Finite Group Databasebeta

← #460500

Subgroups — #460500

Order 512.

1218 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#2101C212yesmaximal, complemented
3#14340C211yesmaximal, characteristic
4#11384C212yesmaximal, complemented
5#11268C212yesmaximal, complemented
6#11207C212yesmaximal, complemented
7#11200C212yesmaximal
8#11199C212yesmaximal
9#10713C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14#2628—28no—
15C2×#400—21no—
16C2×#399—21no—
17C2×#395—21no—
18C2×#393—21no—
19C2×#389C2212yes—
20C2×#384C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#380C2212yes—
24C2×#380C2212yes—
25#2101—28no—
26C2×#340C2212yes—
27C2×#340C2212yes—
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×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—41no—
72C2×C22⋊C8—41no—
73C2×C22⋊C8—41no—
74C2×C22⋊C8—41no—
75#400—28no—
76#399—28no—
77#395—28no—
78#393—28no—
79#389—24no—
80#389D418yes—
81#384—41no—
82#384—41no—
83#382—24no—
84#382—24no—
85#380—24no—
86#380—24no—
87#380—24no—
88#380—24no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—42no—
103#340—42no—
104#340—42no—
105#340—42no—
106#336—24no—
107#336—24no—
108C25C2×D411yescharacteristic
109C22×Q8—24no—
110C22×Q8—42no—
111C22×Q8—42no—
112C22×Q8—82no—
113C22×Q8—82no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4#4112yes—
132C23×C4#4112yes—
133C23×C4C2411yescharacteristic
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
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—
148C4⋊Q8—48no—
149C4⋊Q8—48no—
150#117—48no—
151#117—48no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
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—24no—
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—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4C2×D414yes—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—82no—
201C2×C22⋊C4—82no—
202C2×C22⋊C4—82no—
203C2×C22⋊C4—82no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42C2×D414yes—
211C2×C42#4114yes—
212C22⋊C8—44no—
213C22⋊C8—44no—
214#95—28no—
215#95—24no—
216#95—24no—
217#95—28no—
218#95—42no—
219#95—42no—
220#95—44no—
221#95—82no—
222#95—82no—
223C24—21no—
224C24—21no—
225C24—21no—
226C24—21no—
227C24—21no—
228C24—21no—
229C24—22no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24#12412yes—
234C24C22×D411yescharacteristic
235C2×Q8—48no—
236C2×Q8—88no—
237C2×Q8—88no—
238C2×Q8—84no—
239C2×Q8—84no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
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—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—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
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×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#12012yes—
307C22×C4#12012yes—
308C22×C4#12312yes—
309C22×C4#12312yes—
310C22×C4#12412yes—
311C22×C4C22×D412yes—
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4112yes—
315C22×C4H(5,𝔽2)11yescharacteristic
316C22×C4H(5,𝔽2)11yescharacteristic
317C22×C4H(5,𝔽2)11yescharacteristic
318C22×C42−1+411yescharacteristic
319C22×C42−1+412yes—
320C22×C42−1+412yes—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—48no—
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—84no—
356C4⋊C4—84no—
357C22⋊C4—44no—
358C22⋊C4—44no—
359C22⋊C4—88no—
360C22⋊C4—88no—
361C42—24no—
362C42—24no—
363C42—24no—
364C42—24no—
365C23—24no—
366C23—24no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—41no—
371C23—42no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23C2×#12011yescharacteristic
389C23C2×#12311yescharacteristic
390C23C2×#12411yescharacteristic
391C23#53411yescharacteristic
392C23#53611yescharacteristic
393C23#53611yescharacteristic
394C23#53711yescharacteristic
395C23#57612yes—
396C23#57712yes—
397C23#57712yes—
398C23#57812yes—
399Q8—816no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—82no—
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#53612yes—
460C2×C4#53712yes—
461C2×C4#53811yescharacteristic
462C2×C4#53811yescharacteristic
463C2×C4#53911yescharacteristic
464C2×C4#53911yescharacteristic
465C2×C4#54112yes—
466C2×C4#54212yes—
467C2×C4#54612yes—
468C2×C4#54612yes—
469C2×C4#54812yes—
470C2×C4#54812yes—
471C2×C4#55212yes—
472C2×C4#55212yes—
473C2×C4#56312yes—
474C2×C4#56312yes—
475C8—82no—
476C8—82no—
477C22—44no—
478C22—44no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—82no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—82no—
488C22#240511yescharacteristic
489C22#243611yescharacteristic
490C22#250311yescharacteristic
491C22#261611yescharacteristic
492C22#262511yescharacteristic
493C22#263911yescharacteristic
494C22#281711yescharacteristic
495C22#301212yes—
496C22#301312yes—
497C22#309612yes—
498C22#309712yes—
499C22#313612yes—
500C22#313712yes—
501C22#319911yescharacteristic
502C22#320011yescharacteristic
503C22#320011yescharacteristic
504C22#320211yescharacteristic
505C22#324312yes—
506C22#324312yes—
507C22#324412yes—
508C22#324412yes—
509C22#325712yes—
510C22#325812yes—
511C4—42no—
512C4—42no—
513C4—42no—
514C4—42no—
515C4—42no—
516C4—84no—
517C4—84no—
518C4—162no—
519C4—162no—
520C4#269012yes—
521C4#269012yes—
522C4#270212yes—
523C4#270212yes—
524C2—82no—
525C2#1636611yescharacteristic
526C2#2401511yescharacteristic
527C2#2401611yescharacteristic
528C2#2591612yes—
529C2#2797112yes—
530C2#2970212yes—
531C2#2994512yes—
532C2#3019211yescharacteristic
533C2#3019311yescharacteristic
534C2#3279211yescharacteristic
535C2#3279411yescharacteristic
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×#2101C2semidirect
4. #11384C2semidirect
5. #11268C2semidirect
6. #11207C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#21012
#143402
#113842
#112682
#112072
#112002
#111992
#107132