Finite Group Databasebeta

← #403346

Subgroups — #403346

Order 512.

2526 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2C2×#2026C212yesmaximal
3C2×#2024C211yesmaximal, characteristic
4C2×#1878C212yesmaximal
5C2×#1878C211yesmaximal, characteristic
6#15201C211yesmaximal, characteristic
7#11385C212yesmaximal
8#11384C211yesmaximal, characteristic
9#11380C211yesmaximal, characteristic
10#10789C211yesmaximal, characteristic
11#10789C212yesmaximal
12C23×Q16C2211yescharacteristic
13C2×#557—24no—
14C2×#557—28no—
15C2×#557—24no—
16C2×#557—28no—
17C22×C4⋊Q8—21no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125—22no—
20C22×#125C2212yes—
21C22×C4×Q8—21no—
22C22×C4×Q8C2211yescharacteristic
23C22×C4×Q8C2212yes—
24C22×C4×Q8C2211yescharacteristic
25C22×#103C2212yes—
26C22×#103C2211yescharacteristic
27C22×#103C2212yes—
28C22×#103C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31#2026—28no—
32#2024—24no—
33#1878—28no—
34#1878—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44#1446—21no—
45#1442—21no—
46#1442—22no—
47C4⋊#95C2211yescharacteristic
48C4⋊#95C2212yes—
49C23×Q8—21no—
50C23×Q8D411yescharacteristic
51C23×Q8C2311yescharacteristic
52C23×Q8C2311yescharacteristic
53C22×Q16—22no—
54C22×Q16—24no—
55C22×Q16—24no—
56C22×Q16—22no—
57C22×Q16Q814yescomplemented
58C23×C8C2312yes—
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61#557—264no—
62#557—232no—
63#557—232no—
64C2×C4⋊Q8—216no—
65C2×C4⋊Q8—28no—
66C2×C4⋊Q8—24no—
67C2×C4⋊Q8—24no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8D414yes—
70C2×#125—216no—
71C2×#125—216no—
72C2×#125—216no—
73C2×#125—24no—
74C2×#125—24no—
75C2×#125—24no—
76C2×#125—24no—
77C2×#125—24no—
78C2×#125D418yes—
79C2×C4×Q8—28no—
80C2×C4×Q8—216no—
81C2×C4×Q8—24no—
82C2×C4×Q8—24no—
83C2×C4×Q8—24no—
84C2×C4×Q8—24no—
85C2×C4×Q8—24no—
86C2×C4×Q8—24no—
87C2×C4×Q8—24no—
88C2×C4×Q8—22no—
89C2×C4×Q8—22no—
90C2×C4×Q8—22no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8D418yes—
94C2×C4×Q8D414yes—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4D411yescharacteristic
101C22×C4⋊C4D412yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C42—22no—
107C22×C42—21no—
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110C2×#103—24no—
111C2×#103—22no—
112C2×#103—24no—
113C2×#103—24no—
114C2×#103—24no—
115C2×#103—22no—
116C2×#103—24no—
117C2×#103—24no—
118C2×#103—24no—
119C2×#103—24no—
120#391—24no—
121#391—22no—
122#391—22no—
123#391—24no—
124C2×#95—22no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—22no—
132#340—22no—
133#340—24no—
134#340—24no—
135#340—24no—
136#340—24no—
137#336—24no—
138#336—22no—
139#336—22no—
140#336—24no—
141C22×Q8—28no—
142C22×Q8—24no—
143C22×Q8—22no—
144C22×Q8—24no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—21no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—22no—
155C22×Q8—21no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—21no—
159C22×Q8—22no—
160C22×Q8—21no—
161C22×Q8—42no—
162C22×Q8—44no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8C2×D412yes—
166C22×Q8C2×D414yes—
167C22×Q8C2×Q812yes—
168C22×Q8C2×Q812yes—
169C22×Q8#4114yes—
170C22×Q8#4112yes—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4—21no—
175C23×C4—22no—
176C23×C4—21no—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2411yescharacteristic
182C2×Q16—28no—
183C2×Q16—216no—
184C2×Q16—44no—
185C2×Q16—44no—
186C2×Q16—48no—
187C22×C8—21no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—21no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—21no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—21no—
201C22×C8—22no—
202C22×C8—21no—
203C22×C8C2×Q812yes—
204C4⋊Q8—264no—
205C4⋊Q8—28no—
206C4⋊Q8—28no—
207C4⋊Q8—216no—
208#125—2128no—
209#125—216no—
210#125—216no—
211#125—216no—
212#125—216no—
213C4×Q8—264no—
214C4×Q8—216no—
215C4×Q8—216no—
216C4×Q8—216no—
217C4×Q8—28no—
218C4×Q8—28no—
219C4×Q8—216no—
220C4×Q8—216no—
221C4×Q8—28no—
222C4×Q8—216no—
223C4×Q8—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D414yes—
262C2×C42—28no—
263C2×C42—28no—
264C2×C42—28no—
265C2×C42—28no—
266C2×C42—24no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42C2×D414yes—
274C2×C42C2×D412yes—
275C2×C42C2×D414yes—
276C2×C42C2×D412yes—
277#103—48no—
278#103—44no—
279#103—44no—
280#103—48no—
281#95—44no—
282#95—44no—
283#95—42no—
