Finite Group Databasebeta

← #81449

Subgroups — #81449

Order 384.

410 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 115 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
1D3×C22⋊Q8C211yesmaximal, characteristic
2C2×#897C211yesmaximal, characteristic
3C3×#483C211yesmaximal, characteristic, complemented
4#5050C211yesmaximal, characteristic, complemented
5#4971C211yesmaximal, characteristic, complemented
6#4867C212yesmaximal, complemented
7#4862C212yesmaximal, complemented
8#4816C212yesmaximal, complemented
9#4809C212yesmaximal, complemented
10#4739C211yesmaximal, characteristic, complemented
11#4726C211yesmaximal, characteristic
12#3195—31nomaximal, Sylow
13C2×D3×Q8C2211yescharacteristic
14C22×C4×D3C2211yescharacteristic
15C22×Dic6C2211yescharacteristic
16C3×C22⋊Q8C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18#946C2211yescharacteristic
19C2×#216C2212yescomplemented
20C2×(C2×D3)⋊C4—21no—
21C2×C4⋊Dic3—21no—
22#924C2211yescharacteristic
23C2×Dic12C2212yescomplemented
24C2×ΓL(1,𝔽25)C2212yes—
25#897—21no—
26#897—21no—
27#897C2211yescharacteristic
28#897C2212yes—
29#897C2212yes—
30#896—21no—
31D3×C4⋊C4—21no—
32D3×C4⋊C4C2211yescharacteristic
33D3×C22⋊C4—21no—
34#878—21no—
35C3×#106C2212yescomplemented
36C3×#103C2212yescomplemented
37C3×C22⋊C8C2211yescharacteristic
38#830C2211yescharacteristic
39(C2×D3)⋊C8C2212yes—
40#816C2212yescomplemented
41#810C2212yescomplemented
42#810C2212yescomplemented
43#808C2212yescomplemented
44C2×C22⋊Q8—31no—
45C2×C22⋊Q8—31no—
46#483—31no—
47#483—32no—
48#483D311yescharacteristic, complemented
49#470—32no—
50#470—32no—
51#451—31no—
52#451—32no—
53#451—31no—
54#407—31no—
55C23×D3D411yescharacteristic
56C2×C3×Q8C2311yescharacteristic
57C22×C3×C4—21no—
58C22×C3×C4C2311yescharacteristic
59C22×Dic3—21no—
60C22×Dic3D411yescharacteristic
61D3×Q8—42no—
62C2×C4×D3—21no—
63C2×C4×D3—21no—
64C2×C4×D3—42no—
65C2×C4×D3D411yescharacteristic
66C2×C4×D3D411yescharacteristic
67C2×C4×D3C2311yescharacteristic
68C2×C4×D3C2311yescharacteristic
69C2×Dic6—21no—
70C2×Dic6—21no—
71C2×Dic6—42no—
72C2×Dic6C2311yescharacteristic
73C2×Dic6C2312yes—
74C2×C3×C8C2312yes—
75C3×C4⋊C4—21no—
76C3×C4⋊C4—21no—
77C3×C4⋊C4C2312yes—
78C3×C4⋊C4C2311yescharacteristic
79C3×C22⋊C4—21no—
80#217—21no—
81#216—42no—
82(C2×D3)⋊C4—21no—
83(C2×D3)⋊C4—21no—
84(C2×D3)⋊C4—22no—
85(C2×D3)⋊C4—22no—
86C4⋊Dic3—21no—
87C4⋊Dic3—22no—
88C4⋊Dic3—22no—
89C4⋊Dic3C2311yescharacteristic
90Dic3⋊C4—21no—
91Dic3⋊C4—21no—
92Dic3⋊C4—21no—
93C2×C3⋊C8C2312yes—
94Dic12—42no—
95ΓL(1,𝔽25)—42no—
96C22×Q8—31no—
97C22×Q8—31no—
98C23×C4—31no—
99C2×Q16—32no—
100C2×Q16—32no—
101C2×M16—32no—
102C22⋊Q8—31no—
103C22⋊Q8—31no—
104C22⋊Q8—32no—
105C22⋊Q8—31no—
106C22⋊Q8—32no—
107C22⋊Q8—61no—
108C22⋊Q8—61no—
109C22⋊Q8—61no—
110C22⋊Q8—61no—
111C22⋊Q8C2×D311yescharacteristic
112C2×C4⋊C4—31no—
113C2×C4⋊C4—61no—
114C2×C4⋊C4—61no—
115C2×C4⋊C4C2×D311yescharacteristic
116C2×C22⋊C4—61no—
117C2×C22⋊C4—61no—
118#106—32no—
119#106C2×D312yescomplemented
120#103—32no—
121#103—32no—
122#103—32no—
123#103C2×D312yescomplemented
124C22⋊C8—31no—
125C22⋊C8—32no—
126C22⋊C8C2×D311yescharacteristic
127C23×C3C2×D411yescharacteristic
128C22×D3—22no—
129C22×D3—22no—
130C22×D3—41no—
131C22×D3C2×D411yescharacteristic
132C22×D3C2×D411yescharacteristic
133C3×Q8—22no—
134C2×C3×C4—22no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—42no—
139C2×C3×C4C2×D411yescharacteristic
