Finite Group Databasebeta

← #760299

Subgroups — #760299

Order 512.

4426 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2483C211yesmaximal, characteristic
2C22×#385C212yesmaximal
3C2×#2280C211yesmaximal, characteristic
4#16675C214yesmaximal, complemented
5#14935C214yesmaximal, complemented
6#14907C214yesmaximal, complemented
7#14728C214yesmaximal, complemented
8#14127C211yesmaximal, characteristic
9#14043C214yesmaximal, complemented
10#13972C214yesmaximal, complemented
11#13869C212yesmaximal
12C25×C4C2211yescharacteristic
13C22×#117C2212yes—
14C23×C4⋊C4C2212yes—
15C23×C4⋊C4C2212yes—
16C23×C22⋊C4C2212yes—
17C23×C22⋊C4C2212yes—
18#2483—28no—
19C2×#398C2214yescomplemented
20C2×#397C2214yes—
21C2×#389C2214yescomplemented
22C2×#385—216no—
23C2×#385C2214yes—
24C2×#385C2214yes—
25C2×#385C2214yes—
26C2×#385C2214yes—
27C2×#385C2214yes—
28C2×#385C2214yes—
29#2280—28no—
30C2×#381C2212yes—
31C2×#381C2214yes—
32C2×#381C2214yes—
33C2×#380C2214yes—
34C2×#379C2211yescharacteristic
35C2×#379C2211yescharacteristic
36C2×C4×C22⋊C4C2214yes—
37C22×#95C2212yes—
38C22×#95C2212yes—
39#1444C2218yescomplemented
40#1443C2218yescomplemented
41#1442C2218yescomplemented
42#1441C2218yescomplemented
43#1441C2214yescomplemented
44#1441C2218yescomplemented
45#1440C2218yes—
46#1439C2214yes—
47#1436C2218yescomplemented
48#1434C2214yescomplemented
49C22⋊#95C2214yes—
50C22⋊#95C2214yes—
51C22⋊#95C2214yes—
52C22⋊#95C2214yes—
53C26C2311yescharacteristic
54C24×C4—24no—
55C24×C4—24no—
56C24×C4—44no—
57C24×C4—44no—
58C24×C4D418yes—
59C24×C4C2312yes—
60C24×C4C2311yescharacteristic
61C24×C4C2312yes—
62C24×C4C2312yes—
63C24×C4C2312yes—
64C24×C4C2312yes—
65C24×C4C2312yes—
66C24×C4C2311yescharacteristic
67C2×#117—28no—
68C2×#117—44no—
69C2×#117—44no—
70C22×C4⋊C4—28no—
71C22×C4⋊C4—28no—
72C22×C4⋊C4—48no—
73C22×C4⋊C4—48no—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C22⋊C4—28no—
82C22×C22⋊C4—216no—
83C22×C22⋊C4—28no—
84C22×C22⋊C4—28no—
85C22×C22⋊C4—28no—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C42C2314yes—
95#398—48no—
96#397—216no—
97#389—48no—
98#385—216no—
99#385—216no—
100#385—216no—
101#385—216no—
102#385—216no—
103#385—216no—
104#381—216no—
105#381—28no—
106#381—216no—
107#380—216no—
108#379—24no—
109#379D418yes—
110C4×C22⋊C4—216no—
111C2×#95—28no—
112C2×#95—28no—
113C2×#95C2314yes—
114C2×#95C2318yes—
115C2×#95C2318yes—
116C2×#95C2318yes—
117C2×#95C2318yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2318yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2318yes—
125C2×#95C2318yes—
126C25—28no—
127C25—28no—
128C25—42no—
129C25—42no—
130C25C2×D414yes—
131C25C2×D414yes—
132C25#4114yes—
133C25C2411yescharacteristic
134C25C2412yes—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—232no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—28no—
149C23×C4—28no—
150C23×C4—28no—
151C23×C4—28no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—28no—
157C23×C4—28no—
158C23×C4—28no—
159C23×C4—28no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—416no—
163C23×C4—416no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—432no—
173C23×C4—432no—
174C23×C4C2×D414yes—
175C23×C4C2×D418yes—
176C23×C4C2×D418yes—
177C23×C4C2×D414yes—
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2414yes—
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187#117—432no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—216no—
194C2×C4⋊C4—216no—
195C2×C4⋊C4—216no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—48no—
199C2×C4⋊C4—48no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—432no—
203C2×C4⋊C4—432no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C22⋊C4—264no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—432no—
229C2×C22⋊C4C2×D4116yes—
230C2×C22⋊C4C2×D418yes—
231C2×C22⋊C4#41116yes—
232C2×C42—28no—
233C2×C42—28no—
234C2×C42—44no—
235#95—28no—
236#95—28no—
237#95—28no—
238#95—28no—
239#95—216no—
240#95—28no—
241#95—216no—
242#95—216no—
243#95—216no—
244#95—216no—
245#95—48no—
246#95—44no—
247#95—48no—
248C24—24no—
249C24—24no—
250C24—24no—
251C24—24no—
252C24—216no—
253C24—216no—
254C24—232no—
255C24—232no—
256C24—24no—
257C24—22no—
258C24—22no—
259C24—24no—
260C24—44no—
261C24—44no—
262C24—48no—
263C24—48no—
264C24—416no—
265C24—432no—
266C24—416no—
267C24#121116yes—
