Finite Group Databasebeta

← #79417

Subgroups — #79417

Order 384.

352 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 125 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#5563C211yesmaximal, characteristic
2#5542C211yesmaximal, characteristic, complemented
3C3×C8×D4C211yesmaximal, characteristic, complemented
4#5114C212yesmaximal, complemented
5#5106C212yesmaximal, complemented
6#4850C211yesmaximal, characteristic, complemented
7#4846C211yesmaximal, characteristic, complemented
8#4737C212yesmaximal, complemented
9#4723C212yesmaximal, complemented
10C4×#204C211yesmaximal, characteristic, complemented
11#4680C211yesmaximal, characteristic, complemented
12#3340—31nomaximal, Sylow
13C22×C3×C8C2212yes—
14C3×C4×D4C2211yescharacteristic
15#938—21no—
16#936—21no—
17#930C2212yes—
18#926C2212yes—
19#926C2212yes—
20#924C2212yes—
21C2×#204C2211yescharacteristic
22#892—21no—
23#887—21no—
24#885—22no—
25#879—22no—
26C4×D12C2211yescharacteristic
27#869C2211yescharacteristic
28#869C2211yescharacteristic
29C4×Dic6C2211yescharacteristic
30C3×C4⋊C8C2211yescharacteristic
31C3×C22⋊C8C2212yes—
32C3×C4×C8C2211yescharacteristic
33#821C2211yescharacteristic
34#821C2212yes—
35#818C2212yes—
36#818C2212yes—
37#817C2212yes—
38#817C2212yes—
39#817C2211yescharacteristic
40#816C2211yescharacteristic
41#816C2212yes—
42#556—31no—
43#555—31no—
44#498—31no—
45#485—32no—
46#482—32no—
47#477—31no—
48#474—31no—
49#465—32no—
50C4×SD16—31no—
51C8×D4D311yescharacteristic, complemented
52#427—32no—
53C2×C3×D4D411yescharacteristic
54C22×C3×C4C2312yes—
55C2×#67—22no—
56C2×D12C2311yescharacteristic
57C2×C4×D3—21no—
58C2×Dic6—21no—
59C2×Dic6—21no—
60C2×Dic6C2311yescharacteristic
61C2×C3×C8—24no—
62C2×C3×C8C2311yescharacteristic
63C2×C3×C8C2312yes—
64C2×C3×C8C2311yescharacteristic
65C3×C4⋊C4D411yescharacteristic
66C3×C22⋊C4D412yes—
67C3×C42C2311yescharacteristic
68#217—22no—
69#217—22no—
70#217—22no—
71(C2×D3)⋊C4—21no—
72(C2×D3)⋊C4—22no—
73C4⋊Dic3—21no—
74C4⋊Dic3—21no—
75C4⋊Dic3C2312yes—
76C4⋊Dic3C2311yescharacteristic
77C4⋊Dic3C2312yes—
78C4⋊Dic3C2311yescharacteristic
79Dic3⋊C4—21no—
80Dic3⋊C4—22no—
81C4×Dic3—21no—
82C4×Dic3—22no—
83C4×Dic3—22no—
84#204—22no—
85C2×SD16—31no—
86C22×C8C2×D312yes—
87C4⋊Q8—31no—
88C4⋊Q8—31no—
89#126—62no—
90#124—61no—
91#124—62no—
92C22⋊Q8—32no—
93#121—32no—
94#121—61no—
95C4×Q8—31no—
96C4×Q8—61no—
97C4×D4—31no—
98C4×D4C2×D311yescharacteristic
99#117—32no—
100#117—32no—
101#117—61no—
102#107—32no—
103#107—32no—
104#106—31no—
105#106—32no—
106#106—32no—
107C4⋊C8C2×D311yescharacteristic
108#103—31no—
109#103—32no—
110#102—31no—
111#102—32no—
112C22⋊C8C2×D312yes—
113C4×C8C2×D311yescharacteristic
114C23×C3C2×D412yes—
115C22×D3—21no—
116C3×D4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4C2×D412yes—
120C2×C3×C4C2×D411yescharacteristic
121C2×C3×C4C2×D411yescharacteristic
