Finite Group Databasebeta

← #80184

Subgroups — #80184

Order 384.

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

normal subgroups only

This group has 754 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
1C22×#239C211yesmaximal, characteristic
2C2×#918C212yesmaximal, complemented
3D3×C22⋊Q8C211yesmaximal, characteristic, complemented
4C3×#451C211yesmaximal, characteristic, complemented
5#5228C211yesmaximal, characteristic, complemented
6#5187C212yesmaximal, complemented
7#5046C211yesmaximal, characteristic, complemented
8#4808C212yesmaximal, complemented
9#4803C212yesmaximal, complemented
10#4731C211yesmaximal, characteristic, complemented
11#4726C211yesmaximal, characteristic, complemented
12#3007—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C2×#239—28no—
15C2×#239—21no—
16C2×#239—21no—
17C2×#239C2212yes—
18C2×#239C2212yes—
19C2×D3×Q8C2211yescharacteristic, complemented
20C22×D12—21no—
21C22×D12C2211yescharacteristic
22C22×C4×D3—21no—
23C22×C4×D3C2211yescharacteristic
24C2×C3×Q16C2212yescomplemented
25C3×C22⋊Q8C2211yescharacteristic
26#946C2211yescharacteristic, complemented
27C2×#216C2212yescomplemented
28C2×#215C2212yescomplemented
29#924C2211yescharacteristic
30#918—28no—
31C2×#204C2212yescomplemented
32C2×ΓL(1,𝔽25)C2212yescomplemented
33#897C2211yescharacteristic, complemented
34#896—21no—
35D3×C4⋊C4—21no—
36D3×C4⋊C4C2211yescharacteristic, complemented
37D3×C22⋊C4—21no—
38#878—21no—
39C3×#103C2212yescomplemented
40C3×C22⋊C8C2211yescharacteristic
41#835C2212yescomplemented
42#830C2211yescharacteristic
43#821C2212yescomplemented
44(C2×D3)⋊C8C2212yescomplemented
45#809C2212yescomplemented
46C22×#41—31no—
47C2×#137—32no—
48C2×C22⋊Q8—31no—
49#452—32no—
50#452—32no—
51#451—31no—
52#451D311yescharacteristic, complemented
53#450—31no—
54#450—31no—
55#417—32no—
56#407—31no—
57C23×D3—21no—
58C23×D3D411yescharacteristic
59C2×C3×Q8—24no—
60C2×C3×Q8—21no—
61C2×C3×Q8C2311yescharacteristic
62C2×C3×Q8C2312yes—
63C22×C3×C4—21no—
64C22×C3×C4C2311yescharacteristic
65C22×Dic3D411yescharacteristic
66#239—28no—
67#239—28no—
68#239—24no—
69#239—24no—
70#239—24no—
71#239D418yes—
72D3×Q8—24no—
73C2×D12—28no—
74C2×D12—22no—
75C2×D12—22no—
76C2×D12—21no—
77C2×D12—24no—
78C2×D12C2312yes—
79C2×C4×D3—28no—
80C2×C4×D3—22no—
81C2×C4×D3—22no—
82C2×C4×D3—24no—
83C2×C4×D3—21no—
84C2×C4×D3—21no—
85C2×C4×D3D411yescharacteristic
86C2×C4×D3D411yescharacteristic
87C2×C4×D3C2311yescharacteristic
88C2×C4×D3C2311yescharacteristic
89C2×Dic6—21no—
90C2×Dic6C2311yescharacteristic
91C3×Q16—24no—
92C2×C3×C8C2312yes—
93C3×C4⋊C4—21no—
94C3×C4⋊C4C2311yescharacteristic
95C3×C22⋊C4—21no—
96#217—21no—
97#216—24no—
98#215—24no—
