Finite Group Databasebeta

← #298407

Subgroups — #298407

Order 512.

1201 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 1201 conjugacy classes of subgroups — too many to draw. Only its 208 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#59832C211yesmaximal, characteristic, complemented
2#28441C212yesmaximal
3#28441C211yesmaximal, characteristic
4C2×#1623C214yesmaximal, complemented
5#10230C214yesmaximal, complemented
6#9673C212yesmaximal, complemented
7C4×#333C211yesmaximal, characteristic, complemented
8C2×Q82C2212yes—
9C2×#492—22no—
10C2×#492C2212yes—
11C2×#460—22no—
12C2×#460C2218yes—
13#2867C2211yescharacteristic
14#2734—22no—
15#2582C2212yes—
16#2551—22no—
17#2546—22no—
18#2451—24no—
19#2365—22no—
20C4×C4⋊Q8C2211yescharacteristic
21C4×C4⋊Q8C2212yes—
22#2014—24no—
23#2014—22no—
24#1957—22no—
25#1957C2212yes—
26C4×C4⋊C8C2212yes—
27C4×C4⋊C8C2211yescharacteristic
28#1623—216no—
29C2×#333C2212yes—
30C2×Q8⋊C8C2218yes—
31#1296C2212yes—
32C22×SD16—28no—
33C22×SD16—24no—
34Q82—28no—
35Q82—44no—
36C2×C4⋊Q8—24no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8—24no—
39C2×C4⋊Q8D412yes—
40C2×C4⋊Q8C2311yescharacteristic
41C2×#127—22no—
42C2×#127—21no—
43C2×#127—22no—
44C2×#127—21no—
45C2×#127C2312yes—
46C2×C4×Q8—24no—
47C2×C4×Q8D412yes—
48C2×C4×Q8C2314yes—
49#492—28no—
50#492—48no—
51#460—216no—
52#460—48no—
53#460—48no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8C2314yes—
56C2×#102—28no—
57C2×#102—24no—
58C2×C4×C8—22no—
59C2×C4×C8C2312yes—
60#398—24no—
61#398—24no—
62#395—28no—
63#391—24no—
64#391—24no—
65#391D414yes—
66#391D412yes—
67#389—24no—
68#389—24no—
69#389—22no—
70#389—22no—
71#389—22no—
72#387—28no—
73#384D412yes—
74#384C2311yescharacteristic
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C43C2311yescharacteristic
80#336—22no—
81#336—21no—
82#333D418yescomplemented
83Q8⋊C8—216no—
84C22×Q8—24no—
85C22×Q8C2×D414yes—
86C22×Q8C2×D412yes—
87C22×D4—22no—
88C22×D4—24no—
89C22×D4—41no—
90C22×D4—42no—
91C22×D4—41no—
92C2×SD16—416no—
93C2×SD16—48no—
94C2×SD16—48no—
95C2×SD16—416no—
96C2×SD16—88no—
97C2×SD16—84no—
98C22×C8—24no—
99C22×C8—22no—
100C4⋊Q8—24no—
101C4⋊Q8—22no—
102C4⋊Q8—416no—
103C4⋊Q8—48no—
104C4⋊Q8—48no—
105C4⋊Q8—84no—
106C4⋊Q8SD1618yescomplemented
107C4⋊Q8C2×D414yes—
108#127—24no—
109#127—24no—
110#127—44no—
111#127—44no—
112#127—44no—
113#127—84no—
114#127—82no—
115C4×Q8—28no—
116C4×Q8—28no—
117C4×Q8—44no—
118C4×Q8—416no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4C2×D414yes—
130C2×C4⋊C4C2×D412yes—
131C2×C42—24no—
132C2×C42—22no—
133C2×C42—24no—
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D412yes—
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D412yes—
138C2×C42C2411yescharacteristic
139C4⋊C8—24no—
140C4⋊C8—24no—
141C4⋊C8C2×D418yes—
142#102—416no—
143#102—48no—
144C4×C8—24no—
145C4×C8—44no—
146#95—24no—
147#95—28no—
148#95—24no—
149#95—24no—
150#95—24no—
151C24—42no—
152C2×Q8—22no—
153C2×Q8—24no—
154C2×Q8—28no—
155C2×Q8—28no—
156C2×Q8—48no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—84no—
160C2×Q8—84no—
161C2×Q8#12714yes—
162C2×Q8C2×SD1618yes—
163C2×D4—44no—
164C2×D4—44no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—48no—
168C2×D4—44no—
169C2×D4—44no—
170C2×D4—48no—
171C2×D4—82no—
172C2×D4—84no—
173C2×D4—84no—
174C2×D4—84no—
175C2×D4—84no—
176C2×D4—84no—
177C2×D4—82no—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4#12011yescharacteristic
183C22×C4#12012yes—
184C22×C4#12011yescharacteristic
185C22×C4C22×D411yescharacteristic
186C22×C4C22×D412yes—
187SD16—832no—
188SD16—816no—
189C2×C8—24no—
190C2×C8—28no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—44no—
194C2×C8—44no—
195C4⋊C4—28no—
196C4⋊C4—28no—
197C4⋊C4—28no—
198C4⋊C4—28no—
199C4⋊C4—42no—
200C4⋊C4—416no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—82no—
206C4⋊C4—82no—
207C4⋊C4—84no—
208C42—24no—
209C42—22no—
210C42—41no—
211C42—48no—
212C42—48no—
213C42—44no—
214C42C2×SD1614yes—
215C42#13714yes—
216C42C22×D412yes—
217C42H(5,𝔽2)12yes—
218C23—82no—
219C23—84no—
220C23—84no—
221C23—82no—
222C23—82no—
223C23C2×#12011yescharacteristic
224Q8—216no—
225Q8—44no—
226Q8—48no—
227Q8—82no—
228Q8—168no—
229D4—84no—
230D4—84no—
231D4—88no—
232D4—88no—
233D4—168no—
234D4—164no—
235D4—164no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—24no—
241C2×C4—28no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4#44814yes—
254C2×C4#44814yes—
255C2×C4#45312yes—
256C2×C4C2×#12711yescharacteristic
257C2×C4#53511yescharacteristic
258C2×C4D4214yes—
259C2×C4C22×SD1612yes—
260C2×C4C2×#13712yes—
261C2×C4#57714yes—
262C8—44no—
263C8—48no—
264C22—162no—
265C22—162no—
266C22—162no—
267C22—164no—
268C22—164no—
269C22#243311yescharacteristic
270C22C2×#44812yes—
271C22#301012yes—
272C22C2×#45311yescharacteristic
273C22#302111yescharacteristic
274C4—21no—
275C4—21no—
276C4—24no—
277C4—41no—
278C4—44no—
279C4—48no—
280C4—82no—
281C4—84no—
282C4#162314yescomplemented
283C4#327814yes—
284C4D4×SD1614yes—
285C2—322no—
286C2C2×#162311yescharacteristic
287C2#1871811yescharacteristic
288C2#1891412yes—
289C2#2573911yescharacteristic
290C2#2575612yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radical#2867C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59832C2semidirect
4. C2×#1623C2semidirect
5. #10230C2semidirect
6. #9673C2semidirect
7. C4×#333C2semidirect
82. #333D4semidirect
106. C4⋊Q8SD16semidirect
282. C4#1623semidirect

Maximal subgroups

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

subgroupindex
#598322
#284412
C2×#16232
#102302
#96732
C4×#3332