Finite Group Databasebeta

← #403447

Subgroups — #403447

Order 512.

1482 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2365C211yesmaximal, characteristic, complemented
2C2×#2023C211yesmaximal, characteristic
3C2×#1925C211yesmaximal, characteristic
4#15148C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic, complemented
6#11346C211yesmaximal, characteristic, complemented
7#11328C212yesmaximal, complemented
8#11066C212yesmaximal
9#10931C212yesmaximal
10#10884C212yesmaximal, complemented
11#10806C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×#107C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C2×#399—21no—
19#2365—24no—
20#2365—24no—
21C2×#391—21no—
22C2×#390C2211yescharacteristic
23C2×#390C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#387—22no—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2023—24no—
31#1925—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1439—22no—
43#1439—21no—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×D4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8D414yes—
54C2×C4×Q8—28no—
55C2×C4×Q8—44no—
56C2×C4×Q8—44no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—21no—
71C22×C42—22no—
72C22×C42—21no—
73C22×C42C2312yes—
74C22×C42C2311yescharacteristic
75C2×#107—22no—
76C2×#107—42no—
77C2×#102—22no—
78C2×#102—22no—
79C2×#102—22no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—42no—
84C2×#102Q814yescomplemented
85#399—44no—
86#391—28no—
87#390—24no—
88#390—22no—
89#390—22no—
90#390—42no—
91#389—24no—
92#389—24no—
93#389—24no—
94#389—42no—
95#387—216no—
96C4×C4⋊C4—24no—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4D414yes—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95D412yes—
109#340—22no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114#340—24no—
115#336—22no—
116#336—22no—
117#336—42no—
118#336—42no—
119C25—21no—
120C22×Q8—28no—
121C22×Q8—24no—
122C22×Q8—44no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4C2×Q812yes—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2411yescharacteristic
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8C2×Q812yes—
162C4⋊Q8—28no—
163C4⋊Q8—28no—
164C4⋊Q8—216no—
165C4×Q8—432no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4D814yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D414yes—
200C2×C4⋊C4C2×Q812yes—
201C2×C4⋊C4#4114yes—
202C2×C4⋊C4#4112yes—
203C2×C4⋊C4#4114yes—
204C2×C4⋊C4#4114yes—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4—82no—
215C2×C22⋊C4—82no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—28no—
219C2×C42—24no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—44no—
227C2×C42—44no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232#107—44no—
233#102—28no—
234#102—44no—
235#102—48no—
236#95—28no—
237#95—28no—
238#95—24no—
239#95—24no—
240#95—44no—
241#95—42no—
242#95—81no—
243C24—22no—
244C24—41no—
245C24—42no—
246C24—42no—
247C24—41no—
248C24—41no—
249C24—41no—
250C24—42no—
251C24—41no—
252C24—42no—
253C24—41no—
254C24C22×D411yescharacteristic
255C2×Q8—216no—
256C2×Q8—416no—
257C2×Q8—416no—
258C2×Q8—48no—
259C2×Q8—48no—
260C2×D4—22no—
261C2×D4—24no—
262C2×D4—22no—
263C2×D4—24no—
264C2×D4—42no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—42no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4#12012yes—
329C22×C4#12012yes—
330C22×C4C2×D814yes—
331C22×C4C2×D812yes—
332C22×C4AGL(1,ℤ/8)12yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×Q811yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4112yes—
338C22×C4C2×#4112yes—
339C22×C4C2×#4112yes—
340C22×C4H(5,𝔽2)12yes—
