Finite Group Databasebeta

← #556145

Subgroups — #556145

Order 512.

2866 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2316C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic
3C22×#387C211yesmaximal, characteristic
4#14697C211yesmaximal, characteristic
5#14697C211yesmaximal, characteristic
6#14487C214yesmaximal, complemented
7#14465C214yesmaximal, complemented
8#14441C214yesmaximal, complemented
9#14439C214yesmaximal, complemented
10#14275C211yesmaximal, characteristic
11#14275C211yesmaximal, characteristic
12#14149C212yesmaximal
13#14130C212yesmaximal
14#13962C212yesmaximal
15#13962C212yesmaximal
16C22×#117C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2316C4116yescomplemented
21#2313—28no—
22C2×#389C2214yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2214yes—
26C2×#389C2214yes—
27C2×#387—28no—
28C2×#387C2212yes—
29C2×#387C2214yes—
30C2×#387C2212yes—
31C2×#387C2214yes—
32C2×#387C2212yes—
33C2×#387C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#381C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2212yes—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45C22×#95C2211yescharacteristic
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48C22×#95C2211yescharacteristic
49#1444C2214yes—
50#1444C2212yes—
51#1444C2212yes—
52#1442C2214yes—
53#1442C2212yes—
54#1442C2212yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2214yes—
68C4⋊#95C2214yes—
69C4⋊#95C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72#1434C2214yes—
73#1434C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×#117—44no—
86C2×#117—44no—
87C2×#117C2×C4116yes—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4C2×C418yes—
91C22×C4⋊C4D418yes—
92C22×C4⋊C4Q818yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C42—24no—
113C22×C42—44no—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2314yes—
119#389—44no—
120#389—48no—
121#389—44no—
122#389C2×C4132yescomplemented
123#389C2×C4132yescomplemented
124#387—28no—
125#387—28no—
126#387—28no—
127#387—28no—
128#387—216no—
129#387—216no—
130#385—44no—
131#385—44no—
132#385—44no—
133#385—44no—
134#381—28no—
135#380—24no—
136#380—24no—
137#380C2×C4116yes—
138#380C2×C418yes—
139#380C2×C418yes—
140C4×C22⋊C4—28no—
141C2×#95—24no—
142C2×#95—24no—
143C2×#95—24no—
144C2×#95—24no—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C25C2411yescharacteristic
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4C22×C414yes—
203C23×C4C22×C414yes—
204C23×C4C22×C414yes—
205C23×C4C2×D414yes—
206C23×C4C2×D414yes—
207C23×C4C2×D414yes—
208C23×C4C2×Q814yes—
209C23×C4C2×Q814yes—
210C23×C4C2×Q814yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225#117—432no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—416no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4C4⋊C4164yescomplemented
259C2×C4⋊C4C22×C4116yes—
260C2×C4⋊C4C22×C4116yes—
261C2×C4⋊C4C22×C418yes—
262C2×C4⋊C4C22×C418yes—
263C2×C4⋊C4C22×C418yes—
264C2×C4⋊C4C22×C418yes—
265C2×C4⋊C4C2×D418yes—
266C2×C4⋊C4C2×D418yes—
267C2×C4⋊C4C2×Q818yes—
268C2×C4⋊C4C2×Q818yes—
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—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4C22×C418yes—
280C2×C22⋊C4C22×C418yes—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—416no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42C22×C4116yes—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95C22×C4116yes—
321#95C22×C418yes—
322#95C22×C418yes—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24C23×C412yes—
