Finite Group Databasebeta

← #460471

Subgroups — #460471

Order 512.

1262 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#2085C212yesmaximal, complemented
3#14340C211yesmaximal, characteristic
4#11384C212yesmaximal, complemented
5#11325C212yesmaximal, complemented
6#11207C212yesmaximal, complemented
7#11096C212yesmaximal
8#11095C212yesmaximal
9#10713C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×#107C2212yes—
13C22×#106C2212yes—
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16#2628—28no—
17C2×#400—21no—
18C2×#399—21no—
19C2×#395—21no—
20C2×#393—21no—
21C2×#389C2212yes—
22C2×#384C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#380C2212yes—
26C2×#380C2212yes—
27#2085—28no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1441—21no—
37#1441—21no—
38#1439—21no—
39#1439—21no—
40C4⋊#95C2212yes—
41C23×Q8—21no—
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×C4⋊Q8—24no—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—41no—
49C2×#117—24no—
50C2×#117—24no—
51C2×#117—41no—
52C2×#117—41no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4D411yescharacteristic
66C22×C22⋊C4D411yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×#107—24no—
70C2×#107—42no—
71C2×#107—42no—
72C2×#106—24no—
73C2×#106—42no—
74C2×#106—42no—
75C2×C22⋊C8—41no—
76C2×C22⋊C8—41no—
77C2×C22⋊C8—41no—
78C2×C22⋊C8—41no—
79C2×C22⋊C8Q814yescomplemented
80C2×C22⋊C8Q814yescomplemented
81#400—28no—
82#399—28no—
83#395—28no—
84#393—28no—
85#389—24no—
86#389D418yes—
87#384—41no—
88#384—41no—
89#382—24no—
90#382—24no—
91#380—24no—
92#380—24no—
93#380—42no—
94#380—42no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104#340—24no—
105#340—24no—
106#340—42no—
107#340—42no—
108#340—42no—
109#340—42no—
110#336—24no—
111#336—24no—
112C25C2×D411yescharacteristic
113C22×Q8—24no—
114C22×Q8—42no—
115C22×Q8—42no—
116C22×Q8—82no—
117C22×Q8—82no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×Q812yes—
136C23×C4C2×Q812yes—
137C23×C4C2411yescharacteristic
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8C2×Q814yes—
151C22×C8C2×Q814yes—
152C4⋊Q8—48no—
153C4⋊Q8—48no—
154#117—48no—
155#117—48no—
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—24no—
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—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4C2×D414yes—
195C2×C4⋊C4C2×D414yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—82no—
205C2×C22⋊C4—82no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4—82no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42C2×D414yes—
215C2×C42#4114yes—
216#107—48no—
217#106—48no—
218C22⋊C8—44no—
219C22⋊C8—44no—
220#95—28no—
221#95—24no—
222#95—24no—
223#95—28no—
224#95—44no—
225#95—42no—
226#95—42no—
227#95—82no—
228#95—82no—
229C24—21no—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—21no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24—22no—
239C24C4⋊Q812yes—
240C24C22×D411yescharacteristic
241C2×Q8—48no—
242C2×Q8—88no—
243C2×Q8—88no—
244C2×Q8—84no—
245C2×Q8—84no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
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×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4#12012yes—
313C22×C4#12012yes—
314C22×C4#12312yes—
315C22×C4#12312yes—
316C22×C4C4⋊Q812yes—
317C22×C4C22×D412yes—
318C22×C4C22×Q811yescharacteristic
319C22×C4C22×Q811yescharacteristic
320C22×C4C2×#4112yes—
321C22×C4H(5,𝔽2)11yescharacteristic
322C22×C4H(5,𝔽2)12yes—
323C22×C4H(5,𝔽2)11yescharacteristic
324C22×C4H(5,𝔽2)11yescharacteristic
325C22×C42−1+411yescharacteristic
326C22×C42−1+412yes—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—48no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—88no—
361C4⋊C4—84no—
362C4⋊C4—84no—
363C22⋊C4—44no—
364C22⋊C4—44no—
365C22⋊C4—88no—
366C22⋊C4—88no—
367C42—24no—
368C42—24no—
369C42—24no—
370C42—24no—
371C23—24no—
372C23—24no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—41no—
377C23—42no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23C2×#12011yescharacteristic
395C23C2×#12311yescharacteristic
396C23C2×C4⋊Q811yescharacteristic
397C23#53411yescharacteristic
398C23#53511yescharacteristic
399C23#53611yescharacteristic
400C23#53711yescharacteristic
401C23#57612yes—
402C23#57712yes—
403C23#57712yes—
404C23#57812yes—
405Q8—816no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
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×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4#53512yes—
466C2×C4#53712yes—
467C2×C4#54112yes—
468C2×C4#54212yes—
469C2×C4#54311yescharacteristic
470C2×C4#54311yescharacteristic
471C2×C4#54411yescharacteristic
472C2×C4#54411yescharacteristic
473C2×C4#54612yes—
474C2×C4#54612yes—
475C2×C4D4×Q812yes—
476C2×C4D4×Q812yes—
477C2×C4#55412yes—
478C2×C4#55412yes—
479C2×C4#56212yes—
480C2×C4#56212yes—
481C8—44no—
482C8—44no—
483C22—44no—
484C22—42no—
485C22—42no—
486C22—44no—
487C22—42no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—82no—
493C22—82no—
494C22#240511yescharacteristic
495C22#243411yescharacteristic
496C22#250211yescharacteristic
497C22#261511yescharacteristic
498C22#262511yescharacteristic
499C22#265911yescharacteristic
500C22#281611yescharacteristic
501C22#301212yes—
502C22#301312yes—
503C22#309612yes—
504C22#309712yes—
505C22#316312yes—
506C22#316412yes—
507C22#319911yescharacteristic
508C22#320011yescharacteristic
509C22#320011yescharacteristic
510C22#320211yescharacteristic
511C22#322412yes—
512C22#322512yes—
513C22#324112yes—
514C22#324212yes—
515C22#326212yes—
516C22#326312yes—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—84no—
523C4—84no—
524C4—162no—
525C4—162no—
526C4#269212yes—
527C4#269212yes—
528C4#270412yes—
529C4#270412yes—
530C2—82no—
531C2#1636511yescharacteristic
532C2#2401511yescharacteristic
533C2#2401611yescharacteristic
534C2#2582212yes—
535C2#2795912yes—
536C2#2967712yes—
537C2#2994712yes—
538C2#3030011yescharacteristic
539C2#3030211yescharacteristic
540C2#3277911yescharacteristic
541C2#3278011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2085C2semidirect
4. #11384C2semidirect
5. #11325C2semidirect
6. #11207C2semidirect
79. C2×C22⋊C8Q8semidirect
80. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#20852
#143402
#113842
#113252
#112072
#110962
#110952
#107132