Finite Group Databasebeta

← #360647

Subgroups — #360647

Order 512.

1330 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2356C211yesmaximal, characteristic
2C2×#1835C211yesmaximal, characteristic
3C2×#1805C211yesmaximal, characteristic
4C2×C4×#103C211yesmaximal, characteristic
5#13954C211yesmaximal, characteristic
6#11067C212yesmaximal
7#11067C212yesmaximal
8#10885C212yesmaximal
9#10884C212yesmaximal
10#10850C211yesmaximal, characteristic
11#10834C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2356—24no—
19#2356—24no—
20C2×#391C2212yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#387—22no—
24C2×#383—21no—
25C2×#382C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×#376—21no—
31#1835—24no—
32#1805—24no—
33C4×#103C418yescomplemented
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1444—22no—
44C4⋊#95—21no—
45C4⋊#95—21no—
46C23×Q8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8C2×C414yes—
54C2×C4×Q8D414yes—
55C2×C4×Q8Q814yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4D412yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C42—21no—
66C22×C42—22no—
67C22×C42—21no—
68C22×C42C2312yes—
69C22×C42C2311yescharacteristic
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×#103—22no—
74C2×#103—22no—
75C2×#103—22no—
76C2×#103—22no—
77C2×#103—24no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103C2×C414yes—
81C2×#103C2×C414yes—
82C2×C4×C8—22no—
83C2×C4×C8—22no—
84C2×C4×C8C2×C414yes—
85#391—24no—
86#391—24no—
87#389—24no—
88#389—24no—
89#389—42no—
90#389—42no—
91#387—216no—
92#383—28no—
93#382—22no—
94#382—22no—
95#382—22no—
96#382D414yes—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4C2×C414yes—
103#376—28no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95D411yescharacteristic
115C2×#95D411yescharacteristic
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340C2×C414yes—
122#340C2×C414yes—
123#336—42no—
124#336—42no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8C22×C412yes—
137C22×Q8C2×D412yes—
138C22×Q8C2×Q812yes—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8C22×C412yes—
170C22×C8C22×C412yes—
171C4×Q8—28no—
172C4×Q8—28no—
173C4×Q8—28no—
174C4×Q8C4⋊C4116yescomplemented
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4C22×C412yes—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×Q812yes—
213C2×C4⋊C4#4112yes—
214C2×C4⋊C4#4112yes—
215C2×C4⋊C4#4112yes—
216C2×C4⋊C4#4112yes—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—24no—
220C2×C42—28no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—44no—
229C2×C42—44no—
230C2×C42—82no—
231C2×C42C22×C412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×Q812yes—
235C2×C42#4112yes—
236C2×C42#4112yes—
237C4⋊C8—44no—
238#103—28no—
239#103—28no—
240#103—48no—
241C4×C8—28no—
242#95—28no—
243#95—28no—
244#95—28no—
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—
254C24C22×D411yescharacteristic
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8C2×C4⋊C414yes—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
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—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4C4×D412yes—
337C22×C4C4×D412yes—
338C22×C4#12012yes—
339C22×C4#12012yes—
340C22×C4C22⋊Q812yes—
341C22×C4C22⋊Q812yes—
342C22×C4#13514yes—
343C22×C4C23×C411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×Q811yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4H(5,𝔽2)11yescharacteristic
354C22×C4H(5,𝔽2)11yescharacteristic
355C22×C42−1+411yescharacteristic
356C22×C42−1+411yescharacteristic
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—216no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—48no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—88no—
398C4⋊C4—88no—
399C4⋊C4C2×C4⋊C414yes—
400C4⋊C4#11714yes—
401C4⋊C4#11714yes—
402C42—216no—
403C42—416no—
404C42—42no—
405C42—42no—
406C42—42no—
407C42—42no—
408C42—84no—
409C42C2×C4⋊C414yes—
410C42C4×D414yes—
411C42C22⋊Q814yes—
412C42#12314yes—
413C23C2×C4×D411yescharacteristic
414C23C2×#12011yescharacteristic
415C23C2×C22⋊Q811yescharacteristic
416C23#54611yescharacteristic
417C23#54611yescharacteristic
418C23#54811yescharacteristic
419C23#54811yescharacteristic
420C23C2×#13512yes—
421C23#57512yes—
422C23#57712yes—
423C23#57812yes—
424Q8—24no—
425Q8—24no—
426Q8—42no—
427Q8—42no—
428Q8—42no—
429Q8—42no—
430Q8—42no—
431Q8—42no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
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#38514yes—
491C2×C4#38514yes—
492C2×C4C22×C4⋊C411yescharacteristic
493C2×C4C2×#11711yescharacteristic
494C2×C4C2×#11711yescharacteristic
495C2×C4C2×C4×D411yescharacteristic
496C2×C4#52011yescharacteristic
497C2×C4#52011yescharacteristic
498C2×C4C2×C22⋊Q811yescharacteristic
499C2×C4C2×#12311yescharacteristic
500C2×C4#53211yescharacteristic
501C2×C4#53211yescharacteristic
502C2×C4#53311yescharacteristic
503C2×C4#53311yescharacteristic
504C2×C4#53511yescharacteristic
505C2×C4#53611yescharacteristic
506C2×C4#54411yescharacteristic
507C2×C4#54411yescharacteristic
508C2×C4#54711yescharacteristic
509C2×C4#54711yescharacteristic
510C2×C4#55011yescharacteristic
511C2×C4#55011yescharacteristic
512C2×C4#55211yescharacteristic
513C2×C4#55211yescharacteristic
514C2×C4#55311yescharacteristic
515C2×C4#55311yescharacteristic
516C2×C4#55511yescharacteristic
517C2×C4#55511yescharacteristic
518C2×C4#55611yescharacteristic
519C2×C4#55611yescharacteristic
520C8—42no—
521C8—42no—
522C8—82no—
523C22C2×#38511yescharacteristic
524C22#235211yescharacteristic
525C22#235211yescharacteristic
526C22#243111yescharacteristic
527C22#243211yescharacteristic
528C22#243911yescharacteristic
529C22#243911yescharacteristic
530C22C4×#13511yescharacteristic
531C22#294311yescharacteristic
532C22#294911yescharacteristic
533C22#295011yescharacteristic
534C22#300212yes—
535C22#301312yes—
536C22#307612yes—
537C22#308212yes—
538C22#329612yes—
539C22#330012yes—
540C22#332312yes—
541C22#332712yes—
542C22#336012yes—
543C22#336412yes—
544C22#337612yes—
545C22#337712yes—
546C4—24no—
547C4—24no—
548C4—44no—
549C4—44no—
550C4—44no—
551C4—44no—
552C4—44no—
553C4—84no—
554C4—82no—
555C4—162no—
556C4#228511yescharacteristic
557C4#231711yescharacteristic
558C4#235411yescharacteristic
559C4#235411yescharacteristic
560C4#244111yescharacteristic
561C4#244111yescharacteristic
562C4#249911yescharacteristic
563C4#250011yescharacteristic
564C2#1387911yescharacteristic
565C2#2087211yescharacteristic
566C2#2088011yescharacteristic
567C2#2276011yescharacteristic
568C2#2276111yescharacteristic
569C2#2276811yescharacteristic
570C2#2276911yescharacteristic
571C2#2557612yes—
572C2#2565412yes—
573C2#2602912yes—
574C2#2603112yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×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
33. C4×#103C4semidirect
174. C4×Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23562
C2×#18352
C2×#18052
C2×C4×#1032
#139542
#110672
#108852
#108842
#108502
#108342