Finite Group Databasebeta

← #80003

Subgroups — #80003

Order 384.

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

normal subgroups only

This group has 806 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×#237C211yesmaximal, characteristic, complemented
2C2×#911C212yesmaximal, complemented
3D3×#121C211yesmaximal, characteristic, complemented
4C3×#447C211yesmaximal, characteristic, complemented
5#5217C211yesmaximal, characteristic, complemented
6#5158C212yesmaximal, complemented
7#5040C211yesmaximal, characteristic, complemented
8#4781C212yesmaximal, complemented
9#4771C212yesmaximal, complemented
10#4738C211yesmaximal, characteristic, complemented
11#4726C211yesmaximal, characteristic, complemented
12#3006—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic, complemented
14C22×#67—21no—
15C23×Dic3—21no—
16C2×#237—28no—
17C2×#237—21no—
18C2×#237—21no—
19C2×#237C2212yes—
20C2×#237C2212yescomplemented
21C2×D3×D4—21no—
22C2×D3×D4C2211yescharacteristic, complemented
23C22×C4×D3C2211yescharacteristic
24C22×Dic6C2211yescharacteristic, complemented
25C2×C3×D8C2212yescomplemented
26C3×#121C2211yescharacteristic
27#939—21no—
28#938C2211yescharacteristic, complemented
29C2×#214C2212yescomplemented
30C2×#213C2212yescomplemented
31#930C2211yescharacteristic
32#911—28no—
33C2×#204C2212yescomplemented
34C2×ΓL(1,𝔽25)C2212yescomplemented
35#895C2211yescharacteristic, complemented
36D3×C4⋊C4C2211yescharacteristic, complemented
37#884—21no—
38D3×C22⋊C4—21no—
39C3×#102C2212yescomplemented
40C3×C22⋊C8C2211yescharacteristic, complemented
41#832C2212yescomplemented
42#830C2211yescharacteristic, complemented
43(C2×D3)⋊C8C2212yescomplemented
44#816C2212yescomplemented
45#810C2212yescomplemented
46C22×#41—31no—
47C2×AGL(1,ℤ/8)—32no—
48C2×#121—31no—
49#449—32no—
50#449—32no—
51#448—31no—
52#448—31no—
53#447—31no—
54#447D311yescharacteristic, complemented
55#417—32no—
56#407—31no—
57C24×C3—21no—
58C23×D3—21no—
59C23×D3D411yescharacteristic
60C2×C3×D4—24no—
61C2×C3×D4—21no—
62C2×C3×D4—21no—
63C2×C3×D4C2311yescharacteristic
64C2×C3×D4C2312yes—
65C22×C3×C4C2311yescharacteristic
66C2×#67—28no—
67C2×#67—22no—
68C2×#67—22no—
69C2×#67—21no—
70C2×#67—21no—
71C2×#67—21no—
72C22×Dic3—28no—
73C22×Dic3—21no—
74C22×Dic3—22no—
75C22×Dic3—22no—
76C22×Dic3D411yescharacteristic
77#237—28no—
78#237—28no—
79#237—24no—
80#237—24no—
81#237—24no—
82#237D418yescomplemented
83D3×D4—24no—
84D3×D4—44no—
85C2×D12—21no—
86C2×D12C2311yescharacteristic
87C2×C4×D3—24no—
88C2×C4×D3—21no—
89C2×C4×D3D411yescharacteristic
90C2×C4×D3D411yescharacteristic
91C2×C4×D3C2311yescharacteristic
92C2×C4×D3C2311yescharacteristic
93C2×Dic6—21no—
94C2×Dic6—24no—
95C2×Dic6C2312yes—
96C3×D8—24no—
97C2×C3×C8C2312yes—
98C3×C4⋊C4C2311yescharacteristic
99C3×C22⋊C4—21no—
100#217—21no—
101#214—24no—
102#213—24no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4—21no—
105C4⋊Dic3C2311yescharacteristic
106Dic3⋊C4—21no—
107C2×C3⋊C8C2312yes—
108#204—24no—
