Finite Group Databasebeta

← D4×#216 #83567

Subgroups — D4×#216 #83567

Order 384.

472 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C3×D4×Q8C211yesmaximal, characteristic
2C22×#216C212yesmaximal, complemented
3D4×Dic6C211yesmaximal, characteristic
4#5228C212yesmaximal, complemented
5#5092C211yesmaximal, characteristic, complemented
6#5090C211yesmaximal, characteristic, complemented
7#5051C212yesmaximal, complemented
8#5050C212yesmaximal, complemented
9#5031C211yesmaximal, characteristic, complemented
10C4×#216C211yesmaximal, characteristic, complemented
11D4×C3⋊C8C211yesmaximal, characteristic
12D4×Q16—31nomaximal, Sylow
13C22×C3×Q8C2212yes—
14C22×Dic6C2212yes—
15C3×C4⋊Q8—21no—
16C3×C4⋊Q8C2211yescharacteristic
17C3×C22⋊Q8—22no—
18C3×C22⋊Q8C2212yes—
19C3×C4×Q8C2211yescharacteristic
20C3×C4×D4—21no—
21C3×C4×D4C2211yescharacteristic
22C2×#216—28no—
23C2×#216C2212yes—
24C2×#216C2212yes—
25C2×#216C2212yes—
26C2×#216C2212yes—
27C2×#216C2211yescharacteristic
28D4×Dic3—21no—
29#924C2212yes—
30C22×C3⋊C8C2212yes—
31#888—21no—
32#878—22no—
33#869C2211yescharacteristic
34C4×Dic6C2211yescharacteristic
35#835C2211yescharacteristic
36#835C2212yes—
37#830C2212yes—
38#810C2211yescharacteristic
39#810C2212yes—
40#807C2211yescharacteristic
41C4⋊(C3⋊C8)C2211yescharacteristic
42C4×C3⋊C8C2211yescharacteristic
43C22×Q16—32no—
44D4×Q8—31no—
45D4×Q8D311yescharacteristic
46#494—31no—
47#467—32no—
48#462—31no—
49#462—31no—
50#451—32no—
51#451—32no—
52C4×Q16—31no—
53C8×D4—31no—
54C2×C3×Q8—22no—
55C2×C3×Q8—24no—
56C2×C3×Q8—42no—
57C2×C3×Q8C2312yes—
58C2×C3×Q8C2311yescharacteristic
59C2×C3×D4D411yescharacteristic
60C22×C3×C4—22no—
61C22×C3×C4C2312yes—
62C22×Dic3—22no—
63C2×Dic6—24no—
64C2×Dic6—22no—
65C2×Dic6—42no—
66C2×Dic6C2312yes—
67C2×Dic6C2311yescharacteristic
68C3×C4⋊C4—21no—
69C3×C4⋊C4—21no—
70C3×C4⋊C4—21no—
71C3×C4⋊C4—22no—
72C3×C4⋊C4D411yescharacteristic
73C3×C4⋊C4C2311yescharacteristic
74C3×C22⋊C4—22no—
75C3×C22⋊C4D412yes—
76C3×C42—21no—
77C3×C42C2311yescharacteristic
78#217—22no—
79#216—24no—
80#216—24no—
81#216—24no—
82#216—42no—
83#216D414yesdirect product factor
84C4⋊Dic3—21no—
85C4⋊Dic3—21no—
86C4⋊Dic3C2311yescharacteristic
87Dic3⋊C4—21no—
88Dic3⋊C4—22no—
89C4×Dic3—21no—
90C2×C3⋊C8—24no—
91C2×C3⋊C8C2311yescharacteristic
92C2×C3⋊C8C2312yes—
93C2×C3⋊C8C2311yescharacteristic
94C22×Q8—32no—
95C22×Q8C2×D312yes—
96C2×Q16—31no—
97C2×Q16—32no—
98C2×Q16—32no—
99C2×Q16—32no—
100C2×Q16—32no—
101C2×Q16—68no—
102C22×C8—32no—
103C4⋊Q8—21no—
104C4⋊Q8—31no—
105C4⋊Q8—61no—
106C4⋊Q8C2×D311yescharacteristic
107C22⋊Q8—22no—
108C22⋊Q8—32no—
109C22⋊Q8—62no—
110C22⋊Q8C2×D312yes—
111C4×Q8—31no—
112C4×Q8C2×D311yescharacteristic
113C4×D4—21no—
114C4×D4—61no—
115C4×D4C2×D311yescharacteristic
116#107—31no—
117C4⋊C8—31no—
118#103—31no—
119#103—32no—
120#103—31no—
121#103—32no—
122C22⋊C8—32no—
123C4×C8—31no—
124C23×C3C2×D412yes—
125C3×Q8—21no—
126C3×Q8—22no—
127C3×Q8—22no—
128C3×Q8—44no—
129C3×Q8C2×D412yes—
130C3×D4Q1614yescomplemented
131C2×C3×C4—21no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—21no—
135C2×C3×C4—22no—
136C2×C3×C4—44no—
