Finite Group Databasebeta

← #232399

Subgroups — #232399

Order 512.

1726 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1726 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
2#35011C211yesmaximal, characteristic, complemented
3C22×#328C212yesmaximal, complemented
4#14152C211yesmaximal, characteristic
5#9973C214yesmaximal, complemented
6#9966C214yesmaximal, complemented
7#7907C211yesmaximal, characteristic, complemented
8C22×C22⋊Q8C2212yes—
9C22×#121C2211yescharacteristic
10#3187—28no—
11#3187—28no—
12C2×#481—22no—
13C2×#483C2214yes—
14C2×#465—24no—
15C2×#448—22no—
16C2×#450C2214yes—
17C22×C22⋊C8C2212yes—
18#2857—22no—
19#2659C2214yes—
20#2583—24no—
21#2502—22no—
22#2434—22no—
23#2315—22no—
24C2×#386—22no—
25C2×C4×C22⋊C4C2211yescharacteristic
26C22×#95C2211yescharacteristic
27C2×#328—28no—
28C2×#328C2218yes—
29#1445C2212yes—
30#1438C2212yes—
31#1437—22no—
32#1280C2214yes—
33C23×Q8D412yes—
34C23×D4—21no—
35C23×D4C2311yescharacteristic
36C24×C4—21no—
37C24×C4C2311yescharacteristic
38C22×SD16—24no—
39C23×C8—22no—
40C2×C4⋊Q8—22no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8—24no—
46C2×C22⋊Q8—24no—
47C2×C22⋊Q8D414yes—
48C2×C22⋊Q8C2314yes—
49C2×#121—22no—
50C2×#121—22no—
51C2×#121—24no—
52C2×#121—22no—
53C2×#121D414yes—
54C22×C4⋊C4—22no—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4—22no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—24no—
60C22×C22⋊C4D412yes—
61C22×C42D412yes—
62#483—28no—
63#483—28no—
64#481—216no—
65#465—232no—
66#450—28no—
67#450—28no—
68#448—216no—
69C2×#106—24no—
70C2×#103—24no—
71C2×#102—24no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8D418yes—
75C2×C22⋊C8C2314yes—
76#398—28no—
77#395—28no—
78#391—24no—
79#389—28no—
80#386—48no—
81#385—28no—
82#385—28no—
83#384—22no—
84#380—28no—
85C4×C22⋊C4—22no—
86C4×C22⋊C4—22no—
87C2×#95—24no—
88C2×#95—21no—
89C2×#95—24no—
90C2×#95C2312yes—
91C2×#95C2312yes—
92#328—216no—
93C25—21no—
94C25C2×D411yescharacteristic
95C22×Q8—22no—
96C22×Q8—24no—
97C22×Q8—24no—
98C22×Q8—84no—
99C22×Q8C2×D414yes—
100C22×D4—24no—
101C22×D4—22no—
102C22×D4—22no—
103C22×D4—21no—
104C22×D4—42no—
105C22×D4—42no—
106C22×D4—44no—
107C22×D4—44no—
108C22×D4—82no—
109C22×D4C2×D412yes—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—44no—
115C23×C4—42no—
116C23×C4—44no—
117C23×C4—82no—
118C23×C4—82no—
119C23×C4C2×D414yes—
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2×D414yes—
125C23×C4#4112yes—
126C23×C4C2411yescharacteristic
127C2×SD16—216no—
128C2×SD16—88no—
129C22×C8—24no—
130C22×C8—28no—
131C22×C8—84no—
132C4⋊Q8—84no—
133C22⋊Q8—28no—
134C22⋊Q8—28no—
135C22⋊Q8—24no—
136C22⋊Q8—24no—
137C22⋊Q8—24no—
138C22⋊Q8—24no—
139C22⋊Q8—44no—
140C22⋊Q8—44no—
141C22⋊Q8—44no—
142C22⋊Q8—44no—
143#121—28no—
144#121—28no—
145#121—44no—
146#121—44no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4C2×D414yes—
160C2×C4⋊C4C2×D412yes—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—44no—
167C2×C22⋊C4—44no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—48no—
172C2×C22⋊C4—48no—
173C2×C22⋊C4—84no—
174C2×C22⋊C4SD1618yescomplemented
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—82no—
178#106—216no—
179#103—216no—
180#102—216no—
181C22⋊C8—28no—
182C22⋊C8—24no—
183C22⋊C8C2×D418yes—
184#95—216no—
185#95—24no—
186#95—24no—
187#95—42no—
188#95—48no—
189C24—22no—
190C24—42no—
191C24—42no—
192C24—44no—
193C24—44no—
194C24—41no—
195C24—41no—
196C24—42no—
197C24—42no—
