Finite Group Databasebeta

← #80856

Subgroups — #80856

Order 384.

354 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 109 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#5600C212yesmaximal, complemented
2C3×#469C211yesmaximal, characteristic
3#5153C212yesmaximal, complemented
4#5137C211yesmaximal, characteristic, complemented
5#5118C211yesmaximal, characteristic, complemented
6#5042C212yesmaximal, complemented
7#4857C212yesmaximal, complemented
8#4767C212yesmaximal, complemented
9#4763C212yesmaximal, complemented
10#3256—31nomaximal, Sylow
11C2×C3×D8C2211yescharacteristic
12C2×C3×M16C2211yescharacteristic
13C3×#121C2212yes—
14#936C2212yes—
15#935—22no—
16D4×Dic3—22no—
17D4×Dic3C2212yes—
18C2×#214C2212yes—
19#924C2212yes—
20C2×C4×Dic3C2211yescharacteristic
21C2×#208C2211yescharacteristic
22C2×Dic12C2211yescharacteristic
23#890C2212yes—
24#887C2212yes—
25#879—22no—
26#877—22no—
27C3×#106C2211yescharacteristic
28C3×#102C2212yes—
29#832C2212yes—
30#817C2211yescharacteristic
31#816C2212yes—
32Dic3⋊C8C2212yes—
33Dic3⋊C8C2212yes—
34#810C2212yes—
35#807C2212yes—
36#529—32no—
37#479—32no—
38#478—32no—
39#470—31no—
40#469D311yescharacteristic
41#468—32no—
42#464—32no—
43#463—32no—
44#423—31no—
45C2×C3×D4—22no—
46C2×C3×D4C2312yes—
47C22×C3×C4C2311yescharacteristic
48C22×Dic3—22no—
49C22×Dic3D412yes—
50C2×Dic6C2312yes—
51C3×D8—41no—
52C3×M16—41no—
53C2×C3×C8C2311yescharacteristic
54C2×C3×C8C2311yescharacteristic
55C3×C4⋊C4C2312yes—
56C3×C22⋊C4—22no—
57#217—22no—
58#217—22no—
59#217—22no—
60#217—22no—
61#214—42no—
62C4⋊Dic3—22no—
63C4⋊Dic3C2312yes—
64Dic3⋊C4—22no—
65Dic3⋊C4—22no—
66Dic3⋊C4—22no—
67C4×Dic3—22no—
68C4×Dic3D411yescharacteristic
69C4×Dic3D411yescharacteristic
70C4×Dic3C2312yes—
71#208—41no—
72C2×C3⋊C8C2312yes—
73Dic12—41no—
74C2×Q16—31no—
75C2×SD16—32no—
76C2×D8C2×D311yescharacteristic
77C2×M16—31no—
78C2×M16C2×D311yescharacteristic
79#126—62no—
80#125—32no—
81#124—32no—
82#123—62no—
83C22⋊Q8—32no—
84#121C2×D312yes—
85C4×Q8—32no—
86C4×D4—32no—
87C4×D4—62no—
88#117—62no—
89C2×C42—31no—
90#107—32no—
91#106—31no—
92#106C2×D311yescharacteristic
93C4⋊C8—32no—
94C4⋊C8—32no—
95#103—32no—
96#103—32no—
97#102—32no—
98#102C2×D312yes—
99C23×C3—22no—
100C23×C3C2×D411yescharacteristic
101C3×D4—22no—
102C3×D4—22no—
103C3×D4—44no—
104C2×C3×C4—21no—
105C2×C3×C4—21no—
106C2×C3×C4—22no—
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4C2411yescharacteristic
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3—42no—
112C2×Dic3—44no—
113C2×Dic3C2×D412yes—
114C2×Dic3C2×D412yes—
115Dic6—22no—
116Dic6—22no—
117C3×C8—21no—
118C3×C8—21no—
119C3⋊C8—22no—
120C2×Q8—32no—
121C2×D4—22no—
122C2×D4C22×D312yes—
123C22×C4—32no—
124C22×C4—62no—
125C22×C4C22×D311yescharacteristic
126Q16—121no—
127SD16—122no—
128D8—41no—
129M16—41no—
130M16—121no—
131C2×C8—32no—
132C2×C8C22×D311yescharacteristic
133C2×C8C22×D311yescharacteristic
134C4⋊C4—32no—
135C4⋊C4—62no—
136C4⋊C4—62no—
137C4⋊C4—62no—
138C4⋊C4—62no—
139C4⋊C4C22×D312yes—
140C22⋊C4—22no—
141C22⋊C4—62no—
142C22⋊C4—62no—
143C22⋊C4—62no—
144C22⋊C4—62no—
145C42—31no—
146C42—31no—
147C42—32no—
148C42—62no—
149C22×C3—21no—
150C22×C3—22no—
151C22×C3—42no—
152C22×C3—42no—
153C22×C3—42no—
154C22×C3C22×D411yescharacteristic
155C3×C4—21no—
156C3×C4—42no—
157C3×C4H(5,𝔽2)11yescharacteristic
158C3×C42−1+411yescharacteristic
159Dic3—22no—
160Dic3—22no—
161Dic3—42no—
162Dic3—42no—
163C23—22no—
164C23D3×D411yescharacteristic
165Q8—62no—
166Q8—62no—
167D4—22no—
168D4—22no—
169D4—44no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—32no—
174C2×C4—32no—
175C2×C4—62no—
176C2×C4—62no—
177C2×C4—122no—
178C2×C4—124no—
179C2×C4D3×D411yescharacteristic
180C2×C4C23×D311yescharacteristic
181C8—21no—
182C8—21no—
183C8—62no—
184C2×C3—41no—
185C2×C3—82no—
186C2×C3#53711yescharacteristic
187C2×C3#57512yes—
188C22—22no—
189C22—21no—
190C22—42no—
191C22—42no—
192C22—42no—
193C22C2×D3×D411yescharacteristic
194C4—21no—
195C4—42no—
196C4—62no—
197C4—62no—
198C4—122no—
199C4—122no—
200C4#100411yescharacteristic
201C4#101011yescharacteristic
202C3#325611yescharacteristic, complemented, Sylow
203C2—41no—
204C2—82no—
205C2#559711yescharacteristic
206C2#577312yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC3×#469C2
socleC22×C3C22×D4
hypercenter#469D3
hypocenterC3#3256
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #5600C2semidirect
3. #5153C2semidirect
4. #5137C2semidirect
5. #5118C2semidirect
6. #5042C2semidirect
7. #4857C2semidirect
8. #4767C2semidirect
9. #4763C2semidirect
202. C3#3256semidirect

Maximal subgroups

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

subgroupindex
#56002
C3×#4692
#51532
#51372
#51182
#50422
#48572
#47672
#47632
#32563