328C24C22×D412yes—
329C24C22×Q812yes—
330C24H(5,𝔽2)12yes—
331C24H(5,𝔽2)12yes—
332C242−1+412yes—
333C242−1+412yes—
334C24C2511yescharacteristic
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—84no—
430C22×C4—84no—
431C22×C4—84no—
432C22×C4—84no—
433C22×C4C2×C4⋊C4116yes—
434C22×C4C2×C4⋊C4116yes—
435C22×C4C2×C4⋊C4116yes—
436C22×C4C23×C418yes—
437C22×C4C23×C418yes—
438C22×C4C23×C412yes—
439C22×C4C23×C412yes—
440C22×C4C23×C414yes—
441C22×C4C23×C414yes—
442C22×C4C22×D414yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D414yes—
446C22×C4C22×Q814yes—
447C22×C4C22×Q812yes—
448C22×C4C22×Q812yes—
449C22×C4C22×Q814yes—
450C22×C4H(5,𝔽2)14yes—
451C22×C42−1+414yes—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—416no—
455C4⋊C4—416no—
456C4⋊C4—416no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—88no—
462C4⋊C4—88no—
463C4⋊C4—88no—
464C4⋊C4—88no—
465C4⋊C4C2×C4⋊C4132yes—
466C4⋊C4C2×C4⋊C4132yes—
467C22⋊C4—216no—
468C22⋊C4—216no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C42—416no—
472C42—48no—
473C42—48no—
474C42—84no—
475C42—84no—
476C23—24no—
477C23—24no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23C22×C4⋊C414yes—
493C23#51814yes—
494C23#51914yes—
495C23#53714yes—
496C23#53714yes—
497C23#54414yes—
498C23#54414yes—
499C23C24×C411yescharacteristic
500C23C23×D411yescharacteristic
501C23C23×Q811yescharacteristic
502C23C2×H(5,𝔽2)11yescharacteristic
503C23C2×H(5,𝔽2)11yescharacteristic
504C23C2×H(5,𝔽2)12yes—
505C23C2×2−1+411yescharacteristic
506C23C2×2−1+411yescharacteristic
507C23C2×2−1+412yes—
508C23#58512yes—
509C23#58512yes—
510C2×C4—232no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—416no—
542C2×C4—416no—
543C2×C4—416no—
544C2×C4—416no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—88no—
592C2×C4—88no—
593C2×C4C22×C4⋊C418yes—
594C2×C4C22×C4⋊C414yes—
595C2×C4C22×C4⋊C414yes—
596C2×C4C22×C4⋊C418yes—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—28no—
602C22—28no—
603C22—28no—
604C22—28no—
605C22—24no—
606C22—24no—
607C22#231618yes—
608C22C23×C4⋊C411yescharacteristic
609C22C2×#51811yescharacteristic
610C22C2×#51911yescharacteristic
611C22C4×H(5,𝔽2)12yes—
612C22C4×2−1+412yes—
613C22C2×#53711yescharacteristic
614C22C2×#53711yescharacteristic
615C22C2×#54411yescharacteristic
616C22C2×#54411yescharacteristic
617C22#348411yescharacteristic
618C22#348511yescharacteristic
619C22#348611yescharacteristic
620C22#348711yescharacteristic
621C22#350112yes—
622C22#350212yes—
623C22#353412yes—
624C22#353512yes—
625C22#353612yes—
626C22#353812yes—
627C22#355912yes—
628C22#356012yes—
629C22#356312yes—
630C22#356412yes—
631C4—28no—
632C4—24no—
633C4—24no—
634C4—28no—
635C4—44no—
636C4—44no—
637C4—48no—
638C4—48no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4—82no—
645C2—28no—
646C2C2×#231611yescharacteristic
647C2#3424811yescharacteristic
648C2#3424911yescharacteristic
649C2#3443212yes—
650C2#3443612yes—
651C2#3671911yescharacteristic
652C2#3672211yescharacteristic
653C2#3675112yes—
654C2#3675212yes—
655C2#3696811yescharacteristic
656C2#3697111yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #14487C2semidirect
7. #14465C2semidirect
8. #14441C2semidirect
9. #14439C2semidirect
20. #2316C4semidirect
122. #389C2×C4semidirect
123. #389C2×C4semidirect
258. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23162
C2×#23132
C22×#3872
#146972
#144872
#144652
#144412
#144392
#142752
#141492
#141302
#139622