Finite Group Databasebeta

← #68122

Subgroups — #68122

Order 384.

280 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 63 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×#883C211yesmaximal, characteristic
2#4726C211yesmaximal, characteristic, complemented
3D3×C22⋊C8C211yesmaximal, characteristic, complemented
4C3×#323C211yesmaximal, characteristic, complemented
5#4525C211yesmaximal, characteristic, complemented
6#4469C211yesmaximal, characteristic, complemented
7#4469C211yesmaximal, characteristic, complemented
8#1468—31nomaximal, Sylow
9C22×#67C411yescharacteristic
10C22×C4×D3C2211yescharacteristic
11C2×C3×C22⋊C4C2211yescharacteristic
12C2×#217C411yescharacteristic
13C2×(C2×D3)⋊C4C411yescharacteristic
14C2×C4⋊Dic3C411yescharacteristic
15C2×Dic3⋊C4C2211yescharacteristic
16C2×ΓL(1,𝔽25)—21no—
17C2×D3×C8—21no—
18#883—21no—
19#883—21no—
20#883—21no—
21#883—21no—
22C3×C22⋊C8C2211yescharacteristic
23C3×C22⋊C8C2211yescharacteristic
24#830C2211yescharacteristic
25#830C2211yescharacteristic
26(C2×D3)⋊C8—21no—
27(C2×D3)⋊C8—21no—
28C2×#123—31no—
29#407—31no—
30C2×C22⋊C8—31no—
31#324—31no—
32#324—31no—
33#323—31no—
34#323D311yescharacteristic, complemented
35C24×C3C2×C411yescharacteristic
36C23×D3C2×C411yescharacteristic
37C22×C3×C4C2×C411yescharacteristic
38C22×C3×C4C2311yescharacteristic
39C2×#67—21no—
40C2×#67—21no—
41C2×#67—42no—
42C22×Dic3C2×C411yescharacteristic
43C22×Dic3C2×C411yescharacteristic
44C22×Dic3C2×C411yescharacteristic
45C2×C4×D3—42no—
46C2×C4×D3D411yescharacteristic
47C2×C4×D3D411yescharacteristic
48C2×C4×D3D411yescharacteristic
49C2×C4×D3D411yescharacteristic
50C2×C3×C8—21no—
51C2×C3×C8—21no—
52C3×C22⋊C4—21no—
53C3×C22⋊C4—21no—
54#217—21no—
55#217—21no—
56(C2×D3)⋊C4—21no—
57(C2×D3)⋊C4—21no—
58C4⋊Dic3—21no—
59C4⋊Dic3—21no—
60Dic3⋊C4—21no—
61Dic3⋊C4—21no—
62C2×C3⋊C8—21no—
63C2×C3⋊C8—21no—
64ΓL(1,𝔽25)—42no—
65D3×C8—42no—
66C22×D4—31no—
67C23×C4—31no—
68C2×M16—61no—
69C22×C8—61no—
70#123—61no—
71#123—61no—
72#123—61no—
73#123—61no—
74C2×C4⋊C4—31no—
75C2×C4⋊C4—31no—
76C2×C22⋊C4—31no—
77C2×C22⋊C4—31no—
78C2×C22⋊C4C2×D311yescharacteristic
79C22⋊C8—31no—
80C22⋊C8—31no—
81C22⋊C8—61no—
82C22⋊C8—61no—
83C22⋊C8C2×D311yescharacteristic
84C22⋊C8C2×D311yescharacteristic
85C23×C3—21no—
86C23×C3—21no—
87C23×C3—21no—
88C23×C3—41no—
89C23×C3—41no—
90C23×C3C22×C411yescharacteristic
91C22×D3—21no—
92C22×D3—21no—
93C22×D3—22no—
94C22×D3—41no—
95C22×D3C22⋊C411yescharacteristic
96C22×D3C22⋊C411yescharacteristic
97C2×C3×C4—21no—
98C2×C3×C4—41no—
99C2×C3×C4—41no—
100C2×C3×C4C2×D411yescharacteristic
101C2×C3×C4C2×D411yescharacteristic
102#67—42no—
103#67—42no—
104C2×Dic3—21no—
105C2×Dic3—21no—
106C2×Dic3—21no—
107C2×Dic3—21no—
108C2×Dic3—41no—
109C2×Dic3—41no—
110C2×Dic3C22⋊C411yescharacteristic
111C2×Dic3C22⋊C411yescharacteristic
112C4×D3—22no—
113C4×D3—22no—
114C4×D3—41no—
115C4×D3—41no—
116C3×C8—41no—
117C3×C8—41no—
