Finite Group Databasebeta

← #80339

Subgroups — #80339

Order 384.

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

normal subgroups only

This group has 490 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
1D3×#127C211yesmaximal, characteristic, complemented
2#5557C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic, complemented
4#5156C212yesmaximal, complemented
5#5154C212yesmaximal, complemented
6#5075C211yesmaximal, characteristic, complemented
7#5015C211yesmaximal, characteristic, complemented
8#4848C211yesmaximal, characteristic, complemented
9#4835C211yesmaximal, characteristic, complemented
10#4776C212yesmaximal, complemented
11#4764C212yesmaximal, complemented
12#3217—31nomaximal, Sylow
13C2×D3×D4—21no—
14C2×D3×D4—21no—
15C2×D3×D4C2212yes—
16C2×C3×D8C2212yescomplemented
17C3×#127C2211yescharacteristic
18C3×C4×D4C2211yescharacteristic, complemented
19#940—21no—
20#940—21no—
21#940C2212yes—
22#938C2211yescharacteristic
23#936C2211yescharacteristic, complemented
24C2×#214C2212yescomplemented
25C2×#213C2212yescomplemented
26C4×#67—21no—
27#926—21no—
28C2×#204C2212yescomplemented
29#897C2211yescharacteristic
30#890C2211yescharacteristic, complemented
31#883—21no—
32#879—21no—
33#874C2211yescharacteristic
34C42×D3C2211yescharacteristic
35C4×Dic6C2211yescharacteristic, complemented
36C3×C4⋊C8C2211yescharacteristic, complemented
37C3×#102C2212yescomplemented
38#832C2212yescomplemented
39#821C2212yescomplemented
40(C2×D3)⋊C8C2212yes—
41#815C2212yescomplemented
42#809C2212yescomplemented
43C4⋊(C3⋊C8)C2211yescharacteristic, complemented
44C2×#127—31no—
45#529—31no—
46#496—32no—
47#490—32no—
48#460—31no—
49#460—31no—
50#459—31no—
51#459D311yescharacteristic, complemented
52#450—32no—
53#447—32no—
54#432—31no—
55C23×D3—22no—
56C2×C3×D4—21no—
57C2×C3×D4—21no—
58C2×C3×D4C2312yes—
59C2×C3×D4C2311yescharacteristic
60C22×C3×C4—21no—
61C2×#67—21no—
62C2×#67—22no—
63C2×#67—22no—
64C2×#67—22no—
65D3×D4—42no—
66D3×D4—42no—
67D3×D4—44no—
68D3×D4—44no—
69C2×D12—21no—
70C2×D12—21no—
71C2×D12C2312yes—
72C2×C4×D3D411yescharacteristic
73C2×C4×D3D411yescharacteristic
74C2×C4×D3C2311yescharacteristic
75C2×Dic6C2311yescharacteristic
76C3×D8—42no—
77C2×C3×C8C2312yes—
78C3×C4⋊C4C2311yescharacteristic
79C3×C22⋊C4—21no—
80C3×C42C2311yescharacteristic
81#217—21no—
82#217—21no—
83#217—21no—
84#214—42no—
85#213—42no—
86(C2×D3)⋊C4—21no—
87C4⋊Dic3—21no—
88C4⋊Dic3—21no—
89C4⋊Dic3C2311yescharacteristic
90Dic3⋊C4—21no—
91Dic3⋊C4—21no—
92C4×Dic3—21no—
93C4×Dic3D411yescharacteristic
94C4×Dic3D411yescharacteristic
95C4×Dic3C2311yescharacteristic
96C2×C3⋊C8C2312yes—
97#204—42no—
98C22×D4—32no—
99C22×D4—61no—
100C22×D4—61no—
101C2×SD16—32no—
102C2×SD16—32no—
103C2×D8—32no—
104C2×D8C2×D312yescomplemented
105#127—32no—
106#127—31no—
107#127—61no—
108#127—61no—
109#127C2×D311yescharacteristic
110#126—61no—
111#125—31no—
112#124—31no—
113#123—61no—
114C22⋊Q8—31no—
115#121—31no—
116C4×Q8—31no—
117C4×D4—61no—
118C4×D4C2×D311yescharacteristic, complemented
119#117—61no—
120C2×C42—31no—
121C4⋊C8—31no—
122C4⋊C8C2×D311yescharacteristic, complemented
123#102—32no—
124#102—32no—
125#102—32no—
126#102C2×D312yescomplemented
127C22⋊C8—32no—
128#97—32no—
129C23×C3—21no—
130C23×C3—22no—
131C22×D3—22no—
132C22×D3—44no—
133C22×D3—44no—
134C22×D3—44no—
135C22×D3C2×D411yescharacteristic
136C3×D4—21no—
137C3×D4—22no—
138C3×D4—21no—
139C3×D4—22no—
140C3×D4—42no—
141C3×D4—42no—
142C2×C3×C4—21no—
143C2×C3×C4—42no—
144C2×C3×C4C2×D411yescharacteristic
145C2×C3×C4C2×D411yescharacteristic
146C2×C3×C4C2411yescharacteristic
147#67—44no—
148#67—44no—
149#67—44no—
150#67—42no—
151C2×Dic3—21no—
152C2×Dic3—21no—
153C2×Dic3C2×D411yescharacteristic
154C2×Dic3C2×D411yescharacteristic
155C2×Dic3C2×D411yescharacteristic
