Finite Group Databasebeta

← #109573

Subgroups — #109573

Order 512.

876 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20526C211yesmaximal, characteristic
2#13782C211yesmaximal, characteristic
3C2×#1282C212yesmaximal
4#7907C211yesmaximal, characteristic
5C2×#1280C212yesmaximal
6C22×C22⋊C8C2212yes—
7C22×C22⋊C8C2212yes—
8C2×#385C412yes—
9C2×C4×C22⋊C4C2211yescharacteristic
10C2×C4×C22⋊C4C2211yescharacteristic
11C22×#95C2211yescharacteristic
12#1752—24no—
13#1749—22no—
14#1436—22no—
15C4⋊#95—21no—
16#1434—21no—
17C22⋊#95C412yes—
18#1282C418yescomplemented
19#1280—24no—
20#1280—21no—
21#1280—21no—
22C24×C4C2×C412yes—
23C24×C4C2311yescharacteristic
24C23×C8—22no—
25C23×C8—22no—
26C22×C4⋊C4—22no—
27C22×C22⋊C4C2×C412yes—
28C22×C22⋊C4C2×C411yescharacteristic
29C22×C22⋊C4C2×C411yescharacteristic
30C22×C42—21no—
31C22×C42D412yes—
32C2×C22⋊C8—22no—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8C2×C414yes—
41C2×C22⋊C8C2×C414yes—
42C2×C22⋊C8D414yes—
43C2×C22⋊C8Q814yes—
44C2×#97—24no—
45#385—24no—
46#385—24no—
47C4×C22⋊C4—22no—
48C4×C22⋊C4—22no—
49C4×C22⋊C4—22no—
50C4×C22⋊C4C2×C414yes—
51C2×#95—22no—
52C2×#95—21no—
53C2×#95—22no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—22no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95D411yescharacteristic
60C2×#95D411yescharacteristic
61#336—24no—
62C25C22×C411yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—22no—
66C23×C4—24no—
67C23×C4—22no—
68C23×C4—24no—
69C23×C4—82no—
70C23×C4—82no—
71C23×C4—82no—
72C23×C4C22⋊C412yes—
73C23×C4C22⋊C412yes—
74C23×C4M1614yes—
75C23×C4C22×C412yes—
76C23×C4C2×D412yes—
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C23×C4C2×Q812yes—
80C23×C4#4112yes—
81C23×C4#4112yes—
82C23×C4#4112yes—
83C22×C8—24no—
84C22×C8—24no—
85C22×C8—24no—
86C22×C8—24no—
87C22×C8—42no—
88C22×C8—42no—
89C22×C8—84no—
90C22×C8—84no—
91C2×C4⋊C4—28no—
92C2×C4⋊C4—82no—
93C2×C4⋊C4—82no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4C4212yes—
103C2×C22⋊C4C4212yes—
104C2×C22⋊C4M1614yes—
105C2×C42—22no—
106C2×C42—24no—
107C2×C42—22no—
108C2×C42—82no—
109C2×C42—82no—
110C22⋊C8—24no—
111C22⋊C8—24no—
112C22⋊C8—24no—
113C22⋊C8—24no—
114C22⋊C8—24no—
115C22⋊C8—42no—
116C22⋊C8—42no—
117C22⋊C8C4⋊C418yescomplemented
118#97—84no—
119#95—28no—
120#95—24no—
121#95—22no—
122#95—22no—
123#95—41no—
124#95—42no—
125#95—41no—
126#95—82no—
127#95—82no—
128C24—42no—
129C24—41no—
130C24—41no—
131C24C2×C4211yescharacteristic
132C24C2×C22⋊C411yescharacteristic
133C24#11711yescharacteristic
134C24C4×D412yes—
135C24C4×Q812yes—
136C24C2×M1612yes—
137C24C2×M1612yes—
138C24#13114yes—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—28no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—42no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—41no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—44no—
172C22×C4—82no—
173C22×C4—82no—
174C22×C4—82no—
175C22×C4—82no—
176C22×C4—84no—
177C22×C4—84no—
178C22×C4—84no—
179C22×C4—84no—
180C22×C4—88no—
181C22×C4C2×C4⋊C412yes—
182C22×C4#11712yes—
183C22×C4C4×D412yes—
184C22×C4C4×D412yes—
185C22×C4C4×D412yes—
186C22×C4C4×Q812yes—
187C22×C4#12011yescharacteristic
188C22×C4#12112yes—
189C22×C4C22⋊Q812yes—
190C22×C4C22⋊Q812yes—
191C22×C4#12311yescharacteristic
192C22×C4#12312yes—
193C22×C4#12411yescharacteristic
194C22×C4#12512yes—
195C22×C4#12612yes—
196C22×C4C4⋊Q811yescharacteristic
197C2×C8—28no—
198C2×C8—48no—
199C2×C8—44no—
200C2×C8—84no—
201C2×C8—84no—
202C2×C8—84no—
203C2×C8—84no—
204C2×C8—84no—
205C4⋊C4—88no—
206C22⋊C4—22no—
207C22⋊C4—24no—
208C22⋊C4—24no—
209C22⋊C4—42no—
210C22⋊C4—44no—
211C22⋊C4—44no—
212C22⋊C4#9714yescomplemented
213C42—44no—
214C42—82no—
215C42—84no—
216C23—22no—
217C23—24no—
218C23—21no—
219C23—22no—
220C23—21no—
221C23—22no—
222C23—41no—
223C23—41no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—44no—
228C23—41no—
229C23—41no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—44no—
234C23C4×C22⋊C411yescharacteristic
235C23C4×C4⋊C412yes—
236C23#38611yescharacteristic
237C23#38712yes—
238C23#39111yescharacteristic
239C23C2×#9711yescharacteristic
240C23C4×M1611yescharacteristic
241C23#40512yes—
242C23#40712yes—
243C23#40812yes—
244C23#43212yes—
245C23#43312yes—
246C23#43514yes—
247C23#43614yes—
248C23#44614yes—
249C23#44614yes—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—88no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—88no—
295C2×C4#38512yes—
296C2×C4#38712yes—
297C2×C4#38812yes—
298C2×C4#38912yes—
299C8—48no—
300C8—84no—
301C22—22no—
302C22—22no—
303C22—24no—
304C22—24no—
305C22—22no—
306C22—22no—
307C22—24no—
308C22—24no—
309C22—21no—
310C22—22no—
311C22—22no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—21no—
317C22—41no—
318C22—41no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—44no—
323C22—44no—
324C22#143611yescharacteristic
325C22#174911yescharacteristic
326C22#175011yescharacteristic
327C22#177012yes—
328C22#177112yes—
329C22#187012yes—
330C22#191312yes—
331C22#191411yescharacteristic
332C22#191511yescharacteristic
333C22#191714yes—
334C22#198512yes—
335C4—22no—
336C4—22no—
337C4—22no—
338C4—22no—
339C4—44no—
340C4—82no—
341C4—82no—
342C4—84no—
343C4—82no—
344C2—21no—
345C2—21no—
346C2—24no—
347C2—22no—
348C2—22no—
349C2—22no—
350C2#910912yes—
351C2#911012yes—
352C2#1087511yescharacteristic
353C2#1091511yescharacteristic
354C2#1091611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC24C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #1282C4semidirect
117. C22⋊C8C4⋊C4semidirect
212. C22⋊C4#97semidirect

Maximal subgroups

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

subgroupindex
#205262
#137822
C2×#12822
#79072
C2×#12802