Finite Group Databasebeta

← #398029

Subgroups — #398029

Order 512.

1780 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#474C211yesmaximal, characteristic
2C2×#1877C211yesmaximal, characteristic
3C2×#1877C212yesmaximal
4#15191C211yesmaximal, characteristic, complemented
5D4×#95C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic, complemented
7#11256C212yesmaximal, complemented
8#11255C211yesmaximal, characteristic, complemented
9#11124C212yesmaximal, complemented
10#11077C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C23×SD16C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×D4C2212yes—
15C22×C4×D4C2211yescharacteristic
16C2×#474—28no—
17C2×#474—24no—
18C22×#107C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#102C2212yes—
24C2×#391C2211yescharacteristic
25C2×#391C2212yes—
26C22×#95—21no—
27#1877—28no—
28#1877—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1446—21no—
37#1446—21no—
38#1442—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42C22⋊#95—22no—
43C22⋊#95—21no—
44C4×#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8C2311yescharacteristic
47C23×D4C2311yescharacteristic
48C24×C4—22no—
49C24×C4—21no—
50C22×SD16—24no—
51C22×SD16—22no—
52C22×SD16—22no—
53C22×SD16—24no—
54C22×SD16Q814yescomplemented
55C23×C8C2311yescharacteristic
56C23×C8C2312yes—
57C23×C8C2311yescharacteristic
58C2×C4⋊Q8—24no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—42no—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—24no—
65C2×C4×D4—24no—
66C2×C4×D4—24no—
67C2×C4×D4—22no—
68C2×C4×D4—42no—
69C2×C4×D4—42no—
70C2×C4×D4—42no—
71C2×C4×D4D414yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—21no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81#474—232no—
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—24no—
85C2×C4⋊C8—24no—
86C2×C4⋊C8—22no—
87C2×#103—24no—
88C2×#103—22no—
89C2×#103—24no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—22no—
95C2×#102—24no—
96C2×#102D418yes—
97#391—24no—
98#391—22no—
99#391—22no—
100#391—42no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—44no—
112C2×#95D412yes—
113C2×#95D411yescharacteristic
114C2×#95D411yescharacteristic
115#340—24no—
116#340—22no—
117#340—24no—
118#340—42no—
119#336—24no—
120#336—22no—
121#336—42no—
122C25—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—21no—
132C22×Q8—44no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—82no—
137C22×Q8C2×Q812yes—
138C22×D4—21no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4C2×D414yes—
144C22×D4C2×D412yes—
145C22×D4C2×Q812yes—
146C22×D4#4112yes—
147C22×D4#4114yes—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—28no—
153C23×C4—24no—
154C23×C4—21no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—44no—
162C23×C4—82no—
163C23×C4—82no—
164C23×C4—82no—
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
170C2×SD16—28no—
171C2×SD16—216no—
172C2×SD16—44no—
173C2×SD16—44no—
174C2×SD16—48no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—21no—
188C22×C8—22no—
189C22×C8—21no—
190C22×C8C2×D414yes—
191C22×C8C2×Q812yes—
192C4⋊Q8—28no—
193C4⋊Q8—48no—
194C4⋊Q8—44no—
195C4×D4—28no—
196C4×D4—44no—
197C4×D4—48no—
198C4×D4—48no—
199C4×D4—48no—
200C4×D4—48no—
201C4×D4—48no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4#4114yes—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42C2×D412yes—
238C2×C42#4112yes—
239#107—216no—
240C4⋊C8—28no—
241#103—48no—
242#103—44no—
243#102—216no—
244#102—44no—
245#95—22no—
246#95—22no—
247#95—24no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—42no—
252#95—42no—
253#95—42no—
254#95—82no—
255#95—81no—
256#95—82no—
257#95—82no—
258C24—22no—
259C24—24no—
260C24—24no—
261C24—22no—
262C24—22no—
263C24—41no—
264C24—41no—
265C24—42no—
266C24—42no—
267C24—41no—
268C24—41no—
269C24C22×D411yescharacteristic
270C2×Q8—22no—
271C2×Q8—24no—
272C2×Q8—22no—
273C2×Q8—24no—
274C2×Q8—48no—
275C2×Q8—41no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—88no—
290C2×Q8—88no—
291C2×Q8—84no—
292C2×Q8—84no—
293C2×D4—24no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—22no—
304C2×D4—24no—
305C2×D4—22no—
306C2×D4#12014yes—
307C2×D4C22⋊Q814yes—
