Finite Group Databasebeta

← #78838

Subgroups — #78838

Order 384.

520 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 137 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#5828C211yesmaximal, characteristic, complemented
2C2×#950C211yesmaximal, characteristic, complemented
3C22×D3×C8C211yesmaximal, characteristic, complemented
4#5524C211yesmaximal, characteristic, complemented
5C3×#416C211yesmaximal, characteristic, complemented
6#5113C211yesmaximal, characteristic, complemented
7#4998C211yesmaximal, characteristic, complemented
8#4813C212yesmaximal, complemented
9#4807C212yesmaximal, complemented
10#4778C212yesmaximal, complemented
11#4775C212yesmaximal, complemented
12#3050—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C2×#235C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C22×C3×C8C2211yescharacteristic
17C3×#117C2211yescharacteristic
18#950—24no—
19#946C2212yes—
20C2×#216C2211yescharacteristic, complemented
21C2×#215C2211yescharacteristic, complemented
22#938C2212yes—
23#937—21no—
24#935—21no—
25C2×#214C2211yescharacteristic, complemented
26C2×#213C2211yescharacteristic, complemented
27C4×#67—22no—
28#926C2211yescharacteristic
29C22×C3⋊C8C2211yescharacteristic
30C2×D3×C8—24no—
31C2×D3×C8C2212yes—
32C2×D3×C8C2212yes—
33#897C2212yes—
34#895C2212yes—
35#886—21no—
36#882—21no—
37C4×D12—22no—
38C3×#103C2212yescomplemented
39C3×#102C2212yescomplemented
40#821C2212yescomplemented
41#816C2212yescomplemented
42#808C2212yescomplemented
43#807C2212yescomplemented
44C2×#135—31no—
45C23×C8—31no—
46#525—31no—
47#525—31no—
48#467—32no—
49#466—32no—
50#465—32no—
51#465—32no—
52#427—31no—
53#416—31no—
54#416D311yescharacteristic, complemented
55C23×D3D411yescharacteristic
56C3×#41—22no—
57C3×#41—22no—
58C2×C3×Q8C2311yescharacteristic
59C2×C3×D4—21no—
60C2×C3×D4C2311yescharacteristic
61C22×C3×C4—21no—
62C22×C3×C4C2311yescharacteristic
63C2×#67—21no—
64C2×#67—22no—
65C22×Dic3D411yescharacteristic
66#235—22no—
67#235—22no—
68C2×D12—22no—
69C2×D12C2311yescharacteristic
70C2×C4×D3—22no—
71C2×C4×D3—22no—
72C2×C4×D3—21no—
73C2×C4×D3D412yes—
74C2×C4×D3C2312yes—
75C2×Dic6C2311yescharacteristic
76C2×C3×C8—22no—
77C2×C3×C8C2312yes—
78C3×C4⋊C4C2312yes—
79C3×C22⋊C4—21no—
80C3×C42—21no—
81#217—21no—
82#217—22no—
83#216—22no—
84#215—22no—
85#214—22no—
86#213—22no—
87(C2×D3)⋊C4—22no—
88(C2×D3)⋊C4—22no—
89(C2×D3)⋊C4—22no—
90C4⋊Dic3—22no—
91C4⋊Dic3C2312yes—
92Dic3⋊C4—22no—
93C4×Dic3—21no—
94C2×C3⋊C8D414yescomplemented
95C2×C3⋊C8C2311yescharacteristic
96C2×C3⋊C8C2311yescharacteristic
97D3×C8—24no—
98D3×C8—24no—
99C2×#41—31no—
100C2×#41C2×D311yescharacteristic
101C23×C4—31no—
102#135—64no—
103C2×Q16—31no—
104C2×SD16—31no—
105C2×SD16—31no—
106C2×D8—31no—
107C22×C8—31no—
108C22×C8—32no—
109C22×C8—32no—
110C22×C8—64no—
111C22×C8C2×D311yescharacteristic
112#123—61no—
113#123—61no—
114C22⋊Q8—32no—
115C22⋊Q8—32no—
116#121—32no—
117#121—32no—
118#120—61no—
119#120—61no—
120C4×D4—62no—
121C4×D4—62no—
122#117—31no—
123#117C2×D311yescharacteristic
124#107—32no—
125#106—32no—
126#103—32no—
127#103C2×D312yescomplemented
128#102—32no—
129#102C2×D312yescomplemented
130C23×C3—21no—
131C23×C3C2×D411yescharacteristic
132C22×D3—22no—
133C22×D3—24no—
134C22×D3—21no—
135C22×D3C2×D412yes—
136C3×Q8—21no—
137C3×Q8—21no—
138C3×D4—21no—
139C3×D4—21no—
140C3×D4—42no—
141C2×C3×C4—21no—
142C2×C3×C4—22no—
143C2×C3×C4—42no—
144C2×C3×C4C2×D412yes—
145C2×C3×C4C2×D411yescharacteristic
146C2×C3×C4#4112yes—
147C2×C3×C4C2411yescharacteristic
148#67—42no—
