Finite Group Databasebeta

← #351468

Subgroups — #351468

Order 512.

612 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33210C211yesmaximal, characteristic
2#33153C211yesmaximal, characteristic
3#33114C211yesmaximal, characteristic
4#20480C211yesmaximal, characteristic
5#9741C211yesmaximal, characteristic, complemented
6#9729C211yesmaximal, characteristic, complemented
7#9620C211yesmaximal, characteristic, complemented
8#9615C211yesmaximal, characteristic, complemented
9#9576C211yesmaximal, characteristic, complemented
10#9574C211yesmaximal, characteristic, complemented
11#9567C211yesmaximal, characteristic, complemented
12#9565C211yesmaximal, characteristic, complemented
13#9138C211yesmaximal, characteristic
14#9136C211yesmaximal, characteristic
15#8597C211yesmaximal, characteristic
16C2×#498C2211yescharacteristic
17C2×#485—21no—
18C2×#432C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20#2857C2211yescharacteristic
21#2857C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1990C2211yescharacteristic
24#1990C2211yescharacteristic
25#1987C2211yescharacteristic
26#1987C2211yescharacteristic
27#1981C2211yescharacteristic
28#1981C2211yescharacteristic
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1947C2211yescharacteristic
32#1947C2211yescharacteristic
33#1943—21no—
34#1943—21no—
35#1928—21no—
36#1927—21no—
37#1919—21no—
38#1918—21no—
39#1873—21no—
40#1854—21no—
41#1845C2211yescharacteristic
42#1825—21no—
43#1809—21no—
44#1809—21no—
45#1797C2211yescharacteristic
46#1797C2211yescharacteristic
47#1770C2211yescharacteristic
48#1770C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51#1753—21no—
52#1753—21no—
53#1745—21no—
54C4×#97—21no—
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16D411yescharacteristic
68C22×M16D411yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×C4⋊Q8C2311yescharacteristic
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—21no—
73C2×#117D411yescharacteristic
74C2×#117D411yescharacteristic
75C22×C42C2311yescharacteristic
76#498—22no—
77#498—22no—
78#485—44no—
79#432—22no—
80#432—22no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C4×M16—22no—
103C4×M16—22no—
104C2×#97—21no—
105C2×#97—21no—
106C2×#97—21no—
107C2×#97D411yescharacteristic
108C2×#97D411yescharacteristic
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#391—21no—
114#391—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#380—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131#376—21no—
132C43—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#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#335—22no—
152#334—22no—
153#333—22no—
154#333—22no—
155#332—22no—
156#332—22no—
157C22×Q8—21no—
158C22×Q8—21no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4#4111yescharacteristic
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—41no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C22⋊Q8—42no—
187C22⋊Q8—42no—
188C22⋊Q8—82no—
189#117—22no—
190#117—22no—
191#117—41no—
192#117—41no—
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—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42#4111yescharacteristic
226C2×C42C2411yescharacteristic
227#107—42no—
228#107—42no—
229#107—42no—
230#107—42no—
231#106—42no—
232#106—42no—
233#106—42no—
234#106—42no—
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—21no—
245C4⋊C8#4112yes—
246C4⋊C8#4112yes—
247#103—42no—
248#103—42no—
249#103—42no—
250#103—42no—
251C22⋊C8—24no—
252C22⋊C8—42no—
253C22⋊C8—42no—
254#97—24no—
255#97—22no—
256#97—22no—
257#97—42no—
258#97—42no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—42no—
262C4×C8—42no—
263C4×C8#4112yes—
264C4×C8#4112yes—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273C24#12111yescharacteristic
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—81no—
283C2×Q8—81no—
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—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—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C2×#4111yescharacteristic
314M16—44no—
315M16—44no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—21no—
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—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8#12312yes—
342C2×C8#12312yes—
343C2×C8#13512yes—
344C2×C8#13712yes—
345C2×C8#13712yes—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
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—81no—
369C4⋊C4—81no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4AGL(1,ℤ/8)12yes—
373C4⋊C4#13712yes—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—42no—
385C42—42no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—81no—
391C42—81no—
392C42C2×#4111yescharacteristic
393C42C2×#4111yescharacteristic
394C42C2×#4111yescharacteristic
395C42C2×#4111yescharacteristic
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23C2×#12111yescharacteristic
404Q8—81no—
405Q8—81no—
406Q8—81no—
407Q8—81no—
408Q8—161no—
409Q8—161no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
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—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
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×C4C2×#12311yescharacteristic
456C2×C4#52511yescharacteristic
457C2×C4#52911yescharacteristic
458C2×C4#52911yescharacteristic
459C2×C4#53211yescharacteristic
460C2×C4#53211yescharacteristic
461C2×C4C2×#13511yescharacteristic
462C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
463C2×C4C2×#13711yescharacteristic
464C2×C4C2×#13711yescharacteristic
465C2×C4C2×#13711yescharacteristic
466C2×C4#57511yescharacteristic
467C2×C4#57511yescharacteristic
468C2×C4#57511yescharacteristic
469C8—22no—
470C8—22no—
471C8—41no—
472C8—42no—
473C8—42no—
474C8—81no—
475C8—81no—
476C8—81no—
477C8—81no—
478C8#48514yescomplemented
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22#285411yescharacteristic
487C22#303511yescharacteristic
488C22#303911yescharacteristic
489C22#305511yescharacteristic
490C22#305611yescharacteristic
491C22#306811yescharacteristic
492C22#306911yescharacteristic
493C4—21no—
494C4—21no—
495C4—41no—
496C4—41no—
497C4—42no—
498C4—42no—
499C4—41no—
500C4—81no—
501C4—81no—
502C4—161no—
503C4—161no—
504C4—161no—
505C4—161no—
506C4C2×#48511yescharacteristic
507C4#309511yescharacteristic
508C4#310711yescharacteristic
509C4#311611yescharacteristic
510C4#317111yescharacteristic
511C4#317211yescharacteristic
512C4#318211yescharacteristic
513C4#318311yescharacteristic
514C2—81no—
515C2#1918411yescharacteristic
516C2#1918511yescharacteristic
517C2#1931411yescharacteristic
518C2#1931511yescharacteristic
519C2#3311411yescharacteristic
520C2#3312711yescharacteristic
521C2#3314511yescharacteristic
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. #9741C2semidirect
6. #9729C2semidirect
7. #9620C2semidirect
8. #9615C2semidirect
9. #9576C2semidirect
10. #9574C2semidirect
11. #9567C2semidirect
12. #9565C2semidirect
478. C8#485semidirect

Maximal subgroups

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

subgroupindex
#332102
#331532
#331142
#204802
#97412
#97292
#96202
#96152
#95762
#95742
#95672
#95652
#91382
#91362
#85972