Finite Group Databasebeta

← #380970

Subgroups — #380970

Order 512.

1430 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#1729C211yesmaximal, characteristic
3#14340C211yesmaximal, characteristic
4#11384C212yesmaximal, complemented
5#11358C212yesmaximal, complemented
6#11208C212yesmaximal, complemented
7#11207C212yesmaximal, complemented
8#10722C212yesmaximal
9#10721C212yesmaximal
10C23×M16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14#2628—24no—
15#2628C418yescomplemented
16C2×#400—21no—
17C2×#399—21no—
18C2×#395—21no—
19C2×#393—21no—
20C2×#389C2212yes—
21C2×#384C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34#1729—24no—
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1441—21no—
38#1441—21no—
39#1439—21no—
40#1439—21no—
41C4⋊#95C2212yes—
42C23×Q8—21no—
43C24×C4C2311yescharacteristic
44C22×M16—22no—
45C22×M16—22no—
46C22×M16—41no—
47C22×M16—41no—
48C22×M16—41no—
49C22×M16—41no—
50C22×M16Q814yescomplemented
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—41no—
54C23×C8—41no—
55C23×C8—41no—
56C23×C8—41no—
57C23×C8Q814yescomplemented
58C23×C8C2312yes—
59C23×C8C2312yes—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—41no—
62C2×C4⋊Q8—41no—
63C2×C4⋊Q8—41no—
64C2×C4⋊Q8—41no—
65C2×C4⋊Q8C2×C414yes—
66C2×#117—41no—
67C2×#117—41no—
68C2×#117—41no—
69C2×#117—41no—
70C2×#117C2×C414yes—
71C2×#117D414yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#400—28no—
89#399—28no—
90#395—28no—
91#393—28no—
92#389—24no—
93#389C2×C418yes—
94#384—41no—
95#384—41no—
96#382—22no—
97#382—41no—
98#382—41no—
99#382C2×C414yes—
100#380—24no—
101#380—24no—
102#380—24no—
103#380C2×C418yes—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113#340—42no—
114#340—42no—
115#340—42no—
116#340—42no—
117#340—42no—
118#340—42no—
119#340—42no—
120#340—42no—
121#336—24no—
122#336—24no—
123C25C2×D411yescharacteristic
124C22×Q8—24no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—82no—
128C22×Q8—82no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4C22×C412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×Q812yes—
149C23×C4C2×Q812yes—
150C23×C4#4112yes—
151C23×C4#4112yes—
152C23×C4C2411yescharacteristic
153C2×M16—28no—
154C2×M16—44no—
155C2×M16—44no—
156C2×M16—44no—
157C2×M16—44no—
158C2×M16—44no—
159C2×M16—44no—
160C22×C8—28no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—44no—
174C22×C8—44no—
175C22×C8—44no—
176C22×C8—44no—
177C22×C8—44no—
178C22×C8—44no—
179C22×C8C2×Q814yes—
180C22×C8C2×Q814yes—
181C4⋊Q8—44no—
182C4⋊Q8—44no—
183C4⋊Q8—44no—
184C4⋊Q8—44no—
185#117—44no—
186#117—44no—
187#117—44no—
188#117—44no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4C22⋊C414yes—
226C2×C4⋊C4C22⋊C414yes—
227C2×C4⋊C4C22×C414yes—
228C2×C4⋊C4C22×C414yes—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—82no—
236C2×C22⋊C4—82no—
237C2×C22⋊C4—82no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4C22⋊C414yes—
240C2×C22⋊C4C22⋊C414yes—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42C22×C412yes—
250C2×C42C22×C412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253#95—28no—
254#95—24no—
255#95—24no—
256#95—28no—
257#95—42no—
258#95—42no—
259#95—44no—
260#95—82no—
261#95—82no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24C2×C22⋊C412yes—
271C24#12412yes—
272C24C4⋊Q812yes—
273C24C22×D411yescharacteristic
274C2×Q8—48no—
275C2×Q8—88no—
276C2×Q8—88no—
277C2×Q8—84no—
278C2×Q8—84no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
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—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
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—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C412yes—
356C22×C4C4×Q814yes—
357C22×C4C4×Q814yes—
358C22×C4C22⋊Q814yes—
359C22×C4C22⋊Q814yes—
360C22×C4#12412yes—
361C22×C4C4⋊Q812yes—
362C22×C4C23×C411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×Q811yescharacteristic
365C22×C4C22×Q811yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4H(5,𝔽2)11yescharacteristic
