Finite Group Databasebeta

← #448986

Subgroups — #448986

Order 512.

1282 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2512C211yesmaximal, characteristic
2C2×#2071C211yesmaximal, characteristic
3C2×#1845C211yesmaximal, characteristic, complemented
4#15152C211yesmaximal, characteristic, complemented
5#11357C211yesmaximal, characteristic, complemented
6#11219C212yesmaximal, complemented
7#11123C212yesmaximal
8#11123C212yesmaximal
9#10925C211yesmaximal, characteristic
10#10907C212yesmaximal, complemented
11#10874C211yesmaximal, characteristic, complemented
12C22×#126—21no—
13C22×#107C2212yes—
14C22×#107C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2512—24no—
19#2512—24no—
20C2×#400—21no—
21C2×#397—21no—
22C2×#396—21no—
23C2×#390C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#388—21no—
28C2×#384C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#2071—24no—
33#1845—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1445—21no—
42#1442—21no—
43#1441—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×D4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#126—28no—
51C2×#126—44no—
52C2×#126—44no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#107—24no—
76C2×#107—24no—
77C2×#107—24no—
78C2×#107—24no—
79C2×#107—42no—
80C2×#107—42no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—42no—
86C2×#102—42no—
87C2×C4×C8—24no—
88C2×C4×C8—22no—
89#400—44no—
90#397—28no—
91#396—28no—
92#390—22no—
93#390—22no—
94#390—22no—
95#390—22no—
96#390—22no—
97#390D414yes—
98#389—22no—
99#389—22no—
100#388—28no—
101#384—22no—
102#384—22no—
103#384—22no—
104#384D414yes—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D411yescharacteristic
118C2×#95D411yescharacteristic
119#340—24no—
120#340—24no—
121#340—42no—
122#340—42no—
123#336—24no—
124#336—24no—
125#336—22no—
126C25—21no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4C2×D412yes—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
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×C8—22no—
157C22×C8—22no—
158C22×C8#4114yes—
159C22×C8#4114yes—
160C22×C8#4114yes—
161#126—432no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
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—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4D814yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—82no—
220C2×C22⋊C4—82no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—24no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236#107—48no—
237#107—48no—
238#102—48no—
239#102—48no—
240C4×C8—28no—
241#95—24no—
242#95—24no—
243#95—22no—
244#95—22no—
245#95—42no—
246#95—44no—
247#95—44no—
248#95—42no—
249#95—82no—
250#95—82no—
251C24—22no—
252C24—41no—
253C24—41no—
254C24—42no—
255C24—42no—
256C24—41no—
257C24—41no—
258C24—41no—
259C24—41no—
260C24—42no—
261C24—42no—
262C24C22×D411yescharacteristic
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—41no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
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—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
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#12012yes—
335C22×C4#12012yes—
336C22×C4C2×D812yes—
337C22×C4C2×D812yes—
338C22×C4C2×D812yes—
339C22×C4#13512yes—
340C22×C4AGL(1,ℤ/8)12yes—
341C22×C4#13712yes—
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C2×#4112yes—
345C22×C4C2×#4112yes—
346C22×C4C2×#4112yes—
347C22×C4H(5,𝔽2)12yes—
348C22×C42−1+412yes—
349C22×C42−1+412yes—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8#12618yescomplemented
364C4⋊C4—28no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—48no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—48no—
375C4⋊C4—44no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C22⋊C4—48no—
388C22⋊C4—48no—
389C22⋊C4—48no—
390C22⋊C4—48no—
391C22⋊C4—84no—
392C22⋊C4—84no—
393C42—22no—
394C42—22no—
395C42—24no—
396C42—48no—
397C42—41no—
398C42—41no—
399C42—42no—
400C42—42no—
401C42—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—82no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—81no—
418C23—82no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23C2×#12011yescharacteristic
426C23#53211yescharacteristic
427C23#53611yescharacteristic
428C23#54712yes—
429C23#55012yes—
430C23C22×D811yescharacteristic
431C23C2×#13511yescharacteristic
432C23C2×AGL(1,ℤ/8)11yescharacteristic
433C23C2×#13711yescharacteristic
434C23#57612yes—
435C23#57812yes—
436D4—44no—
437D4—44no—
438D4—82no—
439D4—82no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
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#44714yes—
502C2×C4#45214yes—
503C2×C4#52511yescharacteristic
504C2×C4C2×#12612yes—
505C2×C4#53211yescharacteristic
506C2×C4#53511yescharacteristic
507C2×C4#53611yescharacteristic
508C2×C4#53912yes—
509C2×C4#54112yes—
510C2×C4#54212yes—
511C2×C4#54612yes—
512C2×C4#54712yes—
513C2×C4#54712yes—
514C2×C4#54812yes—
515C2×C4#55212yes—
516C2×C4#55312yes—
517C2×C4#55612yes—
518C2×C4#56312yes—
519C8—28no—
520C8—82no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—162no—
529C22—162no—
530C22—162no—
531C22—162no—
532C22#240611yescharacteristic
533C22#243111yescharacteristic
534C22#243211yescharacteristic
535C22#251212yes—
536C22#272112yes—
537C22C2×#44711yescharacteristic
538C22C2×#45211yescharacteristic
539C22#300811yescharacteristic
540C22#301111yescharacteristic
541C22#316911yescharacteristic
542C22#317512yes—
543C22#318211yescharacteristic
544C22#322611yescharacteristic
545C22#322811yescharacteristic
546C22#324312yes—
547C22#333412yes—
548C22#333912yes—
549C22#334312yes—
550C22#334612yes—
551C22#339112yes—
552C22#339312yes—
553C22#339812yes—
554C22#339912yes—
555C4—24no—
556C4—24no—
557C4—42no—
558C4—42no—
559C4—81no—
560C4—82no—
561C4—82no—
562C4—82no—
563C4—81no—
564C4—162no—
565C4—162no—
566C4#248112yes—
567C4#249612yes—
568C4#271612yes—
569C4#271812yes—
570C2—162no—
571C2#1592811yescharacteristic
572C2#2408811yescharacteristic
573C2#2409211yescharacteristic
574C2#2563111yescharacteristic
575C2#2563211yescharacteristic
576C2#2567311yescharacteristic
577C2#2568111yescharacteristic
578C2#2801412yes—
579C2#2802512yes—
580C2#3237412yes—
581C2#3237912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1845C2semidirect
4. #15152C2semidirect
5. #11357C2semidirect
6. #11219C2semidirect
10. #10907C2semidirect
11. #10874C2semidirect
363. C2×C8#126semidirect

Maximal subgroups

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

subgroupindex
C2×#25122
C2×#20712
C2×#18452
#151522
#113572
#112192
#111232
#109252
#109072
#108742