Finite Group Databasebeta

← #359964

Subgroups — #359964

Order 512.

1642 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic
2C2×#2345C211yesmaximal, characteristic
3C2×#1825C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic
5#13939C211yesmaximal, characteristic
6#11220C212yesmaximal, complemented
7#11219C212yesmaximal, complemented
8#10908C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10#10722C211yesmaximal, characteristic
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#428—24no—
16C2×#428—24no—
17C2×#428C418yescomplemented
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23#2345—24no—
24#2345—24no—
25C2×#389C2212yes—
26C2×#387—22no—
27C2×#380—21no—
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×C4×C4⋊C4C2212yes—
31C2×C4×C22⋊C4—21no—
32C2×C4×C22⋊C4C2211yescharacteristic
33#1825—24no—
34#1823—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1440—21no—
44#1440—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47#1434—22no—
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—24no—
53C22×M16—24no—
54C22×M16C2×C414yes—
55C23×C8C2312yes—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117C2×C414yes—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4C2×C414yes—
70C22×C4⋊C4D414yes—
71C22×C4⋊C4D414yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4D411yescharacteristic
82C22×C22⋊C4D411yescharacteristic
83C22×C42—21no—
84C22×C42—22no—
85C22×C42C2312yes—
86#428—232no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107C2×C414yes—
92C2×#107C2×C414yes—
93C2×#106—22no—
94C2×#106—22no—
95C2×#106—22no—
96C2×#106—22no—
97C2×#106C2×C414yes—
98C2×#106C2×C414yes—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102#389—24no—
103#389—24no—
104#387—216no—
105#380—28no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380D414yes—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—24no—
112C4×C22⋊C4—28no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95D412yes—
124#340—24no—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—24no—
130#336—24no—
131#336—24no—
132C25C2×D411yescharacteristic
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—82no—
147C23×C4—82no—
148C23×C4C22×C412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4#4112yes—
156C23×C4#4112yes—
157C23×C4C2411yescharacteristic
158C2×M16—216no—
159C2×M16—216no—
160C2×M16—28no—
161C2×M16—28no—
162C2×M16—28no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8C22×C412yes—
177C22×C8C22×C412yes—
178C22×C8#4114yes—
179C22×C8#4114yes—
180#117—28no—
181#117—28no—
182#117—44no—
183#117—44no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C22⋊C4116yes—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C2×D414yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—82no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—28no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—82no—
252C2×C42—82no—
253#107—28no—
254#107—28no—
255#106—28no—
256#106—28no—
257C22⋊C8—44no—
258#95—28no—
259#95—28no—
260#95—24no—
261#95—24no—
262#95—44no—
263#95—42no—
264#95—44no—
265#95—82no—
266#95—82no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24C4×D412yes—
274C24#12012yes—
275C24#12112yes—
276C24C22×D411yescharacteristic
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—216no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
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—88no—
342C22×C4—88no—
343C22×C4—84no—
344C22×C4—84no—
345C22×C4C2×C22⋊C414yes—
346C22×C4C4×D414yes—
347C22×C4C4×D412yes—
348C22×C4#12012yes—
349C22×C4#12114yes—
350C22×C4#12112yes—
351C22×C4#12314yes—
352C22×C4#13714yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C2×#4112yes—
358C22×C4C2×#4112yes—
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C42−1+412yes—
362C22×C42−1+412yes—
363M16—232no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8#11714yes—
380C2×C8#11714yes—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—216no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—88no—
405C4⋊C4C2×C22⋊C414yes—
406C4⋊C4C2×C22⋊C414yes—
407C22⋊C4—28no—
408C22⋊C4—44no—
409C22⋊C4—48no—
410C22⋊C4—48no—
411C22⋊C4—84no—
412C42—42no—
413C42—42no—
414C42—42no—
415C42—42no—
416C42—48no—
417C42—88no—
418C23—22no—
419C23—22no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—22no—
425C23—22no—
426C23—24no—
427C23—24no—
428C23—22no—
429C23—22no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23#38614yes—
439C23C2×C4×D411yescharacteristic
440C23C2×#12011yescharacteristic
441C23C2×#12111yescharacteristic
442C23#54712yes—
443C23#54712yes—
444C23C2×#13712yes—
445C23#57512yes—
446C23#57612yes—
447C23#57712yes—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—816no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4#38614yes—
508C2×C4C22×C22⋊C411yescharacteristic
509C2×C4C2×#11711yescharacteristic
510C2×C4C2×#11711yescharacteristic
511C2×C4C2×C4×D411yescharacteristic
512C2×C4#51911yescharacteristic
513C2×C4#51911yescharacteristic
514C2×C4C2×#12111yescharacteristic
515C2×C4C2×#12311yescharacteristic
516C2×C4#52512yes—
517C2×C4#52512yes—
518C2×C4#53612yes—
519C2×C4#53612yes—
520C2×C4#54712yes—
521C2×C4#54712yes—
522C2×C4#55212yes—
523C2×C4#55212yes—
524C2×C4#55212yes—
525C2×C4#55212yes—
526C8—24no—
527C8—24no—
528C8—82no—
529C22—24no—
530C22—24no—
531C22—24no—
532C22—44no—
533C22—44no—
534C22—44no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22C2×#38611yescharacteristic
539C22#234411yescharacteristic
540C22#234411yescharacteristic
541C22#240612yes—
542C22#241312yes—
543C22C4×#13712yes—
544C22#294612yes—
545C22#300411yescharacteristic
546C22#300511yescharacteristic
547C22#301212yes—
548C22#303712yes—
549C22#304111yescharacteristic
550C22#304211yescharacteristic
551C22#334412yes—
552C22#334612yes—
553C22#334712yes—
554C22#335012yes—
555C22#335112yes—
556C22#335512yes—
557C22#335612yes—
558C22#335712yes—
559C4—24no—
560C4—44no—
561C4—44no—
562C4—44no—
563C4—82no—
564C4—84no—
565C4—82no—
566C4—82no—
567C4—162no—
568C4#228011yescharacteristic
569C4#231011yescharacteristic
570C4#234411yescharacteristic
571C4#234411yescharacteristic
572C4#241312yes—
573C4#248412yes—
574C2—44no—
575C2#1387011yescharacteristic
576C2#2092611yescharacteristic
577C2#2092911yescharacteristic
578C2#2262112yes—
579C2#2262312yes—
580C2#2410312yes—
581C2#2410912yes—
582C2#2459012yes—
583C2#2459112yes—
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
6. #11220C2semidirect
7. #11219C2semidirect
8. #10908C2semidirect
9. #10907C2semidirect
17. C2×#428C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C2×#23452
C2×#18252
C2×#18232
#139392
#112202
#112192
#109082
#109072
#107222
#107152