Finite Group Databasebeta

← #354263

Subgroups — #354263

Order 512.

1960 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Q16C211yesmaximal, characteristic
2C2×#2276C211yesmaximal, characteristic
3C2×#1878C212yesmaximal
4C2×#1763C211yesmaximal, characteristic
5C2×#1763C211yesmaximal, characteristic
6#13786C211yesmaximal, characteristic
7#10932C212yesmaximal
8#10907C212yesmaximal
9#10789C212yesmaximal
10C4×#340C211yesmaximal, characteristic
11#10694C211yesmaximal, characteristic
12C23×Q16C2211yescharacteristic
13C22×C4×Q8C2212yes—
14C22×C4×Q8C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C2×C4×Q16—24no—
17C2×C4×Q16—24no—
18C2×C4×Q16C418yescomplemented
19C22×#107C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2211yescharacteristic
23C22×#103C2212yes—
24C22×#97C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C2×#391C2212yes—
27#2276—24no—
28#2276—24no—
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4—22no—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×#376—21no—
35C2×#376C2211yescharacteristic
36#1878C4116yescomplemented
37#1763—24no—
38#1763—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1436—22no—
46C4⋊#95C2212yes—
47#1434—21no—
48#1434—21no—
49C23×Q8C2311yescharacteristic
50C23×Q8C2311yescharacteristic
51C22×Q16—22no—
52C22×Q16—22no—
53C22×Q16—24no—
54C22×Q16C2×C418yes—
55C22×Q16C2×C414yes—
56C23×C8C2311yescharacteristic
57C23×C8C2312yes—
58C23×C8C2311yescharacteristic
59C2×C4×Q8—24no—
60C2×C4×Q8—24no—
61C2×C4×Q8—24no—
62C2×C4×Q8—24no—
63C2×C4×Q8—22no—
64C2×C4×Q8—24no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8C2×C418yes—
73C2×C4×Q8C2×C414yes—
74C2×C4×Q8C2×C414yes—
75C2×C4×Q8D414yes—
76C2×C4×Q8D414yes—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C42—22no—
85C22×C42—21no—
86C22×C42—21no—
87C22×C42—22no—
88C22×C42—21no—
89C22×C42—21no—
90C22×C42C2312yes—
91C22×C42C2311yescharacteristic
92C4×Q16—232no—
93C2×#107—22no—
94C2×#107—22no—
95C2×#107C2×C414yes—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103C2×C418yes—
105C2×#103C2×C418yes—
106C2×#103C2×C414yes—
107C2×#103C2×C414yes—
108C2×#97—22no—
109C2×#97—22no—
110C2×#97—22no—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8C2×C414yes—
114#391—24no—
115#391C2×C418yes—
116C4×C4⋊C4—216no—
117C4×C4⋊C4—28no—
118C4×C4⋊C4—24no—
119C4×C4⋊C4—24no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4D414yes—
124#376—28no—
125#376—22no—
126#376—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95D411yescharacteristic
135C2×#95D411yescharacteristic
136C2×#95D412yes—
137#340—24no—
138#340—24no—
139#340—24no—
140#340C2×C418yes—
141#336—24no—
142#336C2×C418yes—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8C22×C412yes—
157C22×Q8C22×C414yes—
158C22×Q8C22×C412yes—
159C22×Q8C22×C414yes—
160C22×Q8C2×D412yes—
161C22×Q8C2×D412yes—
162C22×Q8#4114yes—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2411yescharacteristic
174C2×Q16—28no—
175C2×Q16—28no—
176C2×Q16—216no—
177C2×Q16—216no—
178C2×Q16C42116yescomplemented
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8C22×C414yes—
193C22×C8C22×C412yes—
194C22×C8C22×C414yes—
195C22×C8C22×C412yes—
196C4×Q8—216no—
197C4×Q8—216no—
198C4×Q8—216no—
199C4×Q8—216no—
200C4×Q8—28no—
201C4×Q8—28no—
202C4×Q8—28no—
203C4×Q8—28no—
204C4×Q8—28no—
205C4×Q8—44no—
206C4×Q8—44no—
207C4×Q8C22⋊C4116yes—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C414yes—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C414yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D414yes—
240C2×C4⋊C4#4114yes—
241C2×C42—28no—
242C2×C42—28no—
243C2×C42—28no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—28no—
253C2×C42—24no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—81no—
258C2×C42—81no—
259C2×C42—81no—
260C2×C42—81no—
261C2×C42—82no—
262C2×C42—82no—
263C2×C42C22×C414yes—
264C2×C42C22×C412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42#4114yes—
269C2×C42#4112yes—
270C2×C42#4112yes—
271#107—28no—
272#103—28no—
273#103—216no—
274#103—28no—
275#103—216no—
276#97—44no—
277C4×C8—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—24no—
282#95—22no—
