Finite Group Databasebeta

← #79442

Subgroups — #79442

Order 384.

386 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 123 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#5551C211yesmaximal, characteristic
2#5542C211yesmaximal, characteristic, complemented
3C3×C8×D4C211yesmaximal, characteristic, complemented
4#5116C212yesmaximal, complemented
5#5113C212yesmaximal, complemented
6#4849C211yesmaximal, characteristic, complemented
7#4846C211yesmaximal, characteristic, complemented
8#4733C212yesmaximal, complemented
9#4723C212yesmaximal, complemented
10C4×#204C211yesmaximal, characteristic, complemented
11#4681C211yesmaximal, characteristic, complemented
12#3302—31nomaximal, Sylow
13C2×#235C2212yes—
14C22×C3×C8C2212yes—
15C3×C4×D4C2211yescharacteristic
16#938—21no—
17#936—21no—
18#926C2212yes—
19#924C2212yes—
20#924C2212yes—
21C2×Dic12C2212yes—
22C2×#204C2211yescharacteristic
23D3×C4⋊C4—21no—
24#887—21no—
25#883—22no—
26#879—22no—
27C4×D12C2211yescharacteristic
28#869C2211yescharacteristic
29#869C2211yescharacteristic
30C4×Dic6C2211yescharacteristic
31C3×C4⋊C8C2211yescharacteristic
32C3×C22⋊C8C2212yes—
33C3×C4×C8C2211yescharacteristic
34#821C2211yescharacteristic
35#821C2212yes—
36#818C2212yes—
37#817C2211yescharacteristic
38#816C2212yes—
39#816C2212yes—
40#816C2211yescharacteristic
41#816C2212yes—
42#556—31no—
43#547—31no—
44#487—31no—
45#485—32no—
46#474—31no—
47#467—32no—
48#462—31no—
49#452—32no—
50C4×SD16—31no—
51C8×D4D311yescharacteristic, complemented
52#416—32no—
53C2×C3×D4D411yescharacteristic
54C22×C3×C4C2312yes—
55C2×#67—22no—
56#235—24no—
57#235—42no—
58C2×D12C2311yescharacteristic
59C2×C4×D3—21no—
60C2×C4×D3—22no—
61C2×Dic6—21no—
62C2×Dic6C2312yes—
63C2×Dic6C2311yescharacteristic
64C2×C3×C8—42no—
65C2×C3×C8C2311yescharacteristic
66C2×C3×C8C2311yescharacteristic
67C2×C3×C8C2312yes—
68C3×C4⋊C4D411yescharacteristic
69C3×C22⋊C4D412yes—
70C3×C42C2311yescharacteristic
71#217—22no—
72#217—22no—
73#217—22no—
74(C2×D3)⋊C4—21no—
75C4⋊Dic3—21no—
76C4⋊Dic3—21no—
77C4⋊Dic3—21no—
78C4⋊Dic3C2311yescharacteristic
79C4⋊Dic3C2312yes—
80C4⋊Dic3C2311yescharacteristic
81Dic3⋊C4—21no—
82Dic3⋊C4—22no—
83Dic3⋊C4—22no—
84C4×Dic3—21no—
85C4×Dic3—22no—
86Dic12—42no—
87#204—22no—
88C2×#41—32no—
89C2×Q16—32no—
90C2×SD16—31no—
91C22×C8C2×D312yes—
92C4⋊Q8—31no—
93C4⋊Q8—31no—
94#126—62no—
95#124—61no—
96#123—62no—
97C22⋊Q8—32no—
98C22⋊Q8—32no—
99#121—61no—
100C4×Q8—31no—
101C4×Q8—61no—
102C4×D4—31no—
103C4×D4C2×D311yescharacteristic
104#117—32no—
105C2×C4⋊C4—61no—
106#107—32no—
107#106—31no—
108C4⋊C8C2×D311yescharacteristic
109#103—32no—
110#103—32no—
111#103—31no—
112#103—32no—
113#102—31no—
114#102—32no—
115C22⋊C8C2×D312yes—
116C4×C8C2×D311yescharacteristic
117C23×C3C2×D412yes—
118C22×D3—21no—
119C3×D4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4C2×D412yes—
123C2×C3×C4C2×D411yescharacteristic
124C2×C3×C4C2×D411yescharacteristic
