Finite Group Databasebeta

← #360740

Subgroups — #360740

Order 512.

1606 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2351C211yesmaximal, characteristic
2C2×#1865C212yesmaximal
3C2×#1863C212yesmaximal
4C2×#1847C211yesmaximal, characteristic
5C2×#1803C211yesmaximal, characteristic
6C2×#1763C211yesmaximal, characteristic
7#13910C211yesmaximal, characteristic
8#11328C212yesmaximal
9#11327C212yesmaximal
10#10850C211yesmaximal, characteristic
11#10835C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×C4⋊C8C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#97C2211yescharacteristic
21#2351—24no—
22#2351—24no—
23C2×#389—21no—
24C2×#389—21no—
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#384—21no—
30C2×#384C2211yescharacteristic
31C2×#382—21no—
32C2×#382C2211yescharacteristic
33C2×C4×C4⋊C4—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1865—28no—
37#1863—28no—
38#1847—24no—
39#1803—24no—
40#1763C418yescomplemented
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47#1444—21no—
48#1444—21no—
49C4⋊#95—21no—
50C23×Q8C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—24no—
61C2×C4×Q8—24no—
62C2×C4×Q8—42no—
63C2×C4×Q8—42no—
64C2×C4×Q8C2×C414yes—
65C2×C4×Q8D414yes—
66C2×C4×Q8Q814yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C42—21no—
84C22×C42—21no—
85C22×C42—21no—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×#107—24no—
91C2×#107—24no—
92C2×#107—42no—
93C2×#106—24no—
94C2×#106—24no—
95C2×#106—42no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103C2×C418yes—
103C2×#103D418yes—
104C2×#103Q818yes—
105C2×#97—22no—
106C2×#97—22no—
107C2×#97C2×C414yes—
108#389—28no—
109#389—28no—
110#389—22no—
111#389—22no—
112#389—22no—
113#389—22no—
114#389—24no—
115#389—24no—
116#389—42no—
117#389—42no—
118#384—22no—
119#384—22no—
120#384—44no—
121#382—28no—
122#382—22no—
123#382D414yes—
124C4×C4⋊C4—28no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4C2×C414yes—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95D411yescharacteristic
137C2×#95D411yescharacteristic
138#340—24no—
139#340—24no—
140#340—24no—
141#340C2×C418yes—
142#336—24no—
143#336—24no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8C22×C412yes—
150C22×Q8C2×D412yes—
151C22×Q8C2×D414yes—
152C22×Q8C2×Q812yes—
153C22×Q8C2×Q814yes—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8C22×C414yes—
179C22×C8C2×D414yes—
180C22×C8C2×Q814yes—
181C4×Q8—28no—
182C4×Q8—28no—
183C4×Q8—28no—
184C4×Q8—48no—
185C4×Q8—48no—
186C4×Q8—44no—
187C4×Q8—44no—
188C4×Q8—44no—
189C4×Q8—44no—
190C4×Q8C4⋊C4116yescomplemented
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—82no—
237C2×C4⋊C4—82no—
238C2×C4⋊C4—82no—
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C22×C414yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×Q812yes—
246C2×C4⋊C4C2×Q814yes—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—44no—
262C2×C42—44no—
263C2×C42—42no—
264C2×C42—42no—
265C2×C42C22×C412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×Q812yes—
269C2×C42#4112yes—
270C2×C42#4112yes—
271#107—48no—
272#106—48no—
273C4⋊C8—44no—
274#103—48no—
275#103C4⋊C4132yescomplemented
276#97—28no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—22no—
283#95—22no—
284#95—44no—
285#95—44no—
286C24C22×D411yescharacteristic
287C2×Q8—22no—
288C2×Q8—24no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8C2×C4⋊C418yes—
301C2×Q8C2×C4⋊C414yes—
302C2×Q8#12714yes—
303C2×Q8C4⋊Q814yes—
304C2×Q8C4⋊Q818yes—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4C4×D412yes—
366C22×C4C4×D412yes—