369C22×C4H(5,𝔽2)11yescharacteristic
370C22×C4H(5,𝔽2)11yescharacteristic
371C22×C4H(5,𝔽2)11yescharacteristic
372C22×C4H(5,𝔽2)11yescharacteristic
373C22×C4H(5,𝔽2)11yescharacteristic
374C22×C42−1+411yescharacteristic
375C22×C42−1+411yescharacteristic
376M16—416no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—416no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—48no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—88no—
412C4⋊C4—84no—
413C4⋊C4—84no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—88no—
417C22⋊C4—88no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C23—22no—
427C23—22no—
428C23—24no—
429C23—24no—
430C23—22no—
431C23—22no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—22no—
437C23—22no—
438C23—42no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—42no—
443C23—41no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23#39114yes—
449C23C22×C22⋊C411yescharacteristic
450C23C2×#12411yescharacteristic
451C23C2×C4⋊Q811yescharacteristic
452C23#53411yescharacteristic
453C23#53411yescharacteristic
454C23#53711yescharacteristic
455C23#53711yescharacteristic
456C23#57612yes—
457C23#57712yes—
458C23#57712yes—
459C23#57812yes—
460Q8—816no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
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—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4#39114yes—
524C2×C4C2×C4×Q811yescharacteristic
525C2×C4C2×C4×Q811yescharacteristic
526C2×C4#51811yescharacteristic
527C2×C4#51811yescharacteristic
528C2×C4#52011yescharacteristic
529C2×C4#52011yescharacteristic
530C2×C4C2×C22⋊Q811yescharacteristic
531C2×C4C2×C22⋊Q811yescharacteristic
532C2×C4#53411yescharacteristic
533C2×C4#53411yescharacteristic
534C2×C4#53511yescharacteristic
535C2×C4#53611yescharacteristic
536C2×C4#53811yescharacteristic
537C2×C4#53811yescharacteristic
538C2×C4#53811yescharacteristic
539C2×C4#53811yescharacteristic
540C2×C4#53911yescharacteristic
541C2×C4#53911yescharacteristic
542C2×C4#53911yescharacteristic
543C2×C4#53911yescharacteristic
544C2×C4#54311yescharacteristic
545C2×C4#54311yescharacteristic
546C2×C4#54311yescharacteristic
547C2×C4#54311yescharacteristic
548C2×C4#54411yescharacteristic
549C2×C4#54411yescharacteristic
550C2×C4#54411yescharacteristic
551C2×C4#54411yescharacteristic
552C8—44no—
553C8—44no—
554C22—28no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—44no—
559C22—42no—
560C22—42no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22—44no—
565C22C2×#39111yescharacteristic
566C22#230811yescharacteristic
567C22#231111yescharacteristic
568C22#263911yescharacteristic
569C22#263911yescharacteristic
570C22#265911yescharacteristic
571C22#265911yescharacteristic
572C22#289712yes—
573C22#289812yes—
574C22#313612yes—
575C22#313712yes—
576C22#316312yes—
577C22#316412yes—
578C22#319812yes—
579C22#319911yescharacteristic
580C22#320011yescharacteristic
581C22#320011yescharacteristic
582C22#320112yes—
583C22#320211yescharacteristic
584C22#325712yes—
585C22#325812yes—
586C22#326212yes—
587C22#326312yes—
588C4—24no—
589C4—42no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—84no—
594C4—84no—
595C4—162no—
596C4—162no—
597C4#232911yescharacteristic
598C4#232911yescharacteristic
599C4#232911yescharacteristic
600C4#232911yescharacteristic
601C4#262611yescharacteristic
602C4#262611yescharacteristic
603C4#262711yescharacteristic
604C4#262711yescharacteristic
605C2—44no—
606C2#1428111yescharacteristic
607C2#2132612yes—
608C2#2177011yescharacteristic
609C2#2177111yescharacteristic
610C2#2189612yes—
611C2#3019211yescharacteristic
612C2#3019311yescharacteristic
613C2#3019512yes—
614C2#3029912yes—
615C2#3030011yescharacteristic
616C2#3030211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11384C2semidirect
5. #11358C2semidirect
6. #11208C2semidirect
7. #11207C2semidirect
15. #2628C4semidirect
50. C22×M16Q8semidirect
57. C23×C8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#17292
#143402
#113842
#113582
#112082
#112072
#107222
#107212