Finite Group Databasebeta

← #464870

Subgroups — #464870

Order 512.

1542 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic, complemented
2C2×#2093C211yesmaximal, characteristic
3C2×#2057C212yesmaximal, complemented
4C2×#2023C212yesmaximal
5C2×#1925C211yesmaximal, characteristic
6#15148C211yesmaximal, characteristic
7#11357C211yesmaximal, characteristic, complemented
8#11346C211yesmaximal, characteristic, complemented
9#11328C212yesmaximal, complemented
10#11066C212yesmaximal
11#10806C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8—21no—
13C22×C4⋊Q8C2212yes—
14C22×#107C2211yescharacteristic
15C22×#107C2212yes—
16C22×#102C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19#2551—24no—
20#2551—24no—
21C2×#399—21no—
22C2×#395—22no—
23C2×#390C2211yescharacteristic
24C2×#390C2212yes—
25C2×#389—21no—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#384C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#2093—24no—
32#2057—28no—
33#2023—28no—
34#1925—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44#1439—22no—
45#1439—21no—
46C4⋊#95C2211yescharacteristic
47C23×Q8—22no—
48C23×D4C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C2×C4⋊Q8—28no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—42no—
58C2×C4⋊Q8—42no—
59C2×C4⋊Q8—44no—
60C2×C4⋊Q8—44no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—21no—
77C22×C42—21no—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—24no—
83C2×#107—22no—
84C2×#107—42no—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—22no—
90C2×#102—24no—
91C2×#102—42no—
92C2×#102Q814yescomplemented
93#399—44no—
94#395—216no—
95#390—24no—
96#390—22no—
97#390—22no—
98#390—42no—
99#389—28no—
100#389—24no—
101#389—24no—
102#389—24no—
103#389—42no—
104#384—22no—
105#384D414yes—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4D414yes—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95D412yes—
116#340—24no—
117#340—22no—
118#340—24no—
119#340—24no—
120#340D418yes—
121#336—24no—
122#336—24no—
123#336—22no—
124#336—22no—
125C25—21no—
126C22×Q8—28no—
127C22×Q8—44no—
128C22×Q8—44no—
129C22×Q8—44no—
130C22×Q8—44no—
131C22×Q8—82no—
132C22×Q8—82no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—21no—
140C22×D4—22no—
141C22×D4—21no—
142C22×D4C2×Q812yes—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2411yescharacteristic
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8C2×D414yes—
169C22×C8C2×Q812yes—
170C4⋊Q8—48no—
171C4⋊Q8—48no—
172C4⋊Q8—48no—
173C4⋊Q8—432no—
174C4⋊Q8—48no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
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—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4D814yes—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D414yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×Q812yes—
219C2×C4⋊C4#4112yes—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4—82no—
230C2×C22⋊C4—82no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244#107—48no—
245#107—44no—
246#102—28no—
247#102—48no—
248#102—44no—
249#95—28no—
250#95—24no—
251#95—24no—
252#95—44no—
253#95—42no—
254#95—81no—
255C24—22no—
256C24—41no—
257C24—42no—
258C24—42no—
259C24—41no—
260C24—41no—
261C24—41no—
262C24—42no—
263C24—42no—
264C24—41no—
265C24—41no—
266C24C22×D411yescharacteristic
267C2×Q8—416no—
268C2×Q8—416no—
269C2×Q8—88no—
270C2×Q8—88no—
271C2×Q8—88no—
272C2×Q8—88no—
273C2×Q8—88no—
274C2×D4—22no—
275C2×D4—24no—
276C2×D4—22no—
277C2×D4—24no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—41no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—41no—
290C2×D4—42no—
291C2×D4—42no—
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—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
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×C4—41no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4#12012yes—
342C22×C4#12012yes—
343C22×C4C2×D814yes—
344C22×C4C2×D812yes—
345C22×C4AGL(1,ℤ/8)12yes—
