Finite Group Databasebeta

← #194857

Subgroups — #194857

Order 512.

1540 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1540 conjugacy classes of subgroups — too many to draw. Only its 266 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2591C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic, complemented
3#21764C212yesmaximal
4C2×#1463C211yesmaximal, characteristic, complemented
5#9022C214yesmaximal, complemented
6#9018C214yesmaximal, complemented
7#8019C212yesmaximal, complemented
8C22×#121C2211yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10C2×#407C2212yes—
11#2591—24no—
12#2591C418yescomplemented
13C2×#400C2212yes—
14C2×#394C2212yes—
15#2313—24no—
16#2313—22no—
17#2313C2214yes—
18C2×#388—22no—
19C2×#387—24no—
20C2×#386—21no—
21C2×#386—21no—
22C2×#381—21no—
23C2×#380C2212yes—
24C2×C4×C22⋊C4—21no—
25C2×C4×C22⋊C4C2211yescharacteristic
26#1895—24no—
27#1874—24no—
28#1824C2214yes—
29#1786C2214yes—
30#1463—24no—
31C2×#324C2214yes—
32#1294C2218yes—
33C23×D4—21no—
34C23×D4C2311yescharacteristic
35C24×C4D411yescharacteristic
36C24×C4D411yescharacteristic
37C24×C4C2311yescharacteristic
38C22×M16—22no—
39C2×#121—22no—
40C2×#121—22no—
41C2×#121—22no—
42C2×#121—24no—
43C2×#121C2×C414yes—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—24no—
47C22×C4⋊C4C2×C414yes—
48C22×C4⋊C4D412yes—
49C22×C4⋊C4D414yes—
50C22×C4⋊C4D414yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C42—22no—
59C22×C42—21no—
60C2×#102—28no—
61#407—24no—
62#407—24no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8C2314yes—
65#400—24no—
66#400C2×C418yescomplemented
67#394—24no—
68#394C2×C418yescomplemented
69#388—216no—
70#387—216no—
71#387—216no—
72#386—28no—
73#386—28no—
74#381—24no—
75#381—24no—
76#380—22no—
77#380—22no—
78#380D414yes—
79#380C2314yes—
80C4×C22⋊C4—24no—
81C4×C22⋊C4—24no—
82C4×C22⋊C4—22no—
83C4×C22⋊C4C2×C414yes—
84C2×#95—22no—
85C2×#95—24no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95C2312yes—
90#340—28no—
91#324—28no—
92C25—21no—
93C25C2×D411yescharacteristic
94C22×D4—24no—
95C22×D4—22no—
96C22×D4—21no—
97C22×D4—22no—
98C22×D4—48no—
99C22×D4—41no—
100C22×D4—41no—
101C22×D4—42no—
102C22×D4—82no—
103C22×D4C22×C412yes—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—24no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—44no—
114C23×C4—82no—
115C23×C4—82no—
116C23×C4C22⋊C414yes—
117C23×C4C22⋊C414yes—
118C23×C4C22×C412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4#4114yes—
126C23×C4C2411yescharacteristic
127C2×M16—44no—
128C2×M16—84no—
129C22×C8—24no—
130#121—28no—
131#121—28no—
132#121—44no—
133#121—44no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—84no—
147C2×C4⋊C4C22⋊C418yes—
148C2×C4⋊C4C22×C412yes—
149C2×C4⋊C4C22×C414yes—
150C2×C4⋊C4C2×D414yes—
151C2×C4⋊C4C2×D418yes—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—28no—
157C2×C22⋊C4—28no—
158C2×C22⋊C4—48no—
159C2×C22⋊C4—44no—
160C2×C22⋊C4—48no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—84no—
164C2×C42—28no—
165C2×C42—24no—
166C2×C42—82no—
167C2×C42—84no—
168#102—88no—
169C22⋊C8—24no—
170C22⋊C8—24no—
171C22⋊C8—44no—
172#95—28no—
173#95—216no—
174#95—28no—
175#95—28no—
176#95—24no—
177#95—24no—
178#95—22no—
179#95C22×C414yes—
180C24—24no—
181C24—24no—
182C24—22no—
183C24—42no—
184C24—44no—
185C24—44no—
186C24—41no—
187C24—41no—
