Finite Group Databasebeta

← #399158

Subgroups — #399158

Order 512.

1712 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic
2C2×#2096C211yesmaximal, characteristic, complemented
3C2×#2095C211yesmaximal, characteristic, complemented
4C2×#1873C212yesmaximal
5C2×#1854C212yesmaximal
6#14702C211yesmaximal, characteristic
7#11373C211yesmaximal, characteristic, complemented
8#11371C211yesmaximal, characteristic, complemented
9#11325C212yesmaximal, complemented
10#11052C212yesmaximal, complemented
11#10712C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117C2212yes—
14C23×C42C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2212yes—
20C2×#391C2211yescharacteristic
21C2×#391C2211yescharacteristic
22#2285—24no—
23#2285—24no—
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#382C2212yes—
27C2×#380—22no—
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4—22no—
30#2096—24no—
31#2095—24no—
32#1873—28no—
33#1854—28no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41#1436—22no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C22⋊#95—22no—
45C23×Q8C2311yescharacteristic
46C24×C4D412yes—
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8Q814yes—
54C2×#117—24no—
55C2×#117—24no—
56C2×#117—24no—
57C2×#117—24no—
58C2×#117—42no—
59C2×#117—42no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—22no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C42—24no—
72C22×C42—22no—
73C22×C42—22no—
74C22×C42—24no—
75C22×C42—22no—
76C22×C42D412yes—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×#107—24no—
80C2×#107—24no—
81C2×#107—42no—
82C2×#106—24no—
83C2×#106—24no—
84C2×#106—42no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—42no—
90C2×#103—42no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8D418yes—
94#391—22no—
95#391—22no—
96#391—24no—
97#384—22no—
98#384—22no—
99#384D414yes—
100#384D414yes—
101#382—24no—
102#382—24no—
103#380—216no—
104#380—24no—
105C4×C4⋊C4—216no—
106C2×#95—24no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—24no—
116#336—22no—
117#336—22no—
118#336—24no—
119C25C2×D411yescharacteristic
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8C2×Q812yes—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D414yes—
153C23×C4C2×Q812yes—
154C23×C4#4114yes—
155C23×C4#4112yes—
156C23×C4C2411yescharacteristic
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8C2×D414yes—
170C22×C8C2×D414yes—
171C22⋊Q8—216no—
172C22⋊Q8—48no—
173#117—48no—
174#117—48no—
175#117—48no—
176#117—48no—
177C2×C4⋊C4—22no—
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—22no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4C2×D414yes—
208C2×C4⋊C4C2×D414yes—
209C2×C4⋊C4C2×Q812yes—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—42no—
218C2×C42—216no—
219C2×C42—216no—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—28no—
229C2×C42—44no—
230C2×C42—44no—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42#4114yes—
234#107—48no—
235#106—48no—
236#103—48no—
237#103—48no—
238C22⋊C8—216no—
239#95—216no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—42no—
244#95—44no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24C22⋊Q812yes—
254C24#12312yes—
255C24#12712yes—
256C24C22×D411yescharacteristic
257C2×Q8—24no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—42no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—216no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4—48no—
309C22×C4—48no—
310C22×C4—48no—
311C22×C4—48no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—42no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4#12014yes—
341C22×C4C22⋊Q818yes—
342C22×C4C22⋊Q812yes—
343C22×C4#12314yes—
344C22×C4#12318yes—
345C22×C4#12312yes—
346C22×C4#12712yes—
347C22×C4#13514yes—
348C22×C4#13714yes—
349C22×C4C22×D412yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×Q811yescharacteristic
353C22×C4C2×#4112yes—
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4112yes—
356C22×C4H(5,𝔽2)12yes—
357C22×C42−1+411yescharacteristic
358C22×C42−1+411yescharacteristic
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—416no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—48no—
385C4⋊C4—416no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C22⋊C4—416no—
391C22⋊C4—48no—
392C22⋊C4—84no—
393C42—216no—
394C42—24no—
395C42—24no—
396C42—24no—
397C42—24no—
398C42—416no—
399C23—24no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23#39914yes—
422C23C2×#12011yescharacteristic
423C23C2×C22⋊Q811yescharacteristic
424C23C2×#12311yescharacteristic
425C23C2×#12311yescharacteristic
426C23C2×#12711yescharacteristic
427C23#53511yescharacteristic
428C23#53611yescharacteristic
429C23C2×#13512yes—
430C23C2×#13712yes—
431C23#57512yes—
432C23#57512yes—
433Q8—44no—
434Q8—44no—
435Q8—82no—
436Q8—82no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—28no—
450C2×C4—28no—
451C2×C4—28no—
452C2×C4—28no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—432no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—48no—
469C2×C4—48no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—81no—
480C2×C4—84no—
481C2×C4—84no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—81no—
493C2×C4#39914yes—
494C2×C4#45218yes—
495C2×C4C2×C22⋊Q812yes—
496C2×C4C2×#12312yes—
497C2×C4#52512yes—
498C2×C4#52512yes—
499C2×C4#53912yes—
500C2×C4#53912yes—
501C2×C4D4212yes—
502C2×C4D4212yes—
503C2×C4#54712yes—
504C2×C4#54712yes—
505C2×C4#54811yescharacteristic
506C2×C4#54811yescharacteristic
507C2×C4D4×Q811yescharacteristic
508C2×C4D4×Q811yescharacteristic
509C2×C4#55312yes—
510C2×C4#55412yes—
511C8—44no—
512C8—44no—
513C22—24no—
514C22—24no—
515C22—24no—
516C22—42no—
517C22—42no—
518C22—44no—
519C22—42no—
520C22—42no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22C2×#39911yescharacteristic
525C22#240911yescharacteristic
526C22#243311yescharacteristic
527C22#248711yescharacteristic
528C22#250111yescharacteristic
529C22#262311yescharacteristic
530C22#284911yescharacteristic
531C22C2×#45212yes—
532C22#300512yes—
533C22#307612yes—
534C22#307812yes—
535C22#309112yes—
536C22#309112yes—
537C22#309512yes—
538C22#309512yes—
539C22#314711yescharacteristic
540C22#315011yescharacteristic
541C22#315211yescharacteristic
542C22#315311yescharacteristic
543C22#321012yes—
544C22#321912yes—
545C22#323312yes—
546C22#323712yes—
547C4—24no—
548C4—28no—
549C4—44no—
550C4—84no—
551C4—84no—
552C4—82no—
553C4—82no—
554C4—161no—
555C4—161no—
556C4#244312yes—
557C4#244312yes—
558C4#244912yes—
559C4#244912yes—
560C2—44no—
561C2#1477711yescharacteristic
562C2#2373011yescharacteristic
563C2#2373211yescharacteristic
564C2#2430312yes—
565C2#2571911yescharacteristic
566C2#2572711yescharacteristic
567C2#2768512yes—
568C2#2791111yescharacteristic
569C2#2791611yescharacteristic
570C2#2991512yes—
571C2#3298412yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2096C2semidirect
3. C2×#2095C2semidirect
7. #11373C2semidirect
8. #11371C2semidirect
9. #11325C2semidirect
10. #11052C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22852
C2×#20962
C2×#20952
C2×#18732
C2×#18542
#147022
#113732
#113712
#113252
#110522
#107122