Finite Group Databasebeta

← #81845

Subgroups — #81845

Order 384.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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×#914C212yesmaximal, complemented
2C2×#911C212yesmaximal, complemented
3D3×#127C211yesmaximal, characteristic, complemented
4#5668C211yesmaximal, characteristic, complemented
5C3×#496C211yesmaximal, characteristic, complemented
6#5175C212yesmaximal, complemented
7#5151C212yesmaximal, complemented
8#5074C211yesmaximal, characteristic, complemented
9#5058C211yesmaximal, characteristic, complemented
10#4712C211yesmaximal, characteristic, complemented
11#4707C211yesmaximal, characteristic, complemented
12#3151—31nomaximal, Sylow
13C2×#239C2211yescharacteristic
14C2×#237C2211yescharacteristic
15C2×D3×D4—22no—
16C2×D3×D4C2212yes—
17C2×C3×SD16C2212yescomplemented
18C2×C3×D8C2212yescomplemented
19C3×#127C2211yescharacteristic
20C3×#124C2211yescharacteristic, complemented
21#944C2211yescharacteristic, complemented
22C2×#215C2212yescomplemented
23#940—22no—
24#940C2212yes—
25#940C2211yescharacteristic, complemented
26C2×#214C2212yescomplemented
27C2×#213C2212yescomplemented
28C2×#213C2212yescomplemented
29#914—28no—
30#911—28no—
31C2×D24C2212yescomplemented
32C2×#204C2212yescomplemented
33C2×ΓL(1,𝔽25)C2212yes—
34#884—22no—
35#67⋊C4—22no—
36#875C2211yescharacteristic, complemented
37#874C2211yescharacteristic
38C42×D3C2211yescharacteristic
39C3×#97C2211yescharacteristic, complemented
40C8×Dic3C2212yescomplemented
41#803C2211yescharacteristic, complemented
42C2×AGL(1,ℤ/8)—32no—
43C2×AGL(1,ℤ/8)—32no—
44#532—31no—
45C2×#127—31no—
46#496—31no—
47#496—31no—
48#496—31no—
49#496D311yescharacteristic, complemented
50#493—32no—
51#492—32no—
52C4×M16—31no—
53C23×D3—22no—
54C2×C3×Q8C2311yescharacteristic
55C2×C3×D4—22no—
56C2×C3×D4C2311yescharacteristic
57C2×C3×D4C2312yes—
58C2×#67—21no—
59C2×#67—22no—
60C2×#67—22no—
61C2×#67—24no—
62C22×Dic3—21no—
63#239—24no—
64#237—24no—
65D3×D4—24no—
66D3×D4—24no—
67D3×D4—48no—
68C2×D12—21no—
69C2×D12—22no—
70C2×D12C2312yes—
71C2×D12C2311yescharacteristic
72C2×C4×D3—21no—
73C2×C4×D3D412yes—
74C2×C4×D3C2311yescharacteristic
75C2×Dic6C2311yescharacteristic
76C3×SD16—24no—
77C3×D8—24no—
78C2×C3×C8C2312yes—
79C3×C22⋊C4—22no—
80C3×C42C2311yescharacteristic
81#215—24no—
82#214—24no—
83#213—24no—
84#213—24no—
85(C2×D3)⋊C4—22no—
86Dic3⋊C4—22no—
87C4×Dic3D412yes—
88C4×Dic3C2311yescharacteristic
89C2×C3⋊C8C2312yes—
90D24—24no—
91#204—24no—
92ΓL(1,𝔽25)D418yescomplemented
93C2×#41—31no—
94C2×#41—31no—
95C22×D4—32no—
96C22×D4—62no—
97AGL(1,ℤ/8)—68no—
98AGL(1,ℤ/8)—68no—
99C2×SD16—32no—
100C2×SD16—32no—
101C2×SD16—32no—
102C2×SD16C2×D312yescomplemented
103C2×D8—32no—
104C2×D8—32no—
105C2×D8—32no—
106C2×D8C2×D312yescomplemented
107C2×M16—32no—
108C4⋊Q8—31no—
109#127—32no—
110#127—31no—
111#127—31no—
112#127—62no—
113#127C2×D311yescharacteristic
114#124—31no—
115#124C2×D311yescharacteristic, complemented
116#121—62no—
117C4×D4—62no—
118C2×C42—31no—
119#97—31no—
120#97C2×D311yescharacteristic, complemented
121C4×C8—32no—
122C23×C3—21no—
123C23×C3—22no—
124C22×D3—21no—
125C22×D3—22no—
126C22×D3—48no—
127C22×D3—44no—
128C22×D3C2×D411yescharacteristic
129C3×Q8—22no—
130C3×D4—22no—
131C3×D4—22no—
132C3×D4—22no—
133C3×D4—44no—
134C2×C3×C4—21no—
135C2×C3×C4C2×D412yes—
136C2×C3×C4C2411yescharacteristic
137#67—42no—
138#67—44no—
139#67—48no—
140#67—44no—
141C2×Dic3—21no—
142C2×Dic3—42no—
143C2×Dic3C2×D411yescharacteristic
