Finite Group Databasebeta

← #505563

Subgroups — #505563

Order 512.

684 conjugacy classes of subgroups.

The subgroup lattice has 323 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1969C211yesmaximal, characteristic
2#20817C211yesmaximal, characteristic
3#13531C212yesmaximal, complemented
4#13481C212yesmaximal, complemented
5#13132C214yesmaximal, complemented
6#12920C212yesmaximal
7#12844C211yesmaximal, characteristic
8#12732C212yesmaximal
9C23×Q16C2211yescharacteristic
10C22×#107C2211yescharacteristic
11C22×#103—21no—
12C22×C4×C8C2211yescharacteristic
13C2×#391—21no—
14#2285—21no—
15C2×#367C2214yes—
16C2×#358C2214yes—
17C2×#358C2214yes—
18C2×C22⋊C16C2212yes—
19#1969—21no—
20#1969—21no—
21#1969C2212yes—
22#1969C2212yes—
23#1845C2211yescharacteristic
24#1845C2211yescharacteristic
25#1834—21no—
26#1808—21no—
27#1808C2212yes—
28C4×#107—21no—
29C4×#107C2212yes—
30#1380C2214yes—
31#1374C2212yes—
32#1374C2212yes—
33C23×Q8—21no—
34C22×Q16—24no—
35C22×Q16—21no—
36C22×Q16—24no—
37C22×Q16—42no—
38C22×Q16C2312yes—
39C23×C8D411yescharacteristic
40C23×C8C2311yescharacteristic
41C2×#117—41no—
42C22×C4⋊C4—21no—
43C22×C42D411yescharacteristic
44C22×C16C2314yes—
45C2×#107—21no—
46C2×#107—21no—
47C2×#107—21no—
48C2×#107—42no—
49C2×#107C2312yes—
50C2×#107C2312yes—
51C2×#107C2312yes—
52C2×#103—22no—
53C2×#103—22no—
54C2×#103—44no—
55C2×C4×C8—24no—
56C2×C4×C8D411yescharacteristic
57C2×C4×C8D411yescharacteristic
58C2×C4×C8C2311yescharacteristic
59C2×C4×C8C2311yescharacteristic
60#391—22no—
61#391—22no—
62#391—41no—
63#391—41no—
64#384—21no—
65#384—21no—
66#382—41no—
67#380—22no—
68#380—41no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—41no—
71C2×#95—41no—
72C2×#95—41no—
73#367—44no—
74#358—28no—
75#358—44no—
76C22⋊C16—24no—
77#340—22no—
78#340—22no—
79C22×Q8—24no—
80C22×Q8—22no—
81C22×Q8—42no—
82C22×Q8—42no—
83C22×Q8—42no—
84C22×Q8—41no—
85C22×Q8—42no—
86C22×Q8—42no—
87C23×C4—41no—
88C23×C4—41no—
89C23×C4D811yescharacteristic
90C23×C4D811yescharacteristic
91C23×C4C2×D411yescharacteristic
92C2×Q16—24no—
93C2×Q16—24no—
94C2×Q16—22no—
95C2×Q16—24no—
96C2×Q16—416no—
97C2×Q16—42no—
98C2×Q16—42no—
99C2×Q16#4114yes—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—41no—
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C22×C8#4114yes—
109C22×C8C2411yescharacteristic
110#117—84no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C22⋊C4—42no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—42no—
125C2×C42D811yescharacteristic
126C2×C42D811yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C16—24no—
130C2×C16—24no—
131C2×C16—24no—
132#107—24no—
133#107—24no—
134#107—24no—
135#107—42no—
136#107—42no—
137#107—42no—
138#103—44no—
139#103—44no—
140C4×C8—22no—
141C4×C8—22no—
142C4×C8—22no—
143C4×C8—22no—
144#95—42no—
145#95—42no—
146#95—42no—
147#95—42no—
148#95—42no—
149#95—42no—
150C24C2×D811yescharacteristic
151C2×Q8—24no—
152C2×Q8—44no—
153C2×Q8—48no—
154C2×Q8—48no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—82no—
161C2×Q8—84no—
162C2×Q8—84no—
163C2×Q8—82no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—41no—
173C22×C4—44no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—82no—
179C22×C4—82no—
180C22×C4—82no—
181C22×C4—82no—
182C22×C4—82no—
183C22×C4SD3214yes—
184C22×C4#12312yes—
185C22×C4C2×D811yescharacteristic
186C22×C4C2×D811yescharacteristic
187C22×C4C2×D811yescharacteristic
188C22×C4C2×D811yescharacteristic
189C22×C4C2×D811yescharacteristic
190C22×C4#13712yes—
191C22×C4C22×D411yescharacteristic
192Q16—28no—
193Q16—48no—
194Q16—44no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—28no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—44no—
204C2×C8#12312yes—
205C2×C8AGL(1,ℤ/8)12yes—
206C2×C8C2×#4112yes—
207C2×C8C2×#4112yes—
208C2×C8H(5,𝔽2)12yes—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—82no—
216C4⋊C4—82no—
217C4⋊C4—82no—
218C4⋊C4—82no—
219C4⋊C4—82no—
220C22⋊C4—84no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—22no—
226C42—22no—
227C42—84no—
228C16—44no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23#48012yes—
234C23C2×SD3212yes—
235C23#51012yes—
236C23C22×D811yescharacteristic
237Q8—44no—
238Q8—44no—
239Q8—88no—
240Q8—82no—
241Q8—82no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—28no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—88no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—164no—
273C2×C4#48012yes—
274C2×C4C2×SD3212yes—
275C2×C4C2×SD3212yes—
276C2×C4#51012yes—
277C2×C4C2×#12311yescharacteristic
278C2×C4#52511yescharacteristic
279C2×C4#53511yescharacteristic
280C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
281C2×C4C2×#13711yescharacteristic
282C2×C4#57511yescharacteristic
283C8—22no—
284C8—22no—
285C8—41no—
286C8#55114yes—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22#223514yes—
294C22C2×#48011yescharacteristic
295C22#310011yescharacteristic
296C22#320511yescharacteristic
297C22C22×SD3211yescharacteristic
298C22C2×#51011yescharacteristic
299C22C2×#51011yescharacteristic
300C22#341411yescharacteristic
301C4—22no—
302C4—22no—
303C4—22no—
304C4—22no—
305C4—41no—
306C4—84no—
307C4—162no—
308C4—162no—
309C4—162no—
310C4#248211yescharacteristic
311C4#309411yescharacteristic
312C4#311011yescharacteristic
313C4#321511yescharacteristic
314C2—24no—
315C2#2758211yescharacteristic
316C2C2×#223511yescharacteristic
317C2#3364911yescharacteristic
318C2#3366311yescharacteristic
319C2#3367311yescharacteristic
320C2#3377211yescharacteristic
321C2#3378111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13531C2semidirect
4. #13481C2semidirect
5. #13132C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19692
#208172
#135312
#134812
#131322
#129202
#128442
#127322