188C24—44no—
189C24C2×C22⋊C412yes—
190C24#12011yescharacteristic
191C24#12011yescharacteristic
192C24#12312yes—
193C24C22×D411yescharacteristic
194C2×D4—24no—
195C2×D4—48no—
196C2×D4—416no—
197C2×D4—44no—
198C2×D4—44no—
199C2×D4—42no—
200C2×D4—82no—
201C2×D4—84no—
202C2×D4—82no—
203C2×D4—88no—
204C2×D4—84no—
205C2×D4—84no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—28no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—48no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—48no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—48no—
231C22×C4—44no—
232C22×C4—48no—
233C22×C4—44no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—44no—
237C22×C4—48no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4—84no—
241C22×C4—84no—
242C22×C4—82no—
243C22×C4—84no—
244C22×C4C2×C22⋊C414yes—
245C22×C4C2×C22⋊C414yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C4×D414yes—
248C22×C4C4×D414yes—
249C22×C4#12012yes—
250C22×C4#12114yes—
251C22×C4#12114yes—
252C22×C4#12312yes—
253C22×C4C23×C411yescharacteristic
254C22×C4C22×D412yes—
255C22×C4C22×D411yescharacteristic
256C22×C4C2×#4112yes—
257M16—88no—
258C2×C8—48no—
259C2×C8—44no—
260C4⋊C4—28no—
261C4⋊C4—28no—
262C4⋊C4—216no—
263C4⋊C4—24no—
264C4⋊C4—48no—
265C4⋊C4—42no—
266C4⋊C4—84no—
267C22⋊C4—416no—
268C22⋊C4—48no—
269C22⋊C4—48no—
270C22⋊C4—48no—
271C22⋊C4—48no—
272C22⋊C4—88no—
273C42—84no—
274C42—88no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—24no—
280C23—24no—
281C23—24no—
282C23—24no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—42no—
287C23—42no—
288C23—416no—
289C23—44no—
290C23—44no—
291C23—44no—
292C23—42no—
293C23—82no—
294C23—82no—
295C23—816no—
296C23—82no—
297C23—82no—
298C23—82no—
299C23—82no—
300C23#38612yes—
301C23#38612yes—
302C23#45414yes—
303C23C22×C22⋊C411yescharacteristic
304C23C2×#12011yescharacteristic
305C23C2×#12311yescharacteristic
306C23#57514yes—
307D4—48no—
308D4—84no—
309D4—88no—
310D4—88no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—48no—
327C2×C4—48no—
328C2×C4—48no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—48no—
338C2×C4—48no—
339C2×C4—48no—
340C2×C4—42no—
341C2×C4—82no—
342C2×C4—84no—
343C2×C4—88no—
344C2×C4—816no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4#38614yes—
348C2×C4C2×C4×D412yes—
349C2×C4C2×#12112yes—
350C8—84no—
351C22—28no—
352C22—28no—
353C22—22no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—21no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—48no—
363C22—48no—
364C22—44no—
365C22—88no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22—164no—
370C22—164no—
371C22#188814yes—
372C22C2×#38611yescharacteristic
373C22#289612yes—
374C22#300512yes—
375C22C2×#45412yes—
376C22#302612yes—
377C22#309312yes—
378C4—22no—
379C4—22no—
380C4—24no—
381C4—44no—
382C4—44no—
383C4—84no—
384C4—88no—
385C4—82no—
386C2—22no—
387C2—22no—
388C2—24no—
389C2—42no—
390C2—162no—
391C2#1832014yes—
392C2#2091011yescharacteristic
393C2C2×#188811yescharacteristic
394C2#2096011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
2. C2×#2313C2semidirect
4. C2×#1463C2semidirect
5. #9022C2semidirect
6. #9018C2semidirect
7. #8019C2semidirect
12. #2591C4semidirect
66. #400C2×C4semidirect
68. #394C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25912
C2×#23132
#217642
C2×#14632
#90222
#90182
#80192