Finite Group Databasebeta

← #124292

Subgroups — #124292

Order 512.

374 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21264C211yesmaximal, characteristic
2#12879C211yesmaximal, characteristic, complemented
3#12653C211yesmaximal, characteristic, complemented
4#7118C212yesmaximal, complemented
5#7102C212yesmaximal
6C2×#431—21no—
7C42×D4C411yescharacteristic
8C2×C4⋊C16—21no—
9C2×C4⋊C16—21no—
10C2×#346—21no—
11#1954C411yescharacteristic
12#1954C2211yescharacteristic
13#1939C411yescharacteristic
14#1939C2211yescharacteristic
15#1923—21no—
16#1923—21no—
17C4×C4⋊C8C411yescharacteristic
18C4×C22⋊C8—21no—
19C42×C8C2211yescharacteristic
20#1374—21no—
21#1374C2212yes—
22#1374C2212yes—
23C2×C4×D4C811yescharacteristic
24C2×C4×D4C811yescharacteristic
25C2×C4×D4C2×C411yescharacteristic
26C22×C42—21no—
27C22×C16—22no—
28C22×C16—22no—
29#431—42no—
30#431—42no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C4⋊C8C2×C411yescharacteristic
35C2×C22⋊C8—21no—
36C2×C22⋊C8—21no—
37C2×C22⋊C8—21no—
38C2×C22⋊C8—21no—
39C2×C4×C8—21no—
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8C2311yescharacteristic
46C4×C4⋊C4C811yescharacteristic
47C4×C4⋊C4C811yescharacteristic
48C4×C4⋊C4C2×C411yescharacteristic
49C4×C22⋊C4—21no—
50C4×C22⋊C4—21no—
51C4×C22⋊C4—21no—
52C43C2×C411yescharacteristic
53C4⋊C16—24no—
54C4⋊C16—42no—
55#346—42no—
56#336—21no—
57#336—21no—
58#336—21no—
59C22×D4C2×C811yescharacteristic
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C22×C8—21no—
64C22×C8—21no—
65C22×C8C22⋊C411yescharacteristic
66C22×C8C22⋊C411yescharacteristic
67C22×C8C2×D411yescharacteristic
68C22×C8C2×D411yescharacteristic
69C4×D4—22no—
70C4×D4—22no—
71C4×D4—42no—
72C4×D4—41no—
73C4×D4—41no—
74C2×C4⋊C4C2×C811yescharacteristic
75C2×C4⋊C4C2×C811yescharacteristic
76C2×C4⋊C4C2×C811yescharacteristic
77C2×C22⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—42no—
84C2×C42—42no—
85C2×C42C22⋊C411yescharacteristic
86C2×C42C22⋊C411yescharacteristic
87C2×C42C2×C811yescharacteristic
88C2×C42C2×C811yescharacteristic
89C2×C42M1611yescharacteristic
90C2×C42M1611yescharacteristic
91C2×C42C22×C411yescharacteristic
92C2×C16—24no—
93C2×C16—42no—
94C2×C16—42no—
95C2×C16—42no—
96C2×C16—42no—
97C2×C16—42no—
98C4⋊C8—41no—
99C4⋊C8—41no—
100C4⋊C8—42no—
101C4⋊C8—41no—
102C4⋊C8—41no—
103C22⋊C8—42no—
104C22⋊C8—42no—
105C22⋊C8—41no—
106C22⋊C8—41no—
107C22⋊C8—41no—
108C22⋊C8—41no—
109C4×C8—21no—
110C4×C8—21no—
111C4×C8—22no—
112C4×C8—41no—
113C4×C8—41no—
114C4×C8—41no—
115C4×C8—42no—
116C4×C8D812yes—
117C4×C8SD1612yes—
118#95—21no—
119#95—21no—
120#95—21no—
121C24—21no—
122C2×D4—21no—
123C2×D4—41no—
124C2×D4—41no—
125C2×D4M3212yes—
126C22×C4—24no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4C22⋊C811yescharacteristic
142C22×C4C22⋊C811yescharacteristic
143C22×C4C22⋊C811yescharacteristic
144C22×C4C22⋊C811yescharacteristic
145C22×C4C2×C22⋊C411yescharacteristic
146C22×C4C22×C811yescharacteristic
147C22×C4C2×M1611yescharacteristic
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8#10212yes—
164C2×C8C2×D811yescharacteristic
165C2×C8C2×SD1611yescharacteristic
166C2×C8AGL(1,ℤ/8)11yescharacteristic
167C2×C8#13711yescharacteristic
168C4⋊C4—21no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4M3212yes—
172C22⋊C4—24no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C42—22no—
176C42—22no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—21no—
183C42—41no—
184C42—42no—
185C42—42no—
186C42—41no—
187C42—41no—
188C42#10212yes—
189C42#10412yes—
190C16—44no—
191C16—82no—
192C23—22no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23C2×C22⋊C811yescharacteristic
200D4—42no—
201D4—42no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—48no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—81no—
226C2×C4—84no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4D4⋊C812yes—
231C2×C4D4⋊C812yes—
232C2×C4#34912yes—
233C2×C4#35012yes—
234C2×C4#41011yescharacteristic
235C2×C4#41311yescharacteristic
236C2×C4C2×#10211yescharacteristic
237C2×C4#41711yescharacteristic
238C2×C4C2×#10411yescharacteristic
239C2×C4#42111yescharacteristic
240C2×C4C2×M3211yescharacteristic
241C2×C4C2×M3211yescharacteristic
242C8—22no—
243C8—21no—
244C8—21no—
245C8—21no—
246C8—21no—
247C8—41no—
248C8—81no—
249C8—81no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—81no—
254C22—81no—
255C22—81no—
256C22—81no—
257C22#145811yescharacteristic
258C22C2×D4⋊C811yescharacteristic
259C22#147711yescharacteristic
260C22#211211yescharacteristic
261C22C2×#34911yescharacteristic
262C22C2×#35011yescharacteristic
263C22#211711yescharacteristic
264C4—22no—
265C4—21no—
266C4—21no—
267C4—42no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—81no—
272C4—81no—
273C4#133014yes—
274C4#150911yescharacteristic
275C4#151711yescharacteristic
276C4#152911yescharacteristic
277C4#154711yescharacteristic
278C2—82no—
279C2#757411yescharacteristic
280C2#1146011yescharacteristic
281C2#1146411yescharacteristic
282C2#1158311yescharacteristic
283C2C2×#133011yescharacteristic
284C2#1162311yescharacteristic
285C2#1162811yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC22×D4C2×C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12879C2semidirect
3. #12653C2semidirect
4. #7118C2semidirect

Maximal subgroups

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

subgroupindex
#212642
#128792
#126532
#71182
#71022