Finite Group Databasebeta

← #188532

Subgroups — #188532

Order 512.

1112 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34224C211yesmaximal, characteristic
2#21955C211yesmaximal, characteristic, complemented
3C2×#1512C211yesmaximal, characteristic
4C2×#1509C211yesmaximal, characteristic
5#13954C211yesmaximal, characteristic, complemented
6#8578C212yesmaximal, complemented
7#8577C212yesmaximal
8#8572C212yesmaximal, complemented
9#8571C212yesmaximal
10#8021C211yesmaximal, characteristic, complemented
11#8017C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15#2373—22no—
16#2370C412yescomplemented
17#2367C412yescomplemented
18#2357C411yescharacteristic, complemented
19#2356C411yescharacteristic, complemented
20#2351C411yescharacteristic, complemented
21#2349C411yescharacteristic, complemented
22#2347—21no—
23#2343—21no—
24#2322—22no—
25#2318—21no—
26C4×C22⋊Q8C2212yes—
27C4×#121C2212yes—
28C2×#389—22no—
29#2283C2211yescharacteristic
30C2×#382C2211yescharacteristic
31#2275—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1940—21no—
35#1940—21no—
36#1940—22no—
37#1797C2212yes—
38#1797C2212yes—
39#1512—24no—
40#1509—24no—
41C2×#328C2211yescharacteristic
42C2×#328C2212yes—
43C2×#328C2211yescharacteristic
44C2×#327C2211yescharacteristic
45C2×#327C2212yes—
46C2×#327C2211yescharacteristic
47#1444—21no—
48#1444—21no—
49C4⋊#95C2212yes—
50C4⋊#95C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53#1294C2212yes—
54#1294C2212yes—
55#529—22no—
56#529—22no—
57#529D414yes—
58#529Q814yes—
59C2×#126—21no—
60C2×#125C2×C411yescharacteristic
61C2×#124C2×C411yescharacteristic
62C2×#123—21no—
63C2×C22⋊Q8C2311yescharacteristic
64C2×#121C2311yescharacteristic
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C412yes—
67C2×C4×D4—21no—
68C2×C4×D4—22no—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C412yes—
71C2×#117—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4D412yes—
77C22×C42—21no—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80#432—22no—
81#432—22no—
82#432—22no—
83#432—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C22⋊C8C2311yescharacteristic
89C2×C22⋊C8C2312yes—
90C2×C22⋊C8C2311yescharacteristic
91C2×#97—21no—
92C2×#97—21no—
93#391C2×C412yes—
94#390C2×C412yes—
95#389—22no—
96#389—22no—
97#389—48no—
98#389C2×C412yes—
99#389C2×C412yes—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#386—22no—
109#385—22no—
110#385—22no—
111#385—22no—
112#383—21no—
113#383—21no—
114#382—21no—
115#382—22no—
116#382—22no—
117#382C2×C411yescharacteristic
118#382C2×C411yescharacteristic
119#381—21no—
120#381—21no—
121#380—21no—
122#380C2312yes—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4C2×C412yes—
129C4×C4⋊C4C2×C412yes—
130C4×C4⋊C4C2×C412yes—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2×C412yes—
134C4×C4⋊C4D411yescharacteristic
135C4×C4⋊C4D411yescharacteristic
136C4×C4⋊C4D412yes—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—22no—
142#376—21no—
143#376—21no—
144C2×#95—21no—
145C2×#95—22no—
146C2×#95—22no—
147C2×#95—21no—
148C2×#95D411yescharacteristic
149C2×#95D411yescharacteristic
150C2×#95C2312yes—
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153#336—22no—
154#336—22no—
155#328—24no—
156#328—22no—
157#328—22no—
158#327—24no—
159#327—22no—
160#327—22no—
161C22×Q8C22×C411yescharacteristic
162C22×D4—21no—
163C22×D4C22×C411yescharacteristic
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2411yescharacteristic
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—21no—
177#126—24no—
178#126—24no—
179#125—22no—
180#125C4⋊C414yescomplemented
181#124—22no—
182#124C4⋊C414yescomplemented
183#123—24no—
184#123—24no—
185C22⋊Q8—21no—
186C22⋊Q8—21no—
187C22⋊Q8C2×D412yes—
188C22⋊Q8C2×Q812yes—
189#121—21no—
190#121—21no—
191#121C2×D412yes—
192#121C2×Q812yes—
193C4×Q8—22no—
194C4×Q8—82no—
195C4×Q8C4⋊C414yescomplemented
196C4×D4—24no—
197C4×D4—24no—
198C4×D4—22no—
199C4×D4—44no—
200C4×D4—44no—
201C4×D4—82no—
202C4×D4C4⋊C414yescomplemented
203#117—24no—
204#117—24no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4C22⋊C412yes—
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C22⋊C412yes—
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C412yes—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—44no—
