Finite Group Databasebeta

← #81339

Subgroups — #81339

Order 384.

472 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#5606C211yesmaximal, characteristic, complemented
2C2×#895C211yesmaximal, characteristic, complemented
3C3×#481C211yesmaximal, characteristic, complemented
4#5036C211yesmaximal, characteristic, complemented
5#4970C211yesmaximal, characteristic, complemented
6#4866C212yesmaximal, complemented
7#4864C212yesmaximal, complemented
8#4781C212yesmaximal, complemented
9#4775C212yesmaximal, complemented
10#4731C211yesmaximal, characteristic, complemented
11D3×C22⋊C8C211yesmaximal, characteristic, complemented
12#3188—31nomaximal, Sylow
13C2×#237C2211yescharacteristic, complemented
14C22×D12—21no—
15C22×D12C2211yescharacteristic
16C22×C4×D3C2211yescharacteristic
17C3×#121C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19#938C2211yescharacteristic, complemented
20#937—21no—
21D4×Dic3—21no—
22C2×#213C2212yescomplemented
23C2×(C2×D3)⋊C4—21no—
24#924C2211yescharacteristic
25C2×#204C2212yescomplemented
26C2×D3×C8C2212yescomplemented
27#897C2211yescharacteristic, complemented
28#895—21no—
29#895—21no—
30#895C2212yes—
31#895C2212yes—
32#892C2211yescharacteristic, complemented
33#883—21no—
34#67⋊C4—21no—
35C3×#106C2212yescomplemented
36C3×#102C2212yescomplemented
37C3×C22⋊C8C2211yescharacteristic
38#830C2211yescharacteristic
39#821C2212yescomplemented
40(C2×D3)⋊C8C2212yescomplemented
41#810C2212yescomplemented
42#809C2212yescomplemented
43#807C2212yescomplemented
44#525—31no—
45C2×#121—31no—
46#484—31no—
47#482—32no—
48#481D311yescharacteristic, complemented
49#466—32no—
50#465—32no—
51#450—31no—
52#449—32no—
53#447—31no—
54C2×C22⋊C8—31no—
55C23×D3—21no—
56C23×D3D411yescharacteristic
57C2×C3×D4—21no—
58C2×C3×D4C2311yescharacteristic
59C22×C3×C4—21no—
60C22×C3×C4C2311yescharacteristic
61C2×#67—21no—
62C2×#67—21no—
63C22×Dic3—21no—
64C22×Dic3D411yescharacteristic
65#237—42no—
66C2×D12—22no—
67C2×D12—22no—
68C2×D12—21no—
69C2×D12—42no—
70C2×D12—44no—
71C2×D12C2312yes—
72C2×C4×D3—42no—
73C2×C4×D3D411yescharacteristic
74C2×C4×D3D411yescharacteristic
75C2×C4×D3C2311yescharacteristic
76C2×C4×D3C2311yescharacteristic
77C2×Dic6C2311yescharacteristic
78C2×C3×C8C2312yes—
79C3×C4⋊C4—21no—
80C3×C4⋊C4C2312yes—
81C3×C4⋊C4C2311yescharacteristic
82C3×C22⋊C4—21no—
83#217—21no—
84#217—21no—
85#213—42no—
86(C2×D3)⋊C4—21no—
87(C2×D3)⋊C4—21no—
88(C2×D3)⋊C4—22no—
89(C2×D3)⋊C4—22no—
90C4⋊Dic3—21no—
91C4⋊Dic3C2311yescharacteristic
92Dic3⋊C4—21no—
93C4×Dic3—21no—
94C2×C3⋊C8C2312yes—
95#204—42no—
96D3×C8—42no—
97C2×#41—31no—
98C22×D4—31no—
99C22×D4—61no—
100C23×C4—31no—
101C2×SD16—32no—
102C2×D8—32no—
103C22×C8—32no—
104#123—61no—
105C22⋊Q8—31no—
106C22⋊Q8—31no—
107#121—31no—
108#121—32no—
109#121—32no—
110#121—61no—
111#121—61no—
112#121C2×D311yescharacteristic
113#120—61no—
114C4×D4—61no—
115C4×D4—61no—
116#117—31no—
117C2×C4⋊C4C2×D311yescharacteristic
118C2×C22⋊C4—61no—
119#107—32no—
120#106C2×D312yescomplemented
121#103—32no—
122#102—32no—
123#102—32no—
124#102C2×D312yescomplemented
125C22⋊C8—31no—
126C22⋊C8—32no—
127C22⋊C8C2×D311yescharacteristic
128C23×C3—21no—
129C23×C3C2×D411yescharacteristic
130C22×D3—22no—
131C22×D3—22no—
132C22×D3—22no—
133C22×D3—41no—
134C22×D3—42no—
135C22×D3—42no—
136C22×D3—42no—
137C22×D3C2×D411yescharacteristic
138C22×D3C2×D411yescharacteristic
139C3×D4—22no—
140C3×D4—42no—
141C2×C3×C4—22no—
142C2×C3×C4—21no—
143C2×C3×C4—22no—
144C2×C3×C4—42no—
145C2×C3×C4C2×D411yescharacteristic
146C2×C3×C4C2411yescharacteristic
147#67—42no—
148#67—42no—
149C2×Dic3—21no—
150C2×Dic3—21no—
151C2×Dic3—41no—
152C2×Dic3—42no—
153C2×Dic3C2×D411yescharacteristic
154C2×Dic3C2×D411yescharacteristic
