Finite Group Databasebeta

← #347855

Subgroups — #347855

Order 512.

839 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 839 conjugacy classes of subgroups — too many to draw. Only its 174 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×#2985C211yesmaximal, characteristic
2#29808C211yesmaximal, characteristic
3#29688C212yesmaximal
4C2×#1690C212yesmaximal, complemented
5#9220C214yesmaximal, complemented
6#9140C212yesmaximal
7#8657C212yesmaximal, complemented
8#8598C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#494C2211yescharacteristic
11C2×#475C2214yes—
12#2985—28no—
13C2×#446—22no—
14C2×#446C2212yes—
15C2×C8×Q8C2212yes—
16C2×#433—22no—
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19#2857C2212yes—
20#2058—22no—
21#2056—22no—
22#2043—22no—
23#2035—24no—
24#2001—22no—
25#1956C2212yes—
26#1834C2211yescharacteristic
27#1803C2214yes—
28#1797C2212yes—
29#1690—28no—
30C2×#334C2212yes—
31C2×#332C2214yes—
32#1296C2214yes—
33#1277C2212yes—
34C22×Q16—22no—
35#533—24no—
36#533—24no—
37C2×C4⋊Q8D411yescharacteristic
38C2×C4⋊Q8D411yescharacteristic
39C2×C4⋊Q8C2312yes—
40C2×#125—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8D412yes—
44C2×C4×Q8—21no—
45C2×C4×Q8C2312yes—
46C2×#117—21no—
47C22×C42C2311yescharacteristic
48#494—24no—
49#475—28no—
50#475—28no—
51#446—216no—
52#446—24no—
53#446D418yes—
54C8×Q8—24no—
55C8×Q8—24no—
56#433—216no—
57#432—24no—
58#431—24no—
59C2×#106—24no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8D412yes—
63C2×C4⋊C8C2314yes—
64C2×C4⋊C8C2312yes—
65C2×#103—24no—
66C2×#103—24no—
67C2×#103—24no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8D412yes—
71C2×#97—22no—
72C2×#97—21no—
73C2×C4×C8—22no—
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#391—22no—
77#389—22no—
78#389—22no—
79#384C2312yes—
80#380—22no—
81#380—22no—
82#340—22no—
83#340—24no—
84#336—22no—
85#334—24no—
86#332—28no—
87C22×Q8—21no—
88C22×Q8—22no—
89C22×Q8C2×D412yes—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4#4111yescharacteristic
93C2×Q16—44no—
94C2×Q16—84no—
95C22×C8—22no—
96C22×C8—24no—
97C22×C8C2×D412yes—
98C22×C8C2×D412yes—
99C4⋊Q8—24no—
100C4⋊Q8—24no—
101C4⋊Q8—24no—
102C4⋊Q8—24no—
103#125—28no—
104C22⋊Q8—28no—
105C22⋊Q8—24no—
106C22⋊Q8—24no—
107C22⋊Q8—84no—
108C4×Q8—24no—
109C4×Q8—24no—
110C4×Q8—22no—
111C4×Q8—22no—
112C4×Q8C2×D414yes—
113C4×Q8#4114yes—
114#117—24no—
115#117—24no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4C2×D412yes—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—42no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—22no—
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42#4111yescharacteristic
136C2×C42C2411yescharacteristic
137#106—48no—
138C4⋊C8—28no—
139C4⋊C8—28no—
140C4⋊C8—24no—
141C4⋊C8—22no—
142C4⋊C8—24no—
143C4⋊C8—24no—
144C4⋊C8C2×D414yes—
145#103—48no—
146#103—84no—
147#103—84no—
148C22⋊C8—24no—
149C22⋊C8—24no—
150C22⋊C8—82no—
151#97—24no—
152#97—28no—
153C4×C8—28no—
154C4×C8—22no—
155C4×C8C2×D414yes—
156#95—42no—
157#95—42no—
158C24#12111yescharacteristic
159C2×Q8—24no—
160C2×Q8—22no—
161C2×Q8—24no—
162C2×Q8—24no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—44no—
166C2×Q8—44no—
167C2×Q8—82no—
168C2×Q8#12114yes—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4#12111yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4#12111yescharacteristic
184C22×C4C22×D411yescharacteristic
185C22×C4C22×D411yescharacteristic
186C22×C4C2×#4111yescharacteristic
187Q16—88no—
188C2×C8—24no—
189C2×C8—22no—
190C2×C8—24no—
191C2×C8—24no—
192C2×C8—44no—
193C2×C8—44no—
194C2×C8—44no—
195C2×C8—44no—
196C2×C8#12014yes—
197C4⋊C4—24no—
198C4⋊C4—24no—
199C4⋊C4—28no—
200C4⋊C4—24no—
201C4⋊C4—22no—
202C4⋊C4—24no—
203C4⋊C4—24no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—42no—
207C4⋊C4—44no—
208C4⋊C4—42no—
209C4⋊C4—82no—
210C4⋊C4—84no—
211C4⋊C4#12114yes—
212C22⋊C4—28no—
213C22⋊C4—84no—
214C42—24no—
215C42—21no—
216C42—22no—
217C42—21no—
218C42—22no—
219C42—24no—
220C42C22×D412yes—
221C42C2×#4112yes—
222C23—21no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23—22no—
227C23C2×#12111yescharacteristic
228Q8—28no—
229Q8—44no—
230Q8—84no—
231Q8—84no—
232Q8—84no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4C2×#12011yescharacteristic
257C2×C4C2×#12112yes—
258C2×C4#52512yes—
259C2×C4#52511yescharacteristic
260C2×C4#57714yes—
261C2×C4#57712yes—
262C2×C4#57812yes—
263C8—24no—
264C8—42no—
265C8—44no—
266C8—84no—
267C22—42no—
268C22—42no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22#284711yescharacteristic
273C22#304312yes—
274C22#305911yescharacteristic
275C22#306011yescharacteristic
276C22#306912yes—
277C4—22no—
278C4—41no—
279C4—41no—
280C4—44no—
281C4—41no—
282C4—44no—
283C4—162no—
284C4—162no—
285C4#301012yes—
286C4#306314yes—
287C4#312612yes—
288C2—81no—
289C2#1952312yes—
290C2#1952412yes—
291C2#3291211yescharacteristic
292C2#3294011yescharacteristic
293C2#3294611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
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
4. C2×#1690C2semidirect
5. #9220C2semidirect
7. #8657C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29852
#298082
#296882
C2×#16902
#92202
#91402
#86572
#85982