Finite Group Databasebeta

← #365495

Subgroups — #365495

Order 512.

1346 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#125C211yesmaximal, characteristic
2C2×#1992C211yesmaximal, characteristic
3C2×#1992C211yesmaximal, characteristic
4C2×#1847C211yesmaximal, characteristic
5#14712C211yesmaximal, characteristic
6#11328C212yesmaximal
7#11327C212yesmaximal
8#10908C212yesmaximal
9#10907C212yesmaximal
10#10843C211yesmaximal, characteristic
11#10693C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×#97C2211yescharacteristic
16C22×#97C2211yescharacteristic
17C4×#125—24no—
18C4×#125—24no—
19C2×#389—21no—
20C2×#389—21no—
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#389C2212yes—
24C2×#387—21no—
25C2×#387—21no—
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C43C2211yescharacteristic
34#1992—24no—
35#1992C418yescomplemented
36#1847—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45C4⋊#95—21no—
46C4⋊#95—21no—
47C4⋊#95C2212yes—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125C2×C414yes—
54C2×#125D414yes—
55C2×#125Q814yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C42—21no—
72C22×C42—21no—
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—42no—
83C2×#106—24no—
84C2×#106—24no—
85C2×#106—42no—
86C2×#97—22no—
87C2×#97—22no—
88C2×#97—22no—
89C2×#97—22no—
90C2×#97—22no—
91C2×#97C2×C414yes—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—42no—
97#389—42no—
98#389—44no—
99#389—44no—
100#387—28no—
101#387—28no—
102#384—22no—
103#384—22no—
104#384—22no—
105#384C2×C414yes—
106#382—22no—
107#382D414yes—
108C4×C4⋊C4—28no—
109C4×C4⋊C4—28no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C43—22no—
124C43—22no—
125#340—24no—
126#340—24no—
127#340—42no—
128#340—42no—
129#340C2×C418yes—
130#340C2×C418yes—
131#336—24no—
132#336Q818yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2411yescharacteristic
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8C22×C414yes—
157C22×C8C2×Q814yes—
158C22×C8C2×Q814yes—
159C22×C8#4114yes—
160#125—28no—
161#125—28no—
162#125—28no—
163#125C4⋊C4116yescomplemented
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
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—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4C22×C414yes—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×Q812yes—
222C2×C4⋊C4C2×Q812yes—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42—41no—
243C2×C42—41no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42C22×C412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×Q814yes—
252C2×C42C2×Q812yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255#107—48no—
256#106—48no—
257#97—28no—
258#97—44no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—44no—
266#95—44no—
267#95—44no—
268C24C22×D411yescharacteristic
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4C4×D412yes—
342C22×C4C4×D412yes—
343C22×C4#12012yes—
344C22×C4#12012yes—
345C22×C4C22⋊Q812yes—
346C22×C4C22⋊Q812yes—
347C22×C4C4⋊Q812yes—
348C22×C4C4⋊Q812yes—
349C22×C4AGL(1,ℤ/8)12yes—
350C22×C4AGL(1,ℤ/8)12yes—
351C22×C4#13712yes—
352C22×C4#13712yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×Q811yescharacteristic
357C22×C4C22×Q812yes—
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4112yes—
360C22×C4C2×#4111yescharacteristic
361C22×C4H(5,𝔽2)12yes—
362C22×C42−1+412yes—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
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×C8C4×Q818yes—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—216no—