341C22×C42−1+411yescharacteristic
342C22×C42−1+411yescharacteristic
343C2×C8—22no—
344C2×C8—24no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—41no—
350C2×C8—42no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C4⋊C4—216no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—28no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—44no—
373C4⋊C4—416no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—88no—
381C4⋊C4C22⋊Q814yes—
382C4⋊C4#12318yes—
383C22⋊C4—88no—
384C22⋊C4—84no—
385C22⋊C4—88no—
386C22⋊C4—84no—
387C42—24no—
388C42—22no—
389C42—22no—
390C42—416no—
391C42—42no—
392C42—42no—
393C42—42no—
394C42—42no—
395C42—48no—
396C23—42no—
397C23—42no—
398C23—44no—
399C23—42no—
400C23—44no—
401C23—82no—
402C23—82no—
403C23—81no—
404C23—81no—
405C23—81no—
406C23—82no—
407C23—82no—
408C23—82no—
409C23—82no—
410C23—81no—
411C23—82no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—82no—
417C23—82no—
418C23—82no—
419C23C2×#12011yescharacteristic
420C23#52512yes—
421C23#54612yes—
422C23#54811yescharacteristic
423C23D4×Q811yescharacteristic
424C23C22×D811yescharacteristic
425C23C2×AGL(1,ℤ/8)11yescharacteristic
426C23#57512yes—
427C23#57712yes—
428C23#57711yescharacteristic
429C23#57811yescharacteristic
430Q8—432no—
431D4—42no—
432D4—42no—
433D4—44no—
434D4—81no—
435D4—81no—
436D4—82no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—28no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4#44714yes—
492C2×C4C2×C22⋊Q811yescharacteristic
493C2×C4C2×#12312yes—
494C2×C4#52511yescharacteristic
495C2×C4#52511yescharacteristic
496C2×C4#52912yes—
497C2×C4#53211yescharacteristic
498C2×C4#53312yes—
499C2×C4#53511yescharacteristic
500C2×C4#53611yescharacteristic
501C2×C4#54212yes—
502C2×C4#54212yes—
503C2×C4#54811yescharacteristic
504C2×C4D4×Q811yescharacteristic
505C2×C4#55012yes—
506C2×C4#55212yes—
507C2×C4#55412yes—
508C2×C4#55612yes—
509C2×C4#55912yes—
510C8—42no—
511C8—81no—
512C8—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—84no—
517C22—84no—
518C22—162no—
519C22—162no—
520C22—161no—
521C22—161no—
522C22—161no—
523C22—161no—
524C22#240911yescharacteristic
525C22#243111yescharacteristic
526C22#245812yes—
527C22#246512yes—
528C22#284911yescharacteristic
529C22C2×#44711yescharacteristic
530C22#300511yescharacteristic
531C22#301011yescharacteristic
532C22#301311yescharacteristic
533C22#310012yes—
534C22#311412yes—
535C22#312112yes—
536C22#312612yes—
537C22#329412yes—
538C22#331112yes—
539C22#331312yes—
540C22#333212yes—
541C22#335811yescharacteristic
542C22#336711yescharacteristic
543C22#337112yes—
544C22Q8×D811yescharacteristic
545C22#338411yescharacteristic
546C22#338812yes—
547C4—24no—
548C4—44no—
549C4—44no—
550C4—48no—
551C4—42no—
552C4—84no—
553C4—82no—
554C4—82no—
555C4—161no—
556C4#239611yescharacteristic
557C4#242312yes—
558C4#244812yes—
559C4#258011yescharacteristic
560C4#260411yescharacteristic
561C4#263211yescharacteristic
562C2—162no—
563C2#1487511yescharacteristic
564C2#2432611yescharacteristic
565C2#2434011yescharacteristic
566C2#2556711yescharacteristic
567C2#2558011yescharacteristic
568C2#2668812yes—
569C2#2670612yes—
570C2#2699812yes—
571C2#2701412yes—
572C2#3300211yescharacteristic
573C2#3301511yescharacteristic
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×#2365C2semidirect
5. #11384C2semidirect
6. #11346C2semidirect
7. #11328C2semidirect
10. #10884C2semidirect
11. #10806C2semidirect
84. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23652
C2×#20232
C2×#19252
#151482
#113842
#113462
#113282
#110662
#109312
#108842
#108062