Finite Group Databasebeta

← #81744

Subgroups — #81744

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#5724C211yesmaximal, characteristic
2#5698C211yesmaximal, characteristic, complemented
3C3×#491C211yesmaximal, characteristic, complemented
4#5092C211yesmaximal, characteristic, complemented
5#5068C211yesmaximal, characteristic, complemented
6#4809C212yesmaximal, complemented
7#4808C212yesmaximal
8#4798C212yesmaximal, complemented
9#4795C212yesmaximal, complemented
10#4715C211yesmaximal, characteristic, complemented
11#4709C211yesmaximal, characteristic, complemented
12#3223—31nomaximal, Sylow
13C2×#239C2211yescharacteristic
14C2×D3×Q8C2211yescharacteristic
15C3×C4⋊Q8C2211yescharacteristic
16C3×#125C2211yescharacteristic
17#947C2211yescharacteristic
18#944C2211yescharacteristic
19C2×#216C2212yes—
20C2×#216C2212yes—
21C2×Dic12C2212yes—
22C2×#204C2212yes—
23#898—22no—
24#897C2212yes—
25#896—22no—
26#894—22no—
27#890—22no—
28#887C2212yes—
29#875C2211yescharacteristic
30#872C2211yescharacteristic
31#869C2211yescharacteristic
32C3×#103C2212yes—
33C3×#103C2212yes—
34C3×#97C2211yescharacteristic
35(C2×D3)⋊C8C2212yes—
36Dic3⋊C8C2212yes—
37#810C2212yes—
38#809C2212yes—
39C4×C3⋊C8C2211yescharacteristic
40#541—31no—
41#536—31no—
42#497—31no—
43#494—31no—
44#491D311yescharacteristic, complemented
45#488—31no—
46#464—32no—
47#462—32no—
48#452—32no—
49#451—32no—
50#433—31no—
51C2×C3×Q8C2311yescharacteristic
52C2×C3×Q8C2311yescharacteristic
53#239—42no—
54D3×Q8—42no—
55C2×D12—21no—
56C2×D12C2311yescharacteristic
57C2×C4×D3—21no—
58C2×C4×D3—21no—
59C2×C4×D3C2311yescharacteristic
60C2×Dic6—21no—
61C2×Dic6C2311yescharacteristic
62C2×Dic6C2312yes—
63C2×C3×C8C2312yes—
64C3×C4⋊C4—22no—
65C3×C4⋊C4—22no—
66C3×C4⋊C4C2312yes—
67C3×C42C2311yescharacteristic
68#216—42no—
69#216—42no—
70(C2×D3)⋊C4—22no—
71(C2×D3)⋊C4—22no—
72(C2×D3)⋊C4—22no—
73(C2×D3)⋊C4D412yes—
74C4⋊Dic3—22no—
75C4⋊Dic3—22no—
76Dic3⋊C4—22no—
77Dic3⋊C4—22no—
78Dic3⋊C4D412yes—
79C4×Dic3—22no—
80C4×Dic3C2311yescharacteristic
81C2×C3⋊C8C2312yes—
82Dic12—42no—
83#204—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⋊Q8C2×D311yescharacteristic
93#126—62no—
94#125—62no—
95#125C2×D311yescharacteristic
96#124—31no—
97#124—31no—
98#123—62no—
99C22⋊Q8—32no—
100C22⋊Q8—62no—
101C4×Q8—32no—
102#117—31no—
103C4⋊C8—32no—
104#103—32no—
105#103C2×D312yes—
106#103C2×D312yes—
107#102—32no—
108C22⋊C8—32no—
109#97C2×D311yescharacteristic
110C4×C8—31no—
111C22×D3—21no—
112C22×D3C2×D411yescharacteristic
113C3×Q8—22no—
114C3×Q8—22no—
115C2×C3×C4—22no—
116C2×C3×C4—21no—
117C2×C3×C4—21no—
118C2×C3×C4C2×D412yes—
119C2×C3×C4C2411yescharacteristic
120C2×Dic3—22no—
121C2×Dic3—21no—
122C2×Dic3C2×D412yes—
123C2×Dic3C2×D411yescharacteristic
124D12—22no—
125D12—42no—
126C4×D3—22no—
127C4×D3—42no—
128C4×D3—42no—
129Dic6—22no—
130Dic6—22no—
131Dic6—22no—
132Dic6—42no—
133C3×C8—22no—
134C3⋊C8—22no—
135#41—122no—
136C2×Q8—31no—
137C2×Q8—32no—
138C2×Q8—61no—
139C2×Q8—122no—
140C2×Q8C22×D311yescharacteristic
141C2×Q8C22×D311yescharacteristic
142C2×D4—31no—
143C2×D4—61no—
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—22no—
155C4⋊C4—32no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4C22×D312yes—
161C22⋊C4—32no—
162C22⋊C4—62no—
163C22⋊C4—62no—
164C22⋊C4—62no—
165C42—31no—
166C42—62no—
167C42C22×D311yescharacteristic
168C22×C3C22×D411yescharacteristic
169C2×D3—21no—
170C2×D3—22no—
171C2×D3—41no—
172C2×D3—42no—
173C3×C4—22no—
174C3×C4—41no—
175C3×C4—41no—
176C3×C4—42no—
177C3×C4H(5,𝔽2)12yes—
178Dic3—21no—
179Dic3—22no—
180Dic3—41no—
181Dic3—42no—
182C23—31no—
183C23—61no—
184Q8—22no—
185Q8—22no—
186Q8—62no—
187Q8—62no—
188Q8—62no—
189Q8—122no—
190D4—62no—
191D4—122no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—32no—
196C2×C4—31no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—61no—
200C2×C4—122no—
201C2×C4—122no—
202C2×C4D3×D412yes—
203C2×C4C23×D311yescharacteristic
204C8—22no—
205C8—62no—
206C2×C3#53511yescharacteristic
207C2×C3#57812yes—
208D3—41no—
209D3—81no—
210C22—61no—
211C22—62no—
212C22—121no—
213C22—122no—
214C22C2×D3×D411yescharacteristic
215C4—22no—
216C4—41no—
217C4—41no—
218C4—42no—
219C4—61no—
220C4—62no—
221C4—121no—
222C4—122no—
223C4#100912yes—
224C3#322311yescharacteristic, complemented, Sylow
225C2—121no—
226C2—241no—
227C2#567411yescharacteristic
228C2#577912yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#491C2
socleC22×C3C22×D4
prime radicalC2×#239C22
hypercenter#491D3
hypocenterC3#3223
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. #5698C2semidirect
3. C3×#491C2semidirect
4. #5092C2semidirect
5. #5068C2semidirect
6. #4809C2semidirect
8. #4798C2semidirect
9. #4795C2semidirect
10. #4715C2semidirect
11. #4709C2semidirect
44. #491D3semidirect
224. C3#3223semidirect

Maximal subgroups

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

subgroupindex
#57242
#56982
C3×#4912
#50922
#50682
#48092
#48082
#47982
#47952
#47152
#47092
#32233