Finite Group Databasebeta

← #360739

Subgroups — #360739

Order 512.

1960 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C4⋊C4C211yesmaximal, characteristic
2C2×#1864C212yesmaximal
3C2×#1862C212yesmaximal
4C2×#1847C211yesmaximal, characteristic, complemented
5C2×#1804C211yesmaximal, characteristic
6C2×#1762C211yesmaximal, characteristic
7#13954C211yesmaximal, characteristic, complemented
8#11328C212yesmaximal, complemented
9#11327C212yesmaximal, complemented
10#10849C211yesmaximal, characteristic, complemented
11#10835C211yesmaximal, characteristic, complemented
12C22×C4×D4C2212yes—
13C22×C4×D4C2211yescharacteristic
14C23×C4⋊C4—21no—
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×C4⋊C8C2211yescharacteristic
18C22×#102C2211yescharacteristic
19C22×#102C2211yescharacteristic
20C22×#102C2212yes—
21C22×#97C2211yescharacteristic
22D4×C4⋊C4—24no—
23D4×C4⋊C4—24no—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#385—22no—
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#380—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1864—28no—
34#1862—28no—
35#1847—24no—
36#1804—24no—
37#1762C418yescomplemented
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1444—21no—
46#1444—21no—
47C4⋊#95—22no—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C24×C4—22no—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C2×C4×D4—24no—
55C2×C4×D4—24no—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—42no—
62C2×C4×D4—42no—
63C2×C4×D4C2×C414yes—
64C2×C4×D4D414yes—
65C2×C4×D4Q814yes—
66C22×C4⋊C4—28no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—48no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C42—21no—
83C22×C42C2312yes—
84C22×C42C2311yescharacteristic
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—42no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—42no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102C2×C418yes—
99C2×#102D418yes—
100C2×#102Q818yes—
101C2×#97—22no—
102C2×#97—22no—
103C2×#97C2×C414yescomplemented
104#389—24no—
105#389—24no—
106#389—24no—
107#389—24no—
108#389—42no—
109#389—42no—
110#385—216no—
111#384—22no—
112#384D414yes—
113#382—22no—
114#382—22no—
115#380—28no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4C2×C414yescomplemented
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—21no—
126C2×#95D412yes—
127#340—22no—
128#340—22no—
129#340—24no—
130#340—24no—
131#340C2×C418yescomplemented
132#336—24no—
133#336—24no—
134C25—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4C22×C412yes—
141C22×D4C2×D412yes—
142C22×D4C2×D414yes—
143C22×D4C2×Q812yes—
144C22×D4C2×Q814yes—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—28no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—44no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4—82no—
161C23×C4—82no—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8C22×C414yes—
182C22×C8C2×D414yes—
183C22×C8C2×Q814yes—
184C4×D4—28no—
185C4×D4—28no—
186C4×D4—28no—
187C4×D4—48no—
188C4×D4—48no—
189C4×D4—48no—
190C4×D4—48no—
191C4×D4C4⋊C4116yescomplemented
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—432no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—81no—
227C2×C4⋊C4—81no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C414yes—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4C2×Q812yes—
236C2×C4⋊C4C2×Q814yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—81no—
251C2×C42—81no—
252C2×C42C22×C412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×Q812yes—
256C2×C42#4112yes—
257C2×C42#4112yes—
258#107—48no—
259#106—48no—
260C4⋊C8—44no—
261#102—44no—
262#102—44no—
263#102C4⋊C4132yescomplemented
264#97—28no—
265#95—28no—
266#95—24no—
267#95—24no—
268#95—44no—
269#95—44no—
270#95—42no—
271#95—42no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—24no—
276C24—24no—
277C24—42no—
278C24—42no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24C22×D411yescharacteristic
284C2×D4—22no—
285C2×D4—24no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4C2×C4⋊C418yes—
298C2×D4C2×C4⋊C414yes—
299C2×D4#12714yes—
300C2×D4C4⋊Q818yes—
301C2×D4C4⋊Q814yes—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—216no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—48no—
330C22×C4—416no—
331C22×C4—416no—
332C22×C4—48no—