137C2×C3×C4C2×D411yescharacteristic
138C2×C3×C4C2×D412yes—
139C2×C3×C4C2×D411yescharacteristic
140C2×C3×C4C2411yescharacteristic
141C2×Dic3—21no—
142C2×Dic3—21no—
143C2×Dic3—22no—
144C2×Dic3—44no—
145Dic6—21no—
146Dic6—22no—
147Dic6—22no—
148Dic6—44no—
149Dic6C2×D412yes—
150C3⋊C8—22no—
151C3⋊C8—21no—
152C3⋊C8C2×D412yes—
153C2×Q8—22no—
154C2×Q8—24no—
155C2×Q8—31no—
156C2×Q8—32no—
157C2×Q8—42no—
158C2×Q8—64no—
159C2×Q8—62no—
160C2×Q8—122no—
161C2×Q8C22×D312yes—
162C2×Q8C22×D311yescharacteristic
163C2×D4#6711yescharacteristic
164C22×C4—22no—
165C22×C4—62no—
166C22×C4C22×D312yes—
167Q16—34no—
168Q16—64no—
169Q16—64no—
170Q16—64no—
171Q16—122no—
172C2×C8—31no—
173C2×C8—31no—
174C2×C8—32no—
175C2×C8—64no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—22no—
180C4⋊C4—31no—
181C4⋊C4—61no—
182C4⋊C4—61no—
183C4⋊C4—62no—
184C4⋊C4—61no—
185C4⋊C4#6711yescharacteristic
186C4⋊C4C22×D311yescharacteristic
187C22⋊C4—22no—
188C22⋊C4—62no—
189C22⋊C4#6712yes—
190C42—21no—
191C42—61no—
192C42C22×D311yescharacteristic
193C22×C3—22no—
194C22×C3—22no—
195C22×C3C2×Q1614yescomplemented
196C22×C3C22×D411yescharacteristic
197C3×C4—22no—
198C3×C4—22no—
199C3×C4—21no—
200C3×C4—42no—
201C3×C4—41no—
202C3×C4C2×Q1612yescomplemented
203C3×C4C22×D411yescharacteristic
204C3×C4H(5,𝔽2)11yescharacteristic
205Dic3—22no—
206Dic3—42no—
207Dic3—41no—
208C23C2×#6712yes—
209Q8—21no—
210Q8—22no—
211Q8—22no—
212Q8—32no—
213Q8—44no—
214Q8—61no—
215Q8—62no—
216Q8—62no—
217Q8—124no—
218Q8D3×D412yes—
219D4#21614yesdirect product factor
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—44no—
226C2×C4—61no—
227C2×C4—61no—
228C2×C4—62no—
229C2×C4—124no—
230C2×C4C2×#6711yescharacteristic
231C2×C4C2×#6712yes—
232C2×C4C2×#6711yescharacteristic
233C2×C4C23×D311yescharacteristic
234C8—32no—
235C8—62no—
236C8—61no—
237C2×C3—24no—
238C2×C3D4211yescharacteristic
239C2×C3C22×Q1611yescharacteristic
240C2×C3#57811yescharacteristic
241C22—22no—
242C22—22no—
243C22C2×#21614yescomplemented
244C22C22×#6711yescharacteristic
245C4—22no—
246C4—22no—
247C4—21no—
248C4—42no—
249C4—41no—
250C4—62no—
251C4—122no—
252C4—121no—
253C4C2×#21612yescomplemented
254C4C2×D3×D411yescharacteristic
255C4#100411yescharacteristic
256C3D4×Q1611yescharacteristic, complemented, Sylow
257C2—24no—
258C2D4×#6711yescharacteristic
259C2C22×#21611yescharacteristic
260C2#583811yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×D4×Q8C2
socleC22×C3C22×D4
prime radicalC2×C3×D4D4
hypercenterD4×Q8D3
hypocenterC3D4×Q16
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
2. C22×#216C2semidirect
4. #5228C2semidirect
5. #5092C2semidirect
6. #5090C2semidirect
7. #5051C2semidirect
8. #5050C2semidirect
9. #5031C2semidirect
10. C4×#216C2semidirect
83. #216D4direct (N × H)
130. C3×D4Q16semidirect
195. C22×C3C2×Q16semidirect
202. C3×C4C2×Q16semidirect
219. D4#216direct (N × H)
243. C22C2×#216semidirect
253. C4C2×#216semidirect
256. C3D4×Q16semidirect

Maximal subgroups

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

subgroupindex
C3×D4×Q82
C22×#2162
D4×Dic62
#52282
#50922
#50902
#50512
#50502
#50312
C4×#2162
D4×C3⋊C82
D4×Q163