Finite Group Databasebeta

← #4453193

Subgroups — #4453193

Order 512.

3010 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#399C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic, complemented
3#16532C218yesmaximal, complemented
4#16531C218yesmaximal, complemented
5#14822C212yesmaximal, complemented
6#14821C212yesmaximal, complemented
7#14820C212yesmaximal
8#14734C212yesmaximal
9#14342C211yesmaximal, characteristic, complemented
10#14128C212yesmaximal, complemented
11#13868C212yesmaximal
12C22×#117C2212yes—
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C2×#400C2214yescomplemented
20C2×#399—28no—
21C2×#399C2218yes—
22C2×#399C2214yes—
23C2×#398C2214yescomplemented
24C2×#397C2214yes—
25#2316—28no—
26C2×#389C2214yescomplemented
27C2×#389C2214yescomplemented
28C2×#385C2214yes—
29C2×#381C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#379C2214yes—
33C22×#95C2211yescharacteristic
34#1445C2218yescomplemented
35#1444C2214yescomplemented
36#1442C2214yescomplemented
37#1442C2218yescomplemented
38#1441C2214yescomplemented
39#1441C2218yescomplemented
40#1441C2214yescomplemented
41#1440C2218yes—
42#1439C2214yes—
43#1439C2214yes—
44#1439C2218yes—
45#1439C2214yes—
46#1439C2214yes—
47#1437C2218yes—
48#1436C2218yescomplemented
49C22⋊#95C2212yes—
50C22⋊#95C2212yes—
51C22⋊#95C2214yes—
52C22⋊#95C2214yes—
53C26C2311yescharacteristic
54C24×C4C2311yescharacteristic
55C24×C4C2311yescharacteristic
56C24×C4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C2×#117—28no—
61C2×#117—48no—
62C22×C4⋊C4—28no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—44no—
66C22×C4⋊C4—44no—
67C22×C4⋊C4D418yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2318yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—24no—
79C22×C22⋊C4—24no—
80C22×C22⋊C4—24no—
81C22×C22⋊C4—28no—
82C22×C22⋊C4—44no—
83C22×C22⋊C4—44no—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2318yes—
90C22×C22⋊C4C2318yes—
91C22×C42C2314yes—
92#400—48no—
93#399—232no—
94#399—48no—
95#398—48no—
96#397—48no—
97#389—216no—
98#389—216no—
99#385—48no—
100#381—28no—
101#380—28no—
102#380—28no—
103#379—48no—
104C2×#95—24no—
105C2×#95C2318yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2318yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2318yes—
116C2×#95C2314yes—
117C2×#95C2318yes—
118C2×#95C2318yes—
119C25—22no—
120C25—24no—
121C25—24no—
122C25—24no—
123C25—24no—
124C25—42no—
125C25—42no—
126C25—42no—
127C25C2412yes—
128C25C2411yescharacteristic
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4C2×D414yes—
150C23×C4C2×D414yes—
151C23×C4C2×D414yes—
152C23×C4#4118yes—
153C23×C4C2414yes—
154C23×C4C2414yes—
155C23×C4C2418yes—
156C23×C4C2412yes—
157C23×C4C2412yes—
158C23×C4C2412yes—
159C23×C4C2412yes—
160C23×C4C2412yes—
161C23×C4C2412yes—
162#117—432no—
163C2×C4⋊C4—232no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—216no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—416no—
182C2×C4⋊C4—416no—
183C2×C4⋊C4—48no—
184C2×C4⋊C4—48no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4C2×D4116yes—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—216no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—416no—
207C2×C22⋊C4—416no—
208C2×C22⋊C4—416no—
209C2×C22⋊C4—48no—
210C2×C22⋊C4—48no—
211C2×C22⋊C4—48no—
212C2×C22⋊C4—48no—
213C2×C22⋊C4—48no—
214C2×C42—28no—
215C2×C42—44no—
216C2×C42—44no—
217#95—28no—
218#95—28no—
219#95—28no—
220#95—28no—
221#95—28no—
222#95—216no—
223#95—48no—
224#95—48no—
225#95—44no—
226#95—44no—
227#95—48no—
228#95—44no—
229#95—44no—
230#95—48no—
231C24—22no—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—28no—
236C24—22no—
237C24—24no—
238C24—24no—
239C24—24no—
240C24—24no—
241C24—22no—
242C24—24no—
243C24—24no—
244C24—416no—
245C24—416no—
246C24—416no—
247C24—44no—
248C24—44no—
249C24—44no—
250C24—44no—
251C24—44no—
252C24—44no—
253C24—416no—
254C24—88no—
255C24—88no—
256C24—88no—
257C24C22×D412yes—
258C24C2×#4114yes—
259C24H(5,𝔽2)14yes—
260C242−1+414yes—
261C242−1+414yes—
262C24C2511yescharacteristic
