Finite Group Databasebeta

← #759743

Subgroups — #759743

Order 512.

4838 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2537C211yesmaximal, characteristic, complemented
2C2×#2416C214yesmaximal, complemented
3C2×C4×#121C212yesmaximal, complemented
4C22×#384C211yesmaximal, characteristic, complemented
5#16514C214yesmaximal, complemented
6#15131C214yesmaximal, complemented
7#15130C214yesmaximal, complemented
8#14719C214yesmaximal, complemented
9#14151C211yesmaximal, characteristic, complemented
10#13906C212yesmaximal, complemented
11#13872C214yesmaximal, complemented
12C22×#121C2214yes—
13C22×#121C2214yes—
14C22×#121C2212yes—
15C22×C4×D4C2214yes—
16C22×C4×D4C2212yes—
17C23×C4⋊C4C2214yes—
18C23×C4⋊C4C2212yes—
19C23×C42C2211yescharacteristic
20#2537—28no—
21#2416—232no—
22C2×#396C2218yes—
23C2×#394C2218yes—
24C2×#394C2214yes—
25C2×#394C2214yes—
26C4×#121—216no—
27C2×#390C2212yes—
28C2×#390C2214yes—
29C2×#389C2212yes—
30C2×#389C2214yes—
31C2×#388C2214yes—
32C2×#386C2218yes—
33C2×#386C2214yes—
34C2×#384—28no—
35C2×#384C2214yes—
36C2×#384C2211yescharacteristic
37C2×#384C2212yes—
38C2×#384C2211yescharacteristic
39C2×#380C2214yes—
40C2×#380C2214yes—
41C2×#380C2212yes—
42C2×#380C2214yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C22⋊C4C2214yes—
45#1441C2214yes—
46#1441C2218yes—
47#1441C2214yes—
48#1439C2218yes—
49#1439C2218yes—
50#1439C2218yes—
51C4⋊#95C2214yes—
52C4⋊#95C2214yes—
53C22⋊#95C2214yes—
54C23×D4C2314yes—
55C23×D4C2312yes—
56C23×D4C2314yes—
57C24×C4C2314yes—
58C24×C4C2311yescharacteristic
59C24×C4C2314yes—
60C24×C4C2312yes—
61C2×#121—28no—
62C2×#121—28no—
63C2×#121—216no—
64C2×#121—232no—
65C2×#121—216no—
66C2×#121—48no—
67C2×#121—48no—
68C2×#121D4116yescomplemented
69C2×C4×D4—216no—
70C2×C4×D4—216no—
71C2×C4×D4—216no—
72C2×C4×D4—28no—
73C2×C4×D4—28no—
74C2×C4×D4—28no—
75C22×C4⋊C4—216no—
76C22×C4⋊C4—216no—
77C22×C4⋊C4—28no—
78C22×C4⋊C4—48no—
79C22×C4⋊C4—44no—
80C22×C4⋊C4—44no—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2318yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2318yes—
97C22×C22⋊C4C2318yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2318yes—
100C22×C42—24no—
101C22×C42—24no—
102C22×C42D418yescomplemented
103C22×C42C2314yes—
104C22×C42C2312yes—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107#396—232no—
108#394—216no—
109#394—232no—
110#394—48no—
111#390—216no—
112#390—28no—
113#389—216no—
114#389—28no—
115#388—216no—
116#386—232no—
117#386—216no—
118#384—216no—
119#384—28no—
120#384—24no—
121#384D418yescomplemented
122#380—28no—
123#380—216no—
124#380—216no—
125#380—48no—
126C4×C4⋊C4—28no—
127C4×C22⋊C4—216no—
128C2×#95C2314yes—
129C2×#95C2318yes—
130C2×#95C2318yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2318yes—
136C25C2414yes—
137C25C2411yescharacteristic
138C22×D4—28no—
139C22×D4—28no—
140C22×D4—28no—
141C22×D4—28no—
142C22×D4—28no—
143C22×D4—28no—
144C22×D4—216no—
145C22×D4—24no—
146C22×D4—24no—
147C22×D4—28no—
148C22×D4—24no—
149C22×D4—24no—
150C22×D4—24no—
151C22×D4—24no—
152C22×D4—48no—
153C22×D4—44no—
154C22×D4—44no—
155C22×D4C2×D4116yes—
156C22×D4C2×D418yes—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—216no—
168C23×C4—28no—
169C23×C4—28no—
170C23×C4—28no—
171C23×C4—28no—
172C23×C4—28no—
173C23×C4—28no—
174C23×C4—28no—
175C23×C4—28no—
176C23×C4—28no—
177C23×C4—44no—
178C23×C4—48no—
179C23×C4C2×D414yes—
