Finite Group Databasebeta

← #398323

Subgroups — #398323

Order 512.

1944 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×#2857C211yesmaximal, characteristic
2C2×#2036C212yesmaximal
3C2×#2016C211yesmaximal, characteristic, complemented
4C2×#2001C212yesmaximal
5C2×#1969C212yesmaximal, complemented
6C2×C4×C22⋊C8C211yesmaximal, characteristic
7#14702C211yesmaximal, characteristic
8#11324C211yesmaximal, characteristic, complemented
9#11050C211yesmaximal, characteristic, complemented
10#11010C211yesmaximal, characteristic, complemented
11#10926C212yesmaximal, complemented
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C22×C22⋊Q8C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C22×#107C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×C22⋊C8C2212yes—
23C22×C4×C8C2212yes—
24#2857—28no—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#391C2212yes—
28C2×#389—22no—
29C2×#384C2211yescharacteristic
30C2×#380—22no—
31C2×#380C2211yescharacteristic
32#2036—28no—
33#2016—24no—
34#2001—28no—
35#1969—28no—
36C4×C22⋊C8—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41#1436—22no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C22⋊#95—22no—
45C23×Q8C2312yes—
46C23×Q8C2311yescharacteristic
47C24×C4D412yes—
48C24×C4C2311yescharacteristic
49C22×Q16—24no—
50C22×Q16—24no—
51C22×Q16—24no—
52C22×Q16—24no—
53C22×Q16—24no—
54C22×Q16—42no—
55C22×Q16—42no—
56C23×C8C2312yes—
57C23×C8C2312yes—
58C2×C4⋊Q8—24no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—42no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—24no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—42no—
70C2×C22⋊Q8—42no—
71C2×C22⋊Q8—42no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—21no—
83C22×C42—24no—
84C22×C42—22no—
85C22×C42—22no—
86C22×C42—42no—
87C22×C42D412yes—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×#107—24no—
91C2×#107—24no—
92C2×#107—42no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—42no—
103C2×#103—42no—
104C2×#103—42no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—24no—
107C2×C22⋊C8D418yes—
108C2×C4×C8—24no—
109C2×C4×C8—24no—
110C2×C4×C8—42no—
111#391—24no—
112#391—24no—
113#391—22no—
114#391—22no—
115#391D418yes—
116#389—216no—
117#384—24no—
118#380—216no—
119#380—24no—
120C2×#95—24no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—21no—
126#340—24no—
127#340—24no—
128#336—22no—
129#336D414yes—
130C25C2×D411yescharacteristic
131C22×Q8—22no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8C2×D414yes—
154C22×Q8#4112yes—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4Q1618yescomplemented
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D414yes—
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2411yescharacteristic
182C2×Q16—216no—
183C2×Q16—48no—
184C2×Q16—48no—
185C2×Q16—48no—
186C2×Q16—48no—
187C2×Q16—48no—
188C2×Q16—48no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8C2×D414yes—
200C22×C8C2×D414yes—
201C22×C8C2×D414yes—
202C22×C8#4114yes—
203C4⋊Q8—48no—
204C4⋊Q8—48no—
205C22⋊Q8—48no—
206C22⋊Q8—48no—
207C22⋊Q8—48no—
208C22⋊Q8—48no—
209C22⋊Q8—48no—
210C22⋊Q8—48no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4C2×D414yes—
240C2×C4⋊C4#4112yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C42—22no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—48no—
257C2×C42—48no—
258C2×C42—42no—
259C2×C42C2×D414yes—
260C2×C42C2×D412yes—
261C2×C42#4112yes—
262#107—48no—
263#103—48no—
264#103—48no—
265#103—48no—
266#103—48no—
267C22⋊C8—216no—
268C4×C8—48no—
269#95—216no—
270#95—44no—
271#95—44no—
272#95—42no—
273#95—44no—
274#95—44no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24#12712yes—
285C24C2×Q1614yes—
286C24C22×D411yescharacteristic
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—216no—
