Finite Group Databasebeta

← #4869159

Subgroups — #4869159

Order 512.

2672 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2598C212yesmaximal, complemented
2#17168C211yesmaximal, characteristic, complemented
3#17102C211yesmaximal, characteristic
4#17088C212yesmaximal, complemented
5#16702C214yesmaximal, complemented
6#16512C214yesmaximal, complemented
7#16414C214yesmaximal, complemented
8#16266C214yesmaximal, complemented
9#15375C214yesmaximal, complemented
10#14806C214yesmaximal, complemented
11#14343C211yesmaximal, characteristic, complemented
12C22×C4×D4C2212yes—
13C22×#117C2211yescharacteristic
14#2598—216no—
15C2×#400C2212yes—
16C2×#399C2212yes—
17C2×#394C2212yes—
18C2×#394C2214yes—
19C2×#394C2214yes—
20C2×#394C2214yes—
21C2×#392C2212yes—
22C2×#392C2214yes—
23C2×#392C2214yes—
24C2×#388C2214yes—
25C2×#386C2214yes—
26C2×#386C2214yes—
27C2×#386C2214yes—
28C2×#386C2214yes—
29C2×#385C2214yes—
30C2×#383C2212yes—
31C2×#380C2212yes—
32#1445C2214yes—
33#1445C2214yes—
34#1443C2214yes—
35#1443C2214yes—
36#1442C2214yes—
37#1441C2214yes—
38#1441C2214yes—
39#1441C2214yes—
40#1440C2212yes—
41#1440C2211yescharacteristic
42#1440C2211yescharacteristic
43#1440C2214yes—
44#1440C2214yes—
45#1440C2214yes—
46#1440C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1438C2214yes—
54#1438C2214yes—
55C4⋊#95C2214yes—
56C22⋊#95C2214yes—
57C22⋊#95C2214yes—
58C23×D4C2312yes—
59C23×D4C2314yes—
60C23×D4C2314yes—
61C24×C4C2311yescharacteristic
62C24×C4C2314yes—
63C2×C4×D4—28no—
64C2×C4×D4—44no—
65C2×C4×D4—44no—
66C2×#117—44no—
67C2×#117D418yescomplemented
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2314yes—
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2312yes—
84C22×C22⋊C4C2312yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2311yescharacteristic
87C22×C22⋊C4C2311yescharacteristic
88C22×C42C2312yes—
89#400—28no—
90#399—28no—
91#394—28no—
92#394—48no—
93#394—48no—
94#394—48no—
95#392—28no—
96#392—48no—
97#392—48no—
98#388—216no—
99#386—216no—
100#386—216no—
101#386—48no—
102#386—48no—
103#385—216no—
104#383—28no—
105#380—44no—
106C2×#95C2312yes—
107C2×#95C2312yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C25C2411yescharacteristic
126C25C2414yes—
127C22×D4—28no—
128C22×D4—28no—
129C22×D4—28no—
130C22×D4—28no—
131C22×D4—28no—
132C22×D4—24no—
133C22×D4—42no—
134C22×D4—42no—
135C22×D4—44no—
136C22×D4—44no—
137C22×D4—44no—
138C22×D4—44no—
139C22×D4—44no—
140C22×D4—44no—
141C22×D4—44no—
142C22×D4—44no—
143C22×D4—44no—
144C22×D4—44no—
145C22×D4—44no—
146C23×C4—24no—
147C23×C4—28no—
148C23×C4—44no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4C2×D414yes—
156C23×C4C2414yes—
157C23×C4C2414yes—
158C23×C4C2414yes—
159C23×C4C2414yes—
160C23×C4C2414yes—
161C23×C4C2411yescharacteristic
162C23×C4C2412yes—
163C23×C4C2412yes—
164C23×C4C2411yescharacteristic
165C4×D4—432no—
166#117—416no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4C2×D418yes—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4C2×D414yes—
222C2×C22⋊C4C2×D414yes—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42C2×D418yes—
226#95—28no—
227#95—28no—
228#95—24no—
229#95—24no—
230#95—28no—
231#95—28no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—44no—
245C24—24no—
246C24—24no—
247C24—24no—
248C24—22no—
249C24—24no—
