Finite Group Databasebeta

← #449006

Subgroups — #449006

Order 512.

1220 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2513C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic, complemented
3C2×#2071C211yesmaximal, characteristic
4#13910C211yesmaximal, characteristic, complemented
5#11346C211yesmaximal, characteristic, complemented
6#11208C212yesmaximal, complemented
7#11066C212yesmaximal
8#11066C212yesmaximal
9#10931C211yesmaximal, characteristic
10#10885C211yesmaximal, characteristic, complemented
11#10885C212yesmaximal, complemented
12C22×#102C2212yes—
13C22×#102C2212yes—
14#2513—24no—
15#2513—24no—
16C2×#397—21no—
17C2×#397—21no—
18C2×#390C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#389C2211yescharacteristic
24C2×#388—21no—
25C2×#388—21no—
26C2×#384—21no—
27C2×#384C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#2099—24no—
33#2071—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1444—21no—
45#1444—21no—
46C4⋊#95—21no—
47C23×D4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C42—21no—
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—42no—
78C2×#102—42no—
79#397—28no—
80#397—28no—
81#390—22no—
82#390—22no—
83#390—22no—
84#390—22no—
85#390—22no—
86#390D414yes—
87#389—22no—
88#389—24no—
89#389—24no—
90#389—42no—
91#389—42no—
92#389D414yes—
93#388—28no—
94#388—28no—
95#384—22no—
96#384—22no—
97#384—44no—
98#382—22no—
99#382—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95D411yescharacteristic
113C2×#95D411yescharacteristic
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#336—22no—
121#336—22no—
122#336—42no—
123#336—42no—
124C25—21no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4C2×D412yes—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2411yescharacteristic
145C22×C8—22no—
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×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
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—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
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—24no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4#4114yes—
200C2×C4⋊C4#4114yes—
201C2×C4⋊C4#4114yes—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4—82no—
215C2×C22⋊C4—82no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—24no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—82no—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233#102—48no—
234#102—48no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—22no—
241#95—22no—
242#95—44no—
243#95—44no—
244C24—22no—
245C24—41no—
246C24—41no—
247C24—42no—
248C24—42no—
249C24—41no—
250C24—41no—
251C24—41no—
252C24—41no—
253C24—42no—
254C24—42no—
255C24C22×D411yescharacteristic
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—42no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—24no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
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#12012yes—
329C22×C4#12012yes—
330C22×C4#13512yes—
331C22×C4AGL(1,ℤ/8)12yes—
332C22×C4#13712yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C2×#4112yes—
336C22×C4C2×#4112yes—
337C22×C4C2×#4112yes—
338C22×C4H(5,𝔽2)12yes—
339C22×C42−1+412yes—
340C22×C42−1+412yes—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—44no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—84no—
373C4⋊C4—84no—
374C4⋊C4—88no—
375C4⋊C4—88no—
376C4⋊C4—84no—
377C4⋊C4#12618yes—
378C22⋊C4—48no—
379C22⋊C4—48no—
380C22⋊C4—84no—
381C22⋊C4—84no—
382C22⋊C4—84no—
383C22⋊C4—84no—
384C42—24no—
385C42—24no—
386C42—48no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—42no—
392C42—42no—
393C42—84no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—81no—
402C23—82no—
403C23—81no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23—82no—
408C23—82no—
409C23—81no—
410C23—81no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23C2×#12011yescharacteristic
418C23#52511yescharacteristic
419C23#53611yescharacteristic
420C23#54612yes—
421C23#54812yes—
422C23C2×#13511yescharacteristic
423C23C2×AGL(1,ℤ/8)11yescharacteristic
424C23C2×#13711yescharacteristic
425C23#57511yescharacteristic
426C23#57612yes—
427C23#57712yes—
428D4—44no—
429D4—44no—
430D4—82no—
431D4—82no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—28no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4#44914yes—
493C2×C4C2×#12612yes—
494C2×C4#53211yescharacteristic
495C2×C4#53211yescharacteristic
496C2×C4#53511yescharacteristic
497C2×C4#53611yescharacteristic
498C2×C4#53912yes—
499C2×C4#54112yes—
500C2×C4#54212yes—
501C2×C4#54712yes—
502C2×C4#54712yes—
503C2×C4#54712yes—
504C2×C4#55012yes—
505C2×C4#55212yes—
506C2×C4#55312yes—
507C2×C4#55612yes—
508C2×C4#56312yes—
509C8—82no—
510C8—82no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—162no—
519C22—162no—
520C22—162no—
521C22—162no—
522C22#240611yescharacteristic
523C22#243111yescharacteristic
524C22#243211yescharacteristic
525C22#248112yes—
526C22#271612yes—
527C22C2×#44911yescharacteristic
528C22#300711yescharacteristic
529C22#300811yescharacteristic
530C22#300911yescharacteristic
531C22#310811yescharacteristic
532C22#311411yescharacteristic
533C22#311912yes—
534C22#323211yescharacteristic
535C22#323811yescharacteristic
536C22#324112yes—
537C22#330512yes—
538C22#330712yes—
539C22#332012yes—
540C22#332512yes—
541C22#336512yes—
542C22#337012yes—
543C22#337312yes—
544C22#337512yes—
545C4—24no—
546C4—42no—
547C4—48no—
548C4—42no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4—81no—
553C4—81no—
554C4—81no—
555C4—162no—
556C4#249612yes—
557C4#251212yes—
558C4#271812yes—
559C4#272112yes—
560C2—162no—
561C2#1592811yescharacteristic
562C2#2406611yescharacteristic
563C2#2406911yescharacteristic
564C2#2557211yescharacteristic
565C2#2558211yescharacteristic
566C2#2565211yescharacteristic
567C2#2566011yescharacteristic
568C2#2755312yes—
569C2#2756812yes—
570C2#3224112yes—
571C2#3224612yes—
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
2. C2×#2099C2semidirect
4. #13910C2semidirect
5. #11346C2semidirect
6. #11208C2semidirect
10. #10885C2semidirect
11. #10885C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25132
C2×#20992
C2×#20712
#139102
#113462
#112082
#110662
#109312
#108852