Finite Group Databasebeta

← #768861

Subgroups — #768861

Order 512.

3518 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2484C212yesmaximal, complemented
2C2×#2482C212yesmaximal
3C22×#396C211yesmaximal, characteristic
4C2×#2316C211yesmaximal, characteristic, complemented
5#15808C218yesmaximal, complemented
6#15765C218yesmaximal, complemented
7#14936C212yesmaximal, complemented
8#14734C212yesmaximal
9#13903C212yesmaximal
10#13880C212yesmaximal, complemented
11#13776C211yesmaximal, characteristic, complemented
12C22×#117C2212yes—
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19#2484—216no—
20#2482—216no—
21C2×#400C2214yescomplemented
22C2×#399C2214yes—
23C2×#399C2214yes—
24C2×#398C2214yescomplemented
25C2×#397C2214yes—
26C2×#396—28no—
27C2×#396C2218yes—
28#2316—28no—
29C2×#389C2214yescomplemented
30C2×#389C2214yescomplemented
31C2×#388C2218yes—
32C2×#387C2218yescomplemented
33C2×#385C2214yes—
34C2×#385C2214yes—
35C2×#384C2214yescomplemented
36C2×#381C2214yes—
37C2×#381C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#379C2214yes—
41C2×C4×C22⋊C4C2212yes—
42C22×#95C2211yescharacteristic
43#1445C2218yescomplemented
44#1445C2214yescomplemented
45#1444C2218yescomplemented
46#1444C2214yescomplemented
47#1439C2214yes—
48#1439C2214yes—
49#1439C2218yes—
50#1439C2214yes—
51#1437C2218yes—
52#1436C2218yescomplemented
53C4⋊#95C2214yescomplemented
54C22⋊#95C2212yes—
55C26C2311yescharacteristic
56C24×C4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C24×C4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C2×#117—28no—
63C2×#117—28no—
64C2×#117—28no—
65C22×C4⋊C4—28no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—28no—
69C22×C4⋊C4—44no—
70C22×C4⋊C4D418yes—
71C22×C4⋊C4C2314yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2318yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2318yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C22⋊C4—24no—
83C22×C22⋊C4—24no—
84C22×C22⋊C4—24no—
85C22×C22⋊C4—28no—
86C22×C22⋊C4—28no—
87C22×C22⋊C4—44no—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2318yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2318yes—
95C22×C42C2314yes—
96C22×C42C2314yes—
97#400—216no—
98#399—216no—
99#399—48no—
100#398—216no—
101#397—216no—
102#396—232no—
103#389—216no—
104#389—216no—
105#388—232no—
106#387—232no—
107#385—216no—
108#385—216no—
109#384—48no—
110#381—216no—
111#381D4116yes—
112#380—28no—
113#380—28no—
114#379—216no—
115C4×C22⋊C4—28no—
116C2×#95—24no—
117C2×#95C2314yes—
118C2×#95C2318yes—
119C2×#95C2318yes—
120C2×#95C2318yes—
121C2×#95C2318yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C25—22no—
130C25—24no—
131C25—24no—
132C25—24no—
133C25—24no—
134C25—42no—
135C25—42no—
136C25—42no—
137C25C2412yes—
138C25C2411yescharacteristic
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4C2×D414yes—
159C23×C4C2×D418yes—
160C23×C4C2×D414yes—
161C23×C4C2×D414yes—
162C23×C4#4118yes—
163C23×C4C2414yes—
164C23×C4C2414yes—
165C23×C4C2412yes—
166C23×C4C2412yes—
167C23×C4C2418yes—
168C23×C4C2412yes—
169C23×C4C2412yes—
170C23×C4C2412yes—
171C23×C4C2412yes—
172#117—432no—
173C2×C4⋊C4—232no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—216no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—216no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—216no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—48no—
196C2×C4⋊C4—416no—
197C2×C4⋊C4—416no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—48no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—48no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4C2×D4116yes—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—216no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—216no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—216no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—48no—
223C2×C22⋊C4—416no—
224C2×C22⋊C4—416no—
225C2×C22⋊C4—416no—
226C2×C22⋊C4—48no—
227C2×C22⋊C4—48no—
228C2×C22⋊C4C2×D4116yes—
229C2×C42—28no—
230C2×C42—28no—
231C2×C42—28no—
232C2×C42—28no—
233C2×C42—44no—
234C2×C42—44no—
235#95—28no—
236#95—28no—
237#95—28no—
238#95—216no—
239#95—216no—
240#95—216no—
241#95—216no—
242#95—28no—
243#95—44no—
244#95—44no—
245#95C2×D4116yes—
246#95C2×D4116yes—
247C24—22no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—28no—
252C24—22no—
253C24—24no—
254C24—24no—
255C24—24no—
256C24—22no—
257C24—24no—
258C24—24no—
259C24—24no—
260C24—416no—
261C24—416no—
262C24—416no—
263C24—44no—
