Finite Group Databasebeta

← #227138

Subgroups — #227138

Order 512.

759 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23069C211yesmaximal, characteristic, complemented
2#23067C211yesmaximal, characteristic
3#22523C211yesmaximal, characteristic
4#22430C211yesmaximal, characteristic
5C2×#1637C211yesmaximal, characteristic
6#9821C211yesmaximal, characteristic, complemented
7#9573C211yesmaximal, characteristic, complemented
8#9569C211yesmaximal, characteristic, complemented
9#9224C211yesmaximal, characteristic, complemented
10#9223C211yesmaximal, characteristic
11#9060C211yesmaximal, characteristic, complemented
12#8895C211yesmaximal, characteristic, complemented
13#8893C211yesmaximal, characteristic
14#8207C211yesmaximal, characteristic, complemented
15#8206C211yesmaximal, characteristic
16C2×D4×Q8C2211yescharacteristic
17C2×#487C2211yescharacteristic
18C2×#477C2211yescharacteristic
19C2×#476C2211yescharacteristic
20C2×#446C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×#419—21no—
23#2369C2211yescharacteristic
24#2355C2211yescharacteristic
25#2354C2211yescharacteristic
26#1993—21no—
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1988C2211yescharacteristic
31#1987—21no—
32#1981C2211yescharacteristic
33#1956C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1936—21no—
37#1935—21no—
38#1927—21no—
39#1925—21no—
40#1919—21no—
41#1918—21no—
42#1894—21no—
43#1873—21no—
44#1834C2211yescharacteristic
45#1804—21no—
46#1803C2211yescharacteristic
47#1802C2211yescharacteristic
48#1787—21no—
49#1776—21no—
50#1776—21no—
51#1770C2211yescharacteristic
52#1770C2211yescharacteristic
53#1762—21no—
54C4×#103C2211yescharacteristic
55C4×#102C2211yescharacteristic
56#1637C418yescomplemented
57C2×#333C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×C82C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C23×Q8—21no—
69C22×M16—21no—
70D4×Q8—22no—
71D4×Q8—22no—
72D4×Q8—41no—
73D4×Q8C2×C414yes—
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2311yescharacteristic
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—41no—
79C2×C4×Q8C2311yescharacteristic
80C2×C4×D4—21no—
81C2×C4×D4C2311yescharacteristic
82C2×#117—21no—
83#487—41no—
84#487C2×C414yes—
85#477—22no—
86#477C2×C414yes—
87#476—22no—
88#476C2×C414yes—
89#446—22no—
90#446—41no—
91#436—22no—
92#436—41no—
93C2×#107—21no—
94C2×#107—21no—
95C2×#106—21no—
96C2×#106—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102#419—44no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—41no—
110C2×#103D411yescharacteristic
111C2×#103D411yescharacteristic
112C2×#102—21no—
113C2×#102—21no—
114C2×#102D411yescharacteristic
115C2×#102D411yescharacteristic
116C2×C22⋊C8—21no—
117C2×#97—21no—
118C2×#97—21no—
119C2×#97—21no—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123#391—21no—
124#391—41no—
125#390—21no—
126#389—21no—
127#389—21no—
128#388—41no—
129#387—41no—
130#384C2311yescharacteristic
131#384C2311yescharacteristic
132#382—21no—
133#380—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2311yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—41no—
154#340D411yescharacteristic
155#340D411yescharacteristic
156#340D411yescharacteristic
157#340D411yescharacteristic
158#336—21no—
159#336—21no—
160#333—41no—
161#332C2×C414yes—
162Q8⋊C8C2×C414yes—
163D4⋊C8C2×C414yes—
164C82—41no—
165C22×Q8—24no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8—41no—
169C22×Q8—42no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×Q8—81no—
173C22×Q8—81no—
174C22×Q8C2×D411yescharacteristic
175C22×D4C2×D411yescharacteristic
176C23×C4—21no—
177C23×C4—41no—
178C2×M16—42no—
179C2×M16—42no—
180C2×M16—81no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C4⋊Q8—24no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—42no—
198C4⋊Q8C22×C412yes—
199C4⋊Q8C22×C412yes—
200C22⋊Q8—24no—
201C22⋊Q8—44no—
202C22⋊Q8—44no—
203C22⋊Q8—42no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8C22×C412yes—
207C4×Q8#4112yes—
208C4×D4—24no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—42no—
212C4×D4C22×C412yes—
213C4×D4#4112yes—
214#117—42no—
215#117—81no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—41no—
243C2×C42—41no—
244C2×C42—41no—
