Finite Group Databasebeta

← #192639

Subgroups — #192639

Order 512.

1044 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34196C211yesmaximal, characteristic
2#21987C211yesmaximal, characteristic
3#20764C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5C2×#1491C211yesmaximal, characteristic
6#8896C212yesmaximal, complemented
7#8895C212yesmaximal, complemented
8#8892C212yesmaximal, complemented
9#8891C212yesmaximal, complemented
10#8044C211yesmaximal, characteristic
11#7895C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103—21no—
16C2×C4×M16C2211yescharacteristic
17#2355—21no—
18#2355—21no—
19#2355C2211yescharacteristic
20#2355C2211yescharacteristic
21Q8×C4⋊C4C2211yescharacteristic
22Q8×C4⋊C4C2211yescharacteristic
23#2344—22no—
24#2344—21no—
25Q8×C22⋊C4—21no—
26#2320—22no—
27#2320—21no—
28#2285—21no—
29#2285C2211yescharacteristic
30C2×#384C2211yescharacteristic
31#2283—21no—
32#2283C2211yescharacteristic
33#2276—21no—
34#1948C2212yes—
35#1947C2212yes—
36#1853—21no—
37#1852—21no—
38#1847—21no—
39#1836—22no—
40#1836—21no—
41#1834C2212yes—
42#1834C2212yes—
43#1803—21no—
44#1803—21no—
45#1803C2212yes—
46#1803C2212yes—
47#1736—22no—
48C2×#340—21no—
49#1491C418yescomplemented
50C2×Q8⋊C8C2212yes—
51C2×Q8⋊C8C2212yes—
52#1296C2212yes—
53#1296C2212yes—
54#1279C2212yes—
55#1279C2212yes—
56C23×Q8D411yescharacteristic
57C22×M16—21no—
58C23×C8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8C2×C414yes—
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8Q814yes—
69C2×C4×Q8C2311yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×#117—22no—
72C2×#117D412yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80#432—22no—
81#432C2×C414yes—
82C2×#107—22no—
83C2×#106—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—44no—
92C2×C22⋊C8—21no—
93C4×M16—22no—
94C4×M16C2×C414yes—
95C2×#97—21no—
96C2×#97—21no—
97C2×C4×C8C2312yes—
98#391—22no—
99#391—22no—
100#391—21no—
101#391—21no—
102#389—21no—
103#389—21no—
104#389—22no—
105#389—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#384—21no—
113#384—21no—
114#384—21no—
115#384—21no—
116#384—21no—
117#384C2312yes—
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#383—22no—
121#383—21no—
122#382—22no—
123#382—21no—
124#382—21no—
125#382D412yes—
126#382D411yescharacteristic
127#382D411yescharacteristic
128#381—22no—
129#381—21no—
130#380—22no—
131#380—21no—
132#380—21no—
133#380D411yescharacteristic
134#380D411yescharacteristic
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4C2311yescharacteristic
147C4×C4⋊C4C2311yescharacteristic
148C4×C22⋊C4—22no—
149C4×C22⋊C4—21no—
150#376—22no—
151#376—21no—
152#340—22no—
153#340—22no—
154#340—24no—
155#340—24no—
156#340—22no—
157#340—22no—
158Q8⋊C8C2×C418yescomplemented
159Q8⋊C8C2×C418yescomplemented
160C22×Q8—21no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—41no—
165C22×Q8—41no—
166C22×Q8C4⋊C414yes—
167C22×Q8C2×D411yescharacteristic
168C22×Q8C2×D411yescharacteristic
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—41no—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C2×M16—24no—
176C2×M16—42no—
177C2×M16—42no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—42no—
182C22×C8—42no—
183C22×C8—42no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—42no—
195C4×Q8—42no—
196C4×Q8—42no—
197C4×Q8—42no—
198C4×Q8C4⋊C414yes—
199C4×Q8C4⋊C414yes—
200C4×Q8C22×C412yes—
201C4×Q8C22×C412yes—
202C4×Q8C2×Q812yes—
203C4×Q8C2×Q812yes—
204#117—44no—
205#117—84no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4C4⋊C414yes—
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—21no—
244C2×C42—24no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—24no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42C22×C412yes—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×Q812yes—
