Finite Group Databasebeta

← #225538

Subgroups — #225538

Order 512.

716 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22758C212yesmaximal
2#22708C211yesmaximal, characteristic, complemented
3#22705C211yesmaximal, characteristic, complemented
4C2×C8×D8C211yesmaximal, characteristic
5#9619C212yesmaximal, complemented
6#8897C212yesmaximal, complemented
7#8894C212yesmaximal, complemented
8#8817C211yesmaximal, characteristic, complemented
9#8648C211yesmaximal, characteristic, complemented
10#8648C211yesmaximal, characteristic, complemented
11#8282C211yesmaximal, characteristic, complemented
12C2×C4×D8C2211yescharacteristic
13C2×C8×D4C2211yescharacteristic
14C2×C8×D4C2211yescharacteristic
15#2354C2212yes—
16D4×C4⋊C4C2212yes—
17#1948C2212yes—
18#1947C2212yes—
19#1936—22no—
20#1921—22no—
21#1879—22no—
22#1854—22no—
23#1852—22no—
24#1834C2212yes—
25#1834C2211yescharacteristic
26#1834C2211yescharacteristic
27#1834C2212yes—
28#1808—21no—
29#1807—21no—
30#1802C2212yes—
31#1802C2212yes—
32C4×#107C2211yescharacteristic
33#1770C2212yes—
34#1762—22no—
35#1735—21no—
36C2×#340—21no—
37C8×D8C418yescomplemented
38C2×#335C2211yescharacteristic
39C2×#335C2211yescharacteristic
40C2×D4⋊C8C2211yescharacteristic
41C2×D4⋊C8C2211yescharacteristic
42C2×C82C2211yescharacteristic
43#1296C2212yes—
44#1296C2212yes—
45#1279C2212yes—
46C22×D8D411yescharacteristic
47C23×C8—21no—
48C23×C8—21no—
49C2×C4×D4—22no—
50C2×C4×D4C2311yescharacteristic
51C2×C4×D4C2311yescharacteristic
52C2×#117—22no—
53C22×C4⋊C4—22no—
54C4×D8C2×C414yes—
55C4×D8Q814yes—
56C8×D4—22no—
57C8×D4—22no—
58C8×D4C2×C414yes—
59C8×D4C2×C414yescomplemented
60C2×#107—22no—
61C2×#107D412yes—
62C2×#107D411yescharacteristic
63C2×#106—22no—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2311yescharacteristic
67C2×#102—22no—
68C2×#102—22no—
69C2×#102D411yescharacteristic
70C2×#102D411yescharacteristic
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×#97—22no—
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77#390—22no—
78#389—22no—
79#388—42no—
80#385—42no—
81#384C2312yes—
82#384C2312yes—
83#380—22no—
84#380—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4C2312yes—
87#340—21no—
88#340—21no—
89#340—21no—
90#340—21no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—42no—
98#340D411yescharacteristic
99#340D411yescharacteristic
100#336—22no—
101#335—22no—
102#335C2×C414yescomplemented
103D4⋊C8C2×C414yes—
104D4⋊C8C2×C414yescomplemented
105C82C2×C414yescomplemented
106C22×D4C2×D411yescharacteristic
107C22×D4C2×D411yescharacteristic
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—42no—
111C2×D8—22no—
112C2×D8—22no—
113C2×D8C4⋊C414yes—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—24no—
118C22×C8—24no—
119C22×C8—22no—
120C22×C8—82no—
121C22×C8—82no—
122C22×C8C2×D411yescharacteristic
123C22×C8C2×D411yescharacteristic
124C22×C8C2×D411yescharacteristic
125C22×C8C2×D411yescharacteristic
126C4×D4—21no—
127C4×D4—21no—
128C4×D4—21no—
129C4×D4—21no—
130C4×D4—44no—
131C4×D4—82no—
132C4×D4C22×C412yes—
133C4×D4C22×C412yes—
134C4×D4C2×Q812yes—
135C4×D4C2×Q812yes—
136#117—44no—
137#117—82no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—82no—
147C2×C4⋊C4C2×D412yes—
148C2×C4⋊C4C2×D412yes—
149C2×C4⋊C4C2×D411yescharacteristic
150C2×C4⋊C4C2×D411yescharacteristic
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—42no—
156C2×C42—22no—
157C2×C42—42no—
158C2×C42C2411yescharacteristic
159#107—42no—
160#107—82no—
161#107C4⋊C414yes—
