Finite Group Databasebeta

← #470109

Subgroups — #470109

Order 512.

1042 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1948C211yesmaximal, characteristic, complemented
2C2×#1925C211yesmaximal, characteristic
3#15244C211yesmaximal, characteristic
4#13921C211yesmaximal, characteristic, complemented
5#11357C211yesmaximal, characteristic, complemented
6#11325C212yesmaximal, complemented
7#11268C212yesmaximal, complemented
8#11255C211yesmaximal, characteristic, complemented
9#11189C212yesmaximal
10#11121C212yesmaximal
11#11076C211yesmaximal, characteristic
12C22×#107C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2211yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#390C2212yes—
20C2×#384C2211yescharacteristic
21C2×#383—21no—
22C2×#382C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24#1948—24no—
25#1925—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34#1443—22no—
35#1439—21no—
36#1439—21no—
37#1439—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41#1434—21no—
42C4×#95C2211yescharacteristic
43C23×D4C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C42—21no—
59C22×C42C2312yes—
60C22×C42C2311yescharacteristic
61C2×#107—22no—
62C2×#107—42no—
63C2×#106—24no—
64C2×#106—42no—
65C2×#106—42no—
66C2×C4⋊C8—24no—
67C2×C4⋊C8—22no—
68C2×#102—22no—
69C2×#102—22no—
70C2×#102—22no—
71C2×#102—24no—
72C2×#102—24no—
73C2×#102—22no—
74C2×#102—42no—
75C2×#102—42no—
76#390—22no—
77#390—22no—
78#390—42no—
79#390—42no—
80#384—22no—
81#384—22no—
82#383—44no—
83#382—42no—
84#382—42no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4D414yes—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95D411yescharacteristic
97C2×#95D411yescharacteristic
98C2×#95D412yes—
99#340—22no—
100#340—24no—
101#340—42no—
102#340—42no—
103#340—42no—
104#336—24no—
105#336—24no—
106#336—22no—
107C25—21no—
108C22×D4—21no—
109C22×D4—21no—
110C22×D4—21no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4#4112yes—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2411yescharacteristic
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8#4112yes—
144C22×C8#4114yes—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
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—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4—81no—
173C2×C4⋊C4—81no—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D412yes—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—82no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4—82no—
186C2×C22⋊C4—82no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—24no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42C2×D412yes—
199C2×C42#4112yes—
200#107—44no—
201#106—48no—
202C4⋊C8—28no—
203#102—28no—
204#102—44no—
205#102—48no—
206#95—24no—
207#95—22no—
208#95—22no—
209#95—42no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—42no—
216#95—81no—
217#95—82no—
218C24—22no—
219C24—41no—
220C24—42no—
221C24—41no—
222C24—42no—
223C24—41no—
224C24—41no—
225C24—42no—
226C24—41no—
227C24—41no—
228C24—42no—
229C24C22×D411yescharacteristic
230C2×D4—22no—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—22no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—41no—
246C2×D4—42no—
247C2×D4—42no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
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—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—41no—
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#12112yes—
300C22×C4#12112yes—
301C22×C4#13512yes—
302C22×C4AGL(1,ℤ/8)12yes—
303C22×C4C22×D411yescharacteristic
304C22×C4C2×#4111yescharacteristic
305C22×C4C2×#4111yescharacteristic
306C22×C4C2×#4112yes—
307C22×C4C2×#4111yescharacteristic
308C22×C4H(5,𝔽2)12yes—
309C22×C4H(5,𝔽2)12yes—
310C22×C42−1+412yes—
311C22×C42−1+411yescharacteristic
312C22×C42−1+411yescharacteristic
313C2×C8—22no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—42no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—84no—
349C4⋊C4—84no—
350C22⋊C4—84no—
351C22⋊C4—88no—
352C22⋊C4—88no—
353C22⋊C4—84no—
354C42—22no—
355C42—22no—
356C42—24no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—48no—
362C23—42no—
363C23—42no—
364C23—44no—
365C23—42no—
366C23—44no—
367C23—82no—
368C23—81no—
369C23—82no—
370C23—82no—
371C23—82no—
372C23—81no—
373C23—82no—
374C23—81no—
375C23—81no—
376C23—82no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23C2×#12111yescharacteristic
386C23#53512yes—
387C23#54712yes—
388C23#54711yescharacteristic
389C23#54811yescharacteristic
390C23C2×#13511yescharacteristic
391C23C2×AGL(1,ℤ/8)11yescharacteristic
392C23#57512yes—
393C23#57611yescharacteristic
394C23#57712yes—
395C23#57711yescharacteristic
396D4—42no—
397D4—42no—
398D4—44no—
399D4—81no—
400D4—81no—
401D4—82no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
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—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—81no—
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—81no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4#46314yes—
457C2×C4#52511yescharacteristic
458C2×C4#52511yescharacteristic
459C2×C4#53511yescharacteristic
460C2×C4#53511yescharacteristic
461C2×C4#53611yescharacteristic
462C2×C4#53611yescharacteristic
463C2×C4#53912yes—
464C2×C4#53912yes—
465C2×C4#54012yes—
466C2×C4#54212yes—
467C2×C4#54212yes—
468C2×C4#55112yes—
469C2×C4#55211yescharacteristic
470C2×C4#55212yes—
471C2×C4#55312yes—
472C2×C4#55611yescharacteristic
473C2×C4#56312yes—
474C2×C4#56312yes—
475C8—44no—
476C8—42no—
477C8—81no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22—84no—
482C22—84no—
483C22—161no—
484C22—161no—
485C22—162no—
486C22—162no—
487C22—161no—
488C22—161no—
489C22#241311yescharacteristic
490C22#241511yescharacteristic
491C22#261911yescharacteristic
492C22#271812yes—
493C22#272312yes—
494C22C2×#46311yescharacteristic
495C22#304011yescharacteristic
496C22#304311yescharacteristic
497C22#304511yescharacteristic
498C22#321212yes—
499C22#321412yes—
500C22#322212yes—
501C22#322412yes—
502C22#333712yes—
503C22#334111yescharacteristic
504C22#334512yes—
505C22#334812yes—
506C22#335012yes—
507C22#335411yescharacteristic
508C22#335512yes—
509C22#336011yescharacteristic
510C22#336611yescharacteristic
511C22#337612yes—
512C4—24no—
513C4—42no—
514C4—82no—
515C4—82no—
516C4—82no—
517C4—82no—
518C4—161no—
519C4—162no—
520C4—162no—
521C4#259811yescharacteristic
522C4#263211yescharacteristic
523C4#263311yescharacteristic
524C4#273812yes—
525C4#277712yes—
526C4#282111yescharacteristic
527C2—162no—
528C2#1667611yescharacteristic
529C2#2458111yescharacteristic
530C2#2459711yescharacteristic
531C2#2467711yescharacteristic
532C2#2469511yescharacteristic
533C2#2979511yescharacteristic
534C2#2980311yescharacteristic
535C2#3228912yes—
536C2#3229912yes—
537C2#3243812yes—
538C2#3245212yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1948C2semidirect
4. #13921C2semidirect
5. #11357C2semidirect
6. #11325C2semidirect
7. #11268C2semidirect
8. #11255C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19482
C2×#19252
#152442
#139212
#113572
#113252
#112682
#112552
#111892
#111212
#110762