Finite Group Databasebeta

← #188325

Subgroups — #188325

Order 512.

1388 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2#20628C212yesmaximal, complemented
3C2×#1465C211yesmaximal, characteristic
4C2×#1459C211yesmaximal, characteristic
5#8561C214yesmaximal, complemented
6#8560C212yesmaximal, complemented
7#8558C212yesmaximal, complemented
8#7915C212yesmaximal, complemented
9C22×C4×D4C2211yescharacteristic
10C22×#117C411yescharacteristic
11C22×#117C2211yescharacteristic
12C2×#424—22no—
13C2×#407C2212yes—
14C22×C22⋊C8C2212yes—
15#2354—22no—
16#2354—24no—
17#2354—22no—
18C2×#390C412yes—
19C2×#388C412yes—
20C2×#388C2212yes—
21C2×#384C411yescharacteristic
22C2×#380C411yescharacteristic
23C2×#380C2211yescharacteristic
24C2×C4×C4⋊C4C411yescharacteristic
25C2×C4×C4⋊C4C2212yes—
26#1798—24no—
27#1797C2214yes—
28#1797C2214yes—
29#1729—22no—
30#1465—24no—
31#1459—24no—
32C2×#324C2211yescharacteristic
33C2×#324C2212yes—
34C2×#324C2211yescharacteristic
35C2×#323C2212yes—
36C2×#323C2212yes—
37#1280C2214yes—
38#1280C2214yes—
39C23×D4C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2311yescharacteristic
42C22×M16—22no—
43C23×C8—22no—
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46C2×C4×D4—24no—
47C2×C4×D4D414yes—
48C2×C4×D4Q814yes—
49C2×#117—21no—
50C2×#117—22no—
51C2×#117—21no—
52C2×#117—44no—
53C2×#117—44no—
54C2×#117D412yes—
55C2×#117D414yes—
56C2×#117D412yes—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2×C412yes—
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2311yescharacteristic
70C22×C42C2×C412yes—
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C412yes—
73C22×C42C2312yes—
74#424—48no—
75C2×C4⋊C8—24no—
76C2×C4⋊C8—24no—
77#407—24no—
78#407—24no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8C2312yes—
84C2×C22⋊C8C2312yes—
85C2×C22⋊C8C2314yes—
86#390—24no—
87#390—24no—
88#388—24no—
89#388—24no—
90#388—24no—
91#388—24no—
92#384—22no—
93#384—22no—
94#380—22no—
95#380—22no—
96#380—22no—
97#380—22no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—24no—
103C2×#95C2×C412yes—
104C2×#95C2×C412yes—
105C2×#95C2312yes—
106#336—24no—
107#336—24no—
108#324—22no—
109#324—22no—
110#324—24no—
111#323—24no—
112#323—24no—
113C25C22×C411yescharacteristic
114C25C22×C411yescharacteristic
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—41no—
120C22×D4—41no—
121C22×D4C4⋊C414yes—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—42no—
126C23×C4—42no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4C4⋊C414yes—
130C23×C4C22×C412yes—
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C412yes—
134C23×C4C22×C412yes—
135C23×C4C22×C412yes—
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×Q812yes—
142C23×C4#4114yes—
143C23×C4C2411yescharacteristic
144C2×M16—44no—
145C2×M16—44no—
146C2×M16—44no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—24no—
150C22×C8—44no—
151C22×C8—44no—
152C22×C8—44no—
153C4×D4—28no—
154C4×D4—28no—
155C4×D4—28no—
156C4×D4—28no—
157#117—28no—
158#117—48no—
159#117—44no—
160#117—44no—
161#117—48no—
162#117—44no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×Q812yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—82no—
196C2×C22⋊C4—82no—
197C2×C22⋊C4C22⋊C414yes—
198C2×C22⋊C4C22⋊C414yes—
199C2×C22⋊C4C4⋊C414yes—
200C2×C22⋊C4C2×D412yes—
201C2×C22⋊C4C2×Q812yes—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—24no—
210C2×C42—44no—
211C2×C42—42no—
212C2×C42—44no—
213C2×C42C4⋊C414yes—
214C2×C42C2×D414yes—
215C2×C42C2×D414yes—
216C4⋊C8—48no—
217C4⋊C8—48no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—24no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—24no—
224C22⋊C8—44no—
225C22⋊C8—44no—
226#95—24no—
227#95—22no—
