Finite Group Databasebeta

← #73762

Subgroups — #73762

Order 384.

556 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#897C211yesmaximal, characteristic
2C2×#872C211yesmaximal, characteristic
3C2×C4×Dic6C211yesmaximal, characteristic, complemented
4C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
5#4987C211yesmaximal, characteristic
6#4984C211yesmaximal, characteristic, complemented
7#4973C211yesmaximal, characteristic, complemented
8#4940C212yesmaximal
9#4928C212yesmaximal, complemented
10#4670C212yesmaximal
11#4653C212yesmaximal, complemented
12#2323—31nomaximal, Sylow
13C22×C4×D3C2211yescharacteristic
14C22×Dic6C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C42C2212yes—
18C2×C3×C42C2211yescharacteristic
19C2×(C2×D3)⋊C4C2211yescharacteristic
20C2×(C2×D3)⋊C4C2211yescharacteristic
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×C4⋊Dic3C2211yescharacteristic
24C2×C4⋊Dic3C2212yes—
25C2×C4⋊Dic3C2212yes—
26C2×Dic3⋊C4C2211yescharacteristic
27C2×Dic3⋊C4C2211yescharacteristic
28C2×Dic3⋊C4C2212yes—
29C2×C4×Dic3C2212yes—
30C2×C4×Dic3C2211yescharacteristic
31#897C418yescomplemented
32#872—24no—
33C4×Dic6—24no—
34C3×#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C2×C22⋊Q8—31no—
40C2×C4×Q8—31no—
41C2×#117—31no—
42#391—31no—
43#389—31no—
44#389—32no—
45#388—32no—
46#388—32no—
47#387—32no—
48#380—31no—
49C4×C4⋊C4D311yescharacteristic, complemented
50C23×D3C2311yescharacteristic
51C22×C3×C4C2312yes—
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2312yes—
60C22×Dic3C2311yescharacteristic
61C2×C4×D3—22no—
62C2×C4×D3—22no—
63C2×C4×D3C2×C414yes—
64C2×Dic6—22no—
65C2×Dic6—22no—
66C2×Dic6C2×C414yes—
67C3×C4⋊C4—22no—
68C3×C4⋊C4C2×C414yes—
69C3×C42—42no—
70C3×C42D414yes—
71(C2×D3)⋊C4—22no—
72(C2×D3)⋊C4—22no—
73(C2×D3)⋊C4—42no—
74(C2×D3)⋊C4C2×C418yes—
75C4⋊Dic3—22no—
76C4⋊Dic3—42no—
77C4⋊Dic3C2×C418yes—
78Dic3⋊C4—24no—
79Dic3⋊C4—22no—
80Dic3⋊C4—22no—
81C4×Dic3—24no—
82C4×Dic3—22no—
83C22×Q8—31no—
84C23×C4—31no—
85C22⋊Q8—38no—
86C4×Q8—64no—
87#117—64no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4—31no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—32no—
94C2×C4⋊C4C2×D311yescharacteristic
95C2×C4⋊C4C2×D311yescharacteristic
96C2×C22⋊C4—32no—
97C2×C22⋊C4—31no—
98C2×C22⋊C4—31no—
99C2×C22⋊C4—32no—
100C2×C42—31no—
101C2×C42—32no—
102C2×C42C2×D312yes—
103C2×C42C2×D311yescharacteristic
104#95—32no—
105#95—32no—
106#95—32no—
107#95—32no—
108#95C2×D312yes—
109C23×C3C2411yescharacteristic
110C22×D3—21no—
111C22×D3—21no—
112C22×D3—21no—
113C22×D3—21no—
114C22×D3—21no—
115C22×D3—21no—
116C22×D3C22×C412yes—
117C2×C3×C4—22no—
118C2×C3×C4—21no—
119C2×C3×C4—21no—
120C2×C3×C4—22no—
121C2×C3×C4—21no—
122C2×C3×C4—21no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4C22×C414yes—
127C2×C3×C4C22×C412yes—
128C2×C3×C4C2×D412yes—
129C2×C3×C4C2×D412yes—
130C2×C3×C4C2×D412yes—
131C2×C3×C4#4112yes—
132C2×Dic3—22no—
133C2×Dic3—21no—
134C2×Dic3—22no—
135C2×Dic3—21no—
136C2×Dic3—22no—
137C2×Dic3—21no—
138C2×Dic3—21no—
139C2×Dic3—21no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—21no—
143C2×Dic3—21no—
144C2×Dic3—21no—
145C2×Dic3C22×C412yes—
146C2×Dic3C22×C414yes—
147C4×D3—28no—
148Dic6—28no—
149C24—31no—
