Finite Group Databasebeta

← #403377

Subgroups — #403377

Order 512.

2540 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic
2C2×#2059C212yesmaximal, complemented
3C2×#2057C212yesmaximal, complemented
4C2×#1854C211yesmaximal, characteristic
5C2×#1852C211yesmaximal, characteristic
6#14264C211yesmaximal, characteristic
7#11385C212yesmaximal, complemented
8#11384C212yesmaximal, complemented
9#10731C211yesmaximal, characteristic
10#10730C211yesmaximal, characteristic
11#10713C211yesmaximal, characteristic
12C2×#544—28no—
13C2×#544—28no—
14C2×#544—24no—
15C2×#544—24no—
16C22×C4⋊Q8—21no—
17C22×C4⋊Q8C2212yes—
18C22×#125—21no—
19C22×#125C2212yes—
20C22×C22⋊Q8—22no—
21C22×#117C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C22×#107C2212yes—
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29C2×#389C2212yes—
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32#2059—28no—
33#2057—28no—
34#1854—24no—
35#1852—24no—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1444—21no—
46#1442—21no—
47#1439—22no—
48C4⋊#95C2212yes—
49C23×Q8—22no—
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55#544—232no—
56#544—232no—
57#544—264no—
58C2×C4⋊Q8—216no—
59C2×C4⋊Q8—28no—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8—24no—
62C2×C4⋊Q8—24no—
63C2×C4⋊Q8—24no—
64C2×C4⋊Q8—24no—
65C2×C4⋊Q8D418yes—
66C2×#125—216no—
67C2×#125—28no—
68C2×#125—24no—
69C2×#125—24no—
70C2×#125—24no—
71C2×#125—24no—
72C2×#125—24no—
73C2×#125D418yes—
74C2×C22⋊Q8—216no—
75C2×C22⋊Q8—216no—
76C2×C22⋊Q8—216no—
77C2×#117—22no—
78C2×#117—22no—
79C2×#117—22no—
80C2×#117—24no—
81C2×#117—24no—
82C2×#117—24no—
83C2×#117—24no—
84C2×#117D414yes—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4D411yescharacteristic
94C22×C4⋊C4D411yescharacteristic
95C22×C4⋊C4D414yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4—21no—
103C22×C22⋊C4—21no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4D412yes—
106C22×C42—21no—
107C22×C42C2312yes—
108C2×#107—24no—
109C2×#107—24no—
110C2×#107—24no—
111C2×#107—24no—
112C2×#107—22no—
113C2×#106—24no—
114C2×#106—22no—
115C2×C22⋊C8—22no—
116C2×C22⋊C8—22no—
117C2×C22⋊C8—22no—
118C2×C22⋊C8—22no—
119C2×C22⋊C8Q814yescomplemented
120C2×C22⋊C8Q814yescomplemented
121#389—24no—
122#389—24no—
123#380—22no—
124#380—22no—
125#380—22no—
126#380Q814yescomplemented
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—24no—
136#340—24no—
137#340—22no—
138#340—22no—
139#336—24no—
140#336—24no—
141C25C2×D411yescharacteristic
142C22×Q8—28no—
143C22×Q8—28no—
144C22×Q8—28no—
145C22×Q8—82no—
146C22×Q8—82no—
147C22×Q8—82no—
148C22×Q8—82no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—28no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—81no—
162C23×C4—82no—
163C23×C4—81no—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×Q812yes—
170C23×C4C2×Q812yes—
171C23×C4C2×Q812yes—
172C23×C4#4112yes—
173C23×C4C2411yescharacteristic
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8C2×Q812yes—
189C22×C8C2×Q814yes—
190C22×C8C2×Q812yes—
191C4⋊Q8—264no—
192C4⋊Q8—216no—
193C4⋊Q8—216no—
194C4⋊Q8—216no—
195C4⋊Q8—216no—
196#125—264no—
197#125—216no—
198#125—216no—
199#125—216no—
200#125—216no—
201C22⋊Q8—2128no—
202#117—216no—
203#117—28no—
204#117—216no—
205#117—28no—
206#117—28no—
207#117—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—216no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—82no—
248C2×C4⋊C4—82no—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×Q814yes—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—81no—
265C2×C22⋊C4—81no—
266C2×C22⋊C4—82no—
267C2×C42—28no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42C2×D414yes—
274C2×C42C2×D414yes—
275#107—48no—
276#107—44no—
277#106—44no—
278C22⋊C8—28no—
279C22⋊C8—28no—
280#95—24no—
281#95—42no—
282#95—44no—
283#95—82no—
284#95—82no—
285C24—22no—
286C24—22no—
287C24—21no—
288C24—22no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—22no—
293C24#12012yes—
294C24C22⋊Q812yes—
295C24C4⋊Q812yes—
296C24C22×D411yescharacteristic
297C2×Q8—232no—
298C2×Q8—88no—
299C2×Q8—88no—
300C2×Q8—88no—
