Finite Group Databasebeta

← #287458

Subgroups — #287458

Order 512.

615 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29148C211yesmaximal, characteristic
2#28511C211yesmaximal, characteristic, complemented
3#28281C211yesmaximal, characteristic
4#23208C211yesmaximal, characteristic
5#10617C211yesmaximal, characteristic, complemented
6#10615C211yesmaximal, characteristic
7#10486C211yesmaximal, characteristic, complemented
8#10484C211yesmaximal, characteristic
9#9896C211yesmaximal, characteristic, complemented
10#9605C211yesmaximal, characteristic, complemented
11#9228C211yesmaximal, characteristic, complemented
12#9227C211yesmaximal, characteristic
13#8653C211yesmaximal, characteristic, complemented
14#8652C211yesmaximal, characteristic
15#8237C211yesmaximal, characteristic, complemented
16C2×#477—21no—
17C2×#477—21no—
18C2×#475C2211yescharacteristic
19C2×#474C2211yescharacteristic
20C2×#446C2211yescharacteristic
21C2×#436C2211yescharacteristic
22#2868C2211yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24C4×#124C2211yescharacteristic
25#2095—21no—
26#2095—21no—
27#2068—21no—
28#2065—21no—
29#2024—21no—
30#2023—21no—
31#1984C2211yescharacteristic
32#1983—21no—
33#1959—21no—
34#1956C2211yescharacteristic
35#1956C2211yescharacteristic
36#1954C2211yescharacteristic
37#1949—21no—
38#1924—21no—
39#1923—21no—
40#1851—21no—
41#1845C2211yescharacteristic
42#1845C2211yescharacteristic
43#1842C2211yescharacteristic
44#1836—21no—
45#1803C2211yescharacteristic
46#1803C2211yescharacteristic
47#1803C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1767C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1753—21no—
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8C2311yescharacteristic
72C2×#124D411yescharacteristic
73C2×C4×Q8—21no—
74C2×C4×Q8C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76#477—44no—
77#477—44no—
78#475—22no—
79#475—41no—
80#474—22no—
81#474—41no—
82#446—22no—
83#446—41no—
84#436—22no—
85#436—41no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#106—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8D411yescharacteristic
96C2×C4⋊C8D411yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×#97—21no—
119C2×#97—21no—
120C2×C4×C8—21no—
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123#391—21no—
124#391D411yescharacteristic
125#390D411yescharacteristic
126#389—21no—
127#389—21no—
128#388—21no—
129#388—21no—
130#384—21no—
131#384—21no—
132#384—21no—
133#384C2311yescharacteristic
134#384C2311yescharacteristic
135#384C2311yescharacteristic
136#382D411yescharacteristic
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C43C2311yescharacteristic
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336D411yescharacteristic
153#336D411yescharacteristic
154#334—41no—
155#334—41no—
156#332—22no—
157#332—22no—
158C22×Q8—21no—
159C22×Q8—41no—
160C22×Q8C2×D411yescharacteristic
161C22×D4C2×D411yescharacteristic
162C23×C4—21no—
163C2×M16—42no—
164C2×M16—42no—
165C2×M16—81no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—42no—
180C4⋊Q8—42no—
181C4⋊Q8—81no—
182C4⋊Q8—81no—
183C4⋊Q8—82no—
184C4⋊Q8#4112yes—
185#124—22no—
186#124—22no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—42no—
192C4×Q8—42no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—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—21no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C42—21no—
221C2×C42—41no—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42#4111yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42C2411yescharacteristic
229#107—42no—
230#107—42no—
231#107—42no—
232#107—42no—
233#107—42no—
234#106—42no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8—81no—
251C4⋊C8#4112yes—
252#103—42no—
253#103—42no—
254#103—42no—
255#103—42no—
256#103—42no—
257#103—81no—
258#103—81no—
259#103—81no—
260#102—42no—
261#102—81no—
262#102—81no—
263#102—81no—
264C22⋊C8—42no—
265C22⋊C8—42no—
266C22⋊C8—81no—
267#97—42no—
268#97—42no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—81no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—41no—
279#95—41no—
280C24—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—22no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—82no—
294C2×Q8—82no—
295C2×Q8—82no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
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#12111yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C2×#4111yescharacteristic
322M16—84no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—22no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—41no—
335C2×C8—41no—
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×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8AGL(1,ℤ/8)12yes—
352C2×C8AGL(1,ℤ/8)12yes—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
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—41no—
374C4⋊C4—41no—
375C4⋊C4—81no—
376C4⋊C4—81no—
377C4⋊C4—81no—
378C4⋊C4—82no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C4⋊C4—82no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—81no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—22no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42—82no—
399C42#13712yes—
400C42#13712yes—
401C42C2×#4111yescharacteristic
402C42C2×#4111yescharacteristic
403C42H(5,𝔽2)11yescharacteristic
404C422−1+411yescharacteristic
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23C2×#12111yescharacteristic
413Q8—41no—
414Q8—41no—
415Q8—41no—
416Q8—41no—
417Q8—42no—
418Q8—42no—
419Q8—81no—
420Q8—81no—
421Q8—164no—
422D4—41no—
423D4—41no—
424D4—41no—
425D4—41no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
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—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4#47012yes—
463C2×C4#47012yes—
464C2×C4#47312yes—
465C2×C4#52911yescharacteristic
466C2×C4#54211yescharacteristic
467C2×C4#54611yescharacteristic
468C2×C4#54611yescharacteristic
469C2×C4#54711yescharacteristic
470C2×C4#54811yescharacteristic
471C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
472C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
473C2×C4C2×#13711yescharacteristic
474C2×C4C2×#13711yescharacteristic
475C2×C4#57611yescharacteristic
476C2×C4#57711yescharacteristic
477C2×C4#57711yescharacteristic
478C2×C4#57811yescharacteristic
479C8—22no—
480C8—22no—
481C8—41no—
482C8—42no—
483C8—41no—
484C8—81no—
485C8—81no—
486C8—81no—
487C8—81no—
488C8—81no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22#242711yescharacteristic
497C22#303811yescharacteristic
498C22#304611yescharacteristic
499C22C2×#47011yescharacteristic
500C22#306211yescharacteristic
501C22C2×#47311yescharacteristic
502C22#306811yescharacteristic
503C4—21no—
504C4—21no—
505C4—22no—
506C4—22no—
507C4—22no—
508C4—41no—
509C4—81no—
510C4—81no—
511C4—81no—
512C4—82no—
513C4—161no—
514C4—161no—
515C4—161no—
516C4#330611yescharacteristic
517C4#330811yescharacteristic
518C4#332911yescharacteristic
519C4#333311yescharacteristic
520C4#335611yescharacteristic
521C4#335711yescharacteristic
522C4#336711yescharacteristic
523C4#337311yescharacteristic
524C2—161no—
525C2#1929311yescharacteristic
526C2#1937611yescharacteristic
527C2#1945311yescharacteristic
528C2#1959011yescharacteristic
529C2#2533411yescharacteristic
530C2#2534911yescharacteristic
531C2#2536011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #28511C2semidirect
5. #10617C2semidirect
7. #10486C2semidirect
9. #9896C2semidirect
10. #9605C2semidirect
11. #9228C2semidirect
13. #8653C2semidirect
15. #8237C2semidirect

Maximal subgroups

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

subgroupindex
#291482
#285112
#282812
#232082
#106172
#106152
#104862
#104842
#98962
#96052
#92282
#92272
#86532
#86522
#82372