Finite Group Databasebeta

← #4462644

Subgroups — #4462644

Order 512.

3333 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2614C211yesmaximal, characteristic, complemented
2C22×#393C211yesmaximal, characteristic, complemented
3#17224C216yesmaximal, complemented
4#17223C212yesmaximal, complemented
5#17217C216yesmaximal, complemented
6#17216C212yesmaximal, complemented
7#14919C213yesmaximal, complemented
8#14918C213yesmaximal, complemented
9#14912C213yesmaximal, complemented
10#14911C213yesmaximal
11#14336C211yesmaximal, characteristic, complemented
12C24×Q8C2211yescharacteristic
13C22×#123C2213yes—
14C22×#120C2211yescharacteristic
15C23×C22⋊C4C2213yes—
16#2614—28no—
17C2×#395C2213yescomplemented
18C2×#395C2211yescharacteristic, complemented
19C2×#393—28no—
20C2×#393C2212yes—
21C2×#393C2216yes—
22C2×#393C2211yescharacteristic, complemented
23C2×#393C2213yescomplemented
24C2×#386C2213yes—
25C2×#385C2213yes—
26C2×#381C2213yes—
27C2×#381C2213yes—
28C2×#381C2213yes—
29C2×#379C2213yes—
30C22×#95C2213yes—
31#1446C2212yescomplemented
32#1446C2216yescomplemented
33#1445C2216yescomplemented
34#1443C2216yescomplemented
35#1443C2216yescomplemented
36#1442C2216yescomplemented
37#1442C2216yescomplemented
38#1441C2216yescomplemented
39#1440C2216yes—
40#1440C2216yes—
41#1440C2216yescomplemented
42#1439C2216yes—
43#1439C2216yes—
44#1439C2216yes—
45#1439C2216yes—
46#1439C2216yescomplemented
47#1438C2216yes—
48#1438C2216yes—
49C22⋊#95C2213yes—
50C22⋊#95C2213yes—
51C26C2311yescharacteristic
52C23×Q8—46no—
53C23×Q8—46no—
54C23×Q8D418yescomplemented
55C23×Q8C2311yescharacteristic
56C23×Q8C2313yes—
57C23×D4C2311yescharacteristic
58C23×D4C2312yes—
59C24×C4C2313yes—
60C24×C4C2313yes—
61C2×#123—212no—
62C2×#123—46no—
63C2×#123—46no—
64C2×#120—46no—
65C22×C4⋊C4C2313yes—
66C22×C4⋊C4C2313yes—
67C22×C4⋊C4C2316yes—
68C22×C22⋊C4—212no—
69C22×C22⋊C4—46no—
70C22×C22⋊C4—46no—
71C22×C22⋊C4C2311yescharacteristic
72C22×C22⋊C4C2313yes—
73C22×C22⋊C4C2313yes—
74C22×C22⋊C4C2316yes—
75C22×C22⋊C4C2316yes—
76C22×C22⋊C4C2313yes—
77C22×C22⋊C4C2313yes—
78C22×C22⋊C4C2316yes—
79C22×C22⋊C4C2312yes—
80C22×C22⋊C4C2316yes—
81C22×C22⋊C4C2316yes—
82#395—212no—
83#395—24no—
84#393—24no—
85#393—212no—
86#393—28no—
87#393—224no—
88#386—46no—
89#385—46no—
90#381—212no—
91#381—46no—
92#381—46no—
93#379—46no—
94C2×#95—212no—
95C2×#95C2316yes—
96C2×#95C2316yes—
97C2×#95C2313yes—
98C2×#95C2313yes—
99C2×#95C2313yes—
100C2×#95C2313yes—
101C2×#95C2316yes—
102C2×#95C2316yes—
103C2×#95C2316yes—
104C2×#95C2316yes—
105C2×#95C2316yes—
106C2×#95C2316yes—
107C2×#95C2316yes—
108C2×#95C2316yes—
109C2×#95C2316yes—
110C2×#95C2316yes—
111C25—22no—
112C25—24no—
113C25—26no—
114C25—26no—
115C25—46no—
116C25C2411yescharacteristic
117C25C2412yes—
118C25C2411yescharacteristic
119C22×Q8—448no—
120C22×Q8—43no—
121C22×Q8—43no—
122C22×Q8—43no—
123C22×Q8—43no—
124C22×Q8—43no—
125C22×Q8—43no—
126C22×Q8—43no—
127C22×Q8—43no—
128C22×Q8—448no—
129C22×Q8—416no—
130C22×Q8C2×D414yes—
131C22×Q8C2×D4112yes—
132C22×D4—22no—
133C22×D4—26no—
134C22×D4—26no—
