Finite Group Databasebeta

← #124244

Subgroups — #124244

Order 512.

658 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 122 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
3#12599C211yesmaximal, 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×#346—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—
54#346—42no—
55C25—21no—
56C22×D4—24no—
57C22×D4—21no—
58C22×D4—42no—
59C22×D4—42no—
60C22×D4C2×C812yes—
61C23×C4—21no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—42no—
65C23×C4—42no—
66C23×C4—42no—
67C23×C4—42no—
68C23×C4—42no—
69C23×C4—42no—
70C23×C4C22⋊C411yescharacteristic
71C23×C4C2×C811yescharacteristic
72C23×C4M1611yescharacteristic
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—22no—
76C22×C8—41no—
77C22×C8—41no—
78C22×C8—41no—
79C22×C8—42no—
80C22×C8—42no—
81C22×C8—42no—
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8D812yes—
85C22×C8SD1612yes—
86C22×C8C2×D411yescharacteristic
87C22×C8C2×D411yescharacteristic
88C4×D4—44no—
89C4×D4—42no—
90C4×D4—42no—
91C4×D4—44no—
92C4×D4—42no—
93C4×D4—42no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4C2×C812yes—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—44no—
100C2×C42—22no—
101C2×C42—22no—
102C2×C42C22⋊C411yescharacteristic
103C2×C42C2×C811yescharacteristic
104C2×C42M1611yescharacteristic
105C2×C42C22×C411yescharacteristic
106C2×C16—24no—
107C2×C16—42no—
108C2×C16—42no—
109C2×C16—42no—
110C2×C16—42no—
111C2×C16—42no—
112C4⋊C8—42no—
113C4⋊C8—42no—
114C22⋊C8—44no—
115C22⋊C8—44no—
116C22⋊C8—42no—
117C22⋊C8—42no—
118C22⋊C8—44no—
119C4×C8—22no—
120C4×C8—22no—
121C4×C8—41no—
122C4×C8—41no—
123C24—24no—
124C24—22no—
125C24—42no—
126C24—42no—
127C24—42no—
128C24—42no—
129C24—42no—
130C24—42no—
131C24C22⋊C811yescharacteristic
132C2×D4—24no—
133C2×D4—22no—
134C2×D4—42no—
135C2×D4—42no—
136C2×D4—416no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4M3214yes—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—41no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—44no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—84no—
164C22×C4—84no—
165C22×C4—84no—
166C22×C4—84no—
167C22×C4C22⋊C811yescharacteristic
168C22×C4C22⋊C811yescharacteristic
169C22×C4C22⋊C811yescharacteristic
170C22×C4#10212yes—
171C22×C4#10412yes—
172C22×C4C2×C22⋊C411yescharacteristic
173C22×C4C22×C811yescharacteristic
174C22×C4C2×M1611yescharacteristic
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—21no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—44no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—84no—
191C2×C8#10212yes—
192C2×C8C2×D811yescharacteristic
193C2×C8C2×SD1611yescharacteristic
194C2×C8AGL(1,ℤ/8)12yes—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C22⋊C4—44no—
198C22⋊C4—44no—
199C42—22no—
200C42—22no—
201C42—21no—
202C42—21no—
203C42#9912yes—
204C42#10012yes—
205C16—44no—
206C16—82no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—24no—
212C23—48no—
213C23—48no—
214C23—48no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—84no—
222C23—84no—
223C23—84no—
224C23—84no—
225C23D4⋊C812yes—
226C23#34912yes—
227C23#35012yes—
228C23C2×C22⋊C811yescharacteristic
229D4—44no—
230D4—48no—
231D4—44no—
232C2×C4—28no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—44no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—88no—
253C2×C4—84no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4#32312yes—
259C2×C4#32312yes—
260C2×C4D4⋊C812yes—
261C2×C4C2×#9911yescharacteristic
262C2×C4C2×#10011yescharacteristic
263C2×C4C2×#10211yescharacteristic
264C2×C4#41811yescharacteristic
265C2×C4C2×#10411yescharacteristic
266C2×C4#42111yescharacteristic
267C2×C4C2×M3212yes—
268C8—22no—
269C8—21no—
270C8—22no—
271C8—21no—
272C8—41no—
273C8—82no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—44no—
281C22—44no—
282C22—44no—
283C22—816no—
284C22—82no—
285C22—82no—
286C22—82no—
287C22—82no—
288C22#133014yes—
289C22C2×#32311yescharacteristic
290C22C2×D4⋊C811yescharacteristic
291C22#147811yescharacteristic
292C22#211211yescharacteristic
293C22C2×#34911yescharacteristic
294C22C2×#35011yescharacteristic
295C22#211711yescharacteristic
296C4—22no—
297C4—22no—
298C4—22no—
299C4—22no—
300C4—41no—
301C4—82no—
302C4#150711yescharacteristic
303C4#151911yescharacteristic
304C4#152711yescharacteristic
305C4#154511yescharacteristic
306C2—24no—
307C2—84no—
308C2#757211yescharacteristic
309C2#1141811yescharacteristic
310C2#1142211yescharacteristic
311C2#1158311yescharacteristic
312C2C2×#133011yescharacteristic
313C2#1163111yescharacteristic
314C2#1163611yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC23×D4C8
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. #12783C2semidirect
3. #12599C2semidirect
4. #7102C2semidirect
18. #1374C22semidirect
19. #1374C22semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
#127832
#125992
#71022