Finite Group Databasebeta

← #81645

Subgroups — #81645

Order 384.

388 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 109 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#5697C211yesmaximal, characteristic, complemented
2#5675C211yesmaximal, characteristic, complemented
3C3×#488C211yesmaximal, characteristic, complemented
4#5069C211yesmaximal, characteristic, complemented
5#5058C211yesmaximal, characteristic, complemented
6#4818C212yesmaximal, complemented
7#4806C212yesmaximal, complemented
8#4787C212yesmaximal, complemented
9#4781C212yesmaximal, complemented
10#4693C211yesmaximal, characteristic, complemented
11#4690C211yesmaximal, characteristic, complemented
12#3221—31nomaximal, Sylow
13C2×D3×Q8C2211yescharacteristic
14C2×#237C2211yescharacteristic
15C3×#125C2211yescharacteristic
16C3×#124C2211yescharacteristic
17#944C2211yescharacteristic
18C2×#215C2212yes—
19#936C2211yescharacteristic
20C2×#213C2212yes—
21C2×#204C2212yes—
22C2×#204C2212yes—
23#895—22no—
24#895C2212yes—
25#894—22no—
26#893C2212yes—
27#883—22no—
28#878—22no—
29#874C2211yescharacteristic
30#872C2211yescharacteristic
31#869C2211yescharacteristic
32C3×#103C2212yes—
33C3×#102C2212yes—
34C3×C4×C8C2211yescharacteristic
35(C2×D3)⋊C8C2212yes—
36Dic3⋊C8C2212yes—
37#810C2212yes—
38#810C2212yes—
39#803C2211yescharacteristic
40#540—31no—
41#536—31no—
42#496—31no—
43#492—31no—
44#491—31no—
45#488D311yescharacteristic, complemented
46#463—32no—
47#461—32no—
48#449—32no—
49#448—32no—
50#433—31no—
51C2×C3×Q8C2311yescharacteristic
52C2×C3×D4C2311yescharacteristic
53C2×#67—21no—
54C22×Dic3—21no—
55D3×Q8—42no—
56#237—42no—
57C2×D12—22no—
58C2×D12—22no—
59C2×D12C2312yes—
60C2×C4×D3—21no—
61C2×C4×D3—22no—
62C2×C4×D3C2311yescharacteristic
63C2×Dic6—21no—
64C2×Dic6C2311yescharacteristic
65C2×Dic6C2311yescharacteristic
66C2×C3×C8C2312yes—
67C3×C4⋊C4—22no—
68C3×C4⋊C4C2312yes—
69C3×C22⋊C4—22no—
70C3×C42C2311yescharacteristic
71#217—22no—
72#215—42no—
73#213—42no—
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4—22no—
76(C2×D3)⋊C4D412yes—
77C4⋊Dic3—22no—
78Dic3⋊C4—22no—
79Dic3⋊C4D412yes—
80C4×Dic3C2311yescharacteristic
81C2×C3⋊C8C2312yes—
82#204—42no—
83#204—42no—
84C2×#41—31no—
85C22×Q8—31no—
86C2×SD16—32no—
87C2×SD16—32no—
88C2×SD16—32no—
89C2×D8—32no—
90C4⋊Q8—31no—
91C4⋊Q8—31no—
92#127—31no—
93#125C2×D311yescharacteristic
94#124—31no—
95#124C2×D311yescharacteristic
96#123—62no—
97#123—62no—
98C22⋊Q8—62no—
99#121—32no—
100#121—62no—
101C4×D4—32no—
102#117—31no—
103C4⋊C8—32no—
104#103—32no—
105#103—32no—
106#103C2×D312yes—
107#102C2×D312yes—
108C22⋊C8—32no—
109#97—31no—
110C4×C8C2×D311yescharacteristic
111C23×C3—21no—
112C22×D3—22no—
113C22×D3C2×D411yescharacteristic
114C3×Q8—22no—
