Finite Group Databasebeta

← #232301

Subgroups — #232301

Order 512.

2000 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 2000 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×#480C212yesmaximal, complemented
2#35001C211yesmaximal, characteristic
3C2×#1519C212yesmaximal, complemented
4#13904C211yesmaximal, characteristic, complemented
5#9970C218yesmaximal, complemented
6#7907C211yesmaximal, characteristic, complemented
7C2×#534—22no—
8C22×#121C2212yes—
9C23×C4⋊C4C2211yescharacteristic
10C2×#480—28no—
11C2×#480—24no—
12C2×#480—28no—
13C2×#480C2218yescomplemented
14C22×#107—22no—
15C22×#102—22no—
16C22×C22⋊C8C2212yes—
17#2856—22no—
18#2656C2214yes—
19#2539—24no—
20#2501—22no—
21#2433—22no—
22C2×#381—22no—
23C2×#380C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25#1519—28no—
26C2×#327C2218yescomplemented
27#1445C2214yescomplemented
28C4⋊#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#480—232no—
63#480—216no—
64#480—216no—
65C2×#107—216no—
66C2×#107—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#381—48no—
79#380—24no—
80C4×C22⋊C4—22no—
81C4×C22⋊C4—22no—
82C2×#95—28no—
83C2×#95C2312yes—
84C2×#95C2312yes—
85#327—216no—
86C25—22no—
87C25C2×D411yescharacteristic
88C22×D4—24no—
89C22×D4—24no—
90C22×D4—28no—
91C22×D4—24no—
92C22×D4—22no—
93C22×D4—24no—
94C22×D4—48no—
95C22×D4—44no—
96C22×D4—44no—
97C22×D4—48no—
98C22×D4—48no—
99C22×D4C2×D414yes—
100C23×C4—22no—
101C23×C4—24no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—44no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4—44no—
108C23×C4—44no—
109C23×C4C2×D412yes—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D414yes—
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D414yes—
115C23×C4#4112yes—
116C23×C4C2411yescharacteristic
117C22×C8—28no—
118C22×C8—24no—
119C22×C8—48no—
120#127—84no—
121#123—432no—
122#121—28no—
123#121—28no—
124#121—24no—
125#121—24no—
126#121—24no—
127#121—24no—
128#121—416no—
129#121—44no—
130#121—44no—
131#121—44no—
132#121—44no—
133#121—416no—
134#120—432no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4C2×D414yes—
153C2×C4⋊C4C2×D414yes—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—216no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—44no—
160C2×C22⋊C4—44no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—44no—
163C2×C22⋊C4D818yes—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—44no—
167#107—232no—
168#102—232no—
169C22⋊C8—28no—
170C22⋊C8—24no—
171C22⋊C8C2×D418yes—
172#95—24no—
173#95—24no—
174#95—416no—
175C24—24no—
176C24—42no—
177C24—42no—
178C24—44no—
179C24—48no—
180C24—84no—
181C24—84no—
182C24#12012yes—
183C24#12112yes—
184C24#12712yes—
185C24C2×D814yes—
186C24AGL(1,ℤ/8)14yes—
187C24C22×D411yescharacteristic
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—48no—
191C2×D4—416no—
192C2×D4—416no—
193C2×D4—44no—
194C2×D4—44no—
195C2×D4—48no—
196C2×D4—48no—
197C2×D4—48no—
198C2×D4—416no—
199C2×D4—44no—
200C2×D4—48no—
201C2×D4—88no—
202C2×D4—88no—
203C2×D4—88no—
204C2×D4—816no—
205C2×D4—88no—
206C2×D4—88no—
207C22×C4—24no—
208C22×C4—28no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—48no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—42no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—48no—
226C22×C4—48no—
227C22×C4—416no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—88no—
233C22×C4—84no—
234C22×C4—84no—
235C22×C4#12014yes—
236C22×C4#12011yescharacteristic
237C22×C4#12014yes—
238C22×C4#12011yescharacteristic
239C22×C4#12114yes—
240C22×C4#12112yes—
241C22×C4#12114yes—
242C22×C4#12712yes—
243C22×C4C2×D818yes—
244C22×C4C22×D412yes—
245C22×C4C22×D411yescharacteristic
246C22×C4C22×D412yes—
247C22×C4C2×#4111yescharacteristic
248C2×C8—28no—
249C2×C8—48no—
250C2×C8—416no—
251C4⋊C4—28no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—416no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—84no—
260C22⋊C4—216no—
261C22⋊C4—432no—
262C22⋊C4—44no—
263C22⋊C4—44no—
264C22⋊C4—48no—
265C22⋊C4—84no—
266C42—44no—
267C42—82no—
268C23—22no—
269C23—24no—
270C23—22no—
271C23—24no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—48no—
277C23—48no—
278C23—44no—
279C23—48no—
280C23—48no—
281C23—48no—
282C23—88no—
283C23—88no—
284C23—88no—
285C23—1616no—
286C23#39214yes—
287C23#44718yes—
288C23#45918yes—
289C23#49314yes—
290C23#49614yes—
291C23C2×#12011yescharacteristic
292C23C2×#12111yescharacteristic
293C23C2×#12711yescharacteristic
294C23C22×D812yes—
295C23C2×AGL(1,ℤ/8)12yes—
296D4—48no—
297D4—84no—
298D4—84no—
299D4—84no—
300D4—88no—
301D4—816no—
302D4—88no—
303D4—1616no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—28no—
308C2×C4—24no—
309C2×C4—28no—
310C2×C4—28no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—48no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—48no—
322C2×C4—48no—
323C2×C4—84no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—84no—
327C2×C4—816no—
328C2×C4—88no—
329C2×C4—82no—
330C2×C4—88no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4#39212yes—
334C2×C4#39212yes—
335C2×C4#45814yes—
336C2×C4C2×#12012yes—
337C2×C4C2×#12112yes—
338C8—48no—
339C22—24no—
340C22—24no—
341C22—216no—
342C22—21no—
343C22—24no—
344C22—21no—
345C22—21no—
346C22—24no—
347C22—21no—
348C22—48no—
349C22—44no—
350C22—44no—
351C22—816no—
352C22—84no—
353C22—84no—
354C22—84no—
355C22—1616no—
356C22#201118yes—
357C22C2×#39211yescharacteristic
358C22C2×#44712yes—
359C22#302212yes—
360C22C2×#45812yes—
361C22C2×#45912yes—
362C22C2×#49311yescharacteristic
363C22C2×#49611yescharacteristic
364C4—22no—
365C4—22no—
366C4—22no—
367C4—48no—
368C4—84no—
369C4—84no—
370C4—162no—
371C4#202214yes—
372C2—21no—
373C2—28no—
374C2—21no—
375C2—22no—
376C2—164no—
377C2#1820214yes—
378C2#2340811yescharacteristic
379C2C2×#201111yescharacteristic
380C2C2×#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×#480C2semidirect
3. C2×#1519C2semidirect
4. #13904C2semidirect
5. #9970C2semidirect
6. #7907C2semidirect
13. C2×#480C22semidirect
26. C2×#327C22semidirect
27. #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×#4802
#350012
C2×#15192
#139042
#99702
#79072