Finite Group Databasebeta

← #386284

Subgroups — #386284

Order 512.

782 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21038C212yesmaximal
2#21037C212yesmaximal
3C2×#1847C211yesmaximal, characteristic
4#11366C212yesmaximal
5#11365C212yesmaximal
6#11210C212yesmaximal
7#10892C211yesmaximal, characteristic
8#10892C211yesmaximal, characteristic
9#10819C212yesmaximal
10C22×#107C2212yes—
11C22×#106C2212yes—
12C22×#97C2211yescharacteristic
13#2316C2212yes—
14C2×#384C2211yescharacteristic
15C2×#382C2211yescharacteristic
16#2274C2211yescharacteristic
17#2274C2211yescharacteristic
18#1921—24no—
19#1920—24no—
20#1919—24no—
21#1918—24no—
22#1847C418yescomplemented
23C2×#343C2211yescharacteristic
24C2×#343C2212yes—
25C2×#343C2211yescharacteristic
26C2×#341C2212yes—
27#1736C2212yes—
28#1736C2212yes—
29#1736C2212yes—
30#1736C2212yes—
31#1735C2212yes—
32C2×#339C2212yes—
33C2×#339C2212yes—
34#1729C2212yes—
35#1729C2212yes—
36C22×M16C2312yes—
37C22×M16C2312yes—
38C23×C8C2312yes—
39C2×#117C2312yes—
40C2×#117C2312yes—
41C22×C4⋊C4—41no—
42C22×C4⋊C4C2312yes—
43C22×C4⋊C4C2311yescharacteristic
44C22×C4⋊C4C2311yescharacteristic
45C22×C42C2311yescharacteristic
46C2×#107—22no—
47C2×#107—22no—
48C2×#107—24no—
49C2×#107C2×C418yes—
50C2×#106—22no—
51C2×#106—22no—
52C2×#106—24no—
53C2×#106C2×C418yes—
54C2×#97—24no—
55C2×#97C2×C414yes—
56#389—24no—
57#389—24no—
58#384—41no—
59#384C2×C414yes—
60#382—41no—
61#382C2×C414yes—
62#380—42no—
63C4×C4⋊C4—24no—
64C4×C4⋊C4—24no—
65C4×C22⋊C4—41no—
66C4×C22⋊C4—41no—
67C2×#95—41no—
68#343—41no—
69#343—41no—
70#343—42no—
71#341—24no—
72#340—24no—
73#340—24no—
74#340—24no—
75#340—24no—
76#340D414yes—
77#340D414yes—
78#339—24no—
79#339—24no—
80#336—22no—
81#336—22no—
82#336—22no—
83#336—22no—
84C23×C4—21no—
85C23×C4—22no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4C2411yescharacteristic
89C2×M16—22no—
90C2×M16—22no—
91C2×M16—22no—
92C2×M16—22no—
93C2×M16—22no—
94C2×M16—22no—
95C2×M16—22no—
96C2×M16—22no—
97C2×M16—22no—
98C2×M16—22no—
99C2×M16—22no—
100C2×M16—22no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8C22×C414yes—
108C22×C8C2×D412yes—
109C22×C8C2×D412yes—
110C22×C8#4114yes—
111#117—22no—
112#117—22no—
113#117—22no—
114#117—22no—
115#117—22no—
116#117—22no—
117#117—22no—
118#117—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—84no—
133C2×C4⋊C4C22×C412yes—
134C2×C4⋊C4C22×C412yes—
135C2×C4⋊C4C22×C414yes—
136C2×C4⋊C4C2×D414yes—
137C2×C4⋊C4C2×D414yes—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C42—21no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—42no—
145C2×C42—42no—
146C2×C42C22×C412yes—
147C2×C42#4114yes—
148#107—28no—
149#107—28no—
150#106—28no—
151#106—28no—
152#97—28no—
153#95—44no—
154#95—42no—
155#95—42no—
156#95—44no—
157C24H(5,𝔽2)11yescharacteristic
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—84no—
174C22×C4—82no—
175C22×C4—82no—
176C22×C4C23×C411yescharacteristic
177C22×C4C22×D412yes—
178C22×C4C2×#4112yes—
179C22×C4C2×#4112yes—
180C22×C4H(5,𝔽2)11yescharacteristic
181C22×C42−1+412yes—
182C22×C42−1+412yes—
183C22×C42−1+412yes—
184M16—42no—
185M16—42no—
186M16—42no—
187M16—42no—
188M16—42no—
189M16—42no—
190M16—42no—
191M16—42no—
192C2×C8—24no—
193C2×C8—24no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—24no—
199C2×C8—24no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8C4×D414yes—
207C2×C8C4×D414yes—
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—24no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—88no—
221C4⋊C4—164no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C42—22no—
227C42—24no—
228C42—42no—
229C42—42no—
230C42—42no—
231C42—42no—
232C42—162no—
233C42—162no—
234C42#11714yes—
235C23—22no—
236C23—22no—
237C23#51811yescharacteristic
238C23#53912yes—
239C23D4211yescharacteristic
240C23#55911yescharacteristic
241C23#56312yes—
242C2×C4—24no—
243C2×C4—42no—
244C2×C4—48no—
245C2×C4—42no—
246C2×C4—44no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—82no—
256C2×C4—84no—
257C2×C4—82no—
258C2×C4—84no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—162no—
262C2×C4—161no—
263C2×C4—161no—
264C2×C4C2×#11711yescharacteristic
265C2×C4C2×C4×D412yes—
266C2×C4#51911yescharacteristic
267C2×C4#51912yes—
268C2×C4#53612yes—
269C2×C4#53612yes—
270C2×C4#53912yes—
271C2×C4#54112yes—
272C2×C4#54112yes—
273C2×C4#54712yes—
274C2×C4#54712yes—
275C2×C4#55012yes—
276C2×C4#55212yes—
277C2×C4#55212yes—
278C2×C4#56311yescharacteristic
279C2×C4#56311yescharacteristic
280C8—24no—
281C8—24no—
282C8—82no—
283C8—82no—
284C22—24no—
285C22—24no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22#231311yescharacteristic
291C22#236611yescharacteristic
292C22#238111yescharacteristic
293C22#268012yes—
294C22#279412yes—
295C22#328812yes—
296C22#328912yes—
297C4—44no—
298C4—84no—
299C4—84no—
300C4—82no—
301C4—84no—
302C4—82no—
303C4#232712yes—
304C4#234412yes—
305C4#272112yes—
306C4#272112yes—
307C2—21no—
308C2—21no—
309C2—22no—
310C2#1442411yescharacteristic
311C2#2320211yescharacteristic
312C2#2320311yescharacteristic
313C2#3111314yes—
subgroupG/H
center, socleC23D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24H(5,𝔽2)
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #1847C4semidirect

Maximal subgroups

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

subgroupindex
#210382
#210372
C2×#18472
#113662
#113652
#112102
#108922
#108192