180C23×C4C2×D418yes—
181C23×C4C2×D418yes—
182C23×C4#4118yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2414yes—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2412yes—
192#121—264no—
193#121—464no—
194#121—464no—
195C4×D4—464no—
196C4×D4—432no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—216no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—216no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—216no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—464no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—48no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—432no—
228C2×C4⋊C4C2×D4116yes—
229C2×C4⋊C4C2×D418yes—
230C2×C4⋊C4C2×D418yes—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—216no—
236C2×C22⋊C4—216no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—216no—
243C2×C22⋊C4—216no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—216no—
250C2×C22⋊C4—48no—
251C2×C22⋊C4—48no—
252C2×C22⋊C4—48no—
253C2×C22⋊C4—48no—
254C2×C22⋊C4C2×D4116yes—
255C2×C42—22no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—232no—
260C2×C42—28no—
261C2×C42—28no—
262C2×C42—28no—
263C2×C42C2×D418yes—
264C2×C42C2×D414yes—
265C2×C42C2×D414yes—
266C2×C42C2×D414yes—
267C2×C42#4114yes—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—216no—
273#95—216no—
274#95—48no—
275#95—44no—
276C24—21no—
277C24—21no—
278C24—22no—
279C24—21no—
280C24—22no—
281C24—21no—
282C24—24no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24C22×D418yes—
297C24C22×D412yes—
298C24C22×D414yes—
299C24C2×#4114yes—
300C24H(5,𝔽2)12yes—
301C242−1+412yes—
302C24C2511yescharacteristic
303C2×D4—232no—
304C2×D4—232no—
305C2×D4—232no—
306C2×D4—216no—
307C2×D4—216no—
308C2×D4—216no—
309C2×D4—432no—
310C2×D4—416no—
311C2×D4—416no—
312C2×D4—416no—
313C2×D4—416no—
314C2×D4—416no—
315C2×D4—48no—
316C2×D4—48no—
317C2×D4—48no—
318C2×D4—48no—
319C2×D4—88no—
320C2×D4—816no—
321C2×D4—88no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—28no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—28no—
340C22×C4—216no—
341C22×C4—28no—
342C22×C4—216no—
343C22×C4—216no—
344C22×C4—216no—
345C22×C4—216no—
346C22×C4—216no—
347C22×C4—24no—
348C22×C4—28no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—28no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—28no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—28no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—432no—
377C22×C4—416no—
378C22×C4—416no—
379C22×C4—416no—
380C22×C4—416no—
381C22×C4—416no—
382C22×C4—416no—
383C22×C4—416no—
384C22×C4—416no—
385C22×C4—416no—
386C22×C4—816no—
387C22×C4—88no—
388C22×C4#121132yescomplemented
389C22×C4C22×D418yes—
390C22×C4C22×D418yes—
391C22×C4C22×D412yes—
392C22×C4C22×D414yes—
393C22×C4C22×D412yes—
394C22×C4C22×D414yes—
395C22×C4C22×D412yes—
396C22×C4C22×D414yes—
397C22×C4C22×D414yes—
398C22×C4C22×D414yes—
399C22×C4C2×#4112yes—
400C22×C4C2×#4114yes—
401C22×C4C2×#4114yes—
402C22×C4C2×#4114yes—
403C22×C4H(5,𝔽2)12yes—
404C22×C4H(5,𝔽2)14yes—
405C22×C42−1+412yes—
406C22×C42−1+414yes—
407C4⋊C4—216no—
408C4⋊C4—232no—
409C4⋊C4—48no—
410C4⋊C4—432no—
411C4⋊C4—416no—
412C4⋊C4—416no—
413C4⋊C4—48no—
414C4⋊C4—416no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—416no—
418C22⋊C4—232no—
419C22⋊C4—432no—
420C22⋊C4—432no—
421C22⋊C4—416no—
422C22⋊C4—416no—
423C22⋊C4—416no—
424C22⋊C4—416no—
425C22⋊C4—816no—
426C42—28no—
427C42—216no—
