Finite Group Databasebeta

← #410201

Subgroups — #410201

Order 512.

1468 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic
2C2×#1851C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic
4C2×#1825C211yesmaximal, characteristic
5#13939C211yesmaximal, characteristic
6#13904C211yesmaximal, characteristic
7#11257C212yesmaximal, complemented
8#11256C212yesmaximal, complemented
9#11232C212yesmaximal, complemented
10#11231C212yesmaximal, complemented
11#10722C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#428—24no—
16C2×#428—24no—
17C2×#428—24no—
18C22×#107C2212yes—
19C22×#106C2212yes—
20C22×C4⋊C8C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C2×#381—22no—
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26#1851—24no—
27#1825—24no—
28#1824—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38#1445—21no—
39#1445—21no—
40#1440—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45#1434—21no—
46#1434—21no—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—24no—
52C22×M16—24no—
53C22×M16D414yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2312yes—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117D414yes—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4D414yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4D412yes—
80C22×C42—21no—
81C22×C42C2312yes—
82#428—232no—
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—24no—
86C2×#106—24no—
87C2×#106—24no—
88C2×#106—24no—
89C2×C4⋊C8—24no—
90C2×C4⋊C8—24no—
91C2×C4⋊C8—24no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95#381—48no—
96#380—22no—
97#380—22no—
98#380—22no—
99#380—22no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95D411yescharacteristic
113C2×#95D411yescharacteristic
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—22no—
122#336—24no—
123C25C2×D411yescharacteristic
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—44no—
136C23×C4—44no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4#4112yes—
146C23×C4#4112yes—
147C23×C4#4112yes—
148C23×C4C2411yescharacteristic
149C2×M16—28no—
150C2×M16—28no—
151C2×M16—216no—
152C2×M16—28no—
153C2×M16—216no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8C2×D414yes—
169C22×C8#4114yes—
170C22×C8#4114yes—
171#117—28no—
172#117—28no—
173#117—44no—
174#117—44no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—81no—
224C2×C22⋊C4—81no—
225C2×C42—24no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—42no—
233C2×C42—42no—
234#107—216no—
235#106—216no—
236C4⋊C8—48no—
237C22⋊C8—44no—
238#95—22no—
239#95—22no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—42no—
244#95—42no—
245#95—44no—
246#95—44no—
247#95—44no—
248#95—82no—
249#95—82no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—21no—
258C24#12112yes—
259C24#12112yes—
260C24#12112yes—
261C24C22×D411yescharacteristic
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—416no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—48no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—84no—
330C22×C4—84no—
331C22×C4—88no—
332C22×C4#12014yes—
333C22×C4#12112yes—
334C22×C4#12112yes—
335C22×C4#12112yes—
336C22×C4#12314yes—
337C22×C4#12314yes—
338C22×C4#12414yes—
339C22×C4AGL(1,ℤ/8)14yes—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4112yes—
345C22×C4C2×#4112yes—
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C42−1+412yes—
349C22×C42−1+412yes—
350M16—232no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—42no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—42no—
367C2×C8#12318yes—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—48no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—42no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C22⋊C4—48no—
392C22⋊C4—48no—
393C22⋊C4—84no—
394C22⋊C4—84no—
395C42—42no—
396C42—42no—
397C42—42no—
398C42—42no—
399C42—48no—
400C23—24no—
401C23—22no—
402C23—22no—
403C23—24no—
404C23—24no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—24no—
412C23—22no—
413C23—22no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23#39414yes—
423C23C2×#12111yescharacteristic
424C23C2×#12111yescharacteristic
425C23C2×#12111yescharacteristic
426C23#54712yes—
427C23#54712yes—
428C23C2×AGL(1,ℤ/8)12yes—
429C23#57512yes—
430C23#57711yescharacteristic
431C23#57711yescharacteristic
432C23#57811yescharacteristic
433C23#57811yescharacteristic
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—28no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—28no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—816no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4#39414yes—
496C2×C4C2×#12011yescharacteristic
497C2×C4C2×#12312yes—
498C2×C4C2×#12311yescharacteristic
499C2×C4C2×#12311yescharacteristic
500C2×C4#52512yes—
501C2×C4#52511yescharacteristic
502C2×C4#52511yescharacteristic
503C2×C4C2×#12411yescharacteristic
504C2×C4#53612yes—
505C2×C4#53611yescharacteristic
506C2×C4#53611yescharacteristic
507C2×C4#53612yes—
508C2×C4#55212yes—
509C2×C4#55212yes—
510C2×C4#55212yes—
511C2×C4#55212yes—
512C2×C4#55212yes—
513C2×C4#55212yes—
514C8—28no—
515C8—81no—
516C8—81no—
517C22—24no—
518C22—24no—
519C22—24no—
520C22—44no—
521C22—44no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—44no—
527C22—44no—
528C22C2×#39411yescharacteristic
529C22#241312yes—
530C22#241311yescharacteristic
531C22#241311yescharacteristic
532C22#248612yes—
533C22#303612yes—
534C22#303811yescharacteristic
535C22#304011yescharacteristic
536C22#304311yescharacteristic
537C22#304411yescharacteristic
538C22#304611yescharacteristic
539C22#304611yescharacteristic
540C22#305512yes—
541C22#306311yescharacteristic
542C22#306311yescharacteristic
543C22#334312yes—
544C22#334512yes—
545C22#334812yes—
546C22#334912yes—
547C22#335112yes—
548C22#335212yes—
549C22#335312yes—
550C22#335412yes—
551C4—24no—
552C4—44no—
553C4—82no—
554C4—82no—
555C4—82no—
556C4—84no—
557C4—162no—
558C4—161no—
559C4—161no—
560C4#248412yes—
561C4#248411yescharacteristic
562C4#248411yescharacteristic
563C4#250312yes—
564C4#261411yescharacteristic
565C4#284911yescharacteristic
566C2—44no—
567C2#1507311yescharacteristic
568C2#2354511yescharacteristic
569C2#2354611yescharacteristic
570C2#2458812yes—
571C2#2458912yes—
572C2#2459211yescharacteristic
573C2#2459311yescharacteristic
574C2#2459411yescharacteristic
575C2#2459511yescharacteristic
576C2#2766912yes—
577C2#2767012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11257C2semidirect
8. #11256C2semidirect
9. #11232C2semidirect
10. #11231C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C2×#18512
C2×#18242
C2×#18252
#139392
#139042
#112572
#112562
#112322
#112312
#107222