Finite Group Databasebeta

← #74026

Subgroups — #74026

Order 384.

540 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#67C211yesmaximal, characteristic
2C2×#926C211yesmaximal, characteristic, complemented
3C3×#380C211yesmaximal, characteristic, complemented
4#4987C211yesmaximal, characteristic, complemented
5#4987C211yesmaximal, characteristic, complemented
6#4984C211yesmaximal, characteristic, complemented
7#4984C211yesmaximal, characteristic, complemented
8#4961C212yesmaximal
9#4953C212yesmaximal, complemented
10#4670C212yesmaximal, complemented
11#4653C212yesmaximal, complemented
12#2485—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C22×#67C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2212yes—
19C2×#217C2211yescharacteristic
20C2×#217C2212yes—
21C4×#67—24no—
22C2×(C2×D3)⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25#926—24no—
26C2×C4⋊Dic3C2212yes—
27C2×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×C4⋊Dic3C2212yes—
30C2×Dic3⋊C4C2211yescharacteristic
31C2×C4×Dic3C2212yes—
32C2×C4×Dic3C2211yescharacteristic
33C3×#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C4×D4—31no—
39C2×#117—31no—
40#397—32no—
41#394—32no—
42#389—31no—
43#389—31no—
44#388—32no—
45#387—32no—
46#380—31no—
47#380—31no—
48#380D311yescharacteristic, complemented
49C24×C3C2311yescharacteristic
50C23×D3C2311yescharacteristic
51C22×C3×C4—22no—
52C22×C3×C4—22no—
53C22×C3×C4D414yes—
54C22×C3×C4C2312yes—
55C22×C3×C4C2312yes—
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C2×#67—22no—
59C2×#67—22no—
60C2×#67—22no—
61C22×Dic3C2311yescharacteristic
62C22×Dic3C2312yes—
63C22×Dic3C2312yes—
64C22×Dic3C2311yescharacteristic
65C22×Dic3C2311yescharacteristic
66C2×C4×D3—22no—
67C2×C4×D3—22no—
68C2×C4×D3—22no—
69C3×C4⋊C4—22no—
70C3×C4⋊C4—22no—
71C3×C22⋊C4—42no—
72#217—24no—
73#217—22no—
74(C2×D3)⋊C4—24no—
75(C2×D3)⋊C4—22no—
76(C2×D3)⋊C4—42no—
77C4⋊Dic3—22no—
78C4⋊Dic3—22no—
79C4⋊Dic3—24no—
80C4⋊Dic3—42no—
81Dic3⋊C4—22no—
82C4×Dic3—24no—
83C4×Dic3—22no—
84C22×D4—31no—
85C23×C4—31no—
86C23×C4C2×D311yescharacteristic
87C4×D4—64no—
88#117—64no—
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×C22⋊C4—32no—
101C2×C22⋊C4C2×D312yes—
102C2×C42—32no—
103C2×C42—31no—
104#95—32no—
105#95—32no—
106#95—32no—
107#95—32no—
108#95C2×D312yes—
109C23×C3—21no—
110C23×C3—21no—
111C23×C3—21no—
112C23×C3—21no—
113C23×C3—21no—
114C23×C3—21no—
115C23×C3C2×D412yes—
116C23×C3C2411yescharacteristic
117C22×D3—21no—
118C22×D3—21no—
119C22×D3—21no—
120C22×D3—21no—
121C22×D3—21no—
122C22×D3—21no—
123C22×D3#4112yes—
124C2×C3×C4—21no—
125C2×C3×C4—21no—
126C2×C3×C4—28no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4C2×D412yes—
134C2×C3×C4C2×D412yes—
135C2×C3×C4#4112yes—
136C2×C3×C4#4112yes—
137#67—28no—
138C2×Dic3—21no—
139C2×Dic3—21no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—21no—
143C2×Dic3—21no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—21no—
147C2×Dic3—21no—
148C2×Dic3—21no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—21no—
152C2×Dic3#4112yes—
153C4×D3—44no—
154C24—31no—
155C24C22×D311yescharacteristic
156C2×D4—62no—
157C2×D4—62no—
158C2×D4—62no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—31no—
162C22×C4—31no—
163C22×C4—31no—
