Finite Group Databasebeta

← #358289

Subgroups — #358289

Order 512.

1794 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34052C211yesmaximal, characteristic
2C2×#1742C211yesmaximal, characteristic
3C2×#1736C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5C2×#1733C211yesmaximal, characteristic
6C2×#1733C211yesmaximal, characteristic
7#13795C211yesmaximal, characteristic
8#11276C218yesmaximal, complemented
9C23×M16C2211yescharacteristic
10C23×M16C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16#2317—22no—
17#2317C2214yes—
18#2316—22no—
19#2315—22no—
20#2296—22no—
21#2296—22no—
22#2285—22no—
23#1742—24no—
24C2×#343C2214yes—
25#1736—22no—
26#1736—22no—
27#1736C2214yes—
28#1736C2214yes—
29C2×#340—22no—
30C2×#340—22no—
31#1733—24no—
32#1733C418yescomplemented
33C2×#339C2214yes—
34C2×#339C2214yes—
35C4⋊#95—22no—
36C4⋊#95C2214yes—
37C22⋊#95—22no—
38C24×C4—21no—
39C24×C4C2311yescharacteristic
40C22×M16—21no—
41C22×M16—21no—
42C22×M16—22no—
43C22×M16—22no—
44C22×M16—22no—
45C22×M16—22no—
46C22×M16—22no—
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—22no—
52C22×M16C2×C414yes—
53C22×M16D414yes—
54C22×M16C2312yes—
55C22×M16C2312yes—
56C23×C8—21no—
57C23×C8—21no—
58C2×#117—21no—
59C2×#117—21no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—21no—
63C2×#117—21no—
64C2×#117—21no—
65C2×#117—22no—
66C2×#117—21no—
67C2×#117—21no—
68C2×#117—21no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117—22no—
74C2×#117—22no—
75C2×#117—22no—
76C2×#117—22no—
77C2×#117—22no—
78C2×#117—22no—
79C2×#117—22no—
80C2×#117—22no—
81C2×#117C2×C414yes—
82C2×#117C2×C414yes—
83C2×#117D412yes—
84C2×#117D412yes—
85C2×#117Q814yes—
86C2×#117C2312yes—
87C2×#117C2312yes—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4D412yes—
98C22×C4⋊C4D412yes—
99C22×C4⋊C4Q814yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C22⋊C4—21no—
103C22×C22⋊C4—21no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4D412yes—
106C22×C42—21no—
107C22×C42—21no—
108C22×C42D412yes—
109C22×C42C2312yes—
110#389—24no—
111#389—24no—
112#389—24no—
113#389—24no—
114#385—28no—
115#385—28no—
116#384—24no—
117#382—24no—
118#380—28no—
119C4×C4⋊C4—28no—
120C2×#95—28no—
121#343—28no—
122#340—28no—
123#340—28no—
124#339—28no—
125#339C2×C4116yescomplemented
126C25C2×D411yescharacteristic
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—41no—
133C23×C4—41no—
134C23×C4—42no—
135C23×C4—84no—
136C23×C4C22×C412yes—
137C23×C4C2×D414yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×Q812yes—
145C23×C4C2×Q812yes—
146C23×C4#4112yes—
147C23×C4#4112yes—
148C23×C4#4112yes—
149C23×C4#4112yes—
150C23×C4#4112yes—
151C23×C4#4112yes—
152C23×C4#4112yes—
153C23×C4#4112yes—
154C23×C4C2411yescharacteristic
155C2×M16—24no—
156C2×M16—24no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—28no—
160C2×M16—28no—
161C2×M16—28no—
162C2×M16—28no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—42no—
172C2×M16—42no—
173C2×M16—44no—
174C2×M16—42no—
175C2×M16—42no—
176C2×M16—44no—
177C2×M16—44no—
178C2×M16—44no—
179C2×M16C22×C414yes—
180C2×M16C2×D414yes—
181C22×C8—24no—
182C22×C8—24no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—42no—
186C22×C8—42no—
187C22×C8—42no—
188C22×C8—42no—
189#117—24no—
190#117—24no—
191#117—24no—
192#117—24no—
193#117—24no—
194#117—22no—
195#117—22no—
196#117—28no—
197#117—28no—
198#117—24no—
199#117—24no—
200#117—22no—
