Finite Group Databasebeta

← #69398

Subgroups — #69398

Order 384.

334 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 77 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#5791C211yesmaximal, characteristic
2C2×C3×#95C211yesmaximal, characteristic, complemented
3#5207C211yesmaximal, characteristic, complemented
4#4452C212yesmaximal, complemented
5#4451C212yesmaximal, complemented
6#1504—31nomaximal, Sylow
7C23×C3×C4—21no—
8C23×C3×C4C2211yescharacteristic
9C2×#235C411yescharacteristic, complemented
10#953—21no—
11#930C411yescharacteristic, complemented
12#930C2211yescharacteristic
13#929—21no—
14C4×#67—21no—
15C4×#67—21no—
16#926C411yescharacteristic, complemented
17#924C411yescharacteristic, complemented
18#924C2211yescharacteristic
19C2×#208—21no—
20C3×#95—21no—
21C3×#95C2212yes—
22#830—21no—
23#830C2212yes—
24#525—31no—
25#407—31no—
26C2×#95D311yescharacteristic, complemented
27#328—32no—
28#327—32no—
29C24×C3D411yescharacteristic
30C22×C3×C4—24no—
31C22×C3×C4—22no—
32C22×C3×C4—42no—
33C22×C3×C4—42no—
34C22×C3×C4—42no—
35C22×C3×C4C2×C411yescharacteristic
36C22×C3×C4C2×C411yescharacteristic
37C22×C3×C4D411yescharacteristic
38C22×C3×C4D411yescharacteristic
39C22×C3×C4D411yescharacteristic
40C22×C3×C4C2311yescharacteristic
41C2×#67—21no—
42C2×#67—21no—
43C2×#67—21no—
44#235—42no—
45C2×D12C2×C411yescharacteristic, complemented
46C2×C4×D3—21no—
47C2×Dic6C2×C411yescharacteristic, complemented
48#217—21no—
49#217—21no—
50#217—21no—
51(C2×D3)⋊C4—21no—
52(C2×D3)⋊C4—21no—
53C4⋊Dic3C2×C411yescharacteristic, complemented
54C4⋊Dic3C2×C411yescharacteristic, complemented
55Dic3⋊C4—21no—
56Dic3⋊C4—21no—
57C4×Dic3—21no—
58#208—42no—
59C2×C3⋊C8—22no—
60C2×#41—31no—
61C23×C4—21no—
62C23×C4C2×D311yescharacteristic
63C2×M16—61no—
64#123—61no—
65C22⋊Q8—31no—
66C22⋊Q8—31no—
67#121—31no—
68#121—31no—
69#120—61no—
70C4×D4—61no—
71C4×D4—61no—
72#117—31no—
73C22⋊C8—32no—
74C22⋊C8—61no—
75#95—21no—
76#95C2×D312yes—
77C23×C3—22no—
78C23×C3—22no—
79C23×C3—21no—
80C23×C3—21no—
81C23×C3C22⋊C411yescharacteristic
82C23×C3C22⋊C411yescharacteristic
83C23×C3C2×D411yescharacteristic
84C22×D3—21no—
85C2×C3×C4—22no—
86C2×C3×C4—22no—
87C2×C3×C4—24no—
88C2×C3×C4—41no—
89C2×C3×C4—41no—
90C2×C3×C4—41no—
91C2×C3×C4—41no—
92C2×C3×C4—48no—
93C2×C3×C4—42no—
94C2×C3×C4—42no—
95C2×C3×C4C22⋊C411yescharacteristic
96C2×C3×C4C22⋊C411yescharacteristic
97C2×C3×C4C22×C411yescharacteristic
98C2×C3×C4C2×D411yescharacteristic
99#67—42no—
100#67—42no—
101#67—41no—
102#67—41no—
103C2×Dic3—21no—
104C2×Dic3—21no—
105C2×Dic3—21no—
106D12—41no—
107C4×D3—42no—
108Dic6—41no—
109C3⋊C8—42no—
110C24#6711yescharacteristic
111#41—122no—
112C2×Q8—31no—
113C2×D4—31no—
114C2×D4—61no—
115C2×D4—61no—
116C2×D4—61no—
117C22×C4—24no—
118C22×C4—22no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—61no—
123C22×C4C4×D311yescharacteristic
