Finite Group Databasebeta

← #80319

Subgroups — #80319

Order 384.

362 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 117 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#5713C211yesmaximal, characteristic, complemented
2#5543C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic
4#5153C212yesmaximal, complemented
5Dic3×D8C212yesmaximal, complemented
6#5070C211yesmaximal, characteristic, complemented
7C4×#214C211yesmaximal, characteristic, complemented
8#4850C211yesmaximal, characteristic, complemented
9#4830C211yesmaximal, characteristic, complemented
10#4768C212yesmaximal, complemented
11#4766C212yesmaximal, complemented
12#3361—31nomaximal, Sylow
13C2×C3×D8C2212yes—
14C3×#127C2211yescharacteristic
15C3×C4×D4C2211yescharacteristic
16#936—21no—
17#936C2212yes—
18#935—22no—
19D4×Dic3—21no—
20D4×Dic3C2212yes—
21D4×Dic3C2212yes—
22C2×#214C2211yescharacteristic
23C2×C4⋊Dic3—21no—
24#924—21no—
25#890C2212yes—
26#878—22no—
27#869C2211yescharacteristic
28C4×Dic6C2211yescharacteristic
29C4×Dic6C2211yescharacteristic
30C3×C4⋊C8C2211yescharacteristic
31C3×#102C2212yes—
32#832C2211yescharacteristic
33#832C2212yes—
34#832C2212yes—
35#818C2212yes—
36#816C2212yes—
37Dic3⋊C8C2212yes—
38C8×Dic3C2212yes—
39#810C2211yescharacteristic
40#808C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#559—31no—
43#554—31no—
44#488—32no—
45#486—31no—
46#478—32no—
47#477—31no—
48#463—32no—
49#459D311yescharacteristic
50C8×Q8—31no—
51C4×SD16—31no—
52C4×D8—32no—
53C2×C3×D4—21no—
54C2×C3×D4—21no—
55C2×C3×D4C2312yes—
56C2×C3×D4C2311yescharacteristic
57C22×C3×C4—21no—
58C22×Dic3—22no—
59C22×Dic3—22no—
60C2×Dic6—21no—
61C2×Dic6D411yescharacteristic
62C2×Dic6C2311yescharacteristic
63C3×D8—24no—
64C2×C3×C8C2312yes—
65C3×C4⋊C4C2311yescharacteristic
66C3×C22⋊C4—21no—
67C3×C42C2311yescharacteristic
68#217—22no—
69#217—22no—
70#217—22no—
71#217—22no—
72#214—22no—
73C4⋊Dic3—21no—
74C4⋊Dic3—22no—
75C4⋊Dic3D411yescharacteristic
76C4⋊Dic3C2311yescharacteristic
77C4⋊Dic3C2312yes—
78Dic3⋊C4—21no—
79Dic3⋊C4—22no—
80Dic3⋊C4D412yes—
81C4×Dic3—21no—
82C4×Dic3C2312yes—
83C2×C3⋊C8C2311yescharacteristic
84C2×C3⋊C8C2311yescharacteristic
85C2×SD16—31no—
86C2×D8C2×D312yes—
87C4⋊Q8—31no—
88#127C2×D311yescharacteristic
89#125—32no—
90#124—32no—
91#124—61no—
92#123—62no—
93C22⋊Q8—62no—
94C22⋊Q8—61no—
95C4×Q8—31no—
96C4×Q8—31no—
97C4×D4—32no—
98C4×D4—32no—
99C4×D4—61no—
100C4×D4C2×D311yescharacteristic
101C2×C4⋊C4—61no—
102#107—32no—
103#106—31no—
104C4⋊C8—32no—
105C4⋊C8C2×D311yescharacteristic
106#103—32no—
107#103—31no—
108#102—31no—
109#102—32no—
110#102—32no—
111#102C2×D312yes—
112C4×C8—32no—
113C4×C8—31no—
114C23×C3—21no—
115C23×C3—22no—
116C3×D4—22no—
117C3×D4—21no—
118C3×D4—22no—
119C3×D4—42no—
120C3×D4—81no—
121C3×D4#4112yes—
122C2×C3×C4—21no—
