Finite Group Databasebeta

← #193201

Subgroups — #193201

Order 512.

1054 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34145C211yesmaximal, characteristic
2#21860C211yesmaximal, characteristic
3#21171C211yesmaximal, characteristic
4C2×#1948C211yesmaximal, characteristic
5C2×#1550C211yesmaximal, characteristic
6#8928C212yesmaximal, complemented
7#8927C212yesmaximal, complemented
8#8925C212yesmaximal, complemented
9#8923C212yesmaximal, complemented
10#7901C211yesmaximal, characteristic
11#7891C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#428—21no—
15C22×#106—21no—
16C2×#423C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18#2370—21no—
19#2370—21no—
20#2369C2211yescharacteristic
21#2369C2211yescharacteristic
22#2368—22no—
23#2365C2211yescharacteristic
24#2365C2211yescharacteristic
25#2360—21no—
26#2360—21no—
27#2296—21no—
28#2295—21no—
29#2295—22no—
30C2×#384C2211yescharacteristic
31C42×Q8—21no—
32C4×#117—21no—
33C4×#117C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1990C2212yes—
36#1956C2212yes—
37#1948—21no—
38#1948—21no—
39#1948—21no—
40#1948C2212yes—
41#1948C2212yes—
42#1948C2212yes—
43#1947—21no—
44#1947C2212yes—
45#1872—22no—
46#1803—22no—
47#1736—21no—
48C2×#340—21no—
49#1550C418yescomplemented
50C2×#333C2212yes—
51C2×#332C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54#1279C2212yes—
55#1279C2212yes—
56C22×M16—21no—
57C23×C8—21no—
58#533—22no—
59#533—22no—
60#533C2×C414yes—
61#533Q814yes—
62C2×C4⋊Q8C2311yescharacteristic
63C2×C4⋊Q8C2311yescharacteristic
64C2×#125—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8D412yes—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8D412yes—
71C2×#117—21no—
72C2×#117D411yescharacteristic
73C2×#117D411yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81#431—22no—
82#431—22no—
83#428—44no—
84C2×#107—22no—
85C2×#106—22no—
86C2×#106—22no—
87C2×#106—22no—
88C2×#106—44no—
89#423—22no—
90#423C2×C414yes—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2×C414yes—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98C2×#103—24no—
99C2×C22⋊C8—21no—
100C2×C4×C8C2312yes—
101#391—22no—
102#391—22no—
103#391—21no—
104#391—21no—
105#389—22no—
106#389—22no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#387—22no—
114#387—22no—
115#387—22no—
116#387—22no—
117#387—22no—
118#387—22no—
119#385—22no—
120#385—22no—
121#384—21no—
122#384—21no—
123#384—21no—
124#384C2312yes—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4D411yescharacteristic
138C4×C4⋊C4D411yescharacteristic
139C4×C4⋊C4D411yescharacteristic
140C4×C4⋊C4D411yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142C4×C4⋊C4C2311yescharacteristic
143C4×C4⋊C4C2311yescharacteristic
144C4×C4⋊C4C2311yescharacteristic
145C4×C22⋊C4—22no—
146C4×C22⋊C4—22no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149#376—22no—
150#376—21no—
151C2×#95—21no—
152C43—22no—
153C43—21no—
154#340—24no—
155#340—22no—
156#340—22no—
157#340—22no—
158#340—22no—
159#333C2×C418yescomplemented
160#332C2×C418yescomplemented
161C22×Q8—21no—
162C22×Q8C2×D412yes—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—41no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C2×M16—24no—
170C2×M16—42no—
171C2×M16—42no—
172C22×C8—24no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—42no—
177C22×C8—42no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8C22×C412yes—
183C4⋊Q8C22×C412yes—
184C4⋊Q8C2×Q812yes—
185C4⋊Q8C2×Q812yes—
186#125—24no—
187#125—24no—
188C22⋊Q8—24no—
189C22⋊Q8—24no—
190C22⋊Q8—24no—
191C22⋊Q8C4⋊C418yes—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—42no—
196C4×Q8—42no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×Q8C4⋊C418yes—
200#117—24no—
201#117—24no—
202#117—22no—
203#117—22no—
204#117—22no—
205#117SD1614yes—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
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⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C22⋊C4—22no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C42—22no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42C22×C412yes—
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×Q812yes—
266C2×C42#4112yes—
