Finite Group Databasebeta

← #360649

Subgroups — #360649

Order 512.

1506 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2357C211yesmaximal, characteristic
2C2×#1836C211yesmaximal, characteristic, complemented
3C2×#1802C211yesmaximal, characteristic
4C2×#1762C211yesmaximal, characteristic
5#13910C211yesmaximal, characteristic, complemented
6#11066C212yesmaximal
7#11066C212yesmaximal
8#10885C212yesmaximal, complemented
9#10884C212yesmaximal, complemented
10#10850C211yesmaximal, characteristic, complemented
11#10835C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#102C2212yes—
15C22×#102C2212yes—
16C22×#97C2211yescharacteristic
17#2357—24no—
18#2357—24no—
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#388—21no—
24C2×#388—21no—
25C2×#384—21no—
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#381—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4—21no—
34#1836—24no—
35#1802—24no—
36#1762C418yescomplemented
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×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4C2×C414yes—
56C2×C4×D4D414yes—
57C2×C4×D4Q814yes—
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×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C42—21no—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—42no—
87C2×#102—42no—
88C2×#102C2×C418yes—
89C2×#97—22no—
90C2×#97—22no—
91C2×#97C2×C414yes—
92#390—22no—
93#390—22no—
94#390—22no—
95#390—22no—
96#389—24no—
97#389—24no—
98#389—42no—
99#389—42no—
100#388—28no—
101#388—28no—
102#384—22no—
103#384—22no—
104#384—44no—
105#382—22no—
106#382D414yes—
107#381—28no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4C2×C414yes—
116C4×C22⋊C4—28no—
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—
134C25—21no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4C22×C412yes—
143C22×D4C2×D412yes—
144C22×D4C2×Q812yes—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—82no—
154C23×C4—82no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
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×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8C22×C414yes—
176C4×D4—28no—
177C4×D4—28no—
178C4×D4—28no—
179C4×D4C4⋊C4116yescomplemented
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
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—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C414yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×Q812yes—
221C2×C4⋊C4#4114yes—
222C2×C4⋊C4#4114yes—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—82no—
235C2×C22⋊C4—82no—
236C2×C42—24no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—82no—
250C2×C42—82no—
251C2×C42C22×C412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×Q812yes—
255C2×C42#4112yes—
256C2×C42#4112yes—
257C4⋊C8—44no—
258#102—216no—
259#102—48no—
260#97—28no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—22no—
267#95—22no—
268#95—44no—
269#95—44no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—42no—
274C24—42no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—42no—
280C24—42no—
281C24C22×D411yescharacteristic
282C2×D4—24no—
283C2×D4—22no—
284C2×D4—24no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—24no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4C2×C4⋊C414yes—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—216no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—88no—
355C22×C4—88no—
356C22×C4—84no—
357C22×C4—84no—
358C22×C4C4×D412yes—
359C22×C4C4×D412yes—
360C22×C4#12012yes—
361C22×C4#12012yes—
362C22×C4C22⋊Q812yes—
363C22×C4C22⋊Q812yes—
364C22×C4AGL(1,ℤ/8)12yes—
365C22×C4#13712yes—
366C22×C4C23×C411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×Q811yescharacteristic
370C22×C4C2×#4112yes—
371C22×C4C2×#4112yes—
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4H(5,𝔽2)12yes—
375C22×C42−1+412yes—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—88no—
409C4⋊C4—84no—
410C4⋊C4—84no—
411C4⋊C4—88no—
412C4⋊C4C2×C4⋊C414yes—
413C4⋊C4#11718yes—
414C22⋊C4—216no—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C22⋊C4—84no—
418C22⋊C4—84no—
419C42—48no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—42no—
425C42—42no—
426C42—84no—
427C42—84no—
428C42C2×C4⋊C414yes—
429C42C4×D414yes—
430C42C22⋊Q814yes—
431C42#12314yes—
432C23—24no—
433C23—42no—
434C23—42no—
435C23—44no—
436C23—44no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—44no—
442C23—44no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23—82no—
455C23C2×C4×D411yescharacteristic
456C23C2×#12011yescharacteristic
457C23C2×C22⋊Q811yescharacteristic
458C23#54612yes—
459C23#54812yes—
460C23C2×AGL(1,ℤ/8)11yescharacteristic
461C23C2×#13711yescharacteristic
462C23#57511yescharacteristic
463C23#57511yescharacteristic
464C23#57612yes—
465C23#57712yes—
466D4—24no—
467D4—24no—
468D4—44no—
469D4—44no—
470D4—42no—
471D4—42no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—28no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—816no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4#38514yes—
533C2×C4#38514yes—
534C2×C4C22×C4⋊C411yescharacteristic
535C2×C4C2×#11712yes—
536C2×C4C2×C4×D411yescharacteristic
537C2×C4#52012yes—
538C2×C4C2×C22⋊Q811yescharacteristic
539C2×C4C2×#12311yescharacteristic
540C2×C4#53211yescharacteristic
541C2×C4#53211yescharacteristic
542C2×C4#53312yes—
543C2×C4#53511yescharacteristic
544C2×C4#53611yescharacteristic
545C2×C4#54412yes—
546C2×C4#54712yes—
547C2×C4#55012yes—
548C2×C4#55212yes—
549C2×C4#55312yes—
550C2×C4#55512yes—
551C2×C4#55612yes—
552C8—44no—
553C8—82no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—82no—
558C22—82no—
559C22—84no—
560C22—84no—
561C22—82no—
562C22—82no—
563C22—84no—
564C22—84no—
565C22C2×#38511yescharacteristic
566C22#235212yes—
567C22#243111yescharacteristic
568C22#243211yescharacteristic
569C22#243912yes—
570C22C4×AGL(1,ℤ/8)11yescharacteristic
571C22C4×#13711yescharacteristic
572C22#294612yes—
573C22#300611yescharacteristic
574C22#300711yescharacteristic
575C22#300811yescharacteristic
576C22#300911yescharacteristic
577C22#307911yescharacteristic
578C22#308011yescharacteristic
579C22#308112yes—
580C22#330712yes—
581C22#330912yes—
582C22#331812yes—
583C22#332012yes—
584C22#337012yes—
585C22#337112yes—
586C22#337312yes—
587C22#337512yes—
588C4—24no—
589C4—24no—
590C4—44no—
591C4—42no—
592C4—42no—
593C4—48no—
594C4—82no—
595C4—82no—
596C4—81no—
597C4—81no—
598C4—162no—
599C4#228511yescharacteristic
600C4#231711yescharacteristic
601C4#235412yes—
602C4#244112yes—
603C4#249911yescharacteristic
604C4#250011yescharacteristic
605C2—84no—
606C2#1387911yescharacteristic
607C2#2087511yescharacteristic
608C2#2087611yescharacteristic
609C2#2277412yes—
610C2#2277512yes—
611C2#2558111yescharacteristic
612C2#2558211yescharacteristic
613C2#2565911yescharacteristic
614C2#2566011yescharacteristic
615C2#2603612yes—
616C2#2603712yes—
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. #13910C2semidirect
8. #10885C2semidirect
9. #10884C2semidirect
10. #10850C2semidirect
11. #10835C2semidirect
36. #1762C4semidirect
179. C4×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23572
C2×#18362
C2×#18022
C2×#17622
#139102
#110662
#108852
#108842
#108502
#108352