Finite Group Databasebeta

← #360106

Subgroups — #360106

Order 512.

2386 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2C22×#382C211yesmaximal, characteristic, complemented
3C2×#1874C212yesmaximal
4C2×#1804C212yesmaximal, complemented
5C2×#1797C211yesmaximal, characteristic, complemented
6#11208C212yesmaximal, complemented
7#11051C212yesmaximal, complemented
8#10850C212yesmaximal, complemented
9#10730C212yesmaximal, complemented
10C22×#121C2211yescharacteristic
11C22×C4×D4—21no—
12C22×C4×D4C2211yescharacteristic
13C23×C4⋊C4C2212yes—
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—
24C2×#382—21no—
25C2×#382—21no—
26C2×#382—28no—
27C2×#382C2212yes—
28C2×#382C2212yes—
29C2×#380C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4—21no—
32C22×#95—21no—
33C22×#95—21no—
34#1874—28no—
35#1804—28no—
36#1797—24no—
37C2×#340C2214yes—
38C2×#340C2214yes—
39C2×#340C2214yes—
40C2×#336C2212yes—
41C23×D4—21no—
42C23×D4C2311yescharacteristic
43C24×C4—22no—
44C24×C4—21no—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8C2314yes—
49C2×#121—22no—
50C2×#121—22no—
51C2×#121—22no—
52C2×#121C2×C414yes—
53C2×#121D414yes—
54C2×#121Q814yes—
55C2×C4×D4—28no—
56C2×C4×D4—22no—
57C2×C4×D4—24no—
58C2×C4×D4—24no—
59C2×C4×D4—48no—
60C2×C4×D4C2×C414yescomplemented
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—28no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—44no—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C42—24no—
77C22×C42—24no—
78C22×C42—21no—
79C22×C42C2×C414yes—
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D414yes—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×C4⋊C8—24no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—24no—
88C2×#102—28no—
89C2×#102—28no—
90C2×#102—44no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94#390—22no—
95#390C2×C414yescomplemented
96#389—22no—
97#389C2×C414yescomplemented
98#388—28no—
99#386—28no—
100#382—28no—
101#382—28no—
102#382—28no—
103#382—24no—
104#382D418yescomplemented
105#380—22no—
106#380C2×C414yes—
107C4×C4⋊C4—22no—
108C4×C4⋊C4C2×C414yescomplemented
109C4×C22⋊C4—28no—
110C2×#95—28no—
111C2×#95—28no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—21no—
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—24no—
145C23×C4—24no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—28no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4C22⋊C414yes—
169C23×C4C22⋊C414yes—
170C23×C4C22×C412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2×Q812yes—
179C23×C4#4112yes—
180C23×C4#4112yes—
181C23×C4C2411yescharacteristic
182C22×C8—24no—
183C22×C8—24no—
184C22×C8—24no—
185C22×C8—24no—
186C22×C8—24no—
187C22×C8—24no—
188C22×C8—24no—
189#121—28no—
190#121—28no—
191#121—28no—
192#121C4⋊C4116yes—
193C4×D4—216no—
194C4×D4—432no—
195C4×D4—48no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—416no—
213C2×C4⋊C4—416no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C2×D418yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×Q812yes—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—216no—
242C2×C42—216no—
243C2×C42—24no—
244C2×C42—44no—
245C2×C42C22⋊C414yes—
246C2×C42C22⋊C414yes—
247C2×C42C22×C412yes—
248C2×C42C22×C412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D414yes—
251C2×C42C2×D412yes—
252C2×C42#4114yes—
253C4⋊C8—48no—
254#102—416no—
255C22⋊C8—48no—
256#95—28no—
257#95—28no—
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—41no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24C2×C22⋊C412yes—
281C24#12112yes—
282C24C22⋊Q812yes—
283C24#13514yes—
284C24AGL(1,ℤ/8)14yes—
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—48no—
298C2×D4—416no—
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—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—28no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4—48no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—416no—
356C22×C4—416no—
357C22×C4—416no—
358C22×C4—41no—
359C22×C4—44no—
360C22×C4—41no—
361C22×C4—44no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—44no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—48no—
373C22×C4—88no—
374C22×C4—88no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4—88no—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C412yes—
384C22×C4C2×C4⋊C414yes—
