Finite Group Databasebeta

← #360102

Subgroups — #360102

Order 512.

2278 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3C2×#1875C212yesmaximal
4C2×#1804C212yesmaximal, complemented
5C2×#1797C211yesmaximal, characteristic, complemented
6#11207C212yesmaximal, complemented
7#11049C212yesmaximal, complemented
8#10850C212yesmaximal, complemented
9#10731C212yesmaximal, complemented
10C22×#121C2211yescharacteristic
11C22×C4×D4—21no—
12C22×C4×D4C2211yescharacteristic
13C22×#117C2212yes—
14C23×C42C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#102C2214yes—
17C22×C22⋊C8C2212yes—
18C4×#121—24no—
19C4×#121C418yescomplemented
20C2×#390C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#388—21no—
23C2×#386—21no—
24#2285—28no—
25C2×#384C2212yes—
26C2×#382C2212yes—
27C2×#380—21no—
28C2×#380—21no—
29C2×#380C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4—21no—
34#1875—28no—
35#1804—28no—
36#1797—24no—
37C2×#340C2214yes—
38C2×#340C2214yes—
39C2×#340C2214yes—
40C2×#336C2212yes—
41C23×D4—21no—
42C23×D4C2311yescharacteristic
43C24×C4—21no—
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8C2314yes—
48C2×#121—22no—
49C2×#121—22no—
50C2×#121—22no—
51C2×#121C2×C414yes—
52C2×#121D414yes—
53C2×#121Q814yes—
54C2×C4×D4—28no—
55C2×C4×D4—22no—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—48no—
59C2×C4×D4C2×C414yes—
60C2×#117—28no—
61C2×#117—24no—
62C2×#117—24no—
63C2×#117—44no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C42—24no—
78C22×C42—24no—
79C22×C42—22no—
80C22×C42—21no—
81C22×C42C2×C414yes—
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D414yes—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C2×C4⋊C8—24no—
88C2×C4⋊C8—24no—
89C2×C4⋊C8—24no—
90C2×#102—28no—
91C2×#102—28no—
92C2×#102—44no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96#390—22no—
97#390C2×C414yes—
98#389—22no—
99#389C2×C414yes—
100#388—28no—
101#386—28no—
102#384—28no—
103#382—24no—
104#382D418yescomplemented
105#380—28no—
106#380—28no—
107#380—22no—
108#380C2×C414yes—
109C4×C4⋊C4—28no—
110C4×C4⋊C4—28no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4C2×C414yes—
113C4×C22⋊C4—28no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—21no—
119#340—28no—
120#340—28no—
121#340—44no—
122#336—24no—
123C25—21no—
124C25C2×D411yescharacteristic
125C22×D4—24no—
126C22×D4—24no—
127C22×D4—24no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—44no—
136C22×D4—44no—
137C22×D4C22×C412yes—
138C22×D4C2×D412yes—
139C22×D4C2×Q812yes—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4C22⋊C414yes—
162C23×C4C22⋊C414yes—
163C23×C4C22×C412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×Q812yes—
172C23×C4#4112yes—
173C23×C4#4112yes—
174C23×C4C2411yescharacteristic
175C22×C8—24no—
176C22×C8—24no—
177C22×C8—24no—
178C22×C8—24no—
179C22×C8—24no—
180C22×C8—24no—
181C22×C8—24no—
182#121—28no—
183#121—28no—
184#121—28no—
185#121C4⋊C4116yescomplemented
186C4×D4—216no—
187C4×D4—432no—
188C4×D4—48no—
189#117—416no—
190#117—416no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C2×D418yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×Q812yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—216no—
241C2×C42—216no—
242C2×C42—28no—
243C2×C42—24no—
244C2×C42—44no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42C22⋊C414yes—
248C2×C42C22⋊C414yes—
249C2×C42C22×C412yes—
250C2×C42C22×C412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D414yes—
253C2×C42C2×D412yes—
254C2×C42#4114yes—
255C4⋊C8—48no—
256#102—416no—
257C22⋊C8—48no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—28no—
262#95—28no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24C2×C22⋊C412yes—
283C24#12112yes—
284C24C22⋊Q812yes—
285C24C22×D411yescharacteristic
286C2×D4—216no—
287C2×D4—22no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—48no—
297C2×D4—416no—
298C2×D4—48no—
299C2×D4—416no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4C2×C4⋊C414yes—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—28no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4—48no—
350C22×C4—48no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—44no—
355C22×C4—41no—
356C22×C4—44no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—42no—
367C22×C4—48no—
368C22×C4—88no—
369C22×C4—88no—
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C412yes—
372C22×C4C2×C22⋊C412yes—