124C22×C4C4×D311yescharacteristic
125C22×C4D1211yescharacteristic
126C22×C4D1211yescharacteristic
127C22×C4#6711yescharacteristic
128C22×C4C22×D311yescharacteristic
129M16—122no—
130C2×C8—62no—
131C4⋊C4—31no—
132C4⋊C4—31no—
133C4⋊C4—61no—
134C4⋊C4—61no—
135C22⋊C4—61no—
136C22⋊C4—61no—
137C22⋊C4—61no—
138C22⋊C4—61no—
139C22⋊C4—61no—
140C42—61no—
141C22×C3—24no—
142C22×C3—24no—
143C22×C3—21no—
144C22×C3—21no—
145C22×C3—21no—
146C22×C3—21no—
147C22×C3—21no—
148C22×C3—21no—
149C22×C3#9912yes—
150C22×C3#10412yes—
151C22×C3C2×C22⋊C411yescharacteristic
152C22×C3AGL(1,ℤ/8)11yescharacteristic
153C22×C3#13711yescharacteristic
154C2×D3—41no—
155C2×D3—41no—
156C2×D3—41no—
157C3×C4—21no—
158C3×C4—21no—
159C3×C4—21no—
160C3×C4—21no—
161C3×C4—44no—
162Dic3—41no—
163Dic3—41no—
164Dic3—41no—
165C23—22no—
166C23—22no—
167C23—21no—
168C23—21no—
169C23—61no—
170C23(C2×D3)⋊C411yescharacteristic
171C23(C2×D3)⋊C411yescharacteristic
172C23C2×#6711yescharacteristic
173Q8—121no—
174D4—121no—
175D4—122no—
176D4—122no—
177D4—121no—
178D4—121no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—48no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—61no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—122no—
193C2×C4(C2×D3)⋊C411yescharacteristic
194C2×C4(C2×D3)⋊C411yescharacteristic
195C2×C4C2×C4×D311yescharacteristic
196C2×C4C2×D1211yescharacteristic
197C8—122no—
198C2×C3—21no—
199C2×C3—22no—
200C2×C3—22no—
201C2×C3—21no—
202C2×C3C2×#9911yescharacteristic
203C2×C3#41711yescharacteristic
204C2×C3C2×#10411yescharacteristic
205D3—81no—
206C22—24no—
207C22—24no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—121no—
215C22—121no—
216C22—121no—
217C22#80512yes—
218C22#80612yes—
219C22C2×(C2×D3)⋊C411yescharacteristic
220C22#93211yescharacteristic
221C22#94211yescharacteristic
222C4—21no—
223C4—21no—
224C4—21no—
225C4—21no—
226C4—44no—
227C4—121no—
228C4—121no—
229C4—121no—
230C3#150411yescharacteristic, complemented, Sylow
231C2—21no—
232C2—22no—
233C2—22no—
234C2—21no—
235C2—241no—
236C2C2×#80511yescharacteristic
237C2C2×#80611yescharacteristic
238C2#496611yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊C4
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC2×C3×#95C2
socleC22×C3C2×C22⋊C4
prime radical#5791C2
cosocleC22×C3×C4C23
hypercenterC2×#95D3
hypocenterC3#1504
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
2. C2×C3×#95C2semidirect
3. #5207C2semidirect
4. #4452C2semidirect
5. #4451C2semidirect
9. C2×#235C4semidirect
11. #930C4semidirect
16. #926C4semidirect
17. #924C4semidirect
26. C2×#95D3semidirect
45. C2×D12C2×C4semidirect
47. C2×Dic6C2×C4semidirect
53. C4⋊Dic3C2×C4semidirect
54. C4⋊Dic3C2×C4semidirect
230. C3#1504semidirect

Maximal subgroups

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

subgroupindex
#57912
C2×C3×#952
#52072
#44522
#44512
#15043