Finite Group Databasebeta

← #227375

Subgroups — #227375

Order 512.

747 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22971C211yesmaximal, characteristic, complemented
2#22963C211yesmaximal, characteristic
3#22522C211yesmaximal, characteristic
4#22487C211yesmaximal, characteristic
5C2×#1632C211yesmaximal, characteristic
6#9839C211yesmaximal, characteristic, complemented
7#9575C211yesmaximal, characteristic, complemented
8#9567C211yesmaximal, characteristic, complemented
9#9224C211yesmaximal, characteristic, complemented
10#9215C211yesmaximal, characteristic
11#9066C211yesmaximal, characteristic, complemented
12#8888C211yesmaximal, characteristic, complemented
13#8886C211yesmaximal, characteristic
14#8213C211yesmaximal, characteristic, complemented
15#8204C211yesmaximal, characteristic
16C2×#554C2211yescharacteristic
17C2×#494C2211yescharacteristic
18C2×#477C2211yescharacteristic
19C2×#476C2211yescharacteristic
20C2×#446C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22#2369C2211yescharacteristic
23Q8×C4⋊C4C2211yescharacteristic
24D4×C4⋊C4C2211yescharacteristic
25#1992—21no—
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1989—21no—
31#1987C2211yescharacteristic
32#1987C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1943—21no—
36#1941—21no—
37#1919—21no—
38#1918—21no—
39#1892—21no—
40#1872—21no—
41#1865—21no—
42#1864—21no—
43#1844C2211yescharacteristic
44#1824—21no—
45#1803C2211yescharacteristic
46#1802—21no—
47#1802C2211yescharacteristic
48#1770C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50#1763—21no—
51#1762—21no—
52C4×#103C2211yescharacteristic
53C4×#102—21no—
54C4×#102C2211yescharacteristic
55C2×#340—21no—
56#1632C418yescomplemented
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×Q16—41no—
69C23×C8—21no—
70#554—22no—
71#554—22no—
72#554—41no—
73#554C2×C414yes—
74C2×C4⋊Q8C2311yescharacteristic
75C2×C4⋊Q8C2311yescharacteristic
76C2×#125—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—41no—
79C2×C4×Q8—21no—
80C2×C4×Q8C2311yescharacteristic
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×C4×D4C2311yescharacteristic
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86#494—41no—
87#494C2×C414yescomplemented
88#477—22no—
89#477C2×C414yescomplemented
90#476—22no—
91#476C2×C414yes—
92#446—22no—
93#446—41no—
94C8×D4—22no—
95C8×D4—41no—
96C2×#107—21no—
97C2×#106—21no—
98C2×#106—21no—
99C2×#106—21no—
100C2×#106—41no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×#103—21no—
107C2×#103—21no—
108C2×#103D411yescharacteristic
109C2×#103D411yescharacteristic
110C2×#102—21no—
111C2×#102—21no—
112C2×#102D411yescharacteristic
113C2×#102D411yescharacteristic
114C2×C22⋊C8—21no—
115C2×#97—21no—
116C2×#97—21no—
117C2×C4×C8—21no—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#391—41no—
122#389—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#389—41no—
127#385—41no—
128#384—21no—
129#384C2311yescharacteristic
130#384C2311yescharacteristic
131#380—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
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—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340D411yescharacteristic
154#340D411yescharacteristic
155#340D411yescharacteristic
156#340D411yescharacteristic
157#336—21no—
158#336—21no—
159#333—41no—
160#333C2×C414yescomplemented
161Q8⋊C8C2×C414yescomplemented
162D4⋊C8C2×C414yes—
163#322—41no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8C2×D411yescharacteristic
167C22×D4C2×D411yescharacteristic
168C23×C4—21no—
169C23×C4—41no—
170C23×C4—41no—
171C2×Q16—44no—
172C2×Q16—82no—
173C2×Q16—82no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—42no—
180C22×C8—42no—
181C22×C8—81no—
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8C22×C412yes—
193C4⋊Q8C22×C412yes—
194#125—24no—
195#125—42no—
196C22⋊Q8—24no—
197C22⋊Q8—44no—
198C22⋊Q8—44no—
199C22⋊Q8—42no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—42no—
203C4×Q8—81no—
204C4×Q8C22×C412yes—
205C4×Q8#4112yes—
206C4×D4—24no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—81no—
212C4×D4C22×C412yes—
213C4×D4#4112yes—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—24no—
