Finite Group Databasebeta

← #4462577

Subgroups — #4462577

Order 512.

2958 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2659C211yesmaximal, characteristic, complemented
2C22×#396C211yesmaximal, characteristic
3#17129C218yesmaximal, complemented
4#17104C218yesmaximal, complemented
5#14822C212yesmaximal, complemented
6#14821C212yesmaximal, complemented
7#14820C212yesmaximal
8#14734C212yesmaximal
9#14720C212yesmaximal, complemented
10#14717C212yesmaximal
11#14349C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8C2212yes—
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19#2659—28no—
20C2×#400C2214yescomplemented
21C2×#399C2214yes—
22C2×#398C2214yescomplemented
23C2×#398C2214yescomplemented
24C2×#397C2214yes—
25C2×#396—28no—
26C2×#396C2218yes—
27C2×#395C2214yescomplemented
28C2×#395C2214yescomplemented
29C2×#393C2214yes—
30C2×#385C2214yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#379C2214yes—
34C22×#95C2211yescharacteristic
35#1444C2218yescomplemented
36#1443C2218yescomplemented
37#1442C2218yescomplemented
38#1441C2214yescomplemented
39#1441C2214yescomplemented
40#1441C2218yescomplemented
41#1441C2214yescomplemented
42#1440C2218yes—
43#1439C2218yes—
44#1439C2214yes—
45#1439C2218yes—
46#1439C2214yes—
47#1439C2214yes—
48C22⋊#95C2212yes—
49C22⋊#95C2212yes—
50C22⋊#95C2212yes—
51C22⋊#95C2214yes—
52C22⋊#95C2214yes—
53C26C2311yescharacteristic
54C23×Q8C2312yes—
55C24×C4C2311yescharacteristic
56C24×C4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C24×C4C2311yescharacteristic
61C2×C22⋊Q8—28no—
62C2×C22⋊Q8—48no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—44no—
67C22×C4⋊C4—44no—
68C22×C4⋊C4—44no—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2314yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2318yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4—24no—
80C22×C22⋊C4—24no—
81C22×C22⋊C4—24no—
82C22×C22⋊C4—44no—
83C22×C22⋊C4—44no—
84C22×C22⋊C4—44no—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2318yes—
91C22×C22⋊C4C2318yes—
92#400—48no—
93#399—48no—
94#398—216no—
95#398—48no—
96#397—48no—
97#396—232no—
98#395—216no—
99#395—48no—
100#393—48no—
101#385—48no—
102#380—28no—
103#380—28no—
104#379—48no—
105C2×#95D418yes—
106C2×#95C2318yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2318yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2318yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2318yes—
117C2×#95C2314yes—
118C2×#95C2318yes—
119C2×#95C2314yes—
120C25—22no—
121C25—24no—
122C25—24no—
123C25—24no—
124C25—24no—
125C25—42no—
126C25—42no—
127C25—42no—
128C25C2412yes—
129C25C2411yescharacteristic
130C22×Q8—24no—
131C22×Q8—44no—
132C22×Q8—44no—
133C22×Q8—44no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4C2×D414yes—
156C23×C4C2×D414yes—
157C23×C4C2×D414yes—
158C23×C4C2414yes—
159C23×C4C2414yes—
160C23×C4C2418yes—
161C23×C4C2412yes—
162C23×C4C2412yes—
163C23×C4C2412yes—
164C23×C4C2412yes—
165C23×C4C2412yes—
166C23×C4C2412yes—
167C22⋊Q8—432no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—216no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—416no—
188C2×C4⋊C4—416no—
189C2×C4⋊C4—416no—
190C2×C4⋊C4—48no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—48no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
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—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—216no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—416no—
213C2×C22⋊C4—416no—
214C2×C22⋊C4—416no—
215C2×C22⋊C4—48no—
216C2×C22⋊C4—48no—
217C2×C22⋊C4—48no—
218C2×C22⋊C4—48no—
219C2×C22⋊C4—48no—
220#95—28no—
221#95—28no—
222#95—216no—
223#95—216no—
224#95—48no—
225#95—44no—
226#95—48no—
227#95—44no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—48no—
232#95—44no—
233#95C2×D4116yes—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24—28no—
239C24—22no—
240C24—24no—
241C24—24no—
242C24—24no—
243C24—24no—
244C24—22no—
245C24—24no—
246C24—24no—
247C24—416no—
248C24—416no—
249C24—416no—
250C24—44no—
251C24—44no—
252C24—44no—
253C24—44no—
