Finite Group Databasebeta

← #394825

Subgroups — #394825

Order 512.

2164 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C22×#389C211yesmaximal, characteristic
3C2×#2057C212yesmaximal, complemented
4C2×#1852C212yesmaximal
5#11328C214yesmaximal, complemented
6#11207C212yesmaximal, complemented
7#10730C212yesmaximal
8#10712C211yesmaximal, characteristic
9C22×C4⋊Q8C2211yescharacteristic
10C22×C22⋊Q8—21no—
11C23×C4⋊C4—21no—
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#107C2214yes—
16C22×C22⋊C8C2212yes—
17#2857—28no—
18C2×#391—21no—
19C2×#389—21no—
20C2×#389—21no—
21C2×#389—21no—
22C2×#389—28no—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#384C2211yescharacteristic
27C2×#380—21no—
28C2×#380C2212yes—
29C22×#95—21no—
30#2057—28no—
31#1852—28no—
32C2×#340C2214yes—
33C2×#340C2214yes—
34C2×#340C2214yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C23×Q8—21no—
38C24×C4—21no—
39C24×C4—21no—
40C24×C4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8C2314yes—
44C2×C4⋊Q8—24no—
45C2×C4⋊Q8—24no—
46C2×C4⋊Q8—42no—
47C2×C22⋊Q8—28no—
48C2×C22⋊Q8—44no—
49C2×C22⋊Q8—44no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—24no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4—28no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—42no—
65C22×C4⋊C4—44no—
66C22×C4⋊C4—44no—
67C22×C4⋊C4Q814yescomplemented
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—22no—
73C22×C42—24no—
74C22×C42—24no—
75C22×C42—42no—
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#107—28no—
81C2×#107—28no—
82C2×#107—44no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8Q818yescomplemented
86#391—28no—
87#389—28no—
88#389—28no—
89#389—28no—
90#389—24no—
91#389—24no—
92#389—24no—
93#389—24no—
94#389—24no—
95#389D418yes—
96#384—24no—
97#380—28no—
98#380—24no—
99#380D418yes—
100C2×#95—28no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105#340—28no—
106#340—28no—
107#340—28no—
108#336—24no—
109#336—42no—
110C25C2×D411yescharacteristic
111C22×Q8—24no—
112C22×Q8—42no—
113C22×Q8—42no—
114C22×Q8—42no—
115C22×Q8—42no—
116C22×Q8—42no—
117C22×Q8—42no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—24no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—41no—
139C23×C4—41no—
140C23×C4—41no—
141C23×C4—41no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4D814yes—
153C23×C4Q1614yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D414yes—
160C23×C4C2×Q814yes—
161C23×C4C2×Q812yes—
162C23×C4#4112yes—
163C23×C4C2411yescharacteristic
164C22×C8—24no—
165C22×C8—24no—
166C22×C8—24no—
167C22×C8—24no—
168C22×C8—24no—
169C22×C8—24no—
170C22×C8C2×Q818yes—
171C4⋊Q8—416no—
172C22⋊Q8—432no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—216no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—216no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—416no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—432no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4C2×D414yes—
217C2×C4⋊C4C2×Q814yes—
218C2×C4⋊C4#4114yes—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—416no—
232C2×C42—41no—
233C2×C42—41no—
234C2×C42C2×D414yes—
235#107—416no—
236C22⋊C8—216no—
237#95—28no—
238#95—28no—
239#95—28no—
240#95—28no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24#12012yes—
250C24C22⋊Q812yes—
251C24C4⋊Q812yes—
252C24C2×D812yes—
253C24C2×Q1612yes—
254C24AGL(1,ℤ/8)12yes—
255C24#13712yes—
256C24C22×D411yescharacteristic
257C2×Q8—48no—
258C2×Q8—48no—
259C2×Q8—48no—
260C2×Q8—84no—
261C2×Q8—84no—
262C2×Q8—84no—
263C2×Q8—84no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—28no—
281C22×C4—28no—
282C22×C4—28no—
283C22×C4—28no—
284C22×C4—216no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—24no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—28no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—48no—
318C22×C4—48no—
319C22×C4—48no—
320C22×C4—48no—
321C22×C4—48no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—48no—
327C22×C4—48no—
328C22×C4—48no—
329C22×C4—48no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—84no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4#12012yes—
353C22×C4#12012yes—
354C22×C4#12012yes—
355C22×C4C22⋊Q818yes—
356C22×C4C22⋊Q812yes—
357C22×C4C4⋊Q818yes—
358C22×C4C4⋊Q812yes—
359C22×C4C2×D812yes—
360C22×C4C2×D812yes—
361C22×C4C2×Q1612yes—
362C22×C4C2×Q1612yes—
363C22×C4AGL(1,ℤ/8)12yes—
