Finite Group Databasebeta

← #192705

Subgroups — #192705

Order 512.

1064 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34243C211yesmaximal, characteristic
2#21867C211yesmaximal, characteristic
3C2×#1948C211yesmaximal, characteristic, complemented
4#20764C211yesmaximal, characteristic, complemented
5C2×#1483C211yesmaximal, characteristic
6#8896C212yesmaximal, complemented
7#8894C212yesmaximal, complemented
8#8893C212yesmaximal, complemented
9#8891C212yesmaximal, complemented
10#8044C211yesmaximal, characteristic, complemented
11#7904C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C22×#106—21no—
15C22×C4⋊C8C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#2370—21no—
18#2367—21no—
19#2359—22no—
20#2355C2211yescharacteristic
21#2354C2211yescharacteristic
22#2352—21no—
23Q8×C4⋊C4C2211yescharacteristic
24D4×C4⋊C4—21no—
25D4×C4⋊C4C2211yescharacteristic
26#2334—22no—
27#2314—21no—
28C4×C22⋊Q8—21no—
29C4×#121—21no—
30#2285C2211yescharacteristic
31C2×#384C2211yescharacteristic
32#2274—21no—
33C2×C4×C4⋊C4C2211yescharacteristic
34#1992—21no—
35#1959—21no—
36#1948—21no—
37#1948—21no—
38#1948C2212yes—
39#1948C2212yes—
40#1874—21no—
41#1872—21no—
42#1836—22no—
43#1834C2212yes—
44#1834C2212yes—
45#1803C2212yes—
46#1802C2212yes—
47#1736—22no—
48C2×#340—21no—
49#1483C418yescomplemented
50C2×Q8⋊C8C2212yes—
51C2×D4⋊C8C2212yes—
52#1296C2212yes—
53#1296C2212yes—
54#1279C2212yes—
55#1279C2212yes—
56C22×M16—21no—
57C23×C8—21no—
58#529—22no—
59#529—22no—
60#529C2×C414yes—
61#529Q814yes—
62C2×#126—21no—
63C2×#125D411yescharacteristic
64C2×#124D411yescharacteristic
65C2×#123—21no—
66C2×C22⋊Q8D411yescharacteristic
67C2×#121D411yescharacteristic
68C2×C4×Q8—21no—
69C2×C4×Q8C2311yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4—22no—
72C2×C4×D4—21no—
73C2×C4×D4C2311yescharacteristic
74C2×#117—21no—
75C2×#117—21no—
76C2×#117D412yes—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4D411yescharacteristic
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85#432—22no—
86#432—22no—
87C2×#106—22no—
88C2×#106—22no—
89C2×#106—44no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8C2×C414yescomplemented
93C2×C4⋊C8C2312yes—
94C2×C4⋊C8C2312yes—
95C2×#103—22no—
96C2×#102—22no—
97C2×C22⋊C8—21no—
98C4×M16—22no—
99C4×M16C2×C414yescomplemented
100C2×#97—21no—
101C2×#97—21no—
102C2×C4×C8C2312yes—
103#391—21no—
104#390—21no—
105#389—22no—
106#389—22no—
107#389—21no—
108#389—21no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#387—22no—
114#387—22no—
115#387—22no—
116#387—22no—
117#386—22no—
118#385—22no—
119#385—22no—
120#385—22no—
121#384—21no—
122#384—21no—
123#384—21no—
124#384—21no—
125#384C2312yes—
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#382—21no—
129#382D412yes—
130#380—21no—
131#380—21no—
132#380—21no—
133#380—21no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
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⋊C4D411yescharacteristic
145C4×C4⋊C4D411yescharacteristic
146C4×C4⋊C4C2311yescharacteristic
147C4×C4⋊C4C2311yescharacteristic
148C4×C22⋊C4—22no—
149C4×C22⋊C4—22no—
150C4×C22⋊C4—22no—
151C2×#95—21no—
152#340—22no—
153#340—22no—
154#340—24no—
155#340—24no—
156#340—22no—
157#340—22no—
158Q8⋊C8C2×C418yescomplemented
159D4⋊C8C2×C418yescomplemented
160C22×Q8C2×D411yescharacteristic
161C22×D4—21no—
162C22×D4C2×D411yescharacteristic
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—41no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C2×M16—24no—
172C2×M16—42no—
173C2×M16—42no—
174C22×C8—24no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—42no—
179C22×C8—42no—
180#126—24no—
181#126—24no—
182#125—22no—
183#125C4⋊C414yes—
184#124—22no—
185#124C4⋊C414yes—
186#123—24no—
187#123—24no—
188C22⋊Q8—22no—
189C22⋊Q8C4⋊C414yes—
190#121—22no—
191#121C4⋊C414yes—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—82no—
195C4×Q8C22×C412yes—
196C4×Q8C2×Q812yes—
197C4×D4—24no—
198C4×D4—24no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—82no—
202C4×D4—84no—
203C4×D4C22×C412yes—
204C4×D4C2×Q812yes—
205#117—24no—
206#117—24no—
207#117—42no—
208#117—42no—
209#117—44no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—82no—
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D412yes—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C42—21no—
257C2×C42—22no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—82no—
