Finite Group Databasebeta

← #366801

Subgroups — #366801

Order 512.

1458 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C211yesmaximal, characteristic
2C2×#1901C211yesmaximal, characteristic
3#20911C211yesmaximal, characteristic
4C2×#1855C211yesmaximal, characteristic
5C2×#1821C211yesmaximal, characteristic
6#20379C211yesmaximal, characteristic
7C2×#1736C211yesmaximal, characteristic
8#11365C212yesmaximal, complemented
9#11128C212yesmaximal, complemented
10#10942C212yesmaximal, complemented
11#10870C212yesmaximal, complemented
12C23×M16C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#484—24no—
17C2×#484—21no—
18C2×#484—21no—
19C2×#484—22no—
20C2×#484C418yescomplemented
21C22×#108C2212yes—
22C22×#107C2212yes—
23C2×#419C2211yescharacteristic
24C2×#419C2211yescharacteristic
25C22×#103C2212yes—
26C22×#101C2212yes—
27C2×#407C2211yescharacteristic
28C2×#407C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C2×#405C2212yes—
31#1901—24no—
32#1873—22no—
33#1872—22no—
34#1855—24no—
35#1821—24no—
36#1750—22no—
37#1749—21no—
38C4×C22⋊C8—21no—
39C2×#343C2212yes—
40C2×#343C2212yes—
41C2×#343C2212yes—
42C2×#341C2212yes—
43#1736—22no—
44#1736C2211yescharacteristic
45#1736C2212yes—
46#1736C2211yescharacteristic
47#1736C2212yes—
48C2×#340—22no—
49#1729C2212yes—
50C23×Q8C2311yescharacteristic
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C22×M16—21no—
54C22×M16—24no—
55C22×M16—22no—
56C22×M16—22no—
57C22×M16—42no—
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2312yes—
61C22×M16C2312yes—
62C23×C8—21no—
63C23×C8C2312yes—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8C2×C414yes—
72C2×#117—21no—
73C2×#117—21no—
74C2×#117—24no—
75C2×#117—42no—
76C2×#117D412yes—
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4C2×C414yes—
89C22×C4⋊C4D412yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C42—21no—
95#484—28no—
96#484—28no—
97#484—216no—
98C2×#108—24no—
99C2×#108—24no—
100C2×#108—42no—
101C2×#107—22no—
102C2×#107—22no—
103C2×#107—24no—
104C2×#107C2×C418yescomplemented
105#419—24no—
106#419—42no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103—24no—
110C2×#103—22no—
111C2×#103—22no—
112C2×#103C2×C418yescomplemented
113C2×#101—24no—
114C2×#101—24no—
115C2×#101—42no—
116#407—22no—
117#407—22no—
118#407—22no—
119#407D414yes—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8C2×C414yes—
125C2×C22⋊C8D414yes—
126C2×C22⋊C8D411yescharacteristic
127C2×C22⋊C8D411yescharacteristic
128C2×C22⋊C8D412yes—
129#405—24no—
130#405—42no—
131C2×#97—22no—
132C2×C4×C8—22no—
133#343—24no—
134#343—24no—
135#343—24no—
136#341—24no—
137#340—22no—
138#340—22no—
139#340—24no—
140#340—24no—
141#336—22no—
142#336—22no—
143C25#4111yescharacteristic
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8C22×C412yes—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—41no—
164C23×C4—41no—
165C23×C4—42no—
166C23×C4—84no—
167C23×C4C22×C412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D412yes—
174C23×C4#4111yescharacteristic
175C23×C4#4112yes—
176C23×C4C2411yescharacteristic
177C2×M16—24no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—48no—
197C2×M16—48no—
198C2×M16—42no—
199C2×M16—42no—
200C2×M16C2×D414yes—
201C2×M16#4114yes—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—24no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—42no—
212C22×C8—42no—
213C22×C8C22×C414yes—
214C22×C8C2×D412yes—
215C22×C8C2×D412yes—
216C22×C8C2×D414yes—
217C22⋊Q8—24no—
218C22⋊Q8—24no—
219C22⋊Q8—24no—
220C22⋊Q8—24no—
221C22⋊Q8—42no—
222C22⋊Q8—42no—
223C22⋊Q8—42no—
224C22⋊Q8—42no—
225#117—22no—
226#117—22no—
227#117—22no—
228#117—24no—
229#117—44no—
230#117—44no—
231#117—42no—
232#117#4114yes—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4C22×C414yes—
263C2×C4⋊C4C22×C412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—42no—
274C2×C42—21no—
275C2×C42—22no—
276C2×C42—21no—
277C2×C42—44no—
278#108—48no—
279#107—28no—
280#107—28no—
281#103—28no—
282#103—28no—
283#103—44no—
284#103—44no—
285#101—48no—
286C22⋊C8—24no—
287C22⋊C8—24no—
288C22⋊C8—42no—
289C22⋊C8—42no—
290C22⋊C8C22⋊C414yes—
291C22⋊C8C22⋊C414yes—
