Finite Group Databasebeta

← #225652

Subgroups — #225652

Order 512.

737 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22807C211yesmaximal, characteristic
2#22713C211yesmaximal, characteristic, complemented
3#22670C211yesmaximal, characteristic
4#22663C211yesmaximal, characteristic, complemented
5C2×#1594C211yesmaximal, characteristic
6#9621C211yesmaximal, characteristic, complemented
7#9611C211yesmaximal, characteristic, complemented
8#8897C211yesmaximal, characteristic, complemented
9#8894C211yesmaximal, characteristic, complemented
10#8889C211yesmaximal, characteristic, complemented
11#8886C211yesmaximal, characteristic, complemented
12#8819C211yesmaximal, characteristic, complemented
13#8654C211yesmaximal, characteristic, complemented
14#8650C211yesmaximal, characteristic, complemented
15#8291C211yesmaximal, characteristic, complemented
16C2×#442—21no—
17C2×C4×D8C2211yescharacteristic
18C2×#436C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×#428—21no—
21#2354C2211yescharacteristic
22#2354C2211yescharacteristic
23D4×C4⋊C4C2211yescharacteristic
24D4×C4⋊C4C2211yescharacteristic
25#1970—21no—
26#1948C2211yescharacteristic
27#1948C2211yescharacteristic
28#1947C2211yescharacteristic
29#1947C2211yescharacteristic
30#1943—21no—
31#1925—21no—
32#1925—21no—
33#1864—21no—
34#1864—21no—
35#1854—21no—
36#1854—21no—
37#1853—21no—
38#1852—21no—
39#1845C2211yescharacteristic
40#1844C2211yescharacteristic
41#1844C2211yescharacteristic
42#1844C2211yescharacteristic
43#1834C2211yescharacteristic
44#1834C2211yescharacteristic
45#1809—21no—
46#1809—21no—
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51C4×#107—21no—
52C4×#107C2211yescharacteristic
53#1770C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55#1736—21no—
56#1594C418yescomplemented
57C2×#335C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68C22×D8D411yescharacteristic
69C22×M16—21no—
70C22×M16—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2311yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×#117—21no—
76C2×#117—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79#442—44no—
80C4×D8C2×C414yes—
81C4×D8Q814yes—
82#436—22no—
83#436—22no—
84#436C2×C414yescomplemented
85#436C2×C414yescomplemented
86#428—44no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#107D411yescharacteristic
92C2×#107D411yescharacteristic
93C2×#107D411yescharacteristic
94C2×#106—21no—
95C2×#106—21no—
96C2×#106—21no—
97C2×#106—21no—
98C2×#106—21no—
99C2×#106—21no—
100C2×#106—41no—
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#102D411yescharacteristic
110C2×#102D411yescharacteristic
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×#97—21no—
114C2×C4×C8—21no—
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#390—21no—
119#390—21no—
120#389—21no—
121#389—21no—
122#388—41no—
123#388—41no—
124#385—41no—
125#385—41no—
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129#384C2311yescharacteristic
130#380—21no—
131#380—21no—
132#380—21no—
133#380—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—41no—
149#340D411yescharacteristic
150#340D411yescharacteristic
151#336—21no—
152#336—21no—
153#335—22no—
154#334C2×C414yescomplemented
155D4⋊C8C2×C414yescomplemented
156D4⋊C8C2×C414yescomplemented
157#322C2×C414yescomplemented
158C22×D4C2×D411yescharacteristic
159C22×D4C2×D411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—41no—
163C23×C4—41no—
164C2×D8—22no—
165C2×D8—22no—
166C2×D8C4⋊C414yes—
167C2×M16—24no—
168C2×M16—24no—
169C2×M16—42no—
170C2×M16—42no—
171C2×M16—81no—
172C2×M16—81no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—42no—
187C4×D4—42no—
188C4×D4—81no—
189C4×D4—81no—
190C4×D4C22×C412yes—
191C4×D4C22×C412yes—
192C4×D4C2×Q812yes—
193C4×D4C2×Q812yes—
194#117—42no—
195#117—42no—
196#117—81no—
197#117—81no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—81no—
215C2×C4⋊C4—81no—
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—41no—
233C2×C42—41no—
234C2×C42C2411yescharacteristic
235#107—22no—
236#107—41no—
237#107—42no—
238#107—42no—
239#107—81no—
240#107—81no—
241#107C4⋊C414yes—
242#106—42no—
