Finite Group Databasebeta

← #449868

Subgroups — #449868

Order 512.

1510 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1824C212yesmaximal, complemented
2#13872C212yesmaximal
3#11377C211yesmaximal, characteristic, complemented
4#11316C212yesmaximal, complemented
5#11256C211yesmaximal, characteristic, complemented
6#11096C212yesmaximal, complemented
7#11096C212yesmaximal, complemented
8#10964C211yesmaximal, characteristic
9#10730C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11C23×C4⋊C4C2212yes—
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C2×#386—22no—
17C2×#380C2212yes—
18C2×#380C2212yes—
19C2×#380C2212yes—
20#1824—28no—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2211yescharacteristic
28C2×#340C2211yescharacteristic
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31#1441—22no—
32#1439—22no—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35C22⋊#95—22no—
36C23×D4D411yescharacteristic
37C24×C4—22no—
38C24×C4C2312yes—
39C23×C8C2312yes—
40C23×C8C2311yescharacteristic
41C23×C8C2311yescharacteristic
42C2×C4×D4—24no—
43C2×C4×D4—24no—
44C2×C4×D4—41no—
45C2×C4×D4—41no—
46C22×C4⋊C4—24no—
47C22×C4⋊C4—24no—
48C22×C4⋊C4—24no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—42no—
52C22×C4⋊C4—42no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4D412yes—
62C22×C42C2311yescharacteristic
63C2×#107—22no—
64C2×#107—41no—
65C2×#107—41no—
66C2×#107—42no—
67C2×#107—41no—
68C2×C22⋊C8—24no—
69C2×C22⋊C8—24no—
70C2×C22⋊C8—24no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8—42no—
73C2×C22⋊C8—42no—
74#386—48no—
75#380—24no—
76#380—24no—
77#380—42no—
78#380—42no—
79#380—42no—
80#380—42no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88#340—24no—
89#340—24no—
90#340—24no—
91#340—24no—
92#340—41no—
93#340—42no—
94#340—42no—
95#340D414yescomplemented
96#336—41no—
97#336—41no—
98C25C2×D412yes—
99C22×D4—24no—
100C22×D4—24no—
101C22×D4—41no—
102C22×D4—41no—
103C22×D4Q1614yes—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—44no—
121C23×C4—44no—
122C23×C4—44no—
123C23×C4—44no—
124C23×C4—44no—
125C23×C4—82no—
126C23×C4—82no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4#4114yes—
131C23×C4#4114yes—
132C23×C4C2411yescharacteristic
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8C2×D412yes—
149C22×C8#4112yes—
150C4×D4—44no—
151C4×D4—44no—
152C4×D4—48no—
153C2×C4⋊C4—216no—
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—24no—
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—48no—
178C2×C4⋊C4—48no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—48no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4C2×D414yes—
185C2×C4⋊C4#4114yes—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—82no—
197C2×C22⋊C4—82no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—21no—
202#107—44no—
203#107—44no—
204C22⋊C8—48no—
205C22⋊C8—48no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—82no—
212#95—82no—
213C24—22no—
214C24—22no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24—22no—
219C24—22no—
220C24—22no—
221C24—22no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24#12314yes—
226C24C2×Q1614yes—
227C24AGL(1,ℤ/8)14yes—
228C24C22×D411yescharacteristic
229C2×D4—216no—
230C2×D4—44no—
231C2×D4—44no—
232C2×D4—48no—
233C2×D4—44no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—42no—
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—416no—
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—88no—
307C22×C4—88no—
308C22×C4—88no—
309C22×C4—88no—
310C22×C4—88no—
311C22×C4—88no—