118C3⋊C8—41no—
119C3⋊C8—41no—
120C24—31no—
121C24C4×D311yescharacteristic
122C2×D4—61no—
123C2×D4—61no—
124C2×D4—122no—
125C22×C4—31no—
126C22×C4—31no—
127C22×C4—31no—
128C22×C4—31no—
129C22×C4—31no—
130C22×C4—31no—
131C22×C4—31no—
132C22×C4—122no—
133C22×C4C4×D311yescharacteristic
134C22×C4C22×D311yescharacteristic
135M16—122no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—61no—
139C2×C8—61no—
140C2×C8—122no—
141C4⋊C4—61no—
142C4⋊C4—61no—
143C4⋊C4—61no—
144C4⋊C4—61no—
145C22⋊C4—21no—
146C22⋊C4—21no—
147C22⋊C4—61no—
148C22⋊C4—61no—
149C22⋊C4—61no—
150C22⋊C4—61no—
151C22×C3—21no—
152C22×C3—21no—
153C22×C3—41no—
154C22×C3—41no—
155C22×C3—41no—
156C22×C3—81no—
157C22×C3—81no—
158C22×C3C2×C22⋊C411yescharacteristic
159C22×C3#13111yescharacteristic
160C22×C3#13111yescharacteristic
161C2×D3—21no—
162C2×D3—21no—
163C2×D3—21no—
164C2×D3—21no—
165C2×D3—21no—
166C2×D3—42no—
167C2×D3—42no—
168C2×D3#9912yes—
169C3×C4—21no—
170C3×C4—21no—
171C3×C4—41no—
172Dic3—21no—
173Dic3—21no—
174Dic3—41no—
175Dic3—41no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23—31no—
180C23—31no—
181C23—41no—
182C23—41no—
183C23—62no—
184C23—61no—
185C23—61no—
186C23—121no—
187C23C2×C4×D311yescharacteristic
188D4—122no—
189D4—122no—
190C2×C4—21no—
191C2×C4—31no—
192C2×C4—31no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4—61no—
198C2×C4—61no—
199C2×C4—61no—
200C2×C4—61no—
201C2×C4—121no—
202C2×C4—121no—
203C2×C4—121no—
204C2×C4—121no—
205C2×C4D3×D411yescharacteristic
206C2×C4D3×D411yescharacteristic
207C8—41no—
208C8—41no—
209C8—121no—
210C8—121no—
211C2×C3—21no—
212C2×C3—21no—
213C2×C3—81no—
214C2×C3#40811yescharacteristic
215C2×C3C2×#9911yescharacteristic
216C2×C3#41311yescharacteristic
217D3—22no—
218D3—41no—
219C22—21no—
220C22—21no—
221C22—32no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—61no—
226C22—61no—
227C22—61no—
228C22—61no—
229C22—61no—
230C22—81no—
231C22—81no—
232C22—122no—
233C22—122no—
234C22D3×C22⋊C411yescharacteristic
235C22#90111yescharacteristic
236C22#90711yescharacteristic
237C4—21no—
238C4—21no—
239C4—41no—
240C4—61no—
241C4—61no—
242C4—121no—
243C4—121no—
244C3#146811yescharacteristic, complemented, Sylow
245C2—21no—
246C2—21no—
247C2—62no—
248C2—81no—
249C2—121no—
250C2#472711yescharacteristic
251C2D3×#9911yescharacteristic
252C2#474511yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC23×C3C22×C4
FrattiniC22×C4C22×D3
FittingC3×#323C2
socleC22×C3C2×C22⋊C4
prime radicalC22×#67C4
cosocleC22×C3×C4C23
hypercenter#323D3
hypocenterC3#1468
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #4726C2semidirect
3. D3×C22⋊C8C2semidirect
4. C3×#323C2semidirect
5. #4525C2semidirect
6. #4469C2semidirect
7. #4469C2semidirect
34. #323D3semidirect
244. C3#1468semidirect

Maximal subgroups

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

subgroupindex
C2×#8832
#47262
D3×C22⋊C82
C3×#3232
#45252
#44692
#14683