Finite Group Databasebeta

← #80172

Subgroups — #80172

Order 384.

384 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 129 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×Q8×Dic3C211yesmaximal, characteristic
2#5622C211yesmaximal, characteristic
3C3×#451C211yesmaximal, characteristic
4#5228C211yesmaximal, characteristic
5#5182C212yesmaximal, complemented
6Dic3×Q16C212yesmaximal, complemented
7#5044C211yesmaximal, characteristic
8#4793C212yesmaximal, complemented
9#4790C212yesmaximal, complemented
10#4722C211yesmaximal, characteristic
11#4718C211yesmaximal, characteristic
12#3104—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C2×C3×Q16C2212yes—
15C3×C22⋊Q8C2211yescharacteristic
16Q8×Dic3—21no—
17Q8×Dic3—21no—
18Q8×Dic3C2211yescharacteristic
19Q8×Dic3C2212yes—
20Q8×Dic3C2212yes—
21#944C2211yescharacteristic
22C2×#216C2212yes—
23C2×C4⋊Dic3—21no—
24C2×C4⋊Dic3C2211yescharacteristic
25#924C2211yescharacteristic
26C2×C4×Dic3—21no—
27C2×C4×Dic3C2211yescharacteristic
28#889—21no—
29#888C2211yescharacteristic
30#887—21no—
31#887C2211yescharacteristic
32#878—21no—
33#877—21no—
34C3×#103C2212yes—
35C3×C22⋊C8C2211yescharacteristic
36#835C2212yes—
37#835C2212yes—
38#830C2211yescharacteristic
39#818C2212yes—
40#816C2212yes—
41Dic3⋊C8C2212yes—
42C8×Dic3C2212yes—
43#807C2212yes—
44#533—31no—
45C2×C4×Q8—31no—
46#484—31no—
47#484—31no—
48#477—32no—
49#462—32no—
50#451—31no—
51#451D311yescharacteristic
52C4×Q16—32no—
53C4×Q16—32no—
54#431—31no—
55C2×C3×Q8—24no—
56C2×C3×Q8—21no—
57C2×C3×Q8C2311yescharacteristic
58C2×C3×Q8C2312yes—
59C22×C3×C4—21no—
60C22×C3×C4C2311yescharacteristic
61C22×Dic3—21no—
62C22×Dic3D411yescharacteristic
63C22×Dic3D411yescharacteristic
64C2×Dic6—21no—
65C2×Dic6C2311yescharacteristic
66C3×Q16—24no—
67C2×C3×C8C2312yes—
68C3×C4⋊C4—21no—
69C3×C4⋊C4C2311yescharacteristic
70C3×C22⋊C4—21no—
71#217—21no—
72#216—24no—
73C4⋊Dic3—21no—
74C4⋊Dic3—21no—
75C4⋊Dic3—22no—
76C4⋊Dic3—22no—
77C4⋊Dic3C2312yes—
78C4⋊Dic3C2311yescharacteristic
79Dic3⋊C4—21no—
80Dic3⋊C4—21no—
81Dic3⋊C4—21no—
82Dic3⋊C4—21no—
83Dic3⋊C4—21no—
84C4×Dic3—21no—
85C4×Dic3—21no—
86C4×Dic3—22no—
87C4×Dic3—22no—
88C4×Dic3D411yescharacteristic
89C4×Dic3D411yescharacteristic
90C4×Dic3C2311yescharacteristic
91C4×Dic3C2311yescharacteristic
92C2×C3⋊C8C2312yes—
93C22×Q8C2×D311yescharacteristic
94C2×Q16—32no—
95C2×Q16C2×D312yes—
96C4⋊Q8—31no—
97C4⋊Q8—31no—
98#125—61no—
99C22⋊Q8—31no—
100C22⋊Q8—61no—
101C22⋊Q8C2×D311yescharacteristic
102C4×Q8—32no—
103C4×Q8—32no—
104C4×Q8—31no—
105C4×Q8—31no—
106C4×Q8—61no—
107C4×Q8—61no—
108C4×Q8—61no—
109#117—61no—
110C2×C4⋊C4—31no—
111C2×C4⋊C4—61no—
112C2×C42—31no—
113C2×C42—61no—
114#107—32no—
115#107—32no—
116C4⋊C8—32no—
117#103—32no—
118#103—32no—
119#103—32no—
120#103C2×D312yes—
121C22⋊C8—31no—
122C22⋊C8C2×D311yescharacteristic
