Finite Group Databasebeta

← #363641

Subgroups — #363641

Order 512.

661 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20368C211yesmaximal, characteristic
2#11299C212yesmaximal
3#11259C212yesmaximal
4#10910C212yesmaximal
5#10867C211yesmaximal, characteristic
6#10824C211yesmaximal, characteristic
7#10824C211yesmaximal, characteristic
8#10814C211yesmaximal, characteristic
9#10803C211yesmaximal, characteristic
10#10799C212yesmaximal
11#10711C211yesmaximal, characteristic
12C22×#108C2212yes—
13C2×#405C2212yes—
14C2×#405C2211yescharacteristic
15C2×#405C2211yescharacteristic
16C4×#117C2211yescharacteristic
17#1746C412yescomplemented
18#1746C412yescomplemented
19#1745C412yescomplemented
20#1744C412yescomplemented
21C2×#344C2212yes—
22C2×#344C2212yes—
23#1742C2211yescharacteristic
24#1742C2211yescharacteristic
25#1740C2211yescharacteristic
26#1740C2211yescharacteristic
27#1735C2212yes—
28C2×#339C2212yes—
29C2×#339C2212yes—
30C2×#338C2212yes—
31C2×H(3,ℤ/4)C2212yes—
32C2×H(3,ℤ/4)C2212yes—
33C2×H(3,ℤ/4)C2212yes—
34#1729C2212yes—
35#1729C2211yescharacteristic
36#1729C2211yescharacteristic
37#1728C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C22×M16C2312yes—
40C22×M16C2312yes—
41C22×M16C2311yescharacteristic
42C22×M16C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C2×#117C2312yes—
46C2×#117C2312yes—
47C2×#117C2311yescharacteristic
48C2×#117C2311yescharacteristic
49C22×C42C2311yescharacteristic
50C2×#108—24no—
51C2×#108—24no—
52C2×#108—42no—
53#405—22no—
54#405—22no—
55#405—24no—
56#405—24no—
57#405—42no—
58C2×#97—22no—
59C2×#97C2×C412yes—
60C2×#97C2×C412yes—
61C2×C4×C8—22no—
62C2×C4×C8C2×C412yes—
63C2×C4×C8C2×C412yes—
64C4×C4⋊C4—21no—
65C4×C4⋊C4—21no—
66C4×C22⋊C4—22no—
67#376C2×C412yes—
68C43C2×C412yes—
69#344—24no—
70#344—42no—
71#343—21no—
72#343—21no—
73#343—21no—
74#343—21no—
75#342—22no—
76#342—22no—
77#340—24no—
78#339—24no—
79#339—24no—
80#338—42no—
81H(3,ℤ/4)—24no—
82H(3,ℤ/4)—42no—
83H(3,ℤ/4)—42no—
84#336—22no—
85#336—22no—
86#336C2×C411yescharacteristic
87#336C2×C411yescharacteristic
88#336C2×C411yescharacteristic
89#336C2×C411yescharacteristic
90#336C2×C412yes—
91#336C2×C412yes—
92#336C2×C412yes—
93#336C2×C412yes—
94#336C2×C412yes—
95#336C2×C412yes—
96C23×C4—21no—
97C23×C4C2411yescharacteristic
98C2×M16—22no—
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—21no—
104C2×M16—21no—
105C2×M16—21no—
106C2×M16—21no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16#4114yes—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8C22×C412yes—
127C22×C8C22×C412yes—
128C22×C8C22×C411yescharacteristic
129C22×C8C22×C411yescharacteristic
130C22×C8C22×C411yescharacteristic
131C22×C8C22×C411yescharacteristic
132C22×C8#4112yes—
133C22×C8#4112yes—
134C22×C8#4114yes—
135#117—22no—
136#117—22no—
137#117—21no—
138#117—21no—
139#117—22no—
140#117—21no—
141#117—21no—
142#117—22no—
143#117—22no—
144#117—22no—
145#117—22no—
146#117—22no—
147#117—22no—
148#117#4114yes—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4#4111yescharacteristic
153C2×C4⋊C4#4111yescharacteristic
154C2×C22⋊C4—22no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4#4112yes—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—21no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42C22×C412yes—
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C411yescharacteristic