135C22×D4—24no—
136C22×D4—46no—
137C22×D4—46no—
138C23×C4—26no—
139C23×C4—26no—
140C23×C4—26no—
141C23×C4—26no—
142C23×C4—26no—
143C23×C4—26no—
144C23×C4—26no—
145C23×C4—26no—
146C23×C4—26no—
147C23×C4—46no—
148C23×C4—46no—
149C23×C4C2×D4112yes—
150C23×C4C2413yes—
151C23×C4C2413yes—
152C23×C4C2416yes—
153C23×C4C2416yes—
154C23×C4C2411yescharacteristic
155C23×C4C2412yes—
156C23×C4C2413yes—
157C23×C4C2413yes—
158#123—448no—
159#120—416no—
160C2×C4⋊C4—26no—
161C2×C4⋊C4—26no—
162C2×C4⋊C4—43no—
163C2×C4⋊C4—43no—
164C2×C4⋊C4—43no—
165C2×C4⋊C4—43no—
166C2×C4⋊C4—46no—
167C2×C4⋊C4—46no—
168C2×C4⋊C4—46no—
169C2×C22⋊C4—26no—
170C2×C22⋊C4—26no—
171C2×C22⋊C4—26no—
172C2×C22⋊C4—26no—
173C2×C22⋊C4—26no—
174C2×C22⋊C4—26no—
175C2×C22⋊C4—26no—
176C2×C22⋊C4—212no—
177C2×C22⋊C4—212no—
178C2×C22⋊C4—26no—
179C2×C22⋊C4—26no—
180C2×C22⋊C4—212no—
181C2×C22⋊C4—46no—
182C2×C22⋊C4—46no—
183C2×C22⋊C4—46no—
184C2×C22⋊C4—46no—
185C2×C22⋊C4—46no—
186C2×C22⋊C4—43no—
187C2×C22⋊C4—43no—
188C2×C22⋊C4—43no—
189C2×C22⋊C4—43no—
190C2×C22⋊C4—448no—
191C2×C22⋊C4—46no—
192C2×C22⋊C4—46no—
193C2×C22⋊C4—46no—
194C2×C22⋊C4—46no—
195C2×C22⋊C4—46no—
196C2×C22⋊C4—46no—
197C2×C22⋊C4—46no—
198C2×C22⋊C4—46no—
199#95—212no—
200#95—212no—
201#95—26no—
202#95—26no—
203#95—26no—
204#95—26no—
205#95—46no—
206#95—46no—
207#95—46no—
208#95—46no—
209#95—46no—
210#95—46no—
211#95—46no—
212#95—46no—
213#95—46no—
214#95—46no—
215C24—22no—
216C24—26no—
217C24—28no—
218C24—22no—
219C24—26no—
220C24—26no—
221C24—22no—
222C24—26no—
223C24—26no—
224C24—21no—
225C24—23no—
226C24—23no—
227C24—22no—
228C24—26no—
229C24—424no—
230C24—424no—
231C24—412no—
232C24—412no—
233C24—412no—
234C24—44no—
235C24—812no—
236C24—812no—
237C24C22×D412yes—
238C24H(5,𝔽2)16yes—
239C24H(5,𝔽2)16yes—
240C242−1+414yes—
241C24C2511yescharacteristic
242C2×Q8—412no—
243C2×Q8—412no—
244C2×Q8—412no—
245C2×Q8—412no—
246C2×Q8—412no—
247C2×Q8—412no—
248C2×Q8—412no—
249C2×Q8—412no—
250C2×Q8—44no—
251C2×Q8—412no—
252C2×Q8—4128no—
253C2×D4—48no—
254C2×D4—412no—
255C2×D4—412no—
256C2×D4—44no—
257C2×D4—812no—
258C2×D4—812no—
259C22×C4—224no—
260C22×C4—224no—
261C22×C4—224no—
262C22×C4—23no—
263C22×C4—23no—
264C22×C4—23no—
265C22×C4—23no—
266C22×C4—23no—
267C22×C4—23no—
268C22×C4—23no—
269C22×C4—23no—
270C22×C4—23no—
271C22×C4—23no—
272C22×C4—23no—
273C22×C4—23no—
274C22×C4—23no—
275C22×C4—23no—
276C22×C4—26no—
277C22×C4—21no—
278C22×C4—26no—
279C22×C4—26no—
280C22×C4—26no—
281C22×C4—23no—
282C22×C4—26no—
283C22×C4—23no—
284C22×C4—26no—
285C22×C4—26no—
286C22×C4—23no—
287C22×C4—23no—
288C22×C4—23no—
289C22×C4—23no—
290C22×C4—26no—
291C22×C4—26no—
292C22×C4—22no—
293C22×C4—26no—
294C22×C4—26no—
295C22×C4—26no—
296C22×C4—26no—
297C22×C4—26no—
298C22×C4—26no—
299C22×C4—26no—
300C22×C4—26no—
301C22×C4—26no—
302C22×C4—26no—
303C22×C4—26no—
304C22×C4—412no—
305C22×C4—412no—
306C22×C4—412no—
307C22×C4—412no—
308C22×C4—412no—
309C22×C4—412no—
310C22×C4—412no—
311C22×C4—812no—
312C22×C4—812no—
313C22×C4C22×D416yes—
314C22×C4C22×D416yes—
315C4⋊C4—412no—
316C4⋊C4—412no—
317C4⋊C4—812no—
