Finite Group Databasebeta

← #274170

Subgroups — #274170

Order 512.

696 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33109C211yesmaximal, characteristic
2#25674C211yesmaximal, characteristic
3#22599C211yesmaximal, characteristic
4#21394C211yesmaximal, characteristic
5#10592C211yesmaximal, characteristic, complemented
6#10590C211yesmaximal, characteristic, complemented
7#10545C211yesmaximal, characteristic, complemented
8#10542C211yesmaximal, characteristic, complemented
9#9736C211yesmaximal, characteristic, complemented
10#9731C211yesmaximal, characteristic, complemented
11#9566C211yesmaximal, characteristic, complemented
12#9561C211yesmaximal, characteristic, complemented
13#9140C211yesmaximal, characteristic
14#8834C211yesmaximal, characteristic
15#8345C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#487C2211yescharacteristic
19C2×#484—21no—
20C2×#432C2211yescharacteristic
21C2×#431—21no—
22#2857C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#2036—21no—
25#2036—21no—
26#1990C2211yescharacteristic
27#1988C2211yescharacteristic
28#1988C2211yescharacteristic
29#1981C2211yescharacteristic
30#1981C2211yescharacteristic
31#1981C2211yescharacteristic
32#1981C2211yescharacteristic
33#1956C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1928—21no—
39#1928—21no—
40#1916—21no—
41#1916—21no—
42#1872—21no—
43#1872—21no—
44#1872—21no—
45#1863—21no—
46#1863—21no—
47#1817—21no—
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1763—21no—
53#1763—21no—
54C4×C22⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#533—22no—
68#533—22no—
69#533—41no—
70#533—41no—
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4⋊Q8C2311yescharacteristic
73C2×C4⋊Q8C2311yescharacteristic
74C2×#125—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8—21no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×#117—21no—
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4D411yescharacteristic
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85#500—22no—
86#500—41no—
87#487—22no—
88#487—41no—
89#484—44no—
90#432—41no—
91#432—41no—
92#431—44no—
93C2×#107—21no—
94C2×#107—21no—
95C2×#107—21no—
96C2×#107—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×#97—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#380—21no—
132#380—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4D411yescharacteristic
136C4×C4⋊C4D411yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139C2×#95—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#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336D411yescharacteristic
155#336D411yescharacteristic
156#335—41no—
157#334—41no—
158#332—22no—
159#332—22no—
160#332—41no—
161#332—41no—
162#332—41no—
163#332—41no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4#4111yescharacteristic
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8#4112yes—
193C4⋊Q8#4112yes—
194#125—42no—
195#125—42no—
196C22⋊Q8—24no—
197C22⋊Q8—42no—
198C22⋊Q8—42no—
199C22⋊Q8—42no—
200C22⋊Q8—42no—
201C22⋊Q8—42no—
202C4×Q8—24no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×Q8—42no—
206#117—42no—
207#117—42no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
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—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—41no—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42#4111yescharacteristic
246C2×C42C2411yescharacteristic
247#107—42no—
248#107—42no—
249#107—42no—
250#107—42no—
251C4⋊C8—21no—
252C4⋊C8—21no—
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—42no—
264C4⋊C8—42no—
265#103—42no—
266#103—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—42no—
276#103—42no—
277C22⋊C8—22no—
278C22⋊C8—41no—
279C22⋊C8—42no—
280C22⋊C8—42no—
281C22⋊C8—42no—
282#97—42no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—42no—
288C4×C8—42no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—41no—
294#95—41no—
295C24#12111yescharacteristic
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C22×C4—24no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—81no—
344C22×C4#12111yescharacteristic
345C22×C4#12111yescharacteristic
346C22×C4#12111yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
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×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×C8AGL(1,ℤ/8)12yes—
380C4⋊C4—24no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—41no—
394C4⋊C4—41no—
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—81no—
406C4⋊C4—82no—
407C4⋊C4#13512yes—
408C4⋊C4#13512yes—
409C4⋊C4AGL(1,ℤ/8)12yes—
410C22⋊C4—44no—
411C22⋊C4—82no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—42no—
419C42—42no—
420C42—44no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42C2×#4111yescharacteristic
426C42C2×#4111yescharacteristic
427C42H(5,𝔽2)11yescharacteristic
428C42H(5,𝔽2)11yescharacteristic
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23#47312yes—
436C23C2×#12111yescharacteristic
437Q8—42no—
438Q8—42no—
439Q8—42no—
440Q8—42no—
441Q8—81no—
442Q8—81no—
443Q8—81no—
444Q8—81no—
445Q8—81no—
446Q8—81no—
447Q8—81no—
448Q8—81no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—48no—
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—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—84no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4#52511yescharacteristic
497C2×C4#53511yescharacteristic
498C2×C4#54611yescharacteristic
499C2×C4#54611yescharacteristic
500C2×C4#55111yescharacteristic
501C2×C4#55111yescharacteristic
502C2×C4C2×#13511yescharacteristic
503C2×C4C2×#13511yescharacteristic
504C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
505C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
506C2×C4#57511yescharacteristic
507C2×C4#57511yescharacteristic
508C2×C4#57811yescharacteristic
509C2×C4#57811yescharacteristic
510C8—22no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—81no—
515C8—81no—
516C8—81no—
517C8—81no—
518C8—81no—
519C8—81no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22#241111yescharacteristic
528C22#303611yescharacteristic
529C22#304411yescharacteristic
530C22#305011yescharacteristic
531C22#305511yescharacteristic
532C22C2×#47311yescharacteristic
533C22#306911yescharacteristic
534C4—21no—
535C4—21no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—81no—
543C4—82no—
544C4—82no—
545C4—161no—
546C4—161no—
547C4#310211yescharacteristic
548C4#310611yescharacteristic
549C4#320711yescharacteristic
550C4#321211yescharacteristic
551C4#330811yescharacteristic
552C4#331511yescharacteristic
553C4#332711yescharacteristic
554C4#333011yescharacteristic
555C2—42no—
556C2#1925911yescharacteristic
557C2#1930411yescharacteristic
558C2#1940711yescharacteristic
559C2#1947611yescharacteristic
560C2#2448911yescharacteristic
561C2#2450411yescharacteristic
562C2#2452211yescharacteristic
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. #10592C2semidirect
6. #10590C2semidirect
7. #10545C2semidirect
8. #10542C2semidirect
9. #9736C2semidirect
10. #9731C2semidirect
11. #9566C2semidirect
12. #9561C2semidirect

Maximal subgroups

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

subgroupindex
#331092
#256742
#225992
#213942
#105922
#105902
#105452
#105422
#97362
#97312
#95662
#95612
#91402
#88342
#83452