168C2×C42C22×C412yes—
169#108—48no—
170#97—24no—
171#97—24no—
172#97—44no—
173C4×C8—24no—
174C4×C8—22no—
175C4×C8—22no—
176C4×C8—44no—
177#95—22no—
178#95—22no—
179C24C2×#4111yescharacteristic
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—81no—
188C22×C4#11712yes—
189C22×C4#11712yes—
190C22×C4C23×C411yescharacteristic
191C22×C4C2×#4112yes—
192C22×C4C2×#4112yes—
193C22×C4C2×#4112yes—
194C22×C4C2×#4112yes—
195C22×C4C2×#4111yescharacteristic
196C22×C4H(5,𝔽2)11yescharacteristic
197C22×C4H(5,𝔽2)11yescharacteristic
198C22×C42−1+411yescharacteristic
199C22×C42−1+411yescharacteristic
200M16—24no—
201M16—24no—
202M16—42no—
203M16—42no—
204M16—42no—
205M16—42no—
206M16—42no—
207M16—42no—
208M16—42no—
209M16—42no—
210M16—42no—
211M16—42no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—24no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—24no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—24no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8#11712yes—
239C2×C8#11712yes—
240C2×C8#11711yescharacteristic
241C2×C8#11711yescharacteristic
242C2×C8#11712yes—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—24no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C42—22no—
259C42—22no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—22no—
265C42—22no—
266C42—24no—
267C42—44no—
268C42—42no—
269C42—42no—
270C42—42no—
271C42—42no—
272C42—82no—
273C23—41no—
274C23—41no—
275C23C2×#11711yescharacteristic
276C23#52912yes—
277C23#52912yes—
278C23#54211yescharacteristic
279C23#54211yescharacteristic
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—41no—
294C2×C4—84no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4C2×#11711yescharacteristic
301C2×C4C2×#11711yescharacteristic
302C2×C4C4×#4111yescharacteristic
303C2×C4C4×#4111yescharacteristic
304C2×C4#52011yescharacteristic
305C2×C4#52011yescharacteristic
306C2×C4#52912yes—
307C2×C4#52912yes—
308C2×C4#54011yescharacteristic
309C2×C4#54111yescharacteristic
310C2×C4#55112yes—
311C2×C4#55112yes—
312C2×C4#55212yes—
313C2×C4#55212yes—
314C2×C4#55212yes—
315C2×C4#55312yes—
316C2×C4#55512yes—
317C2×C4#55612yes—
318C8—22no—
319C8—22no—
320C8—21no—
321C8—22no—
322C8—21no—
323C8—42no—
324C8—42no—
325C8—44no—
326C8—82no—
327C22—21no—
328C22—22no—
329C22—21no—
330C22—22no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22#228311yescharacteristic
335C22#229811yescharacteristic
336C22#231411yescharacteristic
337C22#252712yes—
338C22#252912yes—
339C4—42no—
340C4—42no—
341C4—42no—
342C4—82no—
343C4—81no—
344C4—84no—
345C4—162no—
346C4#234311yescharacteristic
347C4#234311yescharacteristic
348C4#234611yescharacteristic
349C4#234711yescharacteristic
350C4#253112yes—
351C4#253312yes—
352C2—21no—
353C2—21no—
354C2—22no—
355C2—22no—
356C2#1394611yescharacteristic
357C2#2078711yescharacteristic
358C2#2078811yescharacteristic
subgroupG/H
centerC2×C4C2×#117
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2283
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #1729 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1746C4semidirect
18. #1746C4semidirect
19. #1745C4semidirect
20. #1744C4semidirect

Maximal subgroups

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

subgroupindex
#203682
#112992
#112592
#109102
#108672
#108242
#108142
#108032
#107992
#107112