Finite Group Databasebeta

← #362433

Subgroups — #362433

Order 512.

1324 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1834C211yesmaximal, characteristic, complemented
2C2×#1807C211yesmaximal, characteristic
3C2×C8×C22⋊C4C211yesmaximal, characteristic
4#13876C212yesmaximal
5#11323C212yesmaximal, complemented
6#11095C212yesmaximal
7#10908C212yesmaximal, complemented
8#10861C211yesmaximal, characteristic, complemented
9#10820C211yesmaximal, characteristic, complemented
10#10721C211yesmaximal, characteristic
11C4×#340C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#106C2212yes—
14C22×#106C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#388—22no—
19C2×#384C2212yes—
20C2×#380C2212yes—
21C2×#380C2212yes—
22C2×C4×C4⋊C4C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#1834—24no—
25#1807—24no—
26C8×C22⋊C4C418yescomplemented
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1442—22no—
38#1439—22no—
39#1436—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C24×C4C2311yescharacteristic
43C23×C8—22no—
44C23×C8—24no—
45C23×C8—22no—
46C23×C8—42no—
47C23×C8C2×C414yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4D411yescharacteristic
61C22×C22⋊C4D411yescharacteristic
62C22×C42—22no—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×#106—24no—
66C2×#106—24no—
67C2×#106—24no—
68C2×#106—42no—
69C2×#106—42no—
70C2×#106—42no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8C2×C414yes—
73C2×C4×C8—22no—
74C2×C4×C8—22no—
75C2×C4×C8—42no—
76C2×C4×C8C2×C414yes—
77C2×C4×C8C2×C414yes—
78#388—48no—
79#384—24no—
80#384—42no—
81#380—24no—
82#380—24no—
83#380—42no—
84#380—42no—
85C4×C4⋊C4—24no—
86C4×C4⋊C4—42no—
87C4×C22⋊C4C2×C414yes—
88C4×C22⋊C4Q814yes—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95D411yescharacteristic
97C2×#95D411yescharacteristic
98#340—22no—
99#340—22no—
100#340—24no—
101#340—22no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340D414yes—
106#336C2×C414yes—
107#336C2×C414yes—
108C25C2×D411yescharacteristic
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4C22×C412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×Q812yes—
128C23×C4C2411yescharacteristic
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—28no—
142C22×C8—28no—
143C22×C8—44no—
144C22×C8—48no—
145C22×C8—48no—
146C22×C8C22×C412yes—
147C22×C8C22×C412yes—
148C22×C8C22×C412yes—
149C22×C8C22×C412yes—
150C22×C8C2×D412yes—
151C22×C8#4114yes—
152C22×C8#4114yes—
153C22×C8#4112yes—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4C2×D414yes—
183C2×C4⋊C4#4114yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—82no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4C4⋊C414yes—
193C2×C22⋊C4C4⋊C414yes—
194C2×C22⋊C4SD1618yes—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—44no—
202C2×C42—44no—
203C2×C42—82no—
204C2×C42C22×C412yes—
205C2×C42C22×C412yes—
206C2×C42C2×Q812yes—
207C2×C42C2×Q812yes—
208#106—48no—
209#106—48no—
210C22⋊C8—28no—
211C4×C8—28no—
212C4×C8—44no—
213#95—44no—
214#95—44no—
215#95—44no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—82no—
220#95C4⋊C414yes—
221#95C4⋊C414yes—
222C24—22no—
223C24—21no—
224C24—22no—
225C24—21no—
226C24—21no—
227C24—21no—
228C24—21no—
229C24—22no—
230C24—21no—
231C24C2×C4⋊C412yes—
232C24C2×SD1614yes—
233C24C22×D411yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4C2×C4⋊C412yes—
299C22×C4C2×C4⋊C412yes—
300C22×C4C2×C4⋊C412yes—
301C22×C4C2×C4⋊C412yes—
302C22×C4C2×C4⋊C412yes—
303C22×C4C2×SD1614yes—
304C22×C4C2×SD1614yes—
305C22×C4#13512yes—
306C22×C4#13512yes—
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×Q811yescharacteristic