109ΓL(1,𝔽25)—24no—
110C2×#41—32no—
111C2×#41—32no—
112C2×#41—61no—
113C2×#41—61no—
114C2×#41—68no—
115C22×Q8—31no—
116C22×D4—31no—
117C22×D4—61no—
118C22×D4—61no—
119C22×D4C2×D311yescharacteristic, complemented
120C23×C4—31no—
121C23×C4—61no—
122AGL(1,ℤ/8)—68no—
123C2×SD16—32no—
124C2×SD16—32no—
125C2×D8—32no—
126C2×D8C2×D312yescomplemented
127C2×M16—32no—
128#121—31no—
129#121—31no—
130#121—31no—
131#121—61no—
132#121—61no—
133#121C2×D311yescharacteristic
134C2×C4⋊C4—31no—
135C2×C22⋊C4—61no—
136#103—32no—
137#103—32no—
138#102—32no—
139#102C2×D312yescomplemented
140C22⋊C8—31no—
141C22⋊C8—32no—
142C22⋊C8C2×D311yescharacteristic, complemented
143C23×C3—24no—
144C23×C3—22no—
145C23×C3—21no—
146C23×C3—42no—
147C23×C3—42no—
148C23×C3C2×D411yescharacteristic
149C22×D3—22no—
150C22×D3—22no—
151C22×D3—21no—
152C22×D3—41no—
153C22×D3—42no—
154C22×D3—42no—
155C22×D3—42no—
156C22×D3C2×D411yescharacteristic
157C22×D3C2×D411yescharacteristic
158C3×D4—24no—
159C3×D4—22no—
160C3×D4—22no—
161C3×D4—42no—
162C3×D4C2×D414yescomplemented
163C2×C3×C4—22no—
164C2×C3×C4—21no—
165C2×C3×C4C2×D411yescharacteristic
166C2×C3×C4C2411yescharacteristic
167#67—28no—
168#67—28no—
169#67—42no—
170#67—42no—
171#67—42no—
172C2×Dic3—24no—
173C2×Dic3—28no—
174C2×Dic3—22no—
175C2×Dic3—22no—
176C2×Dic3—28no—
177C2×Dic3—21no—
178C2×Dic3C2×D411yescharacteristic
179C2×Dic3C2×D411yescharacteristic
180D12—22no—
181D12—42no—
182C4×D3—22no—
183C4×D3—22no—
184C4×D3—22no—
185C4×D3—42no—
186C4×D3C2×D414yes—
187Dic6—24no—
188Dic6—22no—
189Dic6C2×D414yescomplemented
190C3×C8—22no—
191C3⋊C8—22no—
192C24—21no—
193C24—31no—
194C24—61no—
195#41—38no—
196#41—64no—
197#41—64no—
198#41—68no—
199#41—68no—
200#41—64no—
201C2×Q8—32no—
202C2×Q8—64no—
203C2×Q8—61no—
204C2×D4—24no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—31no—
208C2×D4—61no—
209C2×D4—64no—
210C2×D4—61no—
211C2×D4—62no—
212C2×D4—62no—
213C2×D4—68no—
214C2×D4—61no—
215C2×D4—61no—
216C2×D4—124no—
217C2×D4C22×D311yescharacteristic
218C2×D4C22×D312yes—
219C22×C4—31no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—31no—
223C22×C4—31no—
224C22×C4—64no—
225C22×C4—61no—
226C22×C4—62no—
227C22×C4—62no—
228C22×C4—68no—
229C22×C4—61no—
230C22×C4C22×D311yescharacteristic
231SD16—64no—
232SD16—64no—
233D8—24no—
234D8—64no—
235M16—64no—
236C2×C8—32no—
237C2×C8C22×D312yes—
238C4⋊C4—31no—
239C4⋊C4—61no—
240C4⋊C4C22×D311yescharacteristic
241C22⋊C4—21no—
242C22⋊C4—61no—
243C22⋊C4—61no—
244C22⋊C4—61no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—24no—
248C22×C3—48no—
249C22×C3—42no—
250C22×C3—42no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3AGL(1,ℤ/8)12yescomplemented
255C22×C3C22×D411yescharacteristic
256C2×D3—21no—
257C2×D3—21no—
258C2×D3—21no—
259C2×D3—21no—
260C2×D3—21no—
261C2×D3—44no—
262C2×D3—41no—
263C2×D3—41no—
264C2×D3—41no—
265C2×D3—84no—
266C2×D3#12012yes—
267C3×C4—21no—
268C3×C4—41no—
269C3×C4C22×D412yes—
270Dic3—24no—
271Dic3—21no—
272Dic3—41no—