245C2×C42C2411yescharacteristic
246#107—24no—
247#107—42no—
248#106—24no—
249#106—42no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—42no—
257C4⋊C8C22×C412yes—
258C4⋊C8C22×C412yes—
259#103—24no—
260#103—22no—
261#103—44no—
262#103—41no—
263#103—42no—
264#103—42no—
265#103—42no—
266#103—81no—
267#103—81no—
268#102—24no—
269#102—22no—
270#102—41no—
271#102—42no—
272C22⋊C8—42no—
273#97—42no—
274#97—42no—
275#97—42no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8C22×C412yes—
282#95—21no—
283#95—21no—
284#95—41no—
285#95—41no—
286#95—41no—
287#95—41no—
288#95—41no—
289#95—41no—
290C24—21no—
291C2×Q8—21no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—44no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—48no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—81no—
307C2×Q8—81no—
308C2×Q8—82no—
309C2×Q8—82no—
310C2×Q8—84no—
311C2×Q8—84no—
312C2×Q8—84no—
313C2×Q8C4×D412yes—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4C4×D412yes—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—44no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4H(5,𝔽2)11yescharacteristic
343M16—84no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8AGL(1,ℤ/8)12yes—
370C2×C8#13712yes—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—44no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—81no—
401C4⋊C4—81no—
402C4⋊C4—161no—
403C4⋊C4C4×D412yes—
404C4⋊C4C4×D412yes—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—44no—
408C22⋊C4—82no—
409C22⋊C4—161no—
410C42—22no—
411C42—22no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—82no—
417C42—161no—
418C42—161no—
419C42C23×C411yescharacteristic
420C42C2×#4111yescharacteristic
421C42C2×#4111yescharacteristic
422C42H(5,𝔽2)11yescharacteristic
423C23—22no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23D4211yescharacteristic
431Q8—22no—
432Q8—22no—
433Q8—41no—
434Q8—41no—
435Q8—42no—
436Q8—42no—
437Q8—82no—
438Q8—88no—
439Q8—81no—
440Q8—81no—
441Q8—161no—
442D4—22no—
443D4—22no—
444D4—41no—
445D4—41no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
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—41no—
475C2×C4—88no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4C2×C4×D411yescharacteristic
487C2×C4C2×C4×D411yescharacteristic
488C2×C4#51811yescharacteristic
489C2×C4#54611yescharacteristic
490C2×C4#54611yescharacteristic
491C2×C4#55111yescharacteristic
492C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
493C2×C4C2×#13711yescharacteristic
494C2×C4#57511yescharacteristic
495C2×C4#57511yescharacteristic
496C2×C4#57711yescharacteristic
497C2×C4#57711yescharacteristic
498C2×C4#57811yescharacteristic
499C2×C4#57811yescharacteristic
500C8—22no—
501C8—22no—
502C8—42no—
503C8—42no—
504C8—42no—
505C8—41no—
506C8—41no—
507C8—81no—
508C8—81no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22#235811yescharacteristic
517C22#327211yescharacteristic
518C22#327511yescharacteristic
519C22#327611yescharacteristic
520C22#327711yescharacteristic
521C22#329011yescharacteristic
522C22#329011yescharacteristic
523C4—21no—
524C4—21no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—81no—
529C4—82no—
530C4—81no—
531C4—81no—
532C4—81no—
533C4—82no—
534C4—161no—
535C4—161no—
536C4C4×AGL(1,ℤ/8)11yescharacteristic
537C4C4×#13711yescharacteristic
538C4#294911yescharacteristic
539C4#295011yescharacteristic
540C4#331011yescharacteristic
541C4#331111yescharacteristic
542C4#331211yescharacteristic
543C4#331311yescharacteristic
544C2—82no—
545C2#2029511yescharacteristic
546C2#2029611yescharacteristic
547C2#2031511yescharacteristic
548C2#2031611yescharacteristic
549C2#2286411yescharacteristic
550C2#2286911yescharacteristic
551C2#2289511yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
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. #23069C2semidirect
6. #9821C2semidirect
7. #9573C2semidirect
8. #9569C2semidirect
9. #9224C2semidirect
11. #9060C2semidirect
12. #8895C2semidirect
14. #8207C2semidirect
56. #1637C4semidirect

Maximal subgroups

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

subgroupindex
#230692
#230672
#225232
#224302
C2×#16372
#98212
#95732
#95692
#92242
#92232
#90602
#88952
#88932
#82072
#82062