Finite Group Databasebeta

← #137601

Subgroups — #137601

Order 512.

505 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#346C211yesmaximal, characteristic, complemented
2#21487C211yesmaximal, characteristic
3#12783C211yesmaximal, characteristic, complemented
4C2×#1330C212yesmaximal, complemented
5#7170C212yesmaximal, complemented
6C23×C16—21no—
7C2×#436C2211yescharacteristic
8C2×#435—21no—
9C22×C4×C8C2211yescharacteristic
10C4×#121C411yescharacteristic
11C2×C4⋊C16C2212yes—
12C2×C22⋊C16—21no—
13C2×#346—21no—
14C2×#346—24no—
15C2×#346C2212yes—
16#1954C411yescharacteristic
17#1939C411yescharacteristic
18#1923—21no—
19#1869—21no—
20#1797C411yescharacteristic
21#1770C2211yescharacteristic
22#1753—21no—
23#1374—21no—
24#1330—24no—
25C22×M16—21no—
26C23×C8D411yescharacteristic
27C23×C8D411yescharacteristic
28C2×#121C811yescharacteristic
29C2×C4×D4—21no—
30C2×C4×D4C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C16—24no—
33C22×C16—24no—
34C22×C16—24no—
35C22×C16—22no—
36C22×C16—22no—
37#436—22no—
38#436—22no—
39#435—42no—
40#435—42no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8C2×C411yescharacteristic
43C2×C4⋊C8C2×C411yescharacteristic
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×#97—21no—
49C2×C4×C8—22no—
50C2×C4×C8—22no—
51C2×C4×C8C2×C411yescharacteristic
52C2×C4×C8D411yescharacteristic
53C2×C4×C8D411yescharacteristic
54C2×C4×C8C2311yescharacteristic
55#390C811yescharacteristic
56#389C811yescharacteristic
57#388—21no—
58#386—21no—
59#380C811yescharacteristic
60C4×C4⋊C4C2×C411yescharacteristic
61C4×C22⋊C4—21no—
62C4⋊C16—22no—
63C4⋊C16—22no—
64C22⋊C16—42no—
65#346—24no—
66#346—22no—
67#346D414yescomplemented
68#336—21no—
69#336—21no—
70#336—21no—
71C22×D4—21no—
72C22×D4C2×C811yescharacteristic
73C23×C4—21no—
74C23×C4C22⋊C411yescharacteristic
75C23×C4C2×C811yescharacteristic
76C23×C4M1611yescharacteristic
77C2×M16—42no—
78C2×M16—41no—
79C2×M16—41no—
80C2×M16—42no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—21no—
84C22×C8—22no—
85C22×C8—21no—
86C22×C8—22no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8C22⋊C411yescharacteristic
90C22×C8C22⋊C411yescharacteristic
91C22×C8D812yes—
92C22×C8SD1612yes—
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D411yescharacteristic
95#121—42no—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—42no—
99C4×D4—42no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4C2×C811yescharacteristic
102C2×C4⋊C4C2×C811yescharacteristic
103C2×C4⋊C4C2×C811yescharacteristic
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C42—22no—
110C2×C42—22no—
111C2×C42—21no—
112C2×C42C22⋊C411yescharacteristic
113C2×C42C2×C811yescharacteristic
114C2×C42M1611yescharacteristic
115C2×C42C22×C411yescharacteristic
116C2×C16—216no—
117C2×C16—24no—
118C2×C16—24no—
119C2×C16—24no—
120C2×C16—42no—
121C2×C16—42no—
122C2×C16—42no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—42no—
126C22⋊C8—42no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129C22⋊C8—41no—
130C22⋊C8—41no—
131#97—41no—
132#97—41no—
133C4×C8—21no—
134C4×C8—22no—
135C4×C8—21no—
136C4×C8—22no—
137C4×C8C2×D412yes—
138C4×C8#4112yes—
139#95—21no—
140#95—21no—
141#95—21no—
142C24—21no—
143C24C22⋊C811yescharacteristic
144C2×D4—22no—
145C2×D4—41no—
146C2×D4—41no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—42no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4C22⋊C811yescharacteristic
170C22×C4C22⋊C811yescharacteristic
171C22×C4C22⋊C811yescharacteristic
172C22×C4#10212yes—
173C22×C4#10412yes—
174C22×C4C2×C22⋊C411yescharacteristic
175C22×C4C22×C811yescharacteristic
176C22×C4C2×M1611yescharacteristic
177M16—82no—
178M16—82no—
179C2×C8—24no—
180C2×C8—24no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—24no—
188C2×C8—24no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8#10212yes—
196C2×C8#10412yes—
197C2×C8#12011yescharacteristic
198C2×C8#12112yes—
199C2×C8#12311yescharacteristic
200C2×C8C2×D811yescharacteristic
201C2×C8C2×SD1611yescharacteristic
202C2×C8#13512yes—
203C4⋊C4—22no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C42—21no—
213C42—21no—
214C42—22no—
215C42—22no—
216C42—41no—
217C42—41no—
218C42C4×D412yes—
219C42#13112yes—
220C16—28no—
221C16—82no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23D4⋊C812yes—
234C23#34912yes—
235C23#35012yes—
236C23C2×C22⋊C811yescharacteristic
237D4—82no—
238D4—84no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4D4⋊C812yes—
271C2×C4#38611yescharacteristic
272C2×C4#40811yescharacteristic
273C2×C4C2×#10211yescharacteristic
274C2×C4#41611yescharacteristic
275C2×C4C2×#10411yescharacteristic
276C2×C4C2×#10411yescharacteristic
277C2×C4C8×D412yes—
278C2×C4#43512yes—
279C8—22no—
280C8—21no—
281C8—22no—
282C8—21no—
283C8—81no—
284C8—81no—
285C8#46512yes—
286C8#46612yes—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22#133014yescomplemented
301C22C2×D4⋊C811yescharacteristic
302C22#147611yescharacteristic
303C22#186911yescharacteristic
304C22C2×#34911yescharacteristic
305C22#211511yescharacteristic
306C22C2×#35011yescharacteristic
307C22#211711yescharacteristic
308C4—22no—
309C4—22no—
310C4—22no—
311C4—41no—
312C4—41no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4#149211yescharacteristic
317C4#149411yescharacteristic
318C4#189311yescharacteristic
319C4#189611yescharacteristic
320C2—24no—
321C2—161no—
322C2#907211yescharacteristic
323C2C2×#133011yescharacteristic
324C2#1158811yescharacteristic
325C2#1161811yescharacteristic
326C2#1161911yescharacteristic
327C2#1297811yescharacteristic
328C2#1298511yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC2×#121C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ 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
1. C22×#346C2semidirect
3. #12783C2semidirect
4. C2×#1330C2semidirect
5. #7170C2semidirect
67. #346D4semidirect
300. C22#1330semidirect

Maximal subgroups

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

subgroupindex
C22×#3462
#214872
#127832
C2×#13302
#71702