Finite Group Databasebeta

← #102463

Subgroups — #102463

Order 512.

540 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 98 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#33147C212yesmaximal
2#30242C211yesmaximal, characteristic
3#11346C211yesmaximal, characteristic
4#7668C212yesmaximal
5#7020C211yesmaximal, characteristic
6C2×#467—22no—
7#2857C2212yes—
8C2×C4×C4⋊C4C2211yescharacteristic
9#2095—22no—
10#2016—22no—
11#1988—22no—
12#1948—22no—
13#1919—24no—
14#1853—22no—
15#1824—22no—
16#1797C2212yes—
17C2×#336—22no—
18C2×#336C2211yescharacteristic
19C2×#336C2211yescharacteristic
20#1271—24no—
21C22×Q16—22no—
22C23×C8—21no—
23C23×C8D412yes—
24C2×C4⋊Q8D412yes—
25C2×C22⋊Q8—22no—
26C22×C4⋊C4D411yescharacteristic
27C22×C4⋊C4D411yescharacteristic
28C22×C42—21no—
29C22×C42C2311yescharacteristic
30#467—44no—
31#467—44no—
32C2×#107—22no—
33C2×#106—24no—
34C2×C4⋊C8—22no—
35C2×#103—24no—
36C2×C22⋊C8—22no—
37#391—22no—
38#389—22no—
39#389—22no—
40#384D412yes—
41#380—22no—
42#380—22no—
43C4×C4⋊C4—22no—
44C4×C4⋊C4—22no—
45C2×#95—21no—
46#340—24no—
47#340—24no—
48#340—24no—
49#336—22no—
50#336—22no—
51#336—44no—
52#336D414yes—
53C22×Q8—22no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4C2×D411yescharacteristic
57C23×C4C2×D411yescharacteristic
58C23×C4#4111yescharacteristic
59C2×Q16—44no—
60C2×Q16—82no—
61C2×Q16—82no—
62C22×C8—22no—
63C22×C8—22no—
64C22×C8—22no—
65C22×C8—24no—
66C22×C8—42no—
67C22×C8—42no—
68C22×C8—41no—
69C22×C8—42no—
70C22×C8—41no—
71C22×C8—42no—
72C22×C8SD1614yes—
73C4⋊Q8—44no—
74C22⋊Q8—44no—
75C22⋊Q8—44no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4Q1612yescomplemented
88C2×C22⋊C4—42no—
89C2×C42—42no—
90C2×C42—42no—
91C2×C42—42no—
92C2×C42—41no—
93C2×C42—41no—
94C2×C42C2×D412yes—
95C2×C42C2×D412yes—
96#107—44no—
97#106—48no—
98C4⋊C8—44no—
99#103—48no—
100C22⋊C8—42no—
101C22⋊C8—42no—
102#95—42no—
103#95—42no—
104#95—42no—
105C24#12111yescharacteristic
106C2×Q8—44no—
107C2×Q8—42no—
108C2×Q8—42no—
109C2×Q8—44no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—41no—
123C22×C4—42no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4#12012yes—
131C22×C4#12011yescharacteristic
132C22×C4#12112yes—
133C22×C4#12711yescharacteristic
134C22×C4C2×SD1612yes—
135C22×C4C2×Q1611yescharacteristic
136C22×C4#13511yescharacteristic
137C22×C4AGL(1,ℤ/8)12yes—
138C22×C4#13712yes—
139Q16—88no—
140C2×C8—24no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—44no—
144C2×C8—44no—
145C2×C8—44no—
146C2×C8—42no—
147C2×C8—44no—
148C2×C8—44no—
149C2×C8—82no—
150C2×C8—81no—
151C2×C8—82no—
152C2×C8—81no—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—42no—
156C4⋊C4—44no—
157C4⋊C4—44no—
158C4⋊C4—44no—
159C4⋊C4—42no—
160C4⋊C4—44no—
161C4⋊C4—82no—
162C4⋊C4—82no—
163C4⋊C4—82no—
164C22⋊C4—84no—
165C42—24no—
166C42—42no—
167C42—82no—
168C42—81no—
169C42—81no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23#39211yescharacteristic
175C23#46012yes—
176C23#46711yescharacteristic
177C23#47011yescharacteristic
178C23#47312yes—
179Q8—84no—
180Q8—84no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4#45012yes—
213C2×C4#45112yes—
214C2×C4#45212yes—
215C2×C4#45311yescharacteristic
216C2×C4#45411yescharacteristic
217C2×C4#45812yes—
218C2×C4#49211yescharacteristic
219C2×C4#49611yescharacteristic
220C8—44no—
221C8—84no—
222C8—82no—
223C22—21no—
224C22—21no—
225C22—22no—
226C22—22no—
227C22—21no—
228C22—21no—
229C22—22no—
230C22—22no—
231C22#167012yes—
232C22#169012yes—
233C22#200111yescharacteristic
234C22#200611yescharacteristic
235C22#201011yescharacteristic
236C22#201411yescharacteristic
237C22#201711yescharacteristic
238C22#202011yescharacteristic
239C22#202211yescharacteristic
240C4—22no—
241C4—22no—
242C4—21no—
243C4—21no—
244C4—42no—
245C4—82no—
246C4—82no—
247C4—162no—
248C4—162no—
249C4#161312yes—
250C4#161612yes—
251C2—22no—
252C2—22no—
253C2#996811yescharacteristic
254C2#997911yescharacteristic
255C2#1003011yescharacteristic
256C2#1003411yescharacteristic
257C2#1025811yescharacteristic
258C2#1027611yescharacteristic
259C2#1032511yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C22×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
87. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
#331472
#302422
#113462
#76682
#70202