123C2×C3×C4—42no—
124C2×C3×C4C2×D411yescharacteristic
125C2×C3×C4C2×D411yescharacteristic
126C2×C3×C4C2411yescharacteristic
127C2×Dic3—21no—
128C2×Dic3—21no—
129C2×Dic3—22no—
130C2×Dic3—82no—
131C2×Dic3—82no—
132C2×Dic3C2×D412yes—
133C2×Dic3C2×D412yes—
134Dic6—22no—
135Dic6—21no—
136Dic6—21no—
137Dic6—81no—
138C3×C8—22no—
139C3⋊C8—21no—
140C3⋊C8—21no—
141C2×Q8—31no—
142C2×Q8—31no—
143C2×Q8—61no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4C22×D312yes—
147C2×D4C22×D311yescharacteristic
148C22×C4—21no—
149C22×C4—62no—
150C22×C4—62no—
151SD16—62no—
152D8—24no—
153C2×C8—31no—
154C2×C8—31no—
155C2×C8C22×D312yes—
156C4⋊C4—31no—
157C4⋊C4—32no—
158C4⋊C4—32no—
159C4⋊C4—31no—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—61no—
163C4⋊C4—61no—
164C4⋊C4C22×D311yescharacteristic
165C22⋊C4—21no—
166C22⋊C4—62no—
167C22⋊C4—62no—
168C22⋊C4—62no—
169C22⋊C4—62no—
170C42—32no—
171C42—61no—
172C42C22×D311yescharacteristic
173C22×C3—22no—
174C22×C3—41no—
175C22×C3—41no—
176C22×C3—42no—
177C22×C3—42no—
178C22×C3—42no—
179C22×C3C22×D411yescharacteristic
180C3×C4—21no—
181C3×C4—41no—
182C3×C4#13512yes—
183C3×C4C2×#4111yescharacteristic
184C3×C42−1+411yescharacteristic
185Dic3—22no—
186Dic3—22no—
187Dic3—41no—
188Dic3—42no—
189Dic3—41no—
190C23—21no—
191C23—22no—
192Q8—62no—
193Q8—61no—
194Q8—61no—
195Q8—241no—
196D4—22no—
197D4—21no—
198D4—22no—
199D4—42no—
200D4—81no—
201D4#23712yes—
202C2×C4—21no—
203C2×C4—32no—
204C2×C4—32no—
205C2×C4—42no—
206C2×C4—61no—
207C2×C4—61no—
208C2×C4—62no—
209C2×C4—242no—
210C2×C4—242no—
211C2×C4D3×D411yescharacteristic
212C2×C4D3×D411yescharacteristic
213C2×C4C23×D311yescharacteristic
214C8—22no—
215C8—61no—
216C8—61no—
217C2×C3—42no—
218C2×C3—82no—
219C2×C3#54811yescharacteristic
220C2×C3C2×#13511yescharacteristic
221C2×C3#57611yescharacteristic
222C22—22no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—42no—
227C22—42no—
228C22C2×D3×D411yescharacteristic
229C4—21no—
230C4—41no—
231C4—62no—
232C4—62no—
233C4—121no—
234C4—122no—
235C4—121no—
236C4#91212yes—
237C4C2×#23711yescharacteristic
238C4#101011yescharacteristic
239C3#336111yescharacteristic, complemented, Sylow
240C2—42no—
241C2—82no—
242C2#559811yescharacteristic
243C2C2×#91211yescharacteristic
244C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3361
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #5713C2semidirect
2. #5543C2semidirect
4. #5153C2semidirect
5. Dic3×D8C2semidirect
6. #5070C2semidirect
7. C4×#214C2semidirect
8. #4850C2semidirect
9. #4830C2semidirect
10. #4768C2semidirect
11. #4766C2semidirect
239. C3#3361semidirect

Maximal subgroups

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

subgroupindex
#57132
#55432
C3×#4592
#51532
Dic3×D82
#50702
C4×#2142
#48502
#48302
#47682
#47662
#33613