Finite Group Databasebeta

← #360648

Subgroups — #360648

Order 512.

1600 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2C2×#1836C211yesmaximal, characteristic, complemented
3C2×#1804C211yesmaximal, characteristic
4C2×#1762C211yesmaximal, characteristic
5#13954C211yesmaximal, characteristic, complemented
6#11066C212yesmaximal
7#11066C212yesmaximal
8#10885C212yesmaximal, complemented
9#10884C212yesmaximal, complemented
10#10849C211yesmaximal, characteristic, complemented
11#10835C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117—21no—
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#97C2211yescharacteristic
19#2354—24no—
20#2354—24no—
21C2×#390C2212yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#388—22no—
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#380—21no—
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1836—24no—
32#1804—24no—
33#1762C418yescomplemented
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1444—21no—
44#1444—21no—
45C4⋊#95—22no—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4C2×C414yes—
55C2×C4×D4D414yes—
56C2×C4×D4Q814yes—
57C2×#117—28no—
58C2×#117—48no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C42—22no—
74C22×C42—21no—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—22no—
83C2×#102—22no—
84C2×#102—42no—
85C2×#102—42no—
86C2×#102C2×C418yes—
87C2×#97—22no—
88C2×#97—22no—
89C2×#97C2×C414yes—
90#390—24no—
91#390—24no—
92#389—24no—
93#389—24no—
94#389—42no—
95#389—42no—
96#388—216no—
97#384—22no—
98#384D414yes—
99#382—22no—
100#382—22no—
101#380—28no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—24no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4C2×C414yes—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95D412yes—
115#340—22no—
116#340—22no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340C2×C418yes—
122#336—42no—
123#336—42no—
124C25—21no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4C22×C412yes—
133C22×D4C2×D412yes—
134C22×D4C2×Q812yes—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8C22×C414yes—
167C4×D4—28no—
168C4×D4—28no—
169C4×D4—28no—
170C4×D4C4⋊C4116yescomplemented
171#117—432no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
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—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—81no—
203C2×C4⋊C4—81no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C414yes—
208C2×C4⋊C4C2×D414yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×Q812yes—
211C2×C4⋊C4#4114yes—
212C2×C4⋊C4#4114yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—82no—
221C2×C22⋊C4—82no—
222C2×C42—28no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—44no—
231C2×C42—44no—
232C2×C42—81no—
233C2×C42—81no—
234C2×C42C22×C412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×Q812yes—
238C2×C42#4112yes—
239C2×C42#4112yes—
240C4⋊C8—44no—
241#102—216no—
242#102—44no—
243#102—44no—
244#97—28no—
245#95—28no—
246#95—24no—
247#95—24no—
248#95—44no—
249#95—44no—
250#95—42no—
251#95—42no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24—41no—
260C24—41no—
261C24—41no—
262C24—41no—
263C24C22×D411yescharacteristic
264C2×D4—24no—
265C2×D4—22no—
266C2×D4—24no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—24no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—24no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4C2×C4⋊C414yes—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—216no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—48no—
310C22×C4—416no—
311C22×C4—416no—
312C22×C4—48no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
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—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4C4×D412yes—
336C22×C4C4×D412yes—
337C22×C4#12012yes—
338C22×C4#12012yes—
339C22×C4C22⋊Q812yes—
340C22×C4C22⋊Q812yes—
341C22×C4AGL(1,ℤ/8)14yes—