367C22×C4#12012yes—
368C22×C4#12012yes—
369C22×C4C22⋊Q812yes—
370C22×C4C22⋊Q812yes—
371C22×C4AGL(1,ℤ/8)12yes—
372C22×C4#13712yes—
373C22×C4C23×C411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D412yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×Q811yescharacteristic
378C22×C4C22×Q812yes—
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4H(5,𝔽2)12yes—
382C22×C42−1+412yes—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8C2×C4⋊C418yes—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—216no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—88no—
421C4⋊C4—84no—
422C4⋊C4—84no—
423C4⋊C4—88no—
424C4⋊C4C2×C4⋊C418yes—
425C4⋊C4C2×C4⋊C414yes—
426C42—216no—
427C42—48no—
428C42—48no—
429C42—48no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—42no—
435C42—42no—
436C42C2×C4⋊C414yes—
437C42C4×D414yes—
438C42C22⋊Q814yes—
439C42#12314yes—
440C23C2×C4×D411yescharacteristic
441C23C2×#12011yescharacteristic
442C23C2×C22⋊Q811yescharacteristic
443C23D4212yes—
444C23D4×Q812yes—
445C23C2×AGL(1,ℤ/8)11yescharacteristic
446C23C2×#13711yescharacteristic
447C23#57511yescharacteristic
448C23#57511yescharacteristic
449C23#57712yes—
450C23#57812yes—
451Q8—24no—
452Q8—28no—
453Q8—28no—
454Q8—24no—
455Q8—24no—
456Q8#38418yescomplemented
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—28no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—28no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—28no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4#38514yes—
524C2×C4#38514yes—
525C2×C4C22×C4⋊C411yescharacteristic
526C2×C4C22×C4⋊C412yes—
527C2×C4C2×C4×D411yescharacteristic
528C2×C4#52012yes—
529C2×C4C2×C22⋊Q811yescharacteristic
530C2×C4C2×#12311yescharacteristic
531C2×C4C2×#12711yescharacteristic
532C2×C4C2×C4⋊Q812yes—
533C2×C4C2×C4⋊Q811yescharacteristic
534C2×C4#53511yescharacteristic
535C2×C4#53611yescharacteristic
536C2×C4#54412yes—
537C2×C4D4212yes—
538C2×C4#54712yes—
539C2×C4#54712yes—
540C2×C4D4×Q812yes—
541C2×C4#55412yes—
542C2×C4#55412yes—
543C8—28no—
544C8—82no—
545C22C2×#38511yescharacteristic
546C22D4×C4⋊C412yes—
547C22#243311yescharacteristic
548C22#243411yescharacteristic
549C22#244012yes—
550C22C4×AGL(1,ℤ/8)11yescharacteristic
551C22C4×#13711yescharacteristic
552C22#294712yes—
553C22#300611yescharacteristic
554C22#300711yescharacteristic
555C22#301011yescharacteristic
556C22#301111yescharacteristic
557C22#307911yescharacteristic
558C22#308011yescharacteristic
559C22#308212yes—
560C22#326512yes—
561C22#326912yes—
562C22#327012yes—
563C22#327112yes—
564C22#338612yes—
565C22#338712yes—
566C22#338812yes—
567C22#339012yes—
568C4—28no—
569C4—24no—
570C4—24no—
571C4—44no—
572C4—48no—
573C4—42no—
574C4—42no—
575C4—81no—
576C4—82no—
577C4—81no—
578C4—82no—
579C4—162no—
580C4C2×#38411yescharacteristic
581C4#231711yescharacteristic
582C4D4×C4⋊C412yes—
583C4#244012yes—
584C4#250111yescharacteristic
585C4#250211yescharacteristic
586C2#1388011yescharacteristic
587C2#2087511yescharacteristic
588C2#2087711yescharacteristic
589C2#2267712yes—
590C2#2267912yes—
591C2#2576711yescharacteristic
592C2#2576811yescharacteristic
593C2#2584811yescharacteristic
594C2#2584911yescharacteristic
595C2#2605612yes—
596C2#2605712yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
40. #1763C4semidirect
190. C4×Q8C4⋊C4semidirect
275. #103C4⋊C4semidirect
456. Q8#384semidirect

Maximal subgroups

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

subgroupindex
C2×#23512
C2×#18652
C2×#18632
C2×#18472
C2×#18032
C2×#17632
#139102
#113282
#113272
#108502
#108352