Finite Group Databasebeta

← #363181

Subgroups — #363181

Order 512.

2966 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#529C211yesmaximal, characteristic
2C2×#1989C212yesmaximal, complemented
3C2×#1983C212yesmaximal, complemented
4C2×#1797C211yesmaximal, characteristic, complemented
5C4×#380C211yesmaximal, characteristic, complemented
6#10970C212yesmaximal, complemented
7#10969C212yesmaximal
8#10789C212yesmaximal, complemented
9#10788C212yesmaximal, complemented
10C2×#529—24no—
11C2×#529—28no—
12C2×#529—28no—
13C2×#529C418yescomplemented
14C22×#126—21no—
15C22×#125C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×#123—21no—
18C22×C22⋊Q8C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4—21no—
22C22×C4×D4C2211yescharacteristic
23C22×#117—21no—
24C23×C42C2211yescharacteristic
25C22×C4⋊C8C2212yes—
26C22×#103C2214yes—
27C22×#102C2214yes—
28C22×C22⋊C8C2212yes—
29C2×#380C2212yes—
30C2×C4×C4⋊C4C2212yes—
31C2×C4×C22⋊C4—21no—
32#1989—28no—
33#1983—28no—
34#1797—24no—
35C2×#340C2214yes—
36C2×#340C2214yes—
37C2×#336C2212yes—
38C4⋊#95C2212yes—
39#1434—21no—
40C22⋊#95—21no—
41C4×#95—21no—
42C23×Q8C2311yescharacteristic
43C23×D4—21no—
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8C2314yes—
50#529—264no—
51#529—232no—
52#529—232no—
53C2×#126—216no—
54C2×#126—28no—
55C2×#125—24no—
56C2×#125—22no—
57C2×#125C2×C414yes—
58C2×#125D418yes—
59C2×#124—24no—
60C2×#124—22no—
61C2×#124C2×C414yes—
62C2×#124D418yes—
63C2×#123—216no—
64C2×#123—28no—
65C2×C22⋊Q8—24no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8C2×C414yes—
69C2×#121—24no—
70C2×#121—22no—
71C2×#121—24no—
72C2×#121C2×C414yes—
73C2×C4×Q8—24no—
74C2×C4×Q8—24no—
75C2×C4×Q8—22no—
76C2×C4×Q8C2×C414yes—
77C2×C4×D4—216no—
78C2×C4×D4—24no—
79C2×C4×D4—28no—
80C2×C4×D4—24no—
81C2×C4×D4—22no—
82C2×C4×D4C2×C414yes—
83C2×#117—216no—
84C2×#117—28no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—22no—
98C22×C42—21no—
99C22×C42—24no—
100C22×C42—24no—
101C22×C42—22no—
102C22×C42C2×C414yes—
103C22×C42D411yescharacteristic
104C22×C42D411yescharacteristic
105C22×C42D414yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C2×C4⋊C8—24no—
109C2×C4⋊C8—24no—
110C2×C4⋊C8—24no—
111C2×#103—28no—
112C2×#103—28no—
113C2×#103—44no—
114C2×#102—28no—
115C2×#102—28no—
116C2×#102—44no—
117C2×C22⋊C8—24no—
118C2×C22⋊C8—44no—
119#380—28no—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—24no—
122C4×C22⋊C4—44no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127#340—28no—
128#340—28no—
129#336—24no—
130C25—21no—
131C25C2×D411yescharacteristic
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—24no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8C22×C412yes—
138C22×Q8C2×D414yes—
139C22×D4—28no—
140C22×D4—24no—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—24no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—44no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4C22×C412yes—
150C22×D4C2×D414yes—
151C23×C4—28no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4C22⋊C414yes—
172C23×C4C22⋊C414yes—
173C23×C4C22×C412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2411yescharacteristic
181C22×C8—24no—
182C22×C8—24no—
183C22×C8—24no—
184C22×C8—24no—
185C22×C8—24no—
186C22×C8—24no—
187C22×C8—24no—
188#126—264no—
189#125—28no—
190#125—216no—
191#125C22⋊C4116yescomplemented
192#124—28no—
193#124—216no—
194#124C22⋊C4116yescomplemented
195#123—264no—
196C22⋊Q8—216no—
197C22⋊Q8—28no—
198C22⋊Q8—28no—
199#121—216no—
200#121—28no—
201#121—28no—
202C4×Q8—216no—
203C4×Q8—28no—
204C4×Q8—28no—
205C4×D4—264no—
206C4×D4—216no—
207C4×D4—28no—
208C4×D4—28no—
209#117—264no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C2×D414yes—
230C2×C4⋊C4C2×D414yes—
231C2×C4⋊C4#4118yes—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C42—28no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—28no—
252C2×C42—28no—
253C2×C42—216no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42—44no—
257C2×C42C22⋊C414yes—
258C2×C42C22⋊C414yes—
259C2×C42C22×C412yes—
260C2×C42C22×C412yes—
261C2×C42C2×D412yes—
262C2×C42C2×D414yes—
263C2×C42C2×D414yes—
264C2×C42C2×D412yes—
265C4⋊C8—48no—
266#103—416no—
267#102—416no—
268C22⋊C8—48no—
269#95—28no—
270#95—44no—
271#95—44no—
272#95—44no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—24no—
281C24—42no—
282C24—42no—
283C24—42no—
284C24—41no—
285C24—41no—
286C24—42no—
287C24—42no—
288C24C2×C22⋊C412yes—
289C24AGL(1,ℤ/8)12yes—
290C24#13712yes—