228#95—22no—
229#95—24no—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—21no—
238C24—21no—
239C24—21no—
240C24—41no—
241C24—41no—
242C24—41no—
243C24—41no—
244C24—41no—
245C24—41no—
246C24—41no—
247C24—41no—
248C24C2×C22⋊C412yes—
249C24C2×C4⋊C412yes—
250C24C2×C4⋊C412yes—
251C24#11712yes—
252C24C23×C411yescharacteristic
253C2×D4—28no—
254C2×D4—28no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—44no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—48no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—48no—
298C22×C4—48no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C414yes—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C4⋊C412yes—
315C22×C4C2×C4⋊C412yes—
316C22×C4C2×C4⋊C412yes—
317C22×C4C2×C4⋊C412yes—
318C22×C4#11712yes—
319C22×C4#12114yes—
320C22×C4#12114yes—
321C22×C4C22⋊Q814yes—
322C22×C4C22⋊Q814yes—
323C22×C4#13112yes—
324C22×C4#13112yes—
325C22×C4C22×D412yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×Q811yescharacteristic
328C22×C4C2×#4112yes—
329M16—88no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—44no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—44no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—44no—
339C2×C8—88no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—42no—
346C4⋊C4—48no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—84no—
352C4⋊C4#9914yes—
353C22⋊C4—28no—
354C22⋊C4—24no—
355C22⋊C4—24no—
356C22⋊C4—28no—
357C22⋊C4—44no—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C22⋊C4—84no—
361C22⋊C4—84no—
362C42—24no—
363C42—24no—
364C42—28no—
365C42—44no—
366C42—44no—
367C42—44no—
368C42—44no—
369C42—44no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—81no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23#38014yes—
420C23C22×C22⋊C411yescharacteristic
421C23C22×C4⋊C411yescharacteristic
422C23C2×#11711yescharacteristic
423C23C2×#13111yescharacteristic
424C23C2×#13111yescharacteristic
425C23#56811yescharacteristic
426C23#56811yescharacteristic
427D4—416no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—28no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—28no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—48no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—48no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4#38012yes—
482C2×C4#38012yes—
483C2×C4C2×#9914yes—
484C2×C4C2×#9912yes—
485C2×C4#41012yes—
486C2×C4#42414yescomplemented
487C2×C4C2×#12112yes—
488C2×C4C2×C22⋊Q812yes—
489C8—82no—
490C8—82no—
491C8—84no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—24no—
505C22—24no—
506C22—24no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22C2×#38011yescharacteristic
527C22#288011yescharacteristic
528C22#288011yescharacteristic
529C22C22×#9911yescharacteristic
530C22C2×#41011yescharacteristic
531C22#288811yescharacteristic
532C22#288911yescharacteristic
533C22C2×#42411yescharacteristic
534C22#290511yescharacteristic
535C22#292411yescharacteristic
536C22#292411yescharacteristic
537C4—24no—
538C4—24no—
539C4—22no—
540C4—22no—
541C4—22no—
542C4—42no—
543C4—44no—
544C4—42no—
545C4—42no—
546C4—82no—
547C4—82no—
548C4#179914yes—
549C2—21no—
550C2—21no—
551C2—21no—
552C2—21no—
553C2—44no—
554C2—82no—
555C2#1741611yescharacteristic
556C2#1741811yescharacteristic
557C2#1741911yescharacteristic
558C2#1742011yescharacteristic
559C2#2062811yescharacteristic
560C2C2×#179911yescharacteristic
561C2#2064011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20628C2semidirect
5. #8561C2semidirect
6. #8560C2semidirect
7. #8558C2semidirect
8. #7915C2semidirect
486. C2×C4#424semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
#206282
C2×#14652
C2×#14592
#85612
#85602
#85582
#79152