428C42—416no—
429C42#121116yescomplemented
430C23—22no—
431C23—24no—
432C23—22no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—22no—
446C23—22no—
447C23—24no—
448C23—28no—
449C23—28no—
450C23—28no—
451C23—28no—
452C23—28no—
453C23—28no—
454C23—28no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23C2×#12118yes—
484C23#53714yes—
485C23D4214yes—
486C23#54718yes—
487C23#55514yes—
488C23C23×D411yescharacteristic
489C23C23×D412yes—
490C23C23×D411yescharacteristic
491C23C22×#4111yescharacteristic
492C23C22×#4112yes—
493C23C2×H(5,𝔽2)12yes—
494C23C2×H(5,𝔽2)11yescharacteristic
495C23C2×H(5,𝔽2)11yescharacteristic
496C23C2×2−1+411yescharacteristic
497C23C2×2−1+412yes—
498C23C2×2−1+411yescharacteristic
499D4—464no—
500D4—432no—
501D4—832no—
502C2×C4—232no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—264no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—216no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—48no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—48no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—48no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—832no—
554C2×C4—832no—
555C2×C4C2×#12114yes—
556C2×C4C2×#12118yes—
557C2×C4C2×#12118yes—
558C2×C4C2×#12118yes—
559C2×C4#53714yes—
560C2×C4#53718yes—
561C2×C4D4218yes—
562C2×C4#54718yes—
563C2×C4#54818yes—
564C2×C4#55318yes—
565C22—28no—
566C22—24no—
567C22—24no—
568C22—28no—
569C22—24no—
570C22—28no—
571C22—24no—
572C22—28no—
573C22—24no—
574C22—24no—
575C22—48no—
576C22—48no—
577C22—48no—
578C22—48no—
579C22—48no—
580C22—48no—
581C22—48no—
582C22—84no—
583C22—84no—
584C22—84no—
585C22—84no—
586C22—84no—
587C22—84no—
588C22—84no—
589C22—84no—
590C22#253718yescomplemented
591C22C22×#12112yes—
592C22C22×#12111yescharacteristic
593C22C2×#53711yescharacteristic
594C22C2×#53712yes—
595C22C2×#53711yescharacteristic
596C22C2×D4211yescharacteristic
597C22C2×D4212yes—
598C22C2×#54712yes—
599C22C2×#54712yes—
600C22C2×#54812yes—
601C22C2×#55312yes—
602C22C2×#55511yescharacteristic
603C22#348411yescharacteristic
604C22#348611yescharacteristic
605C22#349211yescharacteristic
606C22#349211yescharacteristic
607C22#349812yes—
608C22#349912yes—
609C22#351112yes—
610C22#351112yes—
611C22#354311yescharacteristic
612C22#354411yescharacteristic
613C22#355212yes—
614C4—28no—
615C4—24no—
616C4—24no—
617C4—48no—
618C4—48no—
619C4—84no—
620C4—84no—
621C4—84no—
622C4#2416116yescomplemented
623C2—28no—
624C2—88no—
625C2C2×#241612yes—
626C2C2×#253711yescharacteristic
627C2#3522311yescharacteristic
628C2#3527312yes—
629C2#3762011yescharacteristic
630C2#5912512yes—
631C2#5912611yescharacteristic
632C2#5913511yescharacteristic
633C2#5930311yescharacteristic
634C2#5930412yes—
635C2#5931311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2537C2semidirect
2. C2×#2416C2semidirect
3. C2×C4×#121C2semidirect
4. C22×#384C2semidirect
5. #16514C2semidirect
6. #15131C2semidirect
7. #15130C2semidirect
8. #14719C2semidirect
9. #14151C2semidirect
10. #13906C2semidirect
11. #13872C2semidirect
68. C2×#121D4semidirect
102. C22×C42D4semidirect
121. #384D4semidirect
388. C22×C4#121semidirect
429. C42#121semidirect
590. C22#2537semidirect
622. C4#2416semidirect

Maximal subgroups

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

subgroupindex
C2×#25372
C2×#24162
C2×C4×#1212
C22×#3842
#165142
#151312
#151302
#147192
#141512
#139062
#138722