Finite Group Databasebeta

← #232280

Subgroups — #232280

Order 512.

1898 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1898 conjugacy classes of subgroups — too many to draw. Only its 218 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×#3094C212yesmaximal, complemented
2#35001C211yesmaximal, characteristic
3C2×#1507C212yesmaximal, complemented
4#13853C211yesmaximal, characteristic, complemented
5#9970C214yesmaximal, complemented
6#9968C214yesmaximal, complemented
7#7911C211yesmaximal, characteristic, complemented
8C2×#534—22no—
9C22×#121C2212yes—
10C22×#117C2211yescharacteristic
11#3094—28no—
12#3094—28no—
13C2×#481C2214yescomplemented
14C2×#482—24no—
15C2×#480C2214yescomplemented
16C2×#428—22no—
17C2×#418—22no—
18C2×#407C2212yes—
19#2856—22no—
20#2656C2214yes—
21#2539—24no—
22#2501—22no—
23#2433—22no—
24C2×C4×C22⋊C4C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26#1507—28no—
27C2×#327C2218yescomplemented
28#1445C2212yescomplemented
29#1445C2212yescomplemented
30#1440—22no—
31C4×#95—22no—
32#1280C2214yescomplemented
33C23×D4—22no—
34C23×D4D412yes—
35C24×C4C2311yescharacteristic
36C22×M16—22no—
37#534—416no—
38C2×#127—22no—
39C2×#123—28no—
40C2×#121—22no—
41C2×#121—28no—
42C2×#121—22no—
43C2×#121—24no—
44C2×#121—24no—
45C2×#121—24no—
46C2×#121—24no—
47C2×#121D414yes—
48C2×#121C2314yes—
49C2×#120—28no—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—24no—
53C2×#117—22no—
54C2×#117D414yescomplemented
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C22⋊C4—21no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4D412yes—
62C22×C42—21no—
63C22×C42—21no—
64C22×C42D412yes—
65#482—232no—
66#481—28no—
67#481—28no—
68#480—28no—
69#480—28no—
70#428—216no—
71C2×#107—24no—
72C2×#106—24no—
73#418—216no—
74C2×#102—24no—
75C2×#102—24no—
76#407—24no—
77#407D418yescomplemented
78C2×C22⋊C8—22no—
79C2×C22⋊C8C2314yes—
80#399—28no—
81#392—28no—
82#390—24no—
83#390—28no—
84#384—22no—
85C4×C22⋊C4—22no—
86C4×C22⋊C4—22no—
87C4×C22⋊C4—24no—
88C2×#95—24no—
89C2×#95—24no—
90C2×#95C2312yes—
91C2×#95C2312yes—
92#327—216no—
93C25—22no—
94C25C2×D411yescharacteristic
95C22×D4—24no—
96C22×D4—24no—
97C22×D4—28no—
98C22×D4—24no—
99C22×D4—22no—
100C22×D4—24no—
101C22×D4—48no—
102C22×D4—416no—
103C22×D4—44no—
104C22×D4—44no—
105C22×D4C2×D414yes—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—44no—
111C23×C4C2×D412yes—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D414yes—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D414yes—
117C23×C4#4112yes—
118C23×C4C2411yescharacteristic
119C2×M16—28no—
120C2×M16—48no—
121C22×C8—24no—
122#127—84no—
123#123—432no—
124#121—28no—
125#121—28no—
126#121—24no—
127#121—24no—
128#121—24no—
129#121—24no—
130#121—416no—
131#121—44no—
132#121—44no—
133#121—44no—
134#121—44no—
135#121—416no—
136#120—432no—
137#117—28no—
138#117—28no—
139#117—44no—
140#117—44no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4C2×D414yes—
152C2×C4⋊C4C2×D412yes—
153C2×C4⋊C4C2×D412yes—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—28no—
156C2×C22⋊C4—216no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—44no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—44no—
165C2×C22⋊C4—44no—
166C2×C42—22no—
167C2×C42—24no—
168C2×C42—22no—
169C2×C42—41no—
170C2×C42—41no—
171C2×C42—42no—
172C2×C42—44no—
173C2×C42—44no—
174C2×C42—42no—
175#107—216no—
176#106—216no—
177#102—216no—
178#102—216no—
179C22⋊C8—28no—
180C22⋊C8—24no—
181C22⋊C8C2×D418yes—
182#95—24no—
183#95—24no—
184#95—48no—