156D12—22no—
157D12—22no—
158D12—42no—
159D12—42no—
160C4×D3—22no—
161C4×D3—22no—
162C4×D3—21no—
163C4×D3—21no—
164Dic6—21no—
165Dic6—21no—
166C3×C8—22no—
167C3⋊C8—22no—
168C24—62no—
169C2×Q8—31no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—32no—
173C2×D4—62no—
174C2×D4—62no—
175C2×D4—62no—
176C2×D4—61no—
177C2×D4—61no—
178C2×D4—61no—
179C2×D4—122no—
180C2×D4—122no—
181C2×D4—124no—
182C2×D4—124no—
183C2×D4C22×D312yes—
184C2×D4C22×D311yescharacteristic
185C22×C4—21no—
186C22×C4—31no—
187C22×C4—31no—
188C22×C4—31no—
189SD16—122no—
190SD16—122no—
191D8—42no—
192D8—122no—
193C2×C8—32no—
194C2×C8C22×D312yes—
195C4⋊C4—31no—
196C4⋊C4—61no—
197C4⋊C4—61no—
198C4⋊C4—61no—
199C4⋊C4—61no—
200C4⋊C4C22×D311yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—61no—
203C22⋊C4—61no—
204C22⋊C4—61no—
205C22⋊C4—61no—
206C42—31no—
207C42—31no—
208C42—31no—
209C42—61no—
210C42C22×D311yescharacteristic
211C22×C3—42no—
212C22×C3—41no—
213C22×C3—42no—
214C22×C3—42no—
215C22×C3—41no—
216C22×C3—41no—
217C22×C3C22×D411yescharacteristic
218C2×D3—21no—
219C2×D3—21no—
220C2×D3—42no—
221C2×D3—42no—
222C2×D3—42no—
223C2×D3—88no—
224C2×D3AGL(1,ℤ/8)12yes—
225C3×C4—21no—
226C3×C4—41no—
227C3×C4AGL(1,ℤ/8)12yes—
228C3×C4H(5,𝔽2)11yescharacteristic
229C3×C4H(5,𝔽2)11yescharacteristic
230Dic3—21no—
231Dic3—21no—
232Dic3—21no—
233Dic3—41no—
234Dic3—41no—
235C23—21no—
236C23—22no—
237C23—31no—
238C23—62no—
239C23—124no—
240C23—124no—
241C23—124no—
242Q8—61no—
243Q8—61no—
244D4—21no—
245D4—22no—
246D4—21no—
247D4—22no—
248D4—42no—
249D4—42no—
250D4—62no—
251D4—62no—
252D4—122no—
253D4—124no—
254D4—124no—
255D4—124no—
256D4—122no—
257D4—122no—
258C2×C4—21no—
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—31no—
262C2×C4—42no—
263C2×C4—62no—
264C2×C4—62no—
265C2×C4—61no—
266C2×C4—61no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4D3×D411yescharacteristic
270C2×C4D3×D411yescharacteristic
271C2×C4C23×D311yescharacteristic
272C8—22no—
273C8—62no—
274C2×C3—82no—
275C2×C3—81no—
276C2×C3#53511yescharacteristic
277C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
278C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
279D3—22no—
280D3—82no—
281C22—32no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—41no—
286C22—42no—
287C22—41no—
288C22—61no—
289C22—61no—
290C22—122no—
291C22—122no—
292C22—122no—
293C22—248no—
294C22C2×D3×D411yescharacteristic
295C4—21no—
296C4—41no—
297C4—61no—
298C4—61no—
299C4—61no—
300C4—121no—
301C4—121no—
302C4#91112yes—
303C4#100411yescharacteristic
304C4#100911yescharacteristic
305C3#321711yescharacteristic, complemented, Sylow
306C2—62no—
307C2—82no—
308C2—81no—
309C2—242no—
310C2#560711yescharacteristic
311C2C2×#91111yescharacteristic
312C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3217
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. D3×#127C2semidirect
2. #5557C2semidirect
3. C3×#459C2semidirect
4. #5156C2semidirect
5. #5154C2semidirect
6. #5075C2semidirect
7. #5015C2semidirect
8. #4848C2semidirect
9. #4835C2semidirect
10. #4776C2semidirect
11. #4764C2semidirect
16. C2×C3×D8C22semidirect
18. C3×C4×D4C22semidirect
23. #936C22semidirect
24. C2×#214C22semidirect
25. C2×#213C22semidirect
28. C2×#204C22semidirect
30. #890C22semidirect
35. C4×Dic6C22semidirect
36. C3×C4⋊C8C22semidirect
37. C3×#102C22semidirect
38. #832C22semidirect
39. #821C22semidirect
41. #815C22semidirect
42. #809C22semidirect
43. C4⋊(C3⋊C8)C22semidirect
51. #459D3semidirect
104. C2×D8C2×D3semidirect
118. C4×D4C2×D3semidirect
122. C4⋊C8C2×D3semidirect
126. #102C2×D3semidirect
305. C3#3217semidirect

Maximal subgroups

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

subgroupindex
D3×#1272
#55572
C3×#4592
#51562
#51542
#50752
#50152
#48482
#48352
#47762
#47642
#32173