Finite Group Databasebeta

← #412356

Subgroups — #412356

Order 512.

1570 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2621C211yesmaximal, characteristic
2C2×#2036C212yesmaximal
3C2×#2035C212yesmaximal
4C2×#1960C211yesmaximal, characteristic, complemented
5C2×#1956C211yesmaximal, characteristic, complemented
6C2×#1846C211yesmaximal, characteristic, complemented
7C2×#1750C211yesmaximal, characteristic
8#14340C211yesmaximal, characteristic
9#11124C212yesmaximal, complemented
10#11122C212yesmaximal, complemented
11#10843C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×#117C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2621—24no—
26#2621—24no—
27C2×#398—22no—
28C2×#397—22no—
29C2×#391C2212yes—
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#380C2212yes—
34#2036—28no—
35#2035—28no—
36#1960—24no—
37#1956—24no—
38#1846—24no—
39#1750—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2211yescharacteristic
44#1441—22no—
45#1439—22no—
46C4⋊#95C2212yes—
47C23×Q8C2312yes—
48C24×C4C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8D414yes—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—42no—
64C2×C22⋊Q8—42no—
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⋊C4—22no—
74C22×C4⋊C4D412yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4D412yes—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—42no—
86C2×#106—24no—
87C2×#106—24no—
88C2×#106—42no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—42no—
100C2×#103—42no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8D414yes—
106C2×C22⋊C8D414yes—
107C2×#97—22no—
108C2×#97—22no—
109C2×#97—22no—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112C2×C4×C8D414yes—
113#398—216no—
114#397—216no—
115#391—24no—
116#391—24no—
117#384—22no—
118#384—22no—
119#382—22no—
120#382—22no—
121#382—22no—
122#382D414yes—
123#380—24no—
124#380—24no—
125C2×#95—22no—
126C2×#95—24no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—24no—
130#340—24no—
131#340—24no—
132#336—22no—
133#336—22no—
134C25C2×D411yescharacteristic
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8C2×D414yes—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2411yescharacteristic
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—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8C2×D412yes—
192C22×C8C2×D412yes—
193C22×C8C2×D412yes—
194C22×C8C2×D412yes—
195C22×C8C2×D412yes—
196C22×C8C2×D412yes—
197C22×C8#4112yes—
198C22×C8#4112yes—
199C4⋊Q8—28no—
200C4⋊Q8—28no—
201C4⋊Q8—44no—
202C4⋊Q8—44no—
203C22⋊Q8—48no—
204C22⋊Q8—48no—
205C22⋊Q8—48no—
206C22⋊Q8—48no—
207#117—44no—
208#117—44no—
209#117—44no—
210#117—44no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
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—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—84no—
239C2×C4⋊C4C2×D414yes—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—84no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42#4112yes—
258C2×C42#4112yes—
259C2×C42#4112yes—
260#107—48no—
261#106—48no—
262#103—48no—
263#103—48no—
264#103—48no—
265#103—48no—
266C22⋊C8—28no—
267C22⋊C8—28no—
268#97—44no—
269C4×C8—28no—
270#95—216no—
271#95—28no—
272#95—44no—
273#95—44no—
274#95—84no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—24no—
282C24—24no—
283C24#12712yes—
284C24C22×D411yescharacteristic
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
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—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4#12014yes—
377C22×C4#12114yes—
378C22×C4#12314yes—
379C22×C4#12712yes—
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4H(5,𝔽2)11yescharacteristic
391C22×C4H(5,𝔽2)11yescharacteristic
392C22×C42−1+411yescharacteristic