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—42no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—81no—
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—41no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—41no—
248C2×C42—41no—
249C2×C42C2411yescharacteristic
250#107—24no—
251#106—24no—
252#106—81no—
253#106—81no—
254#106—82no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—42no—
262C4⋊C8C22×C412yes—
263C4⋊C8C22×C412yes—
264#103—24no—
265#103—22no—
266#103—41no—
267#103—81no—
268#102—24no—
269#102—22no—
270#102—22no—
271#102—42no—
272C22⋊C8—42no—
273#97—42no—
274#97—81no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—81no—
281C4×C8C22×C412yes—
282#95—21no—
283#95—21no—
284#95—41no—
285#95—41no—
286#95—41no—
287#95—41no—
288C24—21no—
289C2×Q8—21no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—81no—
299C2×Q8—81no—
300C2×Q8—81no—
301C2×Q8—81no—
302C2×Q8C4×D412yes—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4C4×D412yes—
307C2×D4#13712yes—
308C2×D4#13712yes—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—44no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—161no—
330C22×C4—161no—
331C22×C4—161no—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4H(5,𝔽2)11yescharacteristic
335Q16—88no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—84no—
364C4⋊C4—22no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—22no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—82no—
394C4⋊C4—81no—
395C4⋊C4—81no—
396C4⋊C4—82no—
397C4⋊C4—161no—
398C4⋊C4—161no—
399C4⋊C4C4×D412yes—
400C4⋊C4C4×D412yes—
401C4⋊C4#13712yes—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—44no—
405C22⋊C4—82no—
406C42—22no—
407C42—22no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—82no—
413C42—161no—
414C42C23×C411yescharacteristic
415C42C2×#4111yescharacteristic
416C42C2×#4111yescharacteristic
417C42H(5,𝔽2)11yescharacteristic
418C23—22no—
419C23—22no—
420C23—22no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23D4211yescharacteristic
426Q8—22no—
427Q8—22no—
428Q8—41no—
429Q8—41no—
430Q8—82no—
431Q8—82no—
432Q8—161no—
433Q8—161no—
434D4—22no—
435D4—22no—
436D4—22no—
437D4#44114yes—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—164no—
478C2×C4—164no—
479C2×C4C2×C4×D411yescharacteristic
480C2×C4C2×C4×D411yescharacteristic
481C2×C4#51811yescharacteristic
482C2×C4#54611yescharacteristic
483C2×C4#54611yescharacteristic
484C2×C4#55111yescharacteristic
485C2×C4C2×#13711yescharacteristic
486C2×C4C2×#13711yescharacteristic
487C2×C4C2×#13711yescharacteristic
488C2×C4#57511yescharacteristic
489C2×C4#57711yescharacteristic
490C2×C4#57711yescharacteristic
491C2×C4#57711yescharacteristic
492C2×C4#57811yescharacteristic
493C8—24no—
494C8—41no—
495C8—42no—
496C8—42no—
497C8—41no—
498C8—41no—
499C8—41no—
500C8—81no—
501C8—81no—
502C22—24no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22#235811yescharacteristic
510C22#326611yescharacteristic
511C22#326611yescharacteristic
512C22#326711yescharacteristic
513C22#327011yescharacteristic
514C22#329111yescharacteristic
515C22#329211yescharacteristic
516C4—21no—
517C4—21no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—81no—
523C4—81no—
524C4—82no—
525C4—82no—
526C4—81no—
527C4—161no—
528C4—161no—
529C4C2×#44111yescharacteristic
530C4C4×#13711yescharacteristic
531C4#294711yescharacteristic
532C4#295211yescharacteristic
533C4#331211yescharacteristic
534C4#331611yescharacteristic
535C4#332311yescharacteristic
536C4#333111yescharacteristic
537C2—44no—
538C2#2004011yescharacteristic
539C2#2004211yescharacteristic
540C2#2004811yescharacteristic
541C2#2011411yescharacteristic
542C2#2285911yescharacteristic
543C2#2288211yescharacteristic
544C2#2289711yescharacteristic
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. #22971C2semidirect
6. #9839C2semidirect
7. #9575C2semidirect
8. #9567C2semidirect
9. #9224C2semidirect
11. #9066C2semidirect
12. #8888C2semidirect
14. #8213C2semidirect
56. #1632C4semidirect
87. #494C2×C4semidirect
89. #477C2×C4semidirect
160. #333C2×C4semidirect
161. Q8⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#229712
#229632
#225222
#224872
C2×#16322
#98392
#95752
#95672
#92242
#92152
#90662
#88882
#88862
#82132
#82042