Finite Group Databasebeta

← #80714

Subgroups — #80714

Order 384.

520 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 133 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×#912C211yesmaximal, characteristic, complemented
2C22×D3×C8C211yesmaximal, characteristic, complemented
3#5606C212yesmaximal, complemented
4C3×#466C211yesmaximal, characteristic, complemented
5#5157C211yesmaximal, characteristic, complemented
6#5116C211yesmaximal, characteristic, complemented
7#5041C212yesmaximal, complemented
8#4887C211yesmaximal, characteristic, complemented
9#4880C211yesmaximal, characteristic, complemented
10#4778C212yesmaximal, complemented
11#4772C212yesmaximal, complemented
12#3050—31nomaximal, Sylow
13C2×#237C2212yes—
14C22×C4×D3C2211yescharacteristic
15C2×C3×D8C2211yescharacteristic, complemented
16C22×C3×C8C2211yescharacteristic
17C3×#121C2212yes—
18#938C2212yes—
19#937—22no—
20D4×Dic3—22no—
21C2×#214C2212yescomplemented
22#924C2212yes—
23C22×C3⋊C8C2211yescharacteristic
24#912—24no—
25C2×Dic12C2211yescharacteristic, complemented
26C2×D3×C8—24no—
27C2×D3×C8C2211yescharacteristic
28C2×D3×C8C2211yescharacteristic
29C2×D3×C8C2211yescharacteristic
30C2×D3×C8C2211yescharacteristic
31#897C2212yes—
32#892C2212yes—
33#883—22no—
34#67⋊C4—22no—
35C3×#107C2211yescharacteristic, complemented
36C3×#102C2212yescomplemented
37#832C2212yescomplemented
38#818C2211yescharacteristic, complemented
39#816C2212yescomplemented
40#810C2212yescomplemented
41#808C2212yescomplemented
42C2×#135—31no—
43C23×C8—31no—
44#525—32no—
45#467—31no—
46#467—31no—
47#466—31no—
48#466D311yescharacteristic, complemented
49#465—32no—
50#465—32no—
51#427—31no—
52#416—32no—
53C23×D3D411yescharacteristic
54C2×C3×D4—22no—
55C2×C3×D4C2312yes—
56C22×C3×C4C2311yescharacteristic
57C2×#67—22no—
58C2×#67—22no—
59C22×Dic3—22no—
60C22×Dic3D411yescharacteristic
61#237—24no—
62#237—24no—
63C2×C4×D3—22no—
64C2×C4×D3—22no—
65C2×C4×D3D411yescharacteristic
66C2×C4×D3D411yescharacteristic
67C2×C4×D3C2311yescharacteristic
68C2×C4×D3C2311yescharacteristic
69C2×Dic6C2312yes—
70C3×D8—22no—
71C2×C3×C8—22no—
72C2×C3×C8C2311yescharacteristic
73C2×C3×C8C2311yescharacteristic
74C3×C4⋊C4C2312yes—
75C3×C22⋊C4—22no—
76#217—22no—
77#217—22no—
78#214—24no—
79(C2×D3)⋊C4—22no—
80(C2×D3)⋊C4—22no—
81C4⋊Dic3—22no—
82C4⋊Dic3C2312yes—
83Dic3⋊C4—22no—
84C4×Dic3—22no—
85C2×C3⋊C8—22no—
86C2×C3⋊C8C2311yescharacteristic
87C2×C3⋊C8C2311yescharacteristic
88Dic12—22no—
89D3×C8—22no—
90D3×C8—22no—
91D3×C8—22no—
92D3×C8D414yescomplemented
93C2×#41—32no—
94C23×C4—31no—
95#135—64no—
96C2×Q16—31no—
97C2×SD16—32no—
98C2×D8C2×D311yescharacteristic, complemented
99C22×C8—31no—
100C22×C8—31no—
101C22×C8—31no—
102C22×C8—31no—
103C22×C8—31no—
104C22×C8—64no—
105C22×C8C2×D311yescharacteristic
106#123—62no—
107C22⋊Q8—32no—
108C22⋊Q8—32no—
109#121—32no—
110#121C2×D312yes—
111#120—62no—
112C4×D4—62no—
113C4×D4—62no—
114#117—32no—
115#107—31no—
116#107C2×D311yescharacteristic, complemented
117#106—32no—
118#103—32no—
119#103—32no—
120#102—32no—
121#102C2×D312yescomplemented
122C23×C3—22no—
123C23×C3C2×D411yescharacteristic
124C22×D3—22no—
125C22×D3—22no—
126C22×D3—22no—
127C22×D3C2×D411yescharacteristic
128C22×D3C2×D411yescharacteristic
129C3×D4—22no—
130C3×D4—22no—
131C3×D4—44no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—22no—
135C2×C3×C4C2×D411yescharacteristic
136C2×C3×C4C2411yescharacteristic
137#67—44no—
138#67—44no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—44no—
143C2×Dic3C2×D411yescharacteristic
144C2×Dic3C2×D411yescharacteristic
