Finite Group Databasebeta

← #285646

Subgroups — #285646

Order 512.

608 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27877C211yesmaximal, characteristic, complemented
2#27626C211yesmaximal, characteristic
3#24701C211yesmaximal, characteristic
4#23083C211yesmaximal, characteristic
5#10651C211yesmaximal, characteristic, complemented
6#10625C211yesmaximal, characteristic, complemented
7#10593C211yesmaximal, characteristic, complemented
8#10480C211yesmaximal, characteristic, complemented
9#9613C211yesmaximal, characteristic, complemented
10#9567C211yesmaximal, characteristic, complemented
11#9215C211yesmaximal, characteristic, complemented
12#9128C211yesmaximal, characteristic, complemented
13#8831C211yesmaximal, characteristic, complemented
14#8644C211yesmaximal, characteristic, complemented
15#8038C211yesmaximal, characteristic, complemented
16C2×#476C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2857C2211yescharacteristic
21C4×#121C2211yescharacteristic
22#2285C2211yescharacteristic
23#2077—21no—
24#2076—21no—
25#2075—21no—
26#2075—21no—
27#2034—21no—
28#2034—21no—
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1987C2211yescharacteristic
32#1954—21no—
33#1954C2211yescharacteristic
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1947C2211yescharacteristic
37#1941—21no—
38#1939C2211yescharacteristic
39#1934—21no—
40#1925—21no—
41#1923—21no—
42#1920—21no—
43#1918—21no—
44#1918—21no—
45#1869—21no—
46#1845C2211yescharacteristic
47#1834C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1797C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C8×C22⋊C4—21no—
54C4×C22⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×C22⋊Q8—21no—
70C2×#121D411yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C22×C42C2311yescharacteristic
75#476—22no—
76#476—41no—
77C8×D4—22no—
78C8×D4—41no—
79#431—22no—
80#431—22no—
81#431—41no—
82#431—41no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8D411yescharacteristic
107C2×C22⋊C8D411yescharacteristic
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#391—21no—
114#390D411yescharacteristic
115#389—21no—
116#389—21no—
117#389D411yescharacteristic
118#388—21no—
119#386—21no—
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#382—21no—
124#380—21no—
125#380—21no—
126#380—21no—
127#380—21no—
128#380D411yescharacteristic
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2311yescharacteristic
132C4×C22⋊C4—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#336—21no—
148#336—21no—
149#336—21no—
150#336D411yescharacteristic
151#336D411yescharacteristic
152#335—41no—
153#334—22no—
154#333—22no—
155#333—41no—
156C22×Q8—21no—
157C22×D4—21no—
158C22×D4C2×D411yescharacteristic
159C23×C4—21no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4#4111yescharacteristic
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—42no—
170C22×C8—42no—
171C22×C8—81no—
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C22⋊Q8—82no—
181#121—22no—
182#121—22no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—81no—
188C4×D4—81no—
189#117—82no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—41no—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42#4111yescharacteristic
222C2×C42C2411yescharacteristic
223#107—42no—
224#107—42no—
225#107—81no—
226#106—42no—
227#106—42no—
228#106—42no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—81no—
240C4⋊C8#4112yes—
241C4⋊C8#4112yes—
242#102—42no—
243#102—42no—
244#102—42no—
245#102—81no—
246C22⋊C8—22no—
247C22⋊C8—41no—
248C22⋊C8—42no—
249C22⋊C8—42no—
250C22⋊C8—42no—
251C22⋊C8—42no—
252C22⋊C8—81no—
253C22⋊C8—81no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—42no—
259C4×C8—42no—
260C4×C8—81no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—41no—
265#95—41no—
266#95—41no—
267#95—41no—
268C24—21no—
269C24#12111yescharacteristic
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—81no—
273C2×Q8—81no—
274C2×D4—24no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—81no—
281C2×D4—81no—
282C2×D4#13512yes—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
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—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
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#12111yescharacteristic
313C22×C4#12111yescharacteristic
314C22×C4#12111yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—84no—
348C2×C8#13512yes—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—81no—
371C4⋊C4—81no—
372C4⋊C4—81no—
373C4⋊C4—81no—
374C4⋊C4—82no—
375C4⋊C4#13512yes—
376C22⋊C4—24no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—42no—
380C22⋊C4—81no—
381C22⋊C4—81no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—41no—
391C42—41no—
392C42—82no—
393C42C2×#4111yescharacteristic
394C42C2×#4111yescharacteristic
395C42H(5,𝔽2)11yescharacteristic
396C422−1+411yescharacteristic
397C23—21no—
398C23—21no—
399C23—21no—
400C23—22no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23C2×#12111yescharacteristic
412Q8—161no—
413Q8—161no—
414D4—22no—
415D4—22no—
416D4—41no—
417D4—41no—
418D4—84no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4#47212yes—
465C2×C4#52911yescharacteristic
466C2×C4#53911yescharacteristic
467C2×C4#54611yescharacteristic
468C2×C4#54611yescharacteristic
469C2×C4#54711yescharacteristic
470C2×C4#54711yescharacteristic
471C2×C4C2×#13511yescharacteristic
472C2×C4C2×#13511yescharacteristic
473C2×C4C2×#13511yescharacteristic
474C2×C4#57511yescharacteristic
475C2×C4#57711yescharacteristic
476C2×C4#57711yescharacteristic
477C2×C4#57711yescharacteristic
478C2×C4#57811yescharacteristic
479C8—22no—
480C8—41no—
481C8—41no—
482C8—42no—
483C8—42no—
484C8—41no—
485C8—81no—
486C8—81no—
487C8—81no—
488C8—81no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—42no—
494C22—41no—
495C22—41no—
496C22—42no—
497C22—41no—
498C22—42no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22#242511yescharacteristic
504C22#303511yescharacteristic
505C22#304211yescharacteristic
506C22#305011yescharacteristic
507C22#306311yescharacteristic
508C22C2×#47211yescharacteristic
509C22#306911yescharacteristic
510C4—21no—
511C4—21no—
512C4—22no—
513C4—41no—
514C4—41no—
515C4—42no—
516C4—81no—
517C4—81no—
518C4—161no—
519C4—161no—
520C4—161no—
521C4—161no—
522C4#329611yescharacteristic
523C4#330111yescharacteristic
524C4#330211yescharacteristic
525C4#331711yescharacteristic
526C4#333611yescharacteristic
527C4#334011yescharacteristic
528C4#334011yescharacteristic
529C4#335011yescharacteristic
530C2—81no—
531C2—82no—
532C2#1912811yescharacteristic
533C2#1924911yescharacteristic
534C2#1939511yescharacteristic
535C2#1956111yescharacteristic
536C2#2520511yescharacteristic
537C2#2523311yescharacteristic
538C2#2526411yescharacteristic
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. #27877C2semidirect
5. #10651C2semidirect
6. #10625C2semidirect
7. #10593C2semidirect
8. #10480C2semidirect
9. #9613C2semidirect
10. #9567C2semidirect
11. #9215C2semidirect
12. #9128C2semidirect
13. #8831C2semidirect
14. #8644C2semidirect
15. #8038C2semidirect

Maximal subgroups

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

subgroupindex
#278772
#276262
#247012
#230832
#106512
#106252
#105932
#104802
#96132
#95672
#92152
#91282
#88312
#86442
#80382