Finite Group Databasebeta

← #80421

Subgroups — #80421

Order 384.

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

normal subgroups only

This group has 360 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#5732C211yesmaximal, characteristic, complemented
2#5542C211yesmaximal, characteristic, complemented
3C3×#461C211yesmaximal, characteristic, complemented
4#5168C212yesmaximal
5#5166C212yesmaximal, complemented
6#5092C211yesmaximal, characteristic, complemented
7C4×#213C211yesmaximal, characteristic
8#4841C211yesmaximal, characteristic, complemented
9#4830C211yesmaximal, characteristic, complemented
10#4798C212yesmaximal, complemented
11#4790C212yesmaximal, complemented
12#3393—31nomaximal, Sylow
13C2×C3×SD16C2212yes—
14C3×C4⋊Q8C2211yescharacteristic
15C3×C4×D4C2211yescharacteristic
16#947C2212yes—
17Q8×Dic3C2212yes—
18C2×#216C2212yes—
19C2×#216C2212yes—
20#936C2212yes—
21C2×#213C2211yescharacteristic
22#926—21no—
23#924—21no—
24C2×#204C2212yes—
25#898—22no—
26#897—21no—
27#892—21no—
28#887C2212yes—
29#879—22no—
30C4×D12C2211yescharacteristic
31#869C2211yescharacteristic
32C4×Dic6C2211yescharacteristic
33C3×C4⋊C8C2211yescharacteristic
34C3×#103C2212yes—
35#832C2211yescharacteristic
36#816C2212yes—
37Dic3⋊C8C2212yes—
38C8×Dic3C2212yes—
39#809C2211yescharacteristic
40#807C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#556—31no—
43#556—31no—
44#495—32no—
45#494—31no—
46#464—32no—
47#464—32no—
48#461—31no—
49#461D311yescharacteristic, complemented
50C8×Q8—31no—
51C4×Q16—32no—
52C4×D8—31no—
53C2×C3×Q8C2312yes—
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56C2×D12C2311yescharacteristic
57C2×C4×D3—21no—
58C2×Dic6D411yescharacteristic
59C2×Dic6C2312yes—
60C3×SD16—42no—
61C2×C3×C8C2312yes—
62C3×C4⋊C4—21no—
63C3×C4⋊C4—21no—
64C3×C4⋊C4C2311yescharacteristic
65C3×C22⋊C4—21no—
66C3×C42C2311yescharacteristic
67#217—22no—
68#217—22no—
69#216—24no—
70#216—42no—
71#213—22no—
72(C2×D3)⋊C4—21no—
73(C2×D3)⋊C4—22no—
74(C2×D3)⋊C4—22no—
75C4⋊Dic3—21no—
76C4⋊Dic3—21no—
77C4⋊Dic3—22no—
78C4⋊Dic3D411yescharacteristic
79C4⋊Dic3C2311yescharacteristic
80Dic3⋊C4—22no—
81Dic3⋊C4D412yes—
82C4×Dic3—22no—
83C4×Dic3—22no—
84C4×Dic3C2312yes—
85C2×C3⋊C8C2311yescharacteristic
86C2×C3⋊C8C2311yescharacteristic
87#204—42no—
88C2×Q16—32no—
89C2×Q16—32no—
90C2×SD16—32no—
91C2×SD16C2×D312yes—
92C2×D8—31no—
93C4⋊Q8—31no—
94C4⋊Q8C2×D311yescharacteristic
95#126—62no—
96#126—62no—
97#124—32no—
98#124—32no—
99C22⋊Q8—61no—
100C22⋊Q8—61no—
101C4×Q8—31no—
102C4×Q8—32no—
103C4×Q8—32no—
104C4×D4—31no—
105C4×D4C2×D311yescharacteristic
106#117—61no—
107#117—61no—
108#107—31no—
109C4⋊C8—32no—
110C4⋊C8C2×D311yescharacteristic
111#103—32no—
112#103C2×D312yes—
113#102—31no—
114#102—31no—
115C4×C8—32no—
116C4×C8—31no—
117C23×C3—21no—
118C22×D3—21no—
119C3×Q8—22no—
120C3×Q8—22no—
121C3×D4—21no—
122C3×D4—21no—
123C2×C3×C4—21no—
