Finite Group Databasebeta

← #349934

Subgroups — #349934

Order 512.

1710 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 1710 conjugacy classes of subgroups — too many to draw. Only its 200 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×#2366C211yesmaximal, characteristic, complemented
2#33106C212yesmaximal, complemented
3#25721C211yesmaximal, characteristic
4#9061C214yesmaximal, complemented
5#8920C214yesmaximal, complemented
6#8706C212yesmaximal, complemented
7#8339C211yesmaximal, characteristic, complemented
8C22×#127C2211yescharacteristic
9C22×C4×D4—22no—
10C2×#493C2212yes—
11C2×#480—22no—
12C2×#447—22no—
13#2856C2211yescharacteristic
14#2366—24no—
15#2366—22no—
16#2366C2214yes—
17C2×#390—22no—
18C2×#386—22no—
19C2×C4×C4⋊C4C2212yes—
20#2011—22no—
21#1982C2214yescomplemented
22#1957C2212yes—
23#1947C2214yescomplemented
24#1925—24no—
25#1874—22no—
26#1797C2212yes—
27C4×C22⋊C8C2211yescharacteristic
28C2×#331C2214yes—
29#1279C2218yescomplemented
30C23×D4—22no—
31C23×D4D412yes—
32C24×C4—22no—
33C22×D8—24no—
34C2×#127—28no—
35C2×#127—22no—
36C2×#127D412yes—
37C2×#127C2311yescharacteristic
38C2×#127C2312yes—
39C2×#121—22no—
40C2×#121—21no—
41C2×C4×D4—28no—
42C2×C4×D4—28no—
43C2×C4×D4—48no—
44C2×C4×D4—48no—
45C22×C4⋊C4D412yes—
46C22×C4⋊C4D412yes—
47C22×C22⋊C4—24no—
48C22×C22⋊C4—22no—
49C22×C42—22no—
50C22×C42C2311yescharacteristic
51#493—28no—
52#480—48no—
53#447—48no—
54C2×#107—24no—
55C2×C4⋊C8C2314yes—
56C2×#102—24no—
57C2×#102—24no—
58C2×#102—24no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—22no—
61C2×C4×C8C2312yes—
62#390—28no—
63#390—28no—
64#390—22no—
65#390—21no—
66#386—28no—
67#386—28no—
68#384C2311yescharacteristic
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4D414yes—
72C4×C4⋊C4C2314yes—
73C2×#95—22no—
74#340—24no—
75#336—22no—
76#336—21no—
77#331—28no—
78C25—22no—
79C22×D4—216no—
80C22×D4—28no—
81C22×D4—24no—
82C22×D4—22no—
83C22×D4—24no—
84C22×D4—21no—
85C22×D4—22no—
86C22×D4—48no—
87C22×D4—44no—
88C22×D4—41no—
89C22×D4—44no—
90C22×D4—44no—
91C22×D4—82no—
92C22×D4C2×D414yes—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—28no—
96C23×C4—416no—
97C23×C4—84no—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D412yes—
100C2×D8—48no—
101C2×D8—48no—
102C2×D8—84no—
103C22×C8—22no—
104C22×C8—24no—
105#127—216no—
106#127—28no—
107#127—24no—
108#127—24no—
109#127#4118yes—
110#121—44no—
111#121—44no—
112#121—44no—
113C4×D4—416no—
114C4×D4—432no—
115C4×D4—416no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4D818yescomplemented
123C2×C4⋊C4C2×D414yes—
124C2×C4⋊C4C2×D414yes—
125C2×C22⋊C4—216no—
126C2×C22⋊C4—28no—
127C2×C22⋊C4—416no—
128C2×C22⋊C4—41no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—84no—
131C2×C42—24no—
132C2×C42—24no—
133C2×C42—24no—
134C2×C42—44no—
135C2×C42—44no—
136C2×C42C2×D412yes—
137C2×C42C2×D411yescharacteristic
138C2×C42C2411yescharacteristic
139#107—48no—
140C4⋊C8—28no—
141#102—48no—
142#102—48no—
143#102—48no—
144C22⋊C8—24no—
145C22⋊C8—44no—
146C4×C8—24no—
147#95—24no—
148#95—24no—
149C24—28no—
150C24—24no—
151C24—44no—
152C24—48no—
153C24—48no—
154C24—44no—
155C24—41no—
156C24#12011yescharacteristic
