Finite Group Databasebeta

← #194790

Subgroups — #194790

Order 512.

1644 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2398C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3#20740C212yesmaximal, complemented
4C2×#1466C211yesmaximal, characteristic
5#9025C214yesmaximal, complemented
6#9013C214yesmaximal, complemented
7#8022C212yesmaximal, complemented
8C2×#518—21no—
9C22×C4×Q8C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11C2×#432C2212yes—
12#2398—24no—
13#2398C418yescomplemented
14C2×#400—21no—
15C2×#397C2212yes—
16C2×#394—21no—
17C2×#393C2212yes—
18#2349—24no—
19D4×C4⋊C4—24no—
20#2316—21no—
21#2315C2214yes—
22#2285C2212yes—
23C2×#380—22no—
24C2×C4×C22⋊C4—22no—
25C2×#376C2211yescharacteristic
26#1823C2214yes—
27#1803—24no—
28#1787C2214yes—
29#1763—24no—
30#1466—24no—
31C2×#323C2214yes—
32#1294C2218yes—
33C23×Q8C2311yescharacteristic
34C23×D4D411yescharacteristic
35C24×C4D412yes—
36#518—48no—
37C2×C4×Q8—22no—
38C2×C4×Q8—22no—
39C2×C4×Q8—22no—
40C2×C4×Q8—24no—
41C2×C4×Q8C2×C414yescomplemented
42C2×C4×D4—28no—
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—24no—
46C2×C4×D4C2×C414yes—
47C2×C4×D4D414yes—
48C2×#117—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C22⋊C4—24no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4C2312yes—
57C22×C42—21no—
58C22×C42—21no—
59C22×C42C2311yescharacteristic
60#432—24no—
61#432—24no—
62C2×C4⋊C8—22no—
63C2×#103—28no—
64C2×C22⋊C8C2314yes—
65C2×#97—22no—
66#400—28no—
67#397—24no—
68#397C2×C418yescomplemented
69#394—28no—
70#393—24no—
71#393C2×C418yescomplemented
72#389—28no—
73#385—216no—
74#384D414yes—
75#382—22no—
76#380—24no—
77#380—24no—
78#380—24no—
79#380—22no—
80#380C2314yes—
81C4×C4⋊C4—22no—
82C4×C4⋊C4D414yes—
83C4×C22⋊C4—216no—
84#376—22no—
85#376C2×C414yescomplemented
86C2×#95—24no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95C2312yes—
91#340—28no—
92#323—28no—
93C25C2×D412yes—
94C22×Q8—21no—
95C22×Q8—22no—
96C22×Q8—22no—
97C22×Q8—44no—
98C22×Q8C22×C412yes—
99C22×D4—24no—
100C22×D4—24no—
101C22×D4—22no—
102C22×D4—22no—
103C22×D4C22⋊C414yes—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—24no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—28no—
111C23×C4—48no—
112C23×C4C22⋊C418yes—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D412yes—
116C23×C4C2411yescharacteristic
117C22×C8—24no—
118C4×Q8—28no—
119C4×Q8—28no—
120C4×Q8—44no—
121C4×Q8—44no—
122C4×D4—28no—
123C4×D4—28no—
124C4×D4—28no—
125C4×D4—28no—
126C4×D4—432no—
127#117—48no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—88no—
140C2×C4⋊C4C22⋊C414yes—
141C2×C4⋊C4C22×C412yes—
142C2×C4⋊C4C2×D418yes—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—28no—
146C2×C22⋊C4—28no—
147C2×C22⋊C4—28no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—48no—
150C2×C22⋊C4—48no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4—48no—
153C2×C22⋊C4—44no—
154C2×C22⋊C4C22×C414yes—
155C2×C22⋊C4C2×D414yes—
156C2×C42—24no—
157C2×C42—24no—
158C2×C42—24no—
159C2×C42—44no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42—42no—
163C2×C42C22×C412yes—
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C2×C42C2×D412yes—
167C2×C42#4114yes—
168C4⋊C8—44no—
169#103—416no—
170C22⋊C8—24no—
171C22⋊C8—24no—
172#97—44no—
173#95—28no—
174#95—24no—
175#95—28no—
176#95—28no—
177#95—24no—
