Finite Group Databasebeta

← #79678

Subgroups — #79678

Order 384.

420 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 125 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
1Q8×D12C211yesmaximal, characteristic, complemented
2#5551C211yesmaximal, characteristic
3C3×C4×SD16C211yesmaximal, characteristic, complemented
4#5170C212yesmaximal, complemented
5#5020C211yesmaximal, characteristic, complemented
6#5019C211yesmaximal, characteristic, complemented
7#4862C212yesmaximal, complemented
8D3×#103C212yesmaximal, complemented
9#4778C212yesmaximal, complemented
10C8×D12C211yesmaximal, characteristic, complemented
11#4681C211yesmaximal, characteristic, complemented
12#3298—31nomaximal, Sylow
13C2×D3×Q8C2212yes—
14C2×#235—21no—
15C2×C3×SD16C2211yescharacteristic
16C3×C4×Q8C2211yescharacteristic
17C3×C4×D4C2211yescharacteristic
18#938C2212yes—
19C2×#214C2212yes—
20#924—21no—
21C2×D3×C8C2212yes—
22#897—22no—
23#897C2212yes—
24D3×C4⋊C4C2212yes—
25#883—22no—
26C4×D12—21no—
27C4×D12C2211yescharacteristic
28#869—21no—
29#869C2211yescharacteristic
30#869C2211yescharacteristic
31C3×#106C2211yescharacteristic
32C3×#103C2211yescharacteristic
33C3×#102C2211yescharacteristic
34C3×C4×C8C2211yescharacteristic
35#835C2212yes—
36(C2×D3)⋊C8C2212yes—
37#816C2212yes—
38#816C2212yes—
39#810C2212yes—
40#808C2212yes—
41C4⋊(C3⋊C8)C2211yescharacteristic
42D4×Q8—31no—
43#547—31no—
44#487—31no—
45#483—32no—
46#475—31no—
47#465—32no—
48#461—31no—
49#448—32no—
50C4×SD16D311yescharacteristic, complemented
51C8×D4—31no—
52C2×#103—32no—
53C2×C3×Q8C2311yescharacteristic
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56C2×#67—22no—
57D3×Q8—24no—
58D3×Q8—42no—
59#235—44no—
60C2×D12D411yescharacteristic
61C2×C4×D3—22no—
62C2×C4×D3—22no—
63C2×C4×D3C2312yes—
64C2×Dic6—21no—
65C2×Dic6—22no—
66C2×Dic6C2312yes—
67C3×SD16—22no—
68C2×C3×C8C2311yescharacteristic
69C2×C3×C8C2311yescharacteristic
70C3×C4⋊C4—21no—
71C3×C4⋊C4C2311yescharacteristic
72C3×C4⋊C4C2311yescharacteristic
73C3×C22⋊C4—21no—
74C3×C42—21no—
75C3×C42C2311yescharacteristic
76#217—22no—
77#214—42no—
78(C2×D3)⋊C4—22no—
79(C2×D3)⋊C4D412yes—
80C4⋊Dic3—21no—
81C4⋊Dic3—21no—
82C4⋊Dic3—21no—
83C4⋊Dic3—22no—
84C4⋊Dic3D411yescharacteristic
85C4⋊Dic3C2312yes—
86Dic3⋊C4—22no—
87C2×C3⋊C8C2312yes—
88D3×C8—42no—
89C2×#41—61no—
90C22×Q8—32no—
91C2×SD16—32no—
92C2×SD16C2×D311yescharacteristic
93C22×C8—32no—
94C4⋊Q8—31no—
95C4⋊Q8—31no—
96C4⋊Q8—61no—
97#123—62no—
98C22⋊Q8—32no—
99C22⋊Q8—62no—
100C22⋊Q8—61no—
101#121—32no—
102C4×Q8C2×D311yescharacteristic
103C4×D4—31no—
104C4×D4—61no—
105C4×D4C2×D311yescharacteristic
106C2×C4⋊C4—32no—
107#106—32no—
108#106C2×D311yescharacteristic
109C4⋊C8—31no—
110#103—32no—
111#103—32no—
112#103—32no—
113#103—32no—
114#103C2×D311yescharacteristic
115#102C2×D311yescharacteristic
116C22⋊C8—32no—
117C4×C8C2×D311yescharacteristic
118C23×C3—21no—
119C22×D3C2×D412yes—
120C3×Q8—21no—
121C3×Q8#4112yescomplemented
122C3×D4—21no—
123C3×D4—21no—
124C2×C3×C4—21no—
125C2×C3×C4—21no—
126C2×C3×C4—21no—
127C2×C3×C4—42no—
128C2×C3×C4C2×D411yescharacteristic
129C2×C3×C4C2×D411yescharacteristic
