Finite Group Databasebeta

← #286397

Subgroups — #286397

Order 512.

562 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27878C211yesmaximal, characteristic
2#27649C211yesmaximal, characteristic
3#24702C211yesmaximal, characteristic
4#23055C211yesmaximal, characteristic
5#10649C211yesmaximal, characteristic, complemented
6#10624C211yesmaximal, characteristic, complemented
7#10597C211yesmaximal, characteristic, complemented
8#10483C211yesmaximal, characteristic, complemented
9#9616C211yesmaximal, characteristic, complemented
10#9568C211yesmaximal, characteristic, complemented
11#9224C211yesmaximal, characteristic, complemented
12#9136C211yesmaximal, characteristic
13#8831C211yesmaximal, characteristic
14#8653C211yesmaximal, characteristic, complemented
15#8046C211yesmaximal, characteristic
16C2×#485—21no—
17C2×#477C2211yescharacteristic
18C2×#446C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#431C2211yescharacteristic
21#2857C2211yescharacteristic
22C4×C22⋊Q8C2211yescharacteristic
23#2285C2211yescharacteristic
24#2085—21no—
25#2076—21no—
26#2076—21no—
27#2076—21no—
28#2074—21no—
29#2036—21no—
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1987C2211yescharacteristic
33#1984C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1942—21no—
38#1939C2211yescharacteristic
39#1933—21no—
40#1924—21no—
41#1923—21no—
42#1921—21no—
43#1921—21no—
44#1917—21no—
45#1845C2211yescharacteristic
46#1834C2211yescharacteristic
47#1825—21no—
48#1803C2211yescharacteristic
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1797C2211yescharacteristic
52#1770—21no—
53C8×C4⋊C4C2211yescharacteristic
54C4×C22⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
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
71C2×#117—21no—
72C22×C42C2311yescharacteristic
73#485—44no—
74#477—22no—
75#477—41no—
76#446—22no—
77#446—41no—
78#432—22no—
79#432—41no—
80#431—22no—
81#431—41no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#106—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8D411yescharacteristic
104C2×C22⋊C8D411yescharacteristic
105C2×#97—21no—
106C2×#97—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#391—21no—
112#391D411yescharacteristic
113#389—21no—
114#389—21no—
115#389D411yescharacteristic
116#387—21no—
117#385—21no—
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#382—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125#380—21no—
126#380D411yescharacteristic
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4C2311yescharacteristic
131C4×C22⋊C4—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
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#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336D411yescharacteristic
152#336D411yescharacteristic
153#335—22no—
154#334—41no—
155#333—22no—
156#333—41no—
157C22×Q8—21no—
158C22×Q8C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4#4111yescharacteristic
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C22⋊Q8—22no—
177C22⋊Q8—41no—
178C22⋊Q8—82no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183#117—42no—
184#117—42no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—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—41no—
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—41no—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42#4111yescharacteristic
217C2×C42C2411yescharacteristic
218#107—42no—
219#107—42no—
220#107—42no—
221#107—42no—
222#107—42no—
223#106—42no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—42no—
235C4⋊C8—42no—
236C4⋊C8#4112yes—
237C4⋊C8#4112yes—
238#103—42no—
239#103—42no—
240#103—42no—
241#103—42no—
242C22⋊C8—22no—
243C22⋊C8—41no—
244C22⋊C8—42no—
245C22⋊C8—42no—
246C22⋊C8—42no—
247#97—42no—
248#97—42no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—42no—
254C4×C8—42no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—41no—
261#95—41no—
262#95—41no—
263#95—41no—
264C24#12111yescharacteristic
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—81no—
274C2×Q8—81no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4#12111yescharacteristic
300C22×C4#12111yescharacteristic
301C22×C4#12111yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C2×#4111yescharacteristic
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8#13512yes—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—42no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—81no—
360C4⋊C4—81no—
361C4⋊C4—81no—
362C4⋊C4—81no—
363C4⋊C4—82no—
364C4⋊C4AGL(1,ℤ/8)12yes—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—82no—
380C42C2×#4111yescharacteristic
381C42C2×#4111yescharacteristic
382C42H(5,𝔽2)11yescharacteristic
383C422−1+411yescharacteristic
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23C2×#12111yescharacteristic
392Q8—41no—
393Q8—41no—
394Q8—41no—
395Q8—41no—
396Q8—161no—
397Q8—161no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4#52911yescharacteristic
444C2×C4#53911yescharacteristic
445C2×C4#54611yescharacteristic
446C2×C4#54611yescharacteristic
447C2×C4#54711yescharacteristic
448C2×C4#54711yescharacteristic
449C2×C4C2×#13511yescharacteristic
450C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
451C2×C4#57511yescharacteristic
452C2×C4#57511yescharacteristic
453C2×C4#57811yescharacteristic
454C2×C4#57811yescharacteristic
455C2×C4#57811yescharacteristic
456C2×C4#57811yescharacteristic
457C8—22no—
458C8—42no—
459C8—41no—
460C8—41no—
461C8—41no—
462C8—42no—
463C8—81no—
464C8—81no—
465C8—81no—
466C8—81no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22#242511yescharacteristic
475C22#303511yescharacteristic
476C22#304411yescharacteristic
477C22#305511yescharacteristic
478C22#305911yescharacteristic
479C22#306711yescharacteristic
480C22#306811yescharacteristic
481C4—21no—
482C4—21no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—42no—
487C4—81no—
488C4—81no—
489C4—81no—
490C4—161no—
491C4—161no—
492C4—161no—
493C4—161no—
494C4#330011yescharacteristic
495C4#331511yescharacteristic
496C4#332711yescharacteristic
497C4#333011yescharacteristic
498C4#334211yescharacteristic
499C4#334911yescharacteristic
500C4#335211yescharacteristic
501C4#335311yescharacteristic
502C2—81no—
503C2#1918011yescharacteristic
504C2#1922611yescharacteristic
505C2#1947911yescharacteristic
506C2#1953111yescharacteristic
507C2#2520111yescharacteristic
508C2#2525611yescharacteristic
509C2#2526711yescharacteristic
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. #10624C2semidirect
7. #10597C2semidirect
8. #10483C2semidirect
9. #9616C2semidirect
10. #9568C2semidirect
11. #9224C2semidirect
14. #8653C2semidirect

Maximal subgroups

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

subgroupindex
#278782
#276492
#247022
#230552
#106492
#106242
#105972
#104832
#96162
#95682
#92242
#91362
#88312
#86532
#80462