262C2×C42#4112yes—
263C2×C42#4112yes—
264C2×C42C2411yescharacteristic
265#107—82no—
266#106—82no—
267C4⋊C8—22no—
268C4⋊C8—22no—
269C4⋊C8—22no—
270C4⋊C8—42no—
271C4⋊C8C22×C414yes—
272#103—44no—
273#103—44no—
274C22⋊C8—24no—
275#97—24no—
276#97—24no—
277C4×C8—22no—
278C4×C8C22×C414yes—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—21no—
288#95—22no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—22no—
295#95—21no—
296#95—21no—
297C24#12011yescharacteristic
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—24no—
303C2×Q8—28no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—44no—
313C2×Q8C2×C4⋊C412yes—
314C2×Q8C2×C4⋊C412yes—
315C2×Q8AGL(1,ℤ/8)12yes—
316C2×Q8#13712yes—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4C2×C4⋊C412yes—
346C22×C4C4×D412yes—
347C22×C4C4×D412yes—
348C22×C4#12011yescharacteristic
349C22×C4#12011yescharacteristic
350C22×C4#12011yescharacteristic
351C22×C4C22⋊Q812yes—
352C22×C4C22⋊Q812yes—
353C22×C4#12312yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357M16—48no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—84no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C4⋊C4C2×C4⋊C412yes—
402C4⋊C4C2×C4⋊C412yes—
403C22⋊C4—42no—
404C22⋊C4—44no—
405C22⋊C4—42no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—24no—
411C42—24no—
412C42—44no—
413C42—44no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—82no—
419C42—82no—
420C42—82no—
421C42C2×C4⋊C412yes—
422C42C4×D412yes—
423C42C4×D412yes—
424C42C22⋊Q812yes—
425C42C22⋊Q812yes—
426C42#12312yes—
427C42C23×C411yescharacteristic
428C42C22×Q811yescharacteristic
429C42C2×#4111yescharacteristic
430C42C2×#4111yescharacteristic
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23#38512yes—
436C23#45412yes—
437C23#45512yes—
438C23C2×#12011yescharacteristic
439Q8—22no—
440Q8—22no—
441Q8—22no—
442Q8—22no—
443Q8—22no—
444Q8—48no—
445Q8—41no—
446Q8—41no—
447Q8#42814yes—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—48no—
471C2×C4—48no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—164no—
500C2×C4#38512yes—
501C2×C4#38512yes—
502C2×C4#38512yes—
503C2×C4C22×C4⋊C411yescharacteristic
504C2×C4C2×C4×D411yescharacteristic
505C2×C4C2×C4×D411yescharacteristic
506C2×C4C2×C22⋊Q811yescharacteristic
507C2×C4C2×C22⋊Q811yescharacteristic
508C2×C4C2×#12311yescharacteristic
509C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
510C2×C4C2×#13711yescharacteristic
511C2×C4#57511yescharacteristic
512C2×C4#57511yescharacteristic
513C2×C4#57712yes—
514C2×C4#57812yes—
515C8—44no—
516C8—44no—
517C8—82no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22#186014yes—
525C22C2×#38511yescharacteristic
526C22#300611yescharacteristic
527C22#300711yescharacteristic
528C22#301011yescharacteristic
529C22#301111yescharacteristic
530C22C2×#45511yescharacteristic
531C22C2×#45411yescharacteristic
532C4—22no—
533C4—24no—
534C4—21no—
535C4—21no—
536C4—22no—
537C4—22no—
538C4—42no—
539C4—42no—
540C4—42no—
541C4—81no—
542C4—82no—
543C4—81no—
544C4—82no—
545C4—81no—
546C4—81no—
547C4—162no—
548C4C2×#42811yescharacteristic
549C4#291111yescharacteristic
550C4#294712yes—
551C4#308212yes—
552C4#309411yescharacteristic
553C4#309511yescharacteristic
554C2—24no—
555C2#1857211yescharacteristic
556C2#1857311yescharacteristic
557C2#1858011yescharacteristic
558C2#1858111yescharacteristic
559C2#2084611yescharacteristic
560C2C2×#186011yescharacteristic
561C2#2087711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8896C2semidirect
7. #8895C2semidirect
8. #8892C2semidirect
9. #8891C2semidirect
49. #1491C4semidirect
158. Q8⋊C8C2×C4semidirect
159. Q8⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341962
#219872
#207642
C2×#18032
C2×#14912
#88962
#88952
#88922
#88912
#80442
#78952