Finite Group Databasebeta

← #399160

Subgroups — #399160

Order 512.

1712 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×#2285C211yesmaximal, characteristic
2C2×#2097C212yesmaximal, complemented
3C2×#1873C212yesmaximal
4C2×#1854C212yesmaximal
5#14702C211yesmaximal, characteristic
6#11372C212yesmaximal, complemented
7#11324C211yesmaximal, characteristic, complemented
8#11323C211yesmaximal, characteristic, complemented
9#11050C211yesmaximal, characteristic, complemented
10#11050C211yesmaximal, characteristic, 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—28no—
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#382C2212yes—
26C2×#380—22no—
27C2×#380C2211yescharacteristic
28C2×C4×C4⋊C4—22no—
29#2097—28no—
30#1873—28no—
31#1854—28no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39#1436—22no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C22⋊#95—22no—
43C23×Q8C2311yescharacteristic
44C24×C4D412yes—
45C24×C4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8Q814yes—
52C2×#117—24no—
53C2×#117—24no—
54C2×#117—24no—
55C2×#117—24no—
56C2×#117—42no—
57C2×#117—42no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C22⋊C4—22no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C42—24no—
70C22×C42—24no—
71C22×C42—22no—
72C22×C42—22no—
73C22×C42—22no—
74C22×C42D412yes—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#107—24no—
78C2×#107—24no—
79C2×#107—42no—
80C2×#106—24no—
81C2×#106—24no—
82C2×#106—42no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—42no—
88C2×#103—42no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8D418yes—
92#391—24no—
93#391—22no—
94#391—22no—
95#384—24no—
96#384—24no—
97#382—24no—
98#382D418yes—
99#380—216no—
100#380—24no—
101C4×C4⋊C4—216no—
102C2×#95—24no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108#340—24no—
109#340—24no—
110#340—24no—
111#340—24no—
112#336—24no—
113#336—22no—
114#336—22no—
115C25C2×D411yescharacteristic
116C22×Q8—21no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8C2×Q812yes—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D414yes—
148C23×C4C2×D412yes—
149C23×C4C2×Q812yes—
150C23×C4#4114yes—
151C23×C4#4112yes—
152C23×C4C2411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
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×C8C2×D414yes—
166C22×C8C2×D414yes—
167C22⋊Q8—216no—
168C22⋊Q8—48no—
169#117—48no—
170#117—48no—
171#117—48no—
172#117—48no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
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—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4C2×D414yes—
205C2×C4⋊C4C2×Q812yes—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—42no—
214C2×C42—216no—
215C2×C42—216no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—28no—
225C2×C42—44no—
226C2×C42—44no—
227C2×C42C2×D414yes—
228C2×C42#4112yes—
229C2×C42#4112yes—
230#107—48no—
231#106—48no—
232#103—48no—
233#103—48no—
234C22⋊C8—216no—
235#95—216no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—42no—
240#95—44no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24C22⋊Q812yes—
250C24#12312yes—
251C24#12712yes—
252C24C22×D411yescharacteristic
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—216no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—48no—
303C22×C4—48no—
304C22×C4—48no—
305C22×C4—48no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4#12014yes—
337C22×C4C22⋊Q818yes—
338C22×C4C22⋊Q812yes—
339C22×C4#12318yes—
340C22×C4#12312yes—
341C22×C4#12314yes—
342C22×C4#12712yes—
343C22×C4#13514yes—
344C22×C4#13714yes—
345C22×C4C22×D412yes—
346C22×C4C22×D412yes—
347C22×C4C22×Q811yescharacteristic
348C22×C4C2×#4112yes—
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4H(5,𝔽2)11yescharacteristic
353C22×C4H(5,𝔽2)11yescharacteristic
354C22×C42−1+412yes—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—416no—
375C4⋊C4—416no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—48no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C22⋊C4—416no—
387C22⋊C4—48no—
388C22⋊C4—84no—
389C42—216no—
390C42—24no—
391C42—24no—
392C42—24no—
393C42—24no—
394C42—416no—
395C23—24no—
396C23—24no—
397C23—24no—
398C23—24no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23#39914yes—
418C23C2×#12011yescharacteristic
419C23C2×C22⋊Q811yescharacteristic
420C23C2×#12311yescharacteristic
421C23C2×#12311yescharacteristic
422C23C2×#12711yescharacteristic
423C23#53511yescharacteristic
424C23#53611yescharacteristic
425C23C2×#13512yes—
426C23C2×#13712yes—
427C23#57512yes—
428C23#57512yes—
429Q8—44no—
430Q8—44no—
431Q8—82no—
432Q8—82no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—28no—
446C2×C4—28no—
447C2×C4—28no—
448C2×C4—28no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—432no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—48no—
465C2×C4—48no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—82no—
472C2×C4—81no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4—84no—
477C2×C4—84no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—81no—
489C2×C4#39914yes—
490C2×C4#48518yes—
491C2×C4C2×C22⋊Q812yes—
492C2×C4C2×#12312yes—
493C2×C4#52512yes—
494C2×C4#52512yes—
495C2×C4#53912yes—
496C2×C4#53912yes—
497C2×C4D4212yes—
498C2×C4D4212yes—
499C2×C4#54712yes—
500C2×C4#54712yes—
501C2×C4#54812yes—
502C2×C4D4×Q812yes—
503C2×C4#55311yescharacteristic
504C2×C4#55311yescharacteristic
505C2×C4#55411yescharacteristic
506C2×C4#55411yescharacteristic
507C8—44no—
508C8—44no—
509C22—24no—
510C22—24no—
511C22—24no—
512C22—42no—
513C22—42no—
514C22—44no—
515C22—42no—
516C22—42no—
517C22—44no—
518C22—44no—
519C22—44no—
520C22C2×#39911yescharacteristic
521C22#240911yescharacteristic
522C22#243311yescharacteristic
523C22#248711yescharacteristic
524C22#250111yescharacteristic
525C22#262311yescharacteristic
526C22#284911yescharacteristic
527C22#300212yes—
528C22#300712yes—
529C22#307612yes—
530C22#307812yes—
531C22C2×#48512yes—
532C22#309112yes—
533C22#309312yes—
534C22#309512yes—
535C22#314711yescharacteristic
536C22#315011yescharacteristic
537C22#315211yescharacteristic
538C22#315311yescharacteristic
539C22#321212yes—
540C22#321612yes—
541C22#323112yes—
542C22#324012yes—
543C4—24no—
544C4—28no—
545C4—44no—
546C4—84no—
547C4—84no—
548C4—82no—
549C4—82no—
550C4—161no—
551C4—161no—
552C4#244312yes—
553C4#244312yes—
554C4#244912yes—
555C4#244912yes—
556C2—44no—
557C2#1477711yescharacteristic
558C2#2373011yescharacteristic
559C2#2373211yescharacteristic
560C2#2430612yes—
561C2#2572311yescharacteristic
562C2#2572811yescharacteristic
563C2#2768212yes—
564C2#2790811yescharacteristic
565C2#2791511yescharacteristic
566C2#2991112yes—
567C2#3298512yes—
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×#2097C2semidirect
6. #11372C2semidirect
7. #11324C2semidirect
8. #11323C2semidirect
9. #11050C2semidirect
10. #11050C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22852
C2×#20972
C2×#18732
C2×#18542
#147022
#113722
#113242
#113232
#110502
#107122