Finite Group Databasebeta

← #411936

Subgroups — #411936

Order 512.

1490 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic, complemented
2C2×#2027C212yesmaximal
3#20646C212yesmaximal
4C4×#390C211yesmaximal, characteristic
5#11392C212yesmaximal, complemented
6#11296C212yesmaximal, complemented
7#11054C212yesmaximal
8#10847C211yesmaximal, characteristic, complemented
9#10839C212yesmaximal, complemented
10C2×#544C2212yes—
11C22×C4×D4C2211yescharacteristic
12C2×#423C2212yes—
13C2×#418C2214yes—
14#2857—22no—
15#2659—22no—
16#2552—22no—
17#2551—22no—
18#2550—24no—
19#2316C2212yes—
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22#2285C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4—22no—
25C2×C43C2211yescharacteristic
26#2027—28no—
27#1804—24no—
28#1802—22no—
29#1802—22no—
30C2×#343C2212yes—
31C2×#343C2212yes—
32#1736C2214yes—
33C2×#339C2214yes—
34C2×#339C2214yes—
35#1437—22no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C23×D4C2311yescharacteristic
39C24×C4—21no—
40C22×M16C2314yes—
41#544—24no—
42#544—24no—
43#544—44no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—24no—
46C2×#125—24no—
47C2×C22⋊Q8—24no—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—24no—
52C2×C4×D4Q814yescomplemented
53C2×#117C2314yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C42—21no—
61C22×C42—21no—
62C22×C42D412yes—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66#423—24no—
67#423—24no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70#418—28no—
71#418—44no—
72C2×#102—24no—
73C2×#102—24no—
74#398—28no—
75#395—28no—
76#391—24no—
77#390—22no—
78#390—22no—
79#390D414yes—
80#390Q814yescomplemented
81#389—24no—
82#389—24no—
83#389—24no—
84#389—24no—
85#384—22no—
86#384D412yes—
87#384D412yes—
88#382D414yes—
89#380—24no—
90#380—22no—
91#380—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C22⋊C4—48no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95D412yes—
100C43—24no—
101#343—24no—
102#343—24no—
103#340—24no—
104#340—24no—
105#339—28no—
106#339—44no—
107C25—21no—
108C22×Q8—24no—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—21no—
112C22×D4—21no—
113C22×D4—21no—
114C22×D4—22no—
115C22×D4—22no—
116C22×D4C2×D412yes—
117C22×D4C2×Q812yes—
118C22×D4C2×Q812yes—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—24no—
123C23×C4—44no—
124C23×C4—44no—
125C23×C4—41no—
126C23×C4—41no—
127C23×C4—42no—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2411yescharacteristic
133C2×M16—24no—
134C2×M16—24no—
135C2×M16—24no—
136C2×M16—24no—
137C2×M16—24no—
138C2×M16—24no—
139C22×C8—24no—
140C4⋊Q8—48no—
141C4⋊Q8—48no—
142C4⋊Q8—84no—
143#125—48no—
144#125—48no—
145C22⋊Q8—48no—
146C22⋊Q8—48no—
147C4×D4—28no—
148C4×D4—28no—
149C4×D4—44no—
150C4×D4—44no—
151#117—24no—
152#117—24no—
153#117—24no—
154#117—24no—
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—24no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C2×D414yes—
171C2×C4⋊C4C2×D414yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×Q812yes—
174C2×C4⋊C4C2×Q812yes—
175C2×C4⋊C4#4112yes—
176C2×C4⋊C4#4112yes—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—24no—
192C2×C42—22no—
193C2×C42—24no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42—44no—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×Q812yes—
207C2×C42C2×Q812yes—
208C2×C42#4114yes—
209C2×C42#4112yes—
210C4⋊C8—44no—
211C4⋊C8—44no—
212#102—48no—
213#102—48no—
214#95—28no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—44no—
219#95—44no—
220C24—22no—
221C24—22no—
222C24—22no—
223C24—42no—
224C24—42no—
225C24—42no—
226C24—42no—
227C24—41no—
228C24—41no—
229C24—41no—
230C24—41no—
231C24C22×D411yescharacteristic
232C2×Q8—48no—
233C2×Q8—84no—
234C2×Q8—84no—
235C2×D4—24no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4C4⋊Q814yescomplemented