283#95—24no—
284#95—22no—
285#95—42no—
286#95—82no—
287#95—82no—
288C24C22×D411yescharacteristic
289C2×Q8—22no—
290C2×Q8—24no—
291C2×Q8—22no—
292C2×Q8—24no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—22no—
296C2×Q8—24no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—24no—
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—22no—
311C2×Q8—24no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8C2×C4214yes—
320C2×Q8C2×C4214yes—
321C2×Q8C2×C22⋊C414yes—
322C2×Q8#11714yes—
323C2×Q8C4×D418yes—
324C2×Q8C4×D418yes—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4C4×D414yes—
382C22×C4C4×D414yes—
383C22×C4C4×D412yes—
384C22×C4C4×D412yes—
385C22×C4#12012yes—
386C22×C4#12012yes—
387C22×C4#12112yes—
388C22×C4#12112yes—
389C22×C4C23×C411yescharacteristic
390C22×C4C23×C412yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C2×#4112yes—
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4112yes—
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4112yes—
399Q16—232no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—22no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—24no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8C2×C4214yes—
417C4⋊C4—232no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—216no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—48no—
434C4⋊C4C2×C22⋊C414yes—
435C4⋊C4C4×D418yes—
436C42—232no—
437C42—24no—
438C42—24no—
439C42—24no—
440C42—24no—
441C42—216no—
442C42—48no—
443C42—84no—
444C42—84no—
445C42—88no—
446C42C2×C22⋊C414yes—
447C42C4×D414yes—
448C42#12114yes—
449C42#12314yes—
450C23C2×C4×D412yes—
451C23C2×C4×D411yescharacteristic
452C23C2×#12011yescharacteristic
453C23C2×#12111yescharacteristic
454C23#52512yes—
455C23#57611yescharacteristic
456C23#57711yescharacteristic
457C23#57711yescharacteristic
458C23#57812yes—
459C23#57812yes—
460C23#57811yescharacteristic
461Q8—24no—
462Q8—24no—
463Q8—24no—
464Q8—28no—
465Q8—28no—
466Q8—24no—
467Q8—24no—
468Q8—42no—
469Q8—42no—
470Q8—44no—
471Q8—44no—
472Q8C4×C22⋊C418yes—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—28no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4#38614yes—
536C2×C4#38614yes—
537C2×C4C22×C4211yescharacteristic
538C2×C4C22×C22⋊C411yescharacteristic
539C2×C4C2×#11711yescharacteristic
540C2×C4C2×C4×D412yes—
541C2×C4C2×C4×D412yes—
542C2×C4C2×C4×D411yescharacteristic
543C2×C4C2×C4×D412yes—
544C2×C4C4×#4111yescharacteristic
545C2×C4C4×#4112yes—
546C2×C4C4×#4111yescharacteristic
547C2×C4C4×#4112yes—
548C2×C4C2×#12111yescharacteristic
549C2×C4C2×#12311yescharacteristic
550C2×C4#52512yes—
551C2×C4#52512yes—
552C2×C4#52912yes—
553C2×C4#52912yes—
554C2×C4#53212yes—
555C8—24no—
556C8—42no—
557C8—44no—
558C22C42×D411yescharacteristic
559C22C2×#38611yescharacteristic
560C22#229011yescharacteristic
561C22C4×#12012yes—
562C22C4×#12112yes—
563C22#294712yes—
564C22#294712yes—
565C22#295012yes—
566C22#295011yescharacteristic
567C22#295011yescharacteristic
568C22#295112yes—
569C22#295111yescharacteristic
570C22#295211yescharacteristic
571C22#301112yes—
572C22#301211yescharacteristic
573C22#301311yescharacteristic
574C22#304412yes—
575C22#304511yescharacteristic
576C22#304611yescharacteristic
577C22#312012yes—
578C22#312112yes—
579C22#312412yes—
580C22#312712yes—
581C4—28no—
582C4—24no—
583C4—24no—
584C4—44no—
585C4—44no—
586C4—44no—
587C4—44no—
588C4—48no—
589C4—84no—
590C4—82no—
591C4C2×C4×C22⋊C411yescharacteristic
592C4C42×D411yescharacteristic
593C4#229011yescharacteristic
594C4#229711yescharacteristic
595C4C4×#12112yes—
596C4C4×#12312yes—
597C2C4×#38611yescharacteristic
598C2#2050511yescharacteristic
599C2#2050711yescharacteristic
600C2#2093511yescharacteristic
601C2#2093711yescharacteristic
602C2#2113211yescharacteristic
603C2#2113511yescharacteristic
604C2#2146312yes—
605C2#2146512yes—
606C2#2151212yes—
607C2#2151612yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
18. C2×C4×Q16C4semidirect
36. #1878C4semidirect
178. C2×Q16C42semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Q162
C2×#22762
C2×#18782
C2×#17632
#137862
#109322
#109072
#107892
C4×#3402
#106942