Finite Group Databasebeta

← #124418

Subgroups — #124418

Order 512.

446 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C8×#121C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic, complemented
3#12544C211yesmaximal, characteristic, complemented
4#7118C212yesmaximal, complemented
5#7102C212yesmaximal, complemented
6C22×M32—21no—
7C2×C8×D4—21no—
8C2×C8×D4C411yescharacteristic
9C22×C4×C8C2211yescharacteristic
10C4×#121C411yescharacteristic
11C2×C4⋊C16—21no—
12C2×C22⋊C16—21no—
13#1954C2211yescharacteristic
14#1939C2211yescharacteristic
15#1923—21no—
16#1869—21no—
17#1797C411yescharacteristic
18C8×C4⋊C4C411yescharacteristic
19C8×C22⋊C4—21no—
20#1374—21no—
21#1374C2212yes—
22#1374C2212yes—
23C23×C8—21no—
24C23×C8D411yescharacteristic
25C23×C8D411yescharacteristic
26C2×#121C811yescharacteristic
27C2×C4×D4—21no—
28C2×C4×D4C2×C411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C2×M32—24no—
31C2×M32—42no—
32C2×M32—42no—
33C22×C16—22no—
34C22×C16—22no—
35C8×D4—42no—
36C8×D4—42no—
37C8×D4—42no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C4⋊C8C2×C411yescharacteristic
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C4×C8—21no—
46C2×C4×C8—42no—
47C2×C4×C8C2×C411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C2×C4×C8C2311yescharacteristic
51#390C811yescharacteristic
52#389C811yescharacteristic
53#388—21no—
54#386—21no—
55#380C811yescharacteristic
56C4×C4⋊C4C2×C411yescharacteristic
57C4×C22⋊C4—21no—
58C4⋊C16—42no—
59C22⋊C16—42no—
60#336—21no—
61#336—21no—
62#336—21no—
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—21no—
74C22×C8—21no—
75C22×C8—41no—
76C22×C8—41no—
77C22×C8—42no—
78C22×C8—42no—
79C22×C8—42no—
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22⋊C411yescharacteristic
82C22×C8D812yes—
83C22×C8SD1612yes—
84C22×C8C2×D411yescharacteristic
85C22×C8C2×D411yescharacteristic
86#121—22no—
87#121—22no—
88C4×D4—41no—
89C4×D4—41no—
90C4×D4—42no—
91C4×D4—42no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4C2×C811yescharacteristic
94C2×C4⋊C4C2×C811yescharacteristic
95C2×C4⋊C4C2×C811yescharacteristic
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C42—21no—
102C2×C42—41no—
103C2×C42—41no—
104C2×C42C22⋊C411yescharacteristic
105C2×C42C2×C811yescharacteristic
106C2×C42M1611yescharacteristic
107C2×C42C22×C411yescharacteristic
108M32—48no—
109C2×C16—24no—
110C2×C16—42no—
111C2×C16—42no—
112C2×C16—42no—
113C2×C16—42no—
114C2×C16—42no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121C22⋊C8—41no—
122C22⋊C8—41no—
123C22⋊C8—41no—
124C22⋊C8—41no—
125C4×C8—22no—
126C4×C8—22no—
127C4×C8—41no—
128C4×C8—41no—
129C4×C8—42no—
130#95—21no—
131#95—21no—
132#95—21no—
133C24—21no—
134C24C22⋊C811yescharacteristic
135C2×D4—24no—
136C2×D4—21no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4M3212yescomplemented
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4C22⋊C811yescharacteristic
163C22×C4C22⋊C811yescharacteristic
164C22×C4C22⋊C811yescharacteristic
165C22×C4#10212yes—
166C22×C4#10412yes—
167C22×C4C2×C22⋊C411yescharacteristic
168C22×C4C22×C811yescharacteristic
169C22×C4C2×M1611yescharacteristic
170C2×C8—24no—
171C2×C8—24no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—84no—
189C2×C8#9912yes—
190C2×C8#10212yes—
191C2×C8C2×D811yescharacteristic
192C2×C8C2×SD1611yescharacteristic
193C2×C8AGL(1,ℤ/8)11yescharacteristic
194C2×C8#13711yescharacteristic
195C4⋊C4—21no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—42no—
199C4⋊C4M3212yescomplemented
200C22⋊C4—24no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—41no—
210C42—41no—
211C42—41no—
212C42—41no—
213C16—44no—
214C16—82no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—22no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23D4⋊C812yes—
229C23C2×C22⋊C811yescharacteristic
230D4—42no—
231D4—42no—
232D4—48no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—44no—
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×C4D4⋊C812yes—
266C2×C4#35012yes—
267C2×C4C2×#9911yescharacteristic
268C2×C4#41311yescharacteristic
269C2×C4C2×#10211yescharacteristic
270C2×C4#41711yescharacteristic
271C2×C4C2×#10411yescharacteristic
272C2×C4#42111yescharacteristic
273C2×C4C2×M3211yescharacteristic
274C2×C4C2×M3211yescharacteristic
275C8—21no—
276C8—22no—
277C8—21no—
278C8—21no—
279C8—22no—
280C8—21no—
281C8—81no—
282C8—81no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22#145911yescharacteristic
298C22C2×D4⋊C811yescharacteristic
299C22#147711yescharacteristic
300C22#211211yescharacteristic
301C22#211511yescharacteristic
302C22C2×#35011yescharacteristic
303C22#211711yescharacteristic
304C4—22no—
305C4—21no—
306C4—21no—
307C4—22no—
308C4—42no—
309C4—41no—
310C4—41no—
311C4—81no—
312C4—81no—
313C4#150411yescharacteristic
314C4#151911yescharacteristic
315C4#153211yescharacteristic
316C4#154711yescharacteristic
317C2—42no—
318C2—82no—
319C2#758611yescharacteristic
320C2#1146611yescharacteristic
321C2#1147011yescharacteristic
322C2#1158311yescharacteristic
323C2#1158611yescharacteristic
324C2#1162311yescharacteristic
325C2#1162711yescharacteristic
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. #12783C2semidirect
3. #12544C2semidirect
4. #7118C2semidirect
5. #7102C2semidirect
141. C2×D4M32semidirect
199. C4⋊C4M32semidirect

Maximal subgroups

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

subgroupindex
C8×#1212
#127832
#125442
#71182
#71022