Finite Group Databasebeta

← #124242

Subgroups — #124242

Order 512.

666 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 138 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
1C2×#1954C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic, complemented
3C4×C22⋊C16C211yesmaximal, characteristic, complemented
4#7102C214yesmaximal, complemented
5C22×C4×D4C411yescharacteristic
6C22×C4⋊C8C411yescharacteristic
7C22×C22⋊C8—21no—
8C22×C22⋊C8—21no—
9C22×C4×C8C2211yescharacteristic
10C2×C4⋊C16—21no—
11C2×C22⋊C16—21no—
12C2×C22⋊C16—21no—
13C2×C4×C16—21no—
14#1954—21no—
15#1954—21no—
16#1954C412yes—
17#1954C2212yes—
18#1374C2212yescomplemented
19#1374C2212yescomplemented
20C23×D4C811yescharacteristic
21C24×C4—21no—
22C23×C8—21no—
23C23×C8D411yescharacteristic
24C23×C8D411yescharacteristic
25C2×C4×D4—21no—
26C2×C4×D4—21no—
27C2×C4×D4—42no—
28C2×C4×D4—42no—
29C2×C4×D4C812yes—
30C2×C4×D4C2×C412yes—
31C22×C4⋊C4C811yescharacteristic
32C22×C22⋊C4—21no—
33C22×C42C2×C411yescharacteristic
34C22×C16—22no—
35C22×C16—22no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—42no—
38C2×C4⋊C8C2×C412yes—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—44no—
44C2×C22⋊C8—42no—
45C2×C22⋊C8—42no—
46C2×C4×C8—42no—
47C2×C4×C8C2×C411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C2×C4×C8C2311yescharacteristic
51C4⋊C16—42no—
52C22⋊C16—24no—
53C22⋊C16—42no—
54C4×C16—42no—
55C25—21no—
56C22×D4—21no—
57C22×D4—22no—
58C22×D4—42no—
59C22×D4—42no—
60C22×D4C1614yescomplemented
61C22×D4C2×C812yes—
62C23×C4—21no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—42no—
66C23×C4—42no—
67C23×C4—42no—
68C23×C4—42no—
69C23×C4—42no—
70C23×C4—42no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C2×C811yescharacteristic
73C23×C4M1611yescharacteristic
74C22×C8—22no—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—41no—
78C22×C8—41no—
79C22×C8—41no—
80C22×C8—42no—
81C22×C8—42no—
82C22×C8—42no—
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22⋊C411yescharacteristic
85C22×C8D812yes—
86C22×C8SD1612yes—
87C22×C8C2×D411yescharacteristic
88C22×C8C2×D411yescharacteristic
89C4×D4—44no—
90C4×D4—42no—
91C4×D4—42no—
92C4×D4—44no—
93C4×D4—42no—
94C4×D4—42no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4C2×C812yes—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—44no—
101C2×C42—22no—
102C2×C42—22no—
103C2×C42C22⋊C411yescharacteristic
104C2×C42C2×C811yescharacteristic
105C2×C42M1611yescharacteristic
106C2×C42C22×C411yescharacteristic
107C2×C16—24no—
108C2×C16—42no—
109C2×C16—42no—
110C2×C16—42no—
111C2×C16—42no—
112C2×C16—42no—
113C4⋊C8—42no—
114C4⋊C8—42no—
115C22⋊C8—44no—
116C22⋊C8—44no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119C22⋊C8—44no—
120C4×C8—22no—
121C4×C8—22no—
122C4×C8—41no—
123C4×C8—41no—
124C24—24no—
125C24—22no—
126C24—42no—
127C24—42no—
128C24—42no—
129C24—42no—
130C24—42no—
131C24—42no—
132C24C22⋊C811yescharacteristic
133C2×D4—24no—
134C2×D4—22no—
135C2×D4—42no—
136C2×D4—42no—
137C2×D4—48no—
138C2×D4—48no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4C2×C1614yes—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—41no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—44no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—84no—
165C22×C4—84no—
166C22×C4—84no—
167C22×C4—84no—
168C22×C4C22⋊C811yescharacteristic
169C22×C4C22⋊C811yescharacteristic
170C22×C4C22⋊C811yescharacteristic
171C22×C4#10212yes—
172C22×C4#10412yes—
173C22×C4C2×C1612yes—
174C22×C4M3212yes—
175C22×C4C2×C22⋊C411yescharacteristic
176C22×C4C22×C811yescharacteristic
177C22×C4C2×M1611yescharacteristic
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—44no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—84no—
194C2×C8#10212yes—
195C2×C8C2×D811yescharacteristic
196C2×C8C2×SD1611yescharacteristic
197C2×C8AGL(1,ℤ/8)12yes—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C22⋊C4—44no—
201C22⋊C4—44no—
202C42—22no—
203C42—22no—
204C42—21no—
205C42—21no—
206C42#9912yes—
207C42#10012yes—
208C16—44no—
209C16—82no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—24no—
215C23—48no—
216C23—48no—
217C23—48no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—84no—
225C23—84no—
226C23—84no—
227C23—84no—
228C23D4⋊C812yes—
229C23C22⋊C1612yes—
230C23#35012yes—
231C23C2×C22⋊C811yescharacteristic
232D4—44no—
233D4—44no—
234D4—48no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—44no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—88no—
255C2×C4—84no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4#32312yes—
261C2×C4#32312yes—
262C2×C4D4⋊C812yes—
263C2×C4C22⋊C1612yes—
264C2×C4#34912yes—
265C2×C4C2×#9911yescharacteristic
266C2×C4C2×#10011yescharacteristic
267C2×C4C2×#10211yescharacteristic
268C2×C4#41811yescharacteristic
269C2×C4C2×#10411yescharacteristic
270C2×C4#42111yescharacteristic
271C2×C4C22×C1611yescharacteristic
272C2×C4C2×M3211yescharacteristic
273C8—22no—
274C8—21no—
275C8—22no—
276C8—21no—
277C8—41no—
278C8—82no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—44no—
286C22—44no—
287C22—44no—
288C22—816no—
289C22—82no—
290C22—82no—
291C22—82no—
292C22—82no—
293C22D4⋊C1614yes—
294C22C2×#32311yescharacteristic
295C22C2×D4⋊C811yescharacteristic
296C22#147811yescharacteristic
297C22C2×C22⋊C1611yescharacteristic
298C22C2×#34911yescharacteristic
299C22C2×#35011yescharacteristic
300C22#211711yescharacteristic
301C4—22no—
302C4—21no—
303C4—21no—
304C4—22no—
305C4—41no—
306C4—82no—
307C4#131414yes—
308C4#150711yescharacteristic
309C4#151911yescharacteristic
310C4#152711yescharacteristic
311C4#154511yescharacteristic
312C2—24no—
313C2—84no—
314C2#757211yescharacteristic
315C2C2×#131411yescharacteristic
316C2#1142211yescharacteristic
317C2C2×D4⋊C1611yescharacteristic
318C2#1158711yescharacteristic
319C2#1162911yescharacteristic
320C2#1163411yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C22×C16
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC23×D4C8
LCSG ▷ C2×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. #12783C2semidirect
3. C4×C22⋊C16C2semidirect
4. #7102C2semidirect
18. #1374C22semidirect
19. #1374C22semidirect
60. C22×D4C16semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
#127832
C4×C22⋊C162
#71022