268C24C22×D412yes—
269C24C22×D412yes—
270C24C22×D414yes—
271C24C22×D414yes—
272C24C2×#4112yes—
273C24C2×#4114yes—
274C24C2×#4118yes—
275C242−1+418yes—
276C24C2511yescharacteristic
277C22×C4—216no—
278C22×C4—216no—
279C22×C4—28no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—28no—
287C22×C4—28no—
288C22×C4—216no—
289C22×C4—216no—
290C22×C4—216no—
291C22×C4—216no—
292C22×C4—28no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—28no—
297C22×C4—28no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—28no—
302C22×C4—24no—
303C22×C4—28no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—216no—
310C22×C4—216no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—28no—
317C22×C4—28no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—48no—
327C22×C4—48no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—432no—
331C22×C4—432no—
332C22×C4—416no—
333C22×C4—416no—
334C22×C4—416no—
335C22×C4—416no—
336C22×C4—4128no—
337C22×C4—48no—
338C22×C4—48no—
339C22×C4—48no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4—416no—
350C22×C4—416no—
351C22×C4—88no—
352C22×C4—88no—
353C22×C4C22×D418yes—
354C22×C4C22×D414yes—
355C22×C4C22×D412yes—
356C22×C4C22×D412yes—
357C22×C4C2×#4118yes—
358C22×C4C2×#4118yes—
359C22×C4H(5,𝔽2)14yes—
360C22×C4H(5,𝔽2)14yes—
361C4⋊C4—416no—
362C4⋊C4—416no—
363C4⋊C4—416no—
364C4⋊C4—416no—
365C4⋊C4—416no—
366C4⋊C4—416no—
367C4⋊C4—88no—
368C22⋊C4—232no—
369C22⋊C4—232no—
370C22⋊C4—232no—
371C22⋊C4—48no—
372C22⋊C4—48no—
373C22⋊C4—416no—
374C22⋊C4—416no—
375C22⋊C4—416no—
376C42—416no—
377C23—24no—
378C23—24no—
379C23—24no—
380C23—24no—
381C23—28no—
382C23—28no—
383C23—216no—
384C23—28no—
385C23—28no—
386C23—216no—
387C23—216no—
388C23—216no—
389C23—24no—
390C23—264no—
391C23—22no—
392C23—22no—
393C23—24no—
394C23—28no—
395C23—28no—
396C23—24no—
397C23—24no—
398C23—416no—
399C23—432no—
400C23—416no—
401C23—416no—
402C23—464no—
403C23—464no—
404C23—44no—
405C23—48no—
406C23—44no—
407C23C2×#12114yes—
408C23C2×#12118yes—
409C23#52518yes—
410C23#53618yes—
411C23#547116yes—
412C23#552116yes—
413C23C23×D411yescharacteristic
414C23C23×D411yescharacteristic
415C23C22×#4111yescharacteristic
416C23C22×#4114yes—
417C23C2×H(5,𝔽2)12yes—
418C23C2×H(5,𝔽2)12yes—
419C23C2×2−1+414yes—
420C2×C4—232no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—28no—
431C2×C4—28no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—216no—
435C2×C4—216no—
436C2×C4—28no—
437C2×C4—216no—
438C2×C4—432no—
439C2×C4—432no—
440C2×C4—432no—
441C2×C4—432no—
442C2×C4—416no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—48no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—48no—
452C2×C4—48no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—48no—
461C2×C4—48no—
462C2×C4—48no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—48no—
466C2×C4—48no—
467C2×C4—48no—
468C2×C4—44no—
469C2×C4—816no—
470C2×C4—816no—
471C2×C4#53414yes—
472C2×C4#53414yes—
473C22—28no—
474C22—28no—
475C22—28no—
476C22—28no—
477C22—232no—
478C22—28no—
479C22—28no—
480C22—232no—
481C22—24no—
482C22—28no—
483C22—464no—
484C22—44no—
485C22—42no—
486C22—42no—
487C22—48no—
488C22—48no—
489C22#2413116yes—
490C22C22×#12111yescharacteristic
491C22C2×#52512yes—
492C22C2×#53411yescharacteristic
493C22C2×#53411yescharacteristic
494C22C2×#53612yes—
495C22C2×#54714yes—
496C22C2×#55214yes—
497C22#348114yes—
498C22#348412yes—
499C22#349012yes—
500C22#352018yes—
501C22#353814yes—
502C4—24no—
503C4—24no—
504C4—48no—
505C4—48no—
506C4—44no—
507C4—48no—
508C4—84no—
509C4—84no—
510C4—84no—
511C2—216no—
512C2—44no—
513C2C2×#241312yes—
514C2#3495611yescharacteristic
515C2#3498111yescharacteristic
516C2#3508412yes—
517C2#3745311yescharacteristic
518C2#5910714yes—
519C2#5928514yes—
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
4. #16675C2semidirect
5. #14935C2semidirect
6. #14907C2semidirect
7. #14728C2semidirect
9. #14043C2semidirect
10. #13972C2semidirect
19. C2×#398C22semidirect
21. C2×#389C22semidirect
39. #1444C22semidirect
40. #1443C22semidirect
41. #1442C22semidirect
42. #1441C22semidirect
43. #1441C22semidirect
44. #1441C22semidirect
47. #1436C22semidirect
48. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24832
C22×#3852
C2×#22802
#166752
#149352
#149072
#147282
#141272
#140432
#139722
#138692