Finite Group Databasebeta

← #4462586

Subgroups — #4462586

Order 512.

3085 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2643C211yesmaximal, characteristic, complemented
2C22×#396C211yesmaximal, characteristic, complemented
3#17213C212yesmaximal, complemented
4#17135C212yesmaximal, complemented
5#17133C216yesmaximal, complemented
6#17096C216yesmaximal, complemented
7#14729C213yesmaximal, complemented
8#14728C213yesmaximal, complemented
9#14726C213yesmaximal, complemented
10#14725C213yesmaximal
11#14353C211yesmaximal, characteristic, complemented
12C22×#120C2211yescharacteristic
13C23×C4⋊C4C2213yes—
14C23×C22⋊C4C2213yes—
15#2643—28no—
16C2×#401C2211yescharacteristic, complemented
17C2×#400C2213yescomplemented
18C2×#398C2216yescomplemented
19C2×#397C2213yescomplemented
20C2×#396—28no—
21C2×#396C2212yes—
22C2×#396C2216yes—
23C2×#396C2216yes—
24C2×#396C2211yescharacteristic, complemented
25C2×#396C2216yescomplemented
26C2×#394C2216yes—
27C2×#386C2213yes—
28C2×#386C2213yes—
29C2×#385C2213yes—
30C2×#385C2213yes—
31C2×#380C2213yes—
32C2×#380C2213yes—
33C2×#379C2213yes—
34C22×#95C2211yescharacteristic
35#1444C2212yescomplemented
36#1444C2216yescomplemented
37#1444C2216yescomplemented
38#1442C2216yescomplemented
39#1441C2216yescomplemented
40#1441C2216yescomplemented
41#1441C2216yescomplemented
42#1440C2216yes—
43#1439C2216yes—
44#1439C2216yes—
45#1439C2216yes—
46#1439C2216yes—
47#1439C2216yescomplemented
48#1438C2216yes—
49C22⋊#95C2213yes—
50C22⋊#95C2213yes—
51C26C2311yescharacteristic
52C23×D4C2311yescharacteristic
53C23×D4C2312yes—
54C24×C4C2313yes—
55C24×C4C2313yes—
56C2×#120—46no—
57C22×C4⋊C4—212no—
58C22×C4⋊C4—412no—
59C22×C4⋊C4C2316yes—
60C22×C4⋊C4C2316yes—
61C22×C4⋊C4C2313yes—
62C22×C4⋊C4C2316yes—
63C22×C4⋊C4C2313yes—
64C22×C4⋊C4C2316yes—
65C22×C4⋊C4C2316yes—
66C22×C22⋊C4—212no—
67C22×C22⋊C4—412no—
68C22×C22⋊C4C2311yescharacteristic
69C22×C22⋊C4C2313yes—
70C22×C22⋊C4C2313yes—
71C22×C22⋊C4C2316yes—
72C22×C22⋊C4C2316yes—
73C22×C22⋊C4C2313yes—
74C22×C22⋊C4C2316yes—
75C22×C22⋊C4C2316yes—
76C22×C22⋊C4C2313yes—
77C22×C22⋊C4C2316yes—
78C22×C22⋊C4C2312yes—
79#401—24no—
80#400—212no—
81#398—412no—
82#397—212no—
83#396—24no—
84#396—28no—
85#396—224no—
86#396—412no—
87#396—412no—
88#394—412no—
89#386—212no—
90#386—46no—
91#385—212no—
92#385—46no—
93#380—46no—
94#380—46no—
95#379—46no—
96C2×#95D418yescomplemented
97C2×#95C2316yes—
98C2×#95C2316yes—
99C2×#95C2313yes—
100C2×#95C2313yes—
101C2×#95C2313yes—
102C2×#95C2313yes—
103C2×#95C2316yes—
104C2×#95C2316yes—
105C2×#95C2316yes—
106C2×#95C2316yes—
107C2×#95C2316yes—
108C2×#95C2316yes—
109C2×#95C2311yescharacteristic
110C2×#95C2313yes—
111C25—22no—
112C25—24no—
113C25—26no—
114C25—26no—
115C25—46no—
116C25C2411yescharacteristic
117C25C2412yes—
118C25C2411yescharacteristic
119C22×D4—22no—
120C22×D4—26no—
121C22×D4—26no—
122C22×D4—24no—
123C22×D4—46no—
124C22×D4—46no—
125C23×C4—26no—