342C22×C4C23×C411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×Q811yescharacteristic
346C22×C4C2×#4112yes—
347C22×C4C2×#4112yes—
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4H(5,𝔽2)12yes—
351C22×C42−1+412yes—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C4⋊C4—28no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—44no—
379C4⋊C4—48no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—84no—
389C4⋊C4—88no—
390C4⋊C4—88no—
391C4⋊C4C2×C4⋊C414yes—
392C4⋊C4#11718yescomplemented
393C22⋊C4—216no—
394C22⋊C4—416no—
395C22⋊C4—88no—
396C42—416no—
397C42—42no—
398C42—42no—
399C42—42no—
400C42—42no—
401C42—84no—
402C42C2×C4⋊C414yes—
403C42C4×D414yes—
404C42C22⋊Q814yes—
405C42#12314yes—
406C23—24no—
407C23—44no—
408C23—42no—
409C23—42no—
410C23—44no—
411C23—42no—
412C23—42no—
413C23—44no—
414C23—42no—
415C23—42no—
416C23—44no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—82no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23—82no—
427C23—82no—
428C23—82no—
429C23C2×C4×D411yescharacteristic
430C23C2×#12011yescharacteristic
431C23C2×C22⋊Q811yescharacteristic
432C23#54612yes—
433C23#54812yes—
434C23C2×AGL(1,ℤ/8)12yes—
435C23#57512yes—
436C23#57611yescharacteristic
437C23#57711yescharacteristic
438C23#57711yescharacteristic
439C23#57811yescharacteristic
440D4—24no—
441D4—24no—
442D4—44no—
443D4—44no—
444D4—42no—
445D4—42no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—28no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—432no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
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—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4#38514yes—
501C2×C4#38514yes—
502C2×C4C22×C4⋊C411yescharacteristic
503C2×C4C2×#11712yes—
504C2×C4C2×C4×D411yescharacteristic
505C2×C4#52012yes—
506C2×C4C2×C22⋊Q811yescharacteristic
507C2×C4C2×#12311yescharacteristic
508C2×C4#53211yescharacteristic
509C2×C4#53211yescharacteristic
510C2×C4#53312yes—
511C2×C4#53511yescharacteristic
512C2×C4#53611yescharacteristic
513C2×C4#54412yes—
514C2×C4#54712yes—
515C2×C4#55012yes—
516C2×C4#55212yes—
517C2×C4#55312yes—
518C2×C4#55512yes—
519C2×C4#55612yes—
520C8—44no—
521C8—81no—
522C8—81no—
523C22—44no—
524C22—44no—
525C22—44no—
526C22—82no—
527C22—82no—
528C22—84no—
529C22—84no—
530C22—82no—
531C22—82no—
532C22—84no—
533C22—84no—
534C22C2×#38511yescharacteristic
535C22#235212yes—
536C22#243111yescharacteristic
537C22#243211yescharacteristic
538C22#243912yes—
539C22C4×AGL(1,ℤ/8)12yes—
540C22#294611yescharacteristic
541C22#294711yescharacteristic
542C22#300311yescharacteristic
543C22#300511yescharacteristic
544C22#301211yescharacteristic
545C22#301311yescharacteristic
546C22#307712yes—
547C22#308311yescharacteristic
548C22#308411yescharacteristic
549C22#330612yes—
550C22#330812yes—
551C22#331912yes—
552C22#332112yes—
553C22#336612yes—
554C22#336812yes—
555C22#337612yes—
556C22#337712yes—
557C4—24no—
558C4—24no—
559C4—44no—
560C4—44no—
561C4—48no—
562C4—84no—
563C4—82no—
564C4—161no—
565C4—161no—
566C4#228511yescharacteristic
567C4#231711yescharacteristic
568C4#235412yes—
569C4#244112yes—
570C4#249911yescharacteristic
571C4#250011yescharacteristic
572C2—84no—
573C2#1387911yescharacteristic
574C2#2087311yescharacteristic
575C2#2087811yescharacteristic
576C2#2278012yes—
577C2#2278112yes—
578C2#2557711yescharacteristic
579C2#2557811yescharacteristic
580C2#2565511yescharacteristic
581C2#2565611yescharacteristic
582C2#2603212yes—
583C2#2603312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
2. C2×#1836C2semidirect
5. #13954C2semidirect
8. #10885C2semidirect
9. #10884C2semidirect
10. #10849C2semidirect
11. #10835C2semidirect
33. #1762C4semidirect
170. C4×D4C4⋊C4semidirect
392. C4⋊C4#117semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×#18362
C2×#18042
C2×#17622
#139542
#110662
#108852
#108842
#108492
#108352