Finite Group Databasebeta

← #359935

Subgroups — #359935

Order 512.

2070 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34178C211yesmaximal, characteristic
2C22×#384C211yesmaximal, characteristic
3#20700C212yesmaximal
4C2×#1821C211yesmaximal, characteristic
5#11210C214yesmaximal, complemented
6#10891C214yesmaximal, complemented
7#10719C212yesmaximal
8C23×C4⋊C4—22no—
9C23×C4⋊C4C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#407C2212yes—
13#2345—24no—
14#2344—24no—
15#2316C2212yes—
16#2313—22no—
17C2×#384—24no—
18C2×#384—22no—
19C2×#384C418yescomplemented
20C2×#384C2212yes—
21C2×#384C2212yes—
22#2280—22no—
23#2280C2212yes—
24#2274C2212yes—
25#1824—24no—
26#1823—24no—
27#1821—24no—
28C2×#343C2212yes—
29C2×#343C2212yes—
30#1736C2218yes—
31C2×#339C2218yes—
32C24×C4—21no—
33C24×C4—21no—
34C24×C4D412yes—
35C24×C4C2311yescharacteristic
36C22×M16C2314yes—
37C2×#117—24no—
38C2×#117C2314yes—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—24no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—24no—
45C22×C4⋊C4—216no—
46C22×C4⋊C4—24no—
47C22×C4⋊C4—24no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—416no—
51C22×C4⋊C4C2×C414yes—
52C22×C4⋊C4C2×C414yes—
53C22×C4⋊C4D412yes—
54C22×C4⋊C4D414yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C42—21no—
59C22×C42—22no—
60C22×C42—28no—
61C22×C42C2×C414yes—
62C22×C42C2312yes—
63#407—24no—
64#407—24no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67#389—28no—
68#387—216no—
69#384—216no—
70#384—24no—
71#384—24no—
72#384C2×C418yes—
73#384C2×C418yes—
74#381—24no—
75#381—22no—
76#380—24no—
77#380—24no—
78#380D414yes—
79#380D414yes—
80#379—22no—
81C4×C4⋊C4—28no—
82C4×C22⋊C4—24no—
83C4×C22⋊C4—24no—
84C4×C22⋊C4—22no—
85C4×C22⋊C4—22no—
86#343—24no—
87#343—24no—
88#340—216no—
89#339—216no—
90C25C2×D411yescharacteristic
91C23×C4—28no—
92C23×C4—28no—
93C23×C4—28no—
94C23×C4—24no—
95C23×C4—22no—
96C23×C4—24no—
97C23×C4—24no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—24no—
101C23×C4—24no—
102C23×C4—22no—
103C23×C4—48no—
104C23×C4—48no—
105C23×C4—41no—
106C23×C4—41no—
107C23×C4—42no—
108C23×C4C22⋊C418yes—
109C23×C4C22×C412yes—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4#4114yes—
117C23×C4C2411yescharacteristic
118C2×M16—24no—
119C2×M16—28no—
120C2×M16—24no—
121C2×M16—28no—
122C22×C8—24no—
123#117—28no—
124#117—28no—
125#117—88no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—28no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—216no—
141C2×C4⋊C4—216no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—464no—
146C2×C4⋊C4—416no—
147C2×C4⋊C4—416no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4C22⋊C418yes—
153C2×C4⋊C4C22×C414yes—
154C2×C4⋊C4C22×C412yes—
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C2×D418yes—
157C2×C4⋊C4C2×D412yes—
158C2×C4⋊C4C2×D412yes—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C42—28no—
163C2×C42—216no—
164C2×C42—24no—
165C2×C42—22no—
166C2×C42—24no—
167C2×C42—216no—
168C2×C42—24no—
169C2×C42—28no—
170C2×C42C22×C414yes—
171C2×C42#4118yes—
172C22⋊C8—44no—
173C22⋊C8—44no—
174#95—28no—
175#95—28no—
176#95—24no—
177#95—24no—
178C24—22no—
179C24—22no—
180C24—24no—
181C24—24no—
182C24C2×C22⋊C412yes—
183C24#12011yescharacteristic
184C24#12011yescharacteristic
185C24#12312yes—
186C24C22×D411yescharacteristic
187C22×C4—216no—
188C22×C4—216no—
189C22×C4—28no—
190C22×C4—264no—