333C22×C4—416no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—416no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—88no—
358C22×C4—88no—
359C22×C4—88no—
360C22×C4—88no—
361C22×C4—88no—
362C22×C4C4×D412yes—
363C22×C4C4×D412yes—
364C22×C4#12012yes—
365C22×C4#12012yes—
366C22×C4C22⋊Q812yes—
367C22×C4C22⋊Q812yes—
368C22×C4AGL(1,ℤ/8)14yes—
369C22×C4C23×C411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D412yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×Q811yescharacteristic
374C22×C4C22×Q812yes—
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4H(5,𝔽2)12yes—
378C22×C42−1+412yes—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8C2×C4⋊C418yescomplemented
396C4⋊C4—28no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—44no—
406C4⋊C4—48no—
407C4⋊C4—416no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—84no—
417C4⋊C4—88no—
418C4⋊C4—88no—
419C4⋊C4C2×C4⋊C414yes—
420C4⋊C4C2×C4⋊C418yescomplemented
421C22⋊C4—216no—
422C22⋊C4—416no—
423C42—42no—
424C42—42no—
425C42—42no—
426C42—42no—
427C42—84no—
428C42C2×C4⋊C414yes—
429C42C4×D414yes—
430C42C22⋊Q814yes—
431C42#12314yes—
432C23—28no—
433C23—24no—
434C23—24no—
435C23—28no—
436C23—24no—
437C23—44no—
438C23—42no—
439C23—42no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—42no—
453C23—42no—
454C23—44no—
455C23C2×C4×D411yescharacteristic
456C23C2×#12011yescharacteristic
457C23C2×C22⋊Q811yescharacteristic
458C23D4212yes—
459C23D4×Q812yes—
460C23C2×AGL(1,ℤ/8)12yes—
461C23#57512yes—
462C23#57611yescharacteristic
463C23#57711yescharacteristic
464C23#57711yescharacteristic
465C23#57811yescharacteristic
466D4—24no—
467D4—28no—
468D4—28no—
469D4—24no—
470D4—24no—
471D4#38418yes—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—432no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—832no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4#38514yes—
528C2×C4#38514yes—
529C2×C4C22×C4⋊C411yescharacteristic
530C2×C4C22×C4⋊C412yes—
531C2×C4C2×C4×D411yescharacteristic
532C2×C4#52012yes—
533C2×C4C2×C22⋊Q811yescharacteristic
534C2×C4C2×#12311yescharacteristic
535C2×C4C2×#12711yescharacteristic
536C2×C4C2×C4⋊Q812yes—
537C2×C4C2×C4⋊Q811yescharacteristic
538C2×C4#53511yescharacteristic
539C2×C4#53611yescharacteristic
540C2×C4#54412yes—
541C2×C4D4212yes—
542C2×C4#54712yes—
543C2×C4#54712yes—
544C2×C4D4×Q812yes—
545C2×C4#55412yes—
546C2×C4#55412yes—
547C8—28no—
548C8—81no—
549C8—81no—
550C22—28no—
551C22—44no—
552C22—44no—
553C22—48no—
554C22—48no—
555C22—44no—
556C22—44no—
557C22—48no—
558C22—48no—
559C22—44no—
560C22—44no—
561C22C2×#38511yescharacteristic
562C22D4×C4⋊C412yes—
563C22#243311yescharacteristic
564C22#243411yescharacteristic
565C22#244012yes—
566C22C4×AGL(1,ℤ/8)12yes—
567C22#294611yescharacteristic
568C22#294711yescharacteristic
569C22#300311yescharacteristic
570C22#300511yescharacteristic
571C22#301211yescharacteristic
572C22#301311yescharacteristic
573C22#307712yes—
574C22#308311yescharacteristic
575C22#308411yescharacteristic
576C22#326412yes—
577C22#326512yes—
578C22#326712yes—
579C22#326912yes—
580C22#338412yes—
581C22#338612yes—
582C22#338812yes—
583C22#338912yes—
584C4—24no—
585C4—24no—
586C4—44no—
587C4—48no—
588C4—44no—
589C4—84no—
590C4—82no—
591C4—161no—
592C4—161no—
593C4C2×#38411yescharacteristic
594C4#231711yescharacteristic
595C4D4×C4⋊C412yes—
596C4#244012yes—
597C4#250111yescharacteristic
598C4#250211yescharacteristic
599C2—48no—
600C2#1388011yescharacteristic
601C2#2087311yescharacteristic
602C2#2087811yescharacteristic
603C2#2267612yes—
604C2#2267712yes—
605C2#2576311yescharacteristic
606C2#2576411yescharacteristic
607C2#2584411yescharacteristic
608C2#2584511yescharacteristic
609C2#2605212yes—
610C2#2605312yes—
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
4. C2×#1847C2semidirect
7. #13954C2semidirect
8. #11328C2semidirect
9. #11327C2semidirect
10. #10849C2semidirect
11. #10835C2semidirect
37. #1762C4semidirect
103. C2×#97C2×C4semidirect
119. C4×C4⋊C4C2×C4semidirect
131. #340C2×C4semidirect
191. C4×D4C4⋊C4semidirect
263. #102C4⋊C4semidirect
395. C2×C8C2×C4⋊C4semidirect
420. C4⋊C4C2×C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C4⋊C42
C2×#18642
C2×#18622
C2×#18472
C2×#18042
C2×#17622
#139542
#113282
#113272
#108492
#108352