Finite Group Databasebeta

← #448406

Subgroups — #448406

Order 512.

1248 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2513C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic, complemented
3C2×#2023C211yesmaximal, characteristic
4#14161C211yesmaximal, characteristic, complemented
5#11346C211yesmaximal, characteristic, complemented
6#11268C212yesmaximal, complemented
7#11189C212yesmaximal
8#11123C212yesmaximal
9#10931C211yesmaximal, characteristic
10#10907C212yesmaximal, complemented
11#10885C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#102C2212yes—
15C22×#102C2212yes—
16#2513—24no—
17#2513—24no—
18C2×#397—21no—
19C2×#397—21no—
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#388—21no—
25C2×#388—21no—
26C2×#382C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2057—24no—
31#2023—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2211yescharacteristic
41#1446—21no—
42#1446—21no—
43C4⋊#95—21no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×D4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C42—21no—
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#107—24no—
76C2×#107—24no—
77C2×#107—42no—
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—42no—
83C2×#102—42no—
84#397—28no—
85#397—28no—
86#390—22no—
87#390—22no—
88#390—22no—
89#390—22no—
90#390—22no—
91#390D414yes—
92#389—22no—
93#389D414yes—
94#388—28no—
95#388—28no—
96#382—22no—
97#382—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—42no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95D411yescharacteristic
113C2×#95D411yescharacteristic
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—42no—
119#336—24no—
120#336—24no—
121#336—22no—
122#336—22no—
123C25—21no—
124C22×Q8—24no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—82no—
128C22×Q8—82no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4C2×D412yes—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
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—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8#4114yes—
163C4⋊Q8—48no—
164C4⋊Q8—48no—
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—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4#4114yes—
199C2×C4⋊C4#4114yes—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—82no—
211C2×C22⋊C4—82no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4—82no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—82no—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231#107—48no—
232#102—48no—
233#102—48no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—22no—
239#95—22no—
240#95—42no—
241#95—82no—
242#95—82no—
243#95—82no—
244#95—82no—
245C24—22no—
246C24—41no—
247C24—41no—
248C24—42no—
249C24—42no—
250C24—41no—
251C24—41no—
252C24—41no—
253C24—41no—
254C24—42no—
255C24—42no—
256C24C22×D411yescharacteristic
257C2×Q8—48no—
258C2×Q8—84no—
259C2×Q8—84no—
260C2×Q8—88no—
261C2×Q8—88no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—24no—
265C2×D4—24no—
266C2×D4—42no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—41no—
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×C4#13512yes—
337C22×C4#13712yes—
338C22×C4#13712yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C2×#4112yes—
342C22×C4C2×#4112yes—
343C22×C4C2×#4112yes—
344C22×C4H(5,𝔽2)12yes—
345C22×C42−1+412yes—
346C22×C42−1+412yes—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—44no—
370C4⋊C4—48no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—44no—
380C4⋊C4—84no—
381C22⋊C4—48no—
382C22⋊C4—48no—
383C22⋊C4—84no—
384C22⋊C4—84no—
385C22⋊C4—84no—
386C22⋊C4—84no—
387C42—24no—
388C42—24no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—42no—
394C42—42no—
395C42—84no—
396C42—84no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23—82no—
408C23—82no—
409C23—81no—
410C23—82no—
411C23—81no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—81no—
416C23—81no—
417C23—82no—
418C23—82no—
419C23—82no—
420C23C2×#12011yescharacteristic
421C23#52511yescharacteristic
422C23#53511yescharacteristic
423C23#54712yes—
424C23#54812yes—
425C23C2×#13511yescharacteristic
426C23C2×#13711yescharacteristic
427C23C2×#13711yescharacteristic
428C23#57511yescharacteristic
429C23#57612yes—
430C23#57612yes—
431Q8—816no—
432D4—44no—
433D4—44no—
434D4—82no—
435D4—82no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
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—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4#45214yes—
497C2×C4#52912yes—
498C2×C4#53211yescharacteristic
499C2×C4#53211yescharacteristic
500C2×C4#53611yescharacteristic
501C2×C4#53611yescharacteristic
502C2×C4#53912yes—
503C2×C4#54112yes—
504C2×C4#54212yes—
505C2×C4#54612yes—
506C2×C4#54612yes—
507C2×C4#54712yes—
508C2×C4#54812yes—
509C2×C4#55112yes—
510C2×C4#55612yes—
511C2×C4#55612yes—
512C2×C4#56312yes—
513C8—44no—
514C8—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—162no—
523C22—162no—
524C22—162no—
525C22—162no—
526C22#240411yescharacteristic
527C22#243211yescharacteristic
528C22#243211yescharacteristic
529C22#246912yes—
530C22#272312yes—
531C22C2×#45211yescharacteristic
532C22#300711yescharacteristic
533C22#300811yescharacteristic
534C22#300811yescharacteristic
535C22#310811yescharacteristic
536C22#311711yescharacteristic
537C22#312212yes—
538C22#321111yescharacteristic
539C22#321811yescharacteristic
540C22#322012yes—
541C22#334112yes—
542C22#334612yes—
543C22#334712yes—
544C22#335712yes—
545C22#336112yes—
546C22#337012yes—
547C22#337512yes—
548C22#337512yes—
549C4—24no—
550C4—42no—
551C4—42no—
552C4—82no—
553C4—82no—
554C4—81no—
555C4—81no—
556C4—81no—
557C4—82no—
558C4—84no—
559C4—84no—
560C4#252112yes—
561C4#256012yes—
562C4#268912yes—
563C4#271612yes—
564C2—162no—
565C2#1591011yescharacteristic
566C2#2393711yescharacteristic
567C2#2393911yescharacteristic
568C2#2564811yescharacteristic
569C2#2566011yescharacteristic
570C2#2570411yescharacteristic
571C2#2571311yescharacteristic
572C2#2713312yes—
573C2#2714812yes—
574C2#3243712yes—
575C2#3244212yes—
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×#2057C2semidirect
4. #14161C2semidirect
5. #11346C2semidirect
6. #11268C2semidirect
10. #10907C2semidirect
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×#20572
C2×#20232
#141612
#113462
#112682
#111892
#111232
#109312
#109072
#108852