Finite Group Databasebeta

← #82859

Subgroups — #82859

Order 384.

708 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#123C211yesmaximal, characteristic, complemented
2C2×#939C211yesmaximal, characteristic
3C2×#935C211yesmaximal, characteristic, complemented
4C2×#929C211yesmaximal, characteristic, complemented
5C22×Dic3⋊C4C211yesmaximal, characteristic, complemented
6C2×#886C211yesmaximal, characteristic, complemented
7#4993C212yesmaximal, complemented
8#4975C212yesmaximal, complemented
9#4957C212yesmaximal, complemented
10#4949C212yesmaximal, complemented
11Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
12#2501—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C22×#67C2211yescharacteristic
16C22×#67C2211yescharacteristic
17C23×Dic3C2211yescharacteristic
18C23×Dic3C2211yescharacteristic
19C3×#123—24no—
20C2×C3×C4⋊C4C2212yes—
21C2×C3×C22⋊C4C2212yes—
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×#217C2211yescharacteristic
24C2×#217C2212yes—
25C2×#217C2211yescharacteristic
26#939—24no—
27#935—24no—
28#929—24no—
29C2×(C2×D3)⋊C4C2212yes—
30C2×(C2×D3)⋊C4C2212yes—
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×C4⋊Dic3C2212yes—
33C2×Dic3⋊C4—24no—
34C2×Dic3⋊C4C2212yes—
35C2×Dic3⋊C4C2212yes—
36C2×Dic3⋊C4C2212yes—
37C2×Dic3⋊C4C2211yescharacteristic
38C2×Dic3⋊C4C2212yes—
39C2×C4×Dic3C2212yes—
40#886—24no—
41C3⋊#95C2212yes—
42C2×#123—31no—
43C2×#123—31no—
44C2×#123—31no—
45C2×#123D311yescharacteristic, complemented
46C2×#121—31no—
47C22×C4⋊C4—31no—
48#399—32no—
49#399—32no—
50#390—32no—
51#384—32no—
52C4×C22⋊C4—31no—
53C24×C3C2311yescharacteristic
54C24×C3C2311yescharacteristic
55C23×D3C2311yescharacteristic
56C2×C3×D4—22no—
57C2×C3×D4—22no—
58C2×C3×D4—22no—
59C22×C3×C4—22no—
60C22×C3×C4—22no—
61C22×C3×C4—22no—
62C22×C3×C4C2312yes—
63C22×C3×C4C2311yescharacteristic
64C22×C3×C4C2312yes—
65C2×#67—22no—
66C2×#67—22no—
67C2×#67—22no—
68C2×#67—22no—
69C2×#67—24no—
70C22×Dic3—24no—
71C22×Dic3—22no—
72C22×Dic3—22no—
73C22×Dic3—22no—
74C22×Dic3D414yes—
75C22×Dic3C2311yescharacteristic
76C22×Dic3C2312yes—
77C22×Dic3C2311yescharacteristic
78C22×Dic3C2312yes—
79C22×Dic3C2311yescharacteristic
80C3×C4⋊C4—24no—
81C3×C22⋊C4—24no—
82C3×C22⋊C4D414yescomplemented
83#217—24no—
84#217—22no—
85#217D414yescomplemented
86(C2×D3)⋊C4—24no—
87(C2×D3)⋊C4—24no—
88(C2×D3)⋊C4—22no—
89C4⋊Dic3—24no—
90Dic3⋊C4—24no—
91Dic3⋊C4—24no—
92Dic3⋊C4—24no—
93Dic3⋊C4—24no—
94Dic3⋊C4—22no—
95C4×Dic3—24no—
96C22×D4—31no—
97C22×D4—31no—
98C22×D4C2×D311yescharacteristic
99C23×C4—31no—
100C23×C4—31no—
101C23×C4C2×D311yescharacteristic
102#123—24no—
103#123—64no—
104#123—64no—
105#123—64no—
106#121—64no—
107C2×C4⋊C4—32no—
108C2×C4⋊C4—32no—
109C2×C4⋊C4—31no—
110C2×C4⋊C4—32no—
111C2×C4⋊C4—32no—
112C2×C4⋊C4—32no—
113C2×C4⋊C4—64no—
114C2×C4⋊C4C2×D312yes—
115C2×C22⋊C4—31no—
116C2×C22⋊C4—31no—
117C2×C22⋊C4—32no—
118C2×C22⋊C4—31no—
119C2×C22⋊C4—32no—
120C2×C22⋊C4—32no—
121C2×C22⋊C4C2×D312yes—
122C2×C22⋊C4C2×D311yescharacteristic
123C2×C42—32no—
124#95—32no—
125C23×C3—22no—
126C23×C3—22no—
127C23×C3—21no—
128C23×C3—21no—
129C23×C3—21no—
130C23×C3—22no—
131C23×C3—22no—
132C23×C3C2×D412yes—
133C23×C3C2×D412yes—
134C23×C3C2×D412yes—
135C23×C3C2411yescharacteristic
136C22×D3—21no—
137C22×D3—21no—
138C22×D3—21no—
139C22×D3—22no—
