Finite Group Databasebeta

← #81685

Subgroups — #81685

Order 384.

434 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 111 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
1#5725C211yesmaximal, characteristic, complemented
2#5668C211yesmaximal, characteristic, complemented
3C3×#489C211yesmaximal, characteristic, complemented
4#5089C211yesmaximal, characteristic, complemented
5#5058C211yesmaximal, characteristic, complemented
6#4818C212yesmaximal, complemented
7#4810C212yesmaximal, complemented
8#4786C212yesmaximal, complemented
9#4781C212yesmaximal, complemented
10#4712C211yesmaximal, characteristic, complemented
11#4705C211yesmaximal, characteristic, complemented
12#3214—31nomaximal, Sylow
13C2×#239—21no—
14C2×#239C2211yescharacteristic
15C2×#237C2211yescharacteristic
16C3×C4⋊Q8C2211yescharacteristic
17C3×#124C2211yescharacteristic
18#947—21no—
19#944C2211yescharacteristic
20C2×#215C2212yes—
21#940C2211yescharacteristic
22C2×#213C2212yes—
23C2×D24C2212yes—
24C2×#204C2212yes—
25#895—21no—
26#895C2212yes—
27#893—21no—
28#893C2212yes—
29#884—21no—
30#884—21no—
31#67⋊C4—21no—
32#67⋊C4—21no—
33#875C2211yescharacteristic
34#874C2211yescharacteristic
35C42×D3C2211yescharacteristic
36C3×#103C2212yes—
37C3×#102C2212yes—
38C3×#97C2211yescharacteristic
39(C2×D3)⋊C8C2212yes—
40Dic3⋊C8C2212yes—
41#810C2212yes—
42#809C2212yes—
43#803C2211yescharacteristic
44#532—31no—
45#532—31no—
46#496—31no—
47#496—31no—
48#489—31no—
49#489D311yescharacteristic, complemented
50#461—32no—
51#459—32no—
52#449—32no—
53#449—32no—
54#432—31no—
55C2×C3×Q8—21no—
56C2×C3×Q8C2311yescharacteristic
57C2×C3×D4C2311yescharacteristic
58C2×#67—21no—
59C2×#67—21no—
60C2×#67—21no—
61C22×Dic3—21no—
62#239—42no—
63#239—44no—
64#237—42no—
65C2×D12—21no—
66C2×D12—22no—
67C2×D12—21no—
68C2×D12—21no—
69C2×D12C2311yescharacteristic
70C2×D12C2312yes—
71C2×C4×D3—21no—
72C2×C4×D3—22no—
73C2×C4×D3D411yescharacteristic
74C2×C4×D3D411yescharacteristic
75C2×C4×D3C2311yescharacteristic
76C2×Dic6C2311yescharacteristic
77C2×C3×C8C2312yes—
78C3×C4⋊C4—21no—
79C3×C4⋊C4C2312yes—
80C3×C22⋊C4—21no—
81C3×C22⋊C4—21no—
82C3×C42C2311yescharacteristic
83#215—42no—
84#213—42no—
85(C2×D3)⋊C4—21no—
86(C2×D3)⋊C4—21no—
87(C2×D3)⋊C4—22no—
88(C2×D3)⋊C4—22no—
89Dic3⋊C4—21no—
90Dic3⋊C4—21no—
91C4×Dic3D411yescharacteristic
92C4×Dic3D411yescharacteristic
93C4×Dic3C2311yescharacteristic
94C2×C3⋊C8C2312yes—
95D24—42no—
96#204—42no—
97C2×#41—31no—
98C2×#41—31no—
99C2×#41—61no—
100C2×SD16—32no—
101C2×SD16—32no—
102C2×D8—32no—
103C2×D8—32no—
104C4⋊Q8—31no—
105C4⋊Q8C2×D311yescharacteristic
106#127—31no—
107#127—31no—
108#124—31no—
109#124—61no—
110#124C2×D311yescharacteristic
111#121—32no—
112#121—61no—
113#121—61no—
114#121—61no—
115C4×D4—32no—
116C4×D4—61no—
117C4×D4—61no—
118C4×D4—61no—
119C2×C42—31no—
120C4⋊C8—32no—
121#103—32no—
122#103C2×D312yes—
123#102—32no—
124#102C2×D312yes—
125C22⋊C8—32no—
126#97—31no—
127#97C2×D311yescharacteristic
128C23×C3—21no—
129C22×D3—22no—
130C22×D3—21no—
131C22×D3C2×D411yescharacteristic
132C3×Q8—22no—
133C3×Q8—42no—
134C3×D4—22no—
135C2×C3×C4—21no—
136C2×C3×C4—22no—
137C2×C3×C4C2×D411yescharacteristic
138C2×C3×C4C2×D411yescharacteristic
139C2×C3×C4C2411yescharacteristic
140#67—42no—
141#67—42no—
142#67—42no—
143C2×Dic3—21no—