149#67—44no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—21no—
153C2×Dic3C2×D412yes—
154D12—21no—
155D12—21no—
156D12—44no—
157C4×D3—24no—
158C4×D3—22no—
159C4×D3—22no—
160C4×D3—42no—
161Dic6—21no—
162Dic6—21no—
163C3×C8—22no—
164C3⋊C8C2×D412yes—
165C3⋊C8C2×D412yes—
166C24—31no—
167#41—22no—
168#41—22no—
169#41—62no—
170#41—62no—
171C2×Q8—31no—
172C2×Q8C22×D311yescharacteristic
173C2×D4—21no—
174C2×D4—31no—
175C2×D4—62no—
176C2×D4—62no—
177C2×D4—61no—
178C2×D4C22×D311yescharacteristic
179C22×C4—21no—
180C22×C4—31no—
181C22×C4—32no—
182C22×C4—32no—
183C22×C4—62no—
184C22×C4—62no—
185C22×C4—61no—
186C22×C4C22×D311yescharacteristic
187Q16—62no—
188SD16—62no—
189SD16—62no—
190D8—62no—
191C2×C8—22no—
192C2×C8—34no—
193C2×C8—31no—
194C2×C8—31no—
195C2×C8—64no—
196C2×C8—64no—
197C2×C8C22×D312yes—
198C4⋊C4—32no—
199C4⋊C4—62no—
200C4⋊C4—62no—
201C4⋊C4C22×D312yes—
202C22⋊C4—21no—
203C22⋊C4—62no—
204C22⋊C4—62no—
205C22⋊C4—62no—
206C22⋊C4—62no—
207C22⋊C4—61no—
208C42—21no—
209C42—61no—
210C22×C3—22no—
211C22×C3—41no—
212C22×C3—42no—
213C22×C3#12112yes—
214C22×C3C22×D411yescharacteristic
215C2×D3—28no—
216C2×D3—22no—
217C2×D3—22no—
218C2×D3—41no—
219C2×D3—42no—
220C2×D3#13514yescomplemented
221C3×C4—41no—
222C3×C4—42no—
223C3×C4#12112yes—
224C3×C4C22×D411yescharacteristic
225C3×C4C2×#4111yescharacteristic
226Dic3—22no—
227Dic3—41no—
228Dic3—42no—
229C23—21no—
230C23—32no—
231C23—64no—
232C23—62no—
233C23—61no—
234C23D3×D411yescharacteristic
235Q8—21no—
236Q8—21no—
237Q8—61no—
238Q8—61no—
239D4—21no—
240D4—21no—
241D4—42no—
242D4—61no—
243D4—61no—
244D4—124no—
245D4—124no—
246D4—122no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—32no—
250C2×C4—42no—
251C2×C4—62no—
252C2×C4—64no—
253C2×C4—62no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4—61no—
257C2×C4—122no—
258C2×C4#23512yes—
259C2×C4D3×D412yes—
260C2×C4D3×D411yescharacteristic
261C2×C4C23×D311yescharacteristic
262C8—22no—
263C8—32no—
264C8—32no—
265C2×C3—22no—
266C2×C3—81no—
267C2×C3C2×#12111yescharacteristic
268C2×C3C2×#13512yes—
269D3—24no—
270D3—81no—
271C22—22no—
272C22—34no—
273C22—41no—
274C22—42no—
275C22—68no—
276C22—62no—
277C22—62no—
278C22—121no—
279C22—122no—
280C22#88412yes—
281C22C2×D3×D411yescharacteristic
282C4—41no—
283C4—42no—
284C4—62no—
285C4—121no—
286C4—122no—
287C4#88412yes—
288C4C2×#23511yescharacteristic
289C4C2×D3×D411yescharacteristic
290C3#305011yescharacteristic, complemented, Sylow
291C2—22no—
292C2—64no—
293C2—81no—
294C2—241no—
295C2C2×#88411yescharacteristic
296C2D3×#13512yes—
subgroupG/H
centerC2×C4D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#416C2
socleC22×C3C22×D4
hypercenter#416D3
hypocenterC3#3050
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #5828C2semidirect
2. C2×#950C2semidirect
3. C22×D3×C8C2semidirect
4. #5524C2semidirect
5. C3×#416C2semidirect
6. #5113C2semidirect
7. #4998C2semidirect
8. #4813C2semidirect
9. #4807C2semidirect
10. #4778C2semidirect
11. #4775C2semidirect
20. C2×#216C22semidirect
21. C2×#215C22semidirect
25. C2×#214C22semidirect
26. C2×#213C22semidirect
38. C3×#103C22semidirect
39. C3×#102C22semidirect
40. #821C22semidirect
41. #816C22semidirect
42. #808C22semidirect
43. #807C22semidirect
54. #416D3semidirect
94. C2×C3⋊C8D4semidirect
127. #103C2×D3semidirect
129. #102C2×D3semidirect
220. C2×D3#135semidirect
290. C3#3050semidirect

Maximal subgroups

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

subgroupindex
#58282
C2×#9502
C22×D3×C82
#55242
C3×#4162
#51132
#49982
#48132
#48072
#47782
#47752
#30503