Finite Group Databasebeta

← #68147

Subgroups — #68147

Order 384.

270 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 65 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×#896C211yesmaximal, characteristic
2#4726C211yesmaximal, characteristic
3D3×C22⋊C8C211yesmaximal, characteristic
4C3×#324C211yesmaximal, characteristic, complemented
5#4528C211yesmaximal, characteristic, complemented
6#4469C211yesmaximal, characteristic, complemented
7#4469C211yesmaximal, characteristic, complemented
8#1472—31nomaximal, Sylow
9C22×C4×D3C2211yescharacteristic
10C22×Dic6C411yescharacteristic
11C2×C3×C4⋊C4C2211yescharacteristic
12C2×(C2×D3)⋊C4C411yescharacteristic
13C2×(C2×D3)⋊C4C411yescharacteristic
14C2×Dic3⋊C4C411yescharacteristic
15C2×Dic3⋊C4C2211yescharacteristic
16C2×ΓL(1,𝔽25)—21no—
17C2×D3×C8—21no—
18#896—21no—
19#896—21no—
20#896—21no—
21#896—21no—
22C3×C22⋊C8C2211yescharacteristic
23C3×C22⋊C8C2211yescharacteristic
24#830C2211yescharacteristic
25#830C2211yescharacteristic
26(C2×D3)⋊C8—21no—
27(C2×D3)⋊C8—21no—
28C2×C22⋊Q8—31no—
29#407—31no—
30C2×C22⋊C8—31no—
31#324—31no—
32#324—31no—
33#324—31no—
34#324D311yescharacteristic, complemented
35C23×D3C2×C411yescharacteristic
36C22×C3×C4C2×C411yescharacteristic
37C22×C3×C4C2×C411yescharacteristic
38C22×C3×C4C2311yescharacteristic
39C22×Dic3C2×C411yescharacteristic
40C22×Dic3C2×C411yescharacteristic
41C22×Dic3C2×C411yescharacteristic
42C2×C4×D3—42no—
43C2×C4×D3D411yescharacteristic
44C2×C4×D3D411yescharacteristic
45C2×C4×D3D411yescharacteristic
46C2×C4×D3D411yescharacteristic
47C2×Dic6—21no—
48C2×Dic6—21no—
49C2×Dic6—42no—
50C2×C3×C8—21no—
51C2×C3×C8—21no—
52C3×C4⋊C4—21no—
53C3×C4⋊C4—21no—
54(C2×D3)⋊C4—21no—
55(C2×D3)⋊C4—21no—
56(C2×D3)⋊C4—21no—
57(C2×D3)⋊C4—21no—
58Dic3⋊C4—21no—
59Dic3⋊C4—21no—
60Dic3⋊C4—21no—
61Dic3⋊C4—21no—
62C2×C3⋊C8—21no—
63C2×C3⋊C8—21no—
64ΓL(1,𝔽25)—42no—
65D3×C8—42no—
66C22×Q8—31no—
67C23×C4—31no—
68C2×M16—61no—
69C22×C8—61no—
70C22⋊Q8—61no—
71C22⋊Q8—61no—
72C22⋊Q8—61no—
73C22⋊Q8—61no—
74C2×C4⋊C4—31no—
75C2×C4⋊C4—31no—
76C2×C4⋊C4C2×D311yescharacteristic
77C2×C22⋊C4—31no—
78C2×C22⋊C4—31no—
79C22⋊C8—31no—
80C22⋊C8—31no—
81C22⋊C8—61no—
82C22⋊C8—61no—
83C22⋊C8C2×D311yescharacteristic
84C22⋊C8C2×D311yescharacteristic
85C23×C3C22×C411yescharacteristic
86C22×D3—21no—
87C22×D3—21no—
88C22×D3—22no—
89C22×D3—41no—
90C22×D3C22⋊C411yescharacteristic
91C22×D3C22⋊C411yescharacteristic
92C2×C3×C4—21no—
93C2×C3×C4—21no—
94C2×C3×C4—21no—
95C2×C3×C4—21no—
96C2×C3×C4—41no—
97C2×C3×C4—41no—
98C2×C3×C4C2×D411yescharacteristic
99C2×C3×C4C2×D411yescharacteristic
100C2×Dic3—22no—
101C2×Dic3—21no—
102C2×Dic3—21no—
103C2×Dic3—21no—
104C2×Dic3—21no—
105C2×Dic3—41no—
106C2×Dic3C22⋊C411yescharacteristic
107C2×Dic3C22⋊C411yescharacteristic
