Finite Group Databasebeta

← #80332

Subgroups — #80332

Order 384.

548 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#911C212yesmaximal, complemented
2#5716C211yesmaximal, characteristic, complemented
3C4×D3×D4C211yesmaximal, characteristic, complemented
4C3×#459C211yesmaximal, characteristic, complemented
5#5153C212yesmaximal, complemented
6#5072C211yesmaximal, characteristic, complemented
7#5019C211yesmaximal, characteristic, complemented
8#4841C211yesmaximal, characteristic, complemented
9#4837C211yesmaximal, characteristic, complemented
10#4787C212yesmaximal, complemented
11#4771C212yesmaximal, complemented
12#3036—31nomaximal, Sylow
13C2×#237C2212yes—
14C2×D3×D4C2211yescharacteristic
15C22×C4×D3—21no—
16C2×C3×D8C2212yes—
17C3×#127C2211yescharacteristic, complemented
18C3×C4×D4C2211yescharacteristic
19#938—21no—
20#938—21no—
21#936C2212yescomplemented
22D4×Dic3—21no—
23D4×Dic3—21no—
24D4×Dic3C2211yescharacteristic
25C2×#214C2212yes—
26C2×#213C2212yes—
27C4×#67—21no—
28#911—28no—
29C2×#204C2212yes—
30C2×ΓL(1,𝔽25)C2212yes—
31#893C2211yescharacteristic
32D3×C4⋊C4C2211yescharacteristic
33#67⋊C4—21no—
34D3×C22⋊C4—21no—
35C4×D12C2211yescharacteristic
36C42×D3C2211yescharacteristic
37#869C2211yescharacteristic, complemented
38C3×C4⋊C8C2211yescharacteristic, complemented
39C3×#102C2212yescomplemented
40#832C2212yescomplemented
41#816C2212yescomplemented
42Dic3⋊C8C2212yescomplemented
43#810C2212yescomplemented
44C4⋊(C3⋊C8)C2211yescharacteristic, complemented
45C2×AGL(1,ℤ/8)—32no—
46#532—31no—
47C2×C4×D4—31no—
48#463—32no—
49#463—32no—
50#461—31no—
51#461—31no—
52#459—31no—
53#459D311yescharacteristic, complemented
54#423—31no—
55#417—32no—
56C23×D3—21no—
57C2×C3×D4—21no—
58C2×C3×D4—21no—
59C2×C3×D4C2312yes—
60C2×C3×D4C2311yescharacteristic
61C22×C3×C4—21no—
62C2×#67—22no—
63C2×#67—21no—
64C22×Dic3—21no—
65C22×Dic3—22no—
66#237—24no—
67#237—24no—
68D3×D4—22no—
69D3×D4D414yescomplemented
70C2×D12C2311yescharacteristic
71C2×C4×D3—21no—
72C2×C4×D3—21no—
73C2×C4×D3—21no—
74C2×C4×D3—44no—
75C2×C4×D3D411yescharacteristic
76C2×C4×D3D411yescharacteristic
77C2×C4×D3C2311yescharacteristic
78C2×Dic6C2312yes—
79C3×D8—24no—
80C2×C3×C8C2312yes—
81C3×C4⋊C4C2311yescharacteristic
82C3×C22⋊C4—21no—
83C3×C42C2311yescharacteristic
84#217—21no—
85#217—22no—
86#217—22no—
87#214—24no—
88#213—24no—
89(C2×D3)⋊C4—21no—
90(C2×D3)⋊C4—21no—
91C4⋊Dic3—21no—
92C4⋊Dic3—21no—
93C4⋊Dic3C2311yescharacteristic
94Dic3⋊C4—21no—
95C4×Dic3D411yescharacteristic
96C4×Dic3D411yescharacteristic
97C4×Dic3C2311yescharacteristic
98C2×C3⋊C8C2312yes—
99#204—24no—
100ΓL(1,𝔽25)—24no—
101C2×#41—32no—
102C22×D4—31no—
103C23×C4—61no—
104AGL(1,ℤ/8)—68no—
105C2×SD16—32no—
106C2×SD16—32no—
107C2×D8—32no—
108C2×D8C2×D312yes—
109C2×M16—32no—
110C4⋊Q8—31no—
111#127C2×D311yescharacteristic, complemented
112#124—32no—
113#121—61no—
114#121—61no—
115C4×D4—31no—
116C4×D4—31no—
117C4×D4—31no—
118C4×D4—61no—
119C4×D4—61no—
120C4×D4—61no—
121C4×D4—61no—
122C4×D4C2×D311yescharacteristic
123C2×C4⋊C4—31no—
124C2×C22⋊C4—61no—
125C2×C42—31no—
126C4⋊C8—32no—
127C4⋊C8—31no—
128C4⋊C8C2×D311yescharacteristic, complemented
129#103—32no—
130#103—32no—
131#102—32no—
132#102C2×D312yescomplemented
133C23×C3—21no—
134C23×C3—22no—
135C22×D3—21no—
136C22×D3—24no—
137C22×D3—42no—
138C22×D3—42no—
139C22×D3C2×D411yescharacteristic
140C3×D4—22no—
141C3×D4—21no—
142C3×D4—22no—
143C3×D4—42no—
144C3×D4—42no—
145C3×D4C2×D412yes—
146C2×C3×C4—21no—
147C2×C3×C4—42no—
148C2×C3×C4C2×D411yescharacteristic
149C2×C3×C4C2×D411yescharacteristic
150C2×C3×C4C2411yescharacteristic
151#67—24no—
152#67—44no—
153C2×Dic3—21no—
154C2×Dic3—22no—
155C2×Dic3—42no—
156C2×Dic3—44no—
157C2×Dic3C2×D411yescharacteristic
158C2×Dic3C2×D411yescharacteristic
159C2×Dic3C2×D411yescharacteristic