364C22×C4#13712yes—
365C22×C4C22×D412yes—
366C22×C4C22×D412yes—
367C22×C4C22×Q811yescharacteristic
368C22×C4C22×Q812yes—
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4112yes—
371C22×C4H(5,𝔽2)12yes—
372C22×C42−1+412yes—
373C2×C8—28no—
374C2×C8—28no—
375C2×C8—28no—
376C2×C8—44no—
377C2×C8—44no—
378C2×C8—44no—
379C2×C8—44no—
380C4⋊C4—216no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—28no—
385C4⋊C4—28no—
386C4⋊C4—28no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C4⋊C4—416no—
390C4⋊C4—48no—
391C4⋊C4—416no—
392C4⋊C4—48no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—416no—
397C4⋊C4C22⋊Q818yescomplemented
398C22⋊C4—416no—
399C42—28no—
400C42—44no—
401C42—44no—
402C42—44no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23#39514yes—
432C23#44714yes—
433C23#45114yes—
434C23#47414yes—
435C23#47714yes—
436C23#50014yes—
437C23#50114yes—
438C23C2×#12011yescharacteristic
439C23C2×#12011yescharacteristic
440C23C2×C22⋊Q811yescharacteristic
441C23#52511yescharacteristic
442C23C2×C4⋊Q811yescharacteristic
443C23#53411yescharacteristic
444C23#53611yescharacteristic
445C23C22×D811yescharacteristic
446C23C22×Q1611yescharacteristic
447C23C2×AGL(1,ℤ/8)11yescharacteristic
448C23C2×AGL(1,ℤ/8)11yescharacteristic
449C23C2×#13711yescharacteristic
450C23C2×#13711yescharacteristic
451C23#57511yescharacteristic
452C23#57511yescharacteristic
453Q8—816no—
454C2×C4—22no—
455C2×C4—22no—
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—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—216no—
473C2×C4—216no—
474C2×C4—28no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—416no—
482C2×C4—416no—
483C2×C4—432no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—44no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—816no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
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#39514yes—
521C2×C4#44714yes—
522C2×C4#45114yes—
523C2×C4C2×C22⋊Q812yes—
524C2×C4C2×C22⋊Q812yes—
525C2×C4#52512yes—
526C2×C4C2×C4⋊Q812yes—
527C2×C4#53912yes—
528C2×C4#54412yes—
529C2×C4D4214yes—
530C2×C4#54714yes—
531C2×C4D4×Q814yes—
532C2×C4#55414yes—
533C8—48no—
534C22—24no—
535C22—24no—
536C22—24no—
537C22—24no—
538C22—24no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22#205718yescomplemented
549C22C2×#39511yescharacteristic
550C22#240311yescharacteristic
551C22#240611yescharacteristic
552C22#243411yescharacteristic
553C22#258311yescharacteristic
554C22#262711yescharacteristic
555C22#284811yescharacteristic
556C22C2×#44711yescharacteristic
557C22C2×#44711yescharacteristic
558C22C2×#45111yescharacteristic
559C22C2×#45111yescharacteristic
560C22#300611yescharacteristic
561C22#300611yescharacteristic
562C22#300711yescharacteristic
563C22#300711yescharacteristic
564C22C2×#47411yescharacteristic
565C22C2×#47711yescharacteristic
566C22#307711yescharacteristic
567C22#307811yescharacteristic
568C22#310011yescharacteristic
569C22#310411yescharacteristic
570C22#311011yescharacteristic
571C22#311211yescharacteristic
572C22C2×#50011yescharacteristic
573C22C2×#50111yescharacteristic
574C22#316511yescharacteristic
575C22#316611yescharacteristic
576C22#318611yescharacteristic
577C22#318911yescharacteristic
578C22#319211yescharacteristic
579C22#319511yescharacteristic
580C22#322611yescharacteristic
581C22#323011yescharacteristic
582C22#323411yescharacteristic
583C22#323611yescharacteristic
584C4—24no—
585C4—24no—
586C4—24no—
587C4—42no—
588C4—42no—
589C4—48no—
590C4—84no—
591C4—84no—
592C4—84no—
593C4#244014yes—
594C4#244914yes—
595C2—28no—
596C2#1472011yescharacteristic
597C2C2×#205711yescharacteristic
598C2#2355411yescharacteristic
599C2#2391811yescharacteristic
600C2#2392111yescharacteristic
601C2#2406011yescharacteristic
602C2#2406411yescharacteristic
603C2#2581311yescharacteristic
604C2#2581911yescharacteristic
605C2#2929611yescharacteristic
606C2#2930211yescharacteristic
607C2#2996611yescharacteristic
608C2#2996911yescharacteristic
609C2#3295311yescharacteristic
610C2#3295711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2057C2semidirect
5. #11328C2semidirect
6. #11207C2semidirect
67. C22×C4⋊C4Q8semidirect
85. C2×C22⋊C8Q8semidirect
397. C4⋊C4C22⋊Q8semidirect
548. C22#2057semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C22×#3892
C2×#20572
C2×#18522
#113282
#112072
#107302
#107122