Finite Group Databasebeta

← #80002

Subgroups — #80002

Order 384.

994 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 143 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×D3×D4C211yesmaximal, characteristic, complemented
2C2×#911C212yesmaximal, complemented
3#5606C211yesmaximal, characteristic, complemented
4C3×#447C211yesmaximal, characteristic, complemented
5#5218C211yesmaximal, characteristic, complemented
6#5156C212yesmaximal, complemented
7#5036C211yesmaximal, characteristic, complemented
8#4776C212yesmaximal, complemented
9#4770C212yesmaximal, complemented
10#4731C211yesmaximal, characteristic, complemented
11#4726C211yesmaximal, characteristic, complemented
12#3003—31nomaximal, Sylow
13C24×D3—21no—
14C22×C3×D4C2211yescharacteristic
15C22×#67—21no—
16C2×#237C2211yescharacteristic, complemented
17C2×D3×D4—28no—
18C2×D3×D4—21no—
19C2×D3×D4—21no—
20C2×D3×D4C2212yes—
21C2×D3×D4C2212yes—
22C22×D12C2211yescharacteristic
23C22×C4×D3C2211yescharacteristic
24C2×C3×D8C2212yescomplemented
25C3×#121C2211yescharacteristic, complemented
26#938C2211yescharacteristic, complemented
27#937—21no—
28D4×Dic3—21no—
29C2×#214C2212yescomplemented
30C2×#213C2212yescomplemented
31#924C2211yescharacteristic, complemented
32#911—28no—
33C2×#204C2212yescomplemented
34C2×ΓL(1,𝔽25)C2212yescomplemented
35#897C2211yescharacteristic, complemented
36#892C2211yescharacteristic, complemented
37#883—21no—
38#67⋊C4—21no—
39C3×#102C2212yescomplemented
40C3×C22⋊C8C2211yescharacteristic, complemented
41#832C2212yescomplemented
42#830C2211yescharacteristic, complemented
43#821C2212yescomplemented
44(C2×D3)⋊C8C2212yescomplemented
45#809C2212yescomplemented
46C23×D4—31no—
47C2×AGL(1,ℤ/8)—32no—
48#525—31no—
49#450—31no—
50#450—32no—
51#450—31no—
52#447—32no—
53#447—31no—
54#447D311yescharacteristic, complemented
55#418—32no—
56#407—31no—
57C24×C3—21no—
58C23×D3—28no—
59C23×D3—21no—
60C23×D3—22no—
61C23×D3—22no—
62C23×D3—44no—
63C23×D3—44no—
64C23×D3D411yescharacteristic
65C2×C3×D4—24no—
66C2×C3×D4—21no—
67C2×C3×D4—21no—
68C2×C3×D4C2312yes—
69C2×C3×D4C2311yescharacteristic, complemented
70C22×C3×C4C2311yescharacteristic
71C2×#67—28no—
72C2×#67—21no—
73C2×#67—21no—
74C2×#67—22no—
75C2×#67—22no—
76C22×Dic3—21no—
77C22×Dic3D411yescharacteristic
78#237—24no—
79D3×D4—28no—
80D3×D4—28no—
81D3×D4—24no—
82D3×D4—24no—
83D3×D4—24no—
84D3×D4D418yescomplemented
85C2×D12—21no—
86C2×D12—24no—
87C2×D12C2312yes—
88C2×C4×D3—24no—
89C2×C4×D3D411yescharacteristic
90C2×C4×D3D411yescharacteristic
91C2×C4×D3C2311yescharacteristic
92C2×C4×D3C2311yescharacteristic
93C2×Dic6C2311yescharacteristic, complemented
94C3×D8—24no—
95C2×C3×C8C2312yescomplemented
96C3×C4⋊C4C2311yescharacteristic, complemented
97C3×C22⋊C4—21no—
98#217—21no—
99#217—21no—
100#214—24no—
101#213—24no—
102(C2×D3)⋊C4—21no—
103(C2×D3)⋊C4—21no—
104C4⋊Dic3—21no—
105C4⋊Dic3C2311yescharacteristic, complemented
106Dic3⋊C4—21no—
107C4×Dic3—21no—
108C2×C3⋊C8C2312yescomplemented
109#204—24no—
110ΓL(1,𝔽25)—24no—
111C25—61no—
112C2×#41—31no—
113C22×D4—32no—
114C22×D4—32no—
115C22×D4—31no—
116C22×D4—61no—
