Finite Group Databasebeta

← #308929

Subgroups — #308929

Order 512.

667 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#28809C211yesmaximal, characteristic, complemented
2#26464C211yesmaximal, characteristic
3#26444C211yesmaximal, characteristic, complemented
4#23182C211yesmaximal, characteristic
5#10598C211yesmaximal, characteristic, complemented
6#10595C211yesmaximal, characteristic
7#10350C211yesmaximal, characteristic, complemented
8#10348C211yesmaximal, characteristic
9#9820C211yesmaximal, characteristic
10#9732C211yesmaximal, characteristic
11#9612C211yesmaximal, characteristic, complemented
12#9593C211yesmaximal, characteristic, complemented
13#8925C211yesmaximal, characteristic, complemented
14#8924C211yesmaximal, characteristic
15C8×#106C211yesmaximal, characteristic, complemented
16C2×#498—21no—
17C2×#492C2211yescharacteristic
18C2×#476—21no—
19C2×C4×SD16—21no—
20#2551C2211yescharacteristic
21#2551C2211yescharacteristic
22#2366C2211yescharacteristic
23#2365C2211yescharacteristic
24#2088—21no—
25#2065—21no—
26#2058—21no—
27#2058—21no—
28#2058—21no—
29#2023—21no—
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1988C2211yescharacteristic
33#1987—21no—
34#1982C2211yescharacteristic
35#1982C2211yescharacteristic
36#1981C2211yescharacteristic
37#1968—21no—
38#1957C2211yescharacteristic
39#1957C2211yescharacteristic
40#1948C2211yescharacteristic
41#1948C2211yescharacteristic
42#1948C2211yescharacteristic
43#1948C2211yescharacteristic
44#1941—21no—
45#1939C2211yescharacteristic
46#1939C2211yescharacteristic
47#1920—21no—
48#1880—21no—
49#1877—21no—
50#1864—21no—
51#1844C2211yescharacteristic
52C4×#106C2211yescharacteristic
53C4×#106C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C8×C4⋊C4C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67C22×SD16—21no—
68C22×SD16—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8C2311yescharacteristic
74C2×#127C2311yescharacteristic
75C2×C4×Q8—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78#498—44no—
79#492—22no—
80#492Q814yescomplemented
81#476—44no—
82C4×SD16—44no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106D411yescharacteristic
90C2×#106D411yescharacteristic
91C2×#106D411yescharacteristic
92C2×#106D411yescharacteristic
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#103—21no—
100C2×#103—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—41no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#395—41no—
114#395—41no—
115#391—21no—
116#390—21no—
117#390—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#387—41no—
124#386—41no—
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127C4×C4⋊C4—21no—
128C4×C4⋊C4C2311yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
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—41no—
141#340D411yescharacteristic
142#340D411yescharacteristic
143#340D411yescharacteristic
144#340D411yescharacteristic
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#334—22no—
150#334—22no—
151#332—22no—
152#331—22no—
153C82—22no—
154C22×Q8—21no—
155C22×Q8—41no—
156C22×Q8—41no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C23×C4—41no—
161C2×SD16—24no—
162C2×SD16—24no—
163C2×SD16—42no—
164C2×SD16—42no—
165C2×SD16—81no—
166C2×SD16—81no—
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—42no—
182C4⋊Q8—42no—
183C4⋊Q8—81no—
184C4⋊Q8—81no—
185C4⋊Q8—81no—
186C4⋊Q8—81no—
187C4⋊Q8C2×Q812yes—
188#127—21no—
189#127—21no—
190#127—21no—
191#127C2×Q812yes—
192C4×Q8—42no—
193C4×Q8—81no—
194C4×D4—42no—
195C4×D4—42no—
196C4×D4—81no—
197C4×D4—81no—
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—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—41no—
227C2×C42C2411yescharacteristic
228#106—22no—
229#106—22no—
230#106—42no—
231#106—42no—
232#106—42no—
233#106—42no—
234#106—81no—
235#106—81no—
236#106SD1614yes—
237#106SD1614yescomplemented
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—42no—
246C4⋊C8—81no—
247C4⋊C8#4112yes—
248#103—42no—
249#103—81no—
250#102—42no—
251#102—42no—
252#102—42no—
253#102—42no—
254#102—81no—
255#102—81no—
