Finite Group Databasebeta

← #80754

Subgroups — #80754

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#5641C212yesmaximal
2C3×#467C211yesmaximal, characteristic, complemented
3#5188C211yesmaximal, characteristic, complemented
4Dic3×Q16C211yesmaximal, characteristic, complemented
5C8×#67C211yesmaximal, characteristic
6#5106C211yesmaximal, characteristic, complemented
7#5045C212yesmaximal, complemented
8#4880C211yesmaximal, characteristic, complemented
9#4874C211yesmaximal, characteristic, complemented
10#4815C212yesmaximal, complemented
11#4791C212yesmaximal, complemented
12#3342—31nomaximal, Sylow
13C2×C3×Q16C2211yescharacteristic
14C22×C3×C8C2211yescharacteristic
15C3×C22⋊Q8C2212yes—
16#946C2212yes—
17Q8×Dic3C2212yes—
18C4×#67C2211yescharacteristic
19#926C2212yes—
20C2×D3×C8C2211yescharacteristic
21#898—22no—
22#892C2212yes—
23#888C2212yes—
24#887—22no—
25#885—22no—
26#879—22no—
27C3×#107C2211yescharacteristic
28C3×#103C2212yes—
29#835C2212yes—
30#835C2212yes—
31#830C2211yescharacteristic
32(C2×D3)⋊C8C2211yescharacteristic
33#818C2211yescharacteristic
34#818C2211yescharacteristic
35#817C2212yes—
36Dic3⋊C8C2211yescharacteristic
37C8×Dic3C2211yescharacteristic
38#808C2212yes—
39#807C2212yes—
40#556—32no—
41#500—31no—
42#485—32no—
43#485—32no—
44#475—32no—
45#467—31no—
46#467D311yescharacteristic, complemented
47C4×Q16—31no—
48C8×D4—31no—
49#427—31no—
50#427—31no—
51C2×C3×Q8C2312yes—
52C22×C3×C4C2311yescharacteristic
53C2×#67D411yescharacteristic
54C2×C4×D3C2311yescharacteristic
55C2×Dic6—22no—
56C3×Q16—22no—
57C2×C3×C8—22no—
58C2×C3×C8C2311yescharacteristic
59C2×C3×C8C2311yescharacteristic
60C3×C4⋊C4—22no—
61C3×C4⋊C4C2312yes—
62C3×C22⋊C4—22no—
63#217—22no—
64#217D411yescharacteristic
65(C2×D3)⋊C4—22no—
66(C2×D3)⋊C4—22no—
67(C2×D3)⋊C4D411yescharacteristic
68C4⋊Dic3—22no—
69C4⋊Dic3C2312yes—
70C4⋊Dic3C2312yes—
71Dic3⋊C4—22no—
72Dic3⋊C4—22no—
73Dic3⋊C4D411yescharacteristic
74C4×Dic3—22no—
75C4×Dic3—22no—
76C4×Dic3—22no—
77C4×Dic3C2311yescharacteristic
78C2×C3⋊C8C2311yescharacteristic
79C2×C3⋊C8C2311yescharacteristic
80D3×C8—22no—
81C2×Q16C2×D311yescharacteristic
82C22×C8—31no—
83C22×C8C2×D311yescharacteristic
84C4⋊Q8—32no—
85#126—62no—
86#126—62no—
87#124—62no—
88C22⋊Q8—32no—
89C22⋊Q8C2×D312yes—
90C4×Q8—32no—
91C4×Q8—62no—
92C4×D4—31no—
93#117—32no—
94#117—32no—
95#107—31no—
96#107—31no—
97#107—32no—
98#107C2×D311yescharacteristic
99#106—32no—
100#106—32no—
101C4⋊C8—31no—
102#103—32no—
103#103—32no—
104#103C2×D312yes—
105C22⋊C8—31no—
106C22⋊C8—31no—
107C4×C8—31no—
108C23×C3C2×D411yescharacteristic
109C22×D3C2×D411yescharacteristic
110C3×Q8—22no—
111C3×Q8—22no—
112C2×C3×C4—21no—
113C2×C3×C4—21no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4C2×D411yescharacteristic
