Finite Group Databasebeta

← #1086333

Subgroups — #1086333

Order 512.

3948 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#543C211yesmaximal, characteristic
2C2×#2421C213yesmaximal, complemented
3#16366C213yesmaximal, complemented
4#15670C216yesmaximal, complemented
5#15354C216yesmaximal, complemented
6#15001C213yesmaximal, complemented
7#14820C213yesmaximal
8#14449C213yesmaximal, complemented
9#13991C213yesmaximal
10C2×#543—224no—
11C22×C22⋊Q8C2216yes—
12C22×C22⋊Q8C2213yes—
13C22×C22⋊Q8C2213yes—
14C22×C4×Q8C2211yescharacteristic, complemented
15C23×C22⋊C4C2213yes—
16#2421—224no—
17C2×#400C2216yescomplemented
18C2×#397C2216yes—
19C2×#395C2212yescomplemented
20C2×#391C2213yescomplemented
21C2×#389C2213yescomplemented
22C2×#388C2216yes—
23C2×#387C2216yescomplemented
24C2×#385C2213yes—
25C2×#382C2213yescomplemented
26C2×#380C2213yes—
27C2×#379C2213yes—
28C2×#379C2211yescharacteristic
29C2×C4×C22⋊C4C2213yes—
30#1446C2216yescomplemented
31#1446C2211yescharacteristic, complemented
32#1446C2213yescomplemented
33#1445C2216yescomplemented
34#1444C2213yescomplemented
35#1442C2213yescomplemented
36#1442C2216yescomplemented
37#1441C2216yescomplemented
38#1440C2216yes—
39#1440C2216yes—
40#1440C2216yes—
41#1439C2216yes—
42#1439C2216yes—
43#1437C2216yes—
44#1436C2216yescomplemented
45C22⋊#95C2213yes—
46C22⋊#95C2216yes—
47C22⋊#95C2213yes—
48C22⋊#95C2213yes—
49C26C2311yescharacteristic
50C23×Q8C2312yes—
51C23×Q8C2311yescharacteristic
52C23×Q8C2311yescharacteristic
53C24×C4C2313yes—
54C24×C4C2313yes—
55#543—2128no—
56C2×C22⋊Q8—224no—
57C2×C22⋊Q8—212no—
58C2×C22⋊Q8—224no—
59C2×C22⋊Q8—224no—
60C2×C22⋊Q8—224no—
61C2×C22⋊Q8—224no—
62C2×C22⋊Q8D4124yes—
63C2×C4×Q8—212no—
64C22×C4⋊C4C2313yes—
65C22×C4⋊C4C2313yes—
66C22×C4⋊C4C2316yes—
67C22×C4⋊C4C2316yes—
68C22×C4⋊C4C2313yes—
69C22×C4⋊C4C2313yes—
70C22×C22⋊C4—212no—
71C22×C22⋊C4—224no—
72C22×C22⋊C4C2313yes—
73C22×C22⋊C4C2316yes—
74C22×C22⋊C4C2316yes—
75C22×C22⋊C4C2313yes—
76C22×C22⋊C4C2316yes—
77C22×C22⋊C4C2313yes—
78C22×C22⋊C4C2316yes—
79C22×C22⋊C4C2313yes—
80C22×C42C2313yes—
81C22×C42C2313yes—
82#400—224no—
83#397—224no—
84#395—44no—
85#391—212no—
86#389—212no—
87#388—224no—
88#387—224no—
89#385—46no—
90#382—212no—
91#380—212no—
92#379—212no—
93#379—42no—
94C4×C22⋊C4—212no—
95C2×#95C2316yes—
96C2×#95C2316yes—
97C2×#95C2316yes—
98C2×#95C2313yes—
99C2×#95C2316yes—
100C2×#95C2316yes—
101C2×#95C2316yes—
102C2×#95C2313yes—
103C2×#95C2316yes—
104C2×#95C2313yes—
105C2×#95C2316yes—
106C2×#95C2316yes—
107C2×#95C2316yes—
108C2×#95C2313yes—
109C2×#95C2316yes—
110C25—26no—
111C25—26no—
112C25—41no—
113C25—43no—
114C25—43no—
115C25—41no—
116C25#4114yes—
117C25C2413yes—
118C22×Q8—26no—
119C22×Q8—212no—
120C22×Q8—43no—
121C22×Q8—41no—
122C22×Q8—41no—
123C22×Q8—43no—
124C22×Q8—46no—
125C22×Q8—42no—
126C22×Q8C2×D4112yes—
127C23×C4—26no—
128C23×C4—212no—
129C23×C4—212no—
