Finite Group Databasebeta

← #124333

Subgroups — #124333

Order 512.

453 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 106 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#21375C211yesmaximal, characteristic
2#12844C211yesmaximal, characteristic, complemented
3C2×#1374C211yesmaximal, characteristic, complemented
4#7128C212yesmaximal, complemented
5#7102C212yesmaximal, complemented
6C2×#529C411yescharacteristic
7C23×C16—21no—
8C2×#432C411yescharacteristic
9C22×C4×C8C2211yescharacteristic
10C2×C22⋊C16—21no—
11#1984C411yescharacteristic
12#1984C2211yescharacteristic
13#1954C411yescharacteristic
14#1954C2211yescharacteristic
15#1869—21no—
16#1869—21no—
17#1750—21no—
18#1750—21no—
19#1374—21no—
20#1374—21no—
21#1374C2212yes—
22#1374C2212yes—
23C23×C8D411yescharacteristic
24C23×C8D411yescharacteristic
25#529—42no—
26#529—42no—
27C2×#126—21no—
28C2×#125C811yescharacteristic
29C2×#124C811yescharacteristic
30C2×#123—21no—
31C2×C22⋊Q8C811yescharacteristic
32C2×#121C811yescharacteristic
33C2×C4×Q8C2×C411yescharacteristic
34C2×C4×D4—21no—
35C2×C4×D4C2×C411yescharacteristic
36C2×#117—21no—
37C22×C42C2×C411yescharacteristic
38C22×C16—24no—
39C22×C16—22no—
40C22×C16—22no—
41C22×C16—42no—
42C22×C16—42no—
43#432—42no—
44C2×C4⋊C8C2×C411yescharacteristic
45C2×C4⋊C8C2×C411yescharacteristic
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×#97—21no—
52C2×C4×C8—42no—
53C2×C4×C8C2×C411yescharacteristic
54C2×C4×C8D411yescharacteristic
55C2×C4×C8D411yescharacteristic
56C2×C4×C8C2311yescharacteristic
57C22⋊C16—42no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62C22×Q8C2×C811yescharacteristic
63C22×D4—21no—
64C22×D4C2×C811yescharacteristic
65C23×C4—21no—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C2×C811yescharacteristic
68C23×C4M1611yescharacteristic
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—41no—
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22⋊C411yescharacteristic
80C22×C8C2×D411yescharacteristic
81C22×C8C2×D411yescharacteristic
82#126—44no—
83#125—42no—
84#124—42no—
85#123—44no—
86C22⋊Q8—42no—
87#121—42no—
88C4×Q8—41no—
89C4×Q8—41no—
90C4×D4—41no—
91C4×D4—41no—
92C4×D4—42no—
93C4×D4—42no—
94#117—42no—
95#117—42no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4C2×C811yescharacteristic
101C2×C4⋊C4C2×C811yescharacteristic
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C42—22no—
108C2×C42—22no—
109C2×C42—21no—
110C2×C42C22⋊C411yescharacteristic
111C2×C42C2×C811yescharacteristic
112C2×C42M1611yescharacteristic
113C2×C42C22×C411yescharacteristic
114C2×C16—24no—
115C2×C16—48no—
116C2×C16—42no—
117C2×C16—42no—
118C2×C16—42no—
119C2×C16—42no—
120C2×C16—42no—
121C4⋊C8—41no—
122C4⋊C8—41no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127C22⋊C8—41no—
128C22⋊C8—41no—
129#97—42no—
130C4×C8—41no—
131C4×C8—41no—
132C4×C8—41no—
133C4×C8—41no—
134C24—21no—
135C24C22⋊C811yescharacteristic
136C2×Q8—22no—
137C2×Q8—41no—
138C2×Q8—41no—
139C2×D4—22no—
140C2×D4—41no—
141C2×D4—41no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—42no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4C22⋊C811yescharacteristic
165C22×C4C22⋊C811yescharacteristic
166C22×C4C22⋊C811yescharacteristic
167C22×C4#9912yes—
168C22×C4#10012yes—
169C22×C4C2×C22⋊C411yescharacteristic
170C22×C4C22×C811yescharacteristic
171C22×C4C2×M1611yescharacteristic
172C2×C8—24no—
173C2×C8—24no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—44no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8#10412yes—
191C2×C8#13512yes—
192C2×C8AGL(1,ℤ/8)11yescharacteristic
193C2×C8#13711yescharacteristic
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C42—22no—
206C42—22no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—41no—
212C42—41no—
213C16—44no—
214C16—82no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23#32312yes—
227C23#35012yes—
228C23#35012yes—
229C23C2×C22⋊C811yescharacteristic
230Q8—82no—
231D4—82no—
232D4—84no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4#32312yes—
266C2×C4C2×#9911yescharacteristic
267C2×C4C2×#10011yescharacteristic
268C2×C4#41611yescharacteristic
269C2×C4#41711yescharacteristic
270C2×C4C2×#10411yescharacteristic
271C2×C4#42111yescharacteristic
272C2×C4#50412yes—
273C8—21no—
274C8—22no—
275C8—21no—
276C8—22no—
277C8—41no—
278C8—81no—
279C8—81no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22#131514yes—
294C22C2×#32311yescharacteristic
295C22#147611yescharacteristic
296C22#147711yescharacteristic
297C22#211311yescharacteristic
298C22#211511yescharacteristic
299C22C2×#35011yescharacteristic
300C22C2×#35011yescharacteristic
301C4—22no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—81no—
307C4—81no—
308C4—81no—
309C4#150411yescharacteristic
310C4#151611yescharacteristic
311C4#153011yescharacteristic
312C4#155111yescharacteristic
313C2—24no—
314C2—161no—
315C2#757811yescharacteristic
316C2#1141911yescharacteristic
317C2C2×#131511yescharacteristic
318C2#1161611yescharacteristic
319C2#1161811yescharacteristic
320C2#1162211yescharacteristic
321C2#1162511yescharacteristic
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. C2×#1374C2semidirect
4. #7128C2semidirect
5. #7102C2semidirect

Maximal subgroups

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

subgroupindex
#213752
#128442
C2×#13742
#71282
#71022