164C22×C4—32no—
165C22×C4—32no—
166C22×C4—62no—
167C22×C4—62no—
168C22×C4—62no—
169C22×C4D1214yes—
170C22×C4C22×D312yes—
171C22×C4C22×D312yes—
172C22×C4C22×D311yescharacteristic
173C22×C4C22×D311yescharacteristic
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—64no—
177C4⋊C4—62no—
178C4⋊C4—62no—
179C4⋊C4—62no—
180C4⋊C4—122no—
181C22⋊C4—42no—
182C22⋊C4—64no—
183C22⋊C4—64no—
184C22⋊C4—62no—
185C22⋊C4—62no—
186C22⋊C4—122no—
187C42—64no—
188C42—62no—
189C22×C3—22no—
190C22×C3—22no—
191C22×C3—22no—
192C22×C3—41no—
193C22×C3—41no—
194C22×C3—41no—
195C22×C3—41no—
196C22×C3C22×D411yescharacteristic
197C22×C3C2×#4111yescharacteristic
198C22×C3C2×#4111yescharacteristic
199C22×C3C2×#4111yescharacteristic
200C22×C3C2×#4111yescharacteristic
201C22×C3H(5,𝔽2)11yescharacteristic
202C22×C32−1+411yescharacteristic
203C2×D3—22no—
204C2×D3—22no—
205C2×D3—22no—
206C2×D3—41no—
207C2×D3—41no—
208C2×D3—41no—
209C2×D3—41no—
210C3×C4—22no—
211C3×C4—42no—
212C3×C4—42no—
213C3×C4#12314yes—
214Dic3—22no—
215Dic3—41no—
216Dic3—42no—
217Dic3—42no—
218Dic3—41no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—32no—
226C23—61no—
227C23—61no—
228C23—61no—
229C23—61no—
230C23—61no—
231C23—61no—
232C23C2×D1212yes—
233C23C23×D311yescharacteristic
234D4—68no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—28no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—32no—
245C2×C4—61no—
246C2×C4—61no—
247C2×C4—61no—
248C2×C4—62no—
249C2×C4—62no—
250C2×C4—61no—
251C2×C4—61no—
252C2×C4—61no—
253C2×C4—62no—
254C2×C4—61no—
255C2×C4—61no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—62no—
259C2×C4—124no—
260C2×C4C2×D1212yes—
261C2×C4C2×D1212yes—
262C2×C4#23712yes—
263C2×C4#23712yes—
264C2×C3—42no—
265C2×C3C2×#12311yescharacteristic
266C2×C3#52511yescharacteristic
267C2×C3#53711yescharacteristic
268C2×C3#55211yescharacteristic
269C2×C3#55311yescharacteristic
270C2×C3#55511yescharacteristic
271C2×C3#55611yescharacteristic
272D3—42no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—62no—
281C22—62no—
282C22—62no—
283C22—121no—
284C22—121no—
285C22—121no—
286C22—121no—
287C22C22×D1211yescharacteristic
288C22C2×#23711yescharacteristic
289C22C2×#23711yescharacteristic
290C22#100411yescharacteristic
291C22#100711yescharacteristic
292C22D3×#4111yescharacteristic
293C22D3×#4111yescharacteristic
294C4—22no—
295C4—42no—
296C4—42no—
297C4—62no—
298C4—121no—
299C4—121no—
300C4—122no—
301C4—122no—
302C4#88614yes—
303C3#248511yescharacteristic, complemented, Sylow
304C2—42no—
305C2—122no—
306C2C2×#88611yescharacteristic
307C2#551011yescharacteristic
308C2#552411yescharacteristic
309C2#561411yescharacteristic
310C2#561611yescharacteristic
311C2#563911yescharacteristic
312C2#564111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#380C2
prime radicalC22×#67C22
hypercenter#380D3
hypocenterC3#2485
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#926C2semidirect
3. C3×#380C2semidirect
4. #4987C2semidirect
5. #4987C2semidirect
6. #4984C2semidirect
7. #4984C2semidirect
9. #4953C2semidirect
10. #4670C2semidirect
11. #4653C2semidirect
48. #380D3semidirect
303. C3#2485semidirect

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
C3×#3802
#49872
#49842
#49612
#49532
#46702
#46532
#24853