264C24—44no—
265C24—44no—
266C24—44no—
267C24—44no—
268C24—44no—
269C24—416no—
270C24—88no—
271C24—88no—
272C24—88no—
273C24C22×D412yes—
274C24C22×D414yes—
275C24C2×#4114yes—
276C242−1+414yes—
277C242−1+414yes—
278C24C2511yescharacteristic
279C22×C4—28no—
280C22×C4—28no—
281C22×C4—232no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—28no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—28no—
290C22×C4—24no—
291C22×C4—28no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—28no—
295C22×C4—28no—
296C22×C4—28no—
297C22×C4—24no—
298C22×C4—28no—
299C22×C4—24no—
300C22×C4—28no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—28no—
313C22×C4—24no—
314C22×C4—28no—
315C22×C4—24no—
316C22×C4—28no—
317C22×C4—28no—
318C22×C4—24no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—416no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—416no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—416no—
334C22×C4—88no—
335C22×C4—88no—
336C22×C4#123116yes—
337C22×C4#127116yes—
338C22×C4C22×D4116yes—
339C22×C4C22×D414yes—
340C22×C4C22×D414yes—
341C22×C4C22×D414yes—
342C22×C4C22×D414yes—
343C22×C4C22×D414yes—
344C22×C4C2×#4114yes—
345C22×C4C2×#4114yes—
346C22×C4H(5,𝔽2)14yes—
347C4⋊C4—232no—
348C4⋊C4—432no—
349C4⋊C4—416no—
350C4⋊C4—416no—
351C4⋊C4—48no—
352C4⋊C4—48no—
353C4⋊C4—84no—
354C4⋊C4—84no—
355C4⋊C4—84no—
356C4⋊C4—84no—
357C4⋊C4—816no—
358C22⋊C4—232no—
359C22⋊C4—416no—
360C22⋊C4—432no—
361C22⋊C4—416no—
362C22⋊C4—416no—
363C22⋊C4—416no—
364C22⋊C4—816no—
365C42—416no—
366C42—416no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—28no—
372C23—28no—
373C23—28no—
374C23—28no—
375C23—28no—
376C23—28no—
377C23—24no—
378C23—28no—
379C23—28no—
380C23—28no—
381C23—416no—
382C23—416no—
383C23—416no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—48no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—48no—
392C23—44no—
393C23—44no—
394C23—48no—
395C23—44no—
396C23—48no—
397C23—864no—
398C23—88no—
399C23—88no—
400C23—88no—
401C23—88no—
402C23C2×#12314yes—
403C23C2×#12714yes—
404C23#53614yes—
405C23#547116yes—
406C23C23×D411yescharacteristic
407C23C23×D412yes—
408C23C22×#4112yes—
409C23C2×H(5,𝔽2)12yes—
410C23C2×2−1+412yes—
411C23C2×2−1+412yes—
412C23#58512yes—
413C23#58512yes—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—232no—
423C2×C4—24no—
424C2×C4—216no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—216no—
429C2×C4—28no—
430C2×C4—216no—
431C2×C4—416no—
432C2×C4—416no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—48no—
437C2×C4—48no—
438C2×C4—48no—
439C2×C4—48no—
440C2×C4—48no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—48no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—48no—
455C2×C4—48no—
456C2×C4—48no—
457C2×C4—48no—
458C2×C4—48no—
459C2×C4—88no—
460C2×C4—88no—
461C2×C4—88no—
462C2×C4C2×#12318yes—
463C2×C4C2×#12718yes—
464C22—24no—
465C22—24no—
466C22—24no—
467C22—24no—
468C22—24no—
469C22—232no—
470C22—24no—
471C22—44no—
472C22—44no—
473C22—44no—
474C22—416no—
475C22—832no—
476C22—82no—
477C22—82no—
478C22—88no—
479C22—82no—
480C22—82no—
481C22#250118yes—
482C22C22×#12311yescharacteristic
483C22C22×#12711yescharacteristic
484C22C2×#53611yescharacteristic
485C22C2×#54714yes—
486C22#348411yescharacteristic
487C22#348511yescharacteristic
488C22#348611yescharacteristic
489C22#348711yescharacteristic
490C22#348814yes—
491C22#348914yes—
492C22#349514yes—
493C22#350514yes—
494C22#353814yes—
495C22#356414yes—
496C4—28no—
497C4—28no—
498C4—44no—
499C4—416no—
500C4—44no—
501C4—84no—
502C4—82no—
503C4—84no—
504C4—82no—
505C2—28no—
506C2—84no—
507C2C2×#250111yescharacteristic
508C2#3535111yescharacteristic
509C2#3556511yescharacteristic
510C2#3563411yescharacteristic
511C2#3567211yescharacteristic
512C2#3670811yescharacteristic
513C2#3696211yescharacteristic
514C2#4393518yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2484C2semidirect
4. C2×#2316C2semidirect
5. #15808C2semidirect
6. #15765C2semidirect
7. #14936C2semidirect
10. #13880C2semidirect
11. #13776C2semidirect
21. C2×#400C22semidirect
24. C2×#398C22semidirect
29. C2×#389C22semidirect
30. C2×#389C22semidirect
32. C2×#387C22semidirect
35. C2×#384C22semidirect
43. #1445C22semidirect
44. #1445C22semidirect
45. #1444C22semidirect
46. #1444C22semidirect
52. #1436C22semidirect
53. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24842
C2×#24822
C22×#3962
C2×#23162
#158082
#157652
#149362
#147342
#139032
#138802
#137762