Finite Group Databasebeta

← #74012

Subgroups — #74012

Order 384.

660 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#67C211yesmaximal, characteristic
2C2×#926C211yesmaximal, characteristic, complemented
3C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×#380C211yesmaximal, characteristic, complemented
5#4987C211yesmaximal, characteristic, complemented
6#4977C211yesmaximal, characteristic, complemented
7Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
8#4956C212yesmaximal
9#4944C212yesmaximal, complemented
10#4666C212yesmaximal, complemented
11#4652C212yesmaximal, complemented
12#2352—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C22×#67C2211yescharacteristic
15C22×C4×D3C2212yes—
16C22×C4×D3C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×#217C2212yes—
21C2×#217C2211yescharacteristic
22C4×#67C418yescomplemented
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2211yescharacteristic
25#926—24no—
26C2×C4⋊Dic3C2211yescharacteristic
27C2×C4⋊Dic3C2211yescharacteristic
28C2×Dic3⋊C4C2212yes—
29C2×Dic3⋊C4C2211yescharacteristic
30C2×C4×Dic3C2212yes—
31C2×C4×Dic3C2212yes—
32C2×C4×Dic3C2211yescharacteristic
33D3×C4⋊C4—24no—
34C3×#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C2×C4×D4—31no—
40C2×#117—31no—
41C22×C4⋊C4—31no—
42#388—32no—
43#387—32no—
44#386—32no—
45#382—31no—
46#380—31no—
47#380D311yescharacteristic, complemented
48C4×C4⋊C4—31no—
49C4×C22⋊C4—32no—
50C24×C3C2311yescharacteristic
51C23×D3C2311yescharacteristic
52C22×C3×C4—22no—
53C22×C3×C4—22no—
54C22×C3×C4C2×C414yescomplemented
55C22×C3×C4C2312yes—
56C22×C3×C4C2312yes—
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C2×#67—22no—
60C2×#67—22no—
61C2×#67C2×C414yes—
62C22×Dic3C2312yes—
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2311yescharacteristic
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2312yes—
67C2×C4×D3—22no—
68C2×C4×D3—22no—
69C2×C4×D3—24no—
70C2×C4×D3—24no—
71C2×C4×D3—42no—
72C2×C4×D3C2×C414yes—
73C3×C4⋊C4—22no—
74C3×C4⋊C4—22no—
75C3×C22⋊C4—42no—
76#217—24no—
77#217C2×C414yescomplemented
78(C2×D3)⋊C4—24no—
79(C2×D3)⋊C4C2×C414yes—
80C4⋊Dic3—22no—
81C4⋊Dic3D414yescomplemented
82Dic3⋊C4—24no—
83Dic3⋊C4C2×C414yescomplemented
84C4×Dic3—24no—
85C4×Dic3—42no—
86C4×Dic3C2×C414yescomplemented
87C22×D4—31no—
88C23×C4—31no—
89C23×C4—32no—
90C23×C4C2×D311yescharacteristic
91C4×D4—38no—
92#117—64no—
93C2×C4⋊C4—31no—
94C2×C4⋊C4—31no—
95C2×C4⋊C4—32no—
96C2×C4⋊C4—31no—
97C2×C4⋊C4—64no—
98C2×C4⋊C4C2×D311yescharacteristic
99C2×C4⋊C4C2×D311yescharacteristic
100C2×C22⋊C4—31no—
101C2×C22⋊C4—31no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4C2×D312yes—
105C2×C42—31no—
106C2×C42—32no—
107C2×C42—32no—
108#95—32no—
109#95—32no—
110#95—32no—
111#95—32no—
112#95C2×D312yes—
113C23×C3—21no—
114C23×C3—21no—
115C23×C3—21no—
116C23×C3—21no—
117C23×C3—21no—
118C23×C3—21no—
119C23×C3C22×C412yes—
120C23×C3C2411yescharacteristic
121C22×D3—21no—
122C22×D3—21no—
123C22×D3—21no—
124C22×D3—21no—
125C22×D3C22×C412yes—
126C22×D3#4112yes—
127C22×D3#4112yes—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—28no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4C22×C412yes—
138C2×C3×C4C22×C412yes—
139C2×C3×C4C2×D412yes—
140C2×C3×C4#4112yes—
141#67—28no—
142C2×Dic3—21no—
143C2×Dic3—21no—
144C2×Dic3—22no—
145C2×Dic3—21no—
146C2×Dic3—21no—
147C2×Dic3—21no—
148C2×Dic3—22no—
149C2×Dic3—21no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3C22×C412yes—
154C2×Dic3C22×C412yes—
155C2×Dic3C22×C412yes—
156C2×Dic3C2×D414yes—
157C4×D3—28no—
158C4×D3—48no—
159C24—31no—
160C24C22×D311yescharacteristic
161C2×D4—34no—
162C2×D4—62no—