247C2×C42—42no—
248C2×C42—82no—
249C2×C42C22×C412yes—
250C2×C42C22×C412yes—
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×Q812yes—
259C2×C42#4112yes—
260C2×C42#4112yes—
261C4⋊C8—42no—
262C4⋊C8—42no—
263C4⋊C8—82no—
264C4⋊C8—82no—
265C22⋊C8—22no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C22⋊C8—21no—
269C22⋊C8—22no—
270C22⋊C8—21no—
271#97—42no—
272#97—42no—
273#95—21no—
274#95—22no—
275#95—21no—
276#95—22no—
277#95—21no—
278#95—21no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—21no—
287#95—21no—
288#95—22no—
289#95—21no—
290#95—21no—
291#95—44no—
292#95—42no—
293#95—44no—
294#95C22⋊C411yescharacteristic
295#95C22⋊C411yescharacteristic
296#95C22⋊C412yes—
297#95C22⋊C411yescharacteristic
298#95C22⋊C411yescharacteristic
299#95C22⋊C412yes—
300C24—21no—
301C24C22×D411yescharacteristic
302C2×Q8—21no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8C2×C4⋊C412yes—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—21no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4C2×C4⋊C412yes—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—44no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C412yes—
360C22×C4#12112yes—
361C22×C4#12112yes—
362C22×C4C22⋊Q812yes—
363C22×C4C22⋊Q812yes—
364C22×C4AGL(1,ℤ/8)14yes—
365C22×C4#13714yes—
366C22×C4C23×C411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×Q811yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C2×C8—41no—
373C2×C8—42no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—42no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—42no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—48no—
395C4⋊C4—44no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4—84no—
399C4⋊C4—88no—
400C4⋊C4—88no—
401C4⋊C4C2×C4⋊C412yes—
402C4⋊C4C2×C4⋊C412yes—
403C22⋊C4—22no—
404C22⋊C4—22no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—44no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C42—24no—
416C42—42no—
417C42—42no—
418C42—42no—
419C42—42no—
420C42—44no—
421C42—82no—
422C42—82no—
423C42—82no—
424C42—84no—
425C42C2×C4⋊C412yes—
426C42C2×C4⋊C412yes—
427C42#11712yes—
428C42#11712yes—
429C42#12112yes—
430C42#12112yes—
431C42C22⋊Q812yes—
432C42C22⋊Q812yes—
433C23—21no—
434C23—21no—
435C23—22no—
436C23—22no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23—42no—
441C23C22×C22⋊C411yescharacteristic
442C23C2×#12111yescharacteristic
443C23C2×C22⋊Q811yescharacteristic
444C23C2×AGL(1,ℤ/8)12yes—
445C23C2×#13712yes—
446Q8—42no—
447Q8—42no—
448D4—42no—
449D4—42no—
450D4—44no—
451D4—42no—
452D4—42no—
453C2×C4—24no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—84no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—164no—
516C2×C4#38012yes—
517C2×C4#38012yes—
518C2×C4#38012yes—
519C2×C4#38012yes—
520C2×C4#41714yes—
521C2×C4#41714yes—
522C2×C4C22×C4⋊C411yescharacteristic
523C2×C4C2×#11711yescharacteristic
524C2×C4C2×#12111yescharacteristic
525C2×C4C2×C22⋊Q811yescharacteristic
526C8—81no—
527C8—81no—
528C8—82no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—82no—
541C22—82no—
542C22C2×#38011yescharacteristic
543C22#288311yescharacteristic
544C22#288411yescharacteristic
545C22C2×#41712yes—
546C22#290112yes—
547C22#303811yescharacteristic
548C22#303911yescharacteristic
549C22#307912yes—
550C4—21no—
551C4—21no—
552C4—21no—
553C4—21no—
554C4—22no—
555C4—22no—
556C4—22no—
557C4—22no—
558C4—24no—
559C4—24no—
560C4—42no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—84no—
566C4—82no—
567C4—82no—
568C4—161no—
569C4—161no—
570C2—21no—
571C2—22no—
572C2—21no—
573C2—82no—
574C2#1762212yes—
575C2#1762812yes—
576C2#2063411yescharacteristic
577C2#2064811yescharacteristic
578C2#2065511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21955C2semidirect
5. #13954C2semidirect
6. #8578C2semidirect
8. #8572C2semidirect
10. #8021C2semidirect
11. #8017C2semidirect
16. #2370C4semidirect
17. #2367C4semidirect
18. #2357C4semidirect
19. #2356C4semidirect
20. #2351C4semidirect
21. #2349C4semidirect
180. #125C4⋊C4semidirect
182. #124C4⋊C4semidirect
195. C4×Q8C4⋊C4semidirect
202. C4×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342242
#219552
C2×#15122
C2×#15092
#139542
#85782
#85772
#85722
#85712
#80212
#80172