191C22×C4—216no—
192C22×C4—28no—
193C22×C4—24no—
194C22×C4—22no—
195C22×C4—28no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—28no—
202C22×C4—28no—
203C22×C4—216no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—28no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—28no—
210C22×C4—24no—
211C22×C4—28no—
212C22×C4—48no—
213C22×C4—416no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—416no—
217C22×C4—48no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—88no—
223C22×C4—84no—
224C22×C4—88no—
225C22×C4—84no—
226C22×C4—84no—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C414yes—
229C22×C4C2×C22⋊C414yes—
230C22×C4C4×D414yes—
231C22×C4C4×D414yes—
232C22×C4#12012yes—
233C22×C4#12114yes—
234C22×C4#12114yes—
235C22×C4#12312yes—
236C22×C4C23×C411yescharacteristic
237C22×C4C22×D412yes—
238C22×C4C22×D411yescharacteristic
239C22×C4C2×#4114yes—
240C22×C4C2×#4112yes—
241C22×C42−1+414yes—
242M16—44no—
243M16—44no—
244M16—48no—
245C2×C8—44no—
246C2×C8—48no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—216no—
252C4⋊C4—28no—
253C4⋊C4—216no—
254C4⋊C4—432no—
255C4⋊C4—432no—
256C4⋊C4—44no—
257C4⋊C4—48no—
258C4⋊C4—88no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—48no—
262C22⋊C4—48no—
263C42—216no—
264C42—24no—
265C42—28no—
266C42—48no—
267C42—88no—
268C42#11718yescomplemented
269C23—216no—
270C23—216no—
271C23—22no—
272C23—24no—
273C23—24no—
274C23—24no—
275C23—24no—
276C23—22no—
277C23—22no—
278C23—24no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23#38612yes—
285C23#38612yes—
286C23#45314yes—
287C23#45614yes—
288C23C22×C22⋊C411yescharacteristic
289C23C2×#12011yescharacteristic
290C23C2×#12311yescharacteristic
291C23#52512yes—
292C23#53612yes—
293C2×C4—28no—
294C2×C4—264no—
295C2×C4—232no—
296C2×C4—28no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—28no—
302C2×C4—28no—
303C2×C4—28no—
304C2×C4—28no—
305C2×C4—28no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—416no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—48no—
314C2×C4—48no—
315C2×C4—48no—
316C2×C4—48no—
317C2×C4—44no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—88no—
321C2×C4—84no—
322C2×C4—88no—
323C2×C4—88no—
324C2×C4—84no—
325C2×C4#38614yes—
326C2×C4C2×#11712yes—
327C2×C4C2×C4×D412yes—
328C2×C4#51912yes—
329C2×C4C2×#12112yes—
330C2×C4#52512yes—
331C2×C4#53612yes—
332C2×C4#54718yes—
333C2×C4#55218yes—
334C8—84no—
335C22—28no—
336C22—28no—
337C22—216no—
338C22—24no—
339C22—24no—
340C22—28no—
341C22—28no—
342C22—22no—
343C22—22no—
344C22—21no—
345C22—21no—
346C22—22no—
347C22—21no—
348C22—21no—
349C22—22no—
350C22#188614yes—
351C22#188714yes—
352C22#228011yescharacteristic
353C22C2×#38611yescharacteristic
354C22#231011yescharacteristic
355C22#240612yes—
356C22#248412yes—
357C22C2×#45312yes—
358C22C2×#45612yes—
359C4—216no—
360C4—24no—
361C4—44no—
362C4—44no—
363C4—88no—
364C4—84no—
365C4#234414yes—
366C4#241314yes—
367C2—21no—
368C2—21no—
369C2—24no—
370C2—24no—
371C2—22no—
372C2#1387011yescharacteristic
373C2C2×#188711yescharacteristic
374C2C2×#188611yescharacteristic
375C2#2407614yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11210C2semidirect
6. #10891C2semidirect
19. C2×#384C4semidirect
268. C42#117semidirect

Maximal subgroups

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

subgroupindex
#341782
C22×#3842
#207002
C2×#18212
#112102
#108912
#107192