115C3×D4—22no—
116C2×C3×C4—21no—
117C2×C3×C4—22no—
118C2×C3×C4C2×D412yes—
119C2×C3×C4C2411yescharacteristic
120#67—42no—
121C2×Dic3—21no—
122C2×Dic3—21no—
123C2×Dic3—42no—
124C2×Dic3C2×D412yes—
125C2×Dic3C2×D411yescharacteristic
126D12—22no—
127D12—22no—
128D12—82no—
129D12—82no—
130C4×D3—22no—
131C4×D3—42no—
132C4×D3—82no—
133Dic6—22no—
134Dic6—22no—
135Dic6—42no—
136C3×C8—22no—
137C3⋊C8—22no—
138#41—122no—
139C2×Q8—31no—
140C2×Q8—31no—
141C2×Q8—61no—
142C2×Q8—122no—
143C2×Q8C22×D311yescharacteristic
144C2×D4—32no—
145C2×D4—62no—
146C2×D4—61no—
147C2×D4—62no—
148C2×D4C22×D311yescharacteristic
149C22×C4—31no—
150C22×C4—62no—
151C22×C4—61no—
152C22×C4—61no—
153SD16—122no—
154SD16—122no—
155SD16—122no—
156D8—122no—
157C2×C8—32no—
158C2×C8C22×D312yes—
159C4⋊C4—22no—
160C4⋊C4—32no—
161C4⋊C4—62no—
162C4⋊C4—62no—
163C4⋊C4C22×D312yes—
164C22⋊C4—22no—
165C22⋊C4—32no—
166C22⋊C4—62no—
167C22⋊C4—62no—
168C22⋊C4—62no—
169C42—31no—
170C42C22×D311yescharacteristic
171C22×C3—41no—
172C22×C3—42no—
173C22×C3C22×D411yescharacteristic
174C2×D3—21no—
175C2×D3—22no—
176C2×D3—42no—
177C2×D3—42no—
178C2×D3—42no—
179C3×C4—22no—
180C3×C4—41no—
181C3×C4—42no—
182C3×C4H(5,𝔽2)12yes—
183Dic3—21no—
184Dic3—22no—
185Dic3—41no—
186Dic3—41no—
187C23—21no—
188C23—31no—
189C23—62no—
190Q8—22no—
191Q8—62no—
192Q8—62no—
193Q8—122no—
194D4—22no—
195D4—62no—
196D4—62no—
197D4—122no—
198D4—242no—
199D4—242no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—32no—
203C2×C4—31no—
204C2×C4—62no—
205C2×C4—61no—
206C2×C4—61no—
207C2×C4—122no—
208C2×C4—122no—
209C2×C4—242no—
210C2×C4D3×D412yes—
211C2×C4C23×D311yescharacteristic
212C8—22no—
213C8—62no—
214C2×C3—81no—
215C2×C3#53511yescharacteristic
216C2×C3#57712yes—
217D3—41no—
218D3—82no—
219C22—41no—
220C22—42no—
221C22—61no—
222C22—62no—
223C22—122no—
224C22—122no—
225C22—122no—
226C22C2×D3×D411yescharacteristic
227C4—22no—
228C4—41no—
229C4—42no—
230C4—61no—
231C4—62no—
232C4—121no—
233C4—121no—
234C4#100912yes—
235C3#322111yescharacteristic, complemented, Sylow
236C2—81no—
237C2—121no—
238C2—242no—
239C2#567411yescharacteristic
240C2#577012yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#488C2
socleC22×C3C22×D4
hypercenter#488D3
hypocenterC3#3221
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #5697C2semidirect
2. #5675C2semidirect
3. C3×#488C2semidirect
4. #5069C2semidirect
5. #5058C2semidirect
6. #4818C2semidirect
7. #4806C2semidirect
8. #4787C2semidirect
9. #4781C2semidirect
10. #4693C2semidirect
11. #4690C2semidirect
45. #488D3semidirect
235. C3#3221semidirect

Maximal subgroups

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

subgroupindex
#56972
#56752
C3×#4882
#50692
#50582
#48182
#48062
#47872
#47812
#46932
#46902
#32213