Finite Group Databasebeta

← #366933

Subgroups — #366933

Order 512.

1398 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic
3C2×#1987C211yesmaximal, characteristic
4C2×#1940C211yesmaximal, characteristic
5C2×#1805C211yesmaximal, characteristic
6C2×#1763C211yesmaximal, characteristic
7#13804C211yesmaximal, characteristic
8#11358C212yesmaximal
9#11077C212yesmaximal
10#10932C212yesmaximal
11#10850C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×#97C2211yescharacteristic
20#2568—24no—
21#2568C418yescomplemented
22C2×#400—21no—
23C2×#400—21no—
24C2×#391—22no—
25C2×#391C2212yes—
26C2×#382C2212yes—
27C2×#382C2211yescharacteristic
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1993—24no—
32#1987—24no—
33#1940—24no—
34#1805—24no—
35#1763—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2211yescharacteristic
43C2×#336C2212yes—
44#1436—22no—
45C4⋊#95C2212yes—
46#1434—21no—
47C4×#95—21no—
48C23×Q8—21no—
49C23×Q8C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yes—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8C2×C414yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4D412yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C42—21no—
74C22×C42—21no—
75C22×C42—22no—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78C2×C4⋊C8—24no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—42no—
84C2×#103—22no—
85C2×#103—22no—
86C2×#103—22no—
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—42no—
92C2×#103Q814yescomplemented
93C2×#97—22no—
94C2×#97—22no—
95C2×#97—22no—
96#400—28no—
97#400—28no—
98#391—216no—
99#391—24no—
100#391C2×C418yes—
101#382—24no—
102#382—22no—
103#382C2×C418yes—
104#382D414yes—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—42no—
108C4×C4⋊C4C2×C414yes—
109C4×C4⋊C4D414yes—
110C4×C4⋊C4D414yes—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95D411yescharacteristic
122C2×#95D411yescharacteristic
123#340—24no—
124#340—24no—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—22no—
129#340—22no—
130#336—24no—
131C22×Q8—24no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—44no—
144C22×Q8—44no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8C22×C412yes—
148C22×Q8C2×Q812yes—
149C22×Q8#4112yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8C2×Q812yes—
181C22×C8#4112yes—
182C4⋊Q8—28no—
183C4⋊Q8—28no—
184C4⋊Q8—44no—
185C4⋊Q8—44no—
186C4×Q8—28no—
187C4×Q8—28no—
188C4×Q8—44no—
189C4×Q8—44no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4C22⋊C418yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C414yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D414yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×Q812yes—
228C2×C4⋊C4#4112yes—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—44no—
238C2×C42—44no—
239C2×C42—82no—
240C2×C42—82no—
241C2×C42C22×C414yes—
242C2×C42C22×C412yes—
243C2×C42C2×D414yes—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42#4112yes—
248C2×C42#4112yes—
249C4⋊C8—48no—
250C4⋊C8—44no—
251#103—28no—
252#103—28no—
253#103—48no—
254#97—44no—
255#95—28no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—44no—
263#95—44no—
264#95—82no—
265#95C22⋊C414yes—
266#95C22⋊C414yes—
267C24C22×D411yescharacteristic
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—416no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—48no—
297C2×Q8—84no—
298C2×Q8—84no—
299C2×Q8—88no—
300C2×Q8C4×Q814yes—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4C2×C22⋊C414yes—
352C22×C4C2×C22⋊C414yes—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C412yes—
355C22×C4#12112yes—
356C22×C4#12112yes—
357C22×C4#12112yes—
