Finite Group Databasebeta

← #513177

Subgroups — #513177

Order 512.

439 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2264C211yesmaximal, characteristic, complemented
2C22×SD64C211yesmaximal, characteristic, complemented
3D4×Q32C211yesmaximal, characteristic, complemented
4#33846C211yesmaximal, characteristic
5#13730C211yesmaximal, characteristic, complemented
6#13699C211yesmaximal, characteristic, complemented
7#13695C211yesmaximal, characteristic, complemented
8#13690C211yesmaximal, characteristic, complemented
9#13687C211yesmaximal, characteristic, complemented
10#13666C211yesmaximal, characteristic, complemented
11#13665C211yesmaximal, characteristic, complemented
12#13664C211yesmaximal, characteristic, complemented
13#13662C211yesmaximal, characteristic, complemented
14#13652C211yesmaximal, characteristic, complemented
15#13645C211yesmaximal, characteristic, complemented
16C2×#508C2211yescharacteristic
17C22×Q32C2211yescharacteristic
18C22×Q32C2211yescharacteristic, complemented
19C22×D16C2211yescharacteristic
20D4×Q16—21no—
21#3280—21no—
22#2264—24no—
23C2×Q64C2211yescharacteristic, complemented
24C2×Q64C2211yescharacteristic, complemented
25C2×Q64C2211yescharacteristic, complemented
26C2×SD64—24no—
27C2×SD64C2211yescharacteristic
28C2×SD64C2211yescharacteristic
29C2×SD64C2211yescharacteristic
30C2×SD64C2211yescharacteristic
31C2×SD64C2211yescharacteristic
32C2×SD64C2211yescharacteristic
33C2×M64C2211yescharacteristic, complemented
34C22×C32C2211yescharacteristic
35#2249C2211yescharacteristic
36#2247C2211yescharacteristic, complemented
37#2216C2211yescharacteristic
38#2216C2211yescharacteristic
39#2216C2211yescharacteristic, complemented
40#2215C2211yescharacteristic
41#2210—21no—
42#2209—21no—
43#2188—21no—
44#2188—21no—
45#2187—21no—
46#2186—21no—
47C4×Q32C2211yescharacteristic, complemented
48C4×D16C2211yescharacteristic
49D4×C16C2211yescharacteristic
50#1423C2211yescharacteristic, complemented
51C4⋊C32C2211yescharacteristic, complemented
52#1416C2211yescharacteristic, complemented
53#1416C2211yescharacteristic, complemented
54#1416C2211yescharacteristic, complemented
55#1416C2211yescharacteristic, complemented
56#1415C2211yescharacteristic
57#1415C2211yescharacteristic
58C22⋊C32C2211yescharacteristic
59C22⋊C32C2211yescharacteristic, complemented
60#1397C2211yescharacteristic, complemented
61C2×#135—21no—
62C22×Q16—21no—
63C22×Q16—21no—
64C22×D8—21no—
65D4×Q8—41no—
66#546—41no—
67#508—22no—
68#508—41no—
69C2×Q32—22no—
70C2×Q32—22no—
71C2×Q32—21no—
72C2×Q32—21no—
73C2×Q32—21no—
74C2×Q32—41no—
75C2×Q32—41no—
76C2×Q32C2311yescharacteristic
77C2×Q32C2311yescharacteristic
78C2×Q32C2311yescharacteristic
79C2×Q32C2311yescharacteristic
80C2×SD32—21no—
81C2×SD32—21no—
82C2×D16—22no—
83C2×D16—21no—
84C2×D16—41no—
85C2×D16C2311yescharacteristic
86C2×D16C2311yescharacteristic
87C22×C16C2311yescharacteristic
88C22×C16C2311yescharacteristic
89#495—21no—
90#494—21no—
91#467—21no—
92#467—21no—
93#467—21no—
94#466—21no—
95#463—41no—
96#462—41no—
97#451—41no—
98#451—41no—
99#450—41no—
100#449—41no—
101C4×Q16—21no—
102C4×D8—21no—
103C8×D4D411yescharacteristic
104Q64—22no—
105Q64—22no—
106Q64—41no—
107SD64—22no—
108SD64—22no—
109SD64—22no—
110SD64—22no—
111SD64—41no—
112SD64D414yescomplemented
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#358—21no—
125#358—21no—
126#357—21no—
127#357—21no—
128C22⋊C16D411yescharacteristic
129C22⋊C16D411yescharacteristic
130C4×C16C2311yescharacteristic
131C2×#41—41no—
132C22×Q8—41no—
133C22×Q8—41no—
134C22×D4—41no—
135#135—42no—
136#135—81no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—42no—
142C2×Q16—42no—
143C2×Q16—41no—
144C2×Q16—41no—
145C2×Q16—41no—
146C2×Q16—81no—
147C2×Q16—81no—
148C2×SD16—41no—
149C2×SD16—41no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—42no—
153C2×D8—41no—
154C2×D8—81no—
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C4⋊Q8—41no—