123C4×C8—32no—
124C23×C3C2×D411yescharacteristic
125C3×Q8—24no—
126C3×Q8—22no—
127C3×Q8—22no—
128C3×Q8#4114yes—
129C2×C3×C4—24no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4C2411yescharacteristic
136C2×Dic3—22no—
137C2×Dic3—21no—
138C2×Dic3—21no—
139C2×Dic3—41no—
140C2×Dic3—42no—
141C2×Dic3Q1614yescomplemented
142C2×Dic3C2×D411yescharacteristic
143C2×Dic3C2×D411yescharacteristic
144C2×Dic3C2×D411yescharacteristic
145C2×Dic3C2×D411yescharacteristic
146Dic6—22no—
147Dic6—42no—
148C3×C8—22no—
149C3⋊C8—22no—
150C2×Q8—24no—
151C2×Q8—21no—
152C2×Q8—31no—
153C2×Q8—61no—
154C2×Q8C22×D311yescharacteristic
155C2×Q8C22×D312yes—
156C22×C4—21no—
157C22×C4—31no—
158C22×C4—31no—
159C22×C4—61no—
160C22×C4C22×D311yescharacteristic
161Q16—24no—
162Q16—64no—
163C2×C8—32no—
164C2×C8C22×D312yes—
165C4⋊C4—21no—
166C4⋊C4—32no—
167C4⋊C4—31no—
168C4⋊C4—62no—
169C4⋊C4—62no—
170C4⋊C4—61no—
171C4⋊C4—61no—
172C4⋊C4—61no—
173C4⋊C4—61no—
174C4⋊C4—61no—
175C4⋊C4—61no—
176C4⋊C4—61no—
177C4⋊C4C22×D311yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—61no—
180C42—31no—
181C42—31no—
182C42—31no—
183C42—31no—
184C42—62no—
185C42—62no—
186C42—61no—
187C42—61no—
188C22×C3—21no—
189C22×C3—21no—
190C22×C3C2×Q1612yes—
191C22×C3C22×D411yescharacteristic
192C3×C4—24no—
193C3×C4—21no—
194C3×C4—41no—
195C3×C4—41no—
196C3×C4C2×#4112yes—
197Dic3—21no—
198Dic3—21no—
199Dic3—41no—
200Dic3—41no—
201Dic3—42no—
202Dic3C2×Q1612yes—
203Dic3C2×Q1612yes—
204C23D3×D411yescharacteristic
205Q8—24no—
206Q8—22no—
207Q8—22no—
208Q8—62no—
209Q8—122no—
210Q8#23714yes—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—34no—
217C2×C4—31no—
218C2×C4—31no—
219C2×C4—31no—
220C2×C4—31no—
221C2×C4—62no—
222C2×C4—61no—
223C2×C4—61no—
224C2×C4—121no—
225C2×C4—122no—
226C2×C4D3×D411yescharacteristic
227C2×C4C23×D311yescharacteristic
228C8—22no—
229C8—62no—
230C2×C3—22no—
231C2×C3#52511yescharacteristic
232C2×C3C22×Q1611yescharacteristic
233C2×C3#57511yescharacteristic
234C22—21no—
235C22—21no—
236C22D3×Q1612yes—
237C22C2×D3×D411yescharacteristic
238C4—24no—
239C4—21no—
240C4—32no—
241C4—32no—
242C4—41no—
243C4—41no—
244C4—61no—
245C4—61no—
246C4—122no—
247C4—121no—
248C4—121no—
249C4C2×#23712yes—
250C3#310411yescharacteristic, complemented, Sylow
251C2—22no—
252C2#558611yescharacteristic
253C2C2×D3×Q1611yescharacteristic
254C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#451C2
socleC22×C3C22×D4
prime radicalC23×C3C2×D4
hypercenter#451D3
hypocenterC3#3104
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
5. #5182C2semidirect
6. Dic3×Q16C2semidirect
8. #4793C2semidirect
9. #4790C2semidirect
141. C2×Dic3Q16semidirect
250. C3#3104semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic32
#56222
C3×#4512
#52282
#51822
Dic3×Q162
#50442
#47932
#47902
#47222
#47182
#31043