Finite Group Databasebeta

← #4930410

Subgroups — #4930410

Order 512.

2162 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2C2×#2584C211yesmaximal, characteristic
3#15256C212yesmaximal
4#15242C212yesmaximal
5#15207C212yesmaximal
6#15197C212yesmaximal
7#15067C214yesmaximal
8#14937C214yesmaximal
9#14283C211yesmaximal, characteristic
10#14160C211yesmaximal, characteristic
11#14054C212yesmaximal
12#14054C212yesmaximal
13#13933C212yesmaximal
14#13926C211yesmaximal, characteristic
15#13918C212yesmaximal
16#13850C211yesmaximal, characteristic
17#13750C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C22×#125C2212yes—
20C22×C4×Q8C2211yescharacteristic
21#2584—28no—
22#2552—28no—
23C2×#398C2212yes—
24C2×#398C2214yes—
25C2×#395C2212yes—
26C2×#391C2212yes—
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#384C2212yes—
36C2×#384C2211yescharacteristic
37C2×#382C2214yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×#376C2211yescharacteristic
44#1446C2212yes—
45#1445C2212yes—
46#1445C2214yes—
47#1445C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2211yescharacteristic
53#1441C2211yescharacteristic
54#1441C2211yescharacteristic
55#1441C2211yescharacteristic
56#1441C2212yes—
57#1441C2212yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2212yes—
61#1441C2212yes—
62#1441C2212yes—
63#1441C2212yes—
64#1441C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2214yes—
71C4⋊#95C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2214yes—
77C4⋊#95C2212yes—
78#1434C2214yes—
79#1434C2214yes—
80C4×#95C2212yes—
81C4×#95C2212yes—
82C4×#95C2214yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C2×#125—28no—
87C2×#125—28no—
88C2×#125—28no—
89C2×#125—24no—
90C2×#125—24no—
91C2×#125—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2314yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2311yescharacteristic
126#398—28no—
127#398—216no—
128#395—28no—
129#391—44no—
130#389—24no—
131#389—24no—
132#389—24no—
133#389—24no—
134#389—28no—
135#389—28no—
136#389—28no—
137#389—28no—
138#384—28no—
139#384—24no—
140#382—48no—
141C4×C4⋊C4—28no—
142C4×C4⋊C4—24no—
143C4×C4⋊C4—24no—
144C4×C4⋊C4D418yes—
145C4×C4⋊C4Q8116yescomplemented
146#376—24no—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2311yescharacteristic
154C2×#95C2311yescharacteristic
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—41no—
186C22×Q8—41no—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204#125—432no—
205#125—416no—
206C4×Q8—416no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
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—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×Q818yes—
264C2×C4⋊C4C2×Q818yes—
265C2×C4⋊C4#4118yes—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—28no—
279C2×C42—44no—
280C2×C42—44no—
281C2×C42—44no—
282C2×C42—42no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42—42no—
286C2×C42C2×D418yes—
287C2×C42C2×D414yes—
288C2×C42C2×Q818yes—
289C2×C42C2×Q8116yes—
290C2×C42#4114yes—
291C2×C42#4114yes—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—28no—
296#95—24no—
297#95—24no—
298#95—22no—
299#95—22no—
300#95—24no—
301#95—28no—
302#95—24no—
303#95—24no—
304#95—28no—
305#95—24no—
306#95—42no—
307#95—42no—
308#95—42no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—42no—
316#95—44no—
317#95—42no—
318#95—42no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95C2×D418yes—
324#95C2×Q8116yes—
325C24C2511yescharacteristic
326C2×Q8—28no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—44no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4C22×D412yes—
419C22×C4C22×D414yes—
420C22×C4C22×D412yes—
421C22×C4C22×D414yes—
422C22×C4C22×D412yes—
423C22×C4C22×Q814yes—
424C22×C4C22×Q818yes—
425C22×C4C22×Q814yes—
426C22×C4C22×Q818yes—
427C22×C4C22×Q814yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4118yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4114yes—
435C22×C4C2×#4112yes—
436C22×C4H(5,𝔽2)12yes—
437C22×C42−1+412yes—
438C4⋊C4—216no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—416no—
448C4⋊C4—416no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—48no—
454C4⋊C4—84no—
455C4⋊C4—84no—
456C4⋊C4—88no—
457C4⋊C4—84no—
458C4⋊C4#124116yes—
459C4⋊C4C4⋊Q8116yes—
460C42—216no—
461C42—48no—
462C42—416no—
463C42—44no—
464C42—44no—
465C42—48no—
466C42—84no—
467C42—88no—
468C42#124116yes—
469C23C23×D411yescharacteristic
470C23C23×Q812yes—
471C23C22×#4111yescharacteristic
472C23C22×#4111yescharacteristic
473C23C22×#4112yes—
474C23C2×H(5,𝔽2)11yescharacteristic
475C23C2×2−1+411yescharacteristic
476C23#58512yes—
477C23#58512yes—
478C23#58511yescharacteristic
479C23#58511yescharacteristic
480Q8—416no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—28no—
494C2×C4—28no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—28no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—44no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4C2×#12414yes—
578C2×C4C2×#12414yes—
579C2×C4C2×#12418yes—
580C2×C4C2×#12414yes—
581C2×C4C2×#12414yes—
582C2×C4C2×C4⋊Q818yes—
583C2×C4C2×C4⋊Q814yes—
584C2×C4#53318yes—
585C2×C4#53318yes—
586C22C22×#12411yescharacteristic
587C22C22×#12411yescharacteristic
588C22C22×C4⋊Q811yescharacteristic
589C22C2×#53312yes—
590C22C2×#53312yes—
591C22#348511yescharacteristic
592C22#348711yescharacteristic
593C22#348811yescharacteristic
594C22#348911yescharacteristic
595C22#349211yescharacteristic
596C22#349311yescharacteristic
597C22#349411yescharacteristic
598C22#349511yescharacteristic
599C22#350112yes—
600C22#350212yes—
601C22#350412yes—
602C22#350412yes—
603C22#351212yes—
604C22#351212yes—
605C22#351512yes—
606C22#351712yes—
607C22#352112yes—
608C22#352412yes—
609C4—24no—
610C4—28no—
611C4—28no—
612C4—24no—
613C4—48no—
614C4—44no—
615C4—44no—
616C4—42no—
617C4—42no—
618C4—81no—
619C4—81no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—84no—
624C4—82no—
625C4#286218yes—
626C2#3554211yescharacteristic
627C2#3561212yes—
628C2#3564212yes—
629C2#3564911yescharacteristic
630C2#3570812yes—
631C2#3595011yescharacteristic
632C2#3595711yescharacteristic
633C2#3603912yes—
634C2#3604211yescharacteristic
635C2#3604911yescharacteristic
636C2C2×#286211yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
145. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#25842
#152562
#152422
#152072
#151972
#150672
#149372
#142832
#141602
#140542
#139332
#139262
#139182
#138502
#137502