Finite Group Databasebeta

← #80040

Subgroups — #80040

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×#239C211yesmaximal, characteristic, complemented
2C2×#914C212yesmaximal, complemented
3D3×#121C211yesmaximal, characteristic, complemented
4C3×#448C211yesmaximal, characteristic, complemented
5#5227C211yesmaximal, characteristic, complemented
6#5172C212yesmaximal, complemented
7#5036C211yesmaximal, characteristic, complemented
8#4810C212yesmaximal, complemented
9#4803C212yesmaximal, complemented
10#4732C211yesmaximal, characteristic, complemented
11#4726C211yesmaximal, characteristic, complemented
12#3006—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic, complemented
14C2×#239—28no—
15C2×#239—21no—
16C2×#239—21no—
17C2×#239C2212yes—
18C2×#239C2212yescomplemented
19C2×D3×D4—21no—
20C2×D3×D4C2211yescharacteristic, complemented
21C22×D12—21no—
22C22×D12C2211yescharacteristic, complemented
23C22×C4×D3—21no—
24C22×C4×D3C2211yescharacteristic
25C2×C3×SD16C2212yescomplemented
26C3×#121C2211yescharacteristic
27C2×#215C2212yescomplemented
28#939—21no—
29#938C2211yescharacteristic, complemented
30C2×#213C2212yescomplemented
31#930C2211yescharacteristic
32#914—28no—
33C2×D24C2212yescomplemented
34C2×ΓL(1,𝔽25)C2212yescomplemented
35#895C2211yescharacteristic, complemented
36D3×C4⋊C4C2211yescharacteristic, complemented
37#884—21no—
38D3×C22⋊C4—21no—
39C3×#103C2212yescomplemented
40C3×C22⋊C8C2211yescharacteristic, complemented
41#835C2212yescomplemented
42#830C2211yescharacteristic, complemented
43#821C2212yescomplemented
44(C2×D3)⋊C8C2212yescomplemented
45#809C2212yescomplemented
46C22×#41—31no—
47C2×AGL(1,ℤ/8)—32no—
48C2×#121—31no—
49#449—32no—
50#449—32no—
51#448—31no—
52#448D311yescharacteristic, complemented
53#447—31no—
54#447—31no—
55#417—32no—
56#407—31no—
57C23×D3—21no—
58C23×D3—21no—
59C23×D3D411yescharacteristic
60C2×C3×Q8—24no—
61C2×C3×Q8—21no—
62C2×C3×Q8C2312yes—
63C2×C3×D4—21no—
64C2×C3×D4C2311yescharacteristic
65C22×C3×C4—21no—
66C22×C3×C4C2311yescharacteristic
67C2×#67—21no—
68C2×#67—21no—
69C2×#67—21no—
70C22×Dic3D411yescharacteristic
71#239—28no—
72#239—28no—
73#239—24no—
74#239—24no—
75#239—24no—
76#239D418yescomplemented
77D3×D4—24no—
78D3×D4—44no—
79C2×D12—28no—
80C2×D12—22no—
81C2×D12—22no—
82C2×D12—21no—
83C2×D12—24no—
84C2×D12—21no—
85C2×D12C2311yescharacteristic
86C2×D12C2312yes—
87C2×C4×D3—28no—
88C2×C4×D3—22no—
89C2×C4×D3—22no—
90C2×C4×D3—24no—
91C2×C4×D3—21no—
92C2×C4×D3D411yescharacteristic
93C2×C4×D3D411yescharacteristic
94C2×C4×D3C2311yescharacteristic
95C2×C4×D3C2311yescharacteristic
96C3×SD16—24no—
97C2×C3×C8C2312yes—
98C3×C4⋊C4C2311yescharacteristic
99C3×C22⋊C4—21no—
100#217—21no—
101#215—24no—
102#213—24no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4—21no—
