Finite Group Databasebeta

← #80134

Subgroups — #80134

Order 384.

444 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 123 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×D4×Dic3C211yesmaximal, characteristic
2#5622C211yesmaximal, characteristic, complemented
3C3×#450C211yesmaximal, characteristic
4#5218C211yesmaximal, characteristic
5#5164C212yesmaximal, complemented
6#5162C212yesmaximal, complemented
7#5044C211yesmaximal, characteristic, complemented
8#4759C212yesmaximal, complemented
9#4757C212yesmaximal, complemented
10#4722C211yesmaximal, characteristic, complemented
11#4719C211yesmaximal, characteristic, complemented
12#3110—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×Dic3—21no—
15C2×C3×SD16C2212yes—
16C3×C22⋊Q8C2211yescharacteristic
17C2×#217—21no—
18Q8×Dic3C2211yescharacteristic, complemented
19#944C2211yescharacteristic, complemented
20D4×Dic3—21no—
21D4×Dic3—21no—
22D4×Dic3C2212yes—
23D4×Dic3C2212yes—
24C2×#214C2212yes—
25C2×C4⋊Dic3C2211yescharacteristic
26#924C2211yescharacteristic
27C2×C4×Dic3C2211yescharacteristic
28#889—21no—
29#888C2211yescharacteristic, complemented
30#887—21no—
31#887C2211yescharacteristic, complemented
32#878—21no—
33#877—21no—
34C3×#102C2212yes—
35C3×C22⋊C8C2211yescharacteristic
36#835C2212yescomplemented
37#832C2212yes—
38#830C2211yescharacteristic
39#818C2212yescomplemented
40#816C2212yescomplemented
41#815C2212yescomplemented
42Dic3⋊C8C2212yescomplemented
43#807C2212yescomplemented
44#533—31no—
45C2×C4×D4—31no—
46#484—31no—
47#484—31no—
48#474—32no—
49#461—32no—
50#450—31no—
51#450D311yescharacteristic
52#440—32no—
53#440—32no—
54#432—31no—
55C24×C3—21no—
56C2×C3×Q8C2311yescharacteristic
57C2×C3×D4—24no—
58C2×C3×D4—21no—
59C2×C3×D4C2312yes—
60C22×C3×C4C2311yescharacteristic
61C22×Dic3—21no—
62C22×Dic3—22no—
63C22×Dic3—22no—
64C22×Dic3—44no—
65C22×Dic3D411yescharacteristic
66C22×Dic3D411yescharacteristic
67C2×Dic6—21no—
68C2×Dic6C2311yescharacteristic
69C3×SD16—24no—
70C2×C3×C8C2312yes—
71C3×C4⋊C4—21no—
72C3×C4⋊C4C2311yescharacteristic
73C3×C22⋊C4—21no—
74#217—21no—
75#217—22no—
76#217—22no—
77#214—24no—
78C4⋊Dic3—21no—
79C4⋊Dic3—21no—
80C4⋊Dic3C2312yes—
81C4⋊Dic3C2311yescharacteristic
82Dic3⋊C4—21no—
83Dic3⋊C4—21no—
84Dic3⋊C4—21no—
85Dic3⋊C4—21no—
86Dic3⋊C4—21no—
87C4×Dic3—21no—
88C4×Dic3—21no—
89C4×Dic3D411yescharacteristic
90C4×Dic3D411yescharacteristic
91C4×Dic3C2311yescharacteristic
92C4×Dic3C2311yescharacteristic
93C2×C3⋊C8C2312yes—
94C22×D4C2×D311yescharacteristic
95C23×C4—61no—
96C2×SD16—32no—
97C2×SD16C2×D312yes—
98C4⋊Q8—31no—
99C4⋊Q8—31no—
100#125—61no—
101C22⋊Q8—31no—
102C22⋊Q8—61no—
103C22⋊Q8C2×D311yescharacteristic
104C4×Q8—31no—
105C4×Q8—31no—
106C4×Q8—61no—
107C4×D4—32no—
108C4×D4—32no—
109C4×D4—61no—
110C4×D4—61no—
111#117—61no—
112C2×C4⋊C4—31no—
113C2×C22⋊C4—61no—
114C2×C42—31no—
115#107—32no—
116#107—32no—
117C4⋊C8—32no—
118#103—32no—
119#103—32no—
120#102—32no—
121#102C2×D312yes—
122C22⋊C8—31no—
123C22⋊C8C2×D311yescharacteristic
124#97—32no—
125C23×C3—24no—
126C23×C3—22no—
127C23×C3—42no—
128C23×C3—42no—
129C23×C3C2×D411yescharacteristic
130C3×Q8—22no—
131C3×D4—24no—
132C3×D4—22no—
133C3×D4#4114yes—
