Finite Group Databasebeta

← #4414123

Subgroups — #4414123

Order 512.

3722 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2817C211yesmaximal, characteristic, complemented
2C2×#2848C212yesmaximal
3C2×#2280C211yesmaximal, characteristic
4#16400C214yesmaximal, complemented
5#16391C214yesmaximal
6#16264C214yesmaximal, complemented
7#14907C214yesmaximal, complemented
8#14841C214yesmaximal, complemented
9#14763C214yesmaximal, complemented
10#14674C211yesmaximal, characteristic, complemented
11#14660C212yesmaximal, complemented
12C25×C4C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×C22⋊Q8C2212yes—
15C22×C22⋊Q8C2212yes—
16C22×#117C2212yes—
17#2848—216no—
18#2817—28no—
19C2×#400C2214yescomplemented
20C2×#398C2214yescomplemented
21C2×#397C2214yes—
22C2×#395C2214yescomplemented
23C2×#389C2214yescomplemented
24C2×#385C2214yes—
25C2×#385C2214yes—
26C2×#385C2214yes—
27#2280—28no—
28C2×#381C2211yescharacteristic
29C2×#381C2211yescharacteristic
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2214yes—
34C2×#379C2214yes—
35C2×#379C2212yes—
36C2×C4×C22⋊C4C2214yes—
37#1446C2214yescomplemented
38#1445C2214yescomplemented
39#1444C2214yescomplemented
40#1442C2214yescomplemented
41#1441C2214yescomplemented
42#1441C2218yescomplemented
43#1441C2214yescomplemented
44#1441C2214yescomplemented
45#1440C2218yes—
46#1440C2218yes—
47#1440C2218yes—
48#1439C2218yes—
49#1439C2214yes—
50#1439C2218yes—
51#1436C2218yescomplemented
52#1434C2214yescomplemented
53C26C2311yescharacteristic
54C23×Q8C2312yes—
55C24×C4—24no—
56C24×C4—24no—
57C24×C4—44no—
58C24×C4—44no—
59C24×C4D418yes—
60C24×C4C2311yescharacteristic
61C24×C4C2312yes—
62C24×C4C2312yes—
63C24×C4C2311yescharacteristic
64C2×C22⋊Q8—28no—
65C2×C22⋊Q8—28no—
66C2×C22⋊Q8—28no—
67C2×C22⋊Q8—28no—
68C2×C22⋊Q8—28no—
69C2×C22⋊Q8—48no—
70C2×C22⋊Q8—48no—
71C2×#117—28no—
72C2×#117—44no—
73C2×#117—44no—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2318yes—
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2312yes—
92C22×C42C2314yes—
93#400—216no—
94#398—48no—
95#397—216no—
96#395—216no—
97#389—48no—
98#385—216no—
99#385—216no—
100#385—216no—
101#381—24no—
102#381D418yes—
103#380—28no—
104#380—28no—
105#380—28no—
106#380—216no—
107#379—216no—
108#379—28no—
109C4×C22⋊C4—216no—
110C2×#95C2318yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2318yes—
118C2×#95C2318yes—
119C2×#95C2318yes—
120C2×#95C2318yes—
121C2×#95C2318yes—
122C2×#95C2314yes—
123C25—22no—
124C25—22no—
125C25—28no—
126C25—42no—
127C25—42no—
128C25—44no—
129C25C2×D414yes—
130C25C2411yescharacteristic
131C25C2412yes—
132C22×Q8—24no—
133C22×Q8—24no—
134C22×Q8—24no—
135C22×Q8—44no—
136C22×Q8—44no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—232no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—28no—
147C23×C4—28no—
148C23×C4—416no—
149C23×C4—416no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—432no—
155C23×C4—432no—
156C23×C4C2×D414yes—
157C23×C4C2×D418yes—
158C23×C4C2×D418yes—
159C23×C4C2×D414yes—
160C23×C4C2×D414yes—
161C23×C4#4114yes—
162C23×C4C2414yes—
163C23×C4C2414yes—
164C23×C4C2414yes—
165C23×C4C2414yes—
166C23×C4C2414yes—
167C23×C4C2414yes—
168C23×C4C2412yes—
169C23×C4C2411yescharacteristic
170C23×C4C2411yescharacteristic
171C22⋊Q8—432no—
172C22⋊Q8—432no—
173C22⋊Q8—432no—
174#117—432no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—216no—
188C2×C4⋊C4—216no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—48no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—216no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—216no—
210C2×C22⋊C4—48no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—48no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—48no—
219C2×C22⋊C4—48no—
220C2×C22⋊C4C2×D418yes—
221C2×C42—28no—
222C2×C42—28no—
223C2×C42—44no—
224#95—28no—
225#95—28no—
226#95—28no—
227#95—28no—
228#95—216no—
229#95—28no—
230#95—48no—
231#95—48no—
232#95—48no—
233#95—48no—
234#95—44no—
235#95—48no—
236#95C2×D4116yes—
237C24—22no—
238C24—22no—
239C24—24no—
240C24—24no—
241C24—22no—
242C24—22no—
243C24—24no—
244C24—24no—
245C24—24no—
246C24—24no—
247C24—24no—
248C24—28no—
249C24—216no—
250C24—44no—
251C24—44no—
252C24—48no—
253C24—416no—
254C24—416no—
255C24—416no—
256C24—48no—
257C24—48no—
258C24—432no—