243#106—42no—
244#106—42no—
245#106—81no—
246#106—81no—
247#106—81no—
248#106—82no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8C22×C412yes—
256C4⋊C8C22×C412yes—
257#102—42no—
258#102—42no—
259#102—81no—
260#102—81no—
261#102C4⋊C414yes—
262#102C4⋊C414yes—
263C22⋊C8—24no—
264C22⋊C8—24no—
265#97—42no—
266C4×C8—21no—
267C4×C8—42no—
268C4×C8C22×C412yes—
269C4×C8C22×C412yes—
270C4×C8C22×C412yes—
271C4×C8C2×Q812yes—
272C4×C8#4112yes—
273#95—21no—
274#95—21no—
275#95—41no—
276#95—41no—
277#95—41no—
278#95—41no—
279C24—21no—
280C24—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4C2×C4⋊C412yes—
292C2×D4C2×C4⋊C412yes—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—161no—
320C22×C4—161no—
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4H(5,𝔽2)11yescharacteristic
324D8—28no—
325M16—48no—
326M16—84no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8C2×C4⋊C412yes—
349C2×C8C2×C4⋊C412yes—
350C2×C8C4×D412yes—
351C2×C8C4×D412yes—
352C2×C8AGL(1,ℤ/8)12yes—
353C2×C8#13712yes—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—81no—
380C4⋊C4—81no—
381C4⋊C4—81no—
382C4⋊C4—81no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4C2×C4⋊C412yes—
386C4⋊C4C2×C4⋊C412yes—
387C4⋊C4AGL(1,ℤ/8)12yes—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C22⋊C4—161no—
397C22⋊C4—161no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42—82no—
403C42—82no—
404C42C23×C411yescharacteristic
405C42C22×Q811yescharacteristic
406C42C2×#4111yescharacteristic
407C422−1+411yescharacteristic
408C23—22no—
409C23—22no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23D4211yescharacteristic
423D4—22no—
424D4—22no—
425D4—22no—
426D4—22no—
427D4—41no—
428D4—41no—
429D4—41no—
430D4—41no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—164no—
472C2×C4—164no—
473C2×C4C22×C4⋊C411yescharacteristic
474C2×C4C2×C4×D411yescharacteristic
475C2×C4#52011yescharacteristic
476C2×C4#54711yescharacteristic
477C2×C4D4×Q811yescharacteristic
478C2×C4#55411yescharacteristic
479C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
480C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
481C2×C4C2×#13711yescharacteristic
482C2×C4#57511yescharacteristic
483C2×C4#57611yescharacteristic
484C2×C4#57711yescharacteristic
485C2×C4#57711yescharacteristic
486C2×C4#57711yescharacteristic
487C8—22no—
488C8—22no—
489C8—22no—
490C8—42no—
491C8—42no—
492C8—42no—
493C8—81no—
494C8—81no—
495C8#42814yescomplemented
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22D4×C4⋊C411yescharacteristic
511C22#326411yescharacteristic
512C22#326711yescharacteristic
513C22#327211yescharacteristic
514C22#327811yescharacteristic
515C22#328811yescharacteristic
516C22#328911yescharacteristic
517C4—21no—
518C4—21no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—81no—
523C4—81no—
524C4—81no—
525C4—161no—
526C4—161no—
527C4—161no—
528C4—161no—
529C4C2×#42811yescharacteristic
530C4#291211yescharacteristic
531C4C4×AGL(1,ℤ/8)11yescharacteristic
532C4#294611yescharacteristic
533C4#335411yescharacteristic
534C4#335611yescharacteristic
535C4#338411yescharacteristic
536C4#338911yescharacteristic
537C2—82no—
538C2—82no—
539C2#2016311yescharacteristic
540C2#2017211yescharacteristic
541C2#2024311yescharacteristic
542C2#2025211yescharacteristic
543C2#2266311yescharacteristic
544C2#2267611yescharacteristic
545C2#2269311yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D8D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22713C2semidirect
4. #22663C2semidirect
6. #9621C2semidirect
7. #9611C2semidirect
8. #8897C2semidirect
9. #8894C2semidirect
10. #8889C2semidirect
11. #8886C2semidirect
12. #8819C2semidirect
13. #8654C2semidirect
14. #8650C2semidirect
15. #8291C2semidirect
56. #1594C4semidirect
84. #436C2×C4semidirect
85. #436C2×C4semidirect
154. #334C2×C4semidirect
155. D4⋊C8C2×C4semidirect
156. D4⋊C8C2×C4semidirect
157. #322C2×C4semidirect
495. C8#428semidirect

Maximal subgroups

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

subgroupindex
#228072
#227132
#226702
#226632
C2×#15942
#96212
#96112
#88972
#88942
#88892
#88862
#88192
#86542
#86502
#82912