Finite Group Databasebeta

← #366285

Subgroups — #366285

Order 512.

2078 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#419C211yesmaximal, characteristic
2C2×#1933C212yesmaximal, complemented
3C2×#1787C211yesmaximal, characteristic
4C2×#1753C211yesmaximal, characteristic
5#14129C211yesmaximal, characteristic
6#13922C211yesmaximal, characteristic
7#11190C212yesmaximal, complemented
8#11067C212yesmaximal, complemented
9#10928C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic
11#10722C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C2×#419—28no—
16C2×#419—24no—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×#97C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#389C2212yes—
27C2×#380C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30#1933—28no—
31#1787C418yescomplemented
32#1753—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1446—22no—
40#1439—22no—
41#1436—22no—
42C4⋊#95C2212yes—
43C22⋊#95—22no—
44C23×Q8—21no—
45C23×Q8—22no—
46C23×Q8—24no—
47C23×Q8—24no—
48C23×Q8—22no—
49C23×Q8—22no—
50C23×Q8—22no—
51C23×Q8—22no—
52C23×Q8—42no—
53C23×Q8—42no—
54C23×Q8C2×C414yes—
55C23×Q8C2312yes—
56C24×C4—21no—
57C24×C4C2311yescharacteristic
58C22×M16—22no—
59C22×M16—22no—
60C22×M16—24no—
61C22×M16—42no—
62C22×M16D414yes—
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—24no—
70C2×#117—42no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4D412yes—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#419—232no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#103C2×C418yes—
93C2×#103C2×C418yes—
94C2×#103D418yes—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8C2×C414yes—
98C2×#97—22no—
99C2×#97—42no—
100C2×C4×C8—24no—
101C2×C4×C8—22no—
102#391—24no—
103#391—24no—
104#391—42no—
105#391—42no—
106#389—24no—
107#389—42no—
108#380—22no—
109#380—22no—
110#380—22no—
111#380C2×C414yes—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95D412yes—
124#340—24no—
125#340—24no—
126#340—42no—
127#340—42no—
128#336—22no—
129#336—22no—
130C25C2×D411yescharacteristic
131C22×Q8—24no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—28no—
136C22×Q8—216no—
137C22×Q8—28no—
138C22×Q8—216no—
139C22×Q8—28no—
140C22×Q8—28no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—44no—
150C22×Q8—44no—
151C22×Q8—48no—
152C22×Q8—48no—
153C22×Q8—48no—
154C22×Q8—48no—
155C22×Q8—48no—
156C22×Q8—48no—
157C22×Q8—48no—
158C22×Q8—44no—
159C22×Q8—44no—
160C22×Q8—48no—
161C22×Q8—42no—
162C22×Q8—44no—
163C22×Q8—42no—
164C22×Q8—44no—
165C22×Q8C22×C414yes—
166C22×Q8C2×D414yes—
167C22×Q8#4114yes—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4C22×C412yes—
182C23×C4C2×D412yes—
183C23×C4C2×D412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D412yes—
186C23×C4#4112yes—
187C23×C4#4112yes—
188C23×C4#4112yes—
189C23×C4#4112yes—
190C23×C4C2411yescharacteristic
191C2×M16—28no—
192C2×M16—28no—
193C2×M16—44no—
194C2×M16—48no—
195C2×M16—48no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—22no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8C22×C412yes—
212C22×C8C22×C412yes—
213C22×C8C2×D412yes—
214C22×C8C2×D414yes—
215C22×C8C2×D412yes—
216#117—28no—
217#117—48no—
218#117—44no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—82no—
238C2×C4⋊C4C22×C414yes—
239C2×C4⋊C4C2×D414yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—82no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—44no—
258C2×C42—42no—
259#103—216no—
260#103—216no—
261#103—216no—
262#103—48no—
263C22⋊C8—28no—
264#97—44no—
265C4×C8—44no—
266#95—24no—
267#95—24no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—42no—
275#95—82no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24C4×D412yes—
283C24#12112yes—
284C24#12412yes—
285C24C22×D411yescharacteristic
