Finite Group Databasebeta

← #238269

Subgroups — #238269

Order 512.

1121 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic, complemented
2C2×#1964C211yesmaximal, characteristic
3C2×#1519C211yesmaximal, characteristic
4C2×#1519C211yesmaximal, characteristic
5#10376C218yesmaximal, complemented
6#8023C211yesmaximal, characteristic, complemented
7C2×#1289C211yesmaximal, characteristic, complemented
8C2×#1282C211yesmaximal, characteristic, complemented
9C22×#121C2211yescharacteristic
10C22×#99—21no—
11C22×#99—21no—
12C22×C22⋊C8C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14#2851—22no—
15#2646—22no—
16#2569—22no—
17#2569C2214yescomplemented
18#2442—22no—
19#2442—22no—
20#2315—21no—
21#2315—21no—
22C2×#380C2211yescharacteristic
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26#1964—22no—
27#1964C2214yes—
28#1519—22no—
29#1519—22no—
30#1519—22no—
31#1519C2214yes—
32C2×#327—22no—
33C2×#327C2214yes—
34#1294C2214yescomplemented
35#1289C2214yescomplemented
36#1282C2214yescomplemented
37C23×D4—21no—
38C23×D4Q811yescharacteristic
39C24×C4C2311yescharacteristic
40C23×C8—21no—
41C23×C8—21no—
42C2×#125—22no—
43C2×#121—21no—
44C2×#121—21no—
45C2×#121—21no—
46C2×#121—21no—
47C2×#121—21no—
48C2×#121—21no—
49C2×#121—21no—
50C2×#121—21no—
51C2×#121—21no—
52C2×#121—21no—
53C2×#121Q812yes—
54C2×#121C2312yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4Q811yescharacteristic
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4D411yescharacteristic
65C22×C42—21no—
66C2×#99—24no—
67C2×#99—24no—
68C2×#99—44no—
69C2×#99—44no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8C2312yes—
81C2×C22⋊C8C2312yes—
82#400—24no—
83#400—24no—
84#398—24no—
85#398—24no—
86#389—22no—
87#389—22no—
88#387—24no—
89#387—24no—
90#385—24no—
91#385—24no—
92#385—24no—
93#385—24no—
94#382—22no—
95#380—24no—
96#380—21no—
97#380—21no—
98#380—21no—
99#380—21no—
100#380—24no—
101#380—21no—
102#380—21no—
103#380—21no—
104#380D412yes—
105#380D412yes—
106#380D412yes—
107#380C2312yes—
108#380C2312yes—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4D412yes—
112C4×C22⋊C4C2312yes—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95D412yes—
116C2×#95C2312yes—
117#327—28no—
118#327—28no—
119C25—21no—
120C25C2×D411yescharacteristic
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—41no—
129C22×D4—41no—
130C22×D4—41no—
131C22×D4—41no—
132C22×D4—42no—
133C22×D4—42no—
134C22×D4—42no—
135C22×D4—42no—
136C22×D4—82no—
137C22×D4C2×Q812yes—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—41no—
144C23×C4—82no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×Q811yescharacteristic
151C23×C4C2×Q811yescharacteristic
152C23×C4#4111yescharacteristic
153C23×C4#4111yescharacteristic
154C23×C4#4111yescharacteristic
155C23×C4#4111yescharacteristic
156C23×C4#4111yescharacteristic
157C23×C4#4111yescharacteristic
158C23×C4C2411yescharacteristic
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—41no—
162C22×C8—41no—
163C22×C8—41no—
164C22×C8—41no—
165C22×C8—82no—
166C22×C8—82no—
167#125—84no—
168#121—22no—
169#121—22no—
170#121—22no—
171#121—22no—
172#121—22no—
173#121—22no—
174#121—22no—
175#121—22no—
176#121—41no—
177#121—41no—
178#121—41no—
179#121—41no—
180#121—41no—
181#121—41no—
182#121—41no—
183#121—41no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C2×Q812yes—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4C2×D412yes—
228C2×C22⋊C4C2×D412yes—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—82no—
232#99—48no—
233#99—48no—
234C22⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C22⋊C8—41no—
239C22⋊C8—41no—
240C22⋊C8—41no—
241C22⋊C8—41no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—41no—
252#95C2×D414yes—
253#95C2×D414yes—
254C24—22no—
255C24—41no—
256C24—41no—
257C24—42no—
258C24—42no—
259C24—81no—
260C24—81no—
261C24—82no—
262C24—82no—
263C24#12011yescharacteristic
264C24#12011yescharacteristic
265C24C22⋊Q811yescharacteristic
266C24C22⋊Q811yescharacteristic
267C24#12411yescharacteristic
268C24#12411yescharacteristic
269C24#13512yes—
270C24#13512yes—
271C24AGL(1,ℤ/8)12yes—
272C24AGL(1,ℤ/8)12yes—
273C24C22×D411yescharacteristic
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—44no—
285C2×D4—44no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—82no—
289C2×D4—82no—
290C2×D4—82no—
