Finite Group Databasebeta

← #491325

Subgroups — #491325

Order 512.

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

normal subgroups only

This group has 410 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#33795C211yesmaximal, characteristic
2#33497C211yesmaximal, characteristic
3#19262C211yesmaximal, characteristic
4C2×#1674C211yesmaximal, characteristic
5#12482C214yesmaximal, complemented
6#12462C214yesmaximal, complemented
7#12032C211yesmaximal, characteristic
8#11971C211yesmaximal, characteristic
9#11949C211yesmaximal, characteristic
10#3170C2211yescharacteristic
11C2×#500C2211yescharacteristic
12#3078—21no—
13C2×#462—21no—
14C2×C8×Q8C2211yescharacteristic
15#2254C2212yes—
16#2241C2212yes—
17#2235—22no—
18#2225—22no—
19#2183—22no—
20#2183C2214yes—
21#2163—22no—
22#2162C2211yescharacteristic
23#2162C2211yescharacteristic
24#1684—22no—
25#1676C2212yes—
26#1674—22no—
27#1674C2214yes—
28#1670C2212yes—
29#1567C2211yescharacteristic
30C2×#335C2211yescharacteristic
31#1550—21no—
32C2×#332—21no—
33#1371C2214yes—
34#1365C2212yes—
35#1364C2214yes—
36#1362C2212yes—
37C22×Q16—41no—
38#533—21no—
39C2×C4⋊Q8—21no—
40C2×C4×Q8D411yescharacteristic
41#500—21no—
42#500—21no—
43#500—21no—
44#500C2312yes—
45#500C2311yescharacteristic
46#494C2311yescharacteristic
47#487—21no—
48#479—42no—
49#477—22no—
50#475—22no—
51#467—21no—
52#467—21no—
53#462—24no—
54#462—42no—
55C8×Q8—21no—
56C8×Q8—21no—
57C8×Q8D412yes—
58C8×Q8C2312yes—
59C4×Q16—21no—
60C4×Q16—21no—
61#428—41no—
62C2×#107—21no—
63C2×#107—21no—
64#423—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8D411yescharacteristic
68#419—41no—
69C2×#103—41no—
70C2×C4×C8—21no—
71C2×C4×C8C2311yescharacteristic
72#367—24no—
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—21no—
92C2×Q16—21no—
93C2×Q16—44no—
94C2×Q16—84no—
95C2×M16—41no—
96C22×C8—21no—
97C22×C8—41no—
98C22×C8C2×D411yescharacteristic
99C22×C8#4111yescharacteristic
100C4⋊Q8—22no—
101C4⋊Q8—21no—
102C4⋊Q8—21no—
103C4⋊Q8—41no—
104C4⋊Q8—41no—
105C4⋊Q8—41no—
106#125—41no—
107C22⋊Q8—41no—
108C22⋊Q8—41no—
109C4×Q8—21no—
110C4×Q8—21no—
111C4×Q8—41no—
112C4×Q8—41no—
113C4×Q8D812yes—
114C4×Q8C2×D412yes—
115#117—41no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4D811yescharacteristic
120C2×C42—21no—
121C2×C42C2×D411yescharacteristic
122C2×C16—22no—
123C2×C16C2×D414yes—
124#107—22no—
125#107—21no—
126#107—22no—
127#107—21no—
128#107—41no—
129#107—41no—
130#107—42no—
131#106—42no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8—21no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8C2×D412yes—
140#103—42no—
141#103—42no—
142#103—41no—
143#103—41no—
144#103—44no—
145C4×C8—22no—
146C4×C8—22no—
147C4×C8C2×D411yescharacteristic
148C4×C8C2×D411yescharacteristic
149C4×C8#4111yescharacteristic
150C4×C8C2411yescharacteristic
151C2×Q8—21no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—41no—
155C2×Q8—42no—
156C2×Q8—41no—
157C2×Q8—81no—
158C2×Q8—82no—
159C2×Q8—82no—
160C2×Q8C2×D812yes—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—81no—
164C22×C4#12111yescharacteristic
165C22×C4C2×D811yescharacteristic
166C22×C4#13511yescharacteristic
167Q16—42no—
168Q16—81no—
169Q16—88no—
170M16—82no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—41no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—81no—
177C2×C8—82no—
178C2×C8#12111yescharacteristic
179C2×C8#12111yescharacteristic
180C2×C8C22×D411yescharacteristic
181C2×C8C2×#4111yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—21no—
185C4⋊C4—41no—
186C4⋊C4—42no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—42no—
190C4⋊C4—41no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C4⋊C4C2×D812yes—
196C22⋊C4—81no—
197C42—22no—
198C42—22no—
199C42—81no—
200C42#12111yescharacteristic
201C42C2×D811yescharacteristic
202C42#13511yescharacteristic
203C42C22×D411yescharacteristic
204C16—24no—
205C16—42no—
206C23#46611yescharacteristic
207Q8—22no—
208Q8—22no—
209Q8—42no—
210Q8—82no—
211Q8—82no—
212Q8—81no—
213Q8—81no—
214Q8—161no—
215Q8—162no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—82no—
226C2×C4—162no—
227C2×C4#46611yescharacteristic
228C2×C4#46611yescharacteristic
229C2×C4#46611yescharacteristic
230C2×C4#50912yes—
231C2×C4#51012yes—
232C2×C4C2×#12111yescharacteristic
233C2×C4C22×D811yescharacteristic
234C2×C4C2×#13511yescharacteristic
235C8—21no—
236C8—21no—
237C8—21no—
238C8—42no—
239C8—82no—
240C8—82no—
241C8#57812yes—
242C22—21no—
243C22—21no—
244C22#222012yes—
245C22C2×#46611yescharacteristic
246C4—21no—
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—162no—
252C4—161no—
253C4—161no—
254C4#222012yes—
255C4#304411yescharacteristic
256C4#306911yescharacteristic
257C4C2×#50911yescharacteristic
258C4C2×#51011yescharacteristic
259C2—22no—
260C2#1940411yescharacteristic
261C2C2×#222011yescharacteristic
262C2#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. #12482C2semidirect
6. #12462C2semidirect

Maximal subgroups

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

subgroupindex
#337952
#334972
#192622
C2×#16742
#124822
#124622
#120322
#119712
#119492