201#117—22no—
202#117—22no—
203#117—22no—
204#117—42no—
205#117—42no—
206#117—42no—
207#117—42no—
208#117—42no—
209#117—42no—
210#117—44no—
211#117—42no—
212#117—42no—
213#117—42no—
214#117—42no—
215#117—42no—
216#117—42no—
217#117—44no—
218#117C4⋊C418yes—
219#117C22×C414yes—
220#117C2×Q814yes—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4C4⋊C418yes—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×Q814yes—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C42—24no—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42C22×C414yes—
291#95—28no—
292#95—28no—
293#95—88no—
294C24—42no—
295C24—42no—
296C24C4×D412yes—
297C24#12011yescharacteristic
298C24#12011yescharacteristic
299C24#12112yes—
300C24C22⋊Q812yes—
301C24C22⋊Q812yes—
302C24#12312yes—
303C24#12412yes—
304C24C22×D411yescharacteristic
305C22×C4—24no—
306C22×C4—28no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—48no—
317C22×C4—44no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—88no—
330C22×C4—84no—
331C22×C4—84no—
332C22×C4—88no—
333C22×C4C2×C4⋊C412yes—
334C22×C4C2×C4⋊C412yes—
335C22×C4#11714yes—
336C22×C4#11714yes—
337C22×C4C4×D412yes—
338C22×C4C4×D412yes—
339C22×C4C4×D414yes—
340C22×C4C4×D412yes—
341C22×C4C4×Q814yes—
342C22×C4#12011yescharacteristic
343C22×C4#12011yescharacteristic
344C22×C4#12112yes—
345C22×C4#12112yes—
346C22×C4#12112yes—
347C22×C4#12112yes—
348C22×C4#12112yes—
349C22×C4C22⋊Q812yes—
350C22×C4C22⋊Q812yes—
351C22×C4C22⋊Q812yes—
352C22×C4C22⋊Q812yes—
353C22×C4C22⋊Q812yes—
354C22×C4C22⋊Q812yes—
355C22×C4C22⋊Q812yes—
356C22×C4C22⋊Q812yes—
357C22×C4#12312yes—
358C22×C4#12312yes—
359C22×C4#12312yes—
360C22×C4#12312yes—
361C22×C4#12314yes—
362C22×C4#12314yes—
363C22×C4#12312yes—
364C22×C4#12412yes—
365C22×C4#12412yes—
366C22×C4#12412yes—
367C22×C4#12412yes—
368C22×C4#12412yes—
369C22×C4#12412yes—
370C22×C4#12412yes—
371C22×C4#12514yes—
372C22×C4#12514yes—
373C22×C4#12514yes—
374C22×C4#12514yes—
375C22×C4#12514yes—
376C22×C4#12614yes—
377C22×C4#12614yes—
378C22×C4#12614yes—
379C22×C4#12614yes—
380C22×C4C4⋊Q814yes—
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×Q811yescharacteristic
386C22×C4C22×Q811yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395M16—24no—
396M16—24no—
397M16—24no—
398M16—24no—
399M16—42no—
400M16—42no—
401M16—42no—
402M16—42no—
403M16—48no—
404M16—48no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—48no—
412C2×C8—48no—
413C4⋊C4—24no—
414C4⋊C4—28no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—48no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—88no—
430C4⋊C4—88no—
431C4⋊C4C2×C4⋊C418yes—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—28no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—48no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C42—24no—
442C42—24no—
443C42—28no—
444C42—24no—
445C42—24no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—44no—
451C42—44no—
452C42—44no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—42no—
460C23—42no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—42no—
465C23—44no—
466C23—42no—
467C23#38512yes—
468C23#38512yes—
469C23#38714yes—
470C23#38814yes—
471C23#39312yes—
472C23#39312yes—
473C23#39412yes—
474C23#39412yes—
475C23#39814yes—
476C23#40014yes—
477C23C2×C4×D411yescharacteristic
478C23C2×#12011yescharacteristic
479C23C2×#12111yescharacteristic
480C23C2×C22⋊Q811yescharacteristic