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—232no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—28no—
269C22×C4—28no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—28no—
285C22×C4—28no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—28no—
290C22×C4—24no—
291C22×C4—28no—
292C22×C4—28no—
293C22×C4—28no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—28no—
298C22×C4—28no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—28no—
303C22×C4—28no—
304C22×C4—28no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—416no—
312C22×C4—416no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—88no—
321C22×C4—88no—
322C22×C4—88no—
323C22×C4#123116yes—
324C22×C4C22×D414yes—
325C22×C4C22×D414yes—
326C22×C4C22×D414yes—
327C22×C4C2×#4114yes—
328C22×C4C2×#4114yes—
329C22×C4H(5,𝔽2)14yes—
330C22×C4H(5,𝔽2)14yes—
331C4⋊C4—232no—
332C4⋊C4—416no—
333C4⋊C4—416no—
334C4⋊C4—48no—
335C4⋊C4—48no—
336C4⋊C4—84no—
337C4⋊C4—84no—
338C4⋊C4—84no—
339C4⋊C4—84no—
340C4⋊C4—816no—
341C22⋊C4—416no—
342C22⋊C4—416no—
343C22⋊C4—416no—
344C22⋊C4—416no—
345C22⋊C4—416no—
346C22⋊C4—816no—
347C22⋊C4—816no—
348C42—416no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—28no—
354C23—28no—
355C23—28no—
356C23—28no—
357C23—28no—
358C23—28no—
359C23—24no—
360C23—28no—
361C23—28no—
362C23—28no—
363C23—416no—
364C23—416no—
365C23—416no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—48no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—44no—
374C23—48no—
375C23—48no—
376C23—48no—
377C23—44no—
378C23—44no—
379C23—864no—
380C23—88no—
381C23—88no—
382C23—88no—
383C23—88no—
384C23C2×#12314yes—
385C23#53514yes—
386C23#53614yes—
387C23#539116yes—
388C23C23×D411yescharacteristic
389C23C22×#4112yes—
390C23C2×H(5,𝔽2)12yes—
391C23C2×H(5,𝔽2)12yes—
392C23C2×H(5,𝔽2)12yes—
393C23C2×2−1+412yes—
394C23C2×2−1+412yes—
395C23#58512yes—
396C2×C4—232no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—28no—
400C2×C4—28no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—28no—
410C2×C4—28no—
411C2×C4—416no—
412C2×C4—416no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—48no—
418C2×C4—48no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—48no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—48no—
432C2×C4—48no—
433C2×C4—48no—
434C2×C4—48no—
435C2×C4—48no—
436C2×C4—48no—
437C2×C4—48no—
438C2×C4—48no—
439C2×C4—48no—
440C2×C4—48no—
441C2×C4—48no—
442C2×C4—88no—
443C2×C4—88no—
444C2×C4—88no—
445C2×C4C2×#12318yes—
446C22—24no—
447C22—24no—
448C22—24no—
449C22—24no—
450C22—24no—
451C22—24no—
452C22—232no—
453C22—44no—
454C22—44no—
455C22—44no—
456C22—416no—
457C22—832no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—88no—
463C22#262318yes—
464C22C22×#12311yescharacteristic
465C22C2×#53511yescharacteristic
466C22C2×#53611yescharacteristic
467C22C2×#53914yes—
468C22#348411yescharacteristic
469C22#348411yescharacteristic
470C22#348611yescharacteristic
471C22#348711yescharacteristic
472C22#349514yes—
473C22#350814yes—
474C22#353414yes—
475C22#353514yes—
476C22#353614yes—
477C22#353814yes—
478C4—28no—
479C4—44no—
480C4—44no—
481C4—44no—
482C4—44no—
483C4—84no—
484C4—84no—
485C4—82no—
486C4—88no—
487C2—28no—
488C2—84no—
489C2C2×#262311yescharacteristic
490C2#3555411yescharacteristic
491C2#3567211yescharacteristic
492C2#3670711yescharacteristic
493C2#3670811yescharacteristic
494C2#3671011yescharacteristic
495C2#3671111yescharacteristic
496C2#4197418yes—
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
2. C2×#2316C2semidirect
3. #16532C2semidirect
4. #16531C2semidirect
5. #14822C2semidirect
6. #14821C2semidirect
9. #14342C2semidirect
10. #14128C2semidirect
19. C2×#400C22semidirect
23. C2×#398C22semidirect
26. C2×#389C22semidirect
27. C2×#389C22semidirect
34. #1445C22semidirect
35. #1444C22semidirect
36. #1442C22semidirect
37. #1442C22semidirect
38. #1441C22semidirect
39. #1441C22semidirect
40. #1441C22semidirect
48. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3992
C2×#23162
#165322
#165312
#148222
#148212
#148202
#147342
#143422
#141282
#138682