284#95—44no—
285#95—42no—
286#95—42no—
287#95—81no—
288#95—82no—
289C24C22×D411yescharacteristic
290C2×Q8—28no—
291C2×Q8—216no—
292C2×Q8—24no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—24no—
296C2×Q8—22no—
297C2×Q8—24no—
298C2×Q8—22no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—22no—
304C2×Q8—24no—
305C2×Q8—22no—
306C2×Q8—24no—
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—48no—
310C2×Q8—48no—
311C2×Q8—416no—
312C2×Q8—44no—
313C2×Q8—48no—
314C2×Q8—416no—
315C2×Q8—41no—
316C2×Q8—42no—
317C2×Q8—41no—
318C2×Q8—42no—
319C2×Q8—44no—
320C2×Q8—48no—
321C2×Q8—42no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—42no—
326C2×Q8—41no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8#12014yes—
332C2×Q8C22⋊Q814yes—
333C2×Q8C22⋊Q818yes—
334C2×Q8#12414yes—
335C2×Q8#12418yes—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4#12014yes—
389C22×C4#12012yes—
390C22×C4#12014yes—
391C22×C4#12012yes—
392C22×C4#13512yes—
393C22×C4#13712yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D412yes—
398C22×C4C22×Q811yescharacteristic
399C22×C4C2×#4112yes—
400C22×C4C2×#4111yescharacteristic
401C22×C42−1+411yescharacteristic
402C22×C42−1+412yes—
403C22×C42−1+411yescharacteristic
404Q16—416no—
405C2×C8—22no—
406C2×C8—24no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—42no—
414C2×C8—41no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C4⋊C4—28no—
423C4⋊C4—216no—
424C4⋊C4—216no—
425C4⋊C4—232no—
426C4⋊C4—232no—
427C4⋊C4—216no—
428C4⋊C4—232no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—42no—
435C4⋊C4—41no—
436C4⋊C4—42no—
437C4⋊C4—41no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4#12014yes—
442C4⋊C4#12018yes—
443C42—232no—
444C42—216no—
445C42—24no—
446C42—24no—
447C42—24no—
448C42—24no—
449C42—22no—
450C42—24no—
451C42—22no—
452C23C2×#12011yescharacteristic
453C23C2×#12012yes—
454C23#54812yes—
455C23#54811yescharacteristic
456C23D4×Q811yescharacteristic
457C23C2×#13511yescharacteristic
458C23C2×#13711yescharacteristic
459C23#57512yes—
460C23#57711yescharacteristic
461C23#57812yes—
462C23#57811yescharacteristic
463Q8—24no—
464Q8—28no—
465Q8—24no—
466Q8—28no—
467Q8—24no—
468Q8—416no—
469Q8—432no—
470Q8—42no—
471Q8—44no—
472Q8—42no—
473Q8—81no—
474Q8—82no—
475Q8—81no—
476Q8#39318yes—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—28no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—81no—
530C2×C4—82no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4#45214yes—
535C2×C4C2×#12011yescharacteristic
536C2×C4C2×#12012yes—
537C2×C4C2×#12011yescharacteristic
538C2×C4C2×C22⋊Q811yescharacteristic
539C2×C4C2×C22⋊Q812yes—
540C2×C4C2×#12411yescharacteristic
541C2×C4C2×#12412yes—
542C2×C4#53611yescharacteristic
543C2×C4#53612yes—
544C2×C4#53611yescharacteristic
545C2×C4#53612yes—
546C2×C4#54812yes—
547C2×C4#54812yes—
548C2×C4#54812yes—
549C2×C4#54811yescharacteristic
550C2×C4#54812yes—
551C2×C4D4×Q811yescharacteristic
552C2×C4D4×Q812yes—
553C8—42no—
554C8—82no—
555C8—81no—
556C22#240911yescharacteristic
557C22#240912yes—
558C22#243611yescharacteristic
559C22#243612yes—
560C22#284611yescharacteristic
561C22C2×#45211yescharacteristic
562C22#300212yes—
563C22#300511yescharacteristic
564C22#300712yes—
565C22#301112yes—
566C22#301111yescharacteristic
567C22#301311yescharacteristic
568C22#301312yes—
569C22#336412yes—
570C22#336511yescharacteristic
571C22#337112yes—
572C22#337411yescharacteristic
573C22#337412yes—
574C22#337712yes—
575C22#337712yes—
576C22#338311yescharacteristic
577C22#338711yescharacteristic
578C22#339012yes—
579C4—28no—
580C4—24no—
581C4—44no—
582C4—48no—
583C4—44no—
584C4—42no—
585C4—44no—
586C4—82no—
587C4—162no—
588C4—161no—
589C4C2×#39311yescharacteristic
590C4#240912yes—
591C4#240911yescharacteristic
592C4#243612yes—
593C4#243611yescharacteristic
594C4#261411yescharacteristic
595C2#1487311yescharacteristic
596C2#2433211yescharacteristic
597C2#2433612yes—
598C2#2434411yescharacteristic
599C2#2434812yes—
600C2#2593111yescharacteristic
601C2#2593712yes—
602C2#2594111yescharacteristic
603C2#2594312yes—
604C2#3288311yescharacteristic
605C2#3288711yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
57. C22×Q16Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#20262
C2×#20242
C2×#18782
#152012
#113852
#113842
#113802
#107892