126C23×C4—26no—
127C23×C4—26no—
128C23×C4—212no—
129C23×C4—26no—
130C23×C4—26no—
131C23×C4—46no—
132C23×C4—46no—
133C23×C4—46no—
134C23×C4C2×D4112yes—
135C23×C4C2413yes—
136C23×C4C2416yes—
137C23×C4C2413yes—
138C23×C4C2412yes—
139C23×C4C2411yescharacteristic
140C23×C4C2413yes—
141C23×C4C2413yes—
142C23×C4C2416yes—
143#120—416no—
144C2×C4⋊C4—212no—
145C2×C4⋊C4—212no—
146C2×C4⋊C4—212no—
147C2×C4⋊C4—212no—
148C2×C4⋊C4—26no—
149C2×C4⋊C4—26no—
150C2×C4⋊C4—212no—
151C2×C4⋊C4—46no—
152C2×C4⋊C4—46no—
153C2×C4⋊C4—448no—
154C2×C4⋊C4—46no—
155C2×C4⋊C4—46no—
156C2×C4⋊C4—46no—
157C2×C4⋊C4—46no—
158C2×C4⋊C4—46no—
159C2×C4⋊C4—46no—
160C2×C4⋊C4—46no—
161C2×C4⋊C4—46no—
162C2×C4⋊C4—46no—
163C2×C4⋊C4—46no—
164C2×C22⋊C4—26no—
165C2×C22⋊C4—26no—
166C2×C22⋊C4—26no—
167C2×C22⋊C4—26no—
168C2×C22⋊C4—26no—
169C2×C22⋊C4—26no—
170C2×C22⋊C4—26no—
171C2×C22⋊C4—212no—
172C2×C22⋊C4—212no—
173C2×C22⋊C4—212no—
174C2×C22⋊C4—212no—
175C2×C22⋊C4—212no—
176C2×C22⋊C4—26no—
177C2×C22⋊C4—26no—
178C2×C22⋊C4—212no—
179C2×C22⋊C4—46no—
180C2×C22⋊C4—46no—
181C2×C22⋊C4—46no—
182C2×C22⋊C4—448no—
183C2×C22⋊C4—46no—
184C2×C22⋊C4—46no—
185C2×C22⋊C4—46no—
186C2×C22⋊C4—46no—
187C2×C22⋊C4—46no—
188C2×C22⋊C4—46no—
189C2×C22⋊C4—46no—
190C2×C22⋊C4—46no—
191C2×C22⋊C4—46no—
192#95—26no—
193#95—26no—
194#95—26no—
195#95—26no—
196#95—212no—
197#95—212no—
198#95—46no—
199#95—46no—
200#95—46no—
201#95—46no—
202#95—46no—
203#95—46no—
204#95C2×D414yes—
205#95C2×D4112yes—
206C24—26no—
207C24—22no—
208C24—28no—
209C24—22no—
210C24—26no—
211C24—26no—
212C24—26no—
213C24—22no—
214C24—26no—
215C24—21no—
216C24—26no—
217C24—23no—
218C24—23no—
219C24—22no—
220C24—424no—
221C24—424no—
222C24—412no—
223C24—412no—
224C24—44no—
225C24—412no—
226C24—812no—
227C24—812no—
228C24C22×D412yes—
229C24H(5,𝔽2)16yes—
230C242−1+414yes—
231C242−1+416yes—
232C24C2511yescharacteristic
233C2×D4—48no—
234C2×D4—412no—
235C2×D4—44no—
236C2×D4—412no—
237C2×D4—812no—
238C2×D4—812no—
239C22×C4—224no—
240C22×C4—26no—
241C22×C4—26no—
242C22×C4—26no—
243C22×C4—23no—
244C22×C4—23no—
245C22×C4—26no—
246C22×C4—26no—
247C22×C4—26no—
248C22×C4—26no—
249C22×C4—26no—
250C22×C4—26no—
251C22×C4—23no—
252C22×C4—23no—
253C22×C4—26no—
254C22×C4—224no—
255C22×C4—21no—
256C22×C4—23no—
257C22×C4—23no—
258C22×C4—26no—
259C22×C4—26no—
260C22×C4—26no—
261C22×C4—26no—
262C22×C4—23no—
263C22×C4—26no—
264C22×C4—26no—
265C22×C4—23no—
266C22×C4—26no—
267C22×C4—26no—
268C22×C4—26no—
269C22×C4—26no—
270C22×C4—23no—
271C22×C4—26no—
272C22×C4—23no—
273C22×C4—23no—
274C22×C4—26no—
275C22×C4—26no—
276C22×C4—23no—
277C22×C4—22no—
278C22×C4—412no—
279C22×C4—412no—
280C22×C4—424no—
281C22×C4—424no—
282C22×C4—412no—
283C22×C4—412no—
284C22×C4—812no—
285C22×C4—812no—
286C22×C4C22×D416yes—
287C22×C4C22×D416yes—
288C22×C4H(5,𝔽2)112yes—
289C4⋊C4—424no—
290C4⋊C4—424no—
291C4⋊C4—86no—
292C4⋊C4—86no—
293C4⋊C4—812no—
294C4⋊C4—812no—
295C4⋊C4—812no—
296C22⋊C4—424no—