256#102—82no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—42no—
262C4×C8—42no—
263C4×C8C2×Q812yes—
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—41no—
274C2×Q8—22no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—82no—
281C2×Q8—82no—
282C2×Q8—82no—
283C2×Q8—82no—
284C2×Q8—161no—
285C2×Q8—161no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—81no—
301C2×D4—81no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—161no—
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4H(5,𝔽2)11yescharacteristic
319SD16—48no—
320SD16—84no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—22no—
326C2×C8—21no—
327C2×C8—22no—
328C2×C8—21no—
329C2×C8—21no—
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—41no—
342C2×C8—41no—
343C2×C8C22⋊Q812yes—
344C2×C8C22⋊Q812yes—
345C2×C8C2×SD1612yes—
346C2×C8C2×SD1612yes—
347C2×C8AGL(1,ℤ/8)12yes—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
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—82no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—81no—
375C4⋊C4—81no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C4⋊C4C2×SD1612yes—
379C4⋊C4C2×SD1612yes—
380C4⋊C4C2×SD1612yes—
381C4⋊C4C2×SD1612yes—
382C4⋊C4#13512yes—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4—161no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42—82no—
395C42C22×Q811yescharacteristic
396C42C2×#4111yescharacteristic
397C42C2×#4111yescharacteristic
398C422−1+411yescharacteristic
399C23—42no—
400C23—81no—
401C23—81no—
402C23—81no—
403C23—81no—
404C23—81no—
405C23—81no—
406C23D4211yescharacteristic
407Q8—42no—
408Q8—42no—
409Q8—81no—
410Q8—81no—
411Q8—164no—
412Q8—164no—
413D4—42no—
414D4—42no—
415D4—42no—
416D4—42no—
417D4—82no—
418D4—81no—
419D4—81no—
420D4—81no—
421D4—81no—
422D4—161no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
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—42no—
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—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—164no—
459C2×C4C2×C22⋊Q811yescharacteristic
460C2×C4#53211yescharacteristic
461C2×C4#54211yescharacteristic
462C2×C4#54811yescharacteristic
463C2×C4D4×Q811yescharacteristic
464C2×C4#55411yescharacteristic
465C2×C4C22×SD1611yescharacteristic
466C2×C4C22×SD1611yescharacteristic
467C2×C4C2×#13511yescharacteristic
468C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
469C2×C4#57611yescharacteristic
470C2×C4#57711yescharacteristic
471C2×C4#57711yescharacteristic
472C2×C4#57711yescharacteristic
473C8—22no—
474C8—22no—
475C8—42no—
476C8—42no—
477C8—41no—
478C8—81no—
479C8—81no—
480C8—81no—
481C8#47614yescomplemented
482C22—82no—
483C22—82no—
484C22—82no—
485C22—161no—
486C22—161no—
487C22—161no—
488C22—161no—
489C22#245211yescharacteristic
490C22#327411yescharacteristic
491C22D4×SD1611yescharacteristic
492C22D4×SD1611yescharacteristic
493C22#328211yescharacteristic
494C22#329011yescharacteristic
495C22#329111yescharacteristic
496C4—21no—
497C4—21no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—81no—
504C4—82no—
505C4—81no—
506C4—81no—
507C4—161no—
508C4—161no—
509C4C2×#47611yescharacteristic
510C4#308111yescharacteristic
511C4#316711yescharacteristic
512C4#317311yescharacteristic
513C4#336211yescharacteristic
514C4#337311yescharacteristic
515C4Q8×SD1611yescharacteristic
516C4#338111yescharacteristic
517C2—162no—
518C2#1989211yescharacteristic
519C2#1991211yescharacteristic
520C2SD16211yescharacteristic
521C2#1998111yescharacteristic
522C2#2646411yescharacteristic
523C2#2648011yescharacteristic
524C2#2650011yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×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. #28809C2semidirect
3. #26444C2semidirect
5. #10598C2semidirect
7. #10350C2semidirect
11. #9612C2semidirect
12. #9593C2semidirect
13. #8925C2semidirect
15. C8×#106C2semidirect
80. #492Q8semidirect
237. #106SD16semidirect
481. C8#476semidirect

Maximal subgroups

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

subgroupindex
#288092
#264642
#264442
#231822
#105982
#105952
#103502
#103482
#98202
#97322
#96122
#95932
#89252
#89242
C8×#1062