384C4⋊C4—216no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—48no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C4⋊C4—88no—
404C4⋊C4C2×C4⋊C414yes—
405C4⋊C4C2×C4⋊C414yes—
406C42—24no—
407C42—24no—
408C42—44no—
409C42—44no—
410C42—44no—
411C42—44no—
412C42—48no—
413C42—48no—
414C42—42no—
415C42—42no—
416C42C2×C4⋊C414yes—
417C42C4×D414yes—
418C42C22⋊Q814yes—
419C42#12314yes—
420C23C2×C4×D411yescharacteristic
421C23C2×#12011yescharacteristic
422C23C2×C22⋊Q811yescharacteristic
423C23C2×C4⋊Q811yescharacteristic
424C23#53211yescharacteristic
425C23#53511yescharacteristic
426C23#53611yescharacteristic
427C23C2×AGL(1,ℤ/8)11yescharacteristic
428C23C2×AGL(1,ℤ/8)11yescharacteristic
429C23C2×#13711yescharacteristic
430C23C2×#13711yescharacteristic
431C23#57511yescharacteristic
432C23#57511yescharacteristic
433C23#57511yescharacteristic
434C23#57511yescharacteristic
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
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#38514yes—
517C2×C4#38514yes—
518C2×C4#50114yes—
519C2×C4C22×C4⋊C411yescharacteristic
520C2×C4C2×C4×D411yescharacteristic
521C2×C4C2×C4×Q812yes—
522C2×C4#52012yes—
523C2×C4C2×C22⋊Q811yescharacteristic
524C2×C4C2×#12311yescharacteristic
525C2×C4#53312yes—
526C2×C4#54212yes—
527C2×C4#54612yes—
528C2×C4D4×Q812yes—
529C2×C4D4×Q812yes—
530C2×C4#55012yes—
531C2×C4#55212yes—
532C2×C4#55412yes—
533C2×C4#55712yes—
534C2×C4Q8212yes—
535C8—28no—
536C8—44no—
537C22C2×#38511yescharacteristic
538C22C4×C4⋊Q811yescharacteristic
539C22#232611yescharacteristic
540C22#243111yescharacteristic
541C22#243411yescharacteristic
542C22#256511yescharacteristic
543C22#256711yescharacteristic
544C22C4×AGL(1,ℤ/8)11yescharacteristic
545C22C4×AGL(1,ℤ/8)11yescharacteristic
546C22C4×#13711yescharacteristic
547C22C4×#13711yescharacteristic
548C22#300611yescharacteristic
549C22#300611yescharacteristic
550C22#300711yescharacteristic
551C22#300711yescharacteristic
552C22#307911yescharacteristic
553C22#307911yescharacteristic
554C22#308011yescharacteristic
555C22#308011yescharacteristic
556C22C2×#50111yescharacteristic
557C22#316211yescharacteristic
558C22#316511yescharacteristic
559C22#316611yescharacteristic
560C22#317111yescharacteristic
561C22#317211yescharacteristic
562C22#318011yescharacteristic
563C22#318111yescharacteristic
564C22#321711yescharacteristic
565C22#321811yescharacteristic
566C22#321911yescharacteristic
567C22#321911yescharacteristic
568C22#323411yescharacteristic
569C22#323511yescharacteristic
570C22#323611yescharacteristic
571C22#323711yescharacteristic
572C4—24no—
573C4—24no—
574C4—24no—
575C4—42no—
576C4—42no—
577C4—44no—
578C4—44no—
579C4—42no—
580C4—82no—
581C4—82no—
582C4—84no—
583C4—162no—
584C4Q8×C4⋊C412yes—
585C4#236012yes—
586C4#242312yes—
587C4#245112yes—
588C2#1397611yescharacteristic
589C2#2087511yescharacteristic
590C2#2087511yescharacteristic
591C2C4×#50111yescharacteristic
592C2#2173111yescharacteristic
593C2#2222511yescharacteristic
594C2#2222611yescharacteristic
595C2#2560411yescharacteristic
596C2#2560511yescharacteristic
597C2#2581811yescharacteristic
598C2#2581911yescharacteristic
599C2#2918311yescharacteristic
600C2#2918411yescharacteristic
601C2#2921311yescharacteristic
602C2#2921411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
35. #1992C4semidirect
163. #125C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1252
C2×#19922
C2×#18472
#147122
#113282
#113272
#109082
#109072
#108432
#106932