Finite Group Databasebeta

← #468080

Subgroups — #468080

Order 512.

1450 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic, complemented
3C2×#1864C211yesmaximal, characteristic
4#15202C211yesmaximal, characteristic, complemented
5#11380C211yesmaximal, characteristic, complemented
6#11357C211yesmaximal, characteristic, complemented
7#11327C212yesmaximal, complemented
8#11293C212yesmaximal, complemented
9#11189C212yesmaximal
10#11066C212yesmaximal
11#10788C211yesmaximal, characteristic
12C22×C22⋊Q8—21no—
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2211yescharacteristic
18C22×#102C2211yescharacteristic
19#2538—24no—
20#2538—24no—
21C2×#396—22no—
22C2×#390C2212yes—
23C2×#389C2212yes—
24C2×#389C2211yescharacteristic
25C2×#389C2212yes—
26C2×#384C2211yescharacteristic
27C2×#380—21no—
28#2099—24no—
29#1864—24no—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1446—21no—
41#1444—22no—
42#1444—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×Q8D411yescharacteristic
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C22⋊Q8—28no—
52C2×C22⋊Q8—44no—
53C2×C22⋊Q8—44no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—42no—
57C2×C4×Q8—42no—
58C2×C4×D4—22no—
59C2×C4×D4—42no—
60C2×C4×D4—42no—
61C2×C4×D4D414yes—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4D412yes—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79C2×#106—22no—
80C2×#106—42no—
81C2×#102—22no—
82C2×#102—22no—
83C2×#102—22no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—42no—
87C2×#102—42no—
88C2×#102Q814yescomplemented
89#396—216no—
90#390—24no—
91#390—24no—
92#389—24no—
93#389—24no—
94#389—22no—
95#389—22no—
96#389—42no—
97#389—42no—
98#384—22no—
99#384D414yes—
100#380—28no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—21no—
108#340—24no—
109#340—24no—
110#340—22no—
111#340—24no—
112#340—24no—
113#340—42no—
114#336—22no—
115#336—22no—
116#336—42no—
117#336—42no—
118C25—21no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—42no—
122C22×Q8—42no—
123C22×Q8SD1614yes—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4C2×D412yes—
132C22×D4C2×Q812yes—
133C22×D4#4112yes—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—24no—
139C23×C4—21no—
140C23×C4—44no—
141C23×C4—44no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8C2×Q812yes—
166C22⋊Q8—432no—
167C4×Q8—44no—
168C4×Q8—44no—
169C4×Q8—48no—
170C4×D4—44no—
171C4×D4—44no—
172C4×D4—48no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
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—21no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4C2×Q812yes—
216C2×C4⋊C4#4114yes—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—82no—
224C2×C22⋊C4—82no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235#106—44no—
236#102—28no—
237#102—48no—
238#102—44no—
239#95—28no—
240#95—44no—
241#95—44no—
242#95—42no—
243#95—42no—
244#95—81no—
245#95—82no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—42no—
250C24—42no—
251C24—41no—
252C24—41no—
253C24—42no—
254C24—42no—
255C24—41no—
256C24—41no—
257C24C22×D411yescharacteristic
258C2×Q8—28no—
259C2×Q8—48no—
260C2×Q8—44no—
261C2×Q8—48no—
262C2×Q8—44no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4C22⋊Q814yes—
281C22×C4—22no—
282C22×C4—22no—
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—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—48no—
305C22×C4—48no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—48no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—88no—
333C22×C4—88no—
334C22×C4#12012yes—
335C22×C4#12012yes—
336C22×C4C2×SD1614yes—
337C22×C4C2×SD1612yes—
