Finite Group Databasebeta

← #11863931

Subgroups — #11863931

Order 1152.

1491 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1491 conjugacy classes of subgroups — too many to draw. Only its 202 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
1C23×D36C211yesmaximal, characteristic, complemented
2#10589893C211yesmaximal, characteristic, complemented
3C2×#63470C212yesmaximal, complemented
4C9×#407C211yesmaximal, characteristic, complemented
5#10588813C212yesmaximal, complemented
6#10588414C214yesmaximal, complemented
7#10588413C214yesmaximal, complemented
8#78637—31nomaximal
9C24×D9—21no—
10C23×C4×C9C2211yescharacteristic
11C2×#3697C2211yescharacteristic, complemented
12C22×D36—22no—
13C22×D36—21no—
14C22×D36—28no—
15C22×D36C2214yes—
16C22×D36C2212yes—
17C2×C9×M16C2212yescomplemented
18#63515—21no—
19#63492C2212yescomplemented
20#63491—21no—
21C4×#611—22no—
22#63488C2211yescharacteristic, complemented
23#63486C2212yescomplemented
24#63470—28no—
25C2×D72C2214yescomplemented
26C2×#3664C2214yescomplemented
27C9×C22⋊C8C2214yescomplemented
28#63380C2218yescomplemented
29C23×D12—31no—
30#5791—31no—
31C2×#908—32no—
32C3×#407D311yescharacteristic
33#5131—32no—
34#4732—34no—
35#4731—34no—
36C24×C9D411yescharacteristic
37C23×D9—28no—
38C23×D9—24no—
39C23×D9—22no—
40C23×D9—48no—
41C22×C4×C9—24no—
42C22×C4×C9D412yes—
43C22×C4×C9D411yescharacteristic
44C22×C4×C9C2312yes—
45C22×C4×C9C2311yescharacteristic
46C2×#611—22no—
47C2×#611—21no—
48#3697—24no—
49C2×D36—28no—
50C2×D36—216no—
51C2×D36—24no—
52C2×D36—22no—
53C2×D36—24no—
54C2×D36—216no—
55C2×D36D418yescomplemented
56C2×D36C2314yes—
57C2×D36C2311yescharacteristic, complemented
58C2×C4×D9—21no—
59C2×Dic18C2311yescharacteristic, complemented
60C9×M16—24no—
61C2×C8×C9C2314yescomplemented
62#3676—21no—
63#3676—22no—
64(C2×D9)⋊C4—22no—
65C4⋊Dic9C2312yescomplemented
66Dic9⋊C4—22no—
67C4×Dic9—21no—
68D72—28no—
69#3664—28no—
70#3003—91noSylow
71C24×D3—61no—
72C23×C3×C4C2×D311yescharacteristic
73C2×#235—31no—
74C22×D12—32no—
75C22×D12—34no—
76C22×D12—68no—
77C22×D12—62no—
78C22×D12—61no—
79C2×C3×M16C2×D312yes—
80#953—61no—
81#930—32no—
82#929—61no—
83C4×#67—62no—
84#926—31no—
85#924—32no—
86#908—68no—
87C2×D24—34no—
88C2×#204—34no—
89C3×C22⋊C8C2×D314yes—
90#821—38no—
91C23×C9—22no—
92C23×C9—24no—
93C23×C9C2×D411yescharacteristic
94C23×C9C2×D412yes—
95C22×D9—216no—
96C22×D9—28no—
97C22×D9—24no—
98C22×D9—21no—
99C22×D9—432no—
100C22×D9—48no—
101C22×D9—48no—
102C22×D9—48no—
103C2×C4×C9—22no—
104C2×C4×C9—24no—
105C2×C4×C9—24no—
106C2×C4×C9C2×D412yes—
107C2×C4×C9C2×D411yescharacteristic
108C2×C4×C9C2×D414yes—
109C2×C4×C9C2411yescharacteristic
110#611—42no—
111#611—44no—
112C2×Dic9—22no—
113C2×Dic9—21no—
