Finite Group Databasebeta

← #124380

Subgroups — #124380

Order 512.

397 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 94 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#21487C211yesmaximal, characteristic
2#12844C211yesmaximal, characteristic, complemented
3#12599C211yesmaximal, characteristic, complemented
4#7118C212yesmaximal, complemented
5#7102C212yesmaximal, complemented
6C2×#436C411yescharacteristic
7C2×#435—21no—
8C22×C4×C8C2211yescharacteristic
9C4×#121C411yescharacteristic
10C2×C22⋊C16—21no—
11C2×C22⋊C16—21no—
12C2×#346—21no—
13#1954C2211yescharacteristic
14#1939C2211yescharacteristic
15#1923—21no—
16#1869—21no—
17#1797C411yescharacteristic
18#1770C411yescharacteristic
19#1753—21no—
20#1374—21no—
21#1374C2212yes—
22#1374C2212yes—
23C22×M16—21no—
24C23×C8D411yescharacteristic
25C23×C8D411yescharacteristic
26C2×#121C811yescharacteristic
27C2×C4×D4—21no—
28C2×C4×D4C2×C411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C16—22no—
31C22×C16—22no—
32#436—42no—
33#435—44no—
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—
41C2×#97—21no—
42C2×C4×C8—42no—
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8C2311yescharacteristic
47#390C811yescharacteristic
48#389C811yescharacteristic
49#388—21no—
50#386—21no—
51#380C811yescharacteristic
52C4×C4⋊C4C2×C411yescharacteristic
53C4×C22⋊C4—21no—
54C22⋊C16—24no—
55C22⋊C16—42no—
56#346—42no—
57#336—21no—
58#336—21no—
59#336—21no—
60C22×D4—21no—
61C22×D4C2×C811yescharacteristic
62C23×C4—21no—
63C23×C4C22⋊C411yescharacteristic
64C23×C4C2×C811yescharacteristic
65C23×C4M1611yescharacteristic
66C2×M16—42no—
67C2×M16—42no—
68C2×M16—42no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—41no—
76C22×C8—41no—
77C22×C8C22⋊C411yescharacteristic
78C22×C8C22⋊C411yescharacteristic
79C22×C8C2×D411yescharacteristic
80C22×C8C2×D411yescharacteristic
81#121—42no—
82C4×D4—41no—
83C4×D4—41no—
84C4×D4—42no—
85C4×D4—42no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4C2×C811yescharacteristic
88C2×C4⋊C4C2×C811yescharacteristic
89C2×C4⋊C4C2×C811yescharacteristic
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C42—21no—
96C2×C42—41no—
97C2×C42—41no—
98C2×C42C22⋊C411yescharacteristic
99C2×C42C2×C811yescharacteristic
100C2×C42M1611yescharacteristic
101C2×C42C22×C411yescharacteristic
102C2×C16—24no—
103C2×C16—42no—
104C2×C16—42no—
105C2×C16—42no—
106C2×C16—42no—
107C2×C16—42no—
108C4⋊C8—41no—
109C4⋊C8—41no—
110C4⋊C8—42no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115#97—42no—
116C4×C8—22no—
117C4×C8—22no—
118C4×C8—41no—
119C4×C8—41no—
120#95—21no—
121#95—21no—
122#95—21no—
123C24—21no—
124C24C22⋊C811yescharacteristic
125C2×D4—22no—
126C2×D4—41no—
127C2×D4—41no—
128C2×D4—42no—
129C2×D4—42no—
130C2×D4—42no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
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—84no—
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—44no—
168C2×C8—44no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8#9912yes—
176C2×C8#10412yes—
177C2×C8#13512yes—
178C2×C8AGL(1,ℤ/8)12yes—
179C4⋊C4—22no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—42no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42—41no—
193C42—41no—
194C42—41no—
195C42—41no—
196C16—44no—
197C16—82no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23#34912yes—
212C23C2×C22⋊C811yescharacteristic
213D4—82no—
214D4—84no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—44no—
230C2×C4—44no—
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—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4C2×#9911yescharacteristic
249C2×C4#41311yescharacteristic
250C2×C4#41611yescharacteristic
251C2×C4#41811yescharacteristic
252C2×C4C2×#10411yescharacteristic
253C2×C4#42111yescharacteristic
254C2×C4#50412yes—
255C8—22no—
256C8—22no—
257C8—22no—
258C8—22no—
259C8—81no—
260C8—81no—
261C22—22no—
262C22—22no—
263C22—22no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22#145911yescharacteristic
276C22#147611yescharacteristic
277C22#147811yescharacteristic
278C22#211311yescharacteristic
279C22C2×#34911yescharacteristic
280C22#211711yescharacteristic
281C22#211711yescharacteristic
282C4—22no—
283C4—41no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—81no—
288C4—81no—
289C4—81no—
290C4#150711yescharacteristic
291C4#151611yescharacteristic
292C4#152911yescharacteristic
293C4#155311yescharacteristic
294C2—42no—
295C2—161no—
296C2#758211yescharacteristic
297C2#1146711yescharacteristic
298C2#1146911yescharacteristic
299C2#1161411yescharacteristic
300C2#1161911yescharacteristic
301C2#1163211yescharacteristic
302C2#1163611yescharacteristic
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 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12844C2semidirect
3. #12599C2semidirect
4. #7118C2semidirect
5. #7102C2semidirect

Maximal subgroups

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

subgroupindex
#214872
#128442
#125992
#71182
#71022