Finite Group Databasebeta

← #432778

Subgroups — #432778

Order 512.

1358 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2442C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic, complemented
3C2×#1992C211yesmaximal, characteristic, complemented
4#15135C211yesmaximal, characteristic
5#11385C211yesmaximal, characteristic, complemented
6#11328C212yesmaximal, complemented
7#11208C212yesmaximal, complemented
8#11096C212yesmaximal
9#10922C211yesmaximal, characteristic
10#10881C211yesmaximal, characteristic, complemented
11#10730C212yesmaximal
12C22×#125C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C22×#107C2212yes—
15C22×C22⋊C8C2212yes—
16C22×#97C2211yescharacteristic
17#2442—24no—
18#2442—24no—
19C2×#400—21no—
20C2×#399—21no—
21C2×#398—21no—
22C2×#389C2211yescharacteristic
23C2×#389C2212yes—
24C2×#389C2211yescharacteristic
25C2×#385—21no—
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#380—21no—
29C2×#380C2211yescharacteristic
30C2×#380C2212yes—
31C2×C4×C22⋊C4C2211yescharacteristic
32#2083—24no—
33#1992—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43#1445—21no—
44#1444—21no—
45#1439—21no—
46C4⋊#95C2211yescharacteristic
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×#125—24no—
52C2×#125—22no—
53C2×#125—42no—
54C2×#125D414yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—42no—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4D411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×#107—24no—
80C2×#107—42no—
81C2×#107—42no—
82C2×C22⋊C8—42no—
83C2×C22⋊C8—42no—
84C2×C22⋊C8Q818yes—
85C2×#97—22no—
86C2×#97—42no—
87#400—28no—
88#399—44no—
89#398—28no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—24no—
94#389—42no—
95#389D414yes—
96#385—28no—
97#384—22no—
98#384—22no—
99#382—22no—
100#382—22no—
101#380—28no—
102#380—22no—
103#380—22no—
104#380—24no—
105#380—42no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D411yescharacteristic
118C2×#95D411yescharacteristic
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—42no—
125#340—42no—
126#336—22no—
127#336—42no—
128C25C2×D411yescharacteristic
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—24no—
142C23×C4—22no—
143C23×C4—44no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—82no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×Q814yes—
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8C2×Q814yes—
167C22×C8C2×Q814yes—
168#125—28no—
169#125—44no—
170#125—48no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
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—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
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—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—48no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4#4114yes—
219C2×C4⋊C4#4114yes—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—82no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240#107—48no—
241C22⋊C8—48no—
242#97—44no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—22no—
249#95—22no—
250#95—44no—
251#95—44no—
252#95—42no—
253#95—42no—
254C24—21no—
255C24—21no—
256C24—22no—
257C24—22no—
258C24—21no—
259C24—21no—
260C24—22no—
261C24—22no—
262C24C4⋊Q812yes—
263C24AGL(1,ℤ/8)12yes—
264C24#13712yes—
265C24C22×D411yescharacteristic
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—48no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—48no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
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—84no—
341C22×C4—88no—
342C22×C4—88no—
343C22×C4#12012yes—
344C22×C4#12012yes—
345C22×C4C4⋊Q812yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×Q812yes—
349C22×C4C2×#4112yes—
350C22×C4C2×#4112yes—
351C22×C4H(5,𝔽2)12yes—
352C22×C4H(5,𝔽2)12yes—
353C22×C42−1+412yes—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—48no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—84no—
391C4⋊C4—84no—
392C4⋊C4—84no—
393C4⋊C4—84no—
394C4⋊C4—88no—
395C4⋊C4#12314yes—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—48no—
399C22⋊C4—88no—
400C42—22no—
401C42—22no—
402C42—24no—
403C42—41no—
404C42—41no—
405C42—42no—
406C23—24no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—24no—
414C23—24no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—41no—
419C23—41no—
420C23—42no—
421C23—42no—
422C23—41no—
423C23—41no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23#50114yes—
429C23C2×#12011yescharacteristic
430C23C2×C4⋊Q811yescharacteristic
431C23#53611yescharacteristic
432C23#54612yes—
433C23#54612yes—
434C23C2×AGL(1,ℤ/8)11yescharacteristic
435C23C2×#13711yescharacteristic
436C23#57511yescharacteristic
437C23#57511yescharacteristic
438C23#57612yes—
439C23#57812yes—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—816no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4C2×#12311yescharacteristic
505C2×C4#53211yescharacteristic
506C2×C4#53312yes—
507C2×C4#53411yescharacteristic
508C2×C4#53511yescharacteristic
509C2×C4#53912yes—
510C2×C4#54212yes—
511C2×C4#54312yes—
512C2×C4#54612yes—
513C2×C4D4×Q812yes—
514C2×C4D4×Q812yes—
515C2×C4#55012yes—
516C2×C4#55212yes—
517C2×C4#55412yes—
518C2×C4#55412yes—
519C2×C4#56212yes—
520C8—44no—
521C8—44no—
522C22—28no—
523C22—24no—
524C22—42no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22#240511yescharacteristic
534C22#243111yescharacteristic
535C22#243411yescharacteristic
536C22#254912yes—
537C22#268712yes—
538C22#300611yescharacteristic
539C22#300711yescharacteristic
540C22#300811yescharacteristic
541C22#301111yescharacteristic
542C22C2×#50111yescharacteristic
543C22#316412yes—
544C22#316611yescharacteristic
545C22#323411yescharacteristic
546C22#323611yescharacteristic
547C22#324312yes—
548C22#330812yes—
549C22#330912yes—
550C22#331212yes—
551C22#331412yes—
552C22#331512yes—
553C22#331812yes—
554C22#332212yes—
555C22#332712yes—
556C4—42no—
557C4—42no—
558C4—44no—
559C4—42no—
560C4—84no—
561C4—81no—
562C4—82no—
563C4—81no—
564C4—82no—
565C4—82no—
566C4—162no—
567C4#242312yes—
568C4#255612yes—
569C4#269212yes—
570C4#273212yes—
571C2—44no—
572C2#1546611yescharacteristic
573C2#2404911yescharacteristic
574C2#2405111yescharacteristic
575C2#2558111yescharacteristic
576C2#2558411yescharacteristic
577C2#2581911yescharacteristic
578C2#2582111yescharacteristic
579C2#2877512yes—
580C2#2878612yes—
581C2#3136612yes—
582C2#3137212yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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×#2083C2semidirect
3. C2×#1992C2semidirect
5. #11385C2semidirect
6. #11328C2semidirect
7. #11208C2semidirect
10. #10881C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24422
C2×#20832
C2×#19922
#151352
#113852
#113282
#112082
#110962
#109222
#108812
#107302