198C24#12012yes—
199C24#12112yes—
200C24#12712yes—
201C24C2×SD1614yes—
202C24#13714yes—
203C24C22×D411yescharacteristic
204C2×Q8—24no—
205C2×Q8—24no—
206C2×Q8—48no—
207C2×Q8—48no—
208C2×Q8—44no—
209C2×Q8—88no—
210C2×Q8—88no—
211C2×D4—24no—
212C2×D4—48no—
213C2×D4—48no—
214C2×D4—416no—
215C2×D4—42no—
216C2×D4—82no—
217C2×D4—84no—
218C2×D4—82no—
219C2×D4—88no—
220C2×D4—84no—
221C2×D4—84no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—48no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—82no—
246C22×C4—84no—
247C22×C4—82no—
248C22×C4—84no—
249C22×C4—84no—
250C22×C4—816no—
251C22×C4—84no—
252C22×C4—84no—
253C22×C4#12011yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12014yes—
256C22×C4#12014yes—
257C22×C4#12114yes—
258C22×C4#12114yes—
259C22×C4#12112yes—
260C22×C4#12712yes—
261C22×C4C2×SD1618yes—
262C22×C4C22×D412yes—
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D412yes—
265C22×C4C2×#4111yescharacteristic
266SD16—816no—
267C2×C8—28no—
268C2×C8—48no—
269C2×C8—88no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—44no—
274C4⋊C4—42no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—48no—
278C4⋊C4—88no—
279C4⋊C4—88no—
280C22⋊C4—216no—
281C22⋊C4—44no—
282C22⋊C4—44no—
283C22⋊C4—48no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—816no—
287C42—44no—
288C42—82no—
289C23—22no—
290C23—24no—
291C23—22no—
292C23—24no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—48no—
297C23—42no—
298C23—48no—
299C23—44no—
300C23—44no—
301C23—42no—
302C23—44no—
303C23—84no—
304C23—82no—
305C23—82no—
306C23—84no—
307C23—88no—
308C23—81no—
309C23—81no—
310C23—82no—
311C23—82no—
312C23—82no—
313C23#39214yes—
314C23#44818yes—
315C23#46118yes—
316C23#49214yes—
317C23#49714yes—
318C23C2×#12011yescharacteristic
319C23C2×#12111yescharacteristic
320C23C2×#12711yescharacteristic
321C23C22×SD1612yes—
322C23C2×#13712yes—
323Q8—48no—
324Q8—84no—
325Q8—88no—
326D4—48no—
327D4—84no—
328D4—816no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—28no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—48no—
345C2×C4—48no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—48no—
351C2×C4—48no—
352C2×C4—44no—
353C2×C4—84no—
354C2×C4—82no—
355C2×C4—84no—
356C2×C4—88no—
357C2×C4—88no—
358C2×C4—816no—
359C2×C4—82no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—82no—
363C2×C4#39212yes—
364C2×C4#39212yes—
365C2×C4C2×#12012yes—
366C2×C4C2×#12112yes—
367C8—48no—
368C22—21no—
369C22—24no—
370C22—21no—
371C22—21no—
372C22—24no—
373C22—21no—
374C22—24no—
375C22—24no—
376C22—216no—
377C22—44no—
378C22—48no—
379C22—44no—
380C22—88no—
381C22—82no—
382C22—82no—
383C22—82no—
384C22—162no—
385C22—162no—
386C22—164no—
387C22#201318yes—
388C22C2×#39211yescharacteristic
389C22C2×#44812yes—
390C22#301812yes—
391C22#302612yes—
392C22C2×#46112yes—
393C22C2×#49211yescharacteristic
394C22C2×#49711yescharacteristic
395C4—22no—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—48no—
400C4—84no—
401C4—84no—
402C4—82no—
403C4—84no—
404C2—21no—
405C2—22no—
406C2—21no—
407C2—28no—
408C2—162no—
409C2#1838214yes—
410C2#2340611yescharacteristic
411C2C2×#201311yescharacteristic
412C2#2344411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
2. #35011C2semidirect
3. C22×#328C2semidirect
5. #9973C2semidirect
6. #9966C2semidirect
7. #7907C2semidirect
174. 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
#350112
C22×#3282
#141522
#99732
#99662
#79072