140C2×C3×C4C2411yescharacteristic
141C2×Dic3—21no—
142C2×Dic3—21no—
143C2×Dic3—22no—
144C2×Dic3—41no—
145C2×Dic3—42no—
146C2×Dic3C2×D411yescharacteristic
147C2×Dic3C2×D411yescharacteristic
148C4×D3—22no—
149C4×D3—22no—
150C4×D3—41no—
151C4×D3—41no—
152C4×D3—42no—
153C4×D3—42no—
154Dic6—22no—
155Dic6—22no—
156Dic6—22no—
157Dic6—42no—
158Dic6—42no—
159C3×C8—22no—
160C3⋊C8—22no—
161C24—31no—
162C2×Q8—32no—
163C2×Q8—31no—
164C2×Q8—61no—
165C2×Q8—61no—
166C2×Q8—122no—
167C2×Q8—122no—
168C2×Q8C22×D311yescharacteristic
169C22×C4—21no—
170C22×C4—31no—
171C22×C4—31no—
172C22×C4—31no—
173C22×C4—31no—
174C22×C4—31no—
175C22×C4—61no—
176C22×C4—61no—
177C22×C4—61no—
178C22×C4—122no—
179C22×C4C22×D311yescharacteristic
180Q16—122no—
181Q16—122no—
182M16—122no—
183C2×C8—32no—
184C2×C8C22×D312yes—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—31no—
188C4⋊C4—61no—
189C4⋊C4—61no—
190C4⋊C4—61no—
191C4⋊C4—62no—
192C4⋊C4—62no—
193C4⋊C4—61no—
194C4⋊C4C22×D311yescharacteristic
195C4⋊C4C22×D312yes—
196C22⋊C4—21no—
197C22⋊C4—61no—
198C22⋊C4—61no—
199C22⋊C4—62no—
200C22⋊C4—62no—
201C22⋊C4—61no—
202C22×C3—21no—
203C22×C3—21no—
204C22×C3#13712yes—
205C22×C3C22×D411yescharacteristic
206C2×D3—21no—
207C2×D3—21no—
208C2×D3—21no—
209C2×D3—21no—
210C2×D3—21no—
211C2×D3—44no—
212C2×D3#13712yes—
213C3×C4—21no—
214C3×C4—41no—
215C3×C4—41no—
216C3×C4—42no—
217C3×C4H(5,𝔽2)12yes—
218Dic3—21no—
219Dic3—21no—
220Dic3—41no—
221Dic3—42no—
222Dic3—41no—
223C23—31no—
224C23—31no—
225C23—62no—
226C23—62no—
227C23—121no—
228C23D3×D411yescharacteristic
229Q8—22no—
230Q8—62no—
231Q8—62no—
232Q8—62no—
233Q8—122no—
234Q8—122no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—31no—
240C2×C4—31no—
241C2×C4—42no—
242C2×C4—62no—
243C2×C4—62no—
244C2×C4—61no—
245C2×C4—62no—
246C2×C4—61no—
247C2×C4—121no—
248C2×C4—121no—
249C2×C4—122no—
250C2×C4—122no—
251C2×C4—121no—
252C2×C4—122no—
253C2×C4D3×D411yescharacteristic
254C2×C4C23×D311yescharacteristic
255C8—22no—
256C8—62no—
257C2×C3—22no—
258C2×C3#53411yescharacteristic
259C2×C3C2×#13711yescharacteristic
260C2×C3C2×#13711yescharacteristic
261D3—22no—
262D3—41no—
263C22—21no—
264C22—21no—
265C22—32no—
266C22—61no—
267C22—61no—
268C22—61no—
269C22—61no—
270C22—61no—
271C22—124no—
272C22#91512yes—
273C22C2×D3×D411yescharacteristic
274C4—21no—
275C4—41no—
276C4—42no—
277C4—41no—
278C4—61no—
279C4—61no—
280C4—121no—
281C4—122no—
282C4—121no—
283C4#100912yes—
284C3#319511yescharacteristic, complemented, Sylow
285C2—22no—
286C2—62no—
287C2—121no—
288C2#565311yescharacteristic
289C2C2×#91511yescharacteristic
290C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#483C2
socleC22×C3C22×D4
prime radicalC23×D3D4
hypercenter#483D3
hypocenterC3#3195
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. C3×#483C2semidirect
4. #5050C2semidirect
5. #4971C2semidirect
6. #4867C2semidirect
7. #4862C2semidirect
8. #4816C2semidirect
9. #4809C2semidirect
10. #4739C2semidirect
19. C2×#216C22semidirect
23. C2×Dic12C22semidirect
35. C3×#106C22semidirect
36. C3×#103C22semidirect
40. #816C22semidirect
41. #810C22semidirect
42. #810C22semidirect
43. #808C22semidirect
48. #483D3semidirect
119. #106C2×D3semidirect
123. #103C2×D3semidirect
284. C3#3195semidirect

Maximal subgroups

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

subgroupindex
D3×C22⋊Q82
C2×#8972
C3×#4832
#50502
#49712
#48672
#48622
#48162
#48092
#47392
#47262
#31953