Finite Group Databasebeta

← #225569

Subgroups — #225569

Order 512.

751 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22740C211yesmaximal, characteristic, complemented
2C4⋊C4×SD16C211yesmaximal, characteristic
3#22671C211yesmaximal, characteristic
4D4×#107C211yesmaximal, characteristic
5C2×SD16⋊C8C211yesmaximal, characteristic
6#9617C211yesmaximal, characteristic, complemented
7#9614C211yesmaximal, characteristic, complemented
8#8892C211yesmaximal, characteristic, complemented
9#8890C211yesmaximal, characteristic
10#8884C211yesmaximal, characteristic, complemented
11#8882C211yesmaximal, characteristic
12#8820C211yesmaximal, characteristic, complemented
13#8648C211yesmaximal, characteristic
14#8645C211yesmaximal, characteristic, complemented
15#8291C211yesmaximal, characteristic, complemented
16C2×C8×Q8C2211yescharacteristic
17C2×#440—21no—
18C2×C4×SD16—21no—
19C2×C4×SD16C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C22×#107—21no—
22Q8×C4⋊C4C2211yescharacteristic
23Q8×C4⋊C4C2211yescharacteristic
24D4×C4⋊C4C2211yescharacteristic
25D4×C4⋊C4C2211yescharacteristic
26#1947—21no—
27#1947—21no—
28#1947C2211yescharacteristic
29#1947C2211yescharacteristic
30#1947C2211yescharacteristic
31#1947C2211yescharacteristic
32#1944—21no—
33#1941—21no—
34#1865—21no—
35#1864—21no—
36#1863—21no—
37#1862—21no—
38#1852—21no—
39#1852—21no—
40#1845C2211yescharacteristic
41#1845C2211yescharacteristic
42#1845C2211yescharacteristic
43#1844C2211yescharacteristic
44#1834—21no—
45#1834C2211yescharacteristic
46#1834C2211yescharacteristic
47#1808—21no—
48#1803C2211yescharacteristic
49#1803C2211yescharacteristic
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52C4×#107—21no—
53C4×#107C2211yescharacteristic
54#1770C2211yescharacteristic
55C8×C4⋊C4C2211yescharacteristic
56SD16⋊C8C418yescomplemented
57C2×#335C2211yescharacteristic
58C2×#335C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68C22×SD16D411yescharacteristic
69C23×C8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2311yescharacteristic
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C8×Q8—22no—
79C8×Q8C2×C414yescomplemented
80#440—44no—
81C4×SD16—44no—
82C4×SD16C2×C414yes—
83C4×SD16Q814yes—
84C8×D4—22no—
85C8×D4C2×C414yes—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#107—21no—
92C2×#107—21no—
93C2×#107—21no—
94C2×#107—44no—
95C2×#107—41no—
96C2×#107D411yescharacteristic
97C2×#107D411yescharacteristic
98C2×#106—21no—
99C2×#106—21no—
100C2×#106D411yescharacteristic
101C2×C4⋊C8—21no—
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×#103—21no—
107C2×#103—21no—
108C2×#103D411yescharacteristic
109C2×#102—21no—
110C2×#102—21no—
111C2×#102D411yescharacteristic
112C2×C22⋊C8—21no—
113C2×#97—21no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—41no—
124#389—41no—
125#385—41no—
126#385—41no—
127#384—21no—
128#384—21no—
129#384C2311yescharacteristic
130#384C2311yescharacteristic
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133#380—21no—
134#380—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2311yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
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—21no—
149#340—41no—
150#340D411yescharacteristic
151#340D411yescharacteristic
152#336—21no—
153#336—21no—
154#335—22no—
155#335C2×C414yescomplemented
156Q8⋊C8C2×C414yescomplemented
157D4⋊C8C2×C414yes—
158#322C2×C414yescomplemented
159C22×Q8C2×D411yescharacteristic
160C22×D4C2×D411yescharacteristic
161C23×C4—21no—
162C23×C4—41no—
163C23×C4—41no—
164C2×SD16C4⋊C414yes—
165C2×SD16D814yes—
166C2×SD16Q1614yes—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—24no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—42no—
174C22×C8—42no—
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×Q8—81no—
184C4×Q8—81no—
185C4×Q8C22×C412yes—
186C4×Q8C2×Q812yes—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—42no—
190C4×D4—42no—
191C4×D4—81no—
192C4×D4—81no—
193C4×D4C22×C412yes—
194C4×D4C2×Q812yes—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
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—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—81no—
217C2×C4⋊C4—81no—
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—41no—
231C2×C42—41no—
232C2×C42C2411yescharacteristic
233#107—22no—
234#107—41no—
235#107—42no—
236#107—42no—
237#107—42no—
238#107—42no—
239#107—42no—
240#107—42no—
241#107—81no—
242#107—81no—
