Finite Group Databasebeta

← #229285

Subgroups — #229285

Order 512.

721 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23216C211yesmaximal, characteristic, complemented
2#23094C211yesmaximal, characteristic, complemented
3#23093C211yesmaximal, characteristic
4#22708C211yesmaximal, characteristic, complemented
5C2×#1681C211yesmaximal, characteristic
6#9822C211yesmaximal, characteristic, complemented
7#9616C211yesmaximal, characteristic, complemented
8#9564C211yesmaximal, characteristic, complemented
9#9562C211yesmaximal, characteristic, complemented
10#8988C211yesmaximal, characteristic, complemented
11#8980C211yesmaximal, characteristic
12#8926C211yesmaximal, characteristic, complemented
13#8924C211yesmaximal, characteristic, complemented
14C8×#107C211yesmaximal, characteristic, complemented
15C8×#102C211yesmaximal, characteristic, complemented
16C2×#486C2211yescharacteristic
17C2×#461C2211yescharacteristic
18C2×#459C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20#2366C2211yescharacteristic
21#2365C2211yescharacteristic
22#2354C2211yescharacteristic
23#2354C2211yescharacteristic
24#1990C2211yescharacteristic
25#1988C2211yescharacteristic
26#1982C2211yescharacteristic
27#1970—21no—
28#1968—21no—
29#1957C2211yescharacteristic
30#1957C2211yescharacteristic
31#1956C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1934—21no—
37#1927—21no—
38#1925—21no—
39#1925—21no—
40#1880—21no—
41#1879—21no—
42#1862—21no—
43#1854—21no—
44#1854—21no—
45#1834C2211yescharacteristic
46#1807—21no—
47C4×#107C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51C4×#103—21no—
52C4×#102—21no—
53C4×#102—21no—
54C4×#102C2211yescharacteristic
55#1735—21no—
56#1681C418yescomplemented
57C2×#335C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×C82C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C22×D8—21no—
70C23×C8—21no—
71C2×C4⋊Q8C2311yescharacteristic
72C2×#127C2311yescharacteristic
73C2×C4×Q8—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4C2311yescharacteristic
77C2×#117—21no—
78C2×#117—21no—
79#486—22no—
80#486C2×C414yes—
81#461—22no—
82#461C2×C414yescomplemented
83#459—22no—
84#459C2×C414yes—
85C8×D4—22no—
86C8×D4C2×C414yescomplemented
87C2×#107—21no—
88C2×#107D411yescharacteristic
89C2×#107D411yescharacteristic
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—41no—
108C2×#102D411yescharacteristic
109C2×#102D411yescharacteristic
110C2×C22⋊C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117#391—21no—
118#390—21no—
119#390—21no—
120#390—21no—
121#390—21no—
122#389—21no—
123#388—41no—
124#388—41no—
125#387—41no—
126#386—41no—
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129#380—21no—
130#380—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2311yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
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—41no—
146#340D411yescharacteristic
147#340D411yescharacteristic
148#340D411yescharacteristic
149#340D411yescharacteristic
150#336—21no—
151#336—21no—
152#336—21no—
153#335C2×C414yescomplemented
154#332C2×C414yescomplemented
155#331C2×C414yes—
156D4⋊C8—22no—
157C82—22no—
158C22×Q8—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4C2×D411yescharacteristic
163C23×C4—21no—
164C23×C4—41no—
165C2×SD16—24no—
166C2×SD16—81no—
167C2×SD16—81no—
168C2×D8—24no—
169C2×D8—81no—
170C2×D8—81no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—24no—
176C22×C8—21no—
177C22×C8—81no—
178C22×C8—81no—
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8C22×C412yes—
187#127—21no—
188#127—21no—
189#127—21no—
190#127C22×C412yes—
191C4×Q8—42no—
192C4×Q8—81no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—42no—
196C4×D4—42no—
197C4×D4—81no—
198C4×D4—81no—
199C4×D4C22×C412yes—
200C4×D4#4112yes—
201#117—42no—
202#117—42no—
203#117—81no—
204#117—81no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—41no—
233C2×C42—41no—
234C2×C42—41no—
235C2×C42C2411yescharacteristic
236#107—22no—
237#107—22no—
238#107—81no—
239#106—81no—
240#106—81no—
