Finite Group Databasebeta

← #102459

Subgroups — #102459

Order 512.

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

normal subgroups only

This group has 668 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#32954C212yesmaximal
2#30242C211yesmaximal, characteristic
3#11380C211yesmaximal, characteristic
4#7668C212yesmaximal
5#7020C211yesmaximal, characteristic
6C22×C4×Q8C2211yescharacteristic
7C2×#475—22no—
8C2×#467—22no—
9#2857C2212yes—
10#2095—22no—
11#2016—22no—
12#1865—24no—
13#1853—22no—
14#1803—22no—
15#1797C2212yes—
16#1787—22no—
17C2×#336—22no—
18C2×#336C2211yescharacteristic
19C2×#336C2211yescharacteristic
20#1271—24no—
21C23×Q8D411yescharacteristic
22C22×Q16—22no—
23C23×C8—21no—
24C23×C8D412yes—
25C2×C4⋊Q8D412yes—
26C2×C22⋊Q8—22no—
27C2×C4×Q8—22no—
28C2×C4×Q8—22no—
29C2×C4×Q8—44no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4D411yescharacteristic
32C22×C42—21no—
33C22×C42C2311yescharacteristic
34#475—48no—
35#467—44no—
36#467—44no—
37C2×#107—22no—
38C2×#106—24no—
39C2×C4⋊C8—22no—
40C2×#103—24no—
41C2×#103—24no—
42C2×#103—24no—
43C2×C22⋊C8—22no—
44#391—22no—
45#389—22no—
46#389—22no—
47#384D412yes—
48#380—22no—
49#380—22no—
50#340—24no—
51#336—22no—
52#336—22no—
53#336—44no—
54#336D414yes—
55C22×Q8—21no—
56C22×Q8—22no—
57C22×Q8—22no—
58C22×Q8—22no—
59C22×Q8—42no—
60C22×Q8—42no—
61C22×Q8Q1612yescomplemented
62C23×C4—21no—
63C23×C4—21no—
64C23×C4C2×D411yescharacteristic
65C23×C4C2×D411yescharacteristic
66C23×C4#4111yescharacteristic
67C2×Q16—44no—
68C2×Q16—82no—
69C2×Q16—82no—
70C22×C8—22no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—24no—
74C22×C8—42no—
75C22×C8—42no—
76C22×C8—41no—
77C22×C8—42no—
78C22×C8—41no—
79C22×C8—42no—
80C22×C8SD1614yes—
81C4⋊Q8—44no—
82C22⋊Q8—44no—
83C22⋊Q8—44no—
84C4×Q8—28no—
85C4×Q8—44no—
86C4×Q8—44no—
87C4×Q8—44no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C42—24no—
101C2×C42—42no—
102C2×C42—42no—
103C2×C42—41no—
104C2×C42—41no—
105C2×C42C2×D412yes—
106C2×C42C2×D412yes—
107#107—44no—
108#106—48no—
109C4⋊C8—44no—
110#103—48no—
111#103—48no—
112#103—48no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115#95—42no—
116#95—42no—
117C24#12111yescharacteristic
118C2×Q8—24no—
119C2×Q8—24no—
120C2×Q8—24no—
121C2×Q8—44no—
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8—44no—
125C2×Q8—48no—
126C2×Q8—42no—
127C2×Q8—44no—
128C2×Q8—42no—
129C2×Q8—42no—
130C2×Q8—44no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—42no—
149C22×C4—41no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4#12012yes—
154C22×C4#12011yescharacteristic
155C22×C4#12112yes—
156C22×C4#12711yescharacteristic
157C22×C4C2×SD1612yes—
158C22×C4C2×Q1611yescharacteristic
159C22×C4#13511yescharacteristic
160C22×C4AGL(1,ℤ/8)12yes—
161C22×C4#13712yes—
162Q16—88no—
163C2×C8—24no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—44no—
167C2×C8—44no—
168C2×C8—44no—
169C2×C8—42no—
170C2×C8—44no—
171C2×C8—44no—
172C2×C8—82no—
173C2×C8—81no—
174C2×C8—82no—
175C2×C8—81no—
176C4⋊C4—24no—
177C4⋊C4—42no—
178C4⋊C4—48no—
179C4⋊C4—44no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—44no—
183C4⋊C4—82no—
184C4⋊C4—82no—
185C4⋊C4—82no—
186C22⋊C4—84no—
187C42—24no—
188C42—42no—
189C42—44no—
190C42—44no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23#39211yescharacteristic
196C23#46012yes—
197C23#46711yescharacteristic
198C23#47011yescharacteristic
199C23#47312yes—
200Q8—28no—
201Q8—44no—
202Q8—44no—
203Q8—44no—
204Q8—84no—
205Q8—84no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—24no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4#45012yes—
240C2×C4#45112yes—
241C2×C4#45212yes—
242C2×C4#45511yescharacteristic
243C2×C4#45611yescharacteristic
244C2×C4B(4,𝔽2)12yes—
245C2×C4#49211yescharacteristic
246C2×C4#49611yescharacteristic
247C8—44no—
248C8—84no—
249C8—82no—
250C22—21no—
251C22—21no—
252C22—22no—
253C22—22no—
254C22—21no—
255C22—21no—
256C22—22no—
257C22—22no—
258C22#167012yes—
259C22#169012yes—
260C22#200111yescharacteristic
261C22#200711yescharacteristic
262C22#201011yescharacteristic
263C22#201411yescharacteristic
264C22#201711yescharacteristic
265C22#202011yescharacteristic
266C22#202111yescharacteristic
267C4—22no—
268C4—22no—
269C4—21no—
270C4—21no—
271C4—44no—
272C4—42no—
273C4—82no—
274C4—162no—
275C4—162no—
276C4#160112yes—
277C4#160412yes—
278C2—22no—
279C2—22no—
280C2#996411yescharacteristic
281C2#997611yescharacteristic
282C2#1003111yescharacteristic
283C2#1003411yescharacteristic
284C2#1025911yescharacteristic
285C2#1027611yescharacteristic
286C2#1032411yescharacteristic
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
61. C22×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#329542
#302422
#113802
#76682
#70202