Finite Group Databasebeta

← #232299

Subgroups — #232299

Order 512.

1972 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1972 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
1C22×#481C212yesmaximal, complemented
2#35001C211yesmaximal, characteristic
3C2×#1519C212yesmaximal, complemented
4#13924C211yesmaximal, characteristic, complemented
5#9968C218yesmaximal, complemented
6#7907C211yesmaximal, characteristic, complemented
7C2×#534—22no—
8C22×#121C2212yes—
9C23×C4⋊C4C2211yescharacteristic
10C2×#481—28no—
11C2×#481—24no—
12C2×#481—28no—
13C2×#481C2218yescomplemented
14C22×#106—22no—
15C22×#102—22no—
16C22×C22⋊C8C2212yes—
17#2856—22no—
18#2656C2214yes—
19#2539—24no—
20#2501—22no—
21#2433—22no—
22C2×#380C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24#1519—28no—
25C2×#327C2218yescomplemented
26#1445C2214yescomplemented
27C4⋊#95—22no—
28C22⋊#95—22no—
29#1280C2214yescomplemented
30C23×D4—22no—
31C23×D4D412yes—
32C24×C4—21no—
33C24×C4C2311yescharacteristic
34C23×C8—22no—
35#534—48no—
36#534—48no—
37C2×#127—22no—
38C2×#123—28no—
39C2×#121—22no—
40C2×#121—28no—
41C2×#121—22no—
42C2×#121—24no—
43C2×#121—24no—
44C2×#121—24no—
45C2×#121—24no—
46C2×#121D414yes—
47C2×#121C2314yes—
48C2×#120—28no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4D414yescomplemented
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4D412yes—
61C22×C42D412yes—
62#481—232no—
63#481—216no—
64#481—216no—
65C2×#106—216no—
66C2×#106—28no—
67C2×#102—216no—
68C2×#102—28no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—24no—
71C2×C22⋊C8D418yescomplemented
72C2×C22⋊C8C2314yes—
73#399—28no—
74#392—28no—
75#390—24no—
76#390—28no—
77#384—22no—
78#380—24no—
79C4×C22⋊C4—22no—
80C4×C22⋊C4—22no—
81C2×#95—28no—
82C2×#95C2312yes—
83C2×#95C2312yes—
84#327—216no—
85C25—22no—
86C25C2×D411yescharacteristic
87C22×D4—24no—
88C22×D4—24no—
89C22×D4—28no—
90C22×D4—24no—
91C22×D4—22no—
92C22×D4—24no—
93C22×D4—48no—
94C22×D4—44no—
95C22×D4—44no—
96C22×D4—48no—
97C22×D4—48no—
98C22×D4C2×D414yes—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—24no—
102C23×C4—22no—
103C23×C4—44no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—84no—
107C23×C4C2×D412yes—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D414yes—
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D414yes—
113C23×C4#4112yes—
114C23×C4C2411yescharacteristic
115C22×C8—28no—
116C22×C8—24no—
117C22×C8—48no—
118#127—84no—
119#123—432no—
120#121—28no—
121#121—28no—
122#121—24no—
123#121—24no—
124#121—24no—
125#121—24no—
126#121—416no—
127#121—44no—
128#121—44no—
129#121—44no—
130#121—44no—
131#121—416no—
132#120—432no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4C2×D414yes—
151C2×C4⋊C4C2×D414yes—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—216no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—44no—
158C2×C22⋊C4—44no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—44no—
161C2×C22⋊C4SD1618yes—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—44no—
165#106—232no—
166#102—232no—
167C22⋊C8—28no—
168C22⋊C8—24no—
169C22⋊C8C2×D418yes—
170#95—24no—
171#95—24no—
172#95—88no—
173C24—24no—
174C24—42no—
175C24—42no—
176C24—44no—
177C24—48no—
178C24—84no—
179C24—84no—
180C24#12012yes—
181C24#12112yes—
182C24#12712yes—
183C24C2×SD1614yes—
184C24AGL(1,ℤ/8)14yes—
185C24C22×D411yescharacteristic
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—48no—