105C4⋊Dic3C2311yescharacteristic
106Dic3⋊C4—21no—
107C2×C3⋊C8C2312yes—
108D24—24no—
109ΓL(1,𝔽25)—24no—
110C2×#41—32no—
111C2×#41—32no—
112C2×#41—61no—
113C2×#41—61no—
114C2×#41—68no—
115C22×Q8C2×D311yescharacteristic, complemented
116C22×D4—31no—
117C22×D4—31no—
118C22×D4—61no—
119C22×D4—61no—
120C23×C4—31no—
121C23×C4—61no—
122AGL(1,ℤ/8)—68no—
123C2×SD16—32no—
124C2×SD16C2×D312yescomplemented
125C2×D8—32no—
126C2×D8—32no—
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#103C2×D312yescomplemented
138#102—32no—
139#102—32no—
140C22⋊C8—31no—
141C22⋊C8—32no—
142C22⋊C8C2×D311yescharacteristic, complemented
143C23×C3—21no—
144C23×C3C2×D411yescharacteristic
145C22×D3—24no—
146C22×D3—22no—
147C22×D3—22no—
148C22×D3—22no—
149C22×D3—21no—
150C22×D3—41no—
151C22×D3—42no—
152C22×D3—42no—
153C22×D3—42no—
154C22×D3—42no—
155C22×D3—42no—
156C22×D3C2×D411yescharacteristic
157C22×D3C2×D411yescharacteristic
158C3×Q8—24no—
159C3×Q8—22no—
160C3×Q8C2×D414yescomplemented
161C3×D4—22no—
162C3×D4—42no—
163C2×C3×C4—24no—
164C2×C3×C4—22no—
165C2×C3×C4—22no—
166C2×C3×C4—21no—
167C2×C3×C4C2×D411yescharacteristic
168C2×C3×C4C2411yescharacteristic
169#67—42no—
170#67—42no—
171#67—42no—
172C2×Dic3—22no—
173C2×Dic3—21no—
174C2×Dic3C2×D411yescharacteristic
175C2×Dic3C2×D411yescharacteristic
176D12—28no—
177D12—24no—
178D12—28no—
179D12—22no—
180D12—22no—
181D12—42no—
182D12C2×D414yescomplemented
183C4×D3—28no—
184C4×D3—22no—
185C4×D3—22no—
186C4×D3—28no—
187C4×D3—22no—
188C4×D3—42no—
189C4×D3C2×D414yes—
190C3×C8—22no—
191C3⋊C8—22no—
192C24—31no—
193C24—61no—
194C24—61no—
195#41—38no—
196#41—64no—
197#41—64no—
198#41—68no—
199#41—68no—
200#41—64no—
201C2×Q8—24no—
202C2×Q8—21no—
203C2×Q8C22×D312yes—
204C2×D4—21no—
205C2×D4—32no—
206C2×D4—31no—
207C2×D4—64no—
208C2×D4—62no—
209C2×D4—62no—
210C2×D4—68no—
211C2×D4—61no—
212C2×D4—64no—
213C2×D4—61no—
214C2×D4—61no—
215C2×D4—61no—
216C2×D4—61no—
217C2×D4—124no—
218C2×D4C22×D311yescharacteristic
219C22×C4—21no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—31no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—64no—
226C22×C4—62no—
227C22×C4—62no—
228C22×C4—68no—
229C22×C4—61no—
230C22×C4C22×D311yescharacteristic
231SD16—24no—
232SD16—64no—
233D8—64no—
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—41no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3AGL(1,ℤ/8)12yescomplemented
251C22×C3C22×D411yescharacteristic
252C2×D3—21no—
253C2×D3—21no—
254C2×D3—21no—
255C2×D3—24no—
256C2×D3—21no—
257C2×D3—21no—
258C2×D3—44no—
259C2×D3—48no—
260C2×D3—42no—
261C2×D3—42no—
262C2×D3—41no—
263C2×D3—41no—
264C2×D3—41no—
265C2×D3—84no—
266C2×D3#12012yes—
267C3×C4—24no—
268C3×C4—21no—
269C3×C4—41no—
270C3×C4C22×D412yes—
271Dic3—21no—
272Dic3—41no—
273Dic3#12012yes—
274C23—21no—