352C22×C4—216no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4#12014yes—
403C22×C4#12114yes—
404C22×C4#12712yes—
405C22×C4C2×Q1614yes—
406C22×C4C2×Q1614yes—
407C22×C4#13514yes—
408C22×C4#13714yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4H(5,𝔽2)11yescharacteristic
417C22×C4H(5,𝔽2)12yes—
418C22×C4H(5,𝔽2)11yescharacteristic
419Q16—432no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8#12118yes—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—416no—
438C4⋊C4—416no—
439C4⋊C4—48no—
440C4⋊C4—416no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C22⋊C4—416no—
449C22⋊C4—48no—
450C22⋊C4—84no—
451C42—24no—
452C42—24no—
453C42—24no—
454C42—24no—
455C42—48no—
456C23—24no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—24no—
463C23—22no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—22no—
468C23—42no—
469C23—41no—
470C23—42no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23#49414yes—
479C23C2×#12011yescharacteristic
480C23C2×#12111yescharacteristic
481C23#52511yescharacteristic
482C23C2×#12711yescharacteristic
483C23#53211yescharacteristic
484C23#53411yescharacteristic
485C23#53511yescharacteristic
486C23C22×Q1612yes—
487C23C2×#13512yes—
488C23C2×#13712yes—
489C23#57512yes—
490Q8—44no—
491Q8—44no—
492Q8—44no—
493Q8—44no—
494Q8—44no—
495Q8—44no—
496Q8—82no—
497Q8—82no—
498Q8—82no—
499Q8—82no—
500Q8—82no—
501Q8—82no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—432no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—48no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—81no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—82no—
542C2×C4—81no—
543C2×C4—82no—
544C2×C4—81no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—81no—
548C2×C4—84no—
549C2×C4—82no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—82no—
556C2×C4—81no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4#45118yes—
561C2×C4#46718yes—
562C2×C4C2×#12112yes—
563C2×C4#52512yes—
564C2×C4#53212yes—
565C2×C4#53212yes—
566C2×C4#53812yes—
567C2×C4#54012yes—
568C2×C4D4212yes—
569C2×C4D4212yes—
570C2×C4D4212yes—
571C2×C4#54611yescharacteristic
572C2×C4#54612yes—
573C2×C4#54612yes—
574C2×C4#54612yes—
575C2×C4#55011yescharacteristic
576C2×C4#55311yescharacteristic
577C2×C4#55511yescharacteristic
578C8—28no—
579C8—44no—
580C22—24no—
581C22—24no—
582C22—24no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22#238811yescharacteristic
592C22#241011yescharacteristic
593C22#241411yescharacteristic
594C22#243311yescharacteristic
595C22#253911yescharacteristic
596C22#266111yescharacteristic
597C22#285611yescharacteristic
598C22C2×#45112yes—
599C22#300212yes—
600C22C2×#46712yes—
601C22#305612yes—
602C22#310412yes—
603C22#310712yes—
604C22C2×#49411yescharacteristic
605C22#314711yescharacteristic
606C22#315211yescharacteristic
607C22#315311yescharacteristic
608C22#317012yes—
609C22#317212yes—
610C22#318912yes—
611C22#319012yes—
612C22#320912yes—
613C22#321212yes—
614C4—28no—
615C4—28no—
616C4—42no—
617C4—84no—
618C4—82no—
619C4—84no—
620C4—82no—
621C4—161no—
622C4—161no—
623C4#242412yes—
624C4#242812yes—
625C4#246412yes—
626C4#255512yes—
627C2—44no—
628C2#1476211yescharacteristic
629C2#2357112yes—
630C2#2438012yes—
631C2#2465212yes—
632C2#2572211yescharacteristic
633C2#2572311yescharacteristic
634C2#2830411yescharacteristic
635C2#2830811yescharacteristic
636C2#3031812yes—
637C2#3317211yescharacteristic
638C2#3318611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
3. C2×#2016C2semidirect
5. C2×#1969C2semidirect
8. #11324C2semidirect
9. #11050C2semidirect
10. #11010C2semidirect
11. #10926C2semidirect
176. C23×C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#20362
C2×#20162
C2×#20012
C2×#19692
C2×C4×C22⋊C82
#147022
#113242
#110502
#110102
#109262