124C2×C3×C4—22no—
125C2×C3×C4—42no—
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4C2×D411yescharacteristic
128C2×C3×C4C2411yescharacteristic
129C2×Dic3—21no—
130C2×Dic3—22no—
131C2×Dic3C2×D412yes—
132C2×Dic3C2×D412yes—
133D12—21no—
134D12—21no—
135C4×D3—42no—
136Dic6—22no—
137Dic6—22no—
138Dic6—22no—
139C3×C8—22no—
140C3⋊C8—21no—
141C3⋊C8#4112yes—
142C2×Q8—31no—
143C2×Q8—32no—
144C2×Q8C22×D312yes—
145C2×D4—31no—
146C2×D4C22×D311yescharacteristic
147C22×C4—21no—
148C22×C4—61no—
149Q16—64no—
150Q16—122no—
151SD16—42no—
152SD16—122no—
153D8—62no—
154C2×C8—31no—
155C2×C8—31no—
156C2×C8C22×D312yes—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—31no—
160C4⋊C4—32no—
161C4⋊C4—31no—
162C4⋊C4—62no—
163C4⋊C4—62no—
164C4⋊C4—61no—
165C4⋊C4—61no—
166C4⋊C4C22×D311yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—62no—
169C22⋊C4—62no—
170C22⋊C4—62no—
171C22⋊C4—62no—
172C22⋊C4—61no—
173C42—32no—
174C42—62no—
175C42—62no—
176C42C22×D311yescharacteristic
177C22×C3—41no—
178C22×C3—41no—
179C22×C3—41no—
180C22×C3C22×D411yescharacteristic
181C2×D3—41no—
182C2×D3—41no—
183C2×D3—41no—
184C3×C4—21no—
185C3×C4—42no—
186C3×C4—41no—
187C3×C4#13512yes—
188C3×C4C2×#4111yescharacteristic
189C3×C4H(5,𝔽2)11yescharacteristic
190Dic3—22no—
191Dic3—22no—
192Dic3—42no—
193Dic3—41no—
194C23—21no—
195C23—61no—
196Q8—22no—
197Q8—22no—
198Q8—62no—
199Q8—62no—
200Q8—62no—
201D4—21no—
202D4—21no—
203D4—61no—
204D4—61no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—32no—
208C2×C4—32no—
209C2×C4—42no—
210C2×C4—61no—
211C2×C4—62no—
212C2×C4—122no—
213C2×C4D3×D411yescharacteristic
214C2×C4D3×D411yescharacteristic
215C2×C4C23×D311yescharacteristic
216C8—22no—
217C8—32no—
218C8—61no—
219C2×C3—81no—
220C2×C3#55011yescharacteristic
221C2×C3C2×#13511yescharacteristic
222C2×C3#57811yescharacteristic
223D3—81no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—121no—
228C22—121no—
229C22—121no—
230C22C2×D3×D411yescharacteristic
231C4—21no—
232C4—41no—
233C4—42no—
234C4—62no—
235C4—62no—
236C4—121no—
237C4—122no—
238C4#91612yes—
239C4#100411yescharacteristic
240C4D3×#4111yescharacteristic
241C3#339311yescharacteristic, complemented, Sylow
242C2—81no—
243C2—241no—
244C2#559911yescharacteristic
245C2C2×#91611yescharacteristic
246C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#461C2
socleC22×C3C22×D4
prime radicalC2×#213C22
hypercenter#461D3
hypocenterC3#3393
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #5732C2semidirect
2. #5542C2semidirect
3. C3×#461C2semidirect
5. #5166C2semidirect
6. #5092C2semidirect
8. #4841C2semidirect
9. #4830C2semidirect
10. #4798C2semidirect
11. #4790C2semidirect
49. #461D3semidirect
241. C3#3393semidirect

Maximal subgroups

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

subgroupindex
#57322
#55422
C3×#4612
#51682
#51662
#50922
C4×#2132
#48412
#48302
#47982
#47902
#33933