259C24—416no—
260C24C22×D412yes—
261C24C22×D412yes—
262C24C22×D414yes—
263C24C2×#4112yes—
264C242−1+418yes—
265C24C2511yescharacteristic
266C2×Q8—48no—
267C2×Q8—48no—
268C2×Q8—48no—
269C2×Q8—88no—
270C2×Q8—88no—
271C22×C4—216no—
272C22×C4—216no—
273C22×C4—28no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—28no—
283C22×C4—28no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—28no—
287C22×C4—28no—
288C22×C4—216no—
289C22×C4—216no—
290C22×C4—216no—
291C22×C4—216no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—28no—
304C22×C4—28no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—28no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—28no—
316C22×C4—28no—
317C22×C4—28no—
318C22×C4—24no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—48no—
324C22×C4—48no—
325C22×C4—416no—
326C22×C4—416no—
327C22×C4—416no—
328C22×C4—416no—
329C22×C4—4128no—
330C22×C4#121116yes—
331C22×C4C22×D418yes—
332C22×C4C22×D414yes—
333C22×C4C22×D414yes—
334C22×C4C22×D412yes—
335C22×C4C22×D412yes—
336C22×C4C2×#4114yes—
337C22×C4H(5,𝔽2)14yes—
338C22×C4H(5,𝔽2)14yes—
339C4⋊C4—416no—
340C4⋊C4—48no—
341C4⋊C4—48no—
342C4⋊C4—416no—
343C4⋊C4—48no—
344C4⋊C4—416no—
345C4⋊C4—416no—
346C4⋊C4—416no—
347C4⋊C4—88no—
348C22⋊C4—48no—
349C22⋊C4—432no—
350C22⋊C4—416no—
351C22⋊C4—416no—
352C22⋊C4—48no—
353C22⋊C4—416no—
354C22⋊C4—416no—
355C22⋊C4—416no—
356C22⋊C4—416no—
357C42—416no—
358C23—24no—
359C23—24no—
360C23—24no—
361C23—22no—
362C23—22no—
363C23—24no—
364C23—24no—
365C23—24no—
366C23—24no—
367C23—24no—
368C23—24no—
369C23—28no—
370C23—28no—
371C23—24no—
372C23—28no—
373C23—28no—
374C23—216no—
375C23—216no—
376C23—416no—
377C23—432no—
378C23—416no—
379C23—432no—
380C23—416no—
381C23—464no—
382C23—464no—
383C23—44no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23—48no—
388C23—44no—
389C23—44no—
390C23—48no—
391C23—48no—
392C23C2×#12114yes—
393C23#53618yes—
394C23C23×D411yescharacteristic
395C23C23×D411yescharacteristic
396C23C22×#4111yescharacteristic
397C23C2×H(5,𝔽2)12yes—
398C23C2×H(5,𝔽2)12yes—
399C23C2×2−1+414yes—
400C23#58514yes—
401Q8—816no—
402C2×C4—232no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—28no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—216no—
417C2×C4—416no—
418C2×C4—432no—
419C2×C4—432no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—48no—
423C2×C4—44no—
424C2×C4—44no—
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—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—48no—
437C2×C4—48no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—48no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—48no—
444C2×C4—48no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—48no—
448C2×C4—48no—
449C2×C4—48no—
450C2×C4C2×#12118yes—
451C2×C4#53414yes—
452C2×C4#53414yes—
453C22—28no—
454C22—28no—
455C22—24no—
456C22—28no—
457C22—28no—
458C22—464no—
459C22—44no—
460C22—42no—
461C22—44no—
462C22—44no—
463C22—42no—
464C22—48no—
465C22—416no—
466C22—48no—
467C22—416no—
468C22—44no—
469C22—44no—
470C22C22×#12111yescharacteristic
471C22C2×#53411yescharacteristic
472C22C2×#53411yescharacteristic
473C22C2×#53612yes—
474C22#344912yes—
475C22#348412yes—
476C22#348914yes—
477C22#349012yes—
478C22#349514yes—
479C22#352914yes—
480C22#353814yes—
481C22#354118yes—
482C4—24no—
483C4—24no—
484C4—28no—
485C4—48no—
486C4—84no—
487C4—84no—
488C4—84no—
489C4—84no—
490C4—84no—
491C2—48no—
492C2—44no—
493C2#3498111yescharacteristic
494C2#3530612yes—
495C2#3665211yescharacteristic
496C2#3743812yes—
497C2#3745311yescharacteristic
498C2#4545314yes—
499C2#4547214yes—
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×#2817C2semidirect
4. #16400C2semidirect
6. #16264C2semidirect
7. #14907C2semidirect
8. #14841C2semidirect
9. #14763C2semidirect
10. #14674C2semidirect
11. #14660C2semidirect
19. C2×#400C22semidirect
20. C2×#398C22semidirect
22. C2×#395C22semidirect
23. C2×#389C22semidirect
37. #1446C22semidirect
38. #1445C22semidirect
39. #1444C22semidirect
40. #1442C22semidirect
41. #1441C22semidirect
42. #1441C22semidirect
43. #1441C22semidirect
44. #1441C22semidirect
51. #1436C22semidirect
52. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28172
C2×#28482
C2×#22802
#164002
#163912
#162642
#149072
#148412
#147632
#146742
#146602