Finite Group Databasebeta

← #232397

Subgroups — #232397

Order 512.

1980 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1980 conjugacy classes of subgroups — too many to draw. Only its 258 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×#3187C212yesmaximal, complemented
2#35001C211yesmaximal, characteristic
3C22×#327C212yesmaximal, complemented
4#14159C211yesmaximal, characteristic, complemented
5#9968C214yesmaximal, complemented
6#9964C214yesmaximal, complemented
7#7907C211yesmaximal, characteristic, complemented
8C2×#534—22no—
9C22×C22⋊Q8C2211yescharacteristic
10C22×#121C2212yes—
11#3187—28no—
12#3187—28no—
13C2×#481C2214yescomplemented
14C2×#483—22no—
15C2×#465—24no—
16C2×#448C2214yescomplemented
17C2×#450—22no—
18C22×C22⋊C8C2212yes—
19#2856—22no—
20#2656C2214yes—
21#2539—24no—
22#2501—22no—
23#2433—22no—
24C2×C4×C22⋊C4C2211yescharacteristic
25C22×#95C2211yescharacteristic
26C2×#327—28no—
27C2×#327C2218yescomplemented
28#1446C2212yescomplemented
29#1445C2212yescomplemented
30#1436—22no—
31C22⋊#95—22no—
32#1280C2214yescomplemented
33C23×Q8C2311yescharacteristic
34C23×D4—22no—
35C23×D4D412yes—
36C24×C4—21no—
37C24×C4C2311yescharacteristic
38C22×SD16—24no—
39C23×C8—22no—
40#534—48no—
41#534—48no—
42C2×#127—22no—
43C2×#123—28no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—24no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8D414yescomplemented
49C2×#121—22no—
50C2×#121—28no—
51C2×#121—22no—
52C2×#121—24no—
53C2×#121—24no—
54C2×#121—24no—
55C2×#121—24no—
56C2×#121D414yes—
57C2×#121C2314yes—
58C2×#120—28no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4D412yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C22⋊C4—21no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4D412yes—
65C22×C42D412yes—
66#483—216no—
67#481—28no—
68#481—28no—
69#465—232no—
70#450—216no—
71#448—28no—
72#448—28no—
73C2×#106—24no—
74C2×#103—24no—
75C2×#102—24no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8D418yescomplemented
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—
87C2×#95—24no—
88C2×#95—24no—
89C2×#95—21no—
90C2×#95—24no—
91C2×#95C2312yes—
92C2×#95C2312yes—
93#327—216no—
94C25—22no—
95C25C2×D411yescharacteristic
96C22×Q8—21no—
97C22×Q8—22no—
98C22×Q8—22no—
99C22×Q8—82no—
100C22×Q8C2×D412yes—
101C22×D4—24no—
102C22×D4—24no—
103C22×D4—28no—
104C22×D4—24no—
105C22×D4—22no—
106C22×D4—24no—
107C22×D4—44no—
108C22×D4—44no—
109C22×D4—48no—
110C22×D4—48no—
111C22×D4—84no—
112C22×D4C2×D414yes—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—44no—
118C23×C4—42no—
119C23×C4—44no—
120C23×C4—82no—
121C23×C4—82no—
122C23×C4C2×D414yes—
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2×D414yes—
128C23×C4#4112yes—
129C23×C4C2411yescharacteristic
130C2×SD16—216no—
131C2×SD16—88no—
132C22×C8—24no—
133C22×C8—28no—
134C22×C8—84no—
135#127—84no—
136#123—432no—
137C22⋊Q8—28no—
138C22⋊Q8—28no—
139C22⋊Q8—44no—
140C22⋊Q8—44no—
141#121—28no—
142#121—28no—
143#121—24no—
144#121—24no—
145#121—24no—
146#121—24no—
147#121—416no—
148#121—44no—
149#121—44no—
150#121—44no—
151#121—44no—
152#121—416no—
153#120—432no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4C2×D414yes—
166C2×C4⋊C4C2×D412yes—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—216no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4SD1618yescomplemented
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—82no—
181#106—216no—
182#103—216no—
183#102—216no—
184C22⋊C8—28no—
185C22⋊C8—24no—
186C22⋊C8C2×D418yes—
187#95—24no—
188#95—24no—
189#95—48no—
190#95—42no—
191#95—84no—
192C24—24no—
193C24—42no—
194C24—42no—
195C24—44no—
196C24—48no—
197C24—84no—
198C24—84no—
199C24#12012yes—
200C24#12112yes—