312C22×C4#12314yes—
313C22×C4C2×Q1612yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C2×#4112yes—
316C22×C4C2×#4112yes—
317C22×C4C2×#4111yescharacteristic
318C22×C4C2×#4112yes—
319C22×C4H(5,𝔽2)11yescharacteristic
320C22×C4H(5,𝔽2)11yescharacteristic
321C22×C4H(5,𝔽2)12yes—
322C22×C4H(5,𝔽2)11yescharacteristic
323C22×C42−1+411yescharacteristic
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
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—416no—
347C4⋊C4—42no—
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—
359C22⋊C4—48no—
360C22⋊C4—88no—
361C22⋊C4—88no—
362C22⋊C4—88no—
363C42—41no—
364C42—41no—
365C42—42no—
366C23—24no—
367C23—24no—
368C23—24no—
369C23—24no—
370C23—24no—
371C23—24no—
372C23—24no—
373C23—24no—
374C23—24no—
375C23—24no—
376C23—24no—
377C23—24no—
378C23—24no—
379C23—24no—
380C23—24no—
381C23—24no—
382C23—24no—
383C23—24no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23#48418yes—
401C23C2×#12312yes—
402C23#53411yescharacteristic
403C23D4211yescharacteristic
404C23#54612yes—
405C23#54711yescharacteristic
406C23C22×Q1611yescharacteristic
407C23C2×AGL(1,ℤ/8)12yes—
408C23#57511yescharacteristic
409C23#57611yescharacteristic
410C23#57712yes—
411C23#57811yescharacteristic
412D4—416no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—832no—
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—82no—
466C2×C4—82no—
467C2×C4#52512yes—
468C2×C4#52912yes—
469C2×C4#53711yescharacteristic
470C2×C4#53811yescharacteristic
471C2×C4#53812yes—
472C2×C4#53912yes—
473C2×C4#54012yes—
474C2×C4#54612yes—
475C2×C4#55011yescharacteristic
476C2×C4#55111yescharacteristic
477C2×C4#55112yes—
478C2×C4#55111yescharacteristic
479C2×C4#55112yes—
480C2×C4#55112yes—
481C2×C4#55212yes—
482C2×C4#55511yescharacteristic
483C2×C4#56011yescharacteristic
484C2×C4#56211yescharacteristic
485C8—42no—
486C8—42no—
487C8—82no—
488C22—28no—
489C22—28no—
490C22—28no—
491C22—44no—
492C22—44no—
493C22—44no—
494C22—44no—
495C22—44no—
496C22—44no—
497C22—44no—
498C22—44no—
499C22—44no—
500C22—44no—
501C22—44no—
502C22—44no—
503C22#241812yes—
504C22#241912yes—
505C22#248311yescharacteristic
506C22#268211yescharacteristic
507C22#269911yescharacteristic
508C22C2×#48412yes—
509C22#309412yes—
510C22#309612yes—
511C22#309712yes—
512C22#318911yescharacteristic
513C22#319211yescharacteristic
514C22#320311yescharacteristic
515C22#320411yescharacteristic
516C22#326512yes—
517C22#327311yescharacteristic
518C22D4×Q1611yescharacteristic
519C22#329912yes—
520C22#330812yes—
521C22#331812yes—
522C22#332312yes—
523C22#333811yescharacteristic
524C22#334512yes—
525C22#335211yescharacteristic
526C4—44no—
527C4—42no—
528C4—84no—
529C4—84no—
530C4—81no—
531C4—162no—
532C4—162no—
533C4#247112yes—
534C4#257211yescharacteristic
535C4#260911yescharacteristic
536C4#269311yescharacteristic
537C4#274312yes—
538C4#276011yescharacteristic
539C2—48no—
540C2#1595211yescharacteristic
541C2#2486712yes—
542C2#2487812yes—
543C2#2493912yes—
544C2#2495012yes—
545C2#2760311yescharacteristic
546C2#2760711yescharacteristic
547C2#3119011yescharacteristic
548C2#3120411yescharacteristic
549C2#3172211yescharacteristic
550C2#3173611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1824C2semidirect
3. #11377C2semidirect
4. #11316C2semidirect
5. #11256C2semidirect
6. #11096C2semidirect
7. #11096C2semidirect
9. #10730C2semidirect
95. #340D4semidirect

Maximal subgroups

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

subgroupindex
C2×#18242
#138722
#113772
#113162
#112562
#110962
#109642
#107302