130C23×C4—46no—
131C23×C4—43no—
132C23×C4—43no—
133C23×C4—43no—
134C23×C4—46no—
135C23×C4—43no—
136C23×C4C2×D4112yes—
137C23×C4C2416yes—
138C23×C4C2413yes—
139C23×C4C2413yes—
140C23×C4C2413yes—
141C23×C4C2416yes—
142C23×C4C2413yes—
143C23×C4C2411yescharacteristic
144C23×C4C2413yes—
145C22⋊Q8—296no—
146C22⋊Q8—2192no—
147C22⋊Q8—296no—
148C4×Q8—416no—
149C2×C4⋊C4—26no—
150C2×C4⋊C4—212no—
151C2×C4⋊C4—212no—
152C2×C4⋊C4—212no—
153C2×C4⋊C4—212no—
154C2×C4⋊C4—26no—
155C2×C4⋊C4—212no—
156C2×C4⋊C4—26no—
157C2×C4⋊C4—212no—
158C2×C4⋊C4—212no—
159C2×C4⋊C4—46no—
160C2×C4⋊C4—43no—
161C2×C4⋊C4—43no—
162C2×C4⋊C4C2×D4112yes—
163C2×C4⋊C4C2×D4124yes—
164C2×C22⋊C4—296no—
165C2×C22⋊C4—212no—
166C2×C22⋊C4—212no—
167C2×C22⋊C4—212no—
168C2×C22⋊C4—212no—
169C2×C22⋊C4—212no—
170C2×C22⋊C4—26no—
171C2×C22⋊C4—26no—
172C2×C22⋊C4—212no—
173C2×C22⋊C4—26no—
174C2×C22⋊C4—26no—
175C2×C22⋊C4—212no—
176C2×C22⋊C4—26no—
177C2×C22⋊C4—212no—
178C2×C22⋊C4—26no—
179C2×C22⋊C4—212no—
180C2×C22⋊C4—26no—
181C2×C22⋊C4—43no—
182C2×C22⋊C4—43no—
183C2×C22⋊C4—43no—
184C2×C22⋊C4—46no—
185C2×C22⋊C4—43no—
186C2×C22⋊C4—46no—
187C2×C22⋊C4—43no—
188C2×C22⋊C4C2×D4124yes—
189C2×C42—26no—
190C2×C42—212no—
191C2×C42—26no—
192C2×C42—46no—
193#95—26no—
194#95—26no—
195#95—212no—
196#95—212no—
197#95—212no—
198#95—212no—
199#95—212no—
200#95—212no—
201#95—46no—
202#95—46no—
203#95—43no—
204#95—46no—
205#95—43no—
206#95—46no—
207#95—46no—
208C24—28no—
209C24—224no—
210C24—26no—
211C24—23no—
212C24—26no—
213C24—23no—
214C24—26no—
215C24—23no—
216C24—23no—
217C24—26no—
218C24—224no—
219C24—44no—
220C24—412no—
221C24—86no—
222C24—86no—
223C24—86no—
224C24—86no—
225C24—86no—
226C24—86no—
227C24—812no—
228C24C22×D416yes—
229C24C2×#4116yes—
230C242−1+416yes—
231C24C2511yescharacteristic
232C2×Q8—28no—
233C2×Q8—216no—
234C2×Q8—412no—
235C2×Q8—412no—
236C2×Q8—86no—
237C2×Q8—86no—
238C2×Q8—812no—
239C2×Q8—812no—
240C2×Q8#120116yes—
241C22×C4—26no—
242C22×C4—23no—
243C22×C4—23no—
244C22×C4—23no—
245C22×C4—26no—
246C22×C4—26no—
247C22×C4—26no—
248C22×C4—26no—
249C22×C4—23no—
250C22×C4—23no—
251C22×C4—26no—
252C22×C4—26no—
253C22×C4—26no—
254C22×C4—23no—
255C22×C4—26no—
256C22×C4—224no—
257C22×C4—224no—
258C22×C4—23no—
259C22×C4—23no—
260C22×C4—26no—
261C22×C4—23no—
262C22×C4—26no—
263C22×C4—26no—
264C22×C4—23no—
265C22×C4—23no—
266C22×C4—26no—
267C22×C4—23no—
268C22×C4—23no—
269C22×C4—26no—
270C22×C4—26no—
271C22×C4—23no—
272C22×C4—26no—
273C22×C4—23no—
274C22×C4—26no—
275C22×C4—26no—
276C22×C4—26no—
277C22×C4—26no—
278C22×C4—26no—
279C22×C4—26no—
280C22×C4—412no—
281C22×C4—412no—
282C22×C4—812no—
283C22×C4—812no—
284C22×C4—812no—
285C22×C4—812no—
286C22×C4—86no—
287C22×C4—86no—
288C22×C4C22×D416yes—
289C22×C4C22×D4112yes—
290C22×C4C22×D4112yes—
291C22×C4C22×D416yes—
292C22×C4H(5,𝔽2)12yes—
293C4⋊C4—224no—
294C4⋊C4—224no—
295C4⋊C4—248no—
296C4⋊C4—424no—