241#106—81no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8C22×C412yes—
248C4⋊C8C22×C412yes—
249C4⋊C8C22×C412yes—
250#103—24no—
251#103—42no—
252#103—81no—
253#102—24no—
254#102—24no—
255#102—22no—
256#102—24no—
257#102—22no—
258#102—42no—
259#102—42no—
260#102—81no—
261#102—81no—
262#102—82no—
263C22⋊C8—24no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—81no—
269C4×C8—81no—
270C4×C8—81no—
271C4×C8C22×C412yes—
272C4×C8#4112yes—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—41no—
277#95—41no—
278#95—41no—
279#95—41no—
280C24—21no—
281C24—41no—
282C2×Q8—22no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—81no—
308C2×D4—81no—
309C2×D4C4×D412yes—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—161no—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4H(5,𝔽2)11yescharacteristic
332SD16—84no—
333D8—84no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—84no—
359C2×C8C4×D412yes—
360C2×C8C4×D412yes—
361C2×C8#13512yes—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—82no—
385C4⋊C4—81no—
386C4⋊C4—81no—
387C4⋊C4C4×D412yes—
388C4⋊C4#13512yes—
389C4⋊C4#13512yes—
390C4⋊C4#13512yes—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C22⋊C4—82no—
397C22⋊C4—161no—
398C22⋊C4—161no—
399C22⋊C4—161no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—82no—
407C42—82no—
408C42—82no—
409C42C23×C411yescharacteristic
410C42C2×#4111yescharacteristic
411C42C2×#4111yescharacteristic
412C422−1+411yescharacteristic
413C23—22no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—42no—
421C23—81no—
422C23—81no—
423C23—81no—
424C23—81no—
425C23—81no—
426C23—81no—
427C23D4211yescharacteristic
428Q8—42no—
429Q8—42no—
430Q8—81no—
431Q8—81no—
432D4—22no—
433D4—22no—
434D4—41no—
435D4—41no—
436D4—42no—
437D4—42no—
438D4—42no—
439D4—42no—
440D4—82no—
441D4—81no—
442D4—81no—
443D4—81no—
444D4—81no—
445D4—161no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—164no—
483C2×C4C2×C4×D411yescharacteristic
484C2×C4C2×C4×D411yescharacteristic
485C2×C4#52011yescharacteristic
486C2×C4#54711yescharacteristic
487C2×C4#54811yescharacteristic
488C2×C4#55311yescharacteristic
489C2×C4C2×#13511yescharacteristic
490C2×C4C2×#13511yescharacteristic
491C2×C4C2×#13511yescharacteristic
492C2×C4C2×#13511yescharacteristic
493C2×C4#57611yescharacteristic
494C2×C4#57611yescharacteristic
495C2×C4#57611yescharacteristic
496C2×C4#57711yescharacteristic
497C8—22no—
498C8—22no—
499C8—22no—
500C8—42no—
501C8—41no—
502C8—42no—
503C8—42no—
504C8—42no—
505C8—81no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—161no—
517C22—161no—
518C22—161no—
519C22—161no—
520C22#236111yescharacteristic
521C22#328011yescharacteristic
522C22#328011yescharacteristic
523C22#328211yescharacteristic
524C22#328711yescharacteristic
525C22#329011yescharacteristic
526C22#329111yescharacteristic
527C4—21no—
528C4—21no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4—82no—
537C4—161no—
538C4—161no—
539C4C4×#13511yescharacteristic
540C4C4×#13511yescharacteristic
541C4#294611yescharacteristic
542C4#294811yescharacteristic
543C4#333911yescharacteristic
544C4#334111yescharacteristic
545C4#336011yescharacteristic
546C4#336111yescharacteristic
547C2—82no—
548C2—162no—
549C2#1998711yescharacteristic
550C2#1999111yescharacteristic
551C2#1999511yescharacteristic
552C2#2000411yescharacteristic
553C2#2299911yescharacteristic
554C2#2302711yescharacteristic
555C2#2304411yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#459C22
LCSG ▷ 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. #23216C2semidirect
2. #23094C2semidirect
4. #22708C2semidirect
6. #9822C2semidirect
7. #9616C2semidirect
8. #9564C2semidirect
9. #9562C2semidirect
10. #8988C2semidirect
12. #8926C2semidirect
13. #8924C2semidirect
14. C8×#107C2semidirect
15. C8×#102C2semidirect
56. #1681C4semidirect
82. #461C2×C4semidirect
86. C8×D4C2×C4semidirect
153. #335C2×C4semidirect
154. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#232162
#230942
#230932
#227082
C2×#16812
#98222
#96162
#95642
#95622
#89882
#89802
#89262
#89242
C8×#1072
C8×#1022