301C2×Q8—88no—
302C2×Q8—88no—
303C2×Q8—88no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
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—216no—
340C22×C4—24no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4—88no—
370C22×C4—88no—
371C22×C4#12014yes—
372C22×C4#12014yes—
373C22×C4#12012yes—
374C22×C4C22⋊Q814yes—
375C22×C4C22⋊Q814yes—
376C22×C4C22⋊Q812yes—
377C22×C4C4⋊Q814yes—
378C22×C4C4⋊Q814yes—
379C22×C4C4⋊Q812yes—
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D412yes—
383C22×C4C22×D412yes—
384C22×C4C22×Q811yescharacteristic
385C22×C4C22×Q811yescharacteristic
386C22×C4C22×Q811yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C42−1+412yes—
389C22×C42−1+412yes—
390C2×C8—22no—
391C2×C8—24no—
392C2×C8—22no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—42no—
400C2×C8—41no—
401C2×C8—42no—
402C2×C8—41no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C4⋊C4—216no—
408C4⋊C4—28no—
409C4⋊C4—28no—
410C4⋊C4—232no—
411C4⋊C4—232no—
412C4⋊C4—232no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—88no—
431C4⋊C4#12014yes—
432C4⋊C4#12018yes—
433C4⋊C4#12014yes—
434C22⋊C4—216no—
435C22⋊C4—216no—
436C22⋊C4—84no—
437C22⋊C4—84no—
438C42—216no—
439C42—24no—
440C42—24no—
441C42—24no—
442C42—24no—
443C23—22no—
444C23—22no—
445C23—24no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—24no—
450C23—22no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23#39514yes—
466C23C2×#12011yescharacteristic
467C23C2×#12012yes—
468C23C2×C22⋊Q811yescharacteristic
469C23C2×C4⋊Q811yescharacteristic
470C23#53612yes—
471C23#57611yescharacteristic
472C23#57612yes—
473C23#57711yescharacteristic
474C23#57711yescharacteristic
475C23#57811yescharacteristic
476C23#57812yes—
477Q8—832no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—28no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—28no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—816no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—82no—
534C2×C4—81no—
535C2×C4—82no—
536C2×C4—81no—
537C2×C4#39514yes—
538C2×C4C2×#12011yescharacteristic
539C2×C4C2×#12011yescharacteristic
540C2×C4C2×#12012yes—
541C2×C4C2×C22⋊Q811yescharacteristic
542C2×C4C2×C22⋊Q811yescharacteristic
543C2×C4C2×C4⋊Q811yescharacteristic
544C2×C4C2×C4⋊Q811yescharacteristic
545C2×C4#53611yescharacteristic
546C2×C4#53611yescharacteristic
547C2×C4#53612yes—
548C2×C4#54812yes—
549C2×C4#54812yes—
550C2×C4D4×Q812yes—
551C2×C4D4×Q812yes—
552C2×C4D4×Q812yes—
553C2×C4D4×Q812yes—
554C2×C4D4×Q812yes—
555C2×C4D4×Q812yes—
556C8—42no—
557C8—42no—
558C8—44no—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22C2×#39511yescharacteristic
571C22#240912yes—
572C22#243412yes—
573C22#261411yescharacteristic
574C22#284611yescharacteristic
575C22#300811yescharacteristic
576C22#300812yes—
577C22#301111yescharacteristic
578C22#301112yes—
579C22#301211yescharacteristic
580C22#301212yes—
581C22#301311yescharacteristic
582C22#301312yes—
583C22#308111yescharacteristic
584C22#308211yescharacteristic
585C22#308312yes—
586C22#316311yescharacteristic
587C22#316412yes—
588C22#316411yescharacteristic
589C22#324112yes—
590C22#324212yes—
591C22#324312yes—
592C22#324312yes—
593C4—24no—
594C4—44no—
595C4—48no—
596C4—44no—
597C4—44no—
598C4—44no—
599C4—84no—
600C4—161no—
601C4—161no—
602C4#240911yescharacteristic
603C4#240911yescharacteristic
604C4#240912yes—
605C4#243411yescharacteristic
606C4#243411yescharacteristic
607C4#243412yes—
608C2—44no—
609C2#1487411yescharacteristic
610C2#2355511yescharacteristic
611C2#2355711yescharacteristic
612C2#2430812yes—
613C2#2431112yes—
614C2#2582112yes—
615C2#2582212yes—
616C2#2964611yescharacteristic
617C2#2964911yescharacteristic
618C2#3288411yescharacteristic
619C2#3288711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2059C2semidirect
3. C2×#2057C2semidirect
7. #11385C2semidirect
8. #11384C2semidirect
119. C2×C22⋊C8Q8semidirect
120. C2×C22⋊C8Q8semidirect
126. #380Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#20592
C2×#20572
C2×#18542
C2×#18522
#142642
#113852
#113842
#107312
#107302
#107132