373C22×C4C2×C22⋊C412yes—
374C22×C4C2×C22⋊C412yes—
375C22×C4C2×C4⋊C414yes—
376C22×C4#11714yes—
377C22×C4#12012yes—
378C22×C4#12012yes—
379C22×C4#12114yes—
380C22×C4#12112yes—
381C22×C4C22⋊Q814yes—
382C22×C4C22⋊Q812yes—
383C22×C4#12312yes—
384C22×C4#12312yes—
385C22×C4#13514yes—
386C22×C4AGL(1,ℤ/8)14yes—
387C22×C4C23×C411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D412yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×Q811yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4112yes—
395C22×C4H(5,𝔽2)12yes—
396C22×C42−1+412yes—
397C2×C8—44no—
398C2×C8—44no—
399C2×C8—44no—
400C2×C8—44no—
401C2×C8—44no—
402C2×C8—44no—
403C2×C8—44no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—28no—
410C4⋊C4—28no—
411C4⋊C4—416no—
412C4⋊C4—416no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—48no—
418C4⋊C4C2×C4⋊C414yes—
419C22⋊C4—216no—
420C22⋊C4—416no—
421C22⋊C4—48no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C42—24no—
426C42—24no—
427C42—28no—
428C42—28no—
429C42—416no—
430C42—48no—
431C42C4×D418yes—
432C42#12118yes—
433C23—24no—
434C23—24no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—24no—
445C23—24no—
446C23—22no—
447C23—24no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—44no—
457C23—44no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—44no—
468C23—44no—
469C23—42no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23#38014yes—
479C23C22×C22⋊C411yescharacteristic
480C23C2×#12011yescharacteristic
481C23C2×#12111yescharacteristic
482C23C2×C22⋊Q811yescharacteristic
483C23C2×#12311yescharacteristic
484C23#53511yescharacteristic
485C23#53611yescharacteristic
486C23C2×#13512yes—
487C23C2×AGL(1,ℤ/8)12yes—
488C23#57512yes—
489C23#57512yes—
490D4—28no—
491D4—432no—
492D4—44no—
493D4—44no—
494D4—44no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—416no—
523C2×C4—416no—
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—42no—
533C2×C4—48no—
534C2×C4—48no—
535C2×C4—48no—
536C2×C4—42no—
537C2×C4—816no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—84no—
541C2×C4—84no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4#38014yes—
547C2×C4#38614yes—
548C2×C4#38614yes—
549C2×C4#41614yes—
550C2×C4#41814yes—
551C2×C4C22×C4⋊C411yescharacteristic
552C2×C4C2×#11711yescharacteristic
553C2×C4C2×C4×D412yes—
554C2×C4#52012yes—
555C2×C4C2×#12111yescharacteristic
556C2×C4C2×#12112yes—
557C2×C4C2×C22⋊Q811yescharacteristic
558C2×C4#53712yes—
559C2×C4D4212yes—
560C2×C4#54612yes—
561C2×C4#54712yes—
562C2×C4#54812yes—
563C2×C4D4×Q812yes—
564C2×C4#55212yes—
565C2×C4#55312yes—
566C2×C4#55412yes—
567C8—84no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—44no—
572C22—44no—
573C22—42no—
574C22—42no—
575C22—44no—
576C22—42no—
577C22—42no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—84no—
583C22—84no—
584C22—82no—
585C22—82no—
586C22—84no—
587C22—82no—
588C22—82no—
589C22—84no—
590C22C2×#38011yescharacteristic
591C22C2×#38611yescharacteristic
592C22#231111yescharacteristic
593C22#240811yescharacteristic
594C22#240911yescharacteristic
595C22#248611yescharacteristic
596C22#248711yescharacteristic
597C22C2×#41611yescharacteristic
598C22C2×#41811yescharacteristic
599C22#289611yescharacteristic
600C22#289611yescharacteristic
601C22#300212yes—
602C22#300612yes—
603C22#303512yes—
604C22#303612yes—
605C22#307612yes—
606C22#307712yes—
607C22#309112yes—
608C22#309412yes—
609C22#320712yes—
610C22#321912yes—
611C22#322812yes—
612C22#324012yes—
613C4—24no—
614C4—24no—
615C4—24no—
616C4—24no—
617C4—24no—
618C4—44no—
619C4—82no—
620C4—88no—
621C4—82no—
622C4D4×C4⋊C412yes—
623C4#235212yes—
624C4#241612yes—
625C4#241712yes—
626C2—44no—
627C2—84no—
628C2#1387211yescharacteristic
629C2#2064511yescharacteristic
630C2#2064611yescharacteristic
631C2#2090811yescharacteristic
632C2#2091211yescharacteristic
633C2#2186311yescharacteristic
634C2#2186811yescharacteristic
635C2#2421412yes—
636C2#2430412yes—
637C2#2765012yes—
638C2#2768312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2285C2semidirect
4. C2×#1804C2semidirect
5. C2×#1797C2semidirect
6. #11207C2semidirect
7. #11049C2semidirect
8. #10850C2semidirect
9. #10731C2semidirect
19. C4×#121C4semidirect
104. #382D4semidirect
185. #121C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
C2×#22852
C2×#18752
C2×#18042
C2×#17972
#112072
#110492
#108502
#107312