130C2×C3×C4C2411yescharacteristic
131#67—44no—
132C2×Dic3—22no—
133C2×Dic3—22no—
134C2×Dic3C2×D412yes—
135D12SD1614yescomplemented
136C4×D3—22no—
137C4×D3—22no—
138C4×D3—44no—
139C4×D3—44no—
140Dic6—22no—
141Dic6—22no—
142Dic6—44no—
143Dic6—42no—
144C3×C8—21no—
145C3×C8—21no—
146C3⋊C8—22no—
147#41—124no—
148C2×Q8—32no—
149C2×Q8—62no—
150C2×Q8—61no—
151C2×Q8—64no—
152C2×Q8—122no—
153C2×Q8C22×D311yescharacteristic
154C2×D4—31no—
155C2×D4—62no—
156C2×D4C22×D311yescharacteristic
157C22×C4—21no—
158C22×C4—32no—
159C22×C4—62no—
160C22×C4—62no—
161SD16—22no—
162SD16—122no—
163C2×C8—32no—
164C2×C8—122no—
165C2×C8C22×D311yescharacteristic
166C2×C8C22×D311yescharacteristic
167C4⋊C4—21no—
168C4⋊C4—31no—
169C4⋊C4—32no—
170C4⋊C4—62no—
171C4⋊C4—62no—
172C4⋊C4—61no—
173C4⋊C4—61no—
174C4⋊C4—61no—
175C4⋊C4C22×D311yescharacteristic
176C4⋊C4C22×D311yescharacteristic
177C22⋊C4—21no—
178C22⋊C4—32no—
179C22⋊C4—62no—
180C22⋊C4—62no—
181C42—21no—
182C42C22×D311yescharacteristic
183C22×C3—41no—
184C22×C3—41no—
185C22×C3—41no—
186C22×C3C22×D411yescharacteristic
187C2×D3—22no—
188C2×D3—22no—
189C2×D3C2×SD1614yes—
190C3×C4—22no—
191C3×C4—21no—
192C3×C4—41no—
193C3×C4—41no—
194C3×C4C2×SD1612yes—
195C3×C4C2×#4111yescharacteristic
196C3×C4H(5,𝔽2)11yescharacteristic
197Dic3—22no—
198Dic3—42no—
199Dic3—42no—
200C23—21no—
201C23—32no—
202Q8—21no—
203Q8—62no—
204Q8—62no—
205Q8—124no—
206Q8—122no—
207Q8#23512yescomplemented
208D4—21no—
209D4—21no—
210D4—34no—
211D4—124no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—32no—
216C2×C4—42no—
217C2×C4—62no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—62no—
221C2×C4—124no—
222C2×C4—124no—
223C2×C4D3×D411yescharacteristic
224C2×C4D3×D411yescharacteristic
225C2×C4C23×D311yescharacteristic
226C8—21no—
227C8—21no—
228C8—62no—
229C2×C3—81no—
230C2×C3#54611yescharacteristic
231C2×C3C22×SD1611yescharacteristic
232C2×C3#57811yescharacteristic
233D3—24no—
234C22—34no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—62no—
239C22—62no—
240C22C2×D3×D411yescharacteristic
241C4—21no—
242C4—22no—
243C4—41no—
244C4—41no—
245C4—62no—
246C4—122no—
247C4—122no—
248C4D3×SD1612yes—
249C4C2×#23511yescharacteristic
250C4#100911yescharacteristic
251C3#329811yescharacteristic, complemented, Sylow
252C2—64no—
253C2—81no—
254C2#555011yescharacteristic
255C2C2×D3×SD1611yescharacteristic
256C2#577111yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×SD16C2
socleC22×C3C22×D4
prime radical#5551C2
hypercenterC4×SD16D3
hypocenterC3#3298
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. Q8×D12C2semidirect
3. C3×C4×SD16C2semidirect
4. #5170C2semidirect
5. #5020C2semidirect
6. #5019C2semidirect
7. #4862C2semidirect
8. D3×#103C2semidirect
9. #4778C2semidirect
10. C8×D12C2semidirect
11. #4681C2semidirect
50. C4×SD16D3semidirect
121. C3×Q8#41semidirect
135. D12SD16semidirect
207. Q8#235semidirect
251. C3#3298semidirect

Maximal subgroups

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

subgroupindex
Q8×D122
#55512
C3×C4×SD162
#51702
#50202
#50192
#48622
D3×#1032
#47782
C8×D122
#46812
#32983