292C22⋊C8C22⋊C418yes—
293#97—44no—
294C4×C8—44no—
295C24—22no—
296C24—22no—
297C24—24no—
298C24—42no—
299C24C4×D412yes—
300C24#12111yescharacteristic
301C24#12111yescharacteristic
302C24#12412yes—
303C24C2×#4111yescharacteristic
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—42no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—24no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—44no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—82no—
362C22×C4—84no—
363C22×C4—84no—
364C22×C4—88no—
365C22×C4—82no—
366C22×C4—84no—
367C22×C4—88no—
368C22×C4C2×C22⋊C412yes—
369C22×C4C2×C22⋊C412yes—
370C22×C4C4×D412yes—
371C22×C4C4×D412yes—
372C22×C4C4×D412yes—
373C22×C4#12112yes—
374C22×C4#12112yes—
375C22×C4#12111yescharacteristic
376C22×C4#12111yescharacteristic
377C22×C4#12412yes—
378C22×C4#12712yes—
379C22×C4#12712yes—
380C22×C4C23×C411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4112yes—
387C22×C4C2×#4112yes—
388C22×C4H(5,𝔽2)12yes—
389C22×C42−1+412yes—
390M16—24no—
391M16—24no—
392M16—24no—
393M16—24no—
394M16—48no—
395M16—42no—
396M16—42no—
397M16—42no—
398M16—42no—
399M16—42no—
400M16—42no—
401M16—42no—
402M16—42no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—24no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—48no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8C2×C22⋊C414yes—
425C2×C8C2×C22⋊C414yes—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—41no—
436C4⋊C4—42no—
437C4⋊C4—41no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4#12314yes—
443C22⋊C4—24no—
444C22⋊C4—44no—
445C22⋊C4—42no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C42—24no—
449C42—44no—
450C42—42no—
451C23—21no—
452C23—21no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—21no—
458C23—21no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—22no—
463C23—22no—
464C23—42no—
465C23—42no—
466C23—48no—
467C23—48no—
468C23#39012yes—
469C23#39012yes—
470C23#47314yes—
471C23C2×C4×D411yescharacteristic
472C23C2×#12111yescharacteristic
473C23C2×#12411yescharacteristic
474C23#55112yes—
475C23#55212yes—
476Q8—42no—
477Q8—44no—
478Q8—42no—
479Q8—82no—
480Q8—81no—
481Q8—81no—
482C2×C4—24no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—88no—
511C2×C4—82no—
512C2×C4—88no—
513C2×C4—84no—
514C2×C4—88no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—84no—
520C2×C4—82no—
521C2×C4—84no—
522C2×C4#39012yes—
523C2×C4#39012yes—
524C2×C4C22×C22⋊C411yescharacteristic
525C2×C4C2×C4×D411yescharacteristic
526C2×C4C4×#4112yes—
527C2×C4#52012yes—
528C2×C4C2×#12111yescharacteristic
529C2×C4C2×#12311yescharacteristic
530C2×C4#52511yescharacteristic
531C2×C4#52912yes—
532C2×C4C2×#12711yescharacteristic
533C2×C4#53411yescharacteristic
534C2×C4#53611yescharacteristic
535C2×C4#53912yes—
536C2×C4#54612yes—
537C2×C4#54612yes—
538C2×C4#54612yes—
539C2×C4#54712yes—
540C2×C4#54712yes—
541C2×C4#54712yes—
542C8—24no—
543C8—24no—
544C8—44no—
545C8—44no—
546C8—81no—
547C8—81no—
548C22—21no—
549C22—21no—
550C22—22no—
551C22—24no—
552C22—24no—
553C22—22no—
554C22—24no—
555C22—24no—
556C22—21no—
557C22—21no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—48no—
562C22—48no—
563C22—42no—
564C22—42no—
565C22#197914yes—
566C22C2×#39011yescharacteristic
567C22#235912yes—
568C22#241211yescharacteristic
569C22#241311yescharacteristic
570C22#248812yes—
571C22C2×#47312yes—
572C22#306812yes—
573C4—22no—
574C4—22no—
575C4—44no—
576C4—82no—
577C4—84no—
578C4—82no—
579C4—82no—
580C4C4×#12311yescharacteristic
581C4#232211yescharacteristic
582C4#233912yes—
583C4#242512yes—
584C4#250111yescharacteristic
585C4#253911yescharacteristic
586C2—22no—
587C2—22no—
588C2—24no—
589C2—42no—
590C2#1399611yescharacteristic
591C2C2×#197911yescharacteristic
592C2#2124911yescharacteristic
593C2#2457212yes—
594C2#2462212yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #11365C2semidirect
9. #11128C2semidirect
10. #10942C2semidirect
11. #10870C2semidirect
20. C2×#484C4semidirect
104. C2×#107C2×C4semidirect
112. C2×#103C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4842
C2×#19012
#209112
C2×#18552
C2×#18212
#203792
C2×#17362
#113652
#111282
#109422
#108702