178#95—22no—
179#95—24no—
180#95C22×C414yes—
181C24—22no—
182C24—22no—
183C24—22no—
184C24—22no—
185C24—44no—
186C24—44no—
187C24C2×C22⋊C414yes—
188C24AGL(1,ℤ/8)14yes—
189C24#13714yes—
190C24C22×D411yescharacteristic
191C2×Q8—28no—
192C2×Q8—24no—
193C2×Q8—44no—
194C2×Q8—44no—
195C2×Q8—44no—
196C2×Q8—48no—
197C2×Q8—42no—
198C2×D4—28no—
199C2×D4—216no—
200C2×D4—28no—
201C2×D4—216no—
202C2×D4—28no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—28no—
211C22×C4—28no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—28no—
215C22×C4—42no—
216C22×C4—416no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—48no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—416no—
231C22×C4—88no—
232C22×C4—84no—
233C22×C4—88no—
234C22×C4—88no—
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C414yes—
238C22×C4#12011yescharacteristic
239C22×C4#12012yes—
240C22×C4#12011yescharacteristic
241C22×C4#12312yes—
242C22×C4#12312yes—
243C22×C4C23×C411yescharacteristic
244C22×C4C22×D412yes—
245C22×C4C22×D411yescharacteristic
246C22×C4C2×#4112yes—
247C2×C8—48no—
248C2×C8—44no—
249C4⋊C4—28no—
250C4⋊C4—24no—
251C4⋊C4—48no—
252C4⋊C4—48no—
253C4⋊C4—42no—
254C4⋊C4—48no—
255C4⋊C4—48no—
256C4⋊C4—88no—
257C22⋊C4—28no—
258C22⋊C4—28no—
259C22⋊C4—416no—
260C22⋊C4—416no—
261C22⋊C4—48no—
262C22⋊C4—416no—
263C42—48no—
264C42—48no—
265C42—44no—
266C42—44no—
267C42C4×D414yes—
268C42C4×D414yes—
269C42#12114yes—
270C42#12114yes—
271C23—22no—
272C23—22no—
273C23—24no—
274C23—24no—
275C23—24no—
276C23—24no—
277C23—24no—
278C23—24no—
279C23—44no—
280C23—44no—
281C23—48no—
282C23—48no—
283C23—44no—
284C23—44no—
285C23—82no—
286C23—84no—
287C23—84no—
288C23—82no—
289C23#41718yes—
290C23C22×C22⋊C411yescharacteristic
291C23C2×#12011yescharacteristic
292C23C2×#12311yescharacteristic
293C23C2×AGL(1,ℤ/8)12yes—
294C23C2×#13712yes—
295Q8—28no—
296Q8—48no—
297Q8—44no—
298D4—232no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—416no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—48no—
324C2×C4—48no—
325C2×C4—48no—
326C2×C4—42no—
327C2×C4—88no—
328C2×C4—82no—
329C2×C4—84no—
330C2×C4—84no—
331C2×C4—82no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—1616no—
335C2×C4#38612yes—
336C2×C4#38614yes—
337C2×C4#38612yes—
338C2×C4#45514yes—
339C2×C4C2×C4×D412yes—
340C2×C4C2×#12112yes—
341C8—84no—
342C22—24no—
343C22—24no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—28no—
349C22—48no—
350C22—48no—
351C22—48no—
352C22—44no—
353C22—44no—
354C22—84no—
355C22—84no—
356C22—84no—
357C22C2×#38611yescharacteristic
358C22C2×#41712yes—
359C22#300311yescharacteristic
360C22#300411yescharacteristic
361C22C2×#45512yes—
362C22#302512yes—
363C22#309212yes—
364C4—24no—
365C4—22no—
366C4—22no—
367C4—22no—
368C4—44no—
369C4—44no—
370C4—84no—
371C4—88no—
372C4—82no—
373C4#188514yes—
374C2—22no—
375C2—22no—
376C2—48no—
377C2#1817912yes—
378C2#1818112yes—
379C2#2090911yescharacteristic
380C2C2×#188511yescharacteristic
381C2#2095911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20740C2semidirect
5. #9025C2semidirect
6. #9013C2semidirect
7. #8022C2semidirect
13. #2398C4semidirect
41. C2×C4×Q8C2×C4semidirect
68. #397C2×C4semidirect
71. #393C2×C4semidirect
85. #376C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23982
#341812
#207402
C2×#14662
#90252
#90132
#80222