Finite Group Databasebeta

← #84121

Subgroups — #84121

Order 384.

724 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C3×#585C211yesmaximal, characteristic, complemented
2#5832C211yesmaximal, characteristic, complemented
3#5822C212yesmaximal, complemented
4#5246C212yesmaximal, complemented
5#5245C212yesmaximal, complemented
6#5242C212yesmaximal, complemented
7#5241C212yesmaximal, complemented
8#5237C211yesmaximal, characteristic, complemented
9C2×#837C212yesmaximal, complemented
10#3018—31nomaximal, Sylow
11C3×2−1+4—22no—
12C3×2−1+4C2212yes—
13C3×H(5,𝔽2)—22no—
14C3×H(5,𝔽2)—22no—
15C3×H(5,𝔽2)C2212yes—
16C2×C3×#41—24no—
17C2×C3×#41—22no—
18C2×C3×#41C2212yes—
19C2×C3×#41C2211yescharacteristic
20C2×#235C2211yescharacteristic, complemented
21#951C2214yescomplemented
22#950—24no—
23#949C2214yescomplemented
24#947C2211yescharacteristic, complemented
25#944C2211yescharacteristic, complemented
26#942—22no—
27#940C2211yescharacteristic, complemented
28#939—22no—
29#936C2211yescharacteristic, complemented
30#932—22no—
31C4×#67—22no—
32C2×C4×Dic3C2211yescharacteristic, complemented
33C2×#208C2212yescomplemented
34#837C2214yescomplemented
35#837C2214yescomplemented
36#836C2212yescomplemented
37#833C2212yescomplemented
38#585D311yescharacteristic, complemented
39#575—32no—
40#532—31no—
41#456—32no—
42#455—32no—
43#454—32no—
44#453—32no—
45C2×#104—32no—
46#413—31no—
47C3×#41—24no—
48C3×#41—28no—
49C3×#41—24no—
50C3×#41—24no—
51C3×#41—24no—
52C3×#41—22no—
53C3×#41—22no—
54C3×#41—24no—
55C3×#41—22no—
56C3×#41—22no—
57C3×#41D414yescomplemented
58C3×#41C2314yes—
59C2×C3×Q8—22no—
60C2×C3×Q8—24no—
61C2×C3×Q8D412yescomplemented
62C2×C3×Q8C2311yescharacteristic
63C2×C3×D4—28no—
64C2×C3×D4—22no—
65C2×C3×D4—24no—
66C2×C3×D4—22no—
67C2×C3×D4—24no—
68C2×C3×D4D412yes—
69C2×C3×D4D412yescomplemented
70C2×C3×D4C2311yescharacteristic
71C22×C3×C4—24no—
72C22×C3×C4—22no—
73C22×C3×C4D412yes—
74C22×C3×C4C2311yescharacteristic
75C2×#67—22no—
76C2×#67—21no—
77C22×Dic3—21no—
78#235—22no—
79#235—22no—
80C2×D12C2311yescharacteristic
81C2×C4×D3—21no—
82C2×Dic6C2311yescharacteristic
83#217—22no—
84#216—24no—
85#215—24no—
86#214—24no—
87#213—24no—
88(C2×D3)⋊C4—22no—
89Dic3⋊C4—22no—
90C4×Dic3—21no—
91C4×Dic3C2311yescharacteristic
92C4×Dic3C2311yescharacteristic
93#208—22no—
94#208C2314yes—
95C2×C3⋊C8—22no—
962−1+4—22no—
972−1+4C2×D312yes—
98H(5,𝔽2)—22no—
99H(5,𝔽2)—22no—
100H(5,𝔽2)C2×D312yes—
101C2×#41—24no—
102C2×#41—22no—
103C2×#41—31no—
104C2×#41C2×D312yes—
105C2×#41C2×D311yescharacteristic
106#137—34no—
107#137—62no—
108AGL(1,ℤ/8)—34no—
109AGL(1,ℤ/8)—62no—
110#135—64no—
111C2×M16—32no—
112C4⋊Q8—31no—
113#127—31no—
114#124—31no—
115#124—31no—
116#121—62no—
117C4×D4—62no—
118C2×C42—31no—
119#104—34no—
120#104—34no—
121#101—32no—
122#100—32no—
123C23×C3—24no—
124C23×C3—22no—
125C23×C3C2×D412yes—
126C23×C3C2×D411yescharacteristic
127C22×D3—21no—
128C3×Q8—24no—
129C3×Q8—22no—
130C3×Q8—22no—
131C3×Q8C2×D414yes—
132C3×D4—28no—
133C3×D4—24no—
134C3×D4—24no—
135C3×D4—24no—
136C3×D4—24no—
137C3×D4—22no—
138C3×D4—22no—
139C3×D4C2×D414yes—
140C2×C3×C4—24no—
141C2×C3×C4—28no—
142C2×C3×C4—22no—
143C2×C3×C4—24no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—24no—
147C2×C3×C4C2×D412yes—
148C2×C3×C4C2×D411yescharacteristic
149C2×C3×C4C2×D412yes—
150C2×C3×C4C2×D412yes—
151C2×C3×C4C2411yescharacteristic
152#67—44no—
153#67—42no—
154C2×Dic3—21no—
155C2×Dic3—21no—
156C2×Dic3—21no—
157C2×Dic3—42no—