393C22×C42−1+411yescharacteristic
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×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8#12114yes—
421C2×C8#12114yes—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—416no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—48no—
428C4⋊C4—416no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—88no—
438C22⋊C4—416no—
439C22⋊C4—48no—
440C22⋊C4—88no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42#12114yes—
446C42#12114yes—
447C42#12314yes—
448C42#12414yes—
449C23—22no—
450C23—24no—
451C23—24no—
452C23—22no—
453C23—22no—
454C23—44no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—41no—
460C23—44no—
461C23—44no—
462C23—41no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23C2×#12011yescharacteristic
467C23C2×#12111yescharacteristic
468C23C2×#12311yescharacteristic
469C23C2×#12711yescharacteristic
470C23#53211yescharacteristic
471C23#53511yescharacteristic
472C23#53611yescharacteristic
473C23#57714yes—
474C23#57814yes—
475Q8—44no—
476Q8—44no—
477Q8—44no—
478Q8—44no—
479Q8—82no—
480Q8—82no—
481Q8—82no—
482Q8—82no—
483C2×C4—28no—
484C2×C4—28no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—48no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—84no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—84no—
525C2×C4—82no—
526C2×C4—84no—
527C2×C4—84no—
528C2×C4—84no—
529C2×C4#39418yes—
530C2×C4C2×#12111yescharacteristic
531C2×C4C2×#12111yescharacteristic
532C2×C4C2×#12111yescharacteristic
533C2×C4C2×#12111yescharacteristic
534C2×C4C2×#12311yescharacteristic
535C2×C4C2×#12411yescharacteristic
536C2×C4#53211yescharacteristic
537C2×C4#53211yescharacteristic
538C2×C4#53711yescharacteristic
539C2×C4#53711yescharacteristic
540C2×C4#54211yescharacteristic
541C2×C4#54211yescharacteristic
542C2×C4D4211yescharacteristic
543C2×C4D4211yescharacteristic
544C2×C4D4211yescharacteristic
545C2×C4D4211yescharacteristic
546C2×C4#54711yescharacteristic
547C2×C4#54711yescharacteristic
548C2×C4#54711yescharacteristic
549C2×C4#54711yescharacteristic
550C2×C4#54711yescharacteristic
551C2×C4#54711yescharacteristic
552C2×C4#54811yescharacteristic
553C2×C4#54811yescharacteristic
554C2×C4#55311yescharacteristic
555C2×C4#55311yescharacteristic
556C2×C4#55511yescharacteristic
557C2×C4#55511yescharacteristic
558C8—24no—
559C8—24no—
560C8—42no—
561C8—42no—
562C22—44no—
563C22—42no—
564C22—44no—
565C22—42no—
566C22—42no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22C2×#39411yescharacteristic
572C22#243211yescharacteristic
573C22#243311yescharacteristic
574C22#250011yescharacteristic
575C22#250111yescharacteristic
576C22#262011yescharacteristic
577C22#285611yescharacteristic
578C22#301314yes—
579C22#305712yes—
580C22#305912yes—
581C22#309714yes—
582C22#315412yes—
583C22#315612yes—
584C22#317412yes—
585C22#317712yes—
586C22#322514yes—
587C22#324214yes—
588C4—28no—
589C4—42no—
590C4—84no—
591C4—84no—
592C4—82no—
593C4—162no—
594C4—162no—
595C4#241611yescharacteristic
596C4#241611yescharacteristic
597C4#242211yescharacteristic
598C4#242211yescharacteristic
599C4#249111yescharacteristic
600C4#249111yescharacteristic
601C4#253711yescharacteristic
602C4#253711yescharacteristic
603C2—82no—
604C2#1513011yescharacteristic
605C2#2348712yes—
606C2#2568312yes—
607C2#2573712yes—
608C2#2790012yes—
609C2#2792212yes—
610C2#2984812yes—
611C2#3317811yescharacteristic
612C2#3318011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
4. C2×#1960C2semidirect
5. C2×#1956C2semidirect
6. C2×#1846C2semidirect
9. #11124C2semidirect
10. #11122C2semidirect
11. #10843C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26212
C2×#20362
C2×#20352
C2×#19602
C2×#19562
C2×#18462
C2×#17502
#143402
#111242
#111222
#108432