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—416no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—88no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4#12011yescharacteristic
293C22×C4#12012yes—
294C22×C4#12011yescharacteristic
295C22×C4#12114yes—
296C22×C4C22⋊Q812yes—
297C22×C4C22⋊Q814yes—
298C22×C4C22⋊Q812yes—
299C22×C4#12412yes—
300C22×C4#12412yes—
301C22×C4C22×D412yes—
302C22×C4C22×D411yescharacteristic
303C22×C4C22×Q811yescharacteristic
304C22×C4C22×Q811yescharacteristic
305C22×C4C2×#4112yes—
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4H(5,𝔽2)12yes—
310C22×C42−1+412yes—
311M16—44no—
312M16—44no—
313M16—44no—
314M16—44no—
315C2×C8—44no—
316C2×C8—44no—
317C2×C8—44no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—48no—
328C4⋊C4—416no—
329C4⋊C4—42no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—84no—
335C4⋊C4—84no—
336C22⋊C4—416no—
337C22⋊C4—416no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—48no—
341C42—24no—
342C42—24no—
343C42—48no—
344C42—48no—
345C42—44no—
346C42—44no—
347C42—42no—
348C42—42no—
349C42—44no—
350C42—82no—
351C42C22⋊Q814yes—
352C42C22⋊Q814yes—
353C42#12414yes—
354C42#12414yes—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—24no—
360C23—44no—
361C23—44no—
362C23—44no—
363C23—44no—
364C23—44no—
365C23—44no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—82no—
373C23—82no—
374C23—82no—
375C23—82no—
376C23—82no—
377C23—82no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23C2×#12011yescharacteristic
383C23C2×#12111yescharacteristic
384C23C2×C22⋊Q811yescharacteristic
385C23C2×C22⋊Q811yescharacteristic
386C23C2×#12411yescharacteristic
387C23#53511yescharacteristic
388C23#53611yescharacteristic
389Q8—168no—
390D4—28no—
391D4—44no—
392D4—44no—
393D4—44no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—88no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—88no—
439C2×C4#39314yes—
440C2×C4#39312yes—
441C2×C4#39312yes—
442C2×C4#45312yes—
443C2×C4#45412yes—
444C2×C4C2×C22⋊Q812yes—
445C2×C4C2×#12412yes—
446C2×C4C2×C4⋊Q811yescharacteristic
447C2×C4#53211yescharacteristic
448C2×C4#53311yescharacteristic
449C2×C4#53311yescharacteristic
450C2×C4#54212yes—
451C2×C4#54412yes—
452C2×C4D4212yes—
453C2×C4#54612yes—
454C2×C4#54712yes—
455C2×C4#54812yes—
456C2×C4D4×Q812yes—
457C2×C4#55412yes—
458C2×C4#55512yes—
459C2×C4#55612yes—
460C8—84no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—48no—
474C22—44no—
475C22—84no—
476C22—84no—
477C22—84no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22C2×#39311yescharacteristic
485C22#240811yescharacteristic
486C22#240911yescharacteristic
487C22#254711yescharacteristic
488C22#256311yescharacteristic
489C22#262811yescharacteristic
490C22#285711yescharacteristic
491C22C2×#45311yescharacteristic
492C22C2×#45411yescharacteristic
493C22#301811yescharacteristic
494C22#301811yescharacteristic
495C4—22no—
496C4—24no—
497C4—44no—
498C4—44no—
499C4—44no—
500C4—44no—
501C4—84no—
502C4—84no—
503C4—82no—
504C4#202714yescomplemented
505C4#243912yes—
506C4#244012yes—
507C4#249112yes—
508C4#249212yes—
509C2—21no—
510C2—21no—
511C2—21no—
512C2—21no—
513C2—84no—
514C2#1512511yescharacteristic
515C2C2×#202711yescharacteristic
516C2#2346111yescharacteristic
517C2#2422211yescharacteristic
518C2#2422311yescharacteristic
519C2#2431211yescharacteristic
520C2#2431311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35167C2semidirect
5. #11392C2semidirect
6. #11296C2semidirect
8. #10847C2semidirect
9. #10839C2semidirect
52. C2×C4×D4Q8semidirect
80. #390Q8semidirect
250. C2×D4C4⋊Q8semidirect
504. C4#2027semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#20272
#206462
C4×#3902
#113922
#112962
#110542
#108472
#108392