Finite Group Databasebeta

← #124604

Subgroups — #124604

Order 512.

354 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C8×C22⋊Q8C211yesmaximal, characteristic
2#12844C211yesmaximal, characteristic
3#12783C211yesmaximal, characteristic
4#7128C212yesmaximal, complemented
5#7118C212yesmaximal, complemented
6C2×C8×Q8C411yescharacteristic
7C22×C4×C8C2211yescharacteristic
8C4×C22⋊Q8C411yescharacteristic
9C2×C4⋊C16—21no—
10C2×C22⋊C16—21no—
11C2×C22⋊C16—21no—
12#1984C2211yescharacteristic
13#1939C2211yescharacteristic
14#1916—21no—
15#1850—21no—
16#1797C411yescharacteristic
17C8×C4⋊C4—21no—
18C8×C4⋊C4C411yescharacteristic
19C8×C22⋊C4—21no—
20#1374—21no—
21#1374C2212yes—
22#1374C2212yes—
23C23×C8D411yescharacteristic
24C23×C8D411yescharacteristic
25C2×C22⋊Q8C811yescharacteristic
26C2×C4×Q8C2×C411yescharacteristic
27C22×C42C2×C411yescharacteristic
28C22×C16—22no—
29C22×C16—22no—
30C8×Q8—42no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C4⋊C8C2×C411yescharacteristic
35C2×C22⋊C8—21no—
36C2×C4×C8—21no—
37C2×C4×C8—21no—
38C2×C4×C8—42no—
39C2×C4×C8C2×C411yescharacteristic
40C2×C4×C8D411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8C2311yescharacteristic
43#391C811yescharacteristic
44#389C811yescharacteristic
45#387—21no—
46#385—21no—
47#380C811yescharacteristic
48C4×C4⋊C4—21no—
49C4×C4⋊C4C2×C411yescharacteristic
50C4×C22⋊C4—21no—
51C4⋊C16—42no—
52C22⋊C16—42no—
53C22⋊C16—42no—
54#336—21no—
55#336—21no—
56#336—21no—
57#336—21no—
58#336—21no—
59C22×Q8C2×C811yescharacteristic
60C23×C4C22⋊C411yescharacteristic
61C23×C4C2×C811yescharacteristic
62C23×C4M1611yescharacteristic
63C22×C8—22no—
64C22×C8—21no—
65C22×C8—22no—
66C22×C8—21no—
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—41no—
70C22×C8—41no—
71C22×C8—41no—
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8C2×D411yescharacteristic
75C22×C8C2×D411yescharacteristic
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78C4×Q8—41no—
79C4×Q8—41no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4C2×C811yescharacteristic
84C2×C4⋊C4C2×C811yescharacteristic
85C2×C4⋊C4C2×C811yescharacteristic
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—41no—
91C2×C42—41no—
92C2×C42C22⋊C411yescharacteristic
93C2×C42C2×C811yescharacteristic
94C2×C42M1611yescharacteristic
95C2×C42C22×C411yescharacteristic
96C2×C16—42no—
97C2×C16—42no—
98C2×C16—42no—
99C2×C16—42no—
100C2×C16—42no—
101C2×C16—42no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104C4⋊C8—42no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107C22⋊C8—42no—
108C4×C8—22no—
109C4×C8—22no—
110C4×C8—41no—
111C4×C8—41no—
112C4×C8—42no—
113C4×C8—42no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119C24C22⋊C811yescharacteristic
120C2×Q8—21no—
121C2×Q8—41no—
122C2×Q8—41no—
123C2×Q8M3212yes—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4C22⋊C811yescharacteristic
142C22×C4C22⋊C811yescharacteristic
143C22×C4C22⋊C811yescharacteristic
144C22×C4C2×C22⋊C411yescharacteristic
145C22×C4C22×C811yescharacteristic
146C22×C4C2×M1611yescharacteristic
147C2×C8—21no—
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—44no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8#9912yes—
166C2×C8AGL(1,ℤ/8)11yescharacteristic
167C2×C8#13711yescharacteristic
168C2×C8#13712yes—
169C4⋊C4—24no—
170C4⋊C4—21no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4M3212yes—
176C22⋊C4—24no—
177C22⋊C4—42no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—41no—
183C42—41no—
184C42—41no—
185C42—41no—
186C42—41no—
187C42—41no—
188C16—82no—
189C16—82no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23#35012yes—
197C23C2×C22⋊C811yescharacteristic
198Q8—42no—
199Q8—42no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4#35012yes—
233C2×C4C2×#9911yescharacteristic
234C2×C4#41311yescharacteristic
235C2×C4#41711yescharacteristic
236C2×C4#41911yescharacteristic
237C2×C4#42111yescharacteristic
238C2×C4#42111yescharacteristic
239C2×C4C2×M3211yescharacteristic
240C2×C4C2×M3211yescharacteristic
241C8—22no—
242C8—22no—
243C8—21no—
244C8—21no—
245C8—41no—
246C8—81no—
247C8—81no—
248C22—22no—
249C22—22no—
250C22—22no—
251C22—41no—
252C22—41no—
253C22—41no—
254C22—41no—
255C22#145911yescharacteristic
256C22#147711yescharacteristic
257C22#147911yescharacteristic
258C22#211211yescharacteristic
259C22#211511yescharacteristic
260C22C2×#35011yescharacteristic
261C22C2×#35011yescharacteristic
262C4—21no—
263C4—21no—
264C4—22no—
265C4—42no—
266C4—41no—
267C4—41no—
268C4—42no—
269C4—41no—
270C4—81no—
271C4—81no—
272C4#151111yescharacteristic
273C4#151311yescharacteristic
274C4#153511yescharacteristic
275C4#155911yescharacteristic
276C2—42no—
277C2#760011yescharacteristic
278C2#1146611yescharacteristic
279C2#1146811yescharacteristic
280C2#1162311yescharacteristic
281C2#1162511yescharacteristic
282C2#1163911yescharacteristic
283C2#1164111yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7128C2semidirect
5. #7118C2semidirect

Maximal subgroups

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

subgroupindex
C8×C22⋊Q82
#128442
#127832
#71282
#71182