189C2×D4—416no—
190C2×D4—416no—
191C2×D4—44no—
192C2×D4—44no—
193C2×D4—48no—
194C2×D4—48no—
195C2×D4—48no—
196C2×D4—416no—
197C2×D4—44no—
198C2×D4—48no—
199C2×D4—88no—
200C2×D4—88no—
201C2×D4—88no—
202C2×D4—816no—
203C2×D4—88no—
204C2×D4—88no—
205C22×C4—24no—
206C22×C4—28no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—48no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—48no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—48no—
223C22×C4—48no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—88no—
229C22×C4—88no—
230C22×C4—88no—
231C22×C4#12014yes—
232C22×C4#12011yescharacteristic
233C22×C4#12014yes—
234C22×C4#12011yescharacteristic
235C22×C4#12114yes—
236C22×C4#12112yes—
237C22×C4#12114yes—
238C22×C4#12712yes—
239C22×C4C2×SD1618yes—
240C22×C4C22×D412yes—
241C22×C4C22×D411yescharacteristic
242C22×C4C22×D412yes—
243C22×C4C2×#4111yescharacteristic
244C2×C8—28no—
245C2×C8—48no—
246C2×C8—416no—
247C4⋊C4—28no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—416no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—84no—
256C22⋊C4—216no—
257C22⋊C4—432no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—48no—
261C22⋊C4—84no—
262C42—44no—
263C42—82no—
264C23—22no—
265C23—24no—
266C23—22no—
267C23—24no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—48no—
273C23—48no—
274C23—44no—
275C23—48no—
276C23—48no—
277C23—48no—
278C23—88no—
279C23—88no—
280C23—88no—
281C23—1616no—
282C23#39214yes—
283C23#45018yes—
284C23#46018yes—
285C23#49214yes—
286C23#49614yes—
287C23C2×#12011yescharacteristic
288C23C2×#12111yescharacteristic
289C23C2×#12711yescharacteristic
290C23C22×SD1612yes—
291C23C2×AGL(1,ℤ/8)12yes—
292D4—48no—
293D4—84no—
294D4—84no—
295D4—84no—
296D4—88no—
297D4—816no—
298D4—88no—
299D4—1616no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—28no—
304C2×C4—24no—
305C2×C4—28no—
306C2×C4—28no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—48no—
315C2×C4—48no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—48no—
319C2×C4—84no—
320C2×C4—84no—
321C2×C4—816no—
322C2×C4—84no—
323C2×C4—88no—
324C2×C4—82no—
325C2×C4—88no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4#39212yes—
329C2×C4#39212yes—
330C2×C4#45814yes—
331C2×C4C2×#12012yes—
332C2×C4C2×#12112yes—
333C8—48no—
334C22—24no—
335C22—24no—
336C22—216no—
337C22—21no—
338C22—21no—
339C22—24no—
340C22—21no—
341C22—24no—
342C22—21no—
343C22—48no—
344C22—44no—
345C22—44no—
346C22—816no—
347C22—84no—
348C22—84no—
349C22—84no—
350C22—1616no—
351C22#201418yes—
352C22C2×#39211yescharacteristic
353C22C2×#45012yes—
354C22#302112yes—
355C22C2×#45812yes—
356C22C2×#46012yes—
357C22C2×#49211yescharacteristic
358C22C2×#49611yescharacteristic
359C4—22no—
360C4—22no—
361C4—22no—
362C4—48no—
363C4—84no—
364C4—84no—
365C4—162no—
366C4#202214yes—
367C2—21no—
368C2—28no—
369C2—21no—
370C2—22no—
371C2—164no—
372C2#1820314yes—
373C2#2340711yescharacteristic
374C2C2×#201411yescharacteristic
375C2C2×#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. C22×#481C2semidirect
3. C2×#1519C2semidirect
4. #13924C2semidirect
5. #9968C2semidirect
6. #7907C2semidirect
13. C2×#481C22semidirect
25. C2×#327C22semidirect
26. #1445C22semidirect
29. #1280C22semidirect
56. C22×C4⋊C4D4semidirect
71. C2×C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4812
#350012
C2×#15192
#139242
#99682
#79072