122C2×C3×C4C2411yescharacteristic
123#67—82no—
124C2×Dic3—21no—
125C2×Dic3—21no—
126C2×Dic3—21no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129D12—21no—
130D12—21no—
131C4×D3—42no—
132Dic6—21no—
133Dic6—21no—
134Dic6—42no—
135Dic6—81no—
136C3×C8—22no—
137C3×C8—21no—
138C3×C8#4112yes—
139C2×Q8—31no—
140C2×Q8—61no—
141C2×Q8—61no—
142C2×D4—31no—
143C2×D4—62no—
144C2×D4D1211yescharacteristic
145C22×C4—61no—
146C22×C4C22×D312yes—
147SD16—62no—
148C2×C8—24no—
149C2×C8C22×D311yescharacteristic
150C2×C8C22×D312yes—
151C2×C8C22×D311yescharacteristic
152C4⋊C4—32no—
153C4⋊C4—31no—
154C4⋊C4—32no—
155C4⋊C4—31no—
156C4⋊C4—61no—
157C4⋊C4—61no—
158C4⋊C4—62no—
159C4⋊C4—61no—
160C4⋊C4D1211yescharacteristic
161C22⋊C4—62no—
162C22⋊C4—62no—
163C22⋊C4—62no—
164C22⋊C4—62no—
165C22⋊C4—61no—
166C22⋊C4D1212yes—
167C42—62no—
168C42—62no—
169C42—61no—
170C42C22×D311yescharacteristic
171C22×C3—22no—
172C22×C3—22no—
173C22×C3—22no—
174C22×C3C22×D411yescharacteristic
175C2×D3—41no—
176C2×D3—41no—
177C2×D3—41no—
178C3×C4—22no—
179C3×C4—21no—
180C3×C4#13512yes—
181C3×C4C2×#4111yescharacteristic
182C3×C42−1+411yescharacteristic
183Dic3—41no—
184Dic3—42no—
185Dic3—41no—
186Dic3—42no—
187Dic3—41no—
188C23—61no—
189C23C2×D1212yes—
190Q8—61no—
191Q8—61no—
192Q8—122no—
193Q8—241no—
194D4—22no—
195D4—61no—
196D4—61no—
197D4—242no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—61no—
201C2×C4—61no—
202C2×C4—62no—
203C2×C4—62no—
204C2×C4—61no—
205C2×C4—122no—
206C2×C4C2×D1212yes—
207C2×C4C2×D1211yescharacteristic
208C2×C4C2×D1211yescharacteristic
209C2×C4C23×D311yescharacteristic
210C8—22no—
211C8—21no—
212C8#23712yes—
213C2×C3—42no—
214C2×C3#54711yescharacteristic
215C2×C3C2×#13511yescharacteristic
216C2×C3#57711yescharacteristic
217D3—81no—
218C22—22no—
219C22—22no—
220C22—22no—
221C22—121no—
222C22—121no—
223C22—121no—
224C22C22×D1211yescharacteristic
225C4—21no—
226C4—22no—
227C4—121no—
228C4—122no—
229C4—122no—
230C4—121no—
231C4—121no—
232C4#90412yes—
233C4C2×#23711yescharacteristic
234C4#101011yescharacteristic
235C3#334011yescharacteristic, complemented, Sylow
236C2—42no—
237C2—241no—
238C2#555511yescharacteristic
239C2C2×#90411yescharacteristic
240C2#575111yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C8×D4C2
socleC22×C3C22×D4
prime radical#5563C2
hypercenterC8×D4D3
hypocenterC3#3340
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
2. #5542C2semidirect
3. C3×C8×D4C2semidirect
4. #5114C2semidirect
5. #5106C2semidirect
6. #4850C2semidirect
7. #4846C2semidirect
8. #4737C2semidirect
9. #4723C2semidirect
10. C4×#204C2semidirect
11. #4680C2semidirect
51. C8×D4D3semidirect
235. C3#3340semidirect

Maximal subgroups

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

subgroupindex
#55632
#55422
C3×C8×D42
#51142
#51062
#48502
#48462
#47372
#47232
C4×#2042
#46802
#33403