Finite Group Databasebeta

← #4458135

Subgroups — #4458135

Order 512.

2226 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2442C212yesmaximal
2#17076C212yesmaximal, complemented
3#17057C212yesmaximal
4#17036C212yesmaximal, complemented
5#16987C212yesmaximal
6#16986C214yesmaximal
7#16907C214yesmaximal, complemented
8#15228C212yesmaximal, complemented
9#15225C212yesmaximal, complemented
10#15221C212yesmaximal, complemented
11#15218C212yesmaximal, complemented
12#14340C211yesmaximal, characteristic
13#13951C212yesmaximal
14#13852C211yesmaximal, characteristic
15#13795C211yesmaximal, characteristic
16C22×#125C2212yes—
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2442—216no—
21C2×#400C2214yes—
22C2×#398C2214yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#385C2214yes—
28C2×#384C2211yescharacteristic
29C2×#382C2212yes—
30C2×#382C2211yescharacteristic
31C2×#380C2212yes—
32C2×#380C2214yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2212yes—
36#1445C2214yes—
37#1445C2214yes—
38#1444C2214yes—
39#1444C2214yes—
40#1444C2214yes—
41#1444C2214yes—
42#1444C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1441C2212yes—
47#1441C2212yes—
48#1441C2212yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2211yescharacteristic
54#1441C2211yescharacteristic
55#1441C2214yes—
56#1441C2214yes—
57#1439C2212yes—
58#1439C2212yes—
59#1439C2212yes—
60#1439C2211yescharacteristic
61#1439C2211yescharacteristic
62#1439C2214yes—
63#1439C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72#1434C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C4×#95C2212yes—
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C2×#125—28no—
79C2×#125—44no—
80C2×#125—44no—
81C2×#117—42no—
82C2×#117—42no—
83C2×#117—42no—
84C2×#117—42no—
85C2×#117Q818yes—
86C2×#117Q818yes—
87C22×C4⋊C4—42no—
88C22×C4⋊C4—42no—
89C22×C4⋊C4D418yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119#400—216no—
120#398—216no—
121#389—28no—
122#389—28no—
123#389—44no—
124#389—44no—
125#385—216no—
126#384—42no—
127#382—28no—
128#382—42no—
129#380—216no—
130#380—28no—
131C4×C4⋊C4—42no—
132C4×C4⋊C4—42no—
133C4×C22⋊C4—28no—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2311yescharacteristic
162C25C2411yescharacteristic
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4C2×D414yes—
172C23×C4C2×D418yes—
173C23×C4C2×Q814yes—
174C23×C4C2×Q814yes—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2414yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2414yes—
182C23×C4C2411yescharacteristic
183C23×C4C2411yescharacteristic
184C23×C4C2411yescharacteristic
185C23×C4C2411yescharacteristic
186C23×C4C2414yes—
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189#125—432no—
190#117—416no—
191#117—416no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—416no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4C2×D418yes—
239C2×C4⋊C4C2×D418yes—
240C2×C4⋊C4C2×Q818yes—
241C2×C4⋊C4C2×Q814yes—
242C2×C4⋊C4C2×Q814yes—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4C2×Q814yes—
259C2×C22⋊C4C2×Q814yes—
260C2×C22⋊C4C2×Q818yes—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42C2×Q814yes—
274C2×C42C2×Q814yes—
275C2×C42C2×Q818yes—
276#95—24no—
277#95—24no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—42no—
288#95—42no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—42no—
295#95—44no—
296#95—42no—
297#95—41no—
298#95—42no—
299#95—41no—
300#95—42no—
301#95—42no—
302#95—44no—
303#95—44no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24C22×D412yes—
313C24C22×Q812yes—
314C24C22×Q812yes—
315C24C2511yescharacteristic
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—28no—
321C22×C4—28no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—48no—
397C22×C4—416no—
398C22×C4—84no—
399C22×C4—84no—
400C22×C4—88no—
401C22×C4C4⋊Q8116yes—
402C22×C4C22×D418yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×Q812yes—
406C22×C4C22×Q812yes—
407C22×C4C22×Q812yes—
408C22×C4C22×Q812yes—
409C22×C4C22×Q818yes—
410C22×C4C22×Q812yes—
411C22×C4C22×Q812yes—
412C22×C4C22×Q812yes—
413C22×C4C22×Q812yes—
414C22×C4H(5,𝔽2)12yes—
415C22×C4H(5,𝔽2)12yes—
416C22×C4H(5,𝔽2)12yes—
417C22×C42−1+412yes—
418C4⋊C4—216no—
419C4⋊C4—216no—
420C4⋊C4—48no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—416no—
424C4⋊C4—416no—
425C4⋊C4—48no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—88no—
431C4⋊C4—88no—
432C4⋊C4—84no—
433C4⋊C4—84no—
434C4⋊C4—88no—
435C4⋊C4—88no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—416no—
442C42—28no—
443C42—28no—
444C42—48no—
445C42—48no—
446C42—48no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—42no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23C2×C4⋊Q814yes—
470C23C23×D411yescharacteristic
471C23C23×Q811yescharacteristic
472C23C23×Q811yescharacteristic
473C23C2×H(5,𝔽2)11yescharacteristic
474C23C2×H(5,𝔽2)11yescharacteristic
475C23C2×H(5,𝔽2)11yescharacteristic
476C23C2×2−1+411yescharacteristic
477C23#58512yes—
478C23#58512yes—
479C23#58512yes—
480C23#58512yes—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—232no—
488C2×C4—28no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—28no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—816no—
565C2×C4C2×C4⋊Q814yes—
566C2×C4C2×C4⋊Q814yes—
567C2×C4#53414yes—
568C2×C4#53714yes—
569C2×C4#54314yes—
570C2×C4#54314yes—
571C2×C4#54414yes—
572C2×C4#54414yes—
573C22—24no—
574C22—28no—
575C22—42no—
576C22—42no—
577C22—44no—
578C22—42no—
579C22—42no—
580C22—44no—
581C22—44no—
582C22—42no—
583C22—42no—
584C22C22×C4⋊Q811yescharacteristic
585C22C2×#53411yescharacteristic
586C22C2×#53711yescharacteristic
587C22C2×#54311yescharacteristic
588C22C2×#54311yescharacteristic
589C22C2×#54411yescharacteristic
590C22C2×#54411yescharacteristic
591C22#348512yes—
592C22#348712yes—
593C22#350412yes—
594C22#350412yes—
595C22#350412yes—
596C22#350412yes—
597C22#353912yes—
598C22#354011yescharacteristic
599C22#354011yescharacteristic
600C22#355111yescharacteristic
601C22#355111yescharacteristic
602C22#355112yes—
603C22#356212yes—
604C22#356212yes—
605C22#356512yes—
606C22#356512yes—
607C4—24no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—48no—
613C4—48no—
614C4—82no—
615C4—84no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—84no—
620C4#265918yes—
621C2—44no—
622C2C2×#265911yescharacteristic
623C2#3564012yes—
624C2#3742511yescharacteristic
625C2#3742611yescharacteristic
626C2#3742912yes—
627C2#4267311yescharacteristic
628C2#4267412yes—
629C2#4288311yescharacteristic
630C2#4295312yes—
631C2#4295511yescharacteristic
632C2#4298111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17076C2semidirect
4. #17036C2semidirect
7. #16907C2semidirect
8. #15228C2semidirect
9. #15225C2semidirect
10. #15221C2semidirect
11. #15218C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24422
#170762
#170572
#170362
#169872
#169862
#169072
#152282
#152252
#152212
#152182
#143402
#139512
#138522
#137952