488C23#57812yes—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—48no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—48no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—48no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—816no—
520C2×C4—816no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—84no—
527C2×C4—82no—
528C2×C4#38418yes—
529C2×C4#38514yes—
530C2×C4#38514yes—
531C2×C4#38518yes—
532C2×C4#38914yes—
533C2×C4#38918yes—
534C2×C4#38918yes—
535C2×C4#39214yes—
536C2×C4#39518yes—
537C2×C4#39514yes—
538C2×C4#39514yes—
539C2×C4#39818yes—
540C2×C4#39818yes—
541C2×C4#39814yes—
542C2×C4#39918yes—
543C2×C4C22×C4⋊C411yescharacteristic
544C2×C4C22×C4⋊C411yescharacteristic
545C2×C4C22×C4⋊C411yescharacteristic
546C2×C4C2×C4×D411yescharacteristic
547C2×C4C2×C4×D411yescharacteristic
548C2×C4C2×C4×Q811yescharacteristic
549C2×C4C2×#12011yescharacteristic
550C2×C4C2×#12111yescharacteristic
551C2×C4C2×#12111yescharacteristic
552C2×C4C2×C22⋊Q811yescharacteristic
553C2×C4C2×C22⋊Q811yescharacteristic
554C2×C4C2×C22⋊Q811yescharacteristic
555C2×C4C2×C22⋊Q811yescharacteristic
556C2×C4C2×C22⋊Q811yescharacteristic
557C2×C4C2×C22⋊Q811yescharacteristic
558C2×C4C2×C22⋊Q811yescharacteristic
559C2×C4C2×#12311yescharacteristic
560C2×C4C2×#12311yescharacteristic
561C2×C4C2×#12311yescharacteristic
562C2×C4C2×#12511yescharacteristic
563C2×C4C2×#12511yescharacteristic
564C2×C4C2×#12511yescharacteristic
565C2×C4C2×#12711yescharacteristic
566C2×C4C2×C4⋊Q811yescharacteristic
567C2×C4C2×C4⋊Q811yescharacteristic
568C2×C4C2×C4⋊Q811yescharacteristic
569C2×C4C2×C4⋊Q811yescharacteristic
570C2×C4C2×C4⋊Q811yescharacteristic
571C8—28no—
572C8—44no—
573C22—24no—
574C22—24no—
575C22—28no—
576C22—24no—
577C22—24no—
578C22—28no—
579C22—24no—
580C22—24no—
581C22—28no—
582C22—28no—
583C22#144418yes—
584C22C2×#38511yescharacteristic
585C22C2×#38511yescharacteristic
586C22C2×#38911yescharacteristic
587C22C2×#39211yescharacteristic
588C22C2×#39511yescharacteristic
589C22C2×#39511yescharacteristic
590C22C2×#39811yescharacteristic
591C22#294612yes—
592C22#294712yes—
593C22#300811yescharacteristic
594C22#300911yescharacteristic
595C22#301011yescharacteristic
596C22#301111yescharacteristic
597C22#301212yes—
598C22#301312yes—
599C22#306212yes—
600C22#306312yes—
601C22#308112yes—
602C22#308212yes—
603C22#308312yes—
604C22#308412yes—
605C22#316312yes—
606C22#316412yes—
607C4—48no—
608C4—48no—
609C4—82no—
610C4—82no—
611C4—84no—
612C4—82no—
613C4—82no—
614C4#144418yes—
615C4C2×#38411yescharacteristic
616C4C2×#38511yescharacteristic
617C4C2×#38911yescharacteristic
618C4C2×#38911yescharacteristic
619C4C2×#39511yescharacteristic
620C4C2×#39811yescharacteristic
621C4C2×#39811yescharacteristic
622C4C2×#39911yescharacteristic
623C2—28no—
624C2C2×#144411yescharacteristic
625C2#2087611yescharacteristic
626C2#2087711yescharacteristic
627C2#2087812yes—
628C2#2116012yes—
629C2#2339311yescharacteristic
630C2#2339411yescharacteristic
631C2#2355511yescharacteristic
632C2#2355611yescharacteristic
633C2#2355712yes—
634C2#2367112yes—
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. #11328C2semidirect
9. #11327C2semidirect
35. #1736C4semidirect
136. #340C2×C4semidirect
178. C2×M16C4⋊C4semidirect
211. #117C4⋊C4semidirect
259. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C2×#23172
C2×#23162
C2×#17362
C2×#17352
C22×#3402
C2×#17292
#113282
#113272