275C23—31no—
276C23—31no—
277C23—62no—
278C23—64no—
279C23—62no—
280C23—62no—
281C23—61no—
282C23—121no—
283C23—122no—
284C23—122no—
285C23—122no—
286C23—122no—
287C23—122no—
288C23D3×D411yescharacteristic
289Q8—24no—
290Q8—22no—
291Q8D3×D414yescomplemented
292D4—22no—
293D4—34no—
294D4—42no—
295D4—68no—
296D4—68no—
297D4—64no—
298D4—62no—
299D4—62no—
300D4—122no—
301D4—122no—
302D4—122no—
303D4—122no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—31no—
309C2×C4—34no—
310C2×C4—31no—
311C2×C4—62no—
312C2×C4—68no—
313C2×C4—62no—
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—81no—
325C2×C3C2×#12011yescharacteristic
326C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
327C2×C3#57511yescharacteristic
328D3—22no—
329D3—41no—
330D3—44no—
331D3—81no—
332C22—21no—
333C22—21no—
334C22—32no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—61no—
339C22—61no—
340C22—61no—
341C22—61no—
342C22—64no—
343C22—61no—
344C22—124no—
345C22—128no—
346C22—122no—
347C22—122no—
348C22—121no—
349C22—121no—
350C22—121no—
351C22—244no—
352C22#91412yescomplemented
353C22C2×D3×D411yescharacteristic
354C4—24no—
355C4—21no—
356C4—32no—
357C4—41no—
358C4—61no—
359C4—121no—
360C4C2×D3×D412yes—
361C3#300611yescharacteristic, complemented, Sylow
362C2—22no—
363C2—62no—
364C2—81no—
365C2—121no—
366C2—124no—
367C2—241no—
368C2D3×#12011yescharacteristic
369C2C2×#91411yescharacteristic
370C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#448C2
socleC22×C3C22×D4
hypercenter#448D3
hypocenterC3#3006
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×#239C2semidirect
2. C2×#914C2semidirect
3. D3×#121C2semidirect
4. C3×#448C2semidirect
5. #5227C2semidirect
6. #5172C2semidirect
7. #5036C2semidirect
8. #4810C2semidirect
9. #4803C2semidirect
10. #4732C2semidirect
11. #4726C2semidirect
13. C22×C3×Q8C22semidirect
18. C2×#239C22semidirect
20. C2×D3×D4C22semidirect
22. C22×D12C22semidirect
25. C2×C3×SD16C22semidirect
27. C2×#215C22semidirect
29. #938C22semidirect
30. C2×#213C22semidirect
33. C2×D24C22semidirect
34. C2×ΓL(1,𝔽25)C22semidirect
35. #895C22semidirect
36. D3×C4⋊C4C22semidirect
39. C3×#103C22semidirect
40. C3×C22⋊C8C22semidirect
41. #835C22semidirect
42. #830C22semidirect
43. #821C22semidirect
44. (C2×D3)⋊C8C22semidirect
45. #809C22semidirect
52. #448D3semidirect
76. #239D4semidirect
115. C22×Q8C2×D3semidirect
124. C2×SD16C2×D3semidirect
137. #103C2×D3semidirect
142. C22⋊C8C2×D3semidirect
160. C3×Q8C2×D4semidirect
182. D12C2×D4semidirect
250. C22×C3AGL(1,ℤ/8)semidirect
291. Q8D3×D4semidirect
352. C22#914semidirect
361. C3#3006semidirect

Maximal subgroups

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

subgroupindex
C22×#2392
C2×#9142
D3×#1212
C3×#4482
#52272
#51722
#50362
#48102
#48032
#47322
#47262
#30063