291C24C22×D411yescharacteristic
292C2×Q8—22no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—28no—
297C2×Q8—28no—
298C2×Q8—42no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8C2×C22⋊C414yes—
304C2×D4—216no—
305C2×D4—22no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—28no—
310C2×D4—28no—
311C2×D4—48no—
312C2×D4—416no—
313C2×D4—416no—
314C2×D4—48no—
315C2×D4—42no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4C2×C22⋊C414yes—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—216no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—28no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—416no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—432no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—48no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C412yes—
384C22×C4C2×C22⋊C412yes—
385C22×C4#12014yes—
386C22×C4#12014yes—
387C22×C4#13514yes—
388C22×C4C23×C411yescharacteristic
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C2×#4114yes—
393C22×C4H(5,𝔽2)12yes—
394C22×C42−1+412yes—
395C2×C8—44no—
396C2×C8—44no—
397C2×C8—44no—
398C2×C8—44no—
399C2×C8—44no—
400C2×C8—44no—
401C2×C8—44no—
402C4⋊C4—216no—
403C4⋊C4—216no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—216no—
409C4⋊C4—216no—
410C4⋊C4—28no—
411C4⋊C4—28no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4C2×C22⋊C414yes—
415C4⋊C4C2×C22⋊C414yes—
416C22⋊C4—216no—
417C22⋊C4—216no—
418C22⋊C4—216no—
419C22⋊C4—216no—
420C22⋊C4—216no—
421C22⋊C4—416no—
422C42—216no—
423C42—22no—
424C42—22no—
425C42—24no—
426C42—28no—
427C42—28no—
428C42—88no—
429C42C2×C22⋊C414yes—
430C42C2×C22⋊C414yes—
431C42#12018yes—
432C23—22no—
433C23—24no—
434C23—24no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—22no—
443C23—22no—
444C23—24no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—48no—
455C23—48no—
456C23—44no—
457C23—44no—
458C23—42no—
459C23—84no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—84no—
465C23#41714yes—
466C23C22×C22⋊C411yescharacteristic
467C23C2×#12012yes—
468C23#52512yes—
469C23#53712yes—
470C23C2×#13512yes—
471C23C2×AGL(1,ℤ/8)11yescharacteristic
472C23C2×#13711yescharacteristic
473C23#57512yes—
474C23#57512yes—
475Q8—28no—
476Q8—44no—
477Q8—44no—
478Q8—44no—
479D4—28no—
480D4—432no—
481D4—44no—
482D4—44no—
483D4—44no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—416no—
504C2×C4—416no—
505C2×C4—432no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—48no—
524C2×C4—48no—
525C2×C4—48no—
526C2×C4—84no—
527C2×C4—84no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4#37914yes—
531C2×C4#37914yes—
532C2×C4#41614yes—
533C2×C4C22×C22⋊C411yescharacteristic
534C2×C4C22×C22⋊C411yescharacteristic
535C2×C4C4×#4112yes—
536C2×C4#52012yes—
537C2×C4C2×#12012yes—
538C2×C4#53212yes—
539C2×C4#53511yescharacteristic
540C2×C4#53611yescharacteristic
541C2×C4#54614yes—
542C2×C4#54714yes—
543C2×C4#54814yes—
544C2×C4#55014yes—
545C8—84no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22—48no—
550C22—48no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—88no—
559C22—88no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22C2×#37911yescharacteristic
565C22#229711yescharacteristic
566C22#231111yescharacteristic
567C22#240714yes—
568C22C2×#41611yescharacteristic
569C22C2×#41711yescharacteristic
570C22#289611yescharacteristic
571C22#289611yescharacteristic
572C22#300214yes—
573C22#300612yes—
574C22#300712yes—
575C22#310214yes—
576C22#311412yes—
577C22#311712yes—
578C22#324712yes—
579C22#324812yes—
580C22#325614yes—
581C4—24no—
582C4—24no—
583C4—24no—
584C4—42no—
585C4—44no—
586C4—44no—
587C4—42no—
588C4—44no—
589C4—88no—
590C4#233912yes—
591C4#234112yes—
592C4#243112yes—
593C4#243212yes—
594C2—44no—
595C2—84no—
596C2#1393711yescharacteristic
597C2#2059711yescharacteristic
598C2#2060111yescharacteristic
599C2#2138711yescharacteristic
600C2#2139111yescharacteristic
601C2#2188711yescharacteristic
602C2#2189511yescharacteristic
603C2#2415914yes—
604C2#2416114yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×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
2. C2×#1989C2semidirect
3. C2×#1983C2semidirect
4. C2×#1797C2semidirect
5. C4×#380C2semidirect
6. #10970C2semidirect
8. #10789C2semidirect
9. #10788C2semidirect
13. C2×#529C4semidirect
191. #125C22⋊C4semidirect
194. #124C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5292
C2×#19892
C2×#19832
C2×#17972
C4×#3802
#109702
#109692
#107892
#107882