Finite Group Databasebeta

← #274697

Subgroups — #274697

Order 512.

708 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32954C211yesmaximal, characteristic
2#27975C211yesmaximal, characteristic
3Q8×C22⋊C8C211yesmaximal, characteristic
4#20734C211yesmaximal, characteristic
5#10626C211yesmaximal, characteristic, complemented
6#10623C211yesmaximal, characteristic, complemented
7#10487C211yesmaximal, characteristic, complemented
8#10486C211yesmaximal, characteristic, complemented
9#9220C211yesmaximal, characteristic, complemented
10#9220C211yesmaximal, characteristic, complemented
11#9130C211yesmaximal, characteristic
12#8831C211yesmaximal, characteristic
13#8727C211yesmaximal, characteristic
14#8649C211yesmaximal, characteristic, complemented
15#8647C211yesmaximal, characteristic, complemented
16C22×C4×Q8C2211yescharacteristic
17C2×#475C2211yescharacteristic
18C2×#475C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2857C2211yescharacteristic
23#2285C2211yescharacteristic
24#2074—21no—
25#2074—21no—
26#1984—21no—
27#1984—21no—
28#1984C2211yescharacteristic
29#1984C2211yescharacteristic
30#1956C2211yescharacteristic
31#1956C2211yescharacteristic
32#1873—21no—
33#1873—21no—
34#1865—21no—
35#1865—21no—
36#1853—21no—
37#1834C2211yescharacteristic
38#1834C2211yescharacteristic
39#1824—21no—
40#1803C2211yescharacteristic
41#1803C2211yescharacteristic
42#1803C2211yescharacteristic
43#1803C2211yescharacteristic
44#1803C2211yescharacteristic
45#1803C2211yescharacteristic
46#1797—21no—
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1787—21no—
50#1787—21no—
51C4×#103—21no—
52C4×#103—21no—
53C4×C22⋊C8—21no—
54C4×C22⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×Q8D411yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×C22⋊Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8—41no—
75C2×C4×Q8—41no—
76C2×C4×Q8D411yescharacteristic
77C2×C4×Q8D411yescharacteristic
78C2×C4×Q8C2311yescharacteristic
79C2×C4×Q8C2311yescharacteristic
80C2×#117—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4D411yescharacteristic
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85#475—22no—
86#475—22no—
87#475—41no—
88#475—41no—
89C8×Q8—22no—
90C8×Q8—22no—
91C8×Q8—41no—
92C8×Q8—41no—
93#431—41no—
94#431—41no—
95C2×#106—21no—
96C2×#106—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112C2×#103—21no—
113C2×#103—21no—
114C2×#103—21no—
115C2×#103—21no—
116C2×#103—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8D411yescharacteristic
121C2×C22⋊C8D411yescharacteristic
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124C2×C4×C8—21no—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127#391—21no—
128#389—21no—
129#389—21no—
130#384C2311yescharacteristic
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133#382—21no—
134#380—21no—
135#380—21no—
136#380—21no—
137#380—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—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#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336D411yescharacteristic
157#336D411yescharacteristic
158#335—41no—
159#334—41no—
160#332—41no—
161#332—41no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—21no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—21no—
168C22×Q8—41no—
169C22×Q8—41no—
170C22×Q8C2×D411yescharacteristic
171C22×Q8C2×D411yescharacteristic
172C23×C4—21no—
173C23×C4—21no—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4#4111yescharacteristic
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C22⋊Q8—42no—
192C22⋊Q8—42no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—41no—
204C4×Q8—41no—
205C4×Q8—42no—
206C4×Q8—42no—
207C4×Q8#4112yes—
208C4×Q8#4112yes—
209#117—81no—
210#117—81no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
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—21no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—41no—