254C24—44no—
255C24—44no—
256C24—416no—
257C24—88no—
258C24—88no—
259C24—88no—
260C24C22×D412yes—
261C24H(5,𝔽2)14yes—
262C242−1+414yes—
263C242−1+414yes—
264C242−1+414yes—
265C24C2511yescharacteristic
266C2×Q8—48no—
267C2×Q8—84no—
268C2×Q8—84no—
269C2×Q8—816no—
270C22×C4—28no—
271C22×C4—28no—
272C22×C4—28no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—28no—
295C22×C4—28no—
296C22×C4—28no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—28no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—28no—
304C22×C4—24no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—28no—
311C22×C4—28no—
312C22×C4—28no—
313C22×C4—28no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—416no—
321C22×C4—416no—
322C22×C4—416no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—88no—
330C22×C4—88no—
331C22×C4—88no—
332C22×C4C22×D414yes—
333C22×C4C22×D414yes—
334C22×C4C22×D414yes—
335C22×C4H(5,𝔽2)14yes—
336C22×C4H(5,𝔽2)14yes—
337C22×C4H(5,𝔽2)14yes—
338C4⋊C4—416no—
339C4⋊C4—416no—
340C4⋊C4—416no—
341C4⋊C4—48no—
342C4⋊C4—416no—
343C4⋊C4—84no—
344C4⋊C4—84no—
345C4⋊C4—84no—
346C4⋊C4—84no—
347C4⋊C4—816no—
348C22⋊C4—416no—
349C22⋊C4—416no—
350C22⋊C4—416no—
351C22⋊C4—416no—
352C22⋊C4—416no—
353C22⋊C4—816no—
354C22⋊C4—816no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—28no—
360C23—28no—
361C23—28no—
362C23—28no—
363C23—28no—
364C23—28no—
365C23—24no—
366C23—28no—
367C23—28no—
368C23—28no—
369C23—416no—
370C23—416no—
371C23—416no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—48no—
379C23—44no—
380C23—48no—
381C23—48no—
382C23—48no—
383C23—44no—
384C23—44no—
385C23—864no—
386C23—88no—
387C23—88no—
388C23—88no—
389C23—88no—
390C23#53514yes—
391C23#53614yes—
392C23#53614yes—
393C23#563116yes—
394C23C23×D411yescharacteristic
395C23C2×H(5,𝔽2)12yes—
396C23C2×H(5,𝔽2)12yes—
397C23C2×H(5,𝔽2)12yes—
398C23C2×H(5,𝔽2)12yes—
399C23C2×2−1+412yes—
400C23C2×2−1+412yes—
401C23C2×2−1+412yes—
402Q8—816no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—28no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—416no—
419C2×C4—416no—
420C2×C4—416no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—48no—
430C2×C4—48no—
431C2×C4—44no—
432C2×C4—48no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—48no—
440C2×C4—48no—
441C2×C4—48no—
442C2×C4—48no—
443C2×C4—48no—
444C2×C4—48no—
445C2×C4—48no—
446C2×C4—48no—
447C2×C4—48no—
448C2×C4—48no—
449C2×C4—48no—
450C2×C4—88no—
451C2×C4—88no—
452C2×C4—88no—
453C22—24no—
454C22—24no—
455C22—24no—
456C22—24no—
457C22—24no—
458C22—24no—
459C22—232no—
460C22—44no—
461C22—44no—
462C22—44no—
463C22—416no—
464C22—832no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22—88no—
470C22#282118yes—
471C22C2×#53511yescharacteristic
472C22C2×#53611yescharacteristic
473C22C2×#53611yescharacteristic
474C22#348411yescharacteristic
475C22#348411yescharacteristic
476C22#348411yescharacteristic
477C22#348611yescharacteristic
478C22C2×#56314yes—
479C22#353614yes—
480C22#353614yes—
481C22#353614yes—
482C22#353714yes—
483C22#353814yes—
484C22#353814yes—
485C4—44no—
486C4—44no—
487C4—44no—
488C4—44no—
489C4—44no—
490C4—44no—
491C4—84no—
492C4—84no—
493C4—88no—
494C2—28no—
495C2—84no—
496C2C2×#282111yescharacteristic
497C2#3670711yescharacteristic
498C2#3670811yescharacteristic
499C2#3670811yescharacteristic
500C2#3671711yescharacteristic
501C2#3671811yescharacteristic
502C2#3671811yescharacteristic
503C2#5247318yes—
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×#2659C2semidirect
3. #17129C2semidirect
4. #17104C2semidirect
5. #14822C2semidirect
6. #14821C2semidirect
9. #14720C2semidirect
11. #14349C2semidirect
20. C2×#400C22semidirect
22. C2×#398C22semidirect
23. C2×#398C22semidirect
27. C2×#395C22semidirect
28. C2×#395C22semidirect
35. #1444C22semidirect
36. #1443C22semidirect
37. #1442C22semidirect
38. #1441C22semidirect
39. #1441C22semidirect
40. #1441C22semidirect
41. #1441C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26592
C22×#3962
#171292
#171042
#148222
#148212
#148202
#147342
#147202
#147172
#143492