160D12—21no—
161D12C2×D412yes—
162C4×D3—22no—
163C4×D3—41no—
164C4×D3—42no—
165C4×D3—42no—
166C4×D3—42no—
167C4×D3C2×D412yes—
168C4×D3#4112yes—
169Dic6—22no—
170Dic6—22no—
171C3×C8—22no—
172C3⋊C8—22no—
173C24—61no—
174#41—64no—
175#41—64no—
176C2×Q8—32no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—34no—
180C2×D4—31no—
181C2×D4—62no—
182C2×D4—61no—
183C2×D4—62no—
184C2×D4C22×D312yes—
185C2×D4C22×D311yescharacteristic
186C22×C4—21no—
187C22×C4—31no—
188C22×C4—31no—
189C22×C4—31no—
190C22×C4—61no—
191C22×C4—61no—
192C22×C4—61no—
193C22×C4—62no—
194C22×C4—61no—
195C22×C4—124no—
196SD16—64no—
197SD16—64no—
198D8—24no—
199D8—64no—
200M16—64no—
201C2×C8—32no—
202C2×C8C22×D312yes—
203C4⋊C4—31no—
204C4⋊C4—61no—
205C4⋊C4—61no—
206C4⋊C4—61no—
207C4⋊C4C22×D311yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—61no—
210C22⋊C4—61no—
211C22⋊C4—62no—
212C22⋊C4—62no—
213C22⋊C4—61no—
214C42—31no—
215C42—31no—
216C42—31no—
217C42C22×D311yescharacteristic
218C22×C3—22no—
219C22×C3—41no—
220C22×C3—41no—
221C22×C3—42no—
222C22×C3—42no—
223C22×C3—42no—
224C22×C3C22×D411yescharacteristic
225C2×D3—21no—
226C2×D3—21no—
227C2×D3—22no—
228C2×D3—48no—
229C2×D3—41no—
230C2×D3—41no—
231C2×D3#12112yes—
232C3×C4—21no—
233C3×C4—41no—
234C3×C4AGL(1,ℤ/8)12yescomplemented
235C3×C4C22×D411yescharacteristic
236C3×C4C2×#4111yescharacteristic
237Dic3—21no—
238Dic3—21no—
239Dic3—42no—
240Dic3—41no—
241Dic3#12112yes—
242C23—21no—
243C23—22no—
244C23—31no—
245C23—64no—
246C23—61no—
247C23—122no—
248C23—122no—
249Q8—62no—
250Q8—62no—
251D4—22no—
252D4—21no—
253D4—22no—
254D4—32no—
255D4—42no—
256D4—42no—
257D4—64no—
258D4—61no—
259D4—124no—
260D4D3×D412yes—
261C2×C4—21no—
262C2×C4—31no—
263C2×C4—32no—
264C2×C4—32no—
265C2×C4—31no—
266C2×C4—31no—
267C2×C4—42no—
268C2×C4—62no—
269C2×C4—61no—
270C2×C4—62no—
271C2×C4—121no—
272C2×C4—124no—
273C2×C4—122no—
274C2×C4—122no—
275C2×C4—122no—
276C2×C4—122no—
277C2×C4D3×D411yescharacteristic
278C2×C4D3×D411yescharacteristic
279C2×C4C23×D311yescharacteristic
280C8—22no—
281C8—62no—
282C2×C3—42no—
283C2×C3—82no—
284C2×C3C2×#12111yescharacteristic
285C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
286C2×C3#57511yescharacteristic
287D3—22no—
288D3—42no—
289C22—22no—
290C22—32no—
291C22—41no—
292C22—41no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—61no—
297C22—61no—
298C22—62no—
299C22—128no—
300C22—121no—
301C22—121no—
302C22C2×D3×D411yescharacteristic
303C4—21no—
304C4—32no—
305C4—41no—
306C4—61no—
307C4—61no—
308C4—121no—
309C4—122no—
310C4#91112yescomplemented
311C4C2×D3×D411yescharacteristic
312C4D3×#4111yescharacteristic
313C3#303611yescharacteristic, complemented, Sylow
314C2—42no—
315C2—62no—
316C2—82no—
317C2—122no—
318C2D3×#12111yescharacteristic
319C2C2×#91111yescharacteristic
320C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3036
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#911C2semidirect
2. #5716C2semidirect
3. C4×D3×D4C2semidirect
4. C3×#459C2semidirect
5. #5153C2semidirect
6. #5072C2semidirect
7. #5019C2semidirect
8. #4841C2semidirect
9. #4837C2semidirect
10. #4787C2semidirect
11. #4771C2semidirect
17. C3×#127C22semidirect
21. #936C22semidirect
37. #869C22semidirect
38. C3×C4⋊C8C22semidirect
39. C3×#102C22semidirect
40. #832C22semidirect
41. #816C22semidirect
42. Dic3⋊C8C22semidirect
43. #810C22semidirect
44. C4⋊(C3⋊C8)C22semidirect
53. #459D3semidirect
69. D3×D4D4semidirect
111. #127C2×D3semidirect
128. C4⋊C8C2×D3semidirect
132. #102C2×D3semidirect
234. C3×C4AGL(1,ℤ/8)semidirect
310. C4#911semidirect
313. C3#3036semidirect

Maximal subgroups

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

subgroupindex
C2×#9112
#57162
C4×D3×D42
C3×#4592
#51532
#50722
#50192
#48412
#48372
#47872
#47712
#30363