Finite Group Databasebeta

← #83379

Subgroups — #83379

Order 384.

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

normal subgroups only

This group has 504 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
1C3×#535C211yesmaximal, characteristic, complemented
2#5833C211yesmaximal, characteristic, complemented
3C2×#948C211yesmaximal, characteristic, complemented
4C2×#932C211yesmaximal, characteristic, complemented
5C2×#832C211yesmaximal, characteristic, complemented
6#5042C212yesmaximal, complemented
7#5041C212yesmaximal, complemented
8#5039C212yesmaximal, complemented
9#5038C212yesmaximal, complemented
10#5002C211yesmaximal, characteristic, complemented
11#4995C211yesmaximal, characteristic, complemented
12#3062—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C22×C3×D4C2211yescharacteristic
15C2×#235C2211yescharacteristic
16C3×#127—21no—
17C3×#124—21no—
18C3×#121C2212yes—
19C3×#121C2212yes—
20C3×#120—22no—
21C3×#117C2211yescharacteristic
22#948—24no—
23#946—21no—
24#939—22no—
25#938—21no—
26C2×#214C2212yes—
27#932—24no—
28C2×#213C2212yes—
29#930C2212yescomplemented
30C2×C4⋊Dic3C2211yescharacteristic
31#924C2212yescomplemented
32C2×#208C2211yescharacteristic
33C2×#208C2212yescomplemented
34C22×C3⋊C8C2211yescharacteristic
35C22×C3⋊C8C2212yescomplemented
36#832C2212yes—
37#832C2212yes—
38#810C2211yescharacteristic, complemented
39#810C2211yescharacteristic, complemented
40#809C2211yescharacteristic, complemented
41#809C2211yescharacteristic, complemented
42#808C2212yescomplemented
43#807C2212yescomplemented
44C2×AGL(1,ℤ/8)—31no—
45C2×#131—31no—
46#537—31no—
47#535D311yescharacteristic, complemented
48#469—32no—
49#468—32no—
50#466—32no—
51#465—32no—
52#428—31no—
53#416—31no—
54C2×#102—31no—
55C24×C3—21no—
56C3×#41—22no—
57C3×#41—22no—
58C2×C3×Q8D411yescharacteristic
59C2×C3×D4—22no—
60C2×C3×D4—22no—
61C2×C3×D4—22no—
62C2×C3×D4—21no—
63C2×C3×D4—22no—
64C2×C3×D4D411yescharacteristic
65C2×C3×D4D412yes—
66C2×C3×D4C2312yes—
67C22×C3×C4C2312yes—
68C22×C3×C4C2311yescharacteristic
69C2×#67—21no—
70C2×#67—22no—
71C22×Dic3—21no—
72#235—22no—
73#235—22no—
74C2×D12C2311yescharacteristic
75C2×C4×D3—21no—
76C2×Dic6C2311yescharacteristic
77C3×C4⋊C4C2311yescharacteristic
78C3×C4⋊C4C2311yescharacteristic
79C3×C22⋊C4—21no—
80C3×C22⋊C4—22no—
81C3×C22⋊C4—22no—
82C3×C42—21no—
83#217—22no—
84#214—24no—
85#213—24no—
86(C2×D3)⋊C4—22no—
87C4⋊Dic3—21no—
88C4⋊Dic3C2312yes—
89Dic3⋊C4—22no—
90#208—24no—
91#208D414yescomplemented
92C2×C3⋊C8—22no—
93C2×C3⋊C8—24no—
94C2×C3⋊C8C2311yescharacteristic
95C2×C3⋊C8C2311yescharacteristic
96C2×C3⋊C8C2312yes—
97C2×#41—31no—
98C2×#41C2×D311yescharacteristic
99C22×D4C2×D311yescharacteristic
100AGL(1,ℤ/8)—64no—
101C2×SD16—32no—
102C2×D8—32no—
103#131—64no—
104C2×M16—31no—
105C2×M16—32no—
106C22×C8—31no—
107C22×C8—32no—
108#127—21no—
109#124—21no—
110C22⋊Q8—32no—
111C22⋊Q8—61no—
112#121—32no—
113#121—61no—
114#121—62no—
115#121C2×D312yes—
116#121C2×D312yes—
117#120—22no—
118#117C2×D311yescharacteristic
119C2×C4⋊C4—31no—
120#107—32no—
121#106—32no—
122#103—31no—
123#103—31no—
124#102—32no—
125#102—32no—
126#102—31no—
127#102—31no—
128C23×C3—22no—
129C23×C3—42no—
130C23×C3—81no—
131C23×C3—81no—
132C23×C3C2×D412yes—
133C23×C3C2×D411yescharacteristic
134C22×D3—21no—
135C3×Q8—22no—
136C3×D4—24no—
137C3×D4—22no—
138C3×D4—22no—
139C3×D4—22no—
140C3×D4—42no—
141C3×D4—82no—
142C3×D4—82no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4C2×D412yes—
148C2×C3×C4C2×D412yes—
149C2×C3×C4C2×D411yescharacteristic