481C23C2×C22⋊Q811yescharacteristic
482C23C2×#12311yescharacteristic
483C23C2×#12411yescharacteristic
484C2×C4—28no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—48no—
488C2×C4—48no—
489C2×C4—48no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—88no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—84no—
502C2×C4—84no—
503C2×C4—88no—
504C2×C4—88no—
505C2×C4—88no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4#38214yes—
509C2×C4#38214yes—
510C2×C4#38512yes—
511C2×C4#38512yes—
512C2×C4#38714yes—
513C2×C4#38714yes—
514C2×C4#38714yes—
515C2×C4#38814yes—
516C2×C4#38814yes—
517C2×C4#38814yes—
518C2×C4#38914yes—
519C2×C4#38914yes—
520C2×C4#39312yes—
521C2×C4#39312yes—
522C2×C4#39412yes—
523C2×C4#39412yes—
524C2×C4#39714yes—
525C2×C4#39714yes—
526C2×C4#39814yes—
527C2×C4#39814yes—
528C2×C4#39814yes—
529C2×C4#40014yes—
530C2×C4#40014yes—
531C2×C4#40014yes—
532C2×C4#40014yes—
533C2×C4#40014yes—
534C2×C4C22×C4⋊C411yescharacteristic
535C2×C4C2×#11711yescharacteristic
536C2×C4C2×#11711yescharacteristic
537C2×C4C2×C4×D411yescharacteristic
538C2×C4C2×C4×D411yescharacteristic
539C2×C4C2×C4×Q811yescharacteristic
540C2×C4C2×#12111yescharacteristic
541C2×C4C2×#12111yescharacteristic
542C2×C4C2×C22⋊Q811yescharacteristic
543C2×C4C2×C22⋊Q811yescharacteristic
544C2×C4C2×C22⋊Q811yescharacteristic
545C2×C4C2×#12311yescharacteristic
546C2×C4C2×#12311yescharacteristic
547C2×C4C2×#12311yescharacteristic
548C2×C4C2×#12311yescharacteristic
549C2×C4C2×#12411yescharacteristic
550C2×C4C2×#12411yescharacteristic
551C2×C4C2×#12411yescharacteristic
552C2×C4C2×#12511yescharacteristic
553C2×C4C2×#12511yescharacteristic
554C2×C4C2×#12511yescharacteristic
555C2×C4C2×#12511yescharacteristic
556C2×C4C2×#12511yescharacteristic
557C2×C4C2×#12611yescharacteristic
558C2×C4C2×#12611yescharacteristic
559C2×C4C2×#12611yescharacteristic
560C2×C4C2×#12611yescharacteristic
561C2×C4C2×C4⋊Q811yescharacteristic
562C8—44no—
563C8—44no—
564C22—22no—
565C22—24no—
566C22—22no—
567C22—24no—
568C22—21no—
569C22—21no—
570C22—22no—
571C22—22no—
572C22—21no—
573C22—21no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—24no—
579C22—24no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22#144214yes—
585C22#144214yes—
586C22C2×#38511yescharacteristic
587C22C2×#38711yescharacteristic
588C22C2×#38811yescharacteristic
589C22C2×#39311yescharacteristic
590C22C2×#39411yescharacteristic
591C22C2×#39811yescharacteristic
592C22C2×#40011yescharacteristic
593C22#302112yes—
594C22#302212yes—
595C4—48no—
596C4—84no—
597C4—84no—
598C4—84no—
599C4—84no—
600C4#144214yes—
601C4#144214yes—
602C4C2×#38211yescharacteristic
603C4C2×#38711yescharacteristic
604C4C2×#38811yescharacteristic
605C4C2×#38911yescharacteristic
606C4C2×#39711yescharacteristic
607C4C2×#39811yescharacteristic
608C4C2×#40011yescharacteristic
609C4C2×#40011yescharacteristic
610C2—21no—
611C2—21no—
612C2—21no—
613C2—24no—
614C2—24no—
615C2—21no—
616C2C2×#144211yescharacteristic
617C2#2086512yes—
618C2#2346311yescharacteristic
619C2#2346411yescharacteristic
620C2#2349512yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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. #11276C2semidirect
32. #1733C4semidirect
125. #339C2×C4semidirect

Maximal subgroups

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

subgroupindex
#340522
C2×#17422
C2×#17362
C2×#17332
#137952
#112762