Finite Group Databasebeta

← #285711

Subgroups — #285711

Order 512.

679 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27961C211yesmaximal, characteristic, complemented
2#27677C211yesmaximal, characteristic
3#24836C211yesmaximal, characteristic
4#22985C211yesmaximal, characteristic
5#10656C211yesmaximal, characteristic, complemented
6#10622C211yesmaximal, characteristic, complemented
7#10593C211yesmaximal, characteristic, complemented
8#10481C211yesmaximal, characteristic, complemented
9#9616C211yesmaximal, characteristic, complemented
10#9568C211yesmaximal, characteristic, complemented
11#9215C211yesmaximal, characteristic, complemented
12#9127C211yesmaximal, characteristic, complemented
13#8829C211yesmaximal, characteristic, complemented
14#8648C211yesmaximal, characteristic, complemented
15C2×#1296C211yesmaximal, characteristic, complemented
16C2×#476C2211yescharacteristic
17C2×C8×D4—21no—
18C2×C8×D4C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21#2857C2211yescharacteristic
22C4×#121C2211yescharacteristic
23C2×#384C2211yescharacteristic
24#2100—21no—
25#2100—21no—
26#2099—21no—
27#2093—21no—
28#2088—21no—
29#2065—21no—
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1987C2211yescharacteristic
33#1954C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1939C2211yescharacteristic
38#1923—21no—
39#1893—21no—
40#1874—21no—
41#1869—21no—
42#1852—21no—
43#1850—21no—
44#1834C2211yescharacteristic
45#1834C2211yescharacteristic
46#1824—21no—
47#1824—21no—
48#1807—21no—
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52#1797C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C4×C22⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C23×C8—21no—
69C23×C8—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×C22⋊Q8—21no—
72C2×#121D411yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4C2311yescharacteristic
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C42C2311yescharacteristic
79#476—22no—
80#476—41no—
81C8×D4—22no—
82C8×D4—41no—
83C8×D4—44no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—41no—
94C2×C4⋊C8—41no—
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8D411yescharacteristic
110C2×C22⋊C8D411yescharacteristic
111C2×C4×C8—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#391—21no—
115#390D411yescharacteristic
116#389—21no—
117#389—21no—
118#389D411yescharacteristic
119#388—21no—
120#386—21no—
121#384—21no—
122#384—21no—
123#384—21no—
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#380—21no—
128#380—21no—
129#380D411yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C22⋊C4—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
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#336—21no—
149#336—21no—
150#336—21no—
151#336D411yescharacteristic
152#336D411yescharacteristic
153#335—22no—
154#335—41no—
155#333—22no—
156#333—41no—
157C22×Q8—21no—
158C22×D4—21no—
159C22×D4C2×D411yescharacteristic
160C23×C4—21no—
161C23×C4—41no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4#4111yescharacteristic
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8—42no—
174C22×C8—42no—
175C22×C8—42no—
176C22×C8—42no—
177C22×C8—81no—
178C22×C8—81no—
179C22×C8—81no—
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C22⋊Q8—82no—
189#121—22no—
190#121Q1614yes—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—42no—
196C4×D4—42no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—81no—
216C2×C4⋊C4—81no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—41no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42#4111yescharacteristic
235C2×C42C2411yescharacteristic
236#107—81no—
237#107—81no—
238#106—42no—
239#106—81no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—42no—
251C4⋊C8—42no—
252C4⋊C8—42no—
253C4⋊C8#4112yes—
254C4⋊C8#4112yes—
255#102—42no—
256#102—81no—
257#102—81no—
258#102—81no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—42no—
262C22⋊C8—42no—
263C22⋊C8—42no—
264C22⋊C8—81no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—42no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—41no—
274#95—41no—
275C24—21no—
276C24#12111yescharacteristic
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—81no—
280C2×Q8—81no—
281C2×D4—24no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4C2×Q1612yes—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4#12111yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4C2×Q1612yes—
322C22×C4#13512yes—
323C22×C4#13512yes—
324C22×C4AGL(1,ℤ/8)12yes—
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
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×C8—84no—
360C2×C8—84no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
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—81no—
384C4⋊C4—81no—
385C4⋊C4—81no—
386C4⋊C4—82no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4—82no—
390C4⋊C4C2×Q1612yes—
391C22⋊C4—24no—
392C22⋊C4—42no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—42no—
396C22⋊C4—81no—
397C22⋊C4—82no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—41no—
405C42—41no—
406C42C2×#4111yescharacteristic
407C42C2×#4111yescharacteristic
408C42H(5,𝔽2)11yescharacteristic
409C422−1+411yescharacteristic
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—22no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23#46312yes—
422C23#46712yes—
423C23#47112yes—
424C23C2×#12111yescharacteristic
425Q8—161no—
426Q8—161no—
427D4—22no—
428D4—22no—
429D4—48no—
430D4—41no—
431D4—41no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—42no—
451C2×C4—42no—
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—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
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—164no—
477C2×C4#46312yes—
478C2×C4#46712yes—
479C2×C4#52911yescharacteristic
480C2×C4#53911yescharacteristic
481C2×C4#54611yescharacteristic
482C2×C4#54611yescharacteristic
483C2×C4#54711yescharacteristic
484C2×C4#54711yescharacteristic
485C2×C4C22×Q1611yescharacteristic
486C2×C4C2×#13511yescharacteristic
487C2×C4C2×#13511yescharacteristic
488C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
489C2×C4#57611yescharacteristic
490C2×C4#57711yescharacteristic
491C2×C4#57711yescharacteristic
492C2×C4#57811yescharacteristic
493C8—41no—
494C8—41no—
495C8—42no—
496C8—41no—
497C8—41no—
498C8—42no—
499C8—81no—
500C8—81no—
501C8—81no—
502C8—81no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22#168314yes—
517C22#242511yescharacteristic
518C22C2×#46311yescharacteristic
519C22#304611yescharacteristic
520C22C2×#46711yescharacteristic
521C22#306211yescharacteristic
522C22C2×#47111yescharacteristic
523C22#306811yescharacteristic
524C4—22no—
525C4—22no—
526C4—21no—
527C4—21no—
528C4—42no—
529C4—41no—
530C4—81no—
531C4—81no—
532C4—161no—
533C4—161no—
534C4—161no—
535C4—161no—
536C4#329711yescharacteristic
537C4#329911yescharacteristic
538C4#330011yescharacteristic
539C4#331011yescharacteristic
540C4#333711yescharacteristic
541C4#333711yescharacteristic
542C4#333811yescharacteristic
543C4#334311yescharacteristic
544C2—24no—
545C2—82no—
546C2C2×#168311yescharacteristic
547C2#1923411yescharacteristic
548C2#1941711yescharacteristic
549C2#1958911yescharacteristic
550C2#2521211yescharacteristic
551C2#2523411yescharacteristic
552C2#2527011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
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
1. #27961C2semidirect
5. #10656C2semidirect
6. #10622C2semidirect
7. #10593C2semidirect
8. #10481C2semidirect
9. #9616C2semidirect
10. #9568C2semidirect
11. #9215C2semidirect
12. #9127C2semidirect
13. #8829C2semidirect
14. #8648C2semidirect
15. C2×#1296C2semidirect

Maximal subgroups

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

subgroupindex
#279612
#276772
#248362
#229852
#106562
#106222
#105932
#104812
#96162
#95682
#92152
#91272
#88292
#86482
C2×#12962