Finite Group Databasebeta

← #124389

Subgroups — #124389

Order 512.

365 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 82 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#21492C211yesmaximal, characteristic
2#12847C211yesmaximal, characteristic, complemented
3#12602C211yesmaximal, characteristic, complemented
4#7124C212yesmaximal, complemented
5#7105C212yesmaximal, complemented
6C2×#435—21no—
7C2×C8×D4C411yescharacteristic
8C2×#405C2211yescharacteristic
9#2392C411yescharacteristic
10C2×#349—21no—
11#2112—21no—
12C4×M32—21no—
13#1955C2211yescharacteristic
14#1940C2211yescharacteristic
15#1923—21no—
16#1869—21no—
17#1798C411yescharacteristic
18#1770C411yescharacteristic
19#1752—21no—
20#1377—21no—
21#1377C2212yes—
22#1377C2212yes—
23C22×M16D411yescharacteristic
24C23×C8—21no—
25C23×C8D411yescharacteristic
26C2×C4×D4—21no—
27C2×C4×D4C2×C411yescharacteristic
28C2×#117C2×C411yescharacteristic
29C2×M32—22no—
30C2×M32—22no—
31#435—44no—
32C8×D4—41no—
33C8×D4—41no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8C2×C411yescharacteristic
36C2×C4⋊C8C2×C411yescharacteristic
37C2×C22⋊C8—21no—
38C2×C22⋊C8—21no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—21no—
41#405—41no—
42#405—41no—
43C2×#97—21no—
44C2×#97D411yescharacteristic
45C2×#97C2311yescharacteristic
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8D411yescharacteristic
48#398C811yescharacteristic
49#397C811yescharacteristic
50#396—21no—
51#394—21no—
52#394C811yescharacteristic
53#392C811yescharacteristic
54C4×C4⋊C4C2×C411yescharacteristic
55C4×C22⋊C4—21no—
56#349—24no—
57C22⋊C16—41no—
58C22⋊C16—41no—
59#346—41no—
60C4×C16—41no—
61#336—21no—
62#336—21no—
63#336—21no—
64C22×D4—21no—
65C22×D4C2×C811yescharacteristic
66C23×C4—21no—
67C23×C4C22⋊C411yescharacteristic
68C2×M16—22no—
69C2×M16—22no—
70C2×M16—41no—
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—22no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—41no—
77C22×C8—42no—
78C22×C8—42no—
79C22×C8—42no—
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22⋊C411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C22×C8C2×D411yescharacteristic
84C4×D4—41no—
85C4×D4—41no—
86C4×D4—42no—
87C4×D4—42no—
88#117—41no—
89#117—41no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4C2×C811yescharacteristic
93C2×C4⋊C4C2×C811yescharacteristic
94C2×C4⋊C4M1611yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4C2×C811yescharacteristic
101C2×C22⋊C4M1611yescharacteristic
102C2×C42—21no—
103C2×C42C22⋊C411yescharacteristic
104C2×C42C22×C411yescharacteristic
105M32—24no—
106M32—42no—
107M32—42no—
108M32—42no—
109C2×C16—42no—
110C2×C16—42no—
111C4⋊C8—41no—
112C4⋊C8—41no—
113C4⋊C8—42no—
114C22⋊C8—42no—
115C22⋊C8—41no—
116C22⋊C8—41no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119#97—41no—
120#97—41no—
121#97—42no—
122C4×C8—21no—
123C4×C8—21no—
124C4×C8—21no—
125C4×C8—21no—
126#95—21no—
127#95—21no—
128#95C2×C811yescharacteristic
129#95C2×C811yescharacteristic
130C24—21no—
131C24C22⋊C811yescharacteristic
132C2×D4—41no—
133C2×D4—42no—
134C2×D4—42no—
135C2×D4—42no—
136C2×D4—81no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4C22⋊C811yescharacteristic
153C22×C4C22⋊C811yescharacteristic
154C22×C4C22⋊C811yescharacteristic
155C22×C4C2×C22⋊C411yescharacteristic
156C22×C4C22×C811yescharacteristic
157C22×C4C2×M1611yescharacteristic
158M16—44no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—84no—
176C2×C8#9911yescharacteristic
177C2×C8#9911yescharacteristic
178C2×C8#10412yes—
179C2×C8#13512yes—
180C2×C8AGL(1,ℤ/8)12yes—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—42no—
185C4⋊C4—81no—
186C22⋊C4—41no—
187C22⋊C4—41no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—81no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—41no—
198C42—41no—
199C16—44no—
200C16—82no—
201C23—22no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23C2×C22⋊C811yescharacteristic
214D4—42no—
215D4—42no—
216D4—84no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
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—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4C2×#9911yescharacteristic
246C2×C4#41311yescharacteristic
247C2×C4#41611yescharacteristic
248C2×C4#41811yescharacteristic
249C2×C4C2×#10411yescharacteristic
250C2×C4#42111yescharacteristic
251C8—22no—
252C8—22no—
253C8—22no—
254C8—21no—
255C8—21no—
256C8—81no—
257C8—81no—
258C22—21no—
259C22—21no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—41no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—81no—
268C22—81no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22#145911yescharacteristic
273C22#147611yescharacteristic
274C22#147811yescharacteristic
275C4—41no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—81no—
281C4—81no—
282C4—81no—
283C4#150711yescharacteristic
284C4#151611yescharacteristic
285C4#152911yescharacteristic
286C4#155311yescharacteristic
287C2—21no—
288C2—21no—
289C2—81no—
290C2—82no—
291C2#758211yescharacteristic
292C2#1148311yescharacteristic
293C2#1148611yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#97C23
socleC22#1459
prime radical#392C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12847C2semidirect
3. #12602C2semidirect
4. #7124C2semidirect
5. #7105C2semidirect

Maximal subgroups

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

subgroupindex
#214922
#128472
#126022
#71242
#71052