108C4×D3—22no—
109C4×D3—22no—
110C4×D3—41no—
111C4×D3—41no—
112Dic6—42no—
113Dic6—42no—
114C3×C8—41no—
115C3×C8—41no—
116C3⋊C8—41no—
117C3⋊C8—41no—
118C24—31no—
119C2×Q8—61no—
120C2×Q8—61no—
121C2×Q8—122no—
122C22×C4—31no—
123C22×C4—31no—
124C22×C4—31no—
125C22×C4—31no—
126C22×C4—31no—
127C22×C4—31no—
128C22×C4—31no—
129C22×C4—122no—
130C22×C4C4×D311yescharacteristic
131C22×C4C4×D311yescharacteristic
132C22×C4C22×D311yescharacteristic
133M16—122no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—61no—
137C2×C8—61no—
138C2×C8—122no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—61no—
142C4⋊C4—61no—
143C4⋊C4—61no—
144C4⋊C4—61no—
145C22⋊C4—61no—
146C22⋊C4—61no—
147C22⋊C4—61no—
148C22⋊C4—61no—
149C22×C3—21no—
150C22×C3—21no—
151C22×C3C2×C22⋊C411yescharacteristic
152C22×C3#13111yescharacteristic
153C22×C3#13111yescharacteristic
154C2×D3—21no—
155C2×D3—21no—
156C2×D3—21no—
157C2×D3—21no—
158C2×D3—21no—
159C2×D3—42no—
160C2×D3—42no—
161C2×D3#10112yes—
162C3×C4—21no—
163C3×C4—21no—
164C3×C4—41no—
165C3×C4—41no—
166Dic3—21no—
167Dic3—41no—
168Dic3—41no—
169Dic3#9912yes—
170C23—31no—
171C23—31no—
172C23—62no—
173C23—61no—
174C23—61no—
175C23—121no—
176C23C2×C4×D311yescharacteristic
177Q8—122no—
178Q8—122no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—31no—
184C2×C4—31no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—61no—
193C2×C4—61no—
194C2×C4—121no—
195C2×C4—121no—
196C2×C4—121no—
197C2×C4D3×D411yescharacteristic
198C2×C4D3×D411yescharacteristic
199C8—41no—
200C8—41no—
201C8—121no—
202C8—121no—
203C2×C3—21no—
204C2×C3—21no—
205C2×C3#40811yescharacteristic
206C2×C3C2×#9911yescharacteristic
207C2×C3C2×#10111yescharacteristic
208D3—22no—
209D3—41no—
210C22—21no—
211C22—21no—
212C22—32no—
213C22—61no—
214C22—61no—
215C22—61no—
216C22—61no—
217C22—61no—
218C22—122no—
219C22—122no—
220C22D3×C22⋊C411yescharacteristic
221C22#90111yescharacteristic
222C22#90711yescharacteristic
223C4—21no—
224C4—21no—
225C4—32no—
226C4—41no—
227C4—41no—
228C4—61no—
229C4—121no—
230C4—121no—
231C3#147211yescharacteristic, complemented, Sylow
232C2—21no—
233C2—21no—
234C2—62no—
235C2—121no—
236C2#472711yescharacteristic
237C2D3×#9911yescharacteristic
238C2D3×#10111yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC23×C3C22×C4
FrattiniC22×C4C22×D3
FittingC3×#324C2
socleC22×C3C2×C22⋊C4
prime radicalC23×D3C2×C4
cosocleC22×C3×C4C23
hypercenter#324D3
hypocenterC3#1472
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
4. C3×#324C2semidirect
5. #4528C2semidirect
6. #4469C2semidirect
7. #4469C2semidirect
34. #324D3semidirect
231. C3#1472semidirect

Maximal subgroups

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

subgroupindex
C2×#8962
#47262
D3×C22⋊C82
C3×#3242
#45282
#44692
#14723