Finite Group Databasebeta

← #136606

Subgroups — #136606

Order 512.

468 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#467C211yesmaximal, characteristic
2C2×#1564C211yesmaximal, characteristic
3C4×C22⋊C16C211yesmaximal, characteristic
4C2×#1332C212yesmaximal, complemented
5#7171C212yesmaximal, complemented
6C2×#467C411yescharacteristic
7C2×C4×Q16C2211yescharacteristic
8C2×#431—21no—
9C22×C4×C8C2211yescharacteristic
10C4×C22⋊Q8—21no—
11C2×C22⋊C16—21no—
12C2×C4×C16C2212yes—
13#1969C411yescharacteristic
14#1942C411yescharacteristic
15#1933—21no—
16#1892—21no—
17#1808C411yescharacteristic
18C4×#107C2211yescharacteristic
19C4×#103—21no—
20#1564—24no—
21C2×#335C2212yes—
22C2×#334—21no—
23#1374—21no—
24#1332—24no—
25C22×Q16C2×C411yescharacteristic
26C23×C8C2×C411yescharacteristic
27C23×C8D411yescharacteristic
28C2×C22⋊Q8—21no—
29C2×C4×Q8—21no—
30C22×C42D411yescharacteristic
31C22×C16—22no—
32#467—22no—
33#467C814yescomplemented
34C4×Q16—22no—
35C4×Q16—22no—
36#431—24no—
37#431—42no—
38C2×#107C2×C411yescharacteristic
39C2×#107C2×C411yescharacteristic
40C2×#107C2×C411yescharacteristic
41C2×C4⋊C8—22no—
42C2×#103—21no—
43C2×#103—21no—
44C2×#103—21no—
45C2×C22⋊C8—41no—
46C2×C4×C8—22no—
47C2×C4×C8—22no—
48C2×C4×C8—41no—
49C2×C4×C8C2×C411yescharacteristic
50C2×C4×C8D411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C2×C4×C8C2311yescharacteristic
53#391—21no—
54#389—21no—
55#387—41no—
56#385—41no—
57#380—21no—
58C4×C4⋊C4—21no—
59C4×C4⋊C4—41no—
60C4×C22⋊C4—41no—
61C22⋊C16—24no—
62C4×C16—22no—
63C4×C16—22no—
64#340—21no—
65#340—21no—
66#340—21no—
67#335—22no—
68#335D414yes—
69#334—24no—
70C22×Q8—21no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4D811yescharacteristic
73C23×C4SD1611yescharacteristic
74C2×Q16—22no—
75C2×Q16—21no—
76C2×Q16—22no—
77C2×Q16C2×C812yes—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—41no—
83C22×C8—42no—
84C22×C8C22⋊C411yescharacteristic
85C22×C8C22⋊C411yescharacteristic
86C22×C8C2×C812yes—
87C22×C8M1612yes—
88C22×C8C22×C411yescharacteristic
89C22×C8C2×D411yescharacteristic
90C22⋊Q8—24no—
91C22⋊Q8—42no—
92C4×Q8—41no—
93C4×Q8—41no—
94C4×Q8—41no—
95C4×Q8—41no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—41no—
109C2×C42—41no—
110C2×C42C22⋊C411yescharacteristic
111C2×C42D811yescharacteristic
112C2×C42SD1611yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C16—24no—
115C2×C16—42no—
116C2×C16—42no—
117#107—21no—
118#107—22no—
119#107—22no—
120#107C2×C812yes—
121C4⋊C8—24no—
122C4⋊C8—42no—
123#103—24no—
124#103—42no—
125#103—42no—
126C22⋊C8—44no—
127C4×C8—22no—
128C4×C8—22no—
129C4×C8—21no—
130C4×C8—21no—
131C4×C8—44no—
132C4×C8C2×D412yes—
133C4×C8#4112yes—
134#95—41no—
135#95—41no—
136#95—41no—
137#95—41no—
138#95—41no—
139C24#10211yescharacteristic
140C2×Q8—22no—
141C2×Q8—41no—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—41no—
145C2×Q8—41no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4C22⋊C812yes—
163C22×C4#10211yescharacteristic
164C22×C4#10211yescharacteristic
165C22×C4#10211yescharacteristic
166C22×C4#10412yes—
167C22×C4C2×C22⋊C411yescharacteristic
168C22×C4C2×D811yescharacteristic
169C22×C4C2×SD1611yescharacteristic
170Q16—24no—
171Q16—42no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—24no—
177C2×C8—24no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—44no—
182C2×C8—44no—
183C2×C8—82no—
184C2×C8—82no—
185C2×C8C22⋊C812yes—
186C2×C8C4×D412yes—
187C2×C8#12112yes—
188C2×C8C22×C811yescharacteristic
189C2×C8C2×M1611yescharacteristic
190C2×C8#13112yes—
191C4⋊C4—22no—
192C4⋊C4—44no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—82no—
199C4⋊C4—81no—
200C4⋊C4—81no—
201C22⋊C4—44no—
202C22⋊C4—82no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—81no—
210C42—81no—
211C42—82no—
212C42#12011yescharacteristic
213C42#12311yescharacteristic
214C42AGL(1,ℤ/8)11yescharacteristic
215C42#13711yescharacteristic
216C16—44no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23D4⋊C812yes—
224C23C2×#10211yescharacteristic
225Q8—42no—
226Q8—42no—
227Q8—81no—
228Q8—81no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4D4⋊C812yes—
259C2×C4#35912yes—
260C2×C4#38611yescharacteristic
261C2×C4C2×C22⋊C811yescharacteristic
262C2×C4#40811yescharacteristic
263C2×C4#41711yescharacteristic
264C2×C4C2×#10411yescharacteristic
265C2×C4#42111yescharacteristic
266C2×C4#44711yescharacteristic
267C2×C4#44811yescharacteristic
268C2×C4#48011yescharacteristic
269C2×C4#48111yescharacteristic
270C2×C4#50812yes—
271C2×C4#51012yes—
272C8—22no—
273C8—22no—
274C8—41no—
275C8—84no—
276C8C8×D412yes—
277C8#43512yes—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22C2×D4⋊C811yescharacteristic
286C22#148011yescharacteristic
287C22#187411yescharacteristic
288C22#213811yescharacteristic
289C22#214111yescharacteristic
290C22C2×#35911yescharacteristic
291C22#214311yescharacteristic
292C4—22no—
293C4—21no—
294C4—21no—
295C4—22no—
296C4—41no—
297C4—82no—
298C4—82no—
299C4—161no—
300C4—161no—
301C4#147711yescharacteristic
302C4#148311yescharacteristic
303C4#186911yescharacteristic
304C4#187111yescharacteristic
305C4#218912yes—
306C4#223712yes—
307C2—42no—
308C2#893911yescharacteristic
309C2#1159111yescharacteristic
310C2#1159411yescharacteristic
311C2#1164711yescharacteristic
312C2#1165011yescharacteristic
313C2#1299011yescharacteristic
314C2#1299411yescharacteristic
subgroupG/H
centerC22×C4C2×D8
commutatorC2×C8C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×#102
prime radicalC24#102
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1332C2semidirect
5. #7171C2semidirect
33. #467C8semidirect

Maximal subgroups

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

subgroupindex
C4×#4672
C2×#15642
C4×C22⋊C162
C2×#13322
#71712