155D12—22no—
156D12—22no—
157D12—42no—
158D12—44no—
159D12—44no—
160C4×D3—22no—
161C4×D3—22no—
162C4×D3—41no—
163C4×D3—41no—
164Dic6—22no—
165C3×C8—22no—
166C3⋊C8—22no—
167C24—31no—
168C24—61no—
169#41—122no—
170C2×Q8—31no—
171C2×D4—21no—
172C2×D4—32no—
173C2×D4—62no—
174C2×D4—62no—
175C2×D4—61no—
176C2×D4—61no—
177C2×D4—61no—
178C2×D4—122no—
179C2×D4—124no—
180C2×D4C22×D311yescharacteristic
181C22×C4—21no—
182C22×C4—31no—
183C22×C4—31no—
184C22×C4—31no—
185C22×C4—31no—
186C22×C4—31no—
187C22×C4—61no—
188C22×C4—122no—
189C22×C4C22×D311yescharacteristic
190SD16—122no—
191D8—122no—
192C2×C8—32no—
193C2×C8—122no—
194C2×C8C22×D312yes—
195C4⋊C4—21no—
196C4⋊C4—31no—
197C4⋊C4—61no—
198C4⋊C4—61no—
199C4⋊C4C22×D311yescharacteristic
200C4⋊C4C22×D312yes—
201C22⋊C4—21no—
202C22⋊C4—62no—
203C22⋊C4—62no—
204C22⋊C4—61no—
205C22⋊C4—61no—
206C22⋊C4—61no—
207C22⋊C4—61no—
208C42—61no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—41no—
212C22×C3—41no—
213C22×C3—41no—
214C22×C3#13512yes—
215C22×C3C22×D411yescharacteristic
216C2×D3—21no—
217C2×D3—21no—
218C2×D3—21no—
219C2×D3—21no—
220C2×D3—21no—
221C2×D3—44no—
222C2×D3—42no—
223C2×D3—42no—
224C2×D3—42no—
225C2×D3—84no—
226C2×D3AGL(1,ℤ/8)12yes—
227C3×C4—21no—
228C3×C4—41no—
229C3×C4—42no—
230C3×C4H(5,𝔽2)12yes—
231Dic3—21no—
232Dic3—21no—
233Dic3—41no—
234Dic3—41no—
235C23—21no—
236C23—31no—
237C23—31no—
238C23—62no—
239C23—62no—
240C23—62no—
241C23—121no—
242C23—122no—
243C23—122no—
244C23—122no—
245C23D3×D411yescharacteristic
246Q8—62no—
247D4—22no—
248D4—42no—
249D4—62no—
250D4—62no—
251D4—122no—
252D4—124no—
253D4—124no—
254D4—122no—
255D4—122no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—42no—
262C2×C4—62no—
263C2×C4—62no—
264C2×C4—61no—
265C2×C4—61no—
266C2×C4—121no—
267C2×C4—121no—
268C2×C4—121no—
269C2×C4—122no—
270C2×C4D3×D411yescharacteristic
271C2×C4C23×D311yescharacteristic
272C8—22no—
273C8—62no—
274C2×C3—22no—
275C2×C3—81no—
276C2×C3#53411yescharacteristic
277C2×C3C2×#13511yescharacteristic
278C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
279D3—22no—
280D3—41no—
281D3—82no—
282C22—21no—
283C22—21no—
284C22—32no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—61no—
289C22—61no—
290C22—61no—
291C22—61no—
292C22—61no—
293C22—124no—
294C22—122no—
295C22—122no—
296C22—122no—
297C22—244no—
298C22#91612yes—
299C22C2×D3×D411yescharacteristic
300C4—21no—
301C4—41no—
302C4—42no—
303C4—61no—
304C4—61no—
305C4—121no—
306C4—121no—
307C4#100912yes—
308C3#318811yescharacteristic, complemented, Sylow
309C2—22no—
310C2—62no—
311C2—81no—
312C2—121no—
313C2—242no—
314C2#565311yescharacteristic
315C2C2×#91611yescharacteristic
316C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#481C2
socleC22×C3C22×D4
hypercenter#481D3
hypocenterC3#3188
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #5606C2semidirect
2. C2×#895C2semidirect
3. C3×#481C2semidirect
4. #5036C2semidirect
5. #4970C2semidirect
6. #4866C2semidirect
7. #4864C2semidirect
8. #4781C2semidirect
9. #4775C2semidirect
10. #4731C2semidirect
11. D3×C22⋊C8C2semidirect
13. C2×#237C22semidirect
19. #938C22semidirect
22. C2×#213C22semidirect
25. C2×#204C22semidirect
26. C2×D3×C8C22semidirect
27. #897C22semidirect
32. #892C22semidirect
35. C3×#106C22semidirect
36. C3×#102C22semidirect
39. #821C22semidirect
40. (C2×D3)⋊C8C22semidirect
41. #810C22semidirect
42. #809C22semidirect
43. #807C22semidirect
48. #481D3semidirect
120. #106C2×D3semidirect
124. #102C2×D3semidirect
308. C3#3188semidirect

Maximal subgroups

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

subgroupindex
#56062
C2×#8952
C3×#4812
#50362
#49702
#48662
#48642
#47812
#47752
#47312
D3×C22⋊C82
#31883