Finite Group Databasebeta

← #491292

Subgroups — #491292

Order 512.

416 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 134 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#33775C211yesmaximal, characteristic
2#33497C211yesmaximal, characteristic
3#19262C211yesmaximal, characteristic
4C2×#1670C211yesmaximal, characteristic
5#12481C214yesmaximal, complemented
6#12443C214yesmaximal, complemented
7#12033C211yesmaximal, characteristic
8#11962C211yesmaximal, characteristic
9#11949C211yesmaximal, characteristic
10#3170C2211yescharacteristic
11C2×#494C2211yescharacteristic
12#3078—21no—
13C2×#475—21no—
14C2×C8×Q8C2211yescharacteristic
15#2247C2212yes—
16#2241C2212yes—
17#2209—22no—
18#2188—22no—
19#2183—22no—
20#2183C2214yes—
21#2163—22no—
22#2162C2211yescharacteristic
23#2162C2211yescharacteristic
24#1684—22no—
25#1676C2212yes—
26#1670—22no—
27#1670C2212yes—
28#1670C2214yes—
29#1567C2211yescharacteristic
30C2×#335C2211yescharacteristic
31#1550—21no—
32C2×#332—21no—
33#1372C2214yes—
34#1365C2212yes—
35#1362C2212yes—
36#1362C2214yes—
37C22×Q16—21no—
38C22×Q16—21no—
39#533—21no—
40C2×C4⋊Q8—21no—
41C2×C4×Q8D411yescharacteristic
42C2×Q32—24no—
43#500C2311yescharacteristic
44#494—21no—
45#494—21no—
46#494—21no—
47#494C2312yes—
48#494C2311yescharacteristic
49#487—21no—
50#479—42no—
51#477—22no—
52#475—24no—
53#475—22no—
54#475—42no—
55#467—21no—
56#467—21no—
57C8×Q8—21no—
58C8×Q8—21no—
59C8×Q8D412yes—
60C8×Q8C2312yes—
61C4×Q16—21no—
62C4×Q16—21no—
63#428—41no—
64C2×#106—41no—
65#423—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8D411yescharacteristic
69#419—41no—
70C2×#103—41no—
71C2×C4×C8—21no—
72C2×C4×C8C2311yescharacteristic
73C4⋊C16—24no—
74C4⋊C16D412yes—
75C4⋊C16C2312yes—
76#358—24no—
77C22⋊C16—21no—
78C22⋊C16D412yes—
79#346—24no—
80C4×C16—21no—
81C4×C16C2312yes—
82#335—21no—
83#335—21no—
84#335C2312yes—
85#335C2312yes—
86#333—22no—
87#332—24no—
88#332—22no—
89C22×Q8—41no—
90C22×Q8D811yescharacteristic
91C2×Q16—22no—
92C2×Q16—21no—
93C2×Q16—22no—
94C2×Q16—21no—
95C2×Q16—41no—
96C2×Q16—41no—
97C2×Q16—82no—
98C2×Q16—82no—
99C2×M16—41no—
100C22×C8—21no—
101C22×C8—41no—
102C22×C8C2×D411yescharacteristic
103C22×C8#4111yescharacteristic
104C4⋊Q8—22no—
105C4⋊Q8—21no—
106C4⋊Q8—21no—
107C4⋊Q8—41no—
108C4⋊Q8—41no—
109C4⋊Q8—41no—
110#125—41no—
111C22⋊Q8—41no—
112C22⋊Q8—41no—
113C4×Q8—21no—
114C4×Q8—21no—
115C4×Q8—41no—
116C4×Q8—41no—
117C4×Q8D812yes—
118C4×Q8C2×D412yes—
119#117—41no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4D811yescharacteristic
124C2×C42—21no—
125C2×C42C2×D411yescharacteristic
126Q32—84no—
127C2×C16—22no—
128C2×C16C2×D414yes—
129#107—21no—
130#107—21no—
131#107—42no—
132#106—44no—
133#106—42no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—21no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8C2×D412yes—
142#103—44no—
143#103—42no—
144#103—42no—
145#103—41no—
146#103—41no—
147C4×C8—22no—
148C4×C8—22no—
149C4×C8C2×D411yescharacteristic
150C4×C8C2×D411yescharacteristic
151C4×C8#4111yescharacteristic
152C4×C8C2411yescharacteristic
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—41no—
157C2×Q8—42no—
158C2×Q8—41no—
159C2×Q8—82no—
160C2×Q8—82no—
161C2×Q8—81no—
162C2×Q8C2×D812yes—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—81no—
166C22×C4#12111yescharacteristic
167C22×C4C2×D811yescharacteristic
168C22×C4#13511yescharacteristic
169Q16—42no—
170Q16—44no—
171Q16—81no—
172Q16—82no—
173Q16—82no—
174Q16—82no—
175M16—82no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—41no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—81no—
182C2×C8—82no—
183C2×C8#12111yescharacteristic
184C2×C8#12111yescharacteristic
185C2×C8C22×D411yescharacteristic
186C2×C8C2×#4111yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—41no—
191C4⋊C4—42no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—41no—
196C4⋊C4—81no—
197C4⋊C4—81no—
198C4⋊C4—81no—
199C4⋊C4—81no—
200C4⋊C4C2×D812yes—
201C22⋊C4—81no—
202C42—22no—
203C42—22no—
204C42—81no—
205C42#12111yescharacteristic
206C42C2×D811yescharacteristic
207C42#13511yescharacteristic
208C42C22×D411yescharacteristic
209C16—24no—
210C16—42no—
211C23#46611yescharacteristic
212Q8—22no—
213Q8—22no—
214Q8—42no—
215Q8—82no—
216Q8—81no—
217Q8—82no—
218Q8—81no—
219Q8—161no—
220Q8—162no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—82no—
231C2×C4—162no—
232C2×C4#46611yescharacteristic
233C2×C4#46611yescharacteristic
234C2×C4#46611yescharacteristic
235C2×C4#51012yes—
236C2×C4#51012yes—
237C2×C4C2×#12111yescharacteristic
238C2×C4C22×D811yescharacteristic
239C2×C4C2×#13511yescharacteristic
240C8—21no—
241C8—21no—
242C8—21no—
243C8—42no—
244C8—82no—
245C8—82no—
246C8#57712yes—
247C22—21no—
248C22—21no—
249C22#221912yes—
250C22C2×#46611yescharacteristic
251C4—21no—
252C4—21no—
253C4—21no—
254C4—42no—
255C4—42no—
256C4—161no—
257C4—162no—
258C4—161no—
259C4#221912yes—
260C4#304311yescharacteristic
261C4#306811yescharacteristic
262C4C2×#51011yescharacteristic
263C4C2×#51011yescharacteristic
264C2—22no—
265C2#1939811yescharacteristic
266C2C2×#221911yescharacteristic
267C2#3361911yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC4×C8C24
prime radicalC23#466
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #12481C2semidirect
6. #12443C2semidirect

Maximal subgroups

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

subgroupindex
#337752
#334972
#192622
C2×#16702
#124812
#124432
#120332
#119622
#119492