Finite Group Databasebeta

← #81623

Subgroups — #81623

Order 384.

340 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 115 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#5730C211yesmaximal, characteristic
2#5721C211yesmaximal, characteristic, complemented
3C3×#487C211yesmaximal, characteristic, complemented
4#5091C211yesmaximal, characteristic, complemented
5#5083C211yesmaximal, characteristic
6#4818C212yesmaximal
7#4817C212yesmaximal
8#4799C212yesmaximal, complemented
9#4797C212yesmaximal, complemented
10C4×#204C211yesmaximal, characteristic
11#4679C211yesmaximal, characteristic, complemented
12#3372—31nomaximal, Sylow
13C3×C4⋊Q8C2211yescharacteristic
14C3×C4⋊Q8C2211yescharacteristic
15#946—22no—
16Q8×Dic3C2212yes—
17#944C2212yes—
18C2×#215C2212yes—
19C2×#204C2211yescharacteristic
20#897—21no—
21#893C2212yes—
22D3×C4⋊C4—21no—
23#890—21no—
24#890C2212yes—
25#889—22no—
26#887—21no—
27C4×D12C2211yescharacteristic
28C4×Dic6C2211yescharacteristic
29C4×Dic6C2211yescharacteristic
30C3×#103C2212yes—
31C3×#103C2212yes—
32C3×C4×C8C2211yescharacteristic
33#835C2212yes—
34#821C2211yescharacteristic
35#817C2211yescharacteristic
36#816C2211yescharacteristic
37#815C2212yes—
38Dic3⋊C8C2212yes—
39#810C2212yes—
40#807C2212yes—
41C4⋊(C3⋊C8)C2211yescharacteristic
42#557—31no—
43#554—31no—
44#491—32no—
45#487D311yescharacteristic, complemented
46#479—32no—
47#477—31no—
48#461—31no—
49#461—32no—
50#446—31no—
51#440—32no—
52C4×SD16—31no—
53C2×C3×Q8—21no—
54C2×C3×Q8C2312yes—
55C2×D12C2311yescharacteristic
56C2×C4×D3—21no—
57C2×C4×D3—22no—
58C2×Dic6D411yescharacteristic
59C2×Dic6C2311yescharacteristic
60C2×C3×C8C2311yescharacteristic
61C2×C3×C8C2311yescharacteristic
62C3×C4⋊C4—21no—
63C3×C4⋊C4—21no—
64C3×C4⋊C4—21no—
65C3×C4⋊C4C2312yes—
66C3×C42C2311yescharacteristic
67#215—24no—
68(C2×D3)⋊C4—21no—
69(C2×D3)⋊C4—22no—
70C4⋊Dic3—22no—
71C4⋊Dic3—22no—
72C4⋊Dic3D411yescharacteristic
73C4⋊Dic3C2311yescharacteristic
74C4⋊Dic3C2311yescharacteristic
75Dic3⋊C4—21no—
76Dic3⋊C4—22no—
77Dic3⋊C4—22no—
78Dic3⋊C4—22no—
79Dic3⋊C4D412yes—
80C4×Dic3—21no—
81C4×Dic3—22no—
82C4×Dic3C2312yes—
83C2×C3⋊C8C2312yes—
84#204—22no—
85C2×SD16—32no—
86C2×SD16—31no—
87C4⋊Q8—32no—
88C4⋊Q8C2×D311yescharacteristic
89C4⋊Q8C2×D311yescharacteristic
90#125—32no—
91#125—62no—
92#125—61no—
93C22⋊Q8—61no—
94C22⋊Q8—62no—
95C4×Q8—31no—
96C4×Q8—32no—
97C4×Q8—31no—
98C4×Q8—61no—
99C4×D4—32no—
100C4×D4—31no—
101C2×C4⋊C4—61no—
102#107—32no—
103#106—31no—
104C4⋊C8—32no—
105C4⋊C8—31no—
106#103—32no—
107#103—31no—
108#103—32no—
109#103C2×D312yes—
110#103C2×D312yes—
111#102—31no—
112#97—32no—
113C4×C8C2×D311yescharacteristic
114C22×D3—21no—
115C3×Q8—22no—