358C22×C4#12112yes—
359C22×C4C23×C411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×Q811yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4H(5,𝔽2)11yescharacteristic
370C22×C42−1+411yescharacteristic
371C22×C42−1+411yescharacteristic
372C22×C42−1+411yescharacteristic
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—48no—
406C4⋊C4—41no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4C22⋊Q814yes—
417C42—24no—
418C42—24no—
419C42—24no—
420C42—24no—
421C42—48no—
422C42—84no—
423C42—88no—
424C42C4×D414yes—
425C42C4×D414yes—
426C42#12414yes—
427C42#12714yes—
428C23C22×C22⋊C411yescharacteristic
429C23C2×#12111yescharacteristic
430C23C2×#12111yescharacteristic
431C23#54611yescharacteristic
432C23#54711yescharacteristic
433C23#54811yescharacteristic
434C23D4×Q811yescharacteristic
435C23#57512yes—
436C23#57512yes—
437C23#57712yes—
438C23#57812yes—
439Q8—24no—
440Q8—24no—
441Q8—42no—
442Q8—42no—
443Q8—42no—
444Q8—42no—
445Q8—42no—
446Q8—42no—
447Q8—816no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—48no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—84no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4#39014yes—
505C2×C4#39014yes—
506C2×C4C2×C4×D411yescharacteristic
507C2×C4C2×C4×D411yescharacteristic
508C2×C4C2×C4×Q811yescharacteristic
509C2×C4C4×#4111yescharacteristic
510C2×C4#51911yescharacteristic
511C2×C4#52011yescharacteristic
512C2×C4C2×C22⋊Q811yescharacteristic
513C2×C4#52511yescharacteristic
514C2×C4C2×#12411yescharacteristic
515C2×C4#52911yescharacteristic
516C2×C4C2×#12711yescharacteristic
517C2×C4#53311yescharacteristic
518C2×C4#53611yescharacteristic
519C2×C4#53711yescharacteristic
520C2×C4#54111yescharacteristic
521C2×C4#54211yescharacteristic
522C2×C4#54611yescharacteristic
523C2×C4#54611yescharacteristic
524C2×C4#54711yescharacteristic
525C2×C4#54711yescharacteristic
526C2×C4#54811yescharacteristic
527C2×C4#54811yescharacteristic
528C2×C4D4×Q811yescharacteristic
529C2×C4D4×Q811yescharacteristic
530C2×C4#55211yescharacteristic
531C2×C4#55211yescharacteristic
532C2×C4#55411yescharacteristic
533C2×C4#55611yescharacteristic
534C8—42no—
535C8—42no—
536C8—82no—
537C22C2×#39011yescharacteristic
538C22#233911yescharacteristic
539C22Q8×C22⋊C411yescharacteristic
540C22#241511yescharacteristic
541C22#241711yescharacteristic
542C22#242711yescharacteristic
543C22#242911yescharacteristic
544C22#289612yes—
545C22#289812yes—
546C22#304011yescharacteristic
547C22#304011yescharacteristic
548C22#304012yes—
549C22#304211yescharacteristic
550C22#304411yescharacteristic
551C22#304612yes—
552C22#331112yes—
553C22#331312yes—
554C22#334912yes—
555C22#335012yes—
556C22#337112yes—
557C22#337712yes—
558C22#338812yes—
559C22#339012yes—
560C4—24no—
561C4—24no—
562C4—44no—
563C4—44no—
564C4—44no—
565C4—44no—
566C4—84no—
567C4—82no—
568C4—82no—
569C4—84no—
570C4C4×C22⋊Q811yescharacteristic
571C4#232311yescharacteristic
572C4#236011yescharacteristic
573C4#236211yescharacteristic
574C4#248911yescharacteristic
575C4#249211yescharacteristic
576C4#254011yescharacteristic
577C4#254211yescharacteristic
578C2#1399711yescharacteristic
579C2#2126111yescharacteristic
580C2#2126211yescharacteristic
581C2#2249412yes—
582C2#2252712yes—
583C2#2469711yescharacteristic
584C2#2469911yescharacteristic
585C2#2483211yescharacteristic
586C2#2483411yescharacteristic
587C2#2533712yes—
588C2#2546812yes—
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
21. #2568C4semidirect
92. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#19932
C2×#19872
C2×#19402
C2×#18052
C2×#17632
#138042
#113582
#110772
#109322
#108502