117C22×D4—61no—
118C22×D4—61no—
119C22×D4—68no—
120C22×D4C2×D311yescharacteristic
121C23×C4—31no—
122AGL(1,ℤ/8)—68no—
123C2×SD16—32no—
124C2×SD16—32no—
125C2×D8—32no—
126C2×D8C2×D312yescomplemented
127C2×M16—32no—
128#123—61no—
129C22⋊Q8—31no—
130C22⋊Q8—31no—
131#121—31no—
132#121C2×D311yescharacteristic, complemented
133#120—61no—
134C4×D4—61no—
135C4×D4—61no—
136#117—31no—
137#102—32no—
138#102—32no—
139#102—32no—
140#102C2×D312yescomplemented
141C22⋊C8—31no—
142C22⋊C8—32no—
143C22⋊C8C2×D311yescharacteristic, complemented
144C23×C3—24no—
145C23×C3—22no—
146C23×C3—21no—
147C23×C3—42no—
148C23×C3—42no—
149C23×C3C2×D411yescharacteristic
150C22×D3—22no—
151C22×D3—22no—
152C22×D3—22no—
153C22×D3—24no—
154C22×D3—28no—
155C22×D3—22no—
156C22×D3—28no—
157C22×D3—432no—
158C22×D3—44no—
159C22×D3—44no—
160C22×D3—44no—
161C22×D3—42no—
162C22×D3—42no—
163C22×D3—44no—
164C22×D3C2×D411yescharacteristic
165C22×D3C2×D411yescharacteristic
166C3×D4—24no—
167C3×D4—22no—
168C3×D4—22no—
169C3×D4—42no—
170C3×D4C2×D414yes—
171C2×C3×C4—22no—
172C2×C3×C4—21no—
173C2×C3×C4C2×D411yescharacteristic
174C2×C3×C4C2411yescharacteristic
175#67—28no—
176#67—28no—
177#67—42no—
178#67—42no—
179C2×Dic3—22no—
180C2×Dic3—21no—
181C2×Dic3—21no—
182C2×Dic3—42no—
183C2×Dic3C2×D411yescharacteristic
184C2×Dic3C2×D411yescharacteristic
185D12—24no—
186D12—22no—
187D12C2×D414yes—
188C4×D3—22no—
189C4×D3—22no—
190C4×D3—22no—
191C4×D3C2×D414yes—
192Dic6—22no—
193C3×C8—22no—
194C3⋊C8—22no—
195C24—21no—
196C24—31no—
197C24—62no—
198C24—62no—
199C24—68no—
200C24—61no—
201C24—124no—
202C24—124no—
203#41—64no—
204C2×Q8—31no—
205C2×D4—24no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—38no—
209C2×D4—32no—
210C2×D4—64no—
211C2×D4—64no—
212C2×D4—62no—
213C2×D4—62no—
214C2×D4—64no—
215C2×D4—68no—
216C2×D4—68no—
217C2×D4—68no—
218C2×D4—64no—
219C2×D4—61no—
220C2×D4—61no—
221C2×D4—61no—
222C2×D4C22×D312yes—
223C2×D4C22×D311yescharacteristic, complemented
224C22×C4—31no—
225C22×C4—31no—
226C22×C4—31no—
227C22×C4—31no—
228C22×C4—31no—
229C22×C4—64no—
230C22×C4—61no—
231C22×C4C22×D311yescharacteristic
232SD16—64no—
233SD16—64no—
234D8—24no—
235D8—64no—
236M16—64no—
237C2×C8—32no—
238C2×C8C22×D312yescomplemented
239C4⋊C4—31no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4C22×D311yescharacteristic, complemented
243C22⋊C4—21no—
244C22⋊C4—61no—
245C22⋊C4—61no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C42—61no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—24no—
252C22×C3—48no—
253C22×C3—42no—
254C22×C3—42no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3—41no—
258C22×C3AGL(1,ℤ/8)12yescomplemented
259C22×C3C22×D411yescharacteristic
260C2×D3—28no—
261C2×D3—21no—
262C2×D3—21no—
263C2×D3—21no—
264C2×D3—21no—
265C2×D3—24no—
266C2×D3—416no—
267C2×D3—416no—
268C2×D3—48no—
269C2×D3—42no—
270C2×D3—42no—
271C2×D3#12012yes—
272C2×D3AGL(1,ℤ/8)12yescomplemented
273C3×C4—21no—
274C3×C4—41no—
275C3×C4C22×D412yes—
276Dic3—21no—
277Dic3—41no—
278Dic3—41no—
279Dic3#12012yes—
280C23—24no—
281C23—22no—
282C23—21no—
283C23—31no—
284C23—31no—
285C23—42no—
286C23—42no—