185#95—48no—
186C24—24no—
187C24—24no—
188C24—42no—
189C24—42no—
190C24—48no—
191C24—44no—
192C24—84no—
193C24—84no—
194C24#12012yes—
195C24#12112yes—
196C24#12712yes—
197C24C22×D411yescharacteristic
198C2×D4—24no—
199C2×D4—24no—
200C2×D4—48no—
201C2×D4—416no—
202C2×D4—416no—
203C2×D4—416no—
204C2×D4—44no—
205C2×D4—44no—
206C2×D4—416no—
207C2×D4—48no—
208C2×D4—44no—
209C2×D4—48no—
210C2×D4—88no—
211C2×D4—88no—
212C2×D4—88no—
213C2×D4—816no—
214C2×D4—816no—
215C22×C4—24no—
216C22×C4—28no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—48no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—48no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4#12014yes—
243C22×C4#12011yescharacteristic
244C22×C4#12014yes—
245C22×C4#12011yescharacteristic
246C22×C4#12114yes—
247C22×C4#12112yes—
248C22×C4#12114yes—
249C22×C4#12712yes—
250C22×C4AGL(1,ℤ/8)18yes—
251C22×C4C22×D412yes—
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D412yes—
254C22×C4C2×#4111yescharacteristic
255M16—416no—
256C2×C8—28no—
257C2×C8—48no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—416no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C22⋊C4—216no—
267C22⋊C4—432no—
268C22⋊C4—48no—
269C22⋊C4—48no—
270C22⋊C4—44no—
271C22⋊C4—44no—
272C22⋊C4—48no—
273C42—44no—
274C42—44no—
275C42—44no—
276C42—82no—
277C42—84no—
278C23—22no—
279C23—22no—
280C23—24no—
281C23—24no—
282C23—24no—
283C23—24no—
284C23—24no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—44no—
290C23—48no—
291C23—48no—
292C23—48no—
293C23—84no—
294C23—84no—
295C23—88no—
296C23—88no—
297C23—88no—
298C23—1616no—
299C23#39214yes—
300C23C2×#12011yescharacteristic
301C23C2×#12111yescharacteristic
302C23C2×#12711yescharacteristic
303C23C2×AGL(1,ℤ/8)14yes—
304D4—48no—
305D4—84no—
306D4—816no—
307D4—88no—
308D4—88no—
309D4—88no—
310D4—1616no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—28no—
315C2×C4—24no—
316C2×C4—28no—
317C2×C4—28no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—48no—
326C2×C4—44no—
327C2×C4—48no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—44no—
332C2×C4—48no—
333C2×C4—44no—
334C2×C4—84no—
335C2×C4—82no—
336C2×C4—84no—
337C2×C4—82no—
338C2×C4—88no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4#39212yes—
344C2×C4#39212yes—
345C2×C4#45814yes—
346C2×C4C2×#12012yes—
347C2×C4C2×#12112yes—
348C8—48no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—24no—
354C22—24no—
355C22—24no—
356C22—24no—
357C22—24no—
358C22—48no—
359C22—84no—
360C22—84no—
361C22—84no—
362C22—84no—
363C22—84no—
364C22—816no—
365C22—1616no—
366C22C2×#39211yescharacteristic
367C22#300312yes—
368C22#301812yes—
369C22C2×#45812yes—
370C22#303812yes—
371C22#315112yes—
372C4—22no—
373C4—22no—
374C4—22no—
375C4—48no—
376C4—84no—
377C4—82no—
378C4—82no—
379C4—84no—
380C4#202214yes—
381C2—22no—
382C2—22no—
383C2—44no—
384C2—164no—
385C2#1817314yes—
386C2#2340611yescharacteristic
387C2#2343011yescharacteristic
388C2C2×#202211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radical#35001C2
LCSG ▷ C23×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
1. C2×#3094C2semidirect
3. C2×#1507C2semidirect
4. #13853C2semidirect
5. #9970C2semidirect
6. #9968C2semidirect
7. #7911C2semidirect
13. C2×#481C22semidirect
15. C2×#480C22semidirect
27. C2×#327C22semidirect
28. #1445C22semidirect
29. #1445C22semidirect
32. #1280C22semidirect
54. C2×#117D4semidirect
77. #407D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30942
#350012
C2×#15072
#138532
#99702
#99682
#79112