Finite Group Databasebeta

← #69979

Subgroups — #69979

Order 384.

286 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 73 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
2C3×#407C211yesmaximal, characteristic, complemented
3#5212C211yesmaximal, characteristic, complemented
4#4470C212yesmaximal, complemented
5#4467C212yesmaximal, complemented
6#1513—31nomaximal, Sylow
7C23×C3×C4C2211yescharacteristic
8C2×#235C411yescharacteristic, complemented
9C2×C3×M16—21no—
10#953—21no—
11C2×#217—21no—
12#930C411yescharacteristic, complemented
13#930C2211yescharacteristic
14#929—21no—
15C4×#67—21no—
16C4×#67—21no—
17#926C411yescharacteristic, complemented
18C2×C4⋊Dic3—21no—
19#924C411yescharacteristic, complemented
20#924C2211yescharacteristic
21C3×C22⋊C8—21no—
22C3×C22⋊C8C2212yes—
23C3⋊#95C2212yes—
24#525—31no—
25#407D311yescharacteristic, complemented
26#380—31no—
27#328—32no—
28#327—32no—
29C24×C3D411yescharacteristic
30C22×C3×C4—42no—
31C22×C3×C4C2×C411yescharacteristic
32C22×C3×C4C2×C411yescharacteristic
33C22×C3×C4D411yescharacteristic
34C22×C3×C4D411yescharacteristic
35C22×C3×C4D411yescharacteristic
36C22×C3×C4C2311yescharacteristic
37C2×#67—21no—
38C2×#67—21no—
39C2×#67—21no—
40C22×Dic3—22no—
41#235—42no—
42C2×D12C2×C411yescharacteristic
43C2×C4×D3—21no—
44C2×Dic6C2×C411yescharacteristic
45C3×M16—42no—
46C2×C3×C8—22no—
47#217—21no—
48#217—21no—
49#217—21no—
50#217—42no—
51(C2×D3)⋊C4—21no—
52(C2×D3)⋊C4—21no—
53C4⋊Dic3—42no—
54C4⋊Dic3C2×C411yescharacteristic
55C4⋊Dic3C2×C411yescharacteristic
56Dic3⋊C4—21no—
57Dic3⋊C4—21no—
58C4×Dic3—21no—
59C2×#41—31no—
60C23×C4C2×D311yescharacteristic
61C2×M16—21no—
62#123—61no—
63C22⋊Q8—31no—
64C22⋊Q8—31no—
65#121—31no—
66#121—31no—
67#120—61no—
68C4×D4—61no—
69C4×D4—61no—
70#117—31no—
71C2×C4⋊C4—61no—
72C2×C22⋊C4—61no—
73C22⋊C8—21no—
74C22⋊C8C2×D312yes—
75#95—32no—
76C23×C3—21no—
77C23×C3—21no—
78C23×C3—41no—
79C23×C3—41no—
80C23×C3C22⋊C411yescharacteristic
81C23×C3C22⋊C411yescharacteristic
82C23×C3C2×D411yescharacteristic
83C22×D3—21no—
84C2×C3×C4—22no—
85C2×C3×C4—22no—
86C2×C3×C4—22no—
87C2×C3×C4—41no—
88C2×C3×C4—41no—
89C2×C3×C4—41no—
90C2×C3×C4C22⋊C411yescharacteristic
91C2×C3×C4C22⋊C411yescharacteristic
92C2×C3×C4C22×C411yescharacteristic
93C2×C3×C4C2×D411yescharacteristic
94#67—42no—
95#67—42no—
96#67—41no—
97#67—41no—
98C2×Dic3—21no—
99C2×Dic3—21no—
100C2×Dic3—21no—
101C2×Dic3—42no—
102C2×Dic3—42no—
103C2×Dic3—42no—
104D12—41no—
105C4×D3—42no—
106Dic6—41no—
107C3×C8—42no—
108C24D1211yescharacteristic
109#41—122no—
110C2×Q8—31no—
111C2×D4—31no—
112C2×D4—61no—
113C2×D4—61no—
114C2×D4—61no—
115C22×C4—42no—
116C22×C4—61no—
117C22×C4—62no—
118C22×C4C4×D311yescharacteristic