338C22×C4AGL(1,ℤ/8)12yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×Q811yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4112yes—
344C22×C4H(5,𝔽2)12yes—
345C22×C4H(5,𝔽2)12yes—
346C22×C42−1+411yescharacteristic
347C22×C42−1+412yes—
348C22×C42−1+411yescharacteristic
349C2×C8—22no—
350C2×C8—24no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—42no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—42no—
357C2×C8—41no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—44no—
375C4⋊C4—48no—
376C4⋊C4—416no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—48no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—42no—
388C4⋊C4—84no—
389C4⋊C4—84no—
390C4⋊C4—88no—
391C22⋊C4—48no—
392C22⋊C4—416no—
393C22⋊C4—88no—
394C42—22no—
395C42—22no—
396C42—24no—
397C42—42no—
398C42—42no—
399C42—42no—
400C42—42no—
401C23—24no—
402C23—42no—
403C23—44no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—44no—
409C23—42no—
410C23—44no—
411C23—44no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—82no—
417C23—82no—
418C23—82no—
419C23—82no—
420C23—82no—
421C23—82no—
422C23—82no—
423C23—82no—
424C23C2×#12011yescharacteristic
425C23#53412yes—
426C23#54811yescharacteristic
427C23#54812yes—
428C23D4×Q811yescharacteristic
429C23C22×SD1611yescharacteristic
430C23C2×AGL(1,ℤ/8)11yescharacteristic
431C23#57512yes—
432C23#57612yes—
433C23#57711yescharacteristic
434C23#57811yescharacteristic
435Q8—416no—
436D4—24no—
437D4—24no—
438D4—42no—
439D4—42no—
440D4—44no—
441D4—44no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—816no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4#45014yes—
496C2×C4C2×C22⋊Q811yescharacteristic
497C2×C4#53211yescharacteristic
498C2×C4#53411yescharacteristic
499C2×C4#53411yescharacteristic
500C2×C4#53512yes—
501C2×C4#53611yescharacteristic
502C2×C4#53611yescharacteristic
503C2×C4#53912yes—
504C2×C4#53912yes—
505C2×C4#54212yes—
506C2×C4#54412yes—
507C2×C4#54712yes—
508C2×C4#54811yescharacteristic
509C2×C4D4×Q811yescharacteristic
510C2×C4#55112yes—
511C2×C4#55412yes—
512C2×C4#56312yes—
513C2×C4#56312yes—
514C8—42no—
515C8—81no—
516C8—82no—
517C22—44no—
518C22—44no—
519C22—44no—
520C22—84no—
521C22—84no—
522C22—82no—
523C22—82no—
524C22—84no—
525C22—84no—
526C22—82no—
527C22—82no—
528C22#240911yescharacteristic
529C22#243211yescharacteristic
530C22#261311yescharacteristic
531C22#272812yes—
532C22#272912yes—
533C22C2×#45011yescharacteristic
534C22#300511yescharacteristic
535C22#300811yescharacteristic
536C22#301311yescharacteristic
537C22#318712yes—
538C22#319412yes—
539C22#319812yes—
540C22#320312yes—
541C22#335911yescharacteristic
542C22#336212yes—
543C22#336612yes—
544C22#337012yes—
545C22#337311yescharacteristic
546C22#337612yes—
547C22#337712yes—
548C22Q8×SD1611yescharacteristic
549C22#338611yescharacteristic
550C22#338912yes—
551C4—24no—
552C4—44no—
553C4—42no—
554C4—82no—
555C4—84no—
556C4—84no—
557C4—82no—
558C4—82no—
559C4—161no—
560C4#256411yescharacteristic
561C4#262711yescharacteristic
562C4#266311yescharacteristic
563C4#272012yes—
564C4#272312yes—
565C4#281711yescharacteristic
566C2—84no—
567C2#1660111yescharacteristic
568C2#2433011yescharacteristic
569C2#2433911yescharacteristic
570C2#2564911yescharacteristic
571C2#2565611yescharacteristic
572C2#2959611yescharacteristic
573C2#2960711yescharacteristic
574C2#3254612yes—
575C2#3255412yes—
576C2#3256812yes—
577C2#3258112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2099C2semidirect
4. #15202C2semidirect
5. #11380C2semidirect
6. #11357C2semidirect
7. #11327C2semidirect
8. #11293C2semidirect
88. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#20992
C2×#18642
#152022
#113802
#113572
#113272
#112932
#111892
#110662
#107882