266C2×C42C22×C412yes—
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×Q812yes—
271C2×C42#4112yes—
272C2×C42#4112yes—
273C2×C42C2411yescharacteristic
274#106—44no—
275#106—44no—
276C4⋊C8—24no—
277C4⋊C8—22no—
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8C22×C414yes—
281#103—82no—
282#102—82no—
283C22⋊C8—42no—
284#97—24no—
285#97—42no—
286C4×C8—22no—
287C4×C8C22×C414yes—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—21no—
295#95—21no—
296#95—22no—
297#95—22no—
298#95—21no—
299#95—21no—
300C24—21no—
301C24#12011yescharacteristic
302C2×Q8—21no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8C2×C4⋊C412yes—
306C2×D4—24no—
307C2×D4—21no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4C2×C4⋊C412yes—
313C22×C4—22no—
314C22×C4—22no—
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—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—81no—
343C22×C4—81no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4C2×C4⋊C412yes—
350C22×C4C4×D412yes—
351C22×C4C4×D412yes—
352C22×C4#12011yescharacteristic
353C22×C4#12011yescharacteristic
354C22×C4#12011yescharacteristic
355C22×C4C22⋊Q812yes—
356C22×C4C22⋊Q812yes—
357C22×C4#12312yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361M16—48no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—48no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4—82no—
404C4⋊C4—82no—
405C4⋊C4—82no—
406C4⋊C4C2×C4⋊C412yes—
407C4⋊C4C2×C4⋊C412yes—
408C4⋊C4AGL(1,ℤ/8)12yes—
409C4⋊C4#13712yes—
410C22⋊C4—24no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—82no—
419C22⋊C4—82no—
420C42—22no—
421C42—22no—
422C42—44no—
423C42—42no—
424C42—42no—
425C42—81no—
426C42—81no—
427C42—82no—
428C42—82no—
429C42—82no—
430C42C2×C4⋊C412yes—
431C42C4×D412yes—
432C42C4×D412yes—
433C42C22⋊Q812yes—
434C42C22⋊Q812yes—
435C42#12312yes—
436C42C23×C411yescharacteristic
437C42C22×Q811yescharacteristic
438C42C2×#4111yescharacteristic
439C42C2×#4111yescharacteristic
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—22no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23#38512yes—
452C23#45412yes—
453C23#45512yes—
454C23C2×#12011yescharacteristic
455Q8—24no—
456Q8—42no—
457D4—24no—
458D4—48no—
459D4—42no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
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—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—84no—
508C2×C4—84no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—164no—
513C2×C4—164no—
514C2×C4#38512yes—
515C2×C4#38512yes—
516C2×C4#38512yes—
517C2×C4C22×C4⋊C411yescharacteristic
518C2×C4C2×C4×D411yescharacteristic
519C2×C4C2×C4×D411yescharacteristic
520C2×C4C2×C22⋊Q811yescharacteristic
521C2×C4C2×C22⋊Q811yescharacteristic
522C2×C4C2×#12311yescharacteristic
523C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
524C2×C4C2×#13711yescharacteristic
525C2×C4#57511yescharacteristic
526C2×C4#57511yescharacteristic
527C2×C4#57712yes—
528C2×C4#57712yes—
529C8—44no—
530C8—44no—
531C8—82no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22#186014yes—
546C22C2×#38511yescharacteristic
547C22#300611yescharacteristic
548C22#300711yescharacteristic
549C22#300911yescharacteristic
550C22#301011yescharacteristic
551C22C2×#45511yescharacteristic
552C22C2×#45411yescharacteristic
553C4—22no—
554C4—21no—
555C4—21no—
556C4—22no—
557C4—22no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—42no—
563C4—81no—
564C4—82no—
565C4—82no—
566C4—81no—
567C4—162no—
568C4#291212yes—
569C4C4×AGL(1,ℤ/8)11yescharacteristic
570C4C4×#13711yescharacteristic
571C4#307911yescharacteristic
572C4#308011yescharacteristic
573C4#309912yes—
574C2—24no—
575C2—82no—
576C2#1857011yescharacteristic
577C2#1857311yescharacteristic
578C2#1857911yescharacteristic
579C2#1858011yescharacteristic
580C2#2084811yescharacteristic
581C2C2×#186011yescharacteristic
582C2#2087511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
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
3. C2×#1948C2semidirect
4. #20764C2semidirect
6. #8896C2semidirect
7. #8894C2semidirect
8. #8893C2semidirect
9. #8891C2semidirect
10. #8044C2semidirect
11. #7904C2semidirect
49. #1483C4semidirect
92. C2×C4⋊C8C2×C4semidirect
99. C4×M16C2×C4semidirect
158. Q8⋊C8C2×C4semidirect
159. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342432
#218672
C2×#19482
#207642
C2×#14832
#88962
#88942
#88932
#88912
#80442
#79042