310C22×C4C2×#4112yes—
311C22×C4C2×#4112yes—
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4112yes—
314C22×C4H(5,𝔽2)12yes—
315C22×C4H(5,𝔽2)11yescharacteristic
316C22×C42−1+411yescharacteristic
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—416no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8#11714yes—
336C2×C8#11714yes—
337C2×C8C4×D414yes—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—88no—
355C4⋊C4—88no—
356C22⋊C4—28no—
357C22⋊C4—88no—
358C22⋊C4—88no—
359C22⋊C4#106116yescomplemented
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—24no—
367C42—88no—
368C23—24no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—24no—
373C23—24no—
374C23—22no—
375C23—22no—
376C23—24no—
377C23—24no—
378C23—24no—
379C23—22no—
380C23—41no—
381C23—41no—
382C23—42no—
383C23—42no—
384C23—41no—
385C23—41no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23C2×#10614yes—
391C23C22×C4⋊C411yescharacteristic
392C23#53212yes—
393C23D4211yescharacteristic
394C23#54612yes—
395C23#54711yescharacteristic
396C23C22×SD1612yes—
397C23C2×#13511yescharacteristic
398C23C2×#13511yescharacteristic
399C23#57611yescharacteristic
400C23#57712yes—
401C23#57811yescharacteristic
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4C2×#10614yes—
461C2×C4C2×#10614yes—
462C2×C4#42714yes—
463C2×C4C2×#11711yescharacteristic
464C2×C4C2×#11711yescharacteristic
465C2×C4C2×C4×D411yescharacteristic
466C2×C4C4×#4111yescharacteristic
467C2×C4#51811yescharacteristic
468C2×C4#52011yescharacteristic
469C2×C4#52912yes—
470C2×C4#53212yes—
471C2×C4#53312yes—
472C2×C4#53812yes—
473C2×C4#53912yes—
474C2×C4#54612yes—
475C2×C4D4×Q811yescharacteristic
476C2×C4#55112yes—
477C2×C4#55212yes—
478C2×C4#55411yescharacteristic
479C2×C4#55412yes—
480C2×C4#55512yes—
481C8—24no—
482C8—24no—
483C8—24no—
484C8—42no—
485C22—24no—
486C22—24no—
487C22—24no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—44no—
492C22—44no—
493C22—42no—
494C22—44no—
495C22—44no—
496C22#228511yescharacteristic
497C22D4×C4⋊C411yescharacteristic
498C22#234911yescharacteristic
499C22#255312yes—
500C22#255512yes—
501C22C22×#10611yescharacteristic
502C22C2×#42711yescharacteristic
503C22#291211yescharacteristic
504C22#291311yescharacteristic
505C22#316712yes—
506C22#316712yes—
507C22#317812yes—
508C22#317912yes—
509C22#328011yescharacteristic
510C22D4×SD1612yes—
511C22#328511yescharacteristic
512C22#329512yes—
513C22#329812yes—
514C22#330312yes—
515C22#330412yes—
516C22#333512yes—
517C22#333611yescharacteristic
518C22#333911yescharacteristic
519C4—24no—
520C4—24no—
521C4—24no—
522C4—42no—
523C4—42no—
524C4—84no—
525C4—84no—
526C4—162no—
527C4—162no—
528C4#229311yescharacteristic
529C4#231811yescharacteristic
530C4#234211yescharacteristic
531C4#234611yescharacteristic
532C4#255612yes—
533C4#257412yes—
534C2—44no—
535C2#1391511yescharacteristic
536C2#2081611yescharacteristic
537C2#2082211yescharacteristic
538C2D4×#10611yescharacteristic
539C2#2265611yescharacteristic
540C2#2270311yescharacteristic
541C2#2270811yescharacteristic
542C2#2884512yes—
543C2#2884712yes—
544C2#2890512yes—
545C2#2890912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#1834C2semidirect
5. #11323C2semidirect
7. #10908C2semidirect
8. #10861C2semidirect
9. #10820C2semidirect
11. C4×#340C2semidirect
26. C8×C22⋊C4C4semidirect
359. C22⋊C4#106semidirect

Maximal subgroups

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

subgroupindex
C2×#18342
C2×#18072
C2×C8×C22⋊C42
#138762
#113232
#110952
#109082
#108612
#108202
#107212
C4×#3402