Finite Group Databasebeta

← #81647

Subgroups — #81647

Order 384.

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

normal subgroups only

This group has 364 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#5698C211yesmaximal, characteristic, complemented
2#5675C211yesmaximal, characteristic
3C3×#488C211yesmaximal, characteristic, complemented
4#5069C211yesmaximal, characteristic, complemented
5#5060C211yesmaximal, characteristic, complemented
6#4809C212yesmaximal, complemented
7#4795C212yesmaximal, complemented
8#4783C212yesmaximal
9#4767C212yesmaximal, complemented
10#4699C211yesmaximal, characteristic, complemented
11#4690C211yesmaximal, characteristic, complemented
12#3223—31nomaximal, Sylow
13C2×D3×Q8C2211yescharacteristic
14C2×#237C2211yescharacteristic
15C3×#125C2211yescharacteristic
16C3×#124C2211yescharacteristic
17#944C2211yescharacteristic
18C2×#216C2212yes—
19#936C2211yescharacteristic
20C2×#214C2212yes—
21C2×Dic12C2212yes—
22C2×Dic12C2212yes—
23#898—22no—
24#897C2212yes—
25#890—22no—
26#887C2212yes—
27#883—22no—
28#878—22no—
29#872C2211yescharacteristic
30#869C2211yescharacteristic
31#869C2211yescharacteristic
32C3×#103C2212yes—
33C3×#102C2212yes—
34C3×C4×C8C2211yescharacteristic
35(C2×D3)⋊C8C2212yes—
36Dic3⋊C8C2212yes—
37#810C2212yes—
38#810C2212yes—
39#803C2211yescharacteristic
40#541—31no—
41#536—31no—
42#497—31no—
43#494—31no—
44#491—31no—
45#488D311yescharacteristic, complemented
46#464—32no—
47#462—32no—
48#452—32no—
49#451—32no—
50#433—31no—
51C2×C3×Q8C2311yescharacteristic
52C2×C3×D4C2311yescharacteristic
53C2×#67—21no—
54C22×Dic3—21no—
55D3×Q8—42no—
56#237—42no—
57C2×C4×D3—21no—
58C2×C4×D3C2311yescharacteristic
59C2×Dic6—21no—
60C2×Dic6C2311yescharacteristic
61C2×Dic6C2311yescharacteristic
62C2×Dic6C2312yes—
63C2×C3×C8C2312yes—
64C3×C4⋊C4—22no—
65C3×C4⋊C4C2312yes—
66C3×C22⋊C4—22no—
67C3×C42C2311yescharacteristic
68#217—22no—
69#216—42no—
70#214—42no—
71(C2×D3)⋊C4—22no—
72(C2×D3)⋊C4D412yes—
73C4⋊Dic3—22no—
74C4⋊Dic3—22no—
75C4⋊Dic3—22no—
76Dic3⋊C4—22no—
77Dic3⋊C4—22no—
78Dic3⋊C4D412yes—
79C4×Dic3—22no—
80C4×Dic3C2311yescharacteristic
81C2×C3⋊C8C2312yes—
82Dic12—42no—
83Dic12—42no—
84C2×#41—31no—
85C22×Q8—31no—
86C2×Q16—32no—
87C2×Q16—32no—
88C2×Q16—32no—
89C2×SD16—32no—
90C4⋊Q8—31no—
91C4⋊Q8—31no—
92C4⋊Q8—31no—
93#126—62no—
94#125—62no—
95#125C2×D311yescharacteristic
96#124—31no—
97#124C2×D311yescharacteristic
98#123—62no—
99C22⋊Q8—32no—
100C22⋊Q8—62no—
101C4×Q8—32no—
102#117—31no—
103C4⋊C8—32no—
104#103—32no—
105#103—32no—
106#103C2×D312yes—
107#102C2×D312yes—
108C22⋊C8—32no—
109#97—31no—
110C4×C8C2×D311yescharacteristic
111C23×C3—21no—
112C22×D3C2×D411yescharacteristic
113C3×Q8—22no—
114C3×D4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—21no—
117C2×C3×C4C2×D412yes—
118C2×C3×C4C2411yescharacteristic
119#67—42no—
120C2×Dic3—22no—
121C2×Dic3—21no—
122C2×Dic3—21no—
123C2×Dic3—42no—
124C2×Dic3C2×D412yes—
125C2×Dic3C2×D411yescharacteristic
126C4×D3—22no—
127C4×D3—42no—
128Dic6—22no—
129Dic6—22no—
130Dic6—22no—
131Dic6—22no—
132Dic6—42no—
133C3×C8—22no—
134C3⋊C8—22no—
135#41—122no—
136C2×Q8—31no—
137C2×Q8—31no—
138C2×Q8—32no—
139C2×Q8—61no—
140C2×Q8—122no—
141C2×Q8C22×D311yescharacteristic
142C2×D4—61no—
143C2×D4C22×D311yescharacteristic
144C22×C4—31no—
145C22×C4—61no—
146C22×C4—61no—
147Q16—122no—
148Q16—122no—
149Q16—122no—
150SD16—122no—
151C2×C8—32no—
152C2×C8C22×D312yes—
153C4⋊C4—22no—
154C4⋊C4—32no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4C22×D312yes—
161C22⋊C4—22no—
162C22⋊C4—32no—
163C22⋊C4—62no—
164C22⋊C4—62no—
165C42—31no—
166C42—62no—
167C42C22×D311yescharacteristic
168C22×C3—41no—
169C22×C3—42no—
170C22×C3C22×D411yescharacteristic
171C2×D3—21no—
172C2×D3—22no—
173C3×C4—22no—
174C3×C4—41no—
175C3×C4—42no—
176C3×C4H(5,𝔽2)12yes—
177Dic3—21no—
178Dic3—22no—
179Dic3—41no—
180Dic3—42no—
181Dic3—41no—
182C23—21no—
183C23—31no—
184Q8—22no—
185Q8—62no—
186Q8—62no—
187Q8—62no—
188Q8—62no—
189Q8—122no—
190D4—22no—
191D4—122no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—32no—
195C2×C4—31no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4—61no—
199C2×C4—61no—
200C2×C4—122no—
201C2×C4—122no—
202C2×C4D3×D412yes—
203C2×C4C23×D311yescharacteristic
204C8—22no—
205C8—62no—
206C2×C3—81no—
207C2×C3#53511yescharacteristic
208C2×C3#57812yes—
209D3—41no—
210C22—41no—
211C22—42no—
212C22—61no—
213C22—62no—
214C22C2×D3×D411yescharacteristic
215C4—22no—
216C4—41no—
217C4—42no—
218C4—61no—
219C4—62no—
220C4—121no—
221C4—121no—
222C4—122no—
223C4#100912yes—
224C3#322311yescharacteristic, complemented, Sylow
225C2—81no—
226C2—121no—
227C2#567411yescharacteristic
228C2#577112yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#488C2
socleC22×C3C22×D4
prime radicalC2×#237C22
hypercenter#488D3
hypocenterC3#3223
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #5698C2semidirect
3. C3×#488C2semidirect
4. #5069C2semidirect
5. #5060C2semidirect
6. #4809C2semidirect
7. #4795C2semidirect
9. #4767C2semidirect
10. #4699C2semidirect
11. #4690C2semidirect
45. #488D3semidirect
224. C3#3223semidirect

Maximal subgroups

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

subgroupindex
#56982
#56752
C3×#4882
#50692
#50602
#48092
#47952
#47832
#47672
#46992
#46902
#32233