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—232no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—416no—
303C2×Q8—432no—
304C2×Q8—48no—
305C2×Q8—416no—
306C2×Q8—48no—
307C2×Q8—416no—
308C2×Q8—416no—
309C2×Q8—432no—
310C2×Q8—416no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—48no—
324C2×Q8C4×D418yes—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—416no—
360C22×C4—48no—
361C22×C4—416no—
362C22×C4—48no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—48no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4#11714yes—
388C22×C4C4×D414yes—
389C22×C4C4×D412yes—
390C22×C4#12112yes—
391C22×C4#12314yes—
392C22×C4#12412yes—
393C22×C4#12614yes—
394C22×C4#13714yes—
395C22×C4C23×C411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D412yes—
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4112yes—
402C22×C4C2×#4111yescharacteristic
403C22×C4H(5,𝔽2)12yes—
404C22×C4H(5,𝔽2)12yes—
405M16—416no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—41no—
418C2×C8—42no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8#12014yes—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—88no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—84no—
442C22⋊C4—84no—
443C42—48no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—42no—
449C42—42no—
450C23—24no—
451C23—24no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—22no—
457C23—22no—
458C23—24no—
459C23—22no—
460C23—24no—
461C23—24no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23#38814yes—
471C23C2×C4×D411yescharacteristic
472C23C2×#12111yescharacteristic
473C23C2×#12411yescharacteristic
474C23D4212yes—
475C23#54612yes—
476C23C2×#13712yes—
477C23#57512yes—
478C23#57712yes—
479C23#57812yes—
480Q8—28no—
481Q8—28no—
482Q8—432no—
483Q8—44no—
484Q8—44no—
485Q8—44no—
486Q8—44no—
487Q8—44no—
488Q8—44no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—28no—
495C2×C4—24no—
496C2×C4—28no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—432no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4#38814yes—
543C2×C4C2×#11711yescharacteristic
544C2×C4C2×C4×D411yescharacteristic
545C2×C4C2×C4×D412yes—
546C2×C4#51812yes—
547C2×C4C2×#12011yescharacteristic
548C2×C4C2×#12311yescharacteristic
549C2×C4#52512yes—
550C2×C4#52511yescharacteristic
551C2×C4C2×#12611yescharacteristic
552C2×C4#53411yescharacteristic
553C2×C4#53511yescharacteristic
554C2×C4#53812yes—
555C2×C4#54612yes—
556C2×C4#54612yes—
557C2×C4#54612yes—
558C2×C4#55112yes—
559C2×C4#55112yes—
560C2×C4#55112yes—
561C8—24no—
562C8—42no—
563C8—42no—
564C8—42no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22C2×#38811yescharacteristic
575C22#235812yes—
576C22#240811yescharacteristic
577C22#247212yes—
578C22#253411yescharacteristic
579C22C4×#13712yes—
580C22#294911yescharacteristic
581C22#295011yescharacteristic
582C22#305612yes—
583C22#306312yes—
584C22#313712yes—
585C22#313911yescharacteristic
586C22#314011yescharacteristic
587C22#326612yes—
588C22#326912yes—
589C22#327012yes—
590C22#327112yes—
591C22#332612yes—
592C22#332712yes—
593C22#332812yes—
594C22#332912yes—
595C4—24no—
596C4—48no—
597C4—44no—
598C4—84no—
599C4—82no—
600C4—84no—
601C4—162no—
602C4C4×#12011yescharacteristic
603C4#232111yescharacteristic
604C4#234212yes—
605C4#240511yescharacteristic
606C4#248012yes—
607C4#248211yescharacteristic
608C2—44no—
609C2#1399111yescharacteristic
610C2#2108811yescharacteristic
611C2#2109411yescharacteristic
612C2#2288112yes—
613C2#2288512yes—
614C2#2427412yes—
615C2#2727512yes—
616C2#2727712yes—
617C2#2811212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1933C2semidirect
7. #11190C2semidirect
8. #11067C2semidirect
9. #10928C2semidirect
31. #1787C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4192
C2×#19332
C2×#17872
C2×#17532
#141292
#139222
#111902
#110672
#109282
#107312
#107222