201C24#12712yes—
202C24C2×SD1614yes—
203C24AGL(1,ℤ/8)14yes—
204C24C22×D411yescharacteristic
205C2×Q8—24no—
206C2×Q8—48no—
207C2×Q8—42no—
208C2×Q8—82no—
209C2×Q8—82no—
210C2×Q8—84no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—48no—
214C2×D4—48no—
215C2×D4—416no—
216C2×D4—44no—
217C2×D4—44no—
218C2×D4—48no—
219C2×D4—48no—
220C2×D4—48no—
221C2×D4—416no—
222C2×D4—44no—
223C2×D4—88no—
224C2×D4—88no—
225C2×D4—88no—
226C2×D4—88no—
227C2×D4—816no—
228C2×D4—88no—
229C2×D4—88no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—42no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—48no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—48no—
255C22×C4—82no—
256C22×C4—84no—
257C22×C4—82no—
258C22×C4—816no—
259C22×C4—84no—
260C22×C4—84no—
261C22×C4#12011yescharacteristic
262C22×C4#12011yescharacteristic
263C22×C4#12014yes—
264C22×C4#12014yes—
265C22×C4#12114yes—
266C22×C4#12114yes—
267C22×C4#12112yes—
268C22×C4#12712yes—
269C22×C4C2×SD1618yes—
270C22×C4C22×D412yes—
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D412yes—
273C22×C4C2×#4111yescharacteristic
274SD16—816no—
275C2×C8—28no—
276C2×C8—48no—
277C2×C8—88no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—416no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C22⋊C4—216no—
287C22⋊C4—432no—
288C22⋊C4—48no—
289C22⋊C4—44no—
290C22⋊C4—44no—
291C22⋊C4—48no—
292C42—44no—
293C42—82no—
294C23—22no—
295C23—24no—
296C23—22no—
297C23—24no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—48no—
302C23—42no—
303C23—48no—
304C23—44no—
305C23—48no—
306C23—48no—
307C23—48no—
308C23—88no—
309C23—88no—
310C23—88no—
311C23—1616no—
312C23#39214yes—
313C23#45018yes—
314C23#46018yes—
315C23#49214yes—
316C23#49614yes—
317C23C2×#12011yescharacteristic
318C23C2×#12111yescharacteristic
319C23C2×#12711yescharacteristic
320C23C22×SD1612yes—
321C23C2×AGL(1,ℤ/8)12yes—
322Q8—48no—
323Q8—84no—
324D4—48no—
325D4—84no—
326D4—88no—
327D4—84no—
328D4—84no—
329D4—88no—
330D4—816no—
331D4—1616no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—28no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—48no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—48no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—48no—
356C2×C4—84no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—84no—
360C2×C4—82no—
361C2×C4—816no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—88no—
366C2×C4#39212yes—
367C2×C4#39212yes—
368C2×C4C2×#12012yes—
369C2×C4C2×#12112yes—
370C8—48no—
371C22—21no—
372C22—21no—
373C22—24no—
374C22—21no—
375C22—21no—
376C22—24no—
377C22—24no—
378C22—24no—
379C22—216no—
380C22—44no—
381C22—48no—
382C22—44no—
383C22—84no—
384C22—84no—
385C22—84no—
386C22—816no—
387C22—1616no—
388C22#201418yes—
389C22C2×#39211yescharacteristic
390C22C2×#45012yes—
391C22#301812yes—
392C22#302612yes—
393C22C2×#46012yes—
394C22C2×#49211yescharacteristic
395C22C2×#49611yescharacteristic
396C4—22no—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—48no—
401C4—84no—
402C4—82no—
403C4—84no—
404C4—82no—
405C2—21no—
406C2—28no—
407C2—21no—
408C2—22no—
409C2—164no—
410C2#1838114yes—
411C2#2340611yescharacteristic
412C2C2×#201411yescharacteristic
413C2#2344411yescharacteristic
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×#3187C2semidirect
3. C22×#327C2semidirect
4. #14159C2semidirect
5. #9968C2semidirect
6. #9964C2semidirect
7. #7907C2semidirect
13. C2×#481C22semidirect
16. C2×#448C22semidirect
27. C2×#327C22semidirect
28. #1446C22semidirect
29. #1445C22semidirect
32. #1280C22semidirect
48. C2×C22⋊Q8D4semidirect
78. C2×C22⋊C8D4semidirect
177. C2×C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#31872
#350012
C22×#3272
#141592
#99682
#99642
#79072