99(C2×D3)⋊C4—21no—
100(C2×D3)⋊C4—21no—
101C4⋊Dic3—21no—
102C4⋊Dic3C2311yescharacteristic
103Dic3⋊C4—21no—
104Dic3⋊C4—21no—
105Dic3⋊C4—21no—
106C2×C3⋊C8C2312yes—
107#204—24no—
108ΓL(1,𝔽25)—24no—
109C2×#41—32no—
110C2×#41—32no—
111C2×#41—61no—
112C2×#41—61no—
113C2×#41—68no—
114C22×Q8—31no—
115C22×Q8C2×D311yescharacteristic
116C22×D4—31no—
117C22×D4—61no—
118C23×C4—31no—
119C23×C4—61no—
120#137—68no—
121C2×Q16—32no—
122C2×Q16C2×D312yescomplemented
123C2×SD16—32no—
124C2×SD16—32no—
125C2×M16—32no—
126C22⋊Q8—31no—
127C22⋊Q8—31no—
128C22⋊Q8—31no—
129C22⋊Q8—61no—
130C22⋊Q8—61no—
131C22⋊Q8C2×D311yescharacteristic
132C2×C4⋊C4—31no—
133C2×C4⋊C4—61no—
134C2×C22⋊C4—61no—
135#103—32no—
136#103C2×D312yescomplemented
137#102—32no—
138#102—32no—
139C22⋊C8—31no—
140C22⋊C8—32no—
141C22⋊C8C2×D311yescharacteristic
142C23×C3C2×D411yescharacteristic
143C22×D3—24no—
144C22×D3—22no—
145C22×D3—22no—
146C22×D3—22no—
147C22×D3—41no—
148C22×D3—42no—
149C22×D3—42no—
150C22×D3C2×D411yescharacteristic
151C22×D3C2×D411yescharacteristic
152C3×Q8—24no—
153C3×Q8—22no—
154C3×Q8—22no—
155C3×Q8C2×D414yes—
156C2×C3×C4—24no—
157C2×C3×C4—22no—
158C2×C3×C4—22no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4C2×D411yescharacteristic
162C2×C3×C4C2411yescharacteristic
163C2×Dic3—22no—
164C2×Dic3—21no—
165C2×Dic3—21no—
166C2×Dic3C2×D411yescharacteristic
167C2×Dic3C2×D411yescharacteristic
168D12—28no—
169D12—24no—
170D12—28no—
171D12—22no—
172D12C2×D414yes—
173C4×D3—28no—
174C4×D3—22no—
175C4×D3—22no—
176C4×D3—28no—
177C4×D3—22no—
178C4×D3—42no—
179C4×D3—42no—
180C4×D3C2×D414yes—
181Dic6—22no—
182Dic6—42no—
183C3×C8—22no—
184C3⋊C8—22no—
185C24—31no—
186C24—61no—
187#41—38no—
188#41—64no—
189#41—64no—
190#41—68no—
191#41—68no—
192#41—64no—
193C2×Q8—24no—
194C2×Q8—21no—
195C2×Q8—31no—
196C2×Q8—61no—
197C2×Q8—64no—
198C2×Q8C22×D311yescharacteristic
199C2×Q8C22×D312yes—
200C2×D4—32no—
201C2×D4—64no—
202C2×D4—62no—
203C2×D4—62no—
204C2×D4—68no—
205C2×D4—61no—
206C22×C4—21no—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—31no—
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—64no—
213C22×C4—62no—
214C22×C4—62no—
215C22×C4—68no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4C22×D311yescharacteristic
219Q16—24no—
220Q16—64no—
221SD16—64no—
222SD16—64no—
223M16—64no—
224C2×C8—32no—
225C2×C8C22×D312yes—
226C4⋊C4—21no—
227C4⋊C4—31no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4C22×D311yescharacteristic
233C22⋊C4—21no—
234C22⋊C4—61no—
235C22⋊C4—61no—
236C22⋊C4—61no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3#13712yescomplemented
240C22×C3C22×D411yescharacteristic
241C2×D3—21no—
242C2×D3—21no—
243C2×D3—21no—
