Finite Group Databasebeta

← #360651

Subgroups — #360651

Order 512.

1354 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2356C211yesmaximal, characteristic
2C2×#1836C211yesmaximal, characteristic
3C2×#1803C211yesmaximal, characteristic
4C2×#1763C211yesmaximal, characteristic
5#13910C211yesmaximal, characteristic
6#11067C212yesmaximal
7#11067C212yesmaximal
8#10885C212yesmaximal
9#10884C212yesmaximal
10#10850C211yesmaximal, characteristic
11#10835C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#97C2211yescharacteristic
17#2356—24no—
18#2356—24no—
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#387—21no—
24C2×#387—21no—
25C2×#384—21no—
26C2×#384C2211yescharacteristic
27C2×#383—21no—
28C2×#382C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×#376—21no—
34#1836—24no—
35#1803—24no—
36#1763C418yescomplemented
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1444—21no—
46#1444—21no—
47C4⋊#95—21no—
48C23×Q8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8C2×C414yes—
55C2×C4×Q8D414yes—
56C2×C4×Q8Q814yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C42—21no—
70C22×C42—21no—
71C22×C42—21no—
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103C2×C418yes—
85C2×#97—22no—
86C2×#97—22no—
87C2×#97C2×C414yes—
88#391—22no—
89#391—22no—
90#391—22no—
91#391—22no—
92#389—24no—
93#389—24no—
94#389—42no—
95#389—42no—
96#387—28no—
97#387—28no—
98#384—22no—
99#384—22no—
100#384—44no—
101#383—28no—
102#382—22no—
103#382D414yes—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4C2×C414yes—
112#376—28no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95D411yescharacteristic
125C2×#95D411yescharacteristic
126#340—24no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340—24no—
131#340C2×C418yes—
132#336—42no—
133#336—42no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8C22×C412yes—
142C22×Q8C2×D412yes—
143C22×Q8C2×Q812yes—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2411yescharacteristic
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×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8C22×C414yes—
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—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—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
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—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C22×C414yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×Q812yes—
219C2×C4⋊C4#4114yes—
220C2×C4⋊C4#4114yes—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42C22×C412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×Q812yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247C4⋊C8—44no—
248#103—216no—
249#103—48no—
250#97—28no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—22no—
261#95—22no—
262#95—44no—
263#95—44no—
264C24C22×D411yescharacteristic
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—24no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—24no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8C2×C4⋊C414yes—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4C4×D412yes—
344C22×C4C4×D412yes—
345C22×C4#12012yes—
346C22×C4#12012yes—
347C22×C4C22⋊Q812yes—
348C22×C4C22⋊Q812yes—
349C22×C4AGL(1,ℤ/8)12yes—
350C22×C4#13712yes—
351C22×C4C23×C411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×Q811yescharacteristic
355C22×C4C2×#4112yes—
356C22×C4C2×#4112yes—
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4H(5,𝔽2)12yes—
360C22×C42−1+412yes—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C4⋊C4—22no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—216no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—88no—
395C4⋊C4—84no—
396C4⋊C4—84no—
397C4⋊C4—88no—
398C4⋊C4C2×C4⋊C414yes—
399C4⋊C4#11718yes—
400C42—216no—
401C42—48no—
402C42—48no—
403C42—48no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—42no—
409C42—42no—
410C42C2×C4⋊C414yes—
411C42C4×D414yes—
412C42C22⋊Q814yes—
413C42#12314yes—
414C23C2×C4×D411yescharacteristic
415C23C2×#12011yescharacteristic
416C23C2×C22⋊Q811yescharacteristic
417C23#54612yes—
418C23#54812yes—
419C23C2×AGL(1,ℤ/8)11yescharacteristic
420C23C2×#13711yescharacteristic
421C23#57511yescharacteristic
422C23#57511yescharacteristic
423C23#57712yes—
424C23#57812yes—
425Q8—24no—
426Q8—24no—
427Q8—44no—
428Q8—44no—
429Q8—42no—
430Q8—42no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—28no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
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—82no—
497C2×C4#38514yes—
498C2×C4#38514yes—
499C2×C4C22×C4⋊C411yescharacteristic
500C2×C4C2×#11712yes—
501C2×C4C2×C4×D411yescharacteristic
502C2×C4#52012yes—
503C2×C4C2×C22⋊Q811yescharacteristic
504C2×C4C2×#12311yescharacteristic
505C2×C4#53211yescharacteristic
506C2×C4#53211yescharacteristic
507C2×C4#53312yes—
508C2×C4#53511yescharacteristic
509C2×C4#53611yescharacteristic
510C2×C4#54412yes—
511C2×C4#54712yes—
512C2×C4#55012yes—
513C2×C4#55212yes—
514C2×C4#55312yes—
515C2×C4#55512yes—
516C2×C4#55612yes—
517C8—44no—
518C8—82no—
519C22C2×#38511yescharacteristic
520C22#235212yes—
521C22#243111yescharacteristic
522C22#243211yescharacteristic
523C22#243912yes—
524C22C4×AGL(1,ℤ/8)11yescharacteristic
525C22C4×#13711yescharacteristic
526C22#294712yes—
527C22#300611yescharacteristic
528C22#300711yescharacteristic
529C22#301011yescharacteristic
530C22#301111yescharacteristic
531C22#307911yescharacteristic
532C22#308011yescharacteristic
533C22#308212yes—
534C22#331012yes—
535C22#331212yes—
536C22#331512yes—
537C22#331712yes—
538C22#336712yes—
539C22#336912yes—
540C22#337612yes—
541C22#337712yes—
542C4—24no—
543C4—24no—
544C4—44no—
545C4—44no—
546C4—42no—
547C4—42no—
548C4—48no—
549C4—82no—
550C4—82no—
551C4—81no—
552C4—81no—
553C4—162no—
554C4#228511yescharacteristic
555C4#231711yescharacteristic
556C4#235412yes—
557C4#244112yes—
558C4#249911yescharacteristic
559C4#250011yescharacteristic
560C2#1387911yescharacteristic
561C2#2087511yescharacteristic
562C2#2087711yescharacteristic
563C2#2277812yes—
564C2#2277912yes—
565C2#2558311yescharacteristic
566C2#2558411yescharacteristic
567C2#2566111yescharacteristic
568C2#2566211yescharacteristic
569C2#2603812yes—
570C2#2603912yes—
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
36. #1763C4semidirect
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×#18362
C2×#18032
C2×#17632
#139102
#110672
#108852
#108842
#108502
#108352