157C2×D4—216no—
158C2×D4—28no—
159C2×D4—24no—
160C2×D4—216no—
161C2×D4—48no—
162C2×D4—432no—
163C2×D4—432no—
164C2×D4—44no—
165C2×D4—44no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—44no—
170C2×D4—44no—
171C2×D4—42no—
172C2×D4—44no—
173C2×D4—42no—
174C2×D4—48no—
175C2×D4—44no—
176C2×D4—48no—
177C2×D4—82no—
178C2×D4—82no—
179C2×D4—816no—
180C2×D4—84no—
181C2×D4—82no—
182C2×D4#12318yes—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—28no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—41no—
191C22×C4—432no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—41no—
198C22×C4—88no—
199C22×C4—832no—
200C22×C4#12011yescharacteristic
201C22×C4#12012yes—
202C22×C4C2×D814yes—
203C22×C4AGL(1,ℤ/8)14yes—
204C22×C4C22×D411yescharacteristic
205C22×C4C22×D412yes—
206D8—816no—
207C2×C8—24no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—44no—
211C2×C8—44no—
212C2×C8—44no—
213C4⋊C4—24no—
214C4⋊C4—28no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—44no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4C2×D818yes—
222C22⋊C4—432no—
223C22⋊C4—82no—
224C22⋊C4—88no—
225C22⋊C4—84no—
226C42—22no—
227C42—24no—
228C42—44no—
229C42—44no—
230C42—48no—
231C42C2×#4114yes—
232C23—21no—
233C23—22no—
234C23—21no—
235C23—28no—
236C23—432no—
237C23—416no—
238C23—48no—
239C23—48no—
240C23—44no—
241C23—44no—
242C23—816no—
243C23—88no—
244C23—88no—
245C23—81no—
246C23—82no—
247C23—82no—
248C23—81no—
249C23—81no—
250C23#44714yes—
251C23#45312yes—
252C23C2×#12011yescharacteristic
253D4—48no—
254D4—432no—
255D4—416no—
256D4—416no—
257D4—84no—
258D4—84no—
259D4—88no—
260D4—84no—
261D4—84no—
262D4—164no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—28no—
268C2×C4—28no—
269C2×C4—21no—
270C2×C4—28no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—44no—
274C2×C4—416no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—832no—
281C2×C4—88no—
282C2×C4—82no—
283C2×C4—81no—
284C2×C4#44714yes—
285C2×C4C2×#12312yes—
286C2×C4#52514yes—
287C2×C4C22×D812yes—
288C2×C4C2×AGL(1,ℤ/8)12yes—
289C2×C4#57514yes—
290C8—44no—
291C8—84no—
292C22—24no—
293C22—22no—
294C22—22no—
295C22—24no—
296C22—48no—
297C22—416no—
298C22—832no—
299C22—84no—
300C22—88no—
301C22—84no—
302C22—161no—
303C22—162no—
304C22—161no—
305C22—161no—
306C22—162no—
307C22#161914yes—
308C22#284911yescharacteristic
309C22C2×#44712yes—
310C22#300512yes—
311C22C2×#45311yescharacteristic
312C22#301911yescharacteristic
313C4—22no—
314C4—24no—
315C4—48no—
316C4—41no—
317C4—84no—
318C4—161no—
319C4#310014yes—
320C4#311414yes—
321C2—24no—
322C2—88no—
323C2—321no—
324C2C2×#161911yescharacteristic
325C2#1852611yescharacteristic
326C2#1876612yes—
327C2#3299011yescharacteristic
328C2#3300212yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radical#25721C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2366C2semidirect
2. #33106C2semidirect
4. #9061C2semidirect
5. #8920C2semidirect
6. #8706C2semidirect
7. #8339C2semidirect
21. #1982C22semidirect
23. #1947C22semidirect
29. #1279C22semidirect
122. C2×C4⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
C2×#23662
#331062
#257212
#90612
#89202
#87062
#83392