158D12—22no—
159C4×D3—41no—
160C4×D3—41no—
161Dic6—22no—
162C3⋊C8—24no—
163#41—24no—
164#41—28no—
165#41—24no—
166#41—24no—
167#41—24no—
168#41—22no—
169#41—22no—
170#41—24no—
171#41—22no—
172#41—22no—
173#41—62no—
174#41—62no—
175#41#6714yescomplemented
176#41C22×D314yes—
177C2×Q8—22no—
178C2×Q8—24no—
179C2×Q8—31no—
180C2×Q8#6712yescomplemented
181C2×Q8C22×D311yescharacteristic
182C2×D4—28no—
183C2×D4—22no—
184C2×D4—24no—
185C2×D4—22no—
186C2×D4—24no—
187C2×D4—31no—
188C2×D4—62no—
189C2×D4—61no—
190C2×D4#6712yes—
191C2×D4#6712yescomplemented
192C2×D4C22×D311yescharacteristic
193C22×C4—24no—
194C22×C4—22no—
195C22×C4—61no—
196C22×C4—61no—
197C22×C4#6712yes—
198C22×C4C22×D311yescharacteristic
199Q16—64no—
200SD16—64no—
201SD16—64no—
202D8—64no—
203M16—34no—
204M16—62no—
205C2×C8—62no—
206C4⋊C4—62no—
207C22⋊C4—62no—
208C22⋊C4—62no—
209C42—31no—
210C42—31no—
211C42—61no—
212C22×C3—22no—
213C22×C3—24no—
214C22×C3—41no—
215C22×C3—42no—
216C22×C3—82no—
217C22×C3—84no—
218C22×C3#12012yes—
219C22×C3C22×D411yescharacteristic
220C2×D3—42no—
221C2×D3—41no—
222C3×C4—24no—
223C3×C4—22no—
224C3×C4#12011yescharacteristic
225C3×C4#12011yescharacteristic
226C3×C4C22×D412yes—
227Dic3—22no—
228Dic3—41no—
229Dic3—41no—
230C23—24no—
231C23—22no—
232C23—61no—
233C23C2×#6712yes—
234C23C2×#6711yescharacteristic
235Q8—24no—
236Q8—22no—
237Q8—22no—
238Q8—62no—
239Q8C2×#6714yes—
240D4—28no—
241D4—24no—
242D4—24no—
243D4—24no—
244D4—24no—
245D4—22no—
246D4—22no—
247D4—62no—
248D4—124no—
249D4—122no—
250D4C2×#6714yes—
251C2×C4—24no—
252C2×C4—28no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—61no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—122no—
262C2×C4—121no—
263C2×C4—121no—
264C2×C4C2×#6712yes—
265C2×C4C2×#6711yescharacteristic
266C2×C4C2×#6712yes—
267C2×C4C2×#6712yes—
268C2×C4C23×D311yescharacteristic
269C8—64no—
270C2×C3—21no—
271C2×C3—22no—
272C2×C3—42no—
273C2×C3—44no—
274C2×C3C2×#12011yescharacteristic
275D3—81no—
276C22—22no—
277C22—24no—
278C22—41no—
279C22—42no—
280C22—82no—
281C22—84no—
282C22—122no—
283C22—121no—
284C22#95312yes—
285C22C22×#6711yescharacteristic
286C4—24no—
287C4—22no—
288C4—62no—
289C4—121no—
290C4—121no—
291C4#95311yescharacteristic
292C4#95311yescharacteristic
293C4C22×#6712yes—
294C3#301811yescharacteristic, complemented, Sylow
295C2—21no—
296C2—22no—
297C2—42no—
298C2—44no—
299C2—241no—
300C2C2×#95311yescharacteristic
subgroupG/H
centerC4#953
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#585C2
socleC2×C3C2×#120
hypercenter#585D3
hypocenterC3#3018
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C3×#585C2semidirect
2. #5832C2semidirect
3. #5822C2semidirect
4. #5246C2semidirect
5. #5245C2semidirect
6. #5242C2semidirect
7. #5241C2semidirect
8. #5237C2semidirect
9. C2×#837C2semidirect
20. C2×#235C22semidirect
21. #951C22semidirect
23. #949C22semidirect
24. #947C22semidirect
25. #944C22semidirect
27. #940C22semidirect
29. #936C22semidirect
32. C2×C4×Dic3C22semidirect
33. C2×#208C22semidirect
34. #837C22semidirect
35. #837C22semidirect
36. #836C22semidirect
37. #833C22semidirect
38. #585D3semidirect
57. C3×#41D4semidirect
61. C2×C3×Q8D4semidirect
69. C2×C3×D4D4semidirect
175. #41#67semidirect
180. C2×Q8#67semidirect
191. C2×D4#67semidirect
294. C3#3018semidirect

Maximal subgroups

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

subgroupindex
C3×#5852
#58322
#58222
#52462
#52452
#52422
#52412
#52372
C2×#8372
#30183