Finite Group Databasebeta

← #286525

Subgroups — #286525

Order 512.

617 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27910C211yesmaximal, characteristic
2#27648C211yesmaximal, characteristic
3#24710C211yesmaximal, characteristic
4#23055C211yesmaximal, characteristic
5#10649C211yesmaximal, characteristic, complemented
6#10618C211yesmaximal, characteristic, complemented
7#10588C211yesmaximal, characteristic, complemented
8#10487C211yesmaximal, characteristic, complemented
9#9614C211yesmaximal, characteristic, complemented
10#9561C211yesmaximal, characteristic, complemented
11#9228C211yesmaximal, characteristic, complemented
12#9140C211yesmaximal, characteristic
13#8834C211yesmaximal, characteristic
14#8653C211yesmaximal, characteristic, complemented
15#8040C211yesmaximal, characteristic
16C2×#484—21no—
17C2×#484—21no—
18C2×#484—21no—
19C2×#475C2211yescharacteristic
20C2×#446C2211yescharacteristic
21C2×#432C2211yescharacteristic
22C2×#432C2211yescharacteristic
23#2857C2211yescharacteristic
24C4×C22⋊Q8C2211yescharacteristic
25C2×#384C2211yescharacteristic
26#2095—21no—
27#2095—21no—
28#2083—21no—
29#2077—21no—
30#2072—21no—
31#2036—21no—
32#1988C2211yescharacteristic
33#1984C2211yescharacteristic
34#1956C2211yescharacteristic
35#1956C2211yescharacteristic
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1947C2211yescharacteristic
39#1943—21no—
40#1939C2211yescharacteristic
41#1935—21no—
42#1924—21no—
43#1923—21no—
44#1917—21no—
45#1845C2211yescharacteristic
46#1845C2211yescharacteristic
47#1824—21no—
48#1803C2211yescharacteristic
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1797C2211yescharacteristic
52#1770—21no—
53C8×C4⋊C4C2211yescharacteristic
54C4×C22⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8D411yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C42C2311yescharacteristic
75#484—44no—
76#484—44no—
77#484—44no—
78#475—22no—
79#475—41no—
80#446—22no—
81#446—41no—
82#432—22no—
83#432—22no—
84#432—41no—
85#432—41no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#107—21no—
92C2×#107—21no—
93C2×#106—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8D411yescharacteristic
114C2×C22⋊C8D411yescharacteristic
115C2×#97—21no—
116C2×#97—21no—
117C2×#97—21no—
118C2×C4×C8—21no—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#391—21no—
122#391D411yescharacteristic
123#389—21no—
124#389—21no—
125#389D411yescharacteristic
126#387—21no—
127#385—21no—
128#384—21no—
129#384—21no—
130#384—21no—
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133#384C2311yescharacteristic
134#380—21no—
135#380—21no—
136#380D411yescharacteristic
137C4×C4⋊C4—21no—
138C4×C4⋊C4C2311yescharacteristic
139C4×C22⋊C4—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#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336D411yescharacteristic
154#336D411yescharacteristic
155#335—22no—
156#334—41no—
157#332—22no—
158#332—41no—
159C22×Q8—21no—
160C22×Q8C2×D411yescharacteristic
161C23×C4—41no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4#4111yescharacteristic
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C22⋊Q8—22no—
180C22⋊Q8—41no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
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—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—41no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42#4111yescharacteristic
225C2×C42C2411yescharacteristic
226#107—42no—
227#107—42no—
228#107—42no—
229#107—42no—
230#107—42no—
231#107—42no—
232#107—42no—
233#106—42no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246C4⋊C8#4112yes—
247C4⋊C8#4112yes—
248#103—42no—
249#103—42no—
250#103—42no—
251#103—42no—
252#103—42no—
253#103—42no—
254#103—42no—
255#103—42no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8—42no—
259C22⋊C8—42no—
260C22⋊C8—42no—
261#97—42no—
262#97—42no—
263#97—42no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—42no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—41no—
275#95—41no—
276C24#12111yescharacteristic
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4#12111yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—21no—
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×C8AGL(1,ℤ/8)12yes—
351C2×C8AGL(1,ℤ/8)12yes—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—42no—
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—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—81no—
377C4⋊C4—81no—
378C4⋊C4—81no—
379C4⋊C4—81no—
380C4⋊C4—82no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C4⋊C4AGL(1,ℤ/8)12yes—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—82no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42C2×#4111yescharacteristic
399C42C2×#4111yescharacteristic
400C42H(5,𝔽2)11yescharacteristic
401C422−1+411yescharacteristic
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23#47312yes—
409C23C2×#12111yescharacteristic
410Q8—41no—
411Q8—41no—
412Q8—41no—
413Q8—41no—
414Q8—81no—
415Q8—81no—
416Q8—81no—
417Q8—81no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
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—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
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—81no—
463C2×C4—164no—
464C2×C4#52911yescharacteristic
465C2×C4#53911yescharacteristic
466C2×C4#54611yescharacteristic
467C2×C4#54611yescharacteristic
468C2×C4#54711yescharacteristic
469C2×C4#54711yescharacteristic
470C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
471C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
472C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
473C2×C4#57511yescharacteristic
474C2×C4#57811yescharacteristic
475C2×C4#57811yescharacteristic
476C2×C4#57811yescharacteristic
477C2×C4#57811yescharacteristic
478C8—22no—
479C8—22no—
480C8—41no—
481C8—42no—
482C8—42no—
483C8—41no—
484C8—81no—
485C8—81no—
486C8—81no—
487C8—81no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22#242511yescharacteristic
496C22#303811yescharacteristic
497C22#304411yescharacteristic
498C22#305511yescharacteristic
499C22#305911yescharacteristic
500C22C2×#47311yescharacteristic
501C22#306811yescharacteristic
502C4—21no—
503C4—21no—
504C4—42no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—81no—
509C4—81no—
510C4—81no—
511C4—161no—
512C4—161no—
513C4—161no—
514C4—161no—
515C4#330811yescharacteristic
516C4#332711yescharacteristic
517C4#333011yescharacteristic
518C4#333011yescharacteristic
519C4#334311yescharacteristic
520C4#335211yescharacteristic
521C4#335311yescharacteristic
522C4#335311yescharacteristic
523C2—42no—
524C2#1931211yescharacteristic
525C2#1935411yescharacteristic
526C2#1947611yescharacteristic
527C2#1953111yescharacteristic
528C2#2521711yescharacteristic
529C2#2525611yescharacteristic
530C2#2526611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10649C2semidirect
6. #10618C2semidirect
7. #10588C2semidirect
8. #10487C2semidirect
9. #9614C2semidirect
10. #9561C2semidirect
11. #9228C2semidirect
14. #8653C2semidirect

Maximal subgroups

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

subgroupindex
#279102
#276482
#247102
#230552
#106492
#106182
#105882
#104872
#96142
#95612
#92282
#91402
#88342
#86532
#80402