Finite Group Databasebeta

← #513176

Subgroups — #513176

Order 512.

447 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2263C211yesmaximal, characteristic, complemented
2C2×#2262C211yesmaximal, characteristic, complemented
3#33853C211yesmaximal, characteristic, complemented
4#33846C211yesmaximal, characteristic, complemented
5#13729C211yesmaximal, characteristic, complemented
6#13699C211yesmaximal, characteristic, complemented
7#13694C211yesmaximal, characteristic, complemented
8#13690C211yesmaximal, characteristic, complemented
9#13685C211yesmaximal, characteristic, complemented
10#13666C211yesmaximal, characteristic, complemented
11#13663C211yesmaximal, characteristic, complemented
12#13663C211yesmaximal, characteristic, complemented
13#13661C211yesmaximal, characteristic, complemented
14#13652C211yesmaximal, characteristic, complemented
15#13645C211yesmaximal, characteristic, complemented
16C2×#508C2211yescharacteristic
17C2×#508C2211yescharacteristic
18C2×#508C2211yescharacteristic, complemented
19C22×D16C2211yescharacteristic
20#3287—21no—
21#3280—21no—
22#2263—24no—
23#2262—24no—
24C2×Q64C2211yescharacteristic, complemented
25C2×SD64C2211yescharacteristic
26C2×SD64C2211yescharacteristic
27C2×SD64C2211yescharacteristic, complemented
28C2×SD64C2211yescharacteristic, complemented
29C2×D32C2211yescharacteristic
30C2×D32C2211yescharacteristic
31C2×D32C2211yescharacteristic
32C2×D32C2211yescharacteristic
33C2×M64C2211yescharacteristic
34C22×C32C2211yescharacteristic, complemented
35#2249C2211yescharacteristic
36#2246C2211yescharacteristic, complemented
37#2216C2211yescharacteristic
38#2215C2211yescharacteristic
39#2215C2211yescharacteristic
40#2215C2211yescharacteristic, complemented
41#2210—21no—
42#2207—21no—
43#2187—21no—
44#2186—21no—
45#2186—21no—
46#2186—21no—
47C4×Q32C2211yescharacteristic, complemented
48C4×D16C2211yescharacteristic
49D4×C16C2211yescharacteristic
50#1423C2211yescharacteristic, complemented
51C4⋊C32C2211yescharacteristic, complemented
52#1416C2211yescharacteristic, complemented
53#1416C2211yescharacteristic, complemented
54#1415C2211yescharacteristic
55#1415C2211yescharacteristic
56#1415C2211yescharacteristic, complemented
57#1415C2211yescharacteristic, complemented
58C22⋊C32C2211yescharacteristic
59C22⋊C32C2211yescharacteristic, complemented
60#1397C2211yescharacteristic, complemented
61C2×#135—21no—
62C2×#135—21no—
63C2×#135—21no—
64C22×D8—21no—
65#550—41no—
66#546—41no—
67#508—22no—
68#508—22no—
69#508—22no—
70#508—41no—
71#508—41no—
72#508—41no—
73C2×Q32C2311yescharacteristic
74C2×Q32C2311yescharacteristic
75C2×SD32—21no—
76C2×SD32—21no—
77C2×SD32—21no—
78C2×SD32—21no—
79C2×D16—22no—
80C2×D16—21no—
81C2×D16—21no—
82C2×D16—41no—
83C2×D16C2311yescharacteristic
84C2×D16C2311yescharacteristic
85C2×D16C2311yescharacteristic
86C2×D16C2311yescharacteristic
87C22×C16C2311yescharacteristic
88C22×C16C2311yescharacteristic
89#495—21no—
90#493—21no—
91#467—21no—
92#466—21no—
93#466—21no—
94#466—21no—
95#463—41no—
96#460—41no—
97#450—41no—
98#449—41no—
99#449—41no—
100#449—41no—
101C4×Q16—21no—
102C4×D8—21no—
103C8×D4D411yescharacteristic
104Q64—22no—
105SD64—22no—
106SD64—22no—
107SD64—41no—
108SD64D414yescomplemented
109D32—22no—
110D32—22no—
111D32—22no—
112D32—41no—
113M64—22no—
114C2×C32—22no—
115C2×C32C2311yescharacteristic
116C2×C32C2311yescharacteristic
117C2×C32C2311yescharacteristic
118C2×C32C2311yescharacteristic
119#366C2311yescharacteristic
120#366C2311yescharacteristic
121C4⋊C16D411yescharacteristic
122#358—21no—
123#358—21no—
124#357—21no—
125#357—21no—
126#357—21no—
127#357—21no—
128C22⋊C16D411yescharacteristic
129C22⋊C16D411yescharacteristic
130C4×C16C2311yescharacteristic
131C2×#41—41no—
132C2×#41—41no—
133C2×#41—41no—
134C22×D4—41no—
135#135—42no—
136#135—42no—
137#135—42no—
138#135—81no—
139#135—81no—
140#135—81no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×SD16—41no—
144C2×SD16—41no—
145C2×SD16—41no—
146C2×SD16—41no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—42no—
152C2×D8—41no—
153C2×D8—41no—
154C2×D8—81no—
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157#127—41no—
158#127—81no—
159#124—41no—
160#123—81no—
161C22⋊Q8—41no—
162#121—41no—
163#121—41no—
164#121—41no—
165#121—81no—
166#121—81no—
167#121—81no—
168#120—81no—
169C4×Q8—41no—
170C4×D4—41no—
171C4×D4—81no—
172C4×D4D811yescharacteristic
173C2×C22⋊C4—81no—
174Q32—21no—
175Q32—21no—
176Q32—21no—
177Q32C2×D412yes—