125C2×C3×C4C2411yescharacteristic
126#67—44no—
127C2×Dic3—21no—
128C2×Dic3—21no—
129C2×Dic3—22no—
130C2×Dic3—21no—
131C2×Dic3—22no—
132D12—21no—
133D12#4112yes—
134C4×D3—44no—
135C4×D3—42no—
136Dic6—21no—
137Dic6—22no—
138Dic6—21no—
139Dic6—22no—
140Dic6—42no—
141C3×C8—21no—
142C3×C8—22no—
143C3×C8—21no—
144#41—64no—
145#41—122no—
146C2×Q8—32no—
147C2×Q8—31no—
148C2×Q8—61no—
149C2×D4—31no—
150C2×D4—62no—
151C2×D4D1211yescharacteristic
152C22×C4—61no—
153C22×C4—62no—
154C22×C4C22×D312yes—
155Q16—122no—
156SD16—62no—
157C2×C8—42no—
158C2×C8C22×D311yescharacteristic
159C2×C8C22×D311yescharacteristic
160C2×C8C22×D312yes—
161C4⋊C4—31no—
162C4⋊C4—32no—
163C4⋊C4—31no—
164C4⋊C4—61no—
165C4⋊C4—62no—
166C4⋊C4—61no—
167C4⋊C4—62no—
168C4⋊C4—61no—
169C4⋊C4—61no—
170C4⋊C4D1211yescharacteristic
171C22⋊C4—62no—
172C22⋊C4—62no—
173C22⋊C4—62no—
174C22⋊C4—61no—
175C22⋊C4D1212yes—
176C42—62no—
177C42—61no—
178C42C22×D311yescharacteristic
179C22×C3—22no—
180C22×C3—22no—
181C22×C3—22no—
182C22×C3C22×D411yescharacteristic
183C2×D3—22no—
184C2×D3—41no—
185C2×D3—41no—
186C3×C4—22no—
187C3×C4—21no—
188C3×C4#13512yes—
189C3×C4C2×#4111yescharacteristic
190C3×C4H(5,𝔽2)11yescharacteristic
191Dic3—42no—
192Dic3—41no—
193Dic3—42no—
194Dic3—41no—
195Dic3—41no—
196C23—61no—
197C23C2×D1212yes—
198Q8—61no—
199Q8—62no—
200Q8—61no—
201Q8—62no—
202Q8—122no—
203D4—22no—
204D4—32no—
205D4—61no—
206D4—124no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—61no—
210C2×C4—62no—
211C2×C4—61no—
212C2×C4—61no—
213C2×C4—62no—
214C2×C4—124no—
215C2×C4—122no—
216C2×C4C2×D1212yes—
217C2×C4C2×D1211yescharacteristic
218C2×C4C2×D1211yescharacteristic
219C2×C4C23×D311yescharacteristic
220C8—21no—
221C8—22no—
222C8—21no—
223C2×C3—42no—
224C2×C3#54611yescharacteristic
225C2×C3C2×#13511yescharacteristic
226C2×C3#57811yescharacteristic
227D3—42no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—62no—
232C22—121no—
233C22—121no—
234C22C22×D1211yescharacteristic
235C4—21no—
236C4—22no—
237C4—121no—
238C4—122no—
239C4—121no—
240C4—122no—
241C4—121no—
242C4#90412yes—
243C4#100411yescharacteristic
244C4D3×#4111yescharacteristic
245C3#330211yescharacteristic, complemented, Sylow
246C2—42no—
247C2—122no—
248C2#555411yescharacteristic
249C2C2×#90411yescharacteristic
250C2#575311yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C8×D4C2
socleC22×C3C22×D4
prime radical#5551C2
hypercenterC8×D4D3
hypocenterC3#3302
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
2. #5542C2semidirect
3. C3×C8×D4C2semidirect
4. #5116C2semidirect
5. #5113C2semidirect
6. #4849C2semidirect
7. #4846C2semidirect
8. #4733C2semidirect
9. #4723C2semidirect
10. C4×#204C2semidirect
11. #4681C2semidirect
51. C8×D4D3semidirect
245. C3#3302semidirect

Maximal subgroups

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

subgroupindex
#55512
#55422
C3×C8×D42
#51162
#51132
#48492
#48462
#47332
#47232
C4×#2042
#46812
#33023