150C2×C3×C4#4112yes—
151C2×C3×C4C2411yescharacteristic
152#67—42no—
153#67—82no—
154C2×Dic3—21no—
155C2×Dic3—22no—
156C2×Dic3—81no—
157D12—21no—
158D12—21no—
159C4×D3—42no—
160Dic6—21no—
161Dic6—21no—
162C3⋊C8—21no—
163C3⋊C8—21no—
164C3⋊C8C2×D414yes—
165C24—21no—
166#41—22no—
167#41—22no—
168#41—62no—
169#41—62no—
170C2×Q8—31no—
171C2×Q8#6711yescharacteristic
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—21no—
177C2×D4—31no—
178C2×D4—61no—
179C2×D4—62no—
180C2×D4#6712yes—
181C2×D4#6711yescharacteristic
182C2×D4C22×D312yes—
183C22×C4—61no—
184C22×C4—61no—
185C22×C4C22×D312yes—
186C22×C4C22×D311yescharacteristic
187SD16—64no—
188D8—64no—
189M16—34no—
190M16—64no—
191C2×C8—32no—
192C2×C8—31no—
193C2×C8—31no—
194C2×C8—64no—
195C2×C8—62no—
196C4⋊C4—32no—
197C4⋊C4—61no—
198C4⋊C4—62no—
199C4⋊C4C22×D311yescharacteristic
200C4⋊C4C22×D311yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—22no—
203C22⋊C4—22no—
204C22⋊C4—62no—
205C22⋊C4—62no—
206C42—21no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—22no—
210C22×C3—22no—
211C22×C3—22no—
212C22×C3—42no—
213C22×C3—42no—
214C22×C3—42no—
215C22×C3—84no—
216C22×C3#12112yes—
217C22×C3C22×D411yescharacteristic
218C2×D3—41no—
219C2×D3—41no—
220C2×D3—41no—
221C3×C4—22no—
222C3×C4—41no—
223C3×C4—41no—
224C3×C4#12112yes—
225C3×C4C22×D411yescharacteristic
226C3×C4C2×#4111yescharacteristic
227Dic3—41no—
228Dic3—42no—
229C23—22no—
230C23—42no—
231C23—61no—
232C23—81no—
233C23—81no—
234C23C2×#6712yes—
235C23C2×#6711yescharacteristic
236Q8—22no—
237Q8—61no—
238Q8—61no—
239D4—24no—
240D4—22no—
241D4—22no—
242D4—22no—
243D4—42no—
244D4—61no—
245D4—61no—
246D4—82no—
247D4—82no—
248D4—122no—
249D4—242no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—62no—
255C2×C4—61no—
256C2×C4—122no—
257C2×C4—241no—
258C2×C4D3×D412yes—
259C2×C4#23712yes—
260C2×C4C2×#6712yes—
261C2×C4C2×#6711yescharacteristic
262C2×C4C23×D311yescharacteristic
263C8—34no—
264C8—61no—
265C8—61no—
266C2×C3—22no—
267C2×C3—42no—
268C2×C3—82no—
269C2×C3C2×#12111yescharacteristic
270C2×C3#57611yescharacteristic
271C2×C3#57711yescharacteristic
272D3—81no—
273C22—21no—
274C22—21no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—84no—
282C22—121no—
283C22—121no—
284C22—121no—
285C22#93912yes—
286C22C22×#6711yescharacteristic
287C4—22no—
288C4—41no—
289C4—41no—
290C4—121no—
291C4—122no—
292C4#93912yes—
293C4C2×D3×D411yescharacteristic
294C4C2×#23711yescharacteristic
295C3#306211yescharacteristic, complemented, Sylow
296C2—22no—
297C2—42no—
298C2—82no—
299C2—241no—
300C2C2×#93911yescharacteristic
301C2#583511yescharacteristic
302C2#583611yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#535C2
socleC22×C3C22×D4
hypercenter#535D3
hypocenterC3#3062
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C3×#535C2semidirect
2. #5833C2semidirect
3. C2×#948C2semidirect
4. C2×#932C2semidirect
5. C2×#832C2semidirect
6. #5042C2semidirect
7. #5041C2semidirect
8. #5039C2semidirect
9. #5038C2semidirect
10. #5002C2semidirect
11. #4995C2semidirect
29. #930C22semidirect
31. #924C22semidirect
33. C2×#208C22semidirect
35. C22×C3⋊C8C22semidirect
38. #810C22semidirect
39. #810C22semidirect
40. #809C22semidirect
41. #809C22semidirect
42. #808C22semidirect
43. #807C22semidirect
47. #535D3semidirect
91. #208D4semidirect
295. C3#3062semidirect

Maximal subgroups

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

subgroupindex
C3×#5352
#58332
C2×#9482
C2×#9322
C2×#8322
#50422
#50412
#50392
#50382
#50022
#49952
#30623