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42#4111yescharacteristic
250C2×C42C2411yescharacteristic
251#106—42no—
252#106—42no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—42no—
266C4⋊C8—42no—
267#103—42no—
268#103—42no—
269#103—42no—
270#103—42no—
271#103—42no—
272#103—42no—
273#103—42no—
274#103—42no—
275#103—81no—
276#103—81no—
277#103—81no—
278#103—81no—
279C22⋊C8—22no—
280C22⋊C8—22no—
281C22⋊C8—42no—
282C22⋊C8—42no—
283C22⋊C8—42no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—42no—
289C4×C8—42no—
290C4×C8—42no—
291#95—41no—
292#95—41no—
293#95—41no—
294#95—41no—
295C24#12111yescharacteristic
296C2×Q8—24no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—44no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8#12312yes—
318C2×Q8#13512yes—
319C2×Q8#13512yes—
320C2×Q8#13712yes—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—24no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—81no—
349C22×C4#12111yescharacteristic
350C22×C4#12111yescharacteristic
351C22×C4#12111yescharacteristic
352C22×C4#13512yes—
353C22×C4#13712yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—81no—
414C4⋊C4—81no—
415C4⋊C4—81no—
416C4⋊C4—82no—
417C4⋊C4#12312yes—
418C22⋊C4—82no—
419C22⋊C4—82no—
420C42—21no—
421C42—21no—
422C42—22no—
423C42—22no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—42no—
429C42—42no—
430C42—41no—
431C42—41no—
432C42—82no—
433C42C2×#4111yescharacteristic
434C42C2×#4111yescharacteristic
435C42H(5,𝔽2)11yescharacteristic
436C42H(5,𝔽2)11yescharacteristic
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23#46412yes—
444C23C2×#12111yescharacteristic
445Q8—22no—
446Q8—22no—
447Q8—22no—
448Q8—22no—
449Q8—22no—
450Q8—48no—
451Q8—41no—
452Q8—41no—
453Q8—81no—
454Q8—81no—
455Q8—81no—
456Q8—81no—
457Q8#48514yes—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—48no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—84no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4#46412yes—
505C2×C4C2×#12311yescharacteristic
506C2×C4#53411yescharacteristic
507C2×C4#54611yescharacteristic
508C2×C4#54611yescharacteristic
509C2×C4#55111yescharacteristic
510C2×C4#55111yescharacteristic
511C2×C4C2×#13511yescharacteristic
512C2×C4C2×#13511yescharacteristic
513C2×C4C2×#13511yescharacteristic
514C2×C4C2×#13711yescharacteristic
515C2×C4C2×#13711yescharacteristic
516C2×C4#57511yescharacteristic
517C2×C4#57711yescharacteristic
518C2×C4#57711yescharacteristic
519C8—41no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—81no—
524C8—81no—
525C8—81no—
526C8—81no—
527C8—81no—
528C8—81no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22#241211yescharacteristic
537C22C2×#46411yescharacteristic
538C22#304311yescharacteristic
539C22#305011yescharacteristic
540C22#305611yescharacteristic
541C22#306711yescharacteristic
542C22#306911yescharacteristic
543C4—22no—
544C4—24no—
545C4—21no—
546C4—21no—
547C4—41no—
548C4—42no—
549C4—41no—
550C4—42no—
551C4—81no—
552C4—161no—
553C4—161no—
554C4—161no—
555C4—161no—
556C4C2×#48511yescharacteristic
557C4#309111yescharacteristic
558C4#319011yescharacteristic
559C4#319111yescharacteristic
560C4#330111yescharacteristic
561C4#330111yescharacteristic
562C4#332011yescharacteristic
563C4#332011yescharacteristic
564C2—42no—
565C2#1912311yescharacteristic
566C2#1917511yescharacteristic
567C2#1940011yescharacteristic
568C2#1947211yescharacteristic
569C2#2453311yescharacteristic
570C2#2455411yescharacteristic
571C2#2457411yescharacteristic
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. #10626C2semidirect
6. #10623C2semidirect
7. #10487C2semidirect
8. #10486C2semidirect
9. #9220C2semidirect
10. #9220C2semidirect
14. #8649C2semidirect
15. #8647C2semidirect

Maximal subgroups

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

subgroupindex
#329542
#279752
Q8×C22⋊C82
#207342
#106262
#106232
#104872
#104862
#92202
#91302
#88312
#87272
#86492
#86472