267C2×C42#4112yes—
268C2×C42C2411yescharacteristic
269#107—44no—
270#106—44no—
271#106—44no—
272#106—44no—
273C4⋊C8—24no—
274C4⋊C8—24no—
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—42no—
278C4⋊C8C22×C414yes—
279C4⋊C8C22×C414yes—
280#103—84no—
281C22⋊C8—42no—
282C4×C8—22no—
283C4×C8—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300C24#12011yescharacteristic
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8C2×C4⋊C414yes—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—81no—
333C22×C4—81no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4C2×C4⋊C412yes—
338C22×C4C4×D412yes—
339C22×C4C4×D412yes—
340C22×C4#12011yescharacteristic
341C22×C4#12011yescharacteristic
342C22×C4#12011yescharacteristic
343C22×C4C22⋊Q812yes—
344C22×C4C22⋊Q812yes—
345C22×C4#12312yes—
346C22×C4C2×SD1612yes—
347C22×C4#13512yes—
348C22×C4#13712yes—
349C22×C4#13712yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353M16—48no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—48no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—28no—
376C4⋊C4—28no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4C2×C4⋊C414yes—
400C4⋊C4C2×SD1612yes—
401C4⋊C4C2×SD1612yes—
402C4⋊C4AGL(1,ℤ/8)12yes—
403C4⋊C4#13712yes—
404C22⋊C4—28no—
405C22⋊C4—24no—
406C22⋊C4—42no—
407C42—28no—
408C42—24no—
409C42—44no—
410C42—44no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—42no—
420C42—42no—
421C42—42no—
422C42—81no—
423C42—81no—
424C42—82no—
425C42—82no—
426C42C2×C4⋊C412yes—
427C42C4×D412yes—
428C42C4×D412yes—
429C42C22⋊Q812yes—
430C42C22⋊Q812yes—
431C42#12312yes—
432C42C23×C411yescharacteristic
433C42C22×Q811yescharacteristic
434C42C2×#4111yescharacteristic
435C42C2×#4111yescharacteristic
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23#38512yes—
441C23#44812yes—
442C23#45212yes—
443C23C2×#12011yescharacteristic
444Q8—44no—
445Q8—42no—
446Q8—42no—
447Q8—42no—
448Q8—42no—
449Q8—82no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
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—84no—
493C2×C4—84no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—164no—
498C2×C4#38512yes—
499C2×C4#38512yes—
500C2×C4#38512yes—
501C2×C4C4×SD1614yes—
502C2×C4#44114yes—
503C2×C4#44812yes—
504C2×C4#45212yes—
505C2×C4#47514yes—
506C2×C4#47914yes—
507C2×C4C22×C4⋊C411yescharacteristic
508C2×C4C2×C4×D411yescharacteristic
509C2×C4C2×C4×D411yescharacteristic
510C2×C4C2×C22⋊Q811yescharacteristic
511C2×C4C2×C22⋊Q811yescharacteristic
512C2×C4C2×#12311yescharacteristic
513C2×C4C22×SD1611yescharacteristic
514C2×C4C2×#13511yescharacteristic
515C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
516C2×C4C2×#13711yescharacteristic
517C2×C4C2×#13711yescharacteristic
518C2×C4C2×#13711yescharacteristic
519C2×C4#57511yescharacteristic
520C2×C4#57511yescharacteristic
521C8—44no—
522C8—44no—
523C8—82no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22#186514yes—
531C22C2×#38511yescharacteristic
532C22C2×#44811yescharacteristic
533C22C2×#45211yescharacteristic
534C22#300611yescharacteristic
535C22#300711yescharacteristic
536C22#302111yescharacteristic
537C22#302211yescharacteristic
538C4—21no—
539C4—21no—
540C4—22no—
541C4—22no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—44no—
547C4—44no—
548C4—81no—
549C4—81no—
550C4—81no—
551C4—81no—
552C4—162no—
553C4#186514yes—
554C4C2×C4×SD1611yescharacteristic
555C4C2×#44111yescharacteristic
556C4C4×AGL(1,ℤ/8)11yescharacteristic
557C4C4×#13711yescharacteristic
558C4C2×#47511yescharacteristic
559C4C2×#47911yescharacteristic
560C4#307911yescharacteristic
561C4#308011yescharacteristic
562C2—24no—
563C2#1878811yescharacteristic
564C2#1879311yescharacteristic
565C2#1880011yescharacteristic
566C2#1880511yescharacteristic
567C2#2085811yescharacteristic
568C2C2×#186511yescharacteristic
569C2#2087511yescharacteristic
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. #8928C2semidirect
7. #8927C2semidirect
8. #8925C2semidirect
9. #8923C2semidirect
49. #1550C4semidirect
159. #333C2×C4semidirect
160. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341452
#218602
#211712
C2×#19482
C2×#15502
#89282
#89272
#89252
#89232
#79012
#78912