144C2×Dic3—42no—
145C2×Dic3C2×D411yescharacteristic
146C2×Dic3C2×D411yescharacteristic
147C2×Dic3C2×D411yescharacteristic
148D12—22no—
149D12—22no—
150D12—22no—
151D12—42no—
152D12—42no—
153D12—42no—
154D12—44no—
155C4×D3—22no—
156C4×D3—22no—
157C4×D3—41no—
158C4×D3—42no—
159C4×D3—44no—
160Dic6—22no—
161C3×C8—22no—
162C3⋊C8—22no—
163#41—122no—
164#41—122no—
165#41—124no—
166C2×Q8—21no—
167C2×Q8—31no—
168C2×Q8C22×D311yescharacteristic
169C2×D4—31no—
170C2×D4—32no—
171C2×D4—62no—
172C2×D4—61no—
173C2×D4—61no—
174C2×D4—61no—
175C2×D4—61no—
176C2×D4—61no—
177C2×D4—61no—
178C2×D4C22×D311yescharacteristic
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—31no—
182C22×C4—62no—
183C22×C4—61no—
184C22×C4—61no—
185SD16—122no—
186SD16—122no—
187D8—122no—
188D8—122no—
189C2×C8—32no—
190C2×C8C22×D312yes—
191C4⋊C4—21no—
192C4⋊C4—61no—
193C4⋊C4—61no—
194C4⋊C4C22×D312yes—
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—62no—
198C22⋊C4—62no—
199C22⋊C4—61no—
200C22⋊C4—61no—
201C42—31no—
202C42—31no—
203C42—31no—
204C42C22×D311yescharacteristic
205C22×C3—41no—
206C22×C3—41no—
207C22×C3—41no—
208C22×C3C22×D411yescharacteristic
209C2×D3—21no—
210C2×D3—21no—
211C2×D3—21no—
212C2×D3—41no—
213C2×D3—42no—
214C2×D3—42no—
215C2×D3—41no—
216C2×D3—42no—
217C2×D3—41no—
218C3×C4—21no—
219C3×C4—21no—
220C3×C4—41no—
221C3×C4—42no—
222C3×C4H(5,𝔽2)12yes—
223Dic3—21no—
224Dic3—21no—
225Dic3—41no—
226Dic3AGL(1,ℤ/8)12yes—
227C23—21no—
228C23—31no—
229C23—61no—
230C23—62no—
231Q8—22no—
232Q8—42no—
233Q8—62no—
234D4—22no—
235D4—62no—
236D4—62no—
237D4—62no—
238D4—122no—
239D4—122no—
240D4—122no—
241D4—122no—
242D4—124no—
243D4—122no—
244D4—122no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—31no—
248C2×C4—31no—
249C2×C4—31no—
250C2×C4—62no—
251C2×C4—62no—
252C2×C4—61no—
253C2×C4—121no—
254C2×C4—124no—
255C2×C4—122no—
256C2×C4—122no—
257C2×C4D3×D411yescharacteristic
258C2×C4D3×D411yescharacteristic
259C2×C4C23×D311yescharacteristic
260C8—22no—
261C8—62no—
262C2×C3—81no—
263C2×C3#53511yescharacteristic
264C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
265C2×C3#57511yescharacteristic
266D3—41no—
267D3—81no—
268D3—82no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—61no—
273C22—61no—
274C22—61no—
275C22—121no—
276C22—122no—
277C22—122no—
278C22—121no—
279C22—122no—
280C22—121no—
281C22C2×D3×D411yescharacteristic
282C4—21no—
283C4—21no—
284C4—32no—
285C4—41no—
286C4—42no—
287C4—61no—
288C4—61no—
289C4—121no—
290C4#100912yes—
291C3#321411yescharacteristic, complemented, Sylow
292C2—81no—
293C2—121no—
294C2—241no—
295C2—242no—
296C2#567411yescharacteristic
297C2D3×AGL(1,ℤ/8)11yescharacteristic
298C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#489C2
socleC22×C3C22×D4
hypercenter#489D3
hypocenterC3#3214
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #5725C2semidirect
2. #5668C2semidirect
3. C3×#489C2semidirect
4. #5089C2semidirect
5. #5058C2semidirect
6. #4818C2semidirect
7. #4810C2semidirect
8. #4786C2semidirect
9. #4781C2semidirect
10. #4712C2semidirect
11. #4705C2semidirect
49. #489D3semidirect
291. C3#3214semidirect

Maximal subgroups

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

subgroupindex
#57252
#56682
C3×#4892
#50892
#50582
#48182
#48102
#47862
#47812
#47122
#47052
#32143