117C2×C3×C4C2411yescharacteristic
118#67—22no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3C2×D411yescharacteristic
122C2×Dic3C2×D411yescharacteristic
123C2×Dic3C2×D411yescharacteristic
124C4×D3—21no—
125C4×D3—21no—
126Dic6—44no—
127C3×C8—21no—
128C3×C8#4112yes—
129C3⋊C8—21no—
130C3⋊C8—21no—
131C2×Q8—62no—
132C2×Q8C22×D312yes—
133C2×D4—31no—
134C22×C4—31no—
135C22×C4C22×D311yescharacteristic
136Q16—22no—
137C2×C8—22no—
138C2×C8—31no—
139C2×C8—31no—
140C2×C8—62no—
141C2×C8C22×D311yescharacteristic
142C2×C8C22×D311yescharacteristic
143C4⋊C4—22no—
144C4⋊C4—31no—
145C4⋊C4—32no—
146C4⋊C4—32no—
147C4⋊C4—62no—
148C4⋊C4—62no—
149C4⋊C4—62no—
150C4⋊C4C22×D312yes—
151C22⋊C4—22no—
152C22⋊C4—31no—
153C22⋊C4—31no—
154C22⋊C4—62no—
155C22⋊C4—62no—
156C22⋊C4—62no—
157C42—31no—
158C42—62no—
159C42—62no—
160C42—62no—
161C22×C3—21no—
162C22×C3—21no—
163C22×C3—21no—
164C22×C3C22×D411yescharacteristic
165C2×D3—21no—
166C2×D3—21no—
167C2×D3—21no—
168C3×C4—21no—
169C3×C4—42no—
170C3×C4—42no—
171C3×C4C2×#4111yescharacteristic
172C3×C42−1+411yescharacteristic
173Dic3—21no—
174Dic3—21no—
175Dic3—42no—
176Dic3—42no—
177Dic3#13512yes—
178C23—31no—
179C23D3×D411yescharacteristic
180Q8—22no—
181Q8—22no—
182Q8—124no—
183D4—62no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—31no—
189C2×C4—31no—
190C2×C4—31no—
191C2×C4—61no—
192C2×C4—61no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4D3×D411yescharacteristic
196C2×C4C23×D311yescharacteristic
197C8—21no—
198C8—61no—
199C8—61no—
200C8#23712yes—
201C2×C3—41no—
202C2×C3#54711yescharacteristic
203C2×C3C2×#13511yescharacteristic
204C2×C3#57811yescharacteristic
205D3—41no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—61no—
210C22—61no—
211C22—61no—
212C22C2×D3×D411yescharacteristic
213C4—21no—
214C4—32no—
215C4—42no—
216C4—42no—
217C4—61no—
218C4—61no—
219C4—122no—
220C4—122no—
221C4C2×#23711yescharacteristic
222C4#101011yescharacteristic
223C3#334211yescharacteristic, complemented, Sylow
224C2—41no—
225C2—121no—
226C2#561111yescharacteristic
227C2#576611yescharacteristic
228C2D3×#13511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#467C2
socleC22×C3C22×D4
prime radicalC2×#67D4
hypercenter#467D3
hypocenterC3#3342
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. C3×#467C2semidirect
3. #5188C2semidirect
4. Dic3×Q16C2semidirect
6. #5106C2semidirect
7. #5045C2semidirect
8. #4880C2semidirect
9. #4874C2semidirect
10. #4815C2semidirect
11. #4791C2semidirect
46. #467D3semidirect
223. C3#3342semidirect

Maximal subgroups

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

subgroupindex
#56412
C3×#4672
#51882
Dic3×Q162
C8×#672
#51062
#50452
#48802
#48742
#48152
#47912
#33423