163C2×D4—62no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—32no—
170C22×C4—34no—
171C22×C4—32no—
172C22×C4—64no—
173C22×C4—64no—
174C22×C4—62no—
175C22×C4—62no—
176C22×C4—122no—
177C22×C4C4×D314yescomplemented
178C22×C4C22×D312yes—
179C22×C4C22×D312yes—
180C22×C4C22×D311yescharacteristic
181C22×C4C22×D311yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—34no—
185C4⋊C4—34no—
186C4⋊C4—64no—
187C4⋊C4—62no—
188C22⋊C4—34no—
189C22⋊C4—34no—
190C22⋊C4—42no—
191C22⋊C4—64no—
192C22⋊C4—64no—
193C42—34no—
194C42—64no—
195C42—122no—
196C22×C3—22no—
197C22×C3—22no—
198C22×C3—22no—
199C22×C3—41no—
200C22×C3—41no—
201C22×C3—41no—
202C22×C3—41no—
203C22×C3C23×C411yescharacteristic
204C22×C3C22×D411yescharacteristic
205C22×C3C2×#4111yescharacteristic
206C22×C3C2×#4111yescharacteristic
207C22×C3C2×#4111yescharacteristic
208C22×C3H(5,𝔽2)11yescharacteristic
209C22×C32−1+411yescharacteristic
210C2×D3—22no—
211C2×D3—22no—
212C2×D3—22no—
213C2×D3—22no—
214C2×D3—22no—
215C2×D3—22no—
216C2×D3#11714yescomplemented
217C3×C4—22no—
218C3×C4—42no—
219C3×C4—42no—
220C3×C4C4×D414yes—
221Dic3—22no—
222Dic3—22no—
223Dic3—22no—
224Dic3—24no—
225Dic3—42no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—32no—
233C23—32no—
234C23—32no—
235C23—61no—
236C23—61no—
237C23—61no—
238C23—61no—
239C23C2×C4×D312yes—
240C23C23×D311yescharacteristic
241D4—68no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—28no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—32no—
252C2×C4—32no—
253C2×C4—32no—
254C2×C4—34no—
255C2×C4—61no—
256C2×C4—61no—
257C2×C4—61no—
258C2×C4—62no—
259C2×C4—61no—
260C2×C4—62no—
261C2×C4—61no—
262C2×C4—61no—
263C2×C4—68no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4—128no—
268C2×C4C2×C4×D312yes—
269C2×C4C2×C4×D312yes—
270C2×C4D3×D412yes—
271C2×C4#23712yes—
272C2×C3—42no—
273C2×C3C2×#11711yescharacteristic
274C2×C3C2×C4×D411yescharacteristic
275C2×C3#52011yescharacteristic
276C2×C3#54611yescharacteristic
277C2×C3#54811yescharacteristic
278C2×C3#55211yescharacteristic
279C2×C3#55311yescharacteristic
280D3—24no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—34no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—62no—
290C22—62no—
291C22—62no—
292C22—62no—
293C22—62no—
294C22—62no—
295C22C22×C4×D311yescharacteristic
296C22C2×D3×D411yescharacteristic
297C22C2×#23711yescharacteristic
298C22#100411yescharacteristic
299C22#100711yescharacteristic
300C22D3×#4111yescharacteristic
301C22D3×#4111yescharacteristic
302C4—22no—
303C4—42no—
304C4—42no—
305C4—62no—
306C4—62no—
307C4—62no—
308C4—64no—
309C4—122no—
310C4#67⋊C414yes—
311C3#235211yescharacteristic, complemented, Sylow
312C2—42no—
313C2—64no—
314C2C2×#67⋊C411yescharacteristic
315C2#550411yescharacteristic
316C2D3×#11711yescharacteristic
317C2#561211yescharacteristic
318C2#561411yescharacteristic
319C2#563311yescharacteristic
320C2#563911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#380C2
socle, cosocleC23×C3C24
prime radicalC22×#67C22
hypercenter#380D3
hypocenterC3#2352
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#926C2semidirect
3. C2×D3×C4⋊C4C2semidirect
4. C3×#380C2semidirect
5. #4987C2semidirect
6. #4977C2semidirect
7. Dic3×C4⋊C4C2semidirect
9. #4944C2semidirect
10. #4666C2semidirect
11. #4652C2semidirect
22. C4×#67C4semidirect
47. #380D3semidirect
54. C22×C3×C4C2×C4semidirect
77. #217C2×C4semidirect
81. C4⋊Dic3D4semidirect
83. Dic3⋊C4C2×C4semidirect
86. C4×Dic3C2×C4semidirect
177. C22×C4C4×D3semidirect
216. C2×D3#117semidirect
311. C3#2352semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#672
C2×#9262
C2×D3×C4⋊C42
C3×#3802
#49872
#49772
Dic3×C4⋊C42
#49562
#49442
#46662
#46522
#23523