Finite Group Databasebeta

← #79990

Subgroups — #79990

Order 384.

488 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 123 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×D4×Dic3C211yesmaximal, characteristic, complemented
2#5596C211yesmaximal, characteristic, complemented
3C3×#447C211yesmaximal, characteristic, complemented
4#5217C211yesmaximal, characteristic, complemented
5#5152C212yesmaximal, complemented
6#5150C212yesmaximal, complemented
7#5034C211yesmaximal, characteristic, complemented
8#4785C212yesmaximal, complemented
9#4761C212yesmaximal, complemented
10#4735C211yesmaximal, characteristic, complemented
11#4719C211yesmaximal, characteristic, complemented
12#3109—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×Dic3—21no—
15C2×#237—21no—
16C2×C3×D8C2212yes—
17C3×#121C2211yescharacteristic
18C2×#217—21no—
19#940C2211yescharacteristic, complemented
20#939—21no—
21D4×Dic3—21no—
22D4×Dic3—21no—
23D4×Dic3C2212yes—
24D4×Dic3C2212yes—
25D4×Dic3C2211yescharacteristic, complemented
26C2×#213C2212yes—
27#930C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×C4×Dic3C2211yescharacteristic
30#893C2211yescharacteristic, complemented
31#888C2211yescharacteristic, complemented
32#885—21no—
33#67⋊C4—21no—
34C3×#102C2212yes—
35C3×C22⋊C8C2211yescharacteristic
36#832C2212yes—
37#832C2212yescomplemented
38#830C2211yescharacteristic
39#821C2212yescomplemented
40#817C2212yescomplemented
41#815C2212yescomplemented
42Dic3⋊C8C2212yescomplemented
43#808C2212yescomplemented
44#532—31no—
45C2×C4×D4—31no—
46#481—31no—
47#481—31no—
48#476—32no—
49#459—32no—
50#447—31no—
51#447D311yescharacteristic, complemented
52#442—32no—
53#442—32no—
54#432—31no—
55C24×C3—21no—
56C2×C3×D4—24no—
57C2×C3×D4—21no—
58C2×C3×D4—21no—
59C2×C3×D4C2311yescharacteristic
60C2×C3×D4C2312yes—
61C22×C3×C4C2311yescharacteristic
62C2×#67—21no—
63C2×#67—21no—
64C2×#67—21no—
65C22×Dic3—21no—
66C22×Dic3—21no—
67C22×Dic3—22no—
68C22×Dic3—22no—
69C22×Dic3—44no—
70C22×Dic3D411yescharacteristic
71C22×Dic3D411yescharacteristic
72#237—44no—
73C2×D12C2311yescharacteristic
74C2×C4×D3—21no—
75C2×Dic6—21no—
76C3×D8—24no—
77C2×C3×C8C2312yes—
78C3×C4⋊C4C2311yescharacteristic
79C3×C22⋊C4—21no—
80#217—21no—
81#217—22no—
82#217—22no—
83#213—24no—
84(C2×D3)⋊C4—21no—
85(C2×D3)⋊C4—21no—
86C4⋊Dic3—21no—
87C4⋊Dic3C2312yes—
88C4⋊Dic3C2311yescharacteristic
89Dic3⋊C4—21no—
90C4×Dic3D411yescharacteristic
91C4×Dic3D411yescharacteristic
92C4×Dic3C2311yescharacteristic
93C4×Dic3C2311yescharacteristic
94C2×C3⋊C8C2312yes—
95C2×#41—61no—
96C22×D4C2×D311yescharacteristic
97C23×C4—61no—
98C2×D8—32no—
99C2×D8C2×D312yes—
100C4⋊Q8—31no—
101#127—31no—
102#124—61no—
103#121—31no—
104#121—61no—
105#121C2×D311yescharacteristic
106C4×D4—32no—
107C4×D4—32no—
108C4×D4—31no—
109C4×D4—31no—
110C4×D4—61no—
111C4×D4—61no—
112C4×D4—61no—
113C2×C4⋊C4—31no—
114C2×C22⋊C4—61no—
115C2×C42—31no—
116#106—32no—
117#106—32no—
118C4⋊C8—32no—
119#102—32no—
120#102—32no—
121#102—32no—
122#102C2×D312yes—
123C22⋊C8—31no—
124C22⋊C8C2×D311yescharacteristic
125#97—32no—
126C23×C3—24no—
127C23×C3—22no—
128C23×C3—21no—
129C23×C3—42no—
130C23×C3—42no—
131C23×C3C2×D411yescharacteristic
132C22×D3—21no—
133C3×D4—24no—
134C3×D4—22no—
135C3×D4—22no—
136C3×D4—42no—
137C3×D4#4114yescomplemented
138C2×C3×C4—22no—
139C2×C3×C4—21no—
140C2×C3×C4C2×D411yescharacteristic
141C2×C3×C4C2411yescharacteristic
142#67—42no—
143#67—42no—
144#67—42no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—21no—
148C2×Dic3—22no—
149C2×Dic3—42no—