291C2×D4—82no—
292C2×D4—82no—
293C2×D4—82no—
294C2×D4—82no—
295C2×D4—82no—
296C2×D4—82no—
297C2×D4—82no—
298C2×D4—82no—
299C2×D4—82no—
300C2×D4—82no—
301C2×D4—82no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—81no—
325C22×C4—81no—
326C22×C4—81no—
327C22×C4—81no—
328C22×C4—81no—
329C22×C4—81no—
330C22×C4—81no—
331C22×C4—81no—
332C22×C4—81no—
333C22×C4—81no—
334C22×C4—81no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4#12011yescharacteristic
343C22×C4#12011yescharacteristic
344C22×C4C22⋊Q811yescharacteristic
345C22×C4C22⋊Q811yescharacteristic
346C22×C4C22⋊Q811yescharacteristic
347C22×C4C22⋊Q811yescharacteristic
348C22×C4C22⋊Q811yescharacteristic
349C22×C4C22⋊Q811yescharacteristic
350C22×C4C22⋊Q811yescharacteristic
351C22×C4C22⋊Q811yescharacteristic
352C22×C4C22⋊Q811yescharacteristic
353C22×C4C22⋊Q811yescharacteristic
354C22×C4#12411yescharacteristic
355C22×C4#12411yescharacteristic
356C22×C4#12411yescharacteristic
357C22×C4#12411yescharacteristic
358C22×C4#12411yescharacteristic
359C22×C4#12411yescharacteristic
360C22×C4#12411yescharacteristic
361C22×C4#12411yescharacteristic
362C22×C4#12411yescharacteristic
363C22×C4#12411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×Q811yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—82no—
373C2×C8—82no—
374C2×C8—82no—
375C2×C8—82no—
376C2×C8—82no—
377C2×C8—82no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—82no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4—84no—
390C4⋊C4—84no—
391C4⋊C4—82no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—22no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—82no—
410C42—44no—
411C42—82no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23—164no—
442C23—164no—
443C23#39311yescharacteristic
444C23#39311yescharacteristic
445C23#39311yescharacteristic
446C23#39311yescharacteristic
447C23#44912yes—
448C23#44912yes—
449C23#44912yes—
450C23#44912yes—
451C23#47812yes—
452C23#47812yes—
453C23#47812yes—
454C23#47812yes—
455C23#48812yes—
456C23#48812yes—
457C23#49012yes—
458C23#49012yes—
459C23C2×#12011yescharacteristic
460C23C2×C22⋊Q811yescharacteristic
461C23C2×#12411yescharacteristic
462C23C2×#13511yescharacteristic
463C23C2×#13511yescharacteristic
464C23C2×AGL(1,ℤ/8)11yescharacteristic
465C23C2×AGL(1,ℤ/8)11yescharacteristic
466D4—82no—
467D4—82no—
468D4—82no—
469D4—82no—
470D4—82no—
471D4—82no—
472D4—82no—
473D4—84no—
474D4—84no—
475D4—82no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—84no—
520C2×C4—84no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—162no—
527C2×C4—162no—
528C2×C4—164no—
529C2×C4—164no—
530C2×C4#39311yescharacteristic
531C2×C4#39311yescharacteristic
532C2×C4#39311yescharacteristic
533C2×C4#39311yescharacteristic
534C2×C4C2×C22⋊Q811yescharacteristic
535C2×C4C2×C22⋊Q811yescharacteristic
536C2×C4C2×#12411yescharacteristic
537C2×C4C2×#12411yescharacteristic
538C8—82no—
539C8—82no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—42no—
555C22—42no—
556C22—41no—
557C22—41no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—41no—
563C22—41no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—162no—
574C22—162no—
575C22—164no—
576C22—164no—
577C22#202512yes—
578C22#202512yes—
579C22#202512yes—
580C22#202512yes—
581C22C2×#39311yescharacteristic
582C22C2×#44911yescharacteristic
583C22C2×#44911yescharacteristic
584C22C2×#47811yescharacteristic
585C22C2×#47811yescharacteristic
586C22C2×#48811yescharacteristic
587C22C2×#49011yescharacteristic
588C4—21no—
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—82no—
596C4—82no—
597C4—84no—
598C4—84no—
599C4—82no—
600C4—162no—
601C4#203212yes—
602C2—21no—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—22no—
608C2—22no—
609C2—21no—
610C2—22no—
611C2—22no—
612C2—162no—
613C2C2×#202511yescharacteristic
614C2#2347211yescharacteristic
615C2C2×#203211yescharacteristic
subgroupG/H
center, socleC22C2×#393
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35020C2semidirect
5. #10376C2semidirect
6. #8023C2semidirect
7. C2×#1289C2semidirect
8. C2×#1282C2semidirect
17. #2569C22semidirect
34. #1294C22semidirect
35. #1289C22semidirect
36. #1282C22semidirect

Maximal subgroups

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

subgroupindex
#350202
C2×#19642
C2×#15192
#103762
#80232
C2×#12892
C2×#12822