158C4⋊Q8—81no—
159#124—41no—
160#123—81no—
161C22⋊Q8—41no—
162C22⋊Q8—41no—
163C22⋊Q8—41no—
164C22⋊Q8—81no—
165C22⋊Q8—81no—
166#121—41no—
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—
177Q32—21no—
178Q32—21no—
179Q32—21no—
180Q32—21no—
181Q32—42no—
182Q32—42no—
183Q32—81no—
184Q32C2×D412yes—
185SD32—42no—
186SD32—81no—
187D16—21no—
188D16—21no—
189D16—21no—
190D16—42no—
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#103—41no—
206#103—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—161no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—82no—
224C2×Q8—82no—
225C2×Q8—81no—
226C2×Q8—81no—
227C2×Q8—161no—
228C2×Q8—161no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—82no—
232C2×D4—81no—
233C2×D4—81no—
234C2×D4—81no—
235C2×D4—161no—
236C2×D4D1611yescharacteristic
237C22×C4—81no—
238C22×C4—81no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4C2×D811yescharacteristic
242C22×C4C2×D811yescharacteristic
243Q16—22no—
244Q16—41no—
245Q16—41no—
246Q16—41no—
247Q16—41no—
248Q16—41no—
249Q16—41no—
250Q16—41no—
251Q16—82no—
252Q16—82no—
253Q16—161no—
254SD16—82no—
255SD16—161no—
256D8—22no—
257D8—41no—
258D8—41no—
259D8—41no—
260D8—82no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8C2×D811yescharacteristic
266C2×C8C2×D811yescharacteristic
267C2×C8C2×D811yescharacteristic
268C2×C8C22×D411yescharacteristic
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—81no—
272C4⋊C4—81no—
273C4⋊C4—81no—
274C4⋊C4—81no—
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—162no—
297C23—162no—
298C23—321no—
299C23C2×D1611yescharacteristic
300C23C2×D1611yescharacteristic
301Q8—42no—
302Q8—81no—
303Q8—81no—
304Q8—81no—
305Q8—81no—
306Q8—81no—
307Q8—81no—
308Q8—81no—
309Q8—162no—
310Q8—162no—
311D4—22no—
312D4—42no—
313D4—81no—
314D4—81no—
315D4—81no—
316D4—162no—
317D4—162no—
318D4—321no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—162no—
330C2×C4—162no—
331C2×C4—162no—
332C2×C4—321no—
333C2×C4C2×D1611yescharacteristic
334C2×C4C2×D1611yescharacteristic
335C2×C4C2×D1611yescharacteristic
336C2×C4C2×D1611yescharacteristic
337C2×C4C22×D811yescharacteristic
338C8—21no—
339C8—21no—
340C8—21no—
341C8D4211yescharacteristic
342C8#57611yescharacteristic
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—82no—
349C22—161no—
350C22—161no—
351C22—161no—
352C22—161no—
353C22—161no—
354C22—324no—
355C22#226412yescomplemented
356C22C22×D1611yescharacteristic
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—82no—
362C4—161no—
363C4—161no—
364C4—161no—
365C4—161no—
366C4—161no—
367C4D4×D811yescharacteristic
368C4#341511yescharacteristic
369C2—22no—
370C2—41no—
371C2—162no—
372C2—321no—
373C2D4×D1611yescharacteristic
374C2C2×#226411yescharacteristic
375C2#3398311yescharacteristic
subgroupG/H
center, socleC22C22×D16
commutator, Frattini, cosocleC2×C16C24
prime radical#33846C2
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×#2264C2semidirect
2. C22×SD64C2semidirect
3. D4×Q32C2semidirect
5. #13730C2semidirect
6. #13699C2semidirect
7. #13695C2semidirect
8. #13690C2semidirect
9. #13687C2semidirect
10. #13666C2semidirect
11. #13665C2semidirect
12. #13664C2semidirect
13. #13662C2semidirect
14. #13652C2semidirect
15. #13645C2semidirect
18. C22×Q32C22semidirect
23. C2×Q64C22semidirect
24. C2×Q64C22semidirect
25. C2×Q64C22semidirect
33. C2×M64C22semidirect
36. #2247C22semidirect
39. #2216C22semidirect
47. C4×Q32C22semidirect
50. #1423C22semidirect
51. C4⋊C32C22semidirect
52. #1416C22semidirect
53. #1416C22semidirect
54. #1416C22semidirect
55. #1416C22semidirect
59. C22⋊C32C22semidirect
60. #1397C22semidirect
112. SD64D4semidirect
355. C22#2264semidirect

Maximal subgroups

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

subgroupindex
C2×#22642
C22×SD642
D4×Q322
#338462
#137302
#136992
#136952
#136902
#136872
#136662
#136652
#136642
#136622
#136522
#136452