Finite Group Databasebeta

← #470110

Subgroups — #470110

Order 512.

1028 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1947C211yesmaximal, characteristic, complemented
2C2×#1926C211yesmaximal, characteristic
3#15244C211yesmaximal, characteristic
4#13936C211yesmaximal, characteristic, complemented
5#11357C211yesmaximal, characteristic, complemented
6#11325C212yesmaximal, complemented
7#11268C212yesmaximal, complemented
8#11255C211yesmaximal, characteristic, complemented
9#11189C212yesmaximal
10#11123C212yesmaximal
11#11076C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2211yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#390C2212yes—
20C2×#384C2211yescharacteristic
21C2×#382C2212yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23#1947—24no—
24#1926—24no—
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33#1443—22no—
34#1439—21no—
35#1439—21no—
36#1439—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40#1434—21no—
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⋊C4C2311yescharacteristic
51C22×C4⋊C4C2312yes—
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—24no—
62C2×#107—42no—
63C2×#107—42no—
64C2×#106—22no—
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#382—42no—
83#382—42no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4D414yes—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95D411yescharacteristic
96C2×#95D411yescharacteristic
97C2×#95D412yes—
98#340—22no—
99#340—24no—
100#340—42no—
101#340—42no—
102#340—42no—
103#336—24no—
104#336—24no—
105#336—22no—
106C25—21no—
107C22×D4—21no—
108C22×D4—21no—
109C22×D4—21no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—21no—
115C22×D4—21no—
116C22×D4#4112yes—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2411yescharacteristic
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8#4112yes—
143C22×C8#4114yes—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
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—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—81no—
172C2×C4⋊C4—81no—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—82no—
183C2×C22⋊C4—82no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4—82no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—24no—
195C2×C42—82no—
196C2×C42C2×D412yes—
197C2×C42#4112yes—
198#107—48no—
199#106—44no—
200C4⋊C8—28no—
201#102—28no—
202#102—44no—
203#102—48no—
204#95—24no—
205#95—22no—
206#95—22no—
207#95—42no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—42no—
212#95—81no—
213#95—82no—
214#95—82no—
215#95—82no—
216C24—22no—
217C24—41no—
218C24—42no—
219C24—41no—
220C24—42no—
221C24—41no—
222C24—41no—
223C24—42no—
224C24—41no—
225C24—42no—
226C24—41no—
227C24C22×D411yescharacteristic
228C2×D4—22no—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—22no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—41no—
244C2×D4—42no—
245C2×D4—42no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
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—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
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—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
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#12112yes—
296C22×C4#12112yes—
297C22×C4#13512yes—
298C22×C4#13712yes—
299C22×C4C22×D411yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C22×C4C2×#4112yes—
303C22×C4C2×#4111yescharacteristic
304C22×C4H(5,𝔽2)12yes—
305C22×C4H(5,𝔽2)12yes—
306C22×C42−1+412yes—
307C22×C42−1+411yescharacteristic
308C22×C42−1+411yescharacteristic
309C2×C8—22no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—42no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—42no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—84no—
345C4⋊C4—84no—
346C22⋊C4—84no—
347C22⋊C4—88no—
348C22⋊C4—88no—
349C22⋊C4—84no—
350C42—22no—
351C42—22no—
352C42—24no—
353C42—42no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—84no—
358C23—42no—
359C23—44no—
360C23—42no—
361C23—42no—
362C23—44no—
363C23—82no—
364C23—81no—
365C23—82no—
366C23—82no—
367C23—82no—
368C23—82no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—82no—
373C23—82no—
374C23—82no—
375C23—82no—
376C23—82no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23C2×#12111yescharacteristic
382C23#53512yes—
383C23#54712yes—
384C23#54711yescharacteristic
385C23#54811yescharacteristic
386C23C2×#13511yescharacteristic
387C23C2×#13711yescharacteristic
388C23#57512yes—
389C23#57612yes—
390C23#57611yescharacteristic
391C23#57711yescharacteristic
392D4—42no—
393D4—42no—
394D4—44no—
395D4—81no—
396D4—81no—
397D4—82no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
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—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—81no—
436C2×C4—82no—
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—81no—
447C2×C4—81no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4#46414yes—
451C2×C4#52511yescharacteristic
452C2×C4#52511yescharacteristic
453C2×C4#53511yescharacteristic
454C2×C4#53511yescharacteristic
455C2×C4#53611yescharacteristic
456C2×C4#53611yescharacteristic
457C2×C4#53912yes—
458C2×C4#53912yes—
459C2×C4#54012yes—
460C2×C4#54212yes—
461C2×C4#54212yes—
462C2×C4#55112yes—
463C2×C4#55211yescharacteristic
464C2×C4#55212yes—
465C2×C4#55312yes—
466C2×C4#55611yescharacteristic
467C2×C4#56312yes—
468C2×C4#56312yes—
469C8—44no—
470C8—42no—
471C8—81no—
472C22—82no—
473C22—82no—
474C22—82no—
475C22—84no—
476C22—84no—
477C22—161no—
478C22—161no—
479C22—162no—
480C22—162no—
481C22—161no—
482C22—161no—
483C22#241311yescharacteristic
484C22#241511yescharacteristic
485C22#261911yescharacteristic
486C22#271812yes—
487C22#272312yes—
488C22C2×#46411yescharacteristic
489C22#304011yescharacteristic
490C22#304111yescharacteristic
491C22#304511yescharacteristic
492C22#321212yes—
493C22#321612yes—
494C22#322012yes—
495C22#322412yes—
496C22#333912yes—
497C22#334011yescharacteristic
498C22#334612yes—
499C22#334712yes—
500C22#334712yes—
501C22#335112yes—
502C22#335711yescharacteristic
503C22#336111yescharacteristic
504C22#337012yes—
505C22#337211yescharacteristic
506C4—24no—
507C4—42no—
508C4—82no—
509C4—82no—
510C4—82no—
511C4—82no—
512C4—161no—
513C4—162no—
514C4—162no—
515C4#259811yescharacteristic
516C4#263211yescharacteristic
517C4#263311yescharacteristic
518C4#273812yes—
519C4#277712yes—
520C4#282111yescharacteristic
521C2—162no—
522C2#1667611yescharacteristic
523C2#2458311yescharacteristic
524C2#2459911yescharacteristic
525C2#2467911yescharacteristic
526C2#2469311yescharacteristic
527C2#2979711yescharacteristic
528C2#2980211yescharacteristic
529C2#3229312yes—
530C2#3229512yes—
531C2#3244012yes—
532C2#3245012yes—
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×#1947C2semidirect
4. #13936C2semidirect
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×#19472
C2×#19262
#152442
#139362
#113572
#113252
#112682
#112552
#111892
#111232
#110762