Finite Group Databasebeta

← #418201

Subgroups — #418201

Order 512.

1482 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1482 conjugacy classes of subgroups — too many to draw. Only its 210 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×#2861C211yesmaximal, characteristic
2#35626C211yesmaximal, characteristic
3#23553C212yesmaximal, complemented
4C2×#1856C212yesmaximal
5#11161C212yesmaximal
6#11136C214yesmaximal, complemented
7#10982C211yesmaximal, characteristic
8#10878C212yesmaximal, complemented
9C22×#125C2211yescharacteristic
10C22×#124C2211yescharacteristic
11C2×#518C2211yescharacteristic
12C2×#428C2214yes—
13C22×#101C2212yes—
14C22×#100C2212yes—
15#2861—28no—
16#2483—24no—
17#2447—28no—
18#2310C2212yes—
19#2309C2211yescharacteristic
20#2296C2212yes—
21C2×#391C2212yes—
22C2×#390C2212yes—
23C2×#388—24no—
24C2×C43C2211yescharacteristic
25#2059—28no—
26#1856—28no—
27#1736C2214yes—
28C2×#339C2214yes—
29C2×#339C2214yes—
30#1728C2212yes—
31C23×Q8C2311yescharacteristic
32C23×D4C2311yescharacteristic
33C22×M16C2314yes—
34C2×#125—24no—
35C2×#125—24no—
36C2×#125—42no—
37C2×#124—24no—
38C2×#124—24no—
39C2×#124—42no—
40#518—24no—
41#518—42no—
42C2×C4×D4—24no—
43C2×#117C2312yes—
44C2×#117C2312yes—
45C22×C4⋊C4—22no—
46C22×C4⋊C4C2312yes—
47C22×C22⋊C4—22no—
48C22×C22⋊C4—24no—
49C22×C22⋊C4D414yes—
50C22×C42—22no—
51C22×C42C2312yes—
52C22×C42C2311yescharacteristic
53#428—28no—
54#428—44no—
55C2×#107—24no—
56C2×#106—24no—
57C2×#101—28no—
58C2×#101—42no—
59C2×#100—24no—
60C2×#100—28no—
61#397—216no—
62#391—24no—
63#391—24no—
64#391—24no—
65#391—24no—
66#390—24no—
67#390—24no—
68#390D418yes—
69#389D418yes—
70#388—232no—
71#385—28no—
72#385—24no—
73#381—28no—
74#381—24no—
75#379—22no—
76C4×C4⋊C4—24no—
77C2×#95—24no—
78C43—24no—
79#340—28no—
80#339—28no—
81#339—28no—
82#336—24no—
83C25C2×D412yes—
84C22×Q8—22no—
85C22×Q8—22no—
86C22×Q8—24no—
87C22×Q8—22no—
88C22×Q8—24no—
89C22×D4—22no—
90C22×D4—24no—
91C22×D4—22no—
92C22×D4—24no—
93C22×D4C2×D414yes—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—24no—
97C23×C4C2×D412yes—
98C23×C4C2×D412yes—
99C23×C4C2411yescharacteristic
100C2×M16—24no—
101C2×M16—24no—
102C2×M16—24no—
103C2×M16—24no—
104C2×M16—24no—
105C2×M16#4118yes—
106C22×C8—24no—
107#125—416no—
108#124—416no—
109C4×D4—48no—
110C4×D4—48no—
111#117—24no—
112#117—24no—
113#117—24no—
114#117—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—28no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4C2×D414yes—
123C2×C4⋊C4C2×D414yes—
124C2×C4⋊C4C2×D414yes—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—216no—
127C2×C22⋊C4—28no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—28no—
131C2×C22⋊C4—28no—
132C2×C22⋊C4—44no—
133C2×C22⋊C4—416no—
134C2×C22⋊C4—44no—
135C2×C22⋊C4—44no—
136C2×C42—22no—
137C2×C42—22no—
138C2×C42—24no—
139C2×C42—24no—
140C2×C42—28no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—48no—
145C2×C42C2×D414yes—
146C2×C42C2×D414yes—
147#107—48no—
148#106—48no—
149#101—48no—
150#100—216no—
151#95—216no—
152#95—216no—
153#95—28no—
154C24—22no—
155C24—22no—
156C24—28no—
157C24—48no—
158C24#12014yes—
159C24C22×D411yescharacteristic
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—44no—
165C2×Q8—44no—
166C2×Q8—44no—
167C2×Q8—44no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×D4—24no—
171C2×D4—24no—
172C2×D4—28no—
173C2×D4—28no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—44no—
177C2×D4—44no—
178C2×D4—42no—
179C2×D4—42no—
180C22×C4—28no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—22no—
190C22×C4—28no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—48no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—48no—
198C22×C4—88no—
199C22×C4#12014yes—
200C22×C4C22×D412yes—
201C22×C4C22×D412yes—
202C22×C4C2×#4114yes—
203C22×C4H(5,𝔽2)12yes—
204C22×C42−1+414yes—
205M16—28no—
206M16—28no—
207M16—44no—
208M16—44no—
209C2×C8—28no—
210C2×C8—44no—
211C2×C8—44no—
212C4⋊C4—24no—
213C4⋊C4—24no—
214C4⋊C4—24no—
215C4⋊C4—28no—
216C4⋊C4—24no—
217C4⋊C4—416no—
218C4⋊C4—44no—
219C4⋊C4—88no—
220C22⋊C4—416no—
221C22⋊C4—432no—
222C22⋊C4—48no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—44no—
226C22⋊C4—88no—
227C42—28no—
228C42—28no—
229C42—416no—
230C42—42no—
231C42—42no—
232C42—44no—
233C42—44no—
234C23—24no—
235C23—24no—
236C23—28no—
237C23—48no—
238C23—44no—
239C23—48no—
240C23—44no—
241C23—84no—
242C23—816no—
243C23C2×#12011yescharacteristic
244C23C2×#12011yescharacteristic
245C23#53411yescharacteristic
246C23#54714yes—
247Q8—84no—
248Q8—84no—
249D4—48no—
250D4—48no—
251C2×C4—28no—
252C2×C4—24no—
253C2×C4—28no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—416no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—48no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—48no—
268C2×C4—48no—
269C2×C4—84no—
270C2×C4—84no—
271C2×C4—168no—
272C2×C4#52512yes—
273C2×C4#53212yes—
274C2×C4#53612yes—
275C2×C4#53612yes—
276C2×C4#54214yes—
277C2×C4D4214yes—
278C2×C4#54714yes—
279C2×C4#54814yes—
280C2×C4#55112yes—
281C2×C4#56312yes—
282C8—48no—
283C22—28no—
284C22—22no—
285C22—22no—
286C22—48no—
287C22—44no—
288C22—88no—
289C22—88no—
290C22—164no—
291C22#240311yescharacteristic
292C22#240612yes—
293C22#243212yes—
294C22#271712yes—
295C22#301814yes—
296C4—48no—
297C4—44no—
298C4—84no—
299C4—84no—
300C4—84no—
301C4—82no—
302C4#246114yes—
303C4#267414yes—
304C2—22no—
305C2—22no—
306C2—84no—
307C2#1526211yescharacteristic
308C2#2390812yes—
309C2#2568914yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #23553C2semidirect
6. #11136C2semidirect
8. #10878C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28612
#356262
#235532
C2×#18562
#111612
#111362
#109822
#108782