Finite Group Databasebeta

← #68336

Subgroups — #68336

Order 384.

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

normal subgroups only

This group has 274 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
1D3×C22⋊Q8C211yesmaximal, characteristic
2#4726C211yesmaximal, characteristic
3#4664C211yesmaximal, characteristic
4C3×#328C211yesmaximal, characteristic, complemented
5#4538C211yesmaximal, characteristic, complemented
6#4467C211yesmaximal, characteristic, complemented
7#4452C211yesmaximal, characteristic, complemented
8#1511—31nomaximal, Sylow
9C2×D3×Q8C411yescharacteristic, complemented
10C22×C4×D3C2211yescharacteristic
11C3×C22⋊Q8C2211yescharacteristic
12#946C411yescharacteristic, complemented
13C2×(C2×D3)⋊C4—21no—
14#924C2211yescharacteristic
15C2×ΓL(1,𝔽25)—21no—
16#897C411yescharacteristic, complemented
17#896—21no—
18D3×C4⋊C4—21no—
19D3×C4⋊C4C411yescharacteristic, complemented
20D3×C22⋊C4—21no—
21#878—21no—
22C3×C22⋊C8C2211yescharacteristic
23C3×#95C2211yescharacteristic
24C3⋊#95—21no—
25C3⋊#95C2211yescharacteristic
26#830C2211yescharacteristic
27(C2×D3)⋊C8—21no—
28C2×C22⋊Q8—31no—
29#407—31no—
30#381—31no—
31#328—31no—
32#328—31no—
33#328—31no—
34#328D311yescharacteristic, complemented
35C23×D3D411yescharacteristic
36C2×C3×Q8C2×C411yescharacteristic
37C22×C3×C4—21no—
38C22×C3×C4C2311yescharacteristic
39C22×Dic3—21no—
40C22×Dic3D411yescharacteristic
41D3×Q8—42no—
42C2×C4×D3—21no—
43C2×C4×D3—21no—
44C2×C4×D3—42no—
45C2×C4×D3C2×C411yescharacteristic
46C2×C4×D3C2×C411yescharacteristic
47C2×C4×D3D411yescharacteristic
48C2×C4×D3D411yescharacteristic
49C2×Dic6—21no—
50C2×Dic6C2×C411yescharacteristic
51C2×C3×C8—21no—
52C3×C4⋊C4—21no—
53C3×C4⋊C4C2×C411yescharacteristic
54C3×C22⋊C4—21no—
55#217—21no—
56(C2×D3)⋊C4—21no—
57(C2×D3)⋊C4—21no—
58(C2×D3)⋊C4—42no—
59C4⋊Dic3—21no—
60C4⋊Dic3C2×C411yescharacteristic
61Dic3⋊C4—21no—
62Dic3⋊C4—21no—
63Dic3⋊C4—21no—
64C2×C3⋊C8—21no—
65ΓL(1,𝔽25)—42no—
66C22×Q8—31no—
67C23×C4—31no—
68C2×M16—61no—
69C22⋊Q8—31no—
70C22⋊Q8—31no—
71C22⋊Q8—31no—
72C22⋊Q8—61no—
73C22⋊Q8—61no—
74C22⋊Q8C2×D311yescharacteristic
75C2×C4⋊C4—31no—
76C2×C4⋊C4—61no—
77C2×C22⋊C4—61no—
78C2×C22⋊C4—61no—
79C22⋊C8—31no—
80C22⋊C8—61no—
81C22⋊C8C2×D311yescharacteristic
82#95—31no—
83#95—61no—
84#95C2×D311yescharacteristic
85C23×C3C2×D411yescharacteristic
86C22×D3—21no—
87C22×D3—21no—
88C22×D3—22no—
89C22×D3—41no—
90C22×D3C22⋊C411yescharacteristic
91C22×D3C22⋊C411yescharacteristic
92C3×Q8—41no—
93C2×C3×C4—21no—
94C2×C3×C4—21no—
95C2×C3×C4—41no—
96C2×C3×C4—41no—
97C2×C3×C4—41no—
98C2×C3×C4—41no—
99C2×C3×C4C22×C411yescharacteristic
100C2×C3×C4C2×D411yescharacteristic
101C2×Dic3—21no—
102C2×Dic3—21no—
103C2×Dic3—41no—
104C2×Dic3—41no—
105C2×Dic3—41no—
106C2×Dic3—41no—
107C2×Dic3C22⋊C411yescharacteristic
108C2×Dic3C22⋊C411yescharacteristic