297C22⋊C4—424no—
298C22⋊C4—48no—
299C22⋊C4—412no—
300C22⋊C4—412no—
301C22⋊C4—44no—
302C22⋊C4—86no—
303C22⋊C4—86no—
304C22⋊C4—812no—
305C22⋊C4—812no—
306C22⋊C4—812no—
307C23—26no—
308C23—22no—
309C23—212no—
310C23—212no—
311C23—212no—
312C23—212no—
313C23—24no—
314C23—212no—
315C23—212no—
316C23—448no—
317C23—46no—
318C23—46no—
319C23—42no—
320C23—46no—
321C23—46no—
322C23—412no—
323C23—46no—
324C23—46no—
325C23—46no—
326C23—43no—
327C23—43no—
328C23—41no—
329C23—41no—
330C23—43no—
331C23—46no—
332C23—46no—
333C23—46no—
334C23—46no—
335C23—816no—
336C23—848no—
337C23—824no—
338C23—88no—
339C23#537112yes—
340C23#56014yes—
341C23#563112yes—
342C23C23×D411yescharacteristic
343C23C2×H(5,𝔽2)13yes—
344C23C2×H(5,𝔽2)16yes—
345C23C2×2−1+413yes—
346C23C2×2−1+412yes—
347D4—816no—
348D4—88no—
349C2×C4—26no—
350C2×C4—26no—
351C2×C4—26no—
352C2×C4—212no—
353C2×C4—26no—
354C2×C4—26no—
355C2×C4—26no—
356C2×C4—212no—
357C2×C4—212no—
358C2×C4—448no—
359C2×C4—46no—
360C2×C4—46no—
361C2×C4—46no—
362C2×C4—46no—
363C2×C4—46no—
364C2×C4—46no—
365C2×C4—46no—
366C2×C4—46no—
367C2×C4—43no—
368C2×C4—43no—
369C2×C4—43no—
370C2×C4—43no—
371C2×C4—43no—
372C2×C4—43no—
373C2×C4—43no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—46no—
377C2×C4—43no—
378C2×C4—46no—
379C2×C4—46no—
380C2×C4—46no—
381C2×C4—46no—
382C2×C4—46no—
383C2×C4—46no—
384C2×C4—46no—
385C2×C4—46no—
386C2×C4—46no—
387C2×C4—46no—
388C2×C4—46no—
389C2×C4—46no—
390C2×C4—824no—
391C22—212no—
392C22—212no—
393C22—28no—
394C22—224no—
395C22—412no—
396C22—44no—
397C22—412no—
398C22—832no—
399C22—82no—
400C22—86no—
401C22—82no—
402C22—86no—
403C22—81no—
404C22—83no—
405C22—83no—
406C22—82no—
407C22—86no—
408C22#282318yes—
409C22C2×#53713yes—
410C22#348413yes—
411C22#348611yescharacteristic
412C22C2×#56011yescharacteristic
413C22C2×#56313yes—
414C22#353616yes—
415C22#353616yes—
416C22#353716yes—
417C22#353816yes—
418C4—412no—
419C4—46no—
420C4—46no—
421C4—81no—
422C4—83no—
423C4—83no—
424C4—86no—
425C4—82no—
426C2—28no—
427C2—84no—
428C2—161no—
429C2C2×#282311yescharacteristic
430C2#3670613yes—
431C2#3671913yes—
432C2#5247212yes—
433C2#5247316yes—
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×#2643C2semidirect
2. C22×#396C2semidirect
3. #17213C2semidirect
4. #17135C2semidirect
5. #17133C2semidirect
6. #17096C2semidirect
7. #14729C2semidirect
8. #14728C2semidirect
9. #14726C2semidirect
11. #14353C2semidirect
16. C2×#401C22semidirect
17. C2×#400C22semidirect
18. C2×#398C22semidirect
19. C2×#397C22semidirect
24. C2×#396C22semidirect
25. C2×#396C22semidirect
35. #1444C22semidirect
36. #1444C22semidirect
37. #1444C22semidirect
38. #1442C22semidirect
39. #1441C22semidirect
40. #1441C22semidirect
41. #1441C22semidirect
47. #1439C22semidirect
96. C2×#95D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26432
C22×#3962
#172132
#171352
#171332
#170962
#147292
#147282
#147262
#147252
#143532