134C2×C3×C4—22no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4C2×D411yescharacteristic
138C2×C3×C4C2411yescharacteristic
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—21no—
142C2×Dic3—21no—
143C2×Dic3—22no—
144C2×Dic3—44no—
145C2×Dic3—42no—
146C2×Dic3—44no—
147C2×Dic3C2×D411yescharacteristic
148C2×Dic3C2×D411yescharacteristic
149C2×Dic3C2×D411yescharacteristic
150C2×Dic3C2×D411yescharacteristic
151Dic6—22no—
152Dic6—42no—
153C3×C8—22no—
154C3⋊C8—22no—
155C24—21no—
156C2×Q8—31no—
157C2×Q8—61no—
158C2×Q8C22×D311yescharacteristic
159C2×D4—24no—
160C2×D4—21no—
161C2×D4C22×D312yes—
162C22×C4—31no—
163C22×C4—31no—
164C22×C4—62no—
165C22×C4—62no—
166C22×C4—61no—
167C22×C4—124no—
168C22×C4C22×D311yescharacteristic
169SD16—24no—
170SD16—64no—
171C2×C8—32no—
172C2×C8C22×D312yes—
173C4⋊C4—21no—
174C4⋊C4—31no—
175C4⋊C4—32no—
176C4⋊C4—61no—
177C4⋊C4—61no—
178C4⋊C4—61no—
179C4⋊C4—61no—
180C4⋊C4—61no—
181C4⋊C4—61no—
182C4⋊C4—61no—
183C4⋊C4C22×D311yescharacteristic
184C22⋊C4—21no—
185C22⋊C4—62no—
186C22⋊C4—62no—
187C22⋊C4—61no—
188C42—31no—
189C42—31no—
190C42—31no—
191C42—31no—
192C42—61no—
193C42—61no—
194C22×C3—21no—
195C22×C3—21no—
196C22×C3—24no—
197C22×C3—48no—
198C22×C3—42no—
199C22×C3—42no—
200C22×C3#13712yes—
201C22×C3C22×D411yescharacteristic
202C3×C4—21no—
203C3×C4—41no—
204C3×C4—41no—
205C3×C4C2×#4112yes—
206Dic3—21no—
207Dic3—21no—
208Dic3—21no—
209Dic3—41no—
210Dic3—41no—
211Dic3—42no—
212Dic3AGL(1,ℤ/8)12yes—
213C23—24no—
214C23—22no—
215C23—42no—
216C23—42no—
217C23D3×D411yescharacteristic
218Q8—22no—
219Q8—62no—
220Q8—122no—
221D4—24no—
222D4—22no—
223D4#23714yes—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—31no—
228C2×C4—31no—
229C2×C4—31no—
230C2×C4—31no—
231C2×C4—62no—
232C2×C4—62no—
233C2×C4—61no—
234C2×C4—61no—
235C2×C4—62no—
236C2×C4—124no—
237C2×C4—124no—
238C2×C4—122no—
239C2×C4D3×D411yescharacteristic
240C2×C4C23×D311yescharacteristic
241C8—22no—
242C8—62no—
243C2×C3—22no—
244C2×C3—44no—
245C2×C3#52511yescharacteristic
246C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
247C2×C3C2×#13711yescharacteristic
248C22—21no—
249C22—21no—
250C22—24no—
251C22—48no—
252C22—42no—
253C22—42no—
254C22#91512yes—
255C22C2×D3×D411yescharacteristic
256C4—21no—
257C4—32no—
258C4—41no—
259C4—41no—
260C4—61no—
261C4—61no—
262C4—61no—
263C4—122no—
264C4—121no—
265C4—121no—
266C4C2×#23712yes—
267C3#311011yescharacteristic, complemented, Sylow
268C2—22no—
269C2—44no—
270C2#558611yescharacteristic
271C2C2×#91511yescharacteristic
272C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#450C2
socleC22×C3C22×D4
prime radicalC22×C3×D4C22
hypercenter#450D3
hypocenterC3#3110
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. #5622C2semidirect
5. #5164C2semidirect
6. #5162C2semidirect
7. #5044C2semidirect
8. #4759C2semidirect
9. #4757C2semidirect
10. #4722C2semidirect
11. #4719C2semidirect
18. Q8×Dic3C22semidirect
19. #944C22semidirect
29. #888C22semidirect
31. #887C22semidirect
36. #835C22semidirect
39. #818C22semidirect
40. #816C22semidirect
41. #815C22semidirect
42. Dic3⋊C8C22semidirect
43. #807C22semidirect
267. C3#3110semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic32
#56222
C3×#4502
#52182
#51642
#51622
#50442
#47592
#47572
#47222
#47192
#31103