114D36—216no—
115D36—28no—
116D36—24no—
117D36—22no—
118D36C2×D418yes—
119C4×D9—42no—
120Dic18—22no—
121C8×C9—24no—
122C23×D4—91no—
123C2×AGL(1,ℤ/8)—92no—
124#525—91no—
125#450—94no—
126#447—94no—
127#418—92no—
128#407D911yescharacteristic, complemented
129C24×C3D1211yescharacteristic
130C23×D3—68no—
131C23×D3—64no—
132C23×D3—62no—
133C23×D3—128no—
134C22×C3×C4—24no—
135C22×C3×C4D1212yes—
136C22×C3×C4D1211yescharacteristic
137C22×C3×C4C22×D312yes—
138C22×C3×C4C22×D311yescharacteristic
139C2×#67—62no—
140C2×#67—61no—
141#235—64no—
142C2×D12—38no—
143C2×D12—34no—
144C2×D12—31no—
145C2×D12—616no—
146C2×D12—68no—
147C2×D12—616no—
148C2×D12—64no—
149C2×D12—62no—
150C2×D12—64no—
151C2×C4×D3—61no—
152C2×Dic6—31no—
153C3×M16—24no—
154C2×C3×C8C22×D314yes—
155#217—62no—
156#217—61no—
157(C2×D3)⋊C4—62no—
158C4⋊Dic3—32no—
159Dic3⋊C4—62no—
160C4×Dic3—61no—
161D24—68no—
162#204—68no—
163C22×C9—28no—
164C22×C9—22no—
165C22×C9—22no—
166C22×C9—22no—
167C22×C9#12012yes—
168C22×C9AGL(1,ℤ/8)14yescomplemented
169C22×C9C22×D411yescharacteristic
170C2×D9—28no—
171C2×D9—416no—
172C2×D9—432no—
173C2×D9—48no—
174C2×D9—41no—
175C2×D9—42no—
176C4×C9—22no—
177C4×C9#12012yes—
178C4×C9C22×D412yes—
179Dic9—41no—
180Dic9—42no—
181C25—181no—
182C2×#41—91no—
183C22×D4—92no—
184C22×D4—94no—
185C22×D4—188no—
186C22×D4—182no—
187C22×D4—181no—
188C23×C4C2×D911yescharacteristic
189AGL(1,ℤ/8)—188no—
190C2×SD16—94no—
191C2×D8—94no—
192C2×M16C2×D912yescomplemented
193#123—181no—
194C22⋊Q8—92no—
195#121—92no—
196#120—181no—
197C4×D4—182no—
198#117—91no—
199#102—98no—
200C22⋊C8C2×D914yescomplemented
201C23×C3—22no—
202C23×C3—24no—
203C23×C3C2×D1211yescharacteristic
204C23×C3C2×D1212yes—
205C22×D3—616no—
206C22×D3—68no—
207C22×D3—64no—
208C22×D3—61no—
209C22×D3—1232no—
210C22×D3—128no—
211C22×D3—128no—
212C22×D3—128no—
213C2×C3×C4—22no—
214C2×C3×C4—24no—
215C2×C3×C4—24no—
216C2×C3×C4C2×D1212yes—
217C2×C3×C4C2×D1211yescharacteristic
218C2×C3×C4D3×D414yes—
219C2×C3×C4C23×D311yescharacteristic
220#67—122no—
221#67—124no—
222C2×Dic3—62no—
223C2×Dic3—61no—
224D12—38no—
225D12—616no—
226D12—68no—
227D12—64no—
228D12—62no—
229C4×D3—122no—
230Dic6—62no—
231C3×C8—24no—
232C2×C9—22no—
233C2×C9—24no—
234C2×C9C2×#12011yescharacteristic
235C2×C9C2×AGL(1,ℤ/8)12yes—
236D9—48no—
237D9—81no—
238C24—188no—
239C24—184no—
240C24—182no—
241C24—368no—
242C24D3611yescharacteristic
243#41—184no—
244C2×Q8—91no—
245C2×D4—98no—
246C2×D4—94no—
247C2×D4—91no—
248C2×D4—1816no—
249C2×D4—188no—
250C2×D4—1816no—
251C2×D4—182no—
252C2×D4—181no—
253C2×D4—184no—
254C2×D4—182no—
255C2×D4—184no—
256C22×C4—24no—
257C22×C4—181no—
258C22×C4D3612yes—
259C22×C4D3611yescharacteristic
260C22×C4C22×D912yes—
261C22×C4C22×D911yescharacteristic