297C4⋊C4—412no—
298C4⋊C4—412no—
299C22⋊C4—248no—
300C22⋊C4—248no—
301C22⋊C4—412no—
302C22⋊C4—412no—
303C22⋊C4—412no—
304C22⋊C4—424no—
305C22⋊C4—812no—
306C22⋊C4—86no—
307C42—412no—
308C42—412no—
309C23—232no—
310C23—26no—
311C23—26no—
312C23—212no—
313C23—26no—
314C23—26no—
315C23—26no—
316C23—212no—
317C23—26no—
318C23—212no—
319C23—26no—
320C23—26no—
321C23—26no—
322C23—26no—
323C23—212no—
324C23—46no—
325C23—46no—
326C23—46no—
327C23—46no—
328C23—46no—
329C23—46no—
330C23—46no—
331C23—46no—
332C23—824no—
333C23—824no—
334C23—88no—
335C23—824no—
336C23—824no—
337C23—88no—
338C23#548112yes—
339C23C23×D413yes—
340C23C22×#4111yescharacteristic
341C23C2×H(5,𝔽2)11yescharacteristic
342C23C2×2−1+413yes—
343C23#58511yescharacteristic
344C23#58513yes—
345C23#58513yes—
346Q8—416no—
347Q8—88no—
348Q8—816no—
349C2×C4—26no—
350C2×C4—26no—
351C2×C4—212no—
352C2×C4—212no—
353C2×C4—26no—
354C2×C4—212no—
355C2×C4—26no—
356C2×C4—26no—
357C2×C4—26no—
358C2×C4—212no—
359C2×C4—212no—
360C2×C4—41no—
361C2×C4—46no—
362C2×C4—46no—
363C2×C4—46no—
364C2×C4—46no—
365C2×C4—43no—
366C2×C4—46no—
367C2×C4—46no—
368C2×C4—43no—
369C2×C4—46no—
370C2×C4—46no—
371C2×C4—43no—
372C2×C4—46no—
373C2×C4—43no—
374C2×C4—43no—
375C2×C4—43no—
376C2×C4—46no—
377C2×C4—43no—
378C2×C4—43no—
379C2×C4—46no—
380C2×C4—43no—
381C2×C4—46no—
382C2×C4—46no—
383C2×C4—46no—
384C2×C4—43no—
385C2×C4—46no—
386C2×C4—46no—
387C2×C4—46no—
388C2×C4—46no—
389C2×C4—46no—
390C2×C4—824no—
391C2×C4—824no—
392C2×C4C2×#120112yes—
393C22—212no—
394C22—212no—
395C22—212no—
396C22—46no—
397C22—412no—
398C22—46no—
399C22—412no—
400C22—46no—
401C22—412no—
402C22—412no—
403C22—46no—
404C22—832no—
405C22C22×#12011yescharacteristic
406C22#345911yescharacteristic
407C22C2×#54813yes—
408C22#348513yes—
409C22#348513yes—
410C22#348613yes—
411C22#348813yes—
412C22#349113yes—
413C22#350013yes—
414C22#356113yes—
415C22#356413yes—
416C22#356416yes—
417C4—212no—
418C4—46no—
419C4—42no—
420C4—412no—
421C4—86no—
422C4—83no—
423C4—83no—
424C4—83no—
425C2—412no—
426C2#3544611yescharacteristic
427C2#3608211yescharacteristic
428C2#3657511yescharacteristic
429C2#3775413yes—
430C2#4349313yes—
431C2#4681513yes—
432C2#4877813yes—
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×#2421C2semidirect
3. #16366C2semidirect
4. #15670C2semidirect
5. #15354C2semidirect
6. #15001C2semidirect
8. #14449C2semidirect
14. C22×C4×Q8C22semidirect
17. C2×#400C22semidirect
19. C2×#395C22semidirect
20. C2×#391C22semidirect
21. C2×#389C22semidirect
23. C2×#387C22semidirect
25. C2×#382C22semidirect
30. #1446C22semidirect
31. #1446C22semidirect
32. #1446C22semidirect
33. #1445C22semidirect
34. #1444C22semidirect
35. #1442C22semidirect
36. #1442C22semidirect
37. #1441C22semidirect
44. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5432
C2×#24212
#163662
#156702
#153542
#150012
#148202
#144492
#139912