Finite Group Databasebeta

← #194740

Subgroups — #194740

Order 512.

1554 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2484C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic
3#20702C212yesmaximal
4C2×#1463C211yesmaximal, characteristic
5#9027C214yesmaximal, complemented
6#9023C214yesmaximal, complemented
7#8015C212yesmaximal
8C22×#117—21no—
9C22×#117C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C2×#407C2212yes—
12#2484—24no—
13#2484C418yescomplemented
14C2×#400C2212yes—
15C2×#398C2212yes—
16#2316—24no—
17#2316—22no—
18#2316C2214yes—
19C2×#389—24no—
20C2×#387—22no—
21C2×#385—21no—
22C2×#381—21no—
23C2×#381C2211yescharacteristic
24C2×#380—21no—
25C2×#380C2212yes—
26#1825—24no—
27#1824C2214yes—
28#1823C2214yes—
29#1736—24no—
30#1463—24no—
31C2×#324C2214yes—
32#1294C2218yes—
33C24×C4D411yescharacteristic
34C24×C4D411yescharacteristic
35C24×C4C2311yescharacteristic
36C22×M16—22no—
37C2×#117—24no—
38C2×#117—24no—
39C2×#117—22no—
40C2×#117—22no—
41C2×#117—22no—
42C2×#117—24no—
43C2×#117—48no—
44C2×#117C2×C414yes—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—24no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4C2×C414yes—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4D414yes—
54C22×C4⋊C4D414yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C42—22no—
62C22×C42—21no—
63#407—24no—
64#407—24no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8C2314yes—
67#400—24no—
68#400C2×C418yescomplemented
69#398—24no—
70#398C2×C418yescomplemented
71#389—216no—
72#389—216no—
73#387—216no—
74#385—28no—
75#381—28no—
76#381—22no—
77#381C2×C414yes—
78#380—24no—
79#380—22no—
80#380—22no—
81#380—24no—
82#380D414yes—
83#380C2314yes—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95C2312yes—
91#340—28no—
92#340—28no—
93#324—28no—
94C25C2×D411yescharacteristic
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4—24no—
99C23×C4—24no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—24no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—44no—
108C23×C4—44no—
109C23×C4C22⋊C414yes—
110C23×C4C22⋊C414yes—
111C23×C4C22×C412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4#4114yes—
119C23×C4C2411yescharacteristic
120C2×M16—48no—
121C2×M16—44no—
122C22×C8—24no—
123#117—28no—
124#117—28no—
125#117—416no—
126#117—48no—
127#117—44no—
128#117—44no—
129#117—48no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—48no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—48no—
152C2×C4⋊C4—84no—
153C2×C4⋊C4C22⋊C418yes—
154C2×C4⋊C4C22×C412yes—
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C22×C414yes—
157C2×C4⋊C4C2×D414yes—
158C2×C4⋊C4C2×D418yes—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—44no—
164C2×C22⋊C4—48no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—42no—
168C2×C42—28no—
169C2×C42—24no—
170C2×C42—48no—
171C2×C42—42no—
172C2×C42—42no—
173C22⋊C8—24no—
174C22⋊C8—24no—
175C22⋊C8—44no—
176#95—28no—
177#95—28no—
178#95—28no—
179#95—24no—
180#95—28no—
181#95—24no—
182#95—24no—
183#95—22no—
184#95C22×C414yes—
185C24—22no—
186C24—22no—
187C24—24no—
188C24—42no—
189C24C2×C22⋊C412yes—
190C24#12011yescharacteristic
191C24#12011yescharacteristic
192C24#12312yes—
193C24C22×D411yescharacteristic
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—216no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—28no—
206C22×C4—216no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—28no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—42no—
219C22×C4—48no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—48no—
223C22×C4—44no—
224C22×C4—48no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—48no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—48no—
238C22×C4—84no—
239C22×C4—82no—
240C22×C4C2×C22⋊C414yes—
241C22×C4C2×C22⋊C414yes—
242C22×C4C2×C22⋊C412yes—
243C22×C4C4×D414yes—
244C22×C4C4×D414yes—
245C22×C4#12012yes—
246C22×C4#12114yes—
247C22×C4#12114yes—
248C22×C4#12312yes—
249C22×C4AGL(1,ℤ/8)14yes—
250C22×C4#13714yes—
251C22×C4C23×C411yescharacteristic
252C22×C4C22×D412yes—
253C22×C4C22×D411yescharacteristic
254C22×C4C2×#4112yes—
255M16—88no—
256C2×C8—48no—
257C2×C8—44no—
258C4⋊C4—28no—
259C4⋊C4—28no—
260C4⋊C4—216no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—48no—
264C4⋊C4—416no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—416no—
268C4⋊C4—84no—
269C4⋊C4—88no—
270C22⋊C4—48no—
271C22⋊C4—48no—
272C22⋊C4—48no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C42—416no—
276C42—48no—
277C23—21no—
278C23—21no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—24no—
283C23—24no—
284C23—24no—
285C23—24no—
286C23—21no—
287C23—21no—
288C23—22no—
289C23—22no—
290C23—42no—
291C23—42no—
292C23—48no—
293C23—48no—
294C23#38612yes—
295C23#38612yes—
296C23#45814yes—
297C23C22×C22⋊C411yescharacteristic
298C23C2×#12011yescharacteristic
299C23C2×#12311yescharacteristic
300C23C2×AGL(1,ℤ/8)12yes—
301C23C2×#13712yes—
302C2×C4—216no—
303C2×C4—216no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—28no—
308C2×C4—28no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—48no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—48no—
331C2×C4—48no—
332C2×C4—48no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—82no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4—82no—
340C2×C4—816no—
341C2×C4—82no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—82no—
345C2×C4#38614yes—
346C2×C4#41718yes—
347C2×C4C2×C4×D412yes—
348C2×C4C2×#12112yes—
349C8—84no—
350C22—22no—
351C22—21no—
352C22—21no—
353C22—22no—
354C22—21no—
355C22—21no—
356C22—24no—
357C22—24no—
358C22—24no—
359C22—24no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—48no—
364C22—48no—
365C22—42no—
366C22—42no—
367C22#189014yes—
368C22C2×#38611yescharacteristic
369C22C2×#41712yes—
370C22#300311yescharacteristic
371C22#300411yescharacteristic
372C22#302012yes—
373C22C2×#45812yes—
374C22#309212yes—
375C4—28no—
376C4—22no—
377C4—22no—
378C4—24no—
379C4—44no—
380C4—84no—
381C4—82no—
382C4—82no—
383C4—88no—
384C2—22no—
385C2—22no—
386C2—24no—
387C2—42no—
388C2#1814712yes—
389C2#1814812yes—
390C2#2090911yescharacteristic
391C2#2095411yescharacteristic
392C2C2×#189011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9027C2semidirect
6. #9023C2semidirect
13. #2484C4semidirect
68. #400C2×C4semidirect
70. #398C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24842
C2×#23162
#207022
C2×#14632
#90272
#90232
#80152