Finite Group Databasebeta

← #403448

Subgroups — #403448

Order 512.

1482 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2365C211yesmaximal, characteristic, complemented
2C2×#2023C211yesmaximal, characteristic
3C2×#1926C211yesmaximal, characteristic
4#15148C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic, complemented
6#11346C211yesmaximal, characteristic, complemented
7#11327C212yesmaximal, complemented
8#11066C212yesmaximal
9#10931C212yesmaximal
10#10885C212yesmaximal, complemented
11#10807C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×#106C2211yescharacteristic
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#1926—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
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—48no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—21no—
70C22×C42—22no—
71C22×C42—21no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74C2×#106—22no—
75C2×#106—42no—
76C2×#102—22no—
77C2×#102—22no—
78C2×#102—22no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—42no—
83C2×#102Q814yescomplemented
84#399—44no—
85#391—28no—
86#390—24no—
87#390—22no—
88#390—22no—
89#390—42no—
90#389—24no—
91#389—24no—
92#389—24no—
93#389—42no—
94#387—216no—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—24no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4D414yes—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95D412yes—
108#340—22no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114#336—22no—
115#336—22no—
116#336—42no—
117#336—42no—
118C25—21no—
119C22×Q8—28no—
120C22×Q8—24no—
121C22×Q8—44no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4C2×Q812yes—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8C2×Q812yes—
161C4⋊Q8—28no—
162C4⋊Q8—28no—
163C4⋊Q8—216no—
164C4×Q8—432no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2×Q812yes—
200C2×C4⋊C4#4114yes—
201C2×C4⋊C4#4112yes—
202C2×C4⋊C4#4114yes—
203C2×C4⋊C4#4114yes—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4—82no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—28no—
218C2×C42—24no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—44no—
226C2×C42—44no—
227C2×C42—44no—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230#106—44no—
231#102—28no—
232#102—44no—
233#102—48no—
234#95—28no—
235#95—28no—
236#95—24no—
237#95—24no—
238#95—44no—
239#95—42no—
240#95—81no—
241C24—22no—
242C24—41no—
243C24—42no—
244C24—42no—
245C24—41no—
246C24—41no—
247C24—41no—
248C24—42no—
249C24—41no—
250C24—42no—
251C24—41no—
252C24C22×D411yescharacteristic
253C2×Q8—216no—
254C2×Q8—48no—
255C2×Q8—416no—
256C2×Q8—416no—
257C2×Q8—48no—
258C2×D4—22no—
259C2×D4—24no—
260C2×D4—22no—
261C2×D4—24no—
262C2×D4—42no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4#12012yes—
325C22×C4#12012yes—
326C22×C4#13512yes—
327C22×C4AGL(1,ℤ/8)14yes—
328C22×C4AGL(1,ℤ/8)12yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×Q811yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4112yes—
334C22×C4C2×#4112yes—
335C22×C4C2×#4112yes—
336C22×C4H(5,𝔽2)12yes—
337C22×C42−1+411yescharacteristic
338C22×C42−1+411yescharacteristic
339C2×C8—22no—
340C2×C8—24no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—41no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C4⋊C4—28no—
357C4⋊C4—216no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—416no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—44no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—88no—
377C4⋊C4C22⋊Q814yes—
378C4⋊C4#12318yes—
379C22⋊C4—88no—
380C22⋊C4—84no—
381C22⋊C4—88no—
382C22⋊C4—84no—
383C42—24no—
384C42—22no—
385C42—22no—
386C42—48no—
387C42—416no—
388C42—42no—
389C42—42no—
390C42—42no—
391C42—42no—
392C23—42no—
393C23—44no—
394C23—42no—
395C23—42no—
396C23—44no—
397C23—81no—
398C23—82no—
399C23—81no—
400C23—82no—
401C23—81no—
402C23—82no—
403C23—82no—
404C23—82no—
405C23—81no—
406C23—82no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23C2×#12011yescharacteristic
416C23#52512yes—
417C23#54612yes—
418C23#54811yescharacteristic
419C23D4×Q811yescharacteristic
420C23C2×#13511yescharacteristic
421C23C2×AGL(1,ℤ/8)12yes—
422C23C2×AGL(1,ℤ/8)11yescharacteristic
423C23#57711yescharacteristic
424C23#57712yes—
425C23#57811yescharacteristic
426Q8—432no—
427D4—42no—
428D4—42no—
429D4—44no—
430D4—81no—
431D4—81no—
432D4—82no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—28no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
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—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4#44914yes—
486C2×C4C2×C22⋊Q811yescharacteristic
487C2×C4C2×#12312yes—
488C2×C4#52511yescharacteristic
489C2×C4#52511yescharacteristic
490C2×C4#52912yes—
491C2×C4#53211yescharacteristic
492C2×C4#53312yes—
493C2×C4#53511yescharacteristic
494C2×C4#53611yescharacteristic
495C2×C4#54212yes—
496C2×C4#54212yes—
497C2×C4#54811yescharacteristic
498C2×C4D4×Q811yescharacteristic
499C2×C4#55012yes—
500C2×C4#55212yes—
501C2×C4#55412yes—
502C2×C4#55612yes—
503C2×C4#55912yes—
504C8—42no—
505C8—81no—
506C8—82no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—84no—
511C22—84no—
512C22—162no—
513C22—162no—
514C22—161no—
515C22—161no—
516C22—161no—
517C22—161no—
518C22#240911yescharacteristic
519C22#243111yescharacteristic
520C22#245812yes—
521C22#246512yes—
522C22#284911yescharacteristic
523C22C2×#44911yescharacteristic
524C22#300311yescharacteristic
525C22#301011yescharacteristic
526C22#301311yescharacteristic
527C22#310512yes—
528C22#310912yes—
529C22#312112yes—
530C22#312612yes—
531C22#330412yes—
532C22#330612yes—
533C22#330812yes—
534C22#332412yes—
535C22#336411yescharacteristic
536C22#336811yescharacteristic
537C22#337312yes—
538C22#338111yescharacteristic
539C22#338412yes—
540C22#338611yescharacteristic
541C4—24no—
542C4—44no—
543C4—44no—
544C4—48no—
545C4—42no—
546C4—84no—
547C4—82no—
548C4—82no—
549C4—161no—
550C4#239611yescharacteristic
551C4#242312yes—
552C4#244812yes—
553C4#258011yescharacteristic
554C4#260411yescharacteristic
555C4#263211yescharacteristic
556C2—162no—
557C2#1487511yescharacteristic
558C2#2432811yescharacteristic
559C2#2433811yescharacteristic
560C2#2557411yescharacteristic
561C2#2557711yescharacteristic
562C2#2669112yes—
563C2#2670512yes—
564C2#2699912yes—
565C2#2701512yes—
566C2#3300411yescharacteristic
567C2#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. #11327C2semidirect
10. #10885C2semidirect
11. #10807C2semidirect
83. 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×#19262
#151482
#113842
#113462
#113272
#110662
#109312
#108852
#108072