150C2×Q8—34no—
151C2×Q8—62no—
152C2×Q8—62no—
153C22×C4—32no—
154C22×C4—31no—
155C22×C4—34no—
156C22×C4—31no—
157C22×C4—31no—
158C22×C4—32no—
159C22×C4—62no—
160C22×C4—62no—
161C22×C4C22×D312yes—
162C22×C4C22×D312yes—
163C22×C4C22×D311yescharacteristic
164C22×C4C22×D311yescharacteristic
165C22×C4C22×D311yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—38no—
168C4⋊C4—62no—
169C4⋊C4—62no—
170C4⋊C4—64no—
171C4⋊C4—62no—
172C4⋊C4—122no—
173C4⋊C4C4×D314yes—
174C22⋊C4—38no—
175C22⋊C4—62no—
176C22⋊C4—62no—
177C22⋊C4—122no—
178C42—42no—
179C42—62no—
180C42—64no—
181C42D1214yes—
182C22×C3C23×C411yescharacteristic
183C22×C3C22×D411yescharacteristic
184C22×C3C2×#4111yescharacteristic
185C22×C3H(5,𝔽2)11yescharacteristic
186C22×C32−1+411yescharacteristic
187C22×C32−1+411yescharacteristic
188C22×C32−1+411yescharacteristic
189C2×D3—22no—
190C2×D3—22no—
191C2×D3—22no—
192C2×D3—41no—
193C2×D3—41no—
194C2×D3—41no—
195C2×D3—41no—
196C3×C4—24no—
197C3×C4—22no—
198C3×C4—22no—
199C3×C4—42no—
200C3×C4C4×D414yes—
201Dic3—24no—
202Dic3—22no—
203Dic3—41no—
204Dic3—42no—
205Dic3—41no—
206C23—32no—
207C23—61no—
208C23—61no—
209C23—61no—
210C23—61no—
211C23—61no—
212C23—61no—
213C23C23×D311yescharacteristic
214Q8—68no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—34no—
225C2×C4—32no—
226C2×C4—62no—
227C2×C4—61no—
228C2×C4—61no—
229C2×C4—62no—
230C2×C4—68no—
231C2×C4—61no—
232C2×C4—61no—
233C2×C4—62no—
234C2×C4—62no—
235C2×C4—61no—
236C2×C4—61no—
237C2×C4—62no—
238C2×C4—61no—
239C2×C4—61no—
240C2×C4C2×C4×D314yes—
241C2×C4C2×C4×D312yes—
242C2×C4C2×D1212yes—
243C2×C4C2×D1212yes—
244C2×C4C2×D1212yes—
245C2×C4#23512yes—
246C2×C3C2×C4×D411yescharacteristic
247C2×C3#51911yescharacteristic
248C2×C3#52011yescharacteristic
249C2×C3#53611yescharacteristic
250C2×C3#53711yescharacteristic
251C2×C3#54111yescharacteristic
252C2×C3#54211yescharacteristic
253D3—42no—
254C22—62no—
255C22—62no—
256C22—62no—
257C22—121no—
258C22—121no—
259C22—121no—
260C22—121no—
261C22C22×C4×D311yescharacteristic
262C22C22×D1211yescharacteristic
263C22C2×#23511yescharacteristic
264C22#100411yescharacteristic
265C22#100711yescharacteristic
266C22#101011yescharacteristic
267C22#101011yescharacteristic
268C4—22no—
269C4—22no—
270C4—24no—
271C4—42no—
272C4—62no—
273C4—64no—
274C4—121no—
275C4—122no—
276C4—121no—
277C4C4×D1214yes—
278C3#232311yescharacteristic, complemented, Sylow
279C2—122no—
280C2C2×C4×D1211yescharacteristic
281C2#551011yescharacteristic
282C2#552611yescharacteristic
283C2#554611yescharacteristic
284C2#556111yescharacteristic
285C2#557211yescharacteristic
286C2#558311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×C4×C4⋊C4C2
socle, cosocleC23×C3C24
prime radicalC23×D3C23
hypercenterC4×C4⋊C4D3
hypocenterC3#2323
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×C4×Dic6C2semidirect
4. C3×C4×C4⋊C4C2semidirect
6. #4984C2semidirect
7. #4973C2semidirect
9. #4928C2semidirect
11. #4653C2semidirect
31. #897C4semidirect
49. C4×C4⋊C4D3semidirect
278. C3#2323semidirect

Maximal subgroups

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

subgroupindex
C2×#8972
C2×#8722
C2×C4×Dic62
C3×C4×C4⋊C42
#49872
#49842
#49732
#49402
#49282
#46702
#46532
#23233