243#107—82no—
244#106—42no—
245#106—81no—
246#106C4⋊C414yes—
247C4⋊C8—24no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8C22×C412yes—
255C4⋊C8C22×C412yes—
256#103—42no—
257#103—42no—
258#103C4⋊C414yes—
259#102—42no—
260#102—42no—
261#102C4⋊C414yes—
262C22⋊C8—24no—
263#97—42no—
264C4×C8—24no—
265C4×C8—21no—
266C4×C8—42no—
267C4×C8C22×C412yes—
268C4×C8C22×C412yes—
269C4×C8C22×C412yes—
270C4×C8C2×Q812yes—
271C4×C8#4112yes—
272#95—21no—
273#95—21no—
274#95—41no—
275#95—41no—
276#95—41no—
277#95—41no—
278C24—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8C2×C4⋊C412yes—
283C2×Q8C2×D812yes—
284C2×Q8C2×Q1612yes—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4C2×C4⋊C412yes—
289C2×D4C2×D812yes—
290C2×D4C2×Q1612yes—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—161no—
313C22×C4—161no—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4H(5,𝔽2)11yescharacteristic
317SD16#107116yescomplemented
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—48no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8C2×C4⋊C412yes—
341C2×C8C2×C4⋊C412yes—
342C2×C8C4×D412yes—
343C2×C8C4×D412yes—
344C2×C8C2×D812yes—
345C2×C8C2×Q1612yes—
346C4⋊C4—22no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—22no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
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—82no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—81no—
377C4⋊C4—81no—
378C4⋊C4—81no—
379C4⋊C4—81no—
380C4⋊C4—82no—
381C4⋊C4—161no—
382C4⋊C4—161no—
383C4⋊C4C2×C4⋊C412yes—
384C4⋊C4C2×C4⋊C412yes—
385C4⋊C4AGL(1,ℤ/8)12yes—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C42—22no—
391C42—22no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42—82no—
397C42—82no—
398C42C23×C411yescharacteristic
399C42C22×Q811yescharacteristic
400C42C2×#4111yescharacteristic
401C422−1+411yescharacteristic
402C23—22no—
403C23—22no—
404C23—22no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23D4211yescharacteristic
410Q8—22no—
411Q8—22no—
412Q8—22no—
413Q8C2×#10714yescomplemented
414D4—22no—
415D4—22no—
416D4—22no—
417D4C2×#10714yes—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—164no—
458C2×C4—164no—
459C2×C4C22×C4⋊C411yescharacteristic
460C2×C4C2×C4×D411yescharacteristic
461C2×C4#52011yescharacteristic
462C2×C4#54711yescharacteristic
463C2×C4D4×Q811yescharacteristic
464C2×C4#55411yescharacteristic
465C2×C4C22×D811yescharacteristic
466C2×C4C22×Q1611yescharacteristic
467C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
468C2×C4#57511yescharacteristic
469C2×C4#57611yescharacteristic
470C2×C4#57711yescharacteristic
471C2×C4#57811yescharacteristic
472C2×C4#57811yescharacteristic
473C8—22no—
474C8—22no—
475C8—22no—
476C8—42no—
477C8—42no—
478C8—42no—
479C8—81no—
480C8—81no—
481C8C2×#10714yescomplemented
482C22—24no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22D4×C4⋊C411yescharacteristic
490C22#326511yescharacteristic
491C22#326911yescharacteristic
492C22D4×D811yescharacteristic
493C22D4×Q1611yescharacteristic
494C22#328811yescharacteristic
495C22#328911yescharacteristic
496C4—24no—
497C4—21no—
498C4—21no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—81no—
503C4—81no—
504C4—81no—
505C4—161no—
506C4—161no—
507C4—161no—
508C4—161no—
509C4C22×#10711yescharacteristic
510C4#291311yescharacteristic
511C4C4×AGL(1,ℤ/8)11yescharacteristic
512C4#294711yescharacteristic
513C4#333411yescharacteristic
514C4#333811yescharacteristic
515C4#338611yescharacteristic
516C4#338811yescharacteristic
517C2—44no—
518C2#1975611yescharacteristic
519C2#1976011yescharacteristic
520C2#1981211yescharacteristic
521C2#1981611yescharacteristic
522C2D4×#10711yescharacteristic
523C2#2267711yescharacteristic
524C2#2269311yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
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
1. #22740C2semidirect
6. #9617C2semidirect
7. #9614C2semidirect
8. #8892C2semidirect
10. #8884C2semidirect
12. #8820C2semidirect
14. #8645C2semidirect
15. #8291C2semidirect
56. SD16⋊C8C4semidirect
79. C8×Q8C2×C4semidirect
155. #335C2×C4semidirect
156. Q8⋊C8C2×C4semidirect
158. #322C2×C4semidirect
317. SD16#107semidirect
413. Q8C2×#107semidirect
481. C8C2×#107semidirect

Maximal subgroups

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

subgroupindex
#227402
C4⋊C4×SD162
#226712
D4×#1072
C2×SD16⋊C82
#96172
#96142
#88922
#88902
#88842
#88822
#88202
#86482
#86452
#82912