150C2×Dic3—44no—
151C2×Dic3—42no—
152C2×Dic3—44no—
153C2×Dic3C2×D411yescharacteristic
154C2×Dic3C2×D411yescharacteristic
155C2×Dic3C2×D411yescharacteristic
156C2×Dic3C2×D411yescharacteristic
157D12—22no—
158C4×D3—42no—
159Dic6—42no—
160C3×C8—22no—
161C3⋊C8—22no—
162C24—21no—
163#41—124no—
164C2×Q8—61no—
165C2×D4—24no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—31no—
169C2×D4—61no—
170C2×D4—61no—
171C2×D4—61no—
172C2×D4C22×D311yescharacteristic
173C2×D4C22×D312yes—
174C22×C4—31no—
175C22×C4—31no—
176C22×C4—61no—
177C22×C4—62no—
178C22×C4—62no—
179C22×C4—61no—
180C22×C4—61no—
181C22×C4—124no—
182C22×C4C22×D311yescharacteristic
183D8—24no—
184D8—64no—
185C2×C8—32no—
186C2×C8C22×D312yes—
187C4⋊C4—32no—
188C4⋊C4—31no—
189C4⋊C4—61no—
190C4⋊C4—61no—
191C4⋊C4C22×D311yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—61no—
194C22⋊C4—62no—
195C22⋊C4—62no—
196C22⋊C4—61no—
197C22⋊C4—61no—
198C42—31no—
199C42—31no—
200C42—31no—
201C42—31no—
202C22×C3—21no—
203C22×C3—21no—
204C22×C3—24no—
205C22×C3—48no—
206C22×C3—42no—
207C22×C3—42no—
208C22×C3—41no—
209C22×C3—41no—
210C22×C3—41no—
211C22×C3AGL(1,ℤ/8)12yes—
212C22×C3C22×D411yescharacteristic
213C2×D3—41no—
214C2×D3—41no—
215C2×D3—41no—
216C3×C4—21no—
217C3×C4—41no—
218C3×C4C2×#4112yes—
219Dic3—21no—
220Dic3—21no—
221Dic3—21no—
222Dic3—41no—
223Dic3—42no—
224Dic3AGL(1,ℤ/8)12yes—
225C23—24no—
226C23—22no—
227C23—21no—
228C23—42no—
229C23—42no—
230C23—61no—
231C23D3×D411yescharacteristic
232Q8—122no—
233D4—24no—
234D4—22no—
235D4—22no—
236D4—42no—
237D4—62no—
238D4—122no—
239D4—122no—
240D4—122no—
241D4#23714yescomplemented
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—31no—
245C2×C4—31no—
246C2×C4—31no—
247C2×C4—31no—
248C2×C4—62no—
249C2×C4—62no—
250C2×C4—62no—
251C2×C4—61no—
252C2×C4—124no—
253C2×C4—122no—
254C2×C4—124no—
255C2×C4—122no—
256C2×C4—122no—
257C2×C4D3×D411yescharacteristic
258C2×C4C23×D311yescharacteristic
259C8—22no—
260C8—62no—
261C2×C3—22no—
262C2×C3—44no—
263C2×C3—81no—
264C2×C3#52511yescharacteristic
265C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
266C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
267D3—81no—
268C22—21no—
269C22—21no—
270C22—24no—
271C22—48no—
272C22—42no—
273C22—42no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—121no—
278C22—121no—
279C22—121no—
280C22#91112yes—
281C22C2×D3×D411yescharacteristic
282C4—21no—
283C4—32no—
284C4—41no—
285C4—61no—
286C4—61no—
287C4—61no—
288C4—121no—
289C4—122no—
290C4C2×#23712yes—
291C3#310911yescharacteristic, complemented, Sylow
292C2—22no—
293C2—44no—
294C2—81no—
295C2—241no—
296C2#558611yescharacteristic
297C2C2×#91111yescharacteristic
298C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#447C2
socleC22×C3C22×D4
hypercenter#447D3
hypocenterC3#3109
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic3C2semidirect
2. #5596C2semidirect
3. C3×#447C2semidirect
4. #5217C2semidirect
5. #5152C2semidirect
6. #5150C2semidirect
7. #5034C2semidirect
8. #4785C2semidirect
9. #4761C2semidirect
10. #4735C2semidirect
11. #4719C2semidirect
19. #940C22semidirect
25. D4×Dic3C22semidirect
30. #893C22semidirect
31. #888C22semidirect
37. #832C22semidirect
39. #821C22semidirect
40. #817C22semidirect
41. #815C22semidirect
42. Dic3⋊C8C22semidirect
43. #808C22semidirect
51. #447D3semidirect
137. C3×D4#41semidirect
241. D4#237semidirect
291. C3#3109semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic32
#55962
C3×#4472
#52172
#51522
#51502
#50342
#47852
#47612
#47352
#47192
#31093