Finite Group Databasebeta

← #68335

Subgroups — #68335

Order 384.

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

normal subgroups only

This group has 294 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
1D3×C22⋊Q8C211yesmaximal, characteristic
2#4726C211yesmaximal, characteristic
3#4666C211yesmaximal, characteristic
4C3×#328C211yesmaximal, characteristic, complemented
5#4538C211yesmaximal, characteristic, complemented
6#4467C211yesmaximal, characteristic, complemented
7#4452C211yesmaximal, characteristic, complemented
8#1508—31nomaximal, Sylow
9C2×D3×Q8C411yescharacteristic, complemented
10C22×C4×D3C2211yescharacteristic
11C3×C22⋊Q8C2211yescharacteristic
12#946C411yescharacteristic, complemented
13C2×(C2×D3)⋊C4—21no—
14#924C2211yescharacteristic
15C2×C4×Dic3—21no—
16C2×ΓL(1,𝔽25)—21no—
17#897C411yescharacteristic, complemented
18#896—21no—
19D3×C4⋊C4—21no—
20D3×C4⋊C4C411yescharacteristic, complemented
21D3×C22⋊C4—21no—
22#878—21no—
23C3×C22⋊C8C2211yescharacteristic
24C3×#95C2211yescharacteristic
25C3⋊#95C2211yescharacteristic
26#830C2211yescharacteristic
27(C2×D3)⋊C8—21no—
28C2×C22⋊Q8—31no—
29#407—31no—
30C4×C22⋊C4—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—24no—
57(C2×D3)⋊C4—21no—
58(C2×D3)⋊C4—21no—
59C4⋊Dic3—21no—
60C4⋊Dic3C2×C411yescharacteristic
61Dic3⋊C4—21no—
62Dic3⋊C4—21no—
63Dic3⋊C4—21no—
64C4×Dic3—42no—
65C2×C3⋊C8—21no—
66ΓL(1,𝔽25)—42no—
67C22×Q8—31no—
68C23×C4—31no—
69C2×M16—61no—
70C22⋊Q8—31no—
71C22⋊Q8—31no—
72C22⋊Q8—31no—
73C22⋊Q8—61no—
74C22⋊Q8—61no—
75C22⋊Q8C2×D311yescharacteristic
76C2×C4⋊C4—31no—
77C2×C4⋊C4—61no—
78C2×C22⋊C4—61no—
79C2×C22⋊C4—61no—
80C2×C42—61no—
81C22⋊C8—31no—
82C22⋊C8—61no—
83C22⋊C8C2×D311yescharacteristic
84#95—31no—
85#95C2×D311yescharacteristic
86C23×C3C2×D411yescharacteristic
87C22×D3—21no—
88C22×D3—21no—
89C22×D3—22no—
90C22×D3—41no—
91C22×D3C22⋊C411yescharacteristic
92C22×D3C22⋊C411yescharacteristic
93C3×Q8—41no—
94C2×C3×C4—22no—
95C2×C3×C4—21no—
96C2×C3×C4—21no—
97C2×C3×C4—41no—
98C2×C3×C4—41no—
99C2×C3×C4—41no—
100C2×C3×C4C22×C411yescharacteristic
101C2×C3×C4C2×D411yescharacteristic
102C2×Dic3—22no—
103C2×Dic3—22no—
104C2×Dic3—21no—
105C2×Dic3—21no—
106C2×Dic3—41no—
107C2×Dic3—41no—
108C2×Dic3C22⋊C411yescharacteristic
109C2×Dic3C22⋊C411yescharacteristic
110C4×D3—22no—
111C4×D3—22no—
112C4×D3—41no—
113C4×D3—41no—
114C4×D3—42no—
115C4×D3—42no—
116Dic6—42no—
117Dic6—41no—
118C3×C8—41no—
119C3⋊C8—41no—
120C24—31no—
121C2×Q8—31no—
122C2×Q8—61no—
123C2×Q8—122no—
124C2×Q8C4×D311yescharacteristic
125C22×C4—21no—
126C22×C4—31no—
127C22×C4—31no—
128C22×C4—31no—
129C22×C4—31no—
130C22×C4—31no—
131C22×C4—61no—
132C22×C4—61no—
133C22×C4—61no—
134C22×C4—122no—
135C22×C4C22×D311yescharacteristic
136M16—122no—
137C2×C8—21no—
138C2×C8—61no—
139C4⋊C4—21no—
140C4⋊C4—31no—
141C4⋊C4—61no—
142C4⋊C4—61no—
143C4⋊C4—61no—
144C4⋊C4—61no—
145C4⋊C4C4×D311yescharacteristic
146C22⋊C4—21no—
147C22⋊C4—61no—
148C22⋊C4—61no—
149C22⋊C4—64no—
150C22⋊C4—61no—
151C42—122no—
152C22×C3—21no—
153C22×C3—21no—
154C22×C3C2×C22⋊C411yescharacteristic
155C22×C3#13711yescharacteristic
156C22×C3#13711yescharacteristic
157C2×D3—21no—
158C2×D3—21no—
159C2×D3—21no—
160C2×D3—21no—
161C2×D3—21no—
162C2×D3—42no—
163C2×D3—42no—
164C2×D3#10412yes—
165C3×C4—21no—
166C3×C4—21no—
167C3×C4—42no—
168C3×C4—41no—
169C3×C4—41no—
170Dic3—21no—
171Dic3—42no—
172Dic3—41no—
173Dic3—41no—
174Dic3#9912yes—
175C23—31no—
176C23—31no—
177C23—62no—
178C23—61no—
179C23—61no—
180C23—121no—
181C23D3×D411yescharacteristic
182Q8—41no—
183Q8—121no—
184Q8—122no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—31no—
189C2×C4—31no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4—61no—
198C2×C4—61no—
199C2×C4—121no—
200C2×C4—121no—
201C2×C4—122no—
202C2×C4—122no—
203C2×C4—121no—
204C2×C4—121no—
205C2×C4C2×C4×D311yescharacteristic
206C2×C4D3×D411yescharacteristic
207C8—41no—
208C8—121no—
209C2×C3—21no—
210C2×C3—21no—
211C2×C3C2×#9911yescharacteristic
212C2×C3#41911yescharacteristic
213C2×C3C2×#10411yescharacteristic
214D3—22no—
215D3—41no—
216C22—21no—
217C22—21no—
218C22—32no—
219C22—61no—
220C22—61no—
221C22—61no—
222C22—61no—
223C22—61no—
224C22—122no—
225C22—122no—
226C22D3×C22⋊C411yescharacteristic
227C22#91511yescharacteristic
228C22#91811yescharacteristic
229C4—21no—
230C4—21no—
231C4—32no—
232C4—42no—
233C4—41no—
234C4—41no—
235C4—61no—
236C4—122no—
237C4—121no—
238C4—121no—
239C3#150811yescharacteristic, complemented, Sylow
240C2—21no—
241C2—21no—
242C2—62no—
243C2—121no—
244C2D3×#9911yescharacteristic
245C2#480211yescharacteristic
246C2D3×#10411yescharacteristic
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#1508
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
17. #897C4semidirect
20. D3×C4⋊C4C4semidirect
34. #328D3semidirect
239. C3#1508semidirect

Maximal subgroups

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

subgroupindex
D3×C22⋊Q82
#47262
#46662
C3×#3282
#45382
#44672
#44522
#15083