Finite Group Databasebeta

← #79217

Subgroups — #79217

Order 384.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 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
1D3×#117C211yesmaximal, characteristic, complemented
2C2×#895C211yesmaximal, characteristic
3C3×#428C211yesmaximal, characteristic, complemented
4#5131C211yesmaximal, characteristic, complemented
5#5126C211yesmaximal, characteristic, complemented
6#5001C211yesmaximal, characteristic, complemented
7#4964C211yesmaximal, characteristic, complemented
8#4885C212yesmaximal, complemented
9#4882C212yesmaximal, complemented
10#4866C212yesmaximal, complemented
11#4863C212yesmaximal, complemented
12#3094—31nomaximal, Sylow
13C22×D12—21no—
14C22×D12C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C2×C3×M16C2211yescharacteristic
17C3×#117C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×(C2×D3)⋊C4—21no—
20#926C2211yescharacteristic
21C2×#208C2211yescharacteristic
22#895—21no—
23#895—21no—
24#895C2212yes—
25#895C2212yes—
26#892C2211yescharacteristic, complemented
27#892C2211yescharacteristic, complemented
28D3×C4⋊C4C2211yescharacteristic, complemented
29D3×C4⋊C4C2211yescharacteristic, complemented
30D3×C22⋊C4—21no—
31#877—21no—
32#872—21no—
33C42×D3—21no—
34C3×#107C2212yescomplemented
35C3×#106C2212yescomplemented
36#821C2212yescomplemented
37#821C2212yescomplemented
38(C2×D3)⋊C8C2212yescomplemented
39(C2×D3)⋊C8C2212yescomplemented
40#809C2212yescomplemented
41#809C2212yescomplemented
42#808C2212yescomplemented
43#807C2212yescomplemented
44C2×#121—31no—
45C2×#117—31no—
46#482—32no—
47#482—32no—
48#481—32no—
49#480—32no—
50#428—31no—
51#428D311yescharacteristic, complemented
52#418—31no—
53#418—31no—
54#407—31no—
55C23×D3—21no—
56C23×D3D411yescharacteristic
57C22×C3×C4—21no—
58C22×C3×C4C2311yescharacteristic
59C22×Dic3D411yescharacteristic
60C2×D12—22no—
61C2×D12—22no—
62C2×D12—22no—
63C2×D12—22no—
64C2×D12—21no—
65C2×D12—44no—
66C2×D12C2312yes—
67C2×C4×D3—22no—
68C2×C4×D3—22no—
69C2×C4×D3—21no—
70C2×C4×D3—21no—
71C2×C4×D3D411yescharacteristic
72C2×C4×D3D411yescharacteristic
73C2×C4×D3C2311yescharacteristic
74C2×C4×D3C2311yescharacteristic
75C3×M16—22no—
76C2×C3×C8C2312yes—
77C3×C4⋊C4—21no—
78C3×C4⋊C4C2312yes—
79C3×C4⋊C4C2311yescharacteristic
80C3×C4⋊C4C2311yescharacteristic
81C3×C22⋊C4—21no—
82C3×C42—21no—
83#217—21no—
84(C2×D3)⋊C4—21no—
85(C2×D3)⋊C4—21no—
86(C2×D3)⋊C4—22no—
87(C2×D3)⋊C4—22no—
88C4⋊Dic3C2311yescharacteristic
89C4⋊Dic3C2311yescharacteristic
90Dic3⋊C4—21no—
91Dic3⋊C4—21no—
92C4×Dic3—21no—
93C4×Dic3—21no—
94C4×Dic3—21no—
95#208—22no—
96C2×C3⋊C8C2312yes—
97C22×D4—31no—
98C22×D4—61no—
99C23×C4—31no—
100C2×M16—31no—
101C2×M16C2×D311yescharacteristic
102#121—32no—
103#121—32no—
104#121—61no—
105#121—61no—
106#117—31no—
107#117—31no—
108#117—31no—
109#117—61no—
110#117—61no—
111#117C2×D311yescharacteristic
112C2×C4⋊C4—31no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4C2×D311yescharacteristic
115C2×C22⋊C4—61no—
116C2×C22⋊C4—61no—
117C2×C42—61no—
118#107—32no—
119#107C2×D312yescomplemented
120#106—32no—
121#106C2×D312yescomplemented
122#102—32no—
123#102—32no—
124#102—32no—
125#102—32no—
126C22⋊C8—32no—
127C22⋊C8—32no—
128C23×C3C2×D411yescharacteristic
129C22×D3—22no—
130C22×D3—22no—
131C22×D3—22no—
132C22×D3—41no—
133C22×D3—42no—
134C22×D3—42no—
135C22×D3—42no—
136C22×D3C2×D411yescharacteristic
137C22×D3C2×D411yescharacteristic
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—22no—
141C2×C3×C4—42no—
142C2×C3×C4C2×D411yescharacteristic
143C2×C3×C4#4112yes—
144C2×C3×C4#4112yes—
145C2×C3×C4C2411yescharacteristic
146C2×Dic3—22no—
147C2×Dic3—21no—
148C2×Dic3—21no—
149C2×Dic3C2×D411yescharacteristic
150C2×Dic3C2×D411yescharacteristic
151D12—22no—
152D12—22no—
153D12—44no—