116C3×Q8—22no—
117C3×Q8—81no—
118C2×C3×C4—22no—
119C2×C3×C4—22no—
120C2×C3×C4C2×D411yescharacteristic
121C2×C3×C4C2×D411yescharacteristic
122C2×C3×C4C2411yescharacteristic
123C2×Dic3—21no—
124C2×Dic3—21no—
125C2×Dic3—21no—
126C2×Dic3C2×D412yes—
127C2×Dic3C2×D412yes—
128D12—21no—
129D12#4112yes—
130C4×D3—42no—
131C4×D3—82no—
132Dic6—22no—
133Dic6—21no—
134Dic6—21no—
135C3×C8—21no—
136C3×C8—21no—
137C3⋊C8—22no—
138C2×Q8—21no—
139C2×Q8—31no—
140C2×Q8—31no—
141C2×Q8C22×D312yes—
142C2×D4—31no—
143C22×C4—62no—
144C22×C4—61no—
145SD16—62no—
146SD16—64no—
147C2×C8—32no—
148C2×C8C22×D311yescharacteristic
149C2×C8C22×D311yescharacteristic
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—31no—
154C4⋊C4—32no—
155C4⋊C4—31no—
156C4⋊C4—31no—
157C4⋊C4—62no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—61no—
163C4⋊C4C22×D312yes—
164C22⋊C4—62no—
165C22⋊C4—61no—
166C42—32no—
167C42—62no—
168C42—61no—
169C42C22×D311yescharacteristic
170C22×C3C22×D411yescharacteristic
171C2×D3—22no—
172C2×D3—41no—
173C2×D3—41no—
174C3×C4—21no—
175C3×C4—42no—
176C3×C4—42no—
177C3×C4#13712yes—
178C3×C4C2×#4111yescharacteristic
179C3×C42−1+411yescharacteristic
180Dic3—22no—
181Dic3—22no—
182Dic3—41no—
183Dic3—41no—
184Dic3—41no—
185C23—61no—
186Q8—22no—
187Q8—22no—
188Q8—62no—
189Q8—61no—
190Q8—61no—
191Q8—81no—
192D4—32no—
193D4—61no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—32no—
197C2×C4—32no—
198C2×C4—61no—
199C2×C4—61no—
200C2×C4—61no—
201C2×C4—122no—
202C2×C4—242no—
203C2×C4D3×D411yescharacteristic
204C2×C4D3×D411yescharacteristic
205C2×C4C23×D311yescharacteristic
206C8—21no—
207C8—21no—
208C8—62no—
209C2×C3#54811yescharacteristic
210C2×C3C2×#13711yescharacteristic
211C2×C3#57711yescharacteristic
212D3—42no—
213C22—62no—
214C22—121no—
215C22—121no—
216C22C2×D3×D411yescharacteristic
217C4—21no—
218C4—42no—
219C4—42no—
220C4—62no—
221C4—62no—
222C4—121no—
223C4—121no—
224C4—121no—
225C4#91812yes—
226C4D3×#4111yescharacteristic
227C4#101011yescharacteristic
228C3#337211yescharacteristic, complemented, Sylow
229C2—122no—
230C2#567711yescharacteristic
231C2#576211yescharacteristic
232C2C2×#91811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#487C2
socleC22×C3C22×D4
prime radicalC2×D12C23
hypercenter#487D3
hypocenterC3#3372
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #5721C2semidirect
3. C3×#487C2semidirect
4. #5091C2semidirect
8. #4799C2semidirect
9. #4797C2semidirect
11. #4679C2semidirect
45. #487D3semidirect
228. C3#3372semidirect

Maximal subgroups

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

subgroupindex
#57302
#57212
C3×#4872
#50912
#50832
#48182
#48172
#47992
#47972
C4×#2042
#46792
#33723