385C22×C4#11714yes—
386C22×C4#12012yes—
387C22×C4#12012yes—
388C22×C4#12114yes—
389C22×C4#12112yes—
390C22×C4C22⋊Q814yes—
391C22×C4C22⋊Q812yes—
392C22×C4#12312yes—
393C22×C4#12312yes—
394C22×C4C23×C411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×Q811yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4112yes—
402C22×C4H(5,𝔽2)12yes—
403C22×C42−1+412yes—
404C2×C8—44no—
405C2×C8—44no—
406C2×C8—44no—
407C2×C8—44no—
408C2×C8—44no—
409C2×C8—44no—
410C2×C8—44no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—28no—
417C4⋊C4—28no—
418C4⋊C4—416no—
419C4⋊C4—48no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4C2×C4⋊C414yes—
425C22⋊C4—216no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C42—24no—
431C42—24no—
432C42—28no—
433C42—28no—
434C42—48no—
435C42C4×D418yes—
436C42#12118yes—
437C23—22no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—22no—
450C23—22no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—22no—
455C23—24no—
456C23—44no—
457C23—42no—
458C23—44no—
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—
479C23#41614yes—
480C23#41814yes—
481C23#46318yes—
482C23#47818yes—
483C23C22×C22⋊C411yescharacteristic
484C23C2×#12011yescharacteristic
485C23C2×#12111yescharacteristic
486C23C2×C22⋊Q811yescharacteristic
487C23C2×#12311yescharacteristic
488C23#53511yescharacteristic
489C23#53611yescharacteristic
490C23C2×#13512yes—
491C23C2×AGL(1,ℤ/8)12yes—
492C23#57512yes—
493C23#57512yes—
494D4—28no—
495D4—432no—
496D4—44no—
497D4—44no—
498D4—44no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
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—22no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—416no—
525C2×C4—416no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—48no—
540C2×C4—48no—
541C2×C4—48no—
542C2×C4—42no—
543C2×C4—816no—
544C2×C4—832no—
545C2×C4—82no—
546C2×C4—84no—
547C2×C4—82no—
548C2×C4—84no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4#38014yes—
554C2×C4#38614yes—
555C2×C4#38614yes—
556C2×C4C22×C4⋊C411yescharacteristic
557C2×C4C2×#11711yescharacteristic
558C2×C4C2×C4×D412yes—
559C2×C4#52012yes—
560C2×C4C2×#12111yescharacteristic
561C2×C4C2×#12112yes—
562C2×C4C2×C22⋊Q811yescharacteristic
563C2×C4#53712yes—
564C2×C4D4212yes—
565C2×C4#54612yes—
566C2×C4#54712yes—
567C2×C4#54812yes—
568C2×C4D4×Q812yes—
569C2×C4#55212yes—
570C2×C4#55312yes—
571C2×C4#55412yes—
572C8—84no—
573C22—28no—
574C22—28no—
575C22—24no—
576C22—24no—
577C22—24no—
578C22—28no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—28no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—84no—
587C22—84no—
588C22—82no—
589C22—82no—
590C22—84no—
591C22—82no—
592C22—82no—
593C22—84no—
594C22#180418yescomplemented
595C22C2×#38011yescharacteristic
596C22C2×#38611yescharacteristic
597C22#231111yescharacteristic
598C22#240811yescharacteristic
599C22#240911yescharacteristic
600C22#248611yescharacteristic
601C22#248711yescharacteristic
602C22C2×#41611yescharacteristic
603C22C2×#41811yescharacteristic
604C22#289611yescharacteristic
605C22#289611yescharacteristic
606C22#300212yes—
607C22#300612yes—
608C22C2×#46312yes—
609C22#304012yes—
610C22C2×#47812yes—
611C22#308012yes—
612C22#309112yes—
613C22#309412yes—
614C22#321112yes—
615C22#321312yes—
616C22#323212yes—
617C22#323412yes—
618C4—24no—
619C4—24no—
620C4—24no—
621C4—44no—
622C4—42no—
623C4—42no—
624C4—82no—
625C4—88no—
626C4—82no—
627C4D4×C4⋊C412yes—
628C4#235212yes—
629C4#241612yes—
630C4#241712yes—
631C2—28no—
632C2—84no—
633C2#1387211yescharacteristic
634C2C2×#180411yescharacteristic
635C2#2064911yescharacteristic
636C2#2090811yescharacteristic
637C2#2091211yescharacteristic
638C2#2186111yescharacteristic
639C2#2186511yescharacteristic
640C2#2421512yes—
641C2#2430512yes—
642C2#2765112yes—
643C2#2768412yes—
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. C22×#382C2semidirect
4. C2×#1804C2semidirect
5. C2×#1797C2semidirect
6. #11208C2semidirect
7. #11051C2semidirect
8. #10850C2semidirect
9. #10730C2semidirect
19. C4×#121C4semidirect
60. C2×C4×D4C2×C4semidirect
95. #390C2×C4semidirect
97. #389C2×C4semidirect
104. #382D4semidirect
108. C4×C4⋊C4C2×C4semidirect
594. C22#1804semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
C22×#3822
C2×#18742
C2×#18042
C2×#17972
#112082
#110512
#108502
#107302