346C22×C4#13714yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D412yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×Q811yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4H(5,𝔽2)12yes—
353C22×C42−1+412yes—
354C22×C42−1+411yescharacteristic
355C22×C42−1+411yescharacteristic
356C22×C42−1+412yes—
357C2×C8—22no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—41no—
363C2×C8—42no—
364C2×C8—41no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—42no—
373C2×C8—42no—
374C4⋊C4—28no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—28no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—416no—
386C4⋊C4—416no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—48no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—42no—
395C4⋊C4—84no—
396C4⋊C4—88no—
397C4⋊C4C22⋊Q814yes—
398C4⋊C4#12714yes—
399C22⋊C4—88no—
400C22⋊C4—88no—
401C22⋊C4—84no—
402C22⋊C4—84no—
403C42—22no—
404C42—22no—
405C42—24no—
406C42—42no—
407C42—42no—
408C42—42no—
409C42—42no—
410C42—48no—
411C23—42no—
412C23—42no—
413C23—44no—
414C23—42no—
415C23—44no—
416C23—81no—
417C23—81no—
418C23—82no—
419C23—81no—
420C23—82no—
421C23—81no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23—81no—
427C23—81no—
428C23—81no—
429C23—82no—
430C23—81no—
431C23—82no—
432C23—82no—
433C23—82no—
434C23C2×#12011yescharacteristic
435C23#53612yes—
436C23D4212yes—
437C23#54811yescharacteristic
438C23D4×Q811yescharacteristic
439C23C22×D811yescharacteristic
440C23C2×AGL(1,ℤ/8)11yescharacteristic
441C23C2×#13712yes—
442C23#57612yes—
443C23#57711yescharacteristic
444C23#57811yescharacteristic
445Q8—832no—
446D4—42no—
447D4—44no—
448D4—42no—
449D4—81no—
450D4—81no—
451D4—82no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4#44714yes—
507C2×C4C2×C22⋊Q811yescharacteristic
508C2×C4C2×#12711yescharacteristic
509C2×C4#53511yescharacteristic
510C2×C4#53611yescharacteristic
511C2×C4#53611yescharacteristic
512C2×C4#53611yescharacteristic
513C2×C4#53612yes—
514C2×C4#53612yes—
515C2×C4#53612yes—
516C2×C4#54212yes—
517C2×C4#54412yes—
518C2×C4D4212yes—
519C2×C4#54712yes—
520C2×C4#54811yescharacteristic
521C2×C4D4×Q811yescharacteristic
522C2×C4D4×Q812yes—
523C2×C4#56312yes—
524C2×C4#56312yes—
525C8—44no—
526C8—42no—
527C8—81no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—84no—
532C22—84no—
533C22—162no—
534C22—162no—
535C22—161no—
536C22—161no—
537C22—161no—
538C22—161no—
539C22#240911yescharacteristic
540C22#243311yescharacteristic
541C22#261411yescharacteristic
542C22#267412yes—
543C22#267712yes—
544C22C2×#44711yescharacteristic
545C22#300411yescharacteristic
546C22#300811yescharacteristic
547C22#301311yescharacteristic
548C22#322612yes—
549C22#323812yes—
550C22#324112yes—
551C22#324312yes—
552C22#326612yes—
553C22#327012yes—
554C22#327412yes—
555C22D4×D812yes—
556C22#335811yescharacteristic
557C22#336711yescharacteristic
558C22#337512yes—
559C22Q8×D811yescharacteristic
560C22#338411yescharacteristic
561C22#338512yes—
562C4—24no—
563C4—44no—
564C4—44no—
565C4—44no—
566C4—44no—
567C4—42no—
568C4—84no—
569C4—84no—
570C4—161no—
571C4#254211yescharacteristic
572C4#262811yescharacteristic
573C4#265811yescharacteristic
574C4#267412yes—
575C4#267712yes—
576C4#282111yescharacteristic
577C2—162no—
578C2#1651111yescharacteristic
579C2#2432611yescharacteristic
580C2#2434111yescharacteristic
581C2#2575711yescharacteristic
582C2#2576611yescharacteristic
583C2#2963811yescharacteristic
584C2#2964611yescharacteristic
585C2#3080512yes—
586C2#3083212yes—
587C2#3094612yes—
588C2#3097012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×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
1. C2×#2551C2semidirect
3. C2×#2057C2semidirect
7. #11357C2semidirect
8. #11346C2semidirect
9. #11328C2semidirect
11. #10806C2semidirect
92. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#20932
C2×#20572
C2×#20232
C2×#19252
#151482
#113572
#113462
#113282
#110662
#108062