145C4×D3—22no—
146C4×D3—21no—
147C4×D3—21no—
148C4×D3—22no—
149C4×D3C2×D412yes—
150C4×D3#4112yes—
151Dic6—22no—
152Dic6—22no—
153C3×C8—21no—
154C3×C8C2×D412yes—
155C3⋊C8—21no—
156C3⋊C8C2×D412yes—
157C24—31no—
158#41—64no—
159#41—64no—
160C2×Q8—32no—
161C2×D4—22no—
162C2×D4—62no—
163C2×D4—62no—
164C2×D4C22×D312yes—
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—62no—
171C22×C4—62no—
172C22×C4—62no—
173C22×C4C22×D311yescharacteristic
174Q16—62no—
175SD16—64no—
176D8—22no—
177C2×C8—22no—
178C2×C8—31no—
179C2×C8—34no—
180C2×C8—31no—
181C2×C8—62no—
182C2×C8—62no—
183C2×C8—62no—
184C2×C8—62no—
185C2×C8C22×D311yescharacteristic
186C2×C8C22×D311yescharacteristic
187C4⋊C4—32no—
188C4⋊C4—62no—
189C4⋊C4—62no—
190C4⋊C4C22×D312yes—
191C22⋊C4—22no—
192C22⋊C4—62no—
193C22⋊C4—62no—
194C22⋊C4—62no—
195C22⋊C4—62no—
196C42—62no—
197C22×C3—21no—
198C22×C3—21no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3—42no—
202C22×C3#13512yescomplemented
203C22×C3C22×D411yescharacteristic
204C2×D3—28no—
205C2×D3—21no—
206C2×D3—21no—
207C2×D3—21no—
208C2×D3—21no—
209C2×D3#12112yes—
210C2×D3#13512yescomplemented
211C3×C4—21no—
212C3×C4—42no—
213C3×C4C22×D411yescharacteristic
214C3×C4C2×#4111yescharacteristic
215Dic3—21no—
216Dic3—42no—
217Dic3—42no—
218Dic3#12112yes—
219C23—22no—
220C23—31no—
221C23—31no—
222C23—62no—
223C23—62no—
224C23—62no—
225C23D3×D411yescharacteristic
226Q8—62no—
227Q8—62no—
228D4—22no—
229D4—22no—
230D4—44no—
231D4—124no—
232D4—124no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—31no—
237C2×C4—32no—
238C2×C4—32no—
239C2×C4—31no—
240C2×C4—62no—
241C2×C4—62no—
242C2×C4—62no—
243C2×C4—61no—
244C2×C4—61no—
245C2×C4—62no—
246C2×C4—62no—
247C2×C4—124no—
248C2×C4D3×D411yescharacteristic
249C2×C4C23×D311yescharacteristic
250C8—21no—
251C8—32no—
252C8—61no—
253C8D3×D412yes—
254C2×C3—22no—
255C2×C3—82no—
256C2×C3C2×#12111yescharacteristic
257C2×C3C2×#13511yescharacteristic
258C2×C3C2×#13511yescharacteristic
259D3—22no—
260D3—22no—
261C22—21no—
262C22—21no—
263C22—32no—
264C22—32no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—68no—
269C22—61no—
270C22—61no—
271C22—61no—
272C22—61no—
273C22#91212yescomplemented
274C22C2×D3×D411yescharacteristic
275C4—21no—
276C4—32no—
277C4—42no—
278C4—61no—
279C4—122no—
280C4—122no—
281C4C2×D3×D411yescharacteristic
282C4D3×#4111yescharacteristic
283C3#305011yescharacteristic, complemented, Sylow
284C2—22no—
285C2—62no—
286C2—62no—
287C2—82no—
288C2D3×#12111yescharacteristic
289C2C2×#91211yescharacteristic
290C2D3×#13511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#466C2
socleC22×C3C22×D4
hypercenter#466D3
hypocenterC3#3050
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#912C2semidirect
2. C22×D3×C8C2semidirect
3. #5606C2semidirect
4. C3×#466C2semidirect
5. #5157C2semidirect
6. #5116C2semidirect
7. #5041C2semidirect
8. #4887C2semidirect
9. #4880C2semidirect
10. #4778C2semidirect
11. #4772C2semidirect
15. C2×C3×D8C22semidirect
21. C2×#214C22semidirect
25. C2×Dic12C22semidirect
35. C3×#107C22semidirect
36. C3×#102C22semidirect
37. #832C22semidirect
38. #818C22semidirect
39. #816C22semidirect
40. #810C22semidirect
41. #808C22semidirect
48. #466D3semidirect
92. D3×C8D4semidirect
98. C2×D8C2×D3semidirect
116. #107C2×D3semidirect
121. #102C2×D3semidirect
202. C22×C3#135semidirect
210. C2×D3#135semidirect
273. C22#912semidirect
283. C3#3050semidirect

Maximal subgroups

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

subgroupindex
C2×#9122
C22×D3×C82
#56062
C3×#4662
#51572
#51162
#50412
#48872
#48802
#47782
#47722
#30503