154D12—44no—
155D12—42no—
156C4×D3—22no—
157C4×D3—22no—
158C4×D3—21no—
159C4×D3—21no—
160C4×D3—21no—
161C4×D3—21no—
162C4×D3—42no—
163C4×D3—42no—
164C3×C8—22no—
165C3⋊C8—22no—
166C24—31no—
167C24—61no—
168C2×D4—32no—
169C2×D4—62no—
170C2×D4—62no—
171C2×D4—62no—
172C2×D4—61no—
173C2×D4—62no—
174C2×D4—124no—
175C22×C4—21no—
176C22×C4—31no—
177C22×C4—31no—
178C22×C4—31no—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—61no—
182C22×C4—61no—
183C22×C4—62no—
184C22×C4—62no—
185C22×C4C22×D311yescharacteristic
186M16—22no—
187M16—62no—
188C2×C8—32no—
189C2×C8C22×D312yes—
190C4⋊C4—21no—
191C4⋊C4—31no—
192C4⋊C4—31no—
193C4⋊C4—61no—
194C4⋊C4—61no—
195C4⋊C4C22×D311yescharacteristic
196C4⋊C4C22×D311yescharacteristic
197C4⋊C4C22×D312yes—
198C22⋊C4—21no—
199C22⋊C4—62no—
200C22⋊C4—62no—
201C22⋊C4—61no—
202C22⋊C4—61no—
203C22⋊C4—61no—
204C42—21no—
205C42—61no—
206C42—61no—
207C42—61no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3#12312yes—
211C22×C3C22×D411yescharacteristic
212C2×D3—21no—
213C2×D3—21no—
214C2×D3—21no—
215C2×D3—21no—
216C2×D3—21no—
217C2×D3—44no—
218C2×D3—42no—
219C2×D3—42no—
220C2×D3—42no—
221C2×D3—84no—
222C2×D3AGL(1,ℤ/8)12yes—
223C3×C4—41no—
224C3×C4—42no—
225C3×C4—41no—
226C3×C4#12312yes—
227C3×C4C2×#4111yescharacteristic
228C3×C4C2×#4111yescharacteristic
229Dic3—21no—
230Dic3—21no—
231Dic3—41no—
232Dic3—41no—
233C23—31no—
234C23—31no—
235C23—62no—
236C23—62no—
237C23—62no—
238C23—121no—
239C23—122no—
240C23—122no—
241C23—122no—
242C23D3×D411yescharacteristic
243D4—62no—
244D4—62no—
245D4—124no—
246D4—124no—
247D4—122no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—31no—
252C2×C4—31no—
253C2×C4—42no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—61no—
258C2×C4—61no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—61no—
262C2×C4—61no—
263C2×C4—122no—
264C2×C4—122no—
265C2×C4#23512yes—
266C2×C4D3×D411yescharacteristic
267C2×C4#23912yes—
268C2×C4C23×D311yescharacteristic
269C8—22no—
270C8—62no—
271C2×C3—22no—
272C2×C3C2×#12311yescharacteristic
273C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
274C2×C3#57511yescharacteristic
275D3—22no—
276D3—41no—
277D3—82no—
278C22—21no—
279C22—21no—
280C22—32no—
281C22—61no—
282C22—61no—
283C22—61no—
284C22—61no—
285C22—61no—
286C22—124no—
287C22—122no—
288C22—122no—
289C22—122no—
290C22—244no—
291C22#89412yes—
292C22C2×D3×D411yescharacteristic
293C4—42no—
294C4—41no—
295C4—41no—
296C4—61no—
297C4—61no—
298C4—121no—
299C4—121no—
300C4#89412yes—
301C4C2×#23511yescharacteristic
302C4C2×#23911yescharacteristic
303C3#309411yescharacteristic, complemented, Sylow
304C2—22no—
305C2—62no—
306C2—121no—
307C2—242no—
308C2C2×#89411yescharacteristic
309C2D3×AGL(1,ℤ/8)11yescharacteristic
310C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#428C2
socleC22×C3C22×D4
prime radicalC2×#895C2
hypercenter#428D3
hypocenterC3#3094
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. D3×#117C2semidirect
3. C3×#428C2semidirect
4. #5131C2semidirect
5. #5126C2semidirect
6. #5001C2semidirect
7. #4964C2semidirect
8. #4885C2semidirect
9. #4882C2semidirect
10. #4866C2semidirect
11. #4863C2semidirect
26. #892C22semidirect
27. #892C22semidirect
28. D3×C4⋊C4C22semidirect
29. D3×C4⋊C4C22semidirect
34. C3×#107C22semidirect
35. C3×#106C22semidirect
36. #821C22semidirect
37. #821C22semidirect
38. (C2×D3)⋊C8C22semidirect
39. (C2×D3)⋊C8C22semidirect
40. #809C22semidirect
41. #809C22semidirect
42. #808C22semidirect
43. #807C22semidirect
51. #428D3semidirect
119. #107C2×D3semidirect
121. #106C2×D3semidirect
303. C3#3094semidirect

Maximal subgroups

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

subgroupindex
D3×#1172
C2×#8952
C3×#4282
#51312
#51262
#50012
#49642
#48852
#48822
#48662
#48632
#30943