262SD16—188no—
263D8—188no—
264M16—24no—
265C2×C8C22×D914yescomplemented
266C4⋊C4—92no—
267C4⋊C4—182no—
268C22⋊C4—181no—
269C22⋊C4—182no—
270C22⋊C4—182no—
271C42—181no—
272C22×C3—28no—
273C22×C3—22no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3#88212yes—
277C22×C3#90814yes—
278C22×C3C22×D1211yescharacteristic
279C2×D3—68no—
280C2×D3—1216no—
281C2×D3—1232no—
282C2×D3—128no—
283C2×D3—121no—
284C2×D3—122no—
285C3×C4—22no—
286C3×C4#88212yes—
287C3×C4C2×D3×D412yes—
288Dic3—121no—
289Dic3—122no—
290C9#300311yescharacteristic, complemented, Sylow
291C23—22no—
292C23—24no—
293C23—1816no—
294C23—188no—
295C23—184no—
296C23—181no—
297C23—3632no—
298C23—368no—
299C23—368no—
300C23—368no—
301C23C2×D3611yescharacteristic
302C23C2×D3612yes—
303Q8—182no—
304D4—98no—
305D4—1816no—
306D4—188no—
307D4—184no—
308D4—182no—
309D4—362no—
310D4—364no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—182no—
315C2×C4—181no—
316C2×C4—362no—
317C2×C4C2×D3612yes—
318C2×C4C2×D3611yescharacteristic
319C2×C4D4×D914yes—
320C2×C4C23×D911yescharacteristic
321C8—24no—
322C2×C3—22no—
323C2×C3—24no—
324C2×C3C2×#88211yescharacteristic
325C2×C3C2×#90812yes—
326D3—128no—
327D3—241no—
328C22—28no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—188no—
333C22—3616no—
334C22—3632no—
335C22—368no—
336C22—361no—
337C22—362no—
338C22#6344412yes—
339C22#6347014yescomplemented
340C22C22×D3611yescharacteristic
341C4—22no—
342C4—361no—
343C4—362no—
344C4#6344412yes—
345C4C2×D4×D912yes—
346C3#7863711yescharacteristic
347C2—22no—
348C2—24no—
349C2—368no—
350C2—721no—
351C2C2×#6344411yescharacteristic
352C2C2×#6347012yes—
subgroupG/H
centerC22C22×D36
commutator, cosocleC2×C4×C9C24
FrattiniC2×C3×C4C23×D3
FittingC9×#407C2
socleC22×C3C22×D12
hypercenter#407D9
hypocenterC9#3003
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C23×D36C2semidirect
2. #10589893C2semidirect
3. C2×#63470C2semidirect
4. C9×#407C2semidirect
5. #10588813C2semidirect
6. #10588414C2semidirect
7. #10588413C2semidirect
11. C2×#3697C22semidirect
17. C2×C9×M16C22semidirect
19. #63492C22semidirect
22. #63488C22semidirect
23. #63486C22semidirect
25. C2×D72C22semidirect
26. C2×#3664C22semidirect
27. C9×C22⋊C8C22semidirect
28. #63380C22semidirect
55. C2×D36D4semidirect
57. C2×D36C23semidirect
59. C2×Dic18C23semidirect
61. C2×C8×C9C23semidirect
65. C4⋊Dic9C23semidirect
128. #407D9semidirect
168. C22×C9AGL(1,ℤ/8)semidirect
192. C2×M16C2×D9semidirect
200. C22⋊C8C2×D9semidirect
265. C2×C8C22×D9semidirect
290. C9#3003semidirect
339. C22#63470semidirect

Maximal subgroups

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

subgroupindex
C23×D362
#105898932
C2×#634702
C9×#4072
#105888132
#105884142
#105884132
#786373