109C4×D3—22no—
110C4×D3—22no—
111C4×D3—41no—
112C4×D3—41no—
113C4×D3—42no—
114C4×D3—42no—
115Dic6—42no—
116Dic6—41no—
117C3×C8—41no—
118C3⋊C8—41no—
119C24—31no—
120C2×Q8—31no—
121C2×Q8—61no—
122C2×Q8—122no—
123C2×Q8C4×D311yescharacteristic
124C22×C4—21no—
125C22×C4—31no—
126C22×C4—31no—
127C22×C4—31no—
128C22×C4—31no—
129C22×C4—31no—
130C22×C4—61no—
131C22×C4—61no—
132C22×C4—61no—
133C22×C4—122no—
134C22×C4C22×D311yescharacteristic
135M16—122no—
136C2×C8—21no—
137C2×C8—61no—
138C4⋊C4—21no—
139C4⋊C4—31no—
140C4⋊C4—61no—
141C4⋊C4—61no—
142C4⋊C4—61no—
143C4⋊C4—61no—
144C4⋊C4C4×D311yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—61no—
147C22⋊C4—61no—
148C22⋊C4—61no—
149C22⋊C4—122no—
150C22×C3—21no—
151C22×C3—21no—
152C22×C3C2×C22⋊C411yescharacteristic
153C22×C3#13711yescharacteristic
154C22×C3#13711yescharacteristic
155C2×D3—21no—
156C2×D3—21no—
157C2×D3—21no—
158C2×D3—21no—
159C2×D3—21no—
160C2×D3—42no—
161C2×D3—42no—
162C2×D3#9912yes—
163C3×C4—21no—
164C3×C4—21no—
165C3×C4—41no—
166C3×C4—41no—
167C3×C4—81no—
168Dic3—21no—
169Dic3—21no—
170Dic3—41no—
171Dic3—41no—
172Dic3—81no—
173C23—31no—
174C23—31no—
175C23—62no—
176C23—61no—
177C23—61no—
178C23—121no—
179C23D3×D411yescharacteristic
180Q8—41no—
181Q8—121no—
182Q8—122no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—61no—
194C2×C4—61no—
195C2×C4—121no—
196C2×C4—121no—
197C2×C4—122no—
198C2×C4—122no—
199C2×C4—121no—
200C2×C4—121no—
201C2×C4—121no—
202C2×C4—121no—
203C2×C4C2×C4×D311yescharacteristic
204C2×C4D3×D411yescharacteristic
205C8—41no—
206C8—121no—
207C2×C3—21no—
208C2×C3—21no—
209C2×C3C2×#9911yescharacteristic
210C2×C3#41911yescharacteristic
211C2×C3#42111yescharacteristic
212D3—22no—
213D3—41no—
214C22—21no—
215C22—21no—
216C22—32no—
217C22—61no—
218C22—61no—
219C22—61no—
220C22—61no—
221C22—61no—
222C22—122no—
223C22—122no—
224C22D3×C22⋊C411yescharacteristic
225C22#91511yescharacteristic
226C22#91811yescharacteristic
227C4—21no—
228C4—21no—
229C4—41no—
230C4—41no—
231C4—61no—
232C4—61no—
233C4—81no—
234C4—121no—
235C4—121no—
236C4—241no—
237C3#151111yescharacteristic, complemented, Sylow
238C2—21no—
239C2—21no—
240C2—62no—
241C2—121no—
242C2D3×#9911yescharacteristic
243C2#480211yescharacteristic
244C2#482111yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#328C2
socleC22×C3C2×C22⋊C4
prime radicalC23×D3D4
cosocleC22×C3×C4C23
hypercenter#328D3
hypocenterC3#1511
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. C3×#328C2semidirect
5. #4538C2semidirect
6. #4467C2semidirect
7. #4452C2semidirect
9. C2×D3×Q8C4semidirect
12. #946C4semidirect
16. #897C4semidirect
19. D3×C4⋊C4C4semidirect
34. #328D3semidirect
237. C3#1511semidirect

Maximal subgroups

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

subgroupindex
D3×C22⋊Q82
#47262
#46642
C3×#3282
#45382
#44672
#44522
#15113