Finite Group Databasebeta

← #232419

Subgroups — #232419

Order 512.

1866 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1866 conjugacy classes of subgroups — too many to draw. Only its 210 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×#3193C212yesmaximal, complemented
2#35001C211yesmaximal, characteristic
3C2×#1510C212yesmaximal, complemented
4#14292C211yesmaximal, characteristic, complemented
5#9970C214yesmaximal, complemented
6#9964C214yesmaximal, complemented
7#7911C211yesmaximal, characteristic, complemented
8C2×#534—22no—
9C22×C22⋊Q8C2211yescharacteristic
10C22×#121C2212yes—
11#3193—28no—
12#3193—28no—
13C2×#484—22no—
14C2×#480C2214yescomplemented
15C2×#468—24no—
16C2×#448C2214yescomplemented
17C2×#450—22no—
18C2×#407C2212yes—
19#2856—22no—
20#2656C2214yes—
21#2539—24no—
22#2501—22no—
23#2433—22no—
24C2×#381C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26#1510—28no—
27C2×#327C2218yescomplemented
28#1446C2212yescomplemented
29#1445C2212yescomplemented
30#1440—22no—
31#1436—22no—
32#1280C2214yescomplemented
33C23×Q8C2311yescharacteristic
34C23×D4—22no—
35C23×D4D412yes—
36C24×C4C2311yescharacteristic
37C22×SD16—24no—
38C22×M16—22no—
39#534—416no—
40C2×#127—22no—
41C2×#123—28no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8D414yescomplemented
47C2×#121—22no—
48C2×#121—28no—
49C2×#121—22no—
50C2×#121—24no—
51C2×#121—24no—
52C2×#121—24no—
53C2×#121—24no—
54C2×#121D414yes—
55C2×#121C2314yes—
56C2×#120—28no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D412yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4D412yes—
64C22×C42D412yes—
65#484—216no—
66#480—28no—
67#480—28no—
68#468—232no—
69#450—216no—
70#448—28no—
71#448—28no—
72C2×#107—24no—
73C2×#103—24no—
74C2×#102—24no—
75#407—24no—
76#407D418yescomplemented
77C2×C22⋊C8—22no—
78C2×C22⋊C8C2314yes—
79#399—28no—
80#392—28no—
81#390—24no—
82#390—28no—
83#384—22no—
84#381—24no—
85C4×C22⋊C4—22no—
86C4×C22⋊C4—22no—
87C2×#95—21no—
88C2×#95—24no—
89C2×#95—24no—
90C2×#95C2312yes—
91C2×#95C2312yes—
92#327—216no—
93C25—22no—
94C25C2×D411yescharacteristic
95C22×Q8—21no—
96C22×Q8—22no—
97C22×Q8—22no—
98C22×Q8—82no—
99C22×Q8C2×D412yes—
100C22×D4—24no—
101C22×D4—24no—
102C22×D4—28no—
103C22×D4—24no—
104C22×D4—22no—
105C22×D4—24no—
106C22×D4—416no—
107C22×D4—44no—
108C22×D4—44no—
109C22×D4—84no—
110C22×D4C2×D414yes—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—82no—
116C23×C4C2×D414yes—
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D412yes—
121C23×C4C2×D414yes—
122C23×C4#4112yes—
123C23×C4C2411yescharacteristic
124C2×SD16—216no—
125C2×SD16—88no—
126C2×M16—28no—
127C2×M16—84no—
128C22×C8—24no—
129#127—84no—
130#123—432no—
131C22⋊Q8—28no—
132C22⋊Q8—28no—
133C22⋊Q8—44no—
134C22⋊Q8—44no—
135#121—28no—
136#121—28no—
137#121—24no—
138#121—24no—
139#121—24no—
140#121—24no—
141#121—416no—
142#121—44no—
143#121—44no—
144#121—44no—
145#121—44no—
146#121—416no—
147#120—432no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4C2×D414yes—
160C2×C4⋊C4C2×D412yes—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—216no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—82no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—82no—
176#107—216no—
177#103—216no—
178#102—216no—
179C22⋊C8—28no—
180C22⋊C8—24no—
181C22⋊C8C2×D418yes—
182#95—24no—
183#95—24no—
184#95—42no—
185#95—48no—