308C2×D4C22⋊Q818yes—
309C2×D4#12414yes—
310C2×D4#12418yes—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—48no—
335C22×C4—416no—
336C22×C4—416no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
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—48no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—84no—
364C22×C4—88no—
365C22×C4—88no—
366C22×C4—88no—
367C22×C4—88no—
368C22×C4—88no—
369C22×C4—88no—
370C22×C4—88no—
371C22×C4#12112yes—
372C22×C4#12112yes—
373C22×C4#13512yes—
374C22×C4AGL(1,ℤ/8)12yes—
375C22×C4#13714yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D412yes—
378C22×C4C22×Q811yescharacteristic
379C22×C4C2×#4112yes—
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4112yes—
383C22×C4H(5,𝔽2)12yes—
384C22×C4H(5,𝔽2)11yescharacteristic
385C22×C42−1+411yescharacteristic
386SD16—416no—
387C2×C8—24no—
388C2×C8—22no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—42no—
403C2×C8—42no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—48no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—41no—
417C4⋊C4—42no—
418C4⋊C4—41no—
419C22⋊C4—48no—
420C22⋊C4—416no—
421C42—22no—
422C42—22no—
423C42—24no—
424C42—42no—
425C42—42no—
426C42—42no—
427C42—42no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—28no—
432C23—28no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—42no—
438C23—44no—
439C23—42no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—44no—
450C23—42no—
451C23C2×#12111yescharacteristic
452C23#52512yes—
453C23D4212yes—
454C23#54611yescharacteristic
455C23D4×Q811yescharacteristic
456C23C2×#13511yescharacteristic
457C23C2×AGL(1,ℤ/8)11yescharacteristic
458C23C2×#13712yes—
459C23#57612yes—
460C23#57711yescharacteristic
461C23#57811yescharacteristic
462Q8—42no—
463Q8—42no—
464Q8—44no—
465Q8—816no—
466Q8—81no—
467Q8—81no—
468Q8—82no—
469D4—24no—
470D4—24no—
471D4—24no—
472D4—28no—
473D4—28no—
474D4#39318yes—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—816no—
510C2×C4—832no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4#46314yes—
530C2×C4C2×#12011yescharacteristic
531C2×C4C2×C22⋊Q811yescharacteristic
532C2×C4C2×C22⋊Q812yes—
533C2×C4C2×#12411yescharacteristic
534C2×C4C2×#12412yes—
535C2×C4#52912yes—
536C2×C4#53211yescharacteristic
537C2×C4#53411yescharacteristic
538C2×C4#53711yescharacteristic
539C2×C4#53812yes—
540C2×C4#53912yes—
541C2×C4#54612yes—
542C2×C4#54612yes—
543C2×C4#54712yes—
544C2×C4#55112yes—
545C2×C4#55211yescharacteristic
546C2×C4#55412yes—
547C2×C4#55411yescharacteristic
548C8—42no—
549C8—44no—
550C8—81no—
551C22—28no—
552C22—44no—
553C22—48no—
554C22—44no—
555C22—44no—
556C22—48no—
557C22—44no—
558C22—44no—
559C22—48no—
560C22—48no—
561C22—44no—
562C22#241011yescharacteristic
563C22#241711yescharacteristic
564C22#244912yes—
565C22#247212yes—
566C22#285011yescharacteristic
567C22C2×#46311yescharacteristic
568C22#303911yescharacteristic
569C22#304111yescharacteristic
570C22#304611yescharacteristic
571C22#310712yes—
572C22#311012yes—
573C22#311812yes—
574C22#312412yes—
575C22#326412yes—
576C22#326612yes—
577C22#327612yes—
578C22#328012yes—
579C22#330211yescharacteristic
580C22#331412yes—
581C22#332411yescharacteristic
582C22#338111yescharacteristic
583C22#338512yes—
584C22#338611yescharacteristic
585C4—24no—
586C4—42no—
587C4—44no—
588C4—82no—
589C4—84no—
590C4—82no—
591C4—84no—
592C4—82no—
593C4—161no—
594C4C2×#39311yescharacteristic
595C4#243011yescharacteristic
596C4#245012yes—
597C4#248812yes—
598C4#256411yescharacteristic
599C4#263911yescharacteristic
600C2—48no—
601C2#1475911yescharacteristic
602C2#2435511yescharacteristic
603C2#2437611yescharacteristic
604C2#2481611yescharacteristic
605C2#2483311yescharacteristic
606C2#2637212yes—
607C2#2638412yes—
608C2#2723712yes—
609C2#2724512yes—
610C2#3303211yescharacteristic
611C2#3303611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #15191C2semidirect
6. #11384C2semidirect
7. #11256C2semidirect
8. #11255C2semidirect
9. #11124C2semidirect
10. #11077C2semidirect
54. C22×SD16Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4742
C2×#18772
#151912
D4×#952
#113842
#112562
#112552
#111242
#110772
#107882