119C22×C4C4×D311yescharacteristic
120C22×C4D1211yescharacteristic
121C22×C4#6711yescharacteristic
122C22×C4#6711yescharacteristic
123C22×C4C22×D311yescharacteristic
124M16—42no—
125C2×C8—22no—
126C4⋊C4—31no—
127C4⋊C4—31no—
128C4⋊C4—61no—
129C4⋊C4—61no—
130C4⋊C4—122no—
131C22⋊C4—61no—
132C22⋊C4—61no—
133C22⋊C4—61no—
134C22⋊C4—61no—
135C22⋊C4—61no—
136C22⋊C4—122no—
137C42—61no—
138C22×C3—21no—
139C22×C3—21no—
140C22×C3—21no—
141C22×C3—21no—
142C22×C3—21no—
143C22×C3—21no—
144C22×C3—21no—
145C22×C3—21no—
146C22×C3—42no—
147C22×C3—42no—
148C22×C3C2×C22⋊C411yescharacteristic
149C22×C3AGL(1,ℤ/8)11yescharacteristic
150C22×C3#13711yescharacteristic
151C2×D3—41no—
152C2×D3—41no—
153C2×D3—41no—
154C3×C4—21no—
155C3×C4—21no—
156C3×C4—21no—
157C3×C4#9912yes—
158Dic3—41no—
159Dic3—41no—
160Dic3—41no—
161Dic3—82no—
162C23—21no—
163C23—21no—
164C23—41no—
165C23—41no—
166C23—61no—
167C23(C2×D3)⋊C411yescharacteristic
168C23(C2×D3)⋊C411yescharacteristic
169C23C2×D1211yescharacteristic
170Q8—121no—
171D4—121no—
172D4—122no—
173D4—122no—
174D4—121no—
175D4—121no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—61no—
183C2×C4—61no—
184C2×C4—61no—
185C2×C4—122no—
186C2×C4—122no—
187C2×C4—122no—
188C2×C4—122no—
189C2×C4(C2×D3)⋊C411yescharacteristic
190C2×C4(C2×D3)⋊C411yescharacteristic
191C2×C4C2×C4×D311yescharacteristic
192C2×C4C2×#6711yescharacteristic
193C8—42no—
194C2×C3—21no—
195C2×C3—21no—
196C2×C3—41no—
197C2×C3—41no—
198C2×C3C2×#9911yescharacteristic
199C2×C3#41711yescharacteristic
200C2×C3#42111yescharacteristic
201D3—81no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—42no—
211C22—42no—
212C22—121no—
213C22—121no—
214C22—121no—
215C22#90811yescharacteristic
216C22#90911yescharacteristic
217C22C2×(C2×D3)⋊C411yescharacteristic
218C4—21no—
219C4—21no—
220C4—21no—
221C4—121no—
222C4—121no—
223C4—121no—
224C4—242no—
225C4#80612yes—
226C3#151311yescharacteristic, complemented, Sylow
227C2—21no—
228C2—21no—
229C2—41no—
230C2—41no—
231C2—241no—
232C2C2×#80611yescharacteristic
233C2#513311yescharacteristic
234C2#513911yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊C4
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#407C2
socleC22×C3C2×C22⋊C4
prime radical#5791C2
cosocleC22×C3×C4C23
hypercenter#407D3
hypocenterC3#1513
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C3×#407C2semidirect
3. #5212C2semidirect
4. #4470C2semidirect
5. #4467C2semidirect
8. C2×#235C4semidirect
12. #930C4semidirect
17. #926C4semidirect
19. #924C4semidirect
25. #407D3semidirect
226. C3#1513semidirect

Maximal subgroups

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

subgroupindex
#57912
C3×#4072
#52122
#44702
#44672
#15133