Finite Group Databasebeta

← #464470

Subgroups — #464470

Order 512.

1280 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2071C211yesmaximal, characteristic
3C2×#2025C212yesmaximal
4C2×#1925C211yesmaximal, characteristic
5#15147C211yesmaximal, characteristic
6#11385C212yesmaximal, complemented
7#11371C211yesmaximal, characteristic, complemented
8#11346C211yesmaximal, characteristic, complemented
9#11219C212yesmaximal, complemented
10#11066C212yesmaximal
11#10806C211yesmaximal, characteristic, complemented
12C22×#125C2212yes—
13C22×#107C2211yescharacteristic
14C22×#102C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2211yescharacteristic
17#2552—24no—
18#2552—24no—
19C2×#398—22no—
20C2×#397—21no—
21C2×#390C2211yescharacteristic
22C2×#390C2212yes—
23C2×#389—21no—
24C2×#389C2212yes—
25C2×#384—21no—
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#2071—24no—
30#2025—28no—
31#1925—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1440—22no—
43#1439—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2212yes—
46C23×D4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×#125—24no—
51C2×#125—24no—
52C2×#125—24no—
53C2×#125—24no—
54C2×#125—42no—
55C2×#125—42no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—21no—
72C22×C42—21no—
73C22×C42C2312yes—
74C22×C42C2311yescharacteristic
75C2×#107—22no—
76C2×#107—24no—
77C2×#102—22no—
78C2×#102—22no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—22no—
82C2×#102—24no—
83C2×#102—42no—
84C2×#102D414yes—
85#398—216no—
86#397—44no—
87#390—24no—
88#390—22no—
89#390—22no—
90#390—42no—
91#389—28no—
92#389—24no—
93#389—24no—
94#384—28no—
95#384—22no—
96#384D414yes—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4D414yes—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95D412yes—
108#340—24no—
109#340—24no—
110#340—22no—
111#340—24no—
112#340—24no—
113#340—42no—
114#336—24no—
115#336—22no—
116#336—22no—
117#336—42no—
118C25—21no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4—22no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4C2×D412yes—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2411yescharacteristic
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8C2×D412yes—
156#125—48no—
157#125—48no—
158#125—48no—
159#125—48no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
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—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4#4112yes—
203C2×C4⋊C4#4114yes—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4—82no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228#107—44no—
229#102—28no—
230#102—48no—
231#102—44no—
232#95—28no—
233#95—24no—
234#95—24no—
235#95—44no—
236#95—42no—
237#95—81no—
238#95—82no—
239C24—22no—
240C24—41no—
241C24—42no—
242C24—41no—
243C24—42no—
244C24—41no—
245C24—41no—
246C24—42no—
247C24—42no—
248C24—41no—
249C24—41no—
250C24C22×D411yescharacteristic
251C2×D4—22no—
252C2×D4—24no—
253C2×D4—22no—
254C2×D4—24no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4#12012yes—
321C22×C4#12012yes—
322C22×C4#13512yes—
323C22×C4AGL(1,ℤ/8)12yes—
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4112yes—
329C22×C4H(5,𝔽2)11yescharacteristic
330C22×C4H(5,𝔽2)11yescharacteristic
331C22×C42−1+412yes—
332C22×C42−1+412yes—
333C22×C42−1+412yes—
334C2×C8—22no—
335C2×C8—24no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—44no—
359C4⋊C4—48no—
360C4⋊C4—416no—
361C4⋊C4—416no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—84no—
373C4⋊C4—88no—
374C4⋊C4#12114yes—
375C4⋊C4#12414yes—
376C22⋊C4—84no—
377C22⋊C4—88no—
378C22⋊C4—88no—
379C22⋊C4—84no—
380C42—22no—
381C42—22no—
382C42—24no—
383C42—42no—
384C42—42no—
385C42—42no—
386C42—42no—
387C42—48no—
388C23—42no—
389C23—44no—
390C23—44no—
391C23—42no—
392C23—42no—
393C23—81no—
394C23—82no—
395C23—82no—
396C23—82no—
397C23—82no—
398C23—81no—
399C23—81no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23—82no—
404C23—82no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—82no—
411C23C2×#12011yescharacteristic
412C23#53612yes—
413C23D4211yescharacteristic
414C23#54611yescharacteristic
415C23#54812yes—
416C23C2×#13511yescharacteristic
417C23C2×AGL(1,ℤ/8)11yescharacteristic
418C23#57512yes—
419C23#57612yes—
420C23#57711yescharacteristic
421C23#57811yescharacteristic
422D4—42no—
423D4—44no—
424D4—42no—
425D4—81no—
426D4—81no—
427D4—82no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4#44914yes—
483C2×C4C2×#12111yescharacteristic
484C2×C4C2×#12411yescharacteristic
485C2×C4#53511yescharacteristic
486C2×C4#53611yescharacteristic
487C2×C4#53611yescharacteristic
488C2×C4#53611yescharacteristic
489C2×C4#53612yes—
490C2×C4#53612yes—
491C2×C4#54112yes—
492C2×C4#54212yes—
493C2×C4#54212yes—
494C2×C4D4211yescharacteristic
495C2×C4#54611yescharacteristic
496C2×C4#54712yes—
497C2×C4#54812yes—
498C2×C4#55612yes—
499C2×C4#56312yes—
500C2×C4#56312yes—
501C8—42no—
502C8—81no—
503C8—82no—
504C22—84no—
505C22—84no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—162no—
510C22—161no—
511C22—161no—
512C22—161no—
513C22—161no—
514C22—162no—
515C22#240811yescharacteristic
516C22#243611yescharacteristic
517C22#261411yescharacteristic
518C22#267412yes—
519C22#271612yes—
520C22C2×#44911yescharacteristic
521C22#300511yescharacteristic
522C22#300811yescharacteristic
523C22#301311yescharacteristic
524C22#323212yes—
525C22#323412yes—
526C22#324112yes—
527C22#324312yes—
528C22#326812yes—
529C22#327211yescharacteristic
530C22#328311yescharacteristic
531C22#330311yescharacteristic
532C22#330912yes—
533C22#333011yescharacteristic
534C22#336112yes—
535C22#336612yes—
536C22#337112yes—
537C22#337712yes—
538C4—24no—
539C4—44no—
540C4—44no—
541C4—42no—
542C4—82no—
543C4—82no—
544C4—84no—
545C4—161no—
546C4—162no—
547C4#254711yescharacteristic
548C4#262111yescharacteristic
549C4#264011yescharacteristic
550C4#267412yes—
551C4#271612yes—
552C4#282111yescharacteristic
553C2—162no—
554C2#1650611yescharacteristic
555C2#2423711yescharacteristic
556C2#2425011yescharacteristic
557C2#2593011yescharacteristic
558C2#2594111yescharacteristic
559C2#2964011yescharacteristic
560C2#2964611yescharacteristic
561C2#3081112yes—
562C2#3083512yes—
563C2#3223712yes—
564C2#3225712yes—
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×#2552C2semidirect
6. #11385C2semidirect
7. #11371C2semidirect
8. #11346C2semidirect
9. #11219C2semidirect
11. #10806C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#20712
C2×#20252
C2×#19252
#151472
#113852
#113712
#113462
#112192
#110662
#108062