273Dic3#12012yes—
274C23—24no—
275C23—22no—
276C23—21no—
277C23—31no—
278C23—31no—
279C23—42no—
280C23—42no—
281C23—62no—
282C23—62no—
283C23—61no—
284C23—121no—
285C23—122no—
286C23—122no—
287C23—122no—
288C23D3×D411yescharacteristic
289Q8—34no—
290Q8—64no—
291Q8—62no—
292D4—24no—
293D4—22no—
294D4—22no—
295D4—42no—
296D4—68no—
297D4—68no—
298D4—62no—
299D4—122no—
300D4—122no—
301D4—122no—
302D4—122no—
303D4D3×D414yescomplemented
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—31no—
307C2×C4—34no—
308C2×C4—31no—
309C2×C4—62no—
310C2×C4—62no—
311C2×C4—62no—
312C2×C4—68no—
313C2×C4—64no—
314C2×C4—68no—
315C2×C4—62no—
316C2×C4—62no—
317C2×C4—61no—
318C2×C4—122no—
319C2×C4D3×D411yescharacteristic
320C2×C4C23×D311yescharacteristic
321C8—22no—
322C8—62no—
323C2×C3—22no—
324C2×C3—44no—
325C2×C3—81no—
326C2×C3C2×#12011yescharacteristic
327C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
328C2×C3#57511yescharacteristic
329D3—22no—
330D3—41no—
331D3—81no—
332C22—21no—
333C22—21no—
334C22—24no—
335C22—32no—
336C22—48no—
337C22—42no—
338C22—42no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—61no—
343C22—61no—
344C22—61no—
345C22—61no—
346C22—61no—
347C22—124no—
348C22—121no—
349C22—121no—
350C22—121no—
351C22—244no—
352C22#91112yescomplemented
353C22C2×D3×D411yescharacteristic
354C4—21no—
355C4—32no—
356C4—41no—
357C4—61no—
358C4—64no—
359C4—121no—
360C4C2×D3×D412yes—
361C3#300611yescharacteristic, complemented, Sylow
362C2—22no—
363C2—44no—
364C2—62no—
365C2—81no—
366C2—121no—
367C2—241no—
368C2D3×#12011yescharacteristic
369C2C2×#91111yescharacteristic
370C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#447C2
socleC22×C3C22×D4
hypercenter#447D3
hypocenterC3#3006
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×#237C2semidirect
2. C2×#911C2semidirect
3. D3×#121C2semidirect
4. C3×#447C2semidirect
5. #5217C2semidirect
6. #5158C2semidirect
7. #5040C2semidirect
8. #4781C2semidirect
9. #4771C2semidirect
10. #4738C2semidirect
11. #4726C2semidirect
13. C22×C3×D4C22semidirect
20. C2×#237C22semidirect
22. C2×D3×D4C22semidirect
24. C22×Dic6C22semidirect
25. C2×C3×D8C22semidirect
28. #938C22semidirect
29. C2×#214C22semidirect
30. C2×#213C22semidirect
33. C2×#204C22semidirect
34. C2×ΓL(1,𝔽25)C22semidirect
35. #895C22semidirect
36. D3×C4⋊C4C22semidirect
39. C3×#102C22semidirect
40. C3×C22⋊C8C22semidirect
41. #832C22semidirect
42. #830C22semidirect
43. (C2×D3)⋊C8C22semidirect
44. #816C22semidirect
45. #810C22semidirect
54. #447D3semidirect
82. #237D4semidirect
119. C22×D4C2×D3semidirect
126. C2×D8C2×D3semidirect
139. #102C2×D3semidirect
142. C22⋊C8C2×D3semidirect
162. C3×D4C2×D4semidirect
189. Dic6C2×D4semidirect
254. C22×C3AGL(1,ℤ/8)semidirect
303. D4D3×D4semidirect
352. C22#911semidirect
361. C3#3006semidirect

Maximal subgroups

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

subgroupindex
C22×#2372
C2×#9112
D3×#1212
C3×#4472
#52172
#51582
#50402
#47812
#47712
#47382
#47262
#30063