162#106—82no—
163C4⋊C8—21no—
164C4⋊C8—21no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8C22×C412yes—
168C4⋊C8C22×C412yes—
169#102—44no—
170#102—82no—
171#102C4⋊C414yes—
172#102C4⋊C414yes—
173C22⋊C8—24no—
174C22⋊C8—24no—
175#97—82no—
176C4×C8—21no—
177C4×C8C22×C412yes—
178C4×C8C22×C412yes—
179C4×C8C22×C412yes—
180C4×C8C2×Q812yes—
181C4×C8#4112yes—
182#95—22no—
183#95—42no—
184#95—42no—
185C24—21no—
186C24—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4C2×C4⋊C412yes—
196C2×D4C2×C4⋊C412yes—
197C2×D4#13512yes—
198C2×D4#13512yes—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—84no—
214C22×C4—84no—
215C22×C4—162no—
216C22×C4C22×D411yescharacteristic
217C22×C4C22×D411yescharacteristic
218C22×C4H(5,𝔽2)11yescharacteristic
219D8—28no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—22no—
223C2×C8—21no—
224C2×C8—22no—
225C2×C8—21no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—84no—
236C2×C8—84no—
237C2×C8C2×C4⋊C412yes—
238C2×C8C2×C4⋊C412yes—
239C2×C8C4×D412yes—
240C2×C8C4×D412yes—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C4⋊C4—84no—
258C4⋊C4C2×C4⋊C412yes—
259C4⋊C4C2×C4⋊C412yes—
260C22⋊C4—22no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—22no—
264C22⋊C4—84no—
265C22⋊C4—84no—
266C22⋊C4—162no—
267C42—42no—
268C42—42no—
269C42—84no—
270C42C23×C411yescharacteristic
271C42C22×Q811yescharacteristic
272C42C2×#4111yescharacteristic
273C422−1+411yescharacteristic
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—41no—
282C23—41no—
283C23—42no—
284C23D4211yescharacteristic
285D4—22no—
286D4—22no—
287D4—22no—
288D4—22no—
289D4—22no—
290D4—41no—
291D4—41no—
292D4#42714yes—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—168no—
317C2×C4C22×C4⋊C411yescharacteristic
318C2×C4C2×C4×D411yescharacteristic
319C2×C4#52011yescharacteristic
320C2×C4#54711yescharacteristic
321C2×C4D4×Q811yescharacteristic
322C2×C4#55411yescharacteristic
323C2×C4C2×#13511yescharacteristic
324C2×C4C2×#13511yescharacteristic
325C2×C4#57512yes—
326C2×C4#57611yescharacteristic
327C2×C4#57612yes—
328C2×C4#57811yescharacteristic
329C8—22no—
330C8—22no—
331C8—22no—
332C8—22no—
333C8—42no—
334C8—42no—
335C8—42no—
336C8—82no—
337C22—24no—
338C22—42no—
339C22—42no—
340C22—44no—
341C22—44no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—82no—
346C22—82no—
347C22D4×C4⋊C411yescharacteristic
348C22#327312yes—
349C22#328011yescharacteristic
350C22#328511yescharacteristic
351C22#329012yes—
352C4—21no—
353C4—21no—
354C4—42no—
355C4—42no—
356C4—82no—
357C4—82no—
358C4—162no—
359C4—162no—
360C4C2×#42711yescharacteristic
361C4#291311yescharacteristic
362C4#294311yescharacteristic
363C4#294811yescharacteristic
364C4#333611yescharacteristic
365C4#333911yescharacteristic
366C4#338912yes—
367C2—44no—
368C2—82no—
369C2#1989912yes—
370C2#1993412yes—
371C2#2265611yescharacteristic
372C2#2267211yescharacteristic
373C2#2269011yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D8D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22708C2semidirect
3. #22705C2semidirect
5. #9619C2semidirect
6. #8897C2semidirect
7. #8894C2semidirect
8. #8817C2semidirect
9. #8648C2semidirect
10. #8648C2semidirect
11. #8282C2semidirect
37. C8×D8C4semidirect
59. C8×D4C2×C4semidirect
102. #335C2×C4semidirect
104. D4⋊C8C2×C4semidirect
105. C82C2×C4semidirect

Maximal subgroups

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

subgroupindex
#227582
#227082
#227052
C2×C8×D82
#96192
#88972
#88942
#88172
#86482
#82822