178SD32—42no—
179SD32—42no—
180SD32—42no—
181SD32—81no—
182D16—21no—
183D16—21no—
184D16—21no—
185D16—21no—
186D16—21no—
187D16—21no—
188D16—21no—
189D16—42no—
190D16—81no—
191D16C2×D412yes—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16C2×D411yescharacteristic
197C2×C16C2×D411yescharacteristic
198C2×C16C2×D411yescharacteristic
199C2×C16C2411yescharacteristic
200#107—21no—
201#107—21no—
202C4⋊C8D811yescharacteristic
203#103—41no—
204#103—41no—
205#102—41no—
206#102—41no—
207#102—41no—
208#102—41no—
209C22⋊C8D811yescharacteristic
210C22⋊C8D811yescharacteristic
211C4×C8C2×D411yescharacteristic
212C32—21no—
213C32—21no—
214C32—21no—
215C32C2×D412yes—
216C24—81no—
217#41—82no—
218#41—82no—
219#41—82no—
220#41—161no—
221#41—161no—
222#41—161no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—82no—
230C2×D4—81no—
231C2×D4—81no—
232C2×D4—81no—
233C2×D4—81no—
234C2×D4—81no—
235C2×D4—81no—
236C2×D4—81no—
237C2×D4—81no—
238C2×D4—81no—
239C2×D4—161no—
240C2×D4D1611yescharacteristic
241C22×C4—81no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4C2×D811yescharacteristic
246C22×C4C2×D811yescharacteristic
247Q16—22no—
248Q16—41no—
249Q16—41no—
250Q16—41no—
251SD16—82no—
252SD16—82no—
253SD16—82no—
254SD16—161no—
255D8—22no—
256D8—41no—
257D8—41no—
258D8—41no—
259D8—41no—
260D8—41no—
261D8—41no—
262D8—41no—
263D8—82no—
264D8—161no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8C2×D811yescharacteristic
270C2×C8C2×D811yescharacteristic
271C2×C8C2×D811yescharacteristic
272C2×C8C22×D411yescharacteristic
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—81no—
276C4⋊C4—81no—
277C4⋊C4D1611yescharacteristic
278C22⋊C4—81no—
279C22⋊C4—81no—
280C22⋊C4—81no—
281C22⋊C4—81no—
282C22⋊C4—81no—
283C22⋊C4—81no—
284C22⋊C4—81no—
285C22⋊C4D1611yescharacteristic
286C22⋊C4D1611yescharacteristic
287C42—81no—
288C42C2×D811yescharacteristic
289C16—21no—
290C16—21no—
291C16—21no—
292C16C22×D411yescharacteristic
293C16H(5,𝔽2)11yescharacteristic
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—162no—
299C23—162no—
300C23—321no—
301C23C2×D1611yescharacteristic
302C23C2×D1611yescharacteristic
303Q8—42no—
304Q8—81no—
305Q8—81no—
306Q8—81no—
307D4—41no—
308D4—42no—
309D4—81no—
310D4—81no—
311D4—81no—
312D4—81no—
313D4—81no—
314D4—81no—
315D4—81no—
316D4—162no—
317D4—162no—
318D4—162no—
319D4—162no—
320D4—162no—
321D4—321no—
322D4—321no—
323D4—321no—
324D4—321no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—162no—
334C2×C4—162no—
335C2×C4—162no—
336C2×C4—321no—
337C2×C4C2×D1611yescharacteristic
338C2×C4C2×D1611yescharacteristic
339C2×C4C2×D1611yescharacteristic
340C2×C4C2×D1611yescharacteristic
341C2×C4C22×D811yescharacteristic
342C8—21no—
343C8—21no—
344C8—21no—
345C8D4211yescharacteristic
346C8#57611yescharacteristic
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—82no—
354C22—161no—
355C22—161no—
356C22—161no—
357C22—161no—
358C22—161no—
359C22—161no—
360C22—161no—
361C22—161no—
362C22—161no—
363C22—161no—
364C22—161no—
365C22—324no—
366C22C22×D1611yescharacteristic
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—82no—
372C4—161no—
373C4—161no—
374C4—161no—
375C4D4×D811yescharacteristic
376C4#341511yescharacteristic
377C2—41no—
378C2—41no—
379C2—162no—
380C2—321no—
381C2—321no—
382C2—321no—
383C2D4×D1611yescharacteristic
384C2#3398111yescharacteristic
385C2#3398211yescharacteristic
subgroupG/H
center, socleC22C22×D16
commutator, Frattini, cosocleC2×C16C24
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16 ◁ G
LECSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#2263C2semidirect
2. C2×#2262C2semidirect
3. #33853C2semidirect
4. #33846C2semidirect
5. #13729C2semidirect
6. #13699C2semidirect
7. #13694C2semidirect
8. #13690C2semidirect
9. #13685C2semidirect
10. #13666C2semidirect
11. #13663C2semidirect
12. #13663C2semidirect
13. #13661C2semidirect
14. #13652C2semidirect
15. #13645C2semidirect
18. C2×#508C22semidirect
24. C2×Q64C22semidirect
27. C2×SD64C22semidirect
28. C2×SD64C22semidirect
34. C22×C32C22semidirect
36. #2246C22semidirect
40. #2215C22semidirect
47. C4×Q32C22semidirect
50. #1423C22semidirect
51. C4⋊C32C22semidirect
52. #1416C22semidirect
53. #1416C22semidirect
56. #1415C22semidirect
57. #1415C22semidirect
59. C22⋊C32C22semidirect
60. #1397C22semidirect
108. SD64D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22632
C2×#22622
#338532
#338462
#137292
#136992
#136942
#136902
#136852
#136662
#136632
#136612
#136522
#136452