Finite Group Databasebeta

← #392923

Subgroups — #392923

Order 512.

1634 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#405C211yesmaximal, characteristic
2C2×#1944C211yesmaximal, characteristic, complemented
3C2×#1942C211yesmaximal, characteristic, complemented
4C2×#1809C211yesmaximal, characteristic
5C2×#1809C211yesmaximal, characteristic
6#14264C212yesmaximal
7#10881C212yesmaximal, complemented
8#10880C212yesmaximal, complemented
9#10863C211yesmaximal, characteristic, complemented
10#10861C211yesmaximal, characteristic, complemented
11#10721C212yesmaximal
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C2×#405—24no—
20C2×#405—24no—
21C2×#405—24no—
22C22×#97C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30#1944—24no—
31#1942—24no—
32#1809—24no—
33#1809C418yescomplemented
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40#1444—22no—
41#1442—22no—
42#1439—22no—
43#1439—22no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
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×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4D411yescharacteristic
82C22×C22⋊C4D411yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#107C2×C414yes—
91C2×#106—22no—
92C2×#106—22no—
93C2×#106—22no—
94C2×#106—22no—
95C2×#106—22no—
96C2×#106C2×C414yes—
97#405—232no—
98C2×#97—22no—
99C2×#97—22no—
100C2×#97—22no—
101C2×#97—22no—
102C2×#97—22no—
103C2×#97D414yes—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8D414yes—
110#389—24no—
111#389—24no—
112#389—24no—
113#389—24no—
114#380—24no—
115#380—24no—
116#380—24no—
117#380C2×C418yes—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340—24no—
131#340C2×C418yes—
132C25C2×D411yescharacteristic
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4C22×C412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×Q812yes—
153C23×C4C2×Q812yes—
154C23×C4#4112yes—
155C23×C4#4112yes—
156C23×C4C2411yescharacteristic
157C2×M16—28no—
158C2×M16—28no—
159C2×M16—28no—
160C2×M16—28no—
161C2×M16—28no—
162C2×M16—28no—
163C2×M16C4⋊C4116yescomplemented
164C22×C8—28no—
165C22×C8—28no—
166C22×C8—28no—
167C22×C8—28no—
168C22×C8—28no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—28no—
182C22×C8—28no—
183C22×C8C22×C412yes—
184C22×C8C22×C412yes—
185C22×C8C2×D412yes—
186C22×C8C2×D412yes—
187C22×C8C2×D412yes—
188C22×C8C2×D412yes—
189C22×C8C2×D412yes—
190C22×C8C2×D412yes—
191C22×C8C2×Q812yes—
192C22×C8C2×Q812yes—
193C22×C8C2×Q812yes—
194C22×C8C2×Q812yes—
195C22×C8#4112yes—
196C22×C8#4112yes—
197C22×C8#4112yes—
198C22×C8#4112yes—
199#117—28no—
200#117—28no—
201#117—28no—
202#117—28no—
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—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—82no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4—82no—
237C2×C4⋊C4C22×C414yes—
238C2×C4⋊C4C22×C414yes—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—82no—
248C2×C22⋊C4—82no—
249C2×C22⋊C4—82no—
250C2×C22⋊C4—82no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42#4112yes—
262C2×C42#4112yes—
263#107—28no—
264#107—44no—
265#106—28no—
266#106—44no—
267#97—28no—
268#97—28no—
269C4×C8—28no—
270C4×C8—28no—
271#95—28no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—82no—
276#95—82no—
277#95—82no—
278#95—82no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24C4×D412yes—
288C24#12112yes—
289C24C4⋊Q812yes—
290C24C22×D411yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
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—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
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×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4C2×C4⋊C414yes—
364C22×C4C4×D412yes—
365C22×C4C4×Q814yes—
366C22×C4#12112yes—
367C22×C4#12112yes—
368C22×C4#12112yes—
369C22×C4#12112yes—
370C22×C4#12112yes—
371C22×C4C22⋊Q814yes—
372C22×C4#12514yes—
373C22×C4C4⋊Q812yes—
374C22×C4C23×C411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×Q811yescharacteristic