144C2×Dic3C2×D412yes—
145D12—22no—
146D12—22no—
147D12—22no—
148D12—44no—
149D12—42no—
150C4×D3—24no—
151C4×D3—42no—
152C4×D3C2×D414yes—
153Dic6—22no—
154C3×C8C2×D414yescomplemented
155C3⋊C8C2×D414yescomplemented
156C24—62no—
157#41—64no—
158#41—64no—
159C2×Q8—31no—
160C2×Q8C22×D311yescharacteristic
161C2×D4—22no—
162C2×D4—31no—
163C2×D4—32no—
164C2×D4—61no—
165C2×D4—64no—
166C2×D4—64no—
167C2×D4—62no—
168C2×D4—64no—
169C2×D4—62no—
170C2×D4—61no—
171C2×D4—62no—
172C2×D4—128no—
173C2×D4C22×D311yescharacteristic
174C2×D4C22×D312yes—
175C22×C4—31no—
176C22×C4—32no—
177C22×C4—61no—
178C22×C4—61no—
179SD16—24no—
180SD16—64no—
181SD16—64no—
182SD16—64no—
183D8—24no—
184D8—64no—
185D8—64no—
186D8—64no—
187M16—38no—
188C2×C8—32no—
189C2×C8C22×D312yes—
190C4⋊C4—62no—
191C22⋊C4—22no—
192C22⋊C4—62no—
193C42—32no—
194C42—31no—
195C42C22×D311yescharacteristic
196C22×C3—41no—
197C22×C3—42no—
198C22×C3—44no—
199C22×C3—42no—
200C22×C3C22×D411yescharacteristic
201C2×D3—22no—
202C2×D3—41no—
203C2×D3—42no—
204C2×D3—44no—
205C2×D3—42no—
206C2×D3—88no—
207C2×D3#12712yes—
208C3×C4—22no—
209C3×C4—41no—
210C3×C4C22×D412yes—
211Dic3—41no—
212Dic3#12712yes—
213Dic3AGL(1,ℤ/8)14yescomplemented
214C23—21no—
215C23—22no—
216C23—31no—
217C23—61no—
218C23—62no—
219C23—124no—
220C23—128no—
221Q8—22no—
222Q8—62no—
223D4—22no—
224D4—22no—
225D4—22no—
226D4—44no—
227D4—62no—
228D4—62no—
229D4—62no—
230D4—124no—
231D4—124no—
232D4—128no—
233D4—124no—
234D4—122no—
235D4—122no—
236C2×C4—21no—
237C2×C4—32no—
238C2×C4—31no—
239C2×C4—34no—
240C2×C4—64no—
241C2×C4—61no—
242C2×C4—122no—
243C2×C4—122no—
244C2×C4D3×D412yes—
245C2×C4C23×D311yescharacteristic
246C8—34no—
247C8D3×D414yescomplemented
248C2×C3—82no—
249C2×C3—81no—
250C2×C3C2×#12711yescharacteristic
251C2×C3C2×AGL(1,ℤ/8)12yes—
252D3—22no—
253D3—81no—
254D3—82no—
255C22—32no—
256C22—41no—
257C22—42no—
258C22—44no—
259C22—42no—
260C22—62no—
261C22—121no—
262C22—122no—
263C22—124no—
264C22—122no—
265C22—248no—
266C22C2×D3×D411yescharacteristic
267C4—22no—
268C4—32no—
269C4—34no—
270C4—41no—
271C4—121no—
272C4C2×D3×D412yes—
273C3#315111yescharacteristic, complemented, Sylow
274C2—62no—
275C2—82no—
276C2—81no—
277C2—242no—
278C2—241no—
279C2D3×#12711yescharacteristic
280C2D3×AGL(1,ℤ/8)12yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#496C2
socleC22×C3C22×D4
hypercenter#496D3
hypocenterC3#3151
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#914C2semidirect
2. C2×#911C2semidirect
3. D3×#127C2semidirect
4. #5668C2semidirect
5. C3×#496C2semidirect
6. #5175C2semidirect
7. #5151C2semidirect
8. #5074C2semidirect
9. #5058C2semidirect
10. #4712C2semidirect
11. #4707C2semidirect
17. C2×C3×SD16C22semidirect
18. C2×C3×D8C22semidirect
20. C3×#124C22semidirect
21. #944C22semidirect
22. C2×#215C22semidirect
25. #940C22semidirect
26. C2×#214C22semidirect
27. C2×#213C22semidirect
28. C2×#213C22semidirect
31. C2×D24C22semidirect
32. C2×#204C22semidirect
36. #875C22semidirect
39. C3×#97C22semidirect
40. C8×Dic3C22semidirect
41. #803C22semidirect
49. #496D3semidirect
92. ΓL(1,𝔽25)D4semidirect
102. C2×SD16C2×D3semidirect
106. C2×D8C2×D3semidirect
115. #124C2×D3semidirect
120. #97C2×D3semidirect
154. C3×C8C2×D4semidirect
155. C3⋊C8C2×D4semidirect
213. Dic3AGL(1,ℤ/8)semidirect
247. C8D3×D4semidirect
273. C3#3151semidirect

Maximal subgroups

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

subgroupindex
C2×#9142
C2×#9112
D3×#1272
#56682
C3×#4962
#51752
#51512
#50742
#50582
#47122
#47072
#31513