244C2×D3—24no—
245C2×D3—21no—
246C2×D3—21no—
247C2×D3—44no—
248C2×D3—48no—
249C2×D3—42no—
250C2×D3—42no—
251C2×D3#12012yes—
252C3×C4—24no—
253C3×C4—21no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4C22×D412yes—
257Dic3—21no—
258Dic3—41no—
259Dic3—41no—
260Dic3#12012yes—
261C23—31no—
262C23—31no—
263C23—62no—
264C23—64no—
265C23—62no—
266C23—62no—
267C23—121no—
268C23—122no—
269C23—122no—
270C23D3×D411yescharacteristic
271Q8—24no—
272Q8—22no—
273Q8—22no—
274Q8—62no—
275Q8—122no—
276Q8D3×D414yes—
277D4—34no—
278D4—68no—
279D4—68no—
280D4—64no—
281D4—62no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—31no—
288C2×C4—34no—
289C2×C4—31no—
290C2×C4—62no—
291C2×C4—68no—
292C2×C4—62no—
293C2×C4—68no—
294C2×C4—62no—
295C2×C4—62no—
296C2×C4—61no—
297C2×C4—61no—
298C2×C4—122no—
299C2×C4—122no—
300C2×C4D3×D411yescharacteristic
301C2×C4C23×D311yescharacteristic
302C8—22no—
303C8—62no—
304C2×C3—22no—
305C2×C3C2×#12011yescharacteristic
306C2×C3C2×#13711yescharacteristic
307C2×C3#57511yescharacteristic
308D3—22no—
309D3—41no—
310D3—44no—
311C22—21no—
312C22—21no—
313C22—32no—
314C22—61no—
315C22—61no—
316C22—61no—
317C22—61no—
318C22—64no—
319C22—61no—
320C22—124no—
321C22—128no—
322C22—122no—
323C22—122no—
324C22#91812yescomplemented
325C22C2×D3×D411yescharacteristic
326C4—24no—
327C4—21no—
328C4—32no—
329C4—41no—
330C4—41no—
331C4—61no—
332C4—121no—
333C4—121no—
334C4C2×D3×D412yes—
335C3#300711yescharacteristic, complemented, Sylow
336C2—22no—
337C2—62no—
338C2—121no—
339C2—124no—
340C2D3×#12011yescharacteristic
341C2C2×#91811yescharacteristic
342C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#451C2
socleC22×C3C22×D4
prime radicalC22×#239C2
hypercenter#451D3
hypocenterC3#3007
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×#918C2semidirect
3. D3×C22⋊Q8C2semidirect
4. C3×#451C2semidirect
5. #5228C2semidirect
6. #5187C2semidirect
7. #5046C2semidirect
8. #4808C2semidirect
9. #4803C2semidirect
10. #4731C2semidirect
11. #4726C2semidirect
19. C2×D3×Q8C22semidirect
24. C2×C3×Q16C22semidirect
26. #946C22semidirect
27. C2×#216C22semidirect
28. C2×#215C22semidirect
31. C2×#204C22semidirect
32. C2×ΓL(1,𝔽25)C22semidirect
33. #897C22semidirect
36. D3×C4⋊C4C22semidirect
39. C3×#103C22semidirect
41. #835C22semidirect
43. #821C22semidirect
44. (C2×D3)⋊C8C22semidirect
45. #809C22semidirect
52. #451D3semidirect
122. C2×Q16C2×D3semidirect
136. #103C2×D3semidirect
239. C22×C3#137semidirect
324. C22#918semidirect
335. C3#3007semidirect

Maximal subgroups

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

subgroupindex
C22×#2392
C2×#9182
D3×C22⋊Q82
C3×#4512
#52282
#51872
#50462
#48082
#48032
#47312
#47262
#30073