318C22⋊C4—424no—
319C22⋊C4—424no—
320C22⋊C4—412no—
321C22⋊C4—412no—
322C22⋊C4—48no—
323C22⋊C4—412no—
324C22⋊C4—412no—
325C22⋊C4—44no—
326C22⋊C4—812no—
327C22⋊C4—812no—
328C22⋊C4—812no—
329C23—22no—
330C23—26no—
331C23—212no—
332C23—212no—
333C23—212no—
334C23—212no—
335C23—24no—
336C23—212no—
337C23—212no—
338C23—448no—
339C23—46no—
340C23—46no—
341C23—46no—
342C23—42no—
343C23—46no—
344C23—46no—
345C23—46no—
346C23—46no—
347C23—46no—
348C23—46no—
349C23—43no—
350C23—41no—
351C23—43no—
352C23—43no—
353C23—41no—
354C23—46no—
355C23—46no—
356C23—46no—
357C23—46no—
358C23—848no—
359C23—816no—
360C23—824no—
361C23—88no—
362C23#534112yes—
363C23#560112yes—
364C23H(3,𝔽4)14yes—
365C23C23×D411yescharacteristic
366C23C2×H(5,𝔽2)13yes—
367C23C2×H(5,𝔽2)13yes—
368C23C2×2−1+412yes—
369C23#58516yes—
370Q8—416no—
371Q8—448no—
372D4—816no—
373D4—88no—
374C2×C4—26no—
375C2×C4—26no—
376C2×C4—26no—
377C2×C4—26no—
378C2×C4—26no—
379C2×C4—26no—
380C2×C4—43no—
381C2×C4—43no—
382C2×C4—43no—
383C2×C4—43no—
384C2×C4—43no—
385C2×C4—43no—
386C2×C4—43no—
387C2×C4—43no—
388C2×C4—448no—
389C2×C4—43no—
390C2×C4—43no—
391C2×C4—46no—
392C2×C4—43no—
393C2×C4—43no—
394C2×C4—43no—
395C2×C4—43no—
396C2×C4—43no—
397C2×C4—46no—
398C2×C4—46no—
399C2×C4—46no—
400C2×C4—43no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—46no—
404C2×C4—46no—
405C2×C4—46no—
406C2×C4—46no—
407C2×C4—46no—
408C2×C4—46no—
409C2×C4—46no—
410C2×C4—46no—
411C2×C4—46no—
412C2×C4—46no—
413C2×C4—46no—
414C2×C4—46no—
415C2×C4—46no—
416C2×C4—46no—
417C2×C4—46no—
418C2×C4—46no—
419C2×C4—824no—
420C22—212no—
421C22—212no—
422C22—224no—
423C22—28no—
424C22—412no—
425C22—412no—
426C22—44no—
427C22—832no—
428C22—82no—
429C22—86no—
430C22—86no—
431C22—82no—
432C22—81no—
433C22—83no—
434C22—83no—
435C22—86no—
436C22—82no—
437C22#281218yes—
438C22C2×#53413yes—
439C22#348611yescharacteristic
440C22#348713yes—
441C22C2×#56013yes—
442C22C2×H(3,𝔽4)11yescharacteristic
443C22#353516yes—
444C22#353516yes—
445C22#354516yes—
446C22#354516yes—
447C4—46no—
448C4—46no—
449C4—81no—
450C4—83no—
451C4—83no—
452C4—86no—
453C4—86no—
454C4—82no—
455C2—28no—
456C2—84no—
457C2—161no—
458C2C2×#281211yescharacteristic
459C2#3718613yes—
460C2#3718713yes—
461C2#5291216yes—
462C2#5292512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
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×#2614C2semidirect
2. C22×#393C2semidirect
3. #17224C2semidirect
4. #17223C2semidirect
5. #17217C2semidirect
6. #17216C2semidirect
7. #14919C2semidirect
8. #14918C2semidirect
9. #14912C2semidirect
11. #14336C2semidirect
17. C2×#395C22semidirect
18. C2×#395C22semidirect
22. C2×#393C22semidirect
23. C2×#393C22semidirect
31. #1446C22semidirect
32. #1446C22semidirect
33. #1445C22semidirect
34. #1443C22semidirect
35. #1443C22semidirect
36. #1442C22semidirect
37. #1442C22semidirect
38. #1441C22semidirect
41. #1440C22semidirect
46. #1439C22semidirect
54. C23×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26142
C22×#3932
#172242
#172232
#172172
#172162
#149192
#149182
#149122
#149112
#143362