379C22×C4C22×Q811yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388M16—232no—
389C2×C8—232no—
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×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8C2×C4⋊C414yes—
409C2×C8C2×C4⋊C414yes—
410C2×C8C4×D414yes—
411C2×C8C4×D414yes—
412C2×C8#12014yes—
413C2×C8C22⋊Q814yes—
414C2×C8C22⋊Q814yes—
415C2×C8C22⋊Q814yes—
416C2×C8C22⋊Q814yes—
417C2×C8#12314yes—
418C4⋊C4—28no—
419C4⋊C4—28no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—88no—
437C4⋊C4—88no—
438C22⋊C4—28no—
439C22⋊C4—28no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—22no—
461C23—22no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—42no—
471C23—42no—
472C23#38914yes—
473C23C2×C4×D411yescharacteristic
474C23C2×#12111yescharacteristic
475C23C2×#12111yescharacteristic
476C23C2×#12111yescharacteristic
477C23C2×C4⋊Q811yescharacteristic
478C23#53211yescharacteristic
479C23#53211yescharacteristic
480C23#57612yes—
481C23#57712yes—
482C23#57712yes—
483C23#57812yes—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
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—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
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—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4#38914yes—
547C2×C4C22×C4⋊C411yescharacteristic
548C2×C4C2×C4×D411yescharacteristic
549C2×C4C2×C4×D411yescharacteristic
550C2×C4C2×C4×Q811yescharacteristic
551C2×C4C4×#4111yescharacteristic
552C2×C4C4×#4111yescharacteristic
553C2×C4C2×#12011yescharacteristic
554C2×C4C2×C22⋊Q811yescharacteristic
555C2×C4C2×C22⋊Q811yescharacteristic
556C2×C4C2×C22⋊Q811yescharacteristic
557C2×C4C2×C22⋊Q811yescharacteristic
558C2×C4C2×C22⋊Q811yescharacteristic
559C2×C4C2×#12311yescharacteristic
560C2×C4#52511yescharacteristic
561C2×C4#52511yescharacteristic
562C2×C4#52511yescharacteristic
563C2×C4#52511yescharacteristic
564C2×C4#52511yescharacteristic
565C2×C4#52511yescharacteristic
566C2×C4C2×#12511yescharacteristic
567C2×C4#52911yescharacteristic
568C2×C4#52911yescharacteristic
569C2×C4#52911yescharacteristic
570C2×C4#52911yescharacteristic
571C2×C4#53311yescharacteristic
572C2×C4#53311yescharacteristic
573C2×C4#53311yescharacteristic
574C2×C4#53311yescharacteristic
575C8—24no—
576C8—24no—
577C8—24no—
578C8#38518yes—
579C22—24no—
580C22—28no—
581C22—42no—
582C22—42no—
583C22—44no—
584C22—44no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—44no—
590C22C2×#38911yescharacteristic
591C22C4×#12111yescharacteristic
592C22C4×#12111yescharacteristic
593C22#284711yescharacteristic
594C22#285411yescharacteristic
595C22#285711yescharacteristic
596C22#285711yescharacteristic
597C22#294612yes—
598C22#294712yes—
599C22#305811yescharacteristic
600C22#305911yescharacteristic
601C22#306011yescharacteristic
602C22#306011yescharacteristic
603C22#306212yes—
604C22#306212yes—
605C22#306312yes—
606C22#306312yes—
607C22#316312yes—
608C22#316412yes—
609C22#317412yes—
610C22#317512yes—
611C22#317812yes—
612C22#317912yes—
613C4—24no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—84no—
619C4—84no—
620C4—162no—
621C4—162no—
622C4C2×#38511yescharacteristic
623C4#229611yescharacteristic
624C4C4×C22⋊Q811yescharacteristic
625C4C4×C22⋊Q811yescharacteristic
626C4#284811yescharacteristic
627C4#285111yescharacteristic
628C4#285111yescharacteristic
629C4#285511yescharacteristic
630C2—44no—
631C2#1469711yescharacteristic
632C2#2116012yes—
633C2#2153611yescharacteristic
634C2#2153711yescharacteristic
635C2#2154312yes—
636C2#3293911yescharacteristic
637C2#3294011yescharacteristic
638C2#3314212yes—
639C2#3321411yescharacteristic
640C2#3321511yescharacteristic
641C2#3321912yes—
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×#1944C2semidirect
3. C2×#1942C2semidirect
7. #10881C2semidirect
8. #10880C2semidirect
9. #10863C2semidirect
10. #10861C2semidirect
33. #1809C4semidirect
163. C2×M16C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4052
C2×#19442
C2×#19422
C2×#18092
#142642
#108812
#108802
#108632
#108612
#107212