186#95—84no—
187C24—24no—
188C24—24no—
189C24—42no—
190C24—42no—
191C24—48no—
192C24—44no—
193C24—84no—
194C24—84no—
195C24#12012yes—
196C24#12112yes—
197C24#12712yes—
198C24C22×D411yescharacteristic
199C2×Q8—24no—
200C2×Q8—48no—
201C2×Q8—42no—
202C2×Q8—82no—
203C2×Q8—82no—
204C2×Q8—84no—
205C2×D4—24no—
206C2×D4—24no—
207C2×D4—48no—
208C2×D4—48no—
209C2×D4—416no—
210C2×D4—416no—
211C2×D4—44no—
212C2×D4—44no—
213C2×D4—416no—
214C2×D4—48no—
215C2×D4—44no—
216C2×D4—88no—
217C2×D4—88no—
218C2×D4—88no—
219C2×D4—88no—
220C2×D4—816no—
221C2×D4—816no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—48no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—82no—
247C22×C4—84no—
248C22×C4—82no—
249C22×C4#12011yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12014yes—
252C22×C4#12014yes—
253C22×C4#12114yes—
254C22×C4#12114yes—
255C22×C4#12112yes—
256C22×C4#12712yes—
257C22×C4AGL(1,ℤ/8)18yes—
258C22×C4C22×D412yes—
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D412yes—
261C22×C4C2×#4111yescharacteristic
262SD16—816no—
263M16—88no—
264C2×C8—28no—
265C2×C8—48no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—416no—
270C4⋊C4—42no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C22⋊C4—216no—
275C22⋊C4—432no—
276C22⋊C4—48no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—48no—
280C22⋊C4—84no—
281C42—44no—
282C42—82no—
283C23—22no—
284C23—22no—
285C23—24no—
286C23—24no—
287C23—24no—
288C23—24no—
289C23—24no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—44no—
295C23—48no—
296C23—48no—
297C23—48no—
298C23—84no—
299C23—84no—
300C23—88no—
301C23—88no—
302C23—88no—
303C23—1616no—
304C23#39214yes—
305C23C2×#12011yescharacteristic
306C23C2×#12111yescharacteristic
307C23C2×#12711yescharacteristic
308C23C2×AGL(1,ℤ/8)14yes—
309Q8—48no—
310Q8—84no—
311D4—48no—
312D4—84no—
313D4—88no—
314D4—816no—
315D4—88no—
316D4—88no—
317D4—1616no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—28no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—48no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—48no—
339C2×C4—44no—
340C2×C4—84no—
341C2×C4—88no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—84no—
349C2×C4—84no—
350C2×C4—82no—
351C2×C4#39212yes—
352C2×C4#39212yes—
353C2×C4C2×#12012yes—
354C2×C4C2×#12112yes—
355C8—48no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—24no—
361C22—24no—
362C22—24no—
363C22—24no—
364C22—24no—
365C22—48no—
366C22—84no—
367C22—84no—
368C22—84no—
369C22—84no—
370C22—84no—
371C22—816no—
372C22—1616no—
373C22C2×#39211yescharacteristic
374C22#300312yes—
375C22#302212yes—
376C22#302612yes—
377C22#303812yes—
378C22#315112yes—
379C4—22no—
380C4—22no—
381C4—22no—
382C4—22no—
383C4—48no—
384C4—82no—
385C4—84no—
386C4—82no—
387C4—162no—
388C2—22no—
389C2—22no—
390C2—44no—
391C2—164no—
392C2#1835114yes—
393C2#2340811yescharacteristic
394C2#2343011yescharacteristic
395C2#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×#3193C2semidirect
3. C2×#1510C2semidirect
4. #14292C2semidirect
5. #9970C2semidirect
6. #9964C2semidirect
7. #7911C2semidirect
14. C2×#480C22semidirect
16. C2×#448C22semidirect
27. C2×#327C22semidirect
28. #1446C22semidirect
29. #1445C22semidirect
32. #1280C22semidirect
46. C2×C22⋊Q8D4semidirect
76. #407D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31932
#350012
C2×#15102
#142922
#99702
#99642
#79112