287C23—62no—
288C23—62no—
289C23—68no—
290C23—64no—
291C23—68no—
292C23—62no—
293C23—62no—
294C23—122no—
295C23—124no—
296C23—1232no—
297C23—124no—
298C23—124no—
299C23—124no—
300C23—122no—
301C23D3×D411yescharacteristic
302Q8—62no—
303D4—24no—
304D4—22no—
305D4—22no—
306D4—34no—
307D4—42no—
308D4—64no—
309D4—68no—
310D4—68no—
311D4—62no—
312D4—122no—
313D4—122no—
314D4D3×D414yes—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—31no—
318C2×C4—34no—
319C2×C4—31no—
320C2×C4—62no—
321C2×C4—62no—
322C2×C4—62no—
323C2×C4—62no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—122no—
327C2×C4D3×D411yescharacteristic
328C2×C4C23×D311yescharacteristic
329C8—22no—
330C8—62no—
331C2×C3—22no—
332C2×C3—44no—
333C2×C3—81no—
334C2×C3C2×#12011yescharacteristic
335C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
336C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
337D3—22no—
338D3—22no—
339D3—44no—
340C22—21no—
341C22—21no—
342C22—24no—
343C22—32no—
344C22—32no—
345C22—48no—
346C22—42no—
347C22—42no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—68no—
352C22—61no—
353C22—61no—
354C22—61no—
355C22—61no—
356C22—64no—
357C22—128no—
358C22—1216no—
359C22—1216no—
360C22—122no—
361C22—122no—
362C22#91112yescomplemented
363C22C2×D3×D411yescharacteristic
364C4—21no—
365C4—32no—
366C4—41no—
367C4—61no—
368C4—121no—
369C4—121no—
370C4C2×D3×D412yes—
371C3#300311yescharacteristic, complemented, Sylow
372C2—22no—
373C2—44no—
374C2—62no—
375C2—62no—
376C2—81no—
377C2—124no—
378C2D3×#12011yescharacteristic
379C2C2×#91111yescharacteristic
380C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#447C2
socleC22×C3C22×D4
hypercenter#447D3
hypocenterC3#3003
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C22×D3×D4C2semidirect
2. C2×#911C2semidirect
3. #5606C2semidirect
4. C3×#447C2semidirect
5. #5218C2semidirect
6. #5156C2semidirect
7. #5036C2semidirect
8. #4776C2semidirect
9. #4770C2semidirect
10. #4731C2semidirect
11. #4726C2semidirect
16. C2×#237C22semidirect
24. C2×C3×D8C22semidirect
25. C3×#121C22semidirect
26. #938C22semidirect
29. C2×#214C22semidirect
30. C2×#213C22semidirect
31. #924C22semidirect
33. C2×#204C22semidirect
34. C2×ΓL(1,𝔽25)C22semidirect
35. #897C22semidirect
36. #892C22semidirect
39. C3×#102C22semidirect
40. C3×C22⋊C8C22semidirect
41. #832C22semidirect
42. #830C22semidirect
43. #821C22semidirect
44. (C2×D3)⋊C8C22semidirect
45. #809C22semidirect
54. #447D3semidirect
69. C2×C3×D4C23semidirect
84. D3×D4D4semidirect
93. C2×Dic6C23semidirect
95. C2×C3×C8C23semidirect
96. C3×C4⋊C4C23semidirect
105. C4⋊Dic3C23semidirect
108. C2×C3⋊C8C23semidirect
126. C2×D8C2×D3semidirect
132. #121C2×D3semidirect
140. #102C2×D3semidirect
143. C22⋊C8C2×D3semidirect
223. C2×D4C22×D3semidirect
238. C2×C8C22×D3semidirect
242. C4⋊C4C22×D3semidirect
258. C22×C3AGL(1,ℤ/8)semidirect
272. C2×D3AGL(1,ℤ/8)semidirect
362. C22#911semidirect
371. C3#3003semidirect

Maximal subgroups

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

subgroupindex
C22×D3×D42
C2×#9112
#56062
C3×#4472
#52182
#51562
#50362
#47762
#47702
#47312
#47262
#30033