Finite Group Databasebeta

← #192132

Subgroups — #192132

Order 512.

1106 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34048C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic, complemented
3#21981C211yesmaximal, characteristic, complemented
4C2×#1458C211yesmaximal, characteristic
5#14340C211yesmaximal, characteristic
6#8858C214yesmaximal
7#8855C214yesmaximal, complemented
8#8026C211yesmaximal, characteristic, complemented
9#8026C211yesmaximal, characteristic, complemented
10C22×#117C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2314—21no—
14#2314—21no—
15#2314—21no—
16#2314—21no—
17#2313—21no—
18#2313C2212yes—
19#2312—21no—
20#2312C2212yes—
21#2298—22no—
22#2298—22no—
23#2285C2211yescharacteristic
24#2285C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27#2280—21no—
28C2×#380—21no—
29C4×#117—21no—
30#1948—22no—
31#1947—22no—
32#1853C2214yes—
33#1852C2214yes—
34#1845—21no—
35#1844—21no—
36#1834—21no—
37#1834—21no—
38#1458C418yescomplemented
39C2×#324C2212yes—
40C2×#323C2212yes—
41#1441—21no—
42#1441C2212yes—
43#1439—21no—
44#1439C2212yes—
45C4⋊#95—21no—
46#1294C2214yes—
47#1294C2214yes—
48C24×C4D411yescharacteristic
49C2×#117—21no—
50C2×#117—21no—
51C2×#117—21no—
52C2×#117—21no—
53C2×#117—21no—
54C2×#117—21no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117C2×C414yes—
58C2×#117Q814yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—22no—
66C22×C22⋊C4D411yescharacteristic
67C22×C22⋊C4C2311yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42C2311yescharacteristic
70#431—22no—
71#431—22no—
72#431C2×C414yes—
73#431C2×C414yes—
74C2×#107—24no—
75C2×#106—24no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C22⋊C8C2312yes—
79C2×C22⋊C8C2312yes—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82#388—22no—
83#388—22no—
84#388—22no—
85#388—22no—
86#388—22no—
87#388—22no—
88#388—22no—
89#388—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384D411yescharacteristic
103#384D411yescharacteristic
104#384D411yescharacteristic
105#384D411yescharacteristic
106#382—21no—
107#382—21no—
108#382—22no—
109#382—22no—
110#381—22no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—44no—
118#380C2312yes—
119#380C2312yes—
120#379—22no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D412yes—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133#376—22no—
134C2×#95—21no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95C2312yes—
139C43—22no—
140#340—24no—
141#340—24no—
142#324C2×C418yes—
143#323C2×C418yes—
144C25C2×D411yescharacteristic
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4C4⋊C414yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
161C22×C8—22no—
162C22×C8—22no—
163#117—24no—
164#117—24no—
165#117—24no—
166#117—24no—
167#117—24no—
168#117—24no—
169#117—24no—
170#117—24no—
171#117—42no—
172#117—42no—
173#117—42no—
174#117—42no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4C4⋊C414yes—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×Q812yes—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—82no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4C4⋊C414yes—
227C2×C22⋊C4C22×C412yes—
228C2×C22⋊C4C2×Q812yes—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42C4⋊C414yes—
239C2×C42C22×C412yes—
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×Q812yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247#107—84no—
248#106—84no—
249C4⋊C8—24no—
250C4⋊C8—24no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8C22×C414yes—
254C22⋊C8C22×C414yes—
255C4×C8—24no—
256C4×C8—24no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—44no—
272#95—42no—
273#95—82no—
274#95—82no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24C2×C4⋊C412yes—
286C24C22×D411yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—28no—
300C22×C4—28no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4C2×C4⋊C412yes—
334C22×C4C2×C4⋊C412yes—
335C22×C4C2×C4⋊C412yes—
336C22×C4C2×C4⋊C412yes—
337C22×C4C2×C4⋊C412yes—
338C22×C4#12011yescharacteristic
339C22×C4#12011yescharacteristic
340C22×C4#12011yescharacteristic
341C22×C4#12011yescharacteristic
342C22×C4#12312yes—
343C22×C4#12312yes—
344C22×C4#13512yes—
345C22×C4#13512yes—
346C22×C4AGL(1,ℤ/8)12yes—
347C22×C4#13712yes—
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×Q811yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—81no—
377C4⋊C4—81no—
378C4⋊C4—81no—
379C4⋊C4—81no—
380C4⋊C4—82no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C4⋊C4—84no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—24no—
390C22⋊C4—24no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—82no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C22⋊C4—88no—
400C42—24no—
401C42—24no—
402C42—44no—
403C42—44no—
404C42—82no—
405C42—82no—
406C42—82no—
407C42—82no—
408C42C4×D412yes—
409C42C4×D412yes—
410C42C4×D412yes—
411C42C4×D412yes—
412C42C22⋊Q812yes—
413C42C22⋊Q812yes—
414C42C22⋊Q812yes—
415C42C22⋊Q812yes—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23C22×C4⋊C411yescharacteristic
444C23C2×#12011yescharacteristic
445C23C2×#12311yescharacteristic
446C23C2×#13511yescharacteristic
447C23C2×#13511yescharacteristic
448C23C2×AGL(1,ℤ/8)11yescharacteristic
449C23C2×#13711yescharacteristic
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—48no—
478C2×C4—48no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4#38512yes—
510C2×C4#38512yes—
511C2×C4#38512yes—
512C2×C4#38512yes—
513C2×C4#42714yes—
514C2×C4#42814yes—
515C2×C4#44912yes—
516C2×C4#44912yes—
517C2×C4#45212yes—
518C2×C4#45212yes—
519C2×C4#48214yes—
520C2×C4#48514yes—
521C2×C4C2×C4×D411yescharacteristic
522C2×C4C2×C4×D411yescharacteristic
523C2×C4C2×C22⋊Q811yescharacteristic
524C2×C4C2×C22⋊Q811yescharacteristic
525C8—44no—
526C8—44no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—44no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22C2×#38511yescharacteristic
551C22C2×#42711yescharacteristic
552C22C2×#42811yescharacteristic
553C22C2×#44911yescharacteristic
554C22C2×#45211yescharacteristic
555C22#302111yescharacteristic
556C22#302211yescharacteristic
557C22#302711yescharacteristic
558C22#302811yescharacteristic
559C22C2×#48211yescharacteristic
560C22C2×#48511yescharacteristic
561C4—22no—
562C4—22no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—42no—
568C4—42no—
569C4—42no—
570C4—82no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4—162no—
575C4—162no—
576C4#185414yes—
577C4#185414yes—
578C2—21no—
579C2—21no—
580C2—21no—
581C2—21no—
582C2—82no—
583C2#1830411yescharacteristic
584C2#1830511yescharacteristic
585C2#1831211yescharacteristic
586C2#1831311yescharacteristic
587C2C2×#185411yescharacteristic
588C2#2085811yescharacteristic
589C2#2089011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21982C2semidirect
3. #21981C2semidirect
7. #8855C2semidirect
8. #8026C2semidirect
9. #8026C2semidirect
38. #1458C4semidirect

Maximal subgroups

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

subgroupindex
#340482
#219822
#219812
C2×#14582
#143402
#88582
#88552
#80262