250C24—24no—
251C24—24no—
252C24—24no—
253C24—24no—
254C24—24no—
255C24—24no—
256C24—24no—
257C24—24no—
258C24—24no—
259C24—24no—
260C24—24no—
261C24—24no—
262C24—24no—
263C24C22×D412yes—
264C24H(5,𝔽2)18yes—
265C24C2511yescharacteristic
266C2×D4—416no—
267C2×D4—48no—
268C2×D4—48no—
269C2×D4—416no—
270C2×D4—48no—
271C2×D4—416no—
272C2×D4—416no—
273C2×D4—416no—
274C2×D4—416no—
275C2×D4—88no—
276C2×D4—88no—
277C2×D4—88no—
278C2×D4—88no—
279C2×D4—88no—
280C2×D4—88no—
281C2×D4—88no—
282C2×D4—88no—
283C2×D4—88no—
284C2×D4—88no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—28no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—48no—
333C22×C4—416no—
334C22×C4—416no—
335C22×C4—88no—
336C22×C4—88no—
337C22×C4—88no—
338C22×C4—88no—
339C22×C4—88no—
340C22×C4—88no—
341C22×C4C22×D414yes—
342C22×C4C22×D412yes—
343C22×C4C22×D414yes—
344C22×C4C22×D412yes—
345C22×C4H(5,𝔽2)14yes—
346C4⋊C4—48no—
347C4⋊C4—48no—
348C4⋊C4—88no—
349C4⋊C4—88no—
350C4⋊C4—88no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—416no—
354C22⋊C4—416no—
355C22⋊C4—416no—
356C22⋊C4—88no—
357C22⋊C4—88no—
358C22⋊C4—88no—
359C22⋊C4—88no—
360C22⋊C4—88no—
361C22⋊C4—88no—
362C22⋊C4—88no—
363C22⋊C4—88no—
364C22⋊C4—84no—
365C22⋊C4—84no—
366C42—216no—
367C23—22no—
368C23—24no—
369C23—28no—
370C23—28no—
371C23—28no—
372C23—28no—
373C23—28no—
374C23—28no—
375C23—28no—
376C23—28no—
377C23—44no—
378C23—44no—
379C23—44no—
380C23—44no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23—44no—
393C23—44no—
394C23—44no—
395C23—42no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—42no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23C23×D411yescharacteristic
414C23C2×H(5,𝔽2)12yes—
415C23C2×H(5,𝔽2)14yes—
416C23#58514yes—
417C23#58514yes—
418D4—432no—
419D4—832no—
420D4—832no—
421C2×C4—24no—
422C2×C4—28no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—28no—
428C2×C4—28no—
429C2×C4—416no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
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—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—832no—
473C2×C4#53414yes—
474C22—48no—
475C22—48no—
476C22—48no—
477C22—48no—
478C22—48no—
479C22—48no—
480C22—42no—
481C22—48no—
482C22—44no—
483C22—48no—
484C22—84no—
485C22—84no—
486C22—84no—
487C22—84no—
488C22—84no—
489C22—84no—
490C22—84no—
491C22—84no—
492C22—84no—
493C22—84no—
494C22C2×#53411yescharacteristic
495C22#344912yes—
496C22#348814yes—
497C22#353314yes—
498C22#353314yes—
499C22#353414yes—
500C22#353712yes—
501C22#354012yes—
502C22#354214yes—
503C22#356014yes—
504C22#356614yes—
505C4—24no—
506C4—42no—
507C4—44no—
508C4—44no—
509C4—84no—
510C4—84no—
511C4—84no—
512C4—84no—
513C4—84no—
514C2—82no—
515C2—88no—
516C2#3726912yes—
517C2#3745411yescharacteristic
518C2#4199514yes—
519C2#5509614yes—
520C2#5551614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#17102C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2598C2semidirect
2. #17168C2semidirect
4. #17088C2semidirect
5. #16702C2semidirect
6. #16512C2semidirect
7. #16414C2semidirect
8. #16266C2semidirect
9. #15375C2semidirect
10. #14806C2semidirect
11. #14343C2semidirect
67. C2×#117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25982
#171682
#171022
#170882
#167022
#165122
#164142
#162662
#153752
#148062
#143432