140C22×D3—22no—
141C3×D4—44no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—21no—
146C2×C3×C4—22no—
147C2×C3×C4—21no—
148C2×C3×C4—22no—
149C2×C3×C4—21no—
150C2×C3×C4—44no—
151C2×C3×C4C2×D414yes—
152#67—44no—
153#67—44no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—28no—
158C2×Dic3—22no—
159C2×Dic3—21no—
160C2×Dic3—22no—
161C2×Dic3—21no—
162C2×Dic3—21no—
163C2×Dic3—22no—
164C2×Dic3—44no—
165C2×Dic3C2×D414yes—
166C2×Dic3C2×D412yes—
167C2×Dic3C2×D412yes—
168C2×Dic3#4114yes—
169C24—31no—
170C24C22×D311yescharacteristic
171C24C22×D311yescharacteristic
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—62no—
176C2×D4—64no—
177C2×D4—62no—
178C2×D4—62no—
179C2×D4—62no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—31no—
184C22×C4—34no—
185C22×C4—31no—
186C22×C4—32no—
187C22×C4—32no—
188C22×C4—31no—
189C22×C4—64no—
190C22×C4—62no—
191C22×C4—62no—
192C22×C4—62no—
193C22×C4C22×D312yes—
194C22×C4C22×D311yescharacteristic
195C22×C4C22×D312yes—
196C4⋊C4—24no—
197C4⋊C4—64no—
198C4⋊C4—64no—
199C4⋊C4—62no—
200C4⋊C4—64no—
201C4⋊C4—64no—
202C4⋊C4—64no—
203C22⋊C4—24no—
204C22⋊C4—34no—
205C22⋊C4—62no—
206C22⋊C4—64no—
207C22⋊C4—62no—
208C22⋊C4—64no—
209C22⋊C4—64no—
210C22⋊C4#6714yescomplemented
211C42—64no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—24no—
215C22×C3—24no—
216C22×C3—41no—
217C22×C3—41no—
218C22×C3—41no—
219C22×C3—42no—
220C22×C3—42no—
221C22×C3#12714yes—
222C22×C3C22×D411yescharacteristic
223C22×C3C22×D411yescharacteristic
224C22×C3C22×D411yescharacteristic
225C22×C3C2×#4112yes—
226C22×C32−1+412yes—
227C2×D3—41no—
228C2×D3—41no—
229C2×D3—41no—
230C2×D3—42no—
231C2×D3—42no—
232C3×C4—24no—
233C3×C4—42no—
234C3×C4—41no—
235Dic3—22no—
236Dic3—24no—
237Dic3—42no—
238Dic3—41no—
239Dic3#12314yescomplemented
240C23—22no—
241C23—22no—
242C23—21no—
243C23—21no—
244C23—22no—
245C23—22no—
246C23—21no—
247C23—61no—
248C23—61no—
249C23—62no—
250C23—62no—
251C23—61no—
252C23D3×D412yes—
253C23D3×D412yes—
254C23C2×#6712yes—
255C23C23×D311yescharacteristic
256D4—44no—
257D4—124no—
258D4—124no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—32no—
268C2×C4—34no—
269C2×C4—34no—
270C2×C4—32no—
271C2×C4—44no—
272C2×C4—68no—
273C2×C4—62no—
274C2×C4—62no—
275C2×C4—62no—
276C2×C4—62no—
277C2×C4—61no—
278C2×C4—62no—
279C2×C4—61no—
280C2×C4—62no—
281C2×C4—61no—
282C2×C4—124no—
283C2×C4C2×#6714yes—
284C2×C3—24no—
285C2×C3—81no—
286C2×C3C2×#12311yescharacteristic
287C2×C3C2×#12711yescharacteristic
288C2×C3#53611yescharacteristic
289C2×C3#54712yes—
290C2×C3#54712yes—
291D3—81no—
292C22—22no—
293C22—22no—
294C22—24no—
295C22—24no—
296C22—41no—
297C22—41no—
298C22—42no—
299C22—41no—
300C22—42no—
301C22—121no—
302C22—121no—
303C22—122no—
304C22—121no—
305C22—122no—
306C22#94014yes—
307C22C2×D3×D411yescharacteristic
308C22C2×D3×D411yescharacteristic
309C22D3×#4112yes—
310C22#101012yes—
311C22C22×#6711yescharacteristic
312C4—24no—
313C4—34no—
314C4—42no—
315C4—41no—
316C4—62no—
317C4—64no—
318C4—122no—
319C4—121no—
320C3#250111yescharacteristic, complemented, Sylow
321C2—24no—
322C2—81no—
323C2—241no—
324C2#564811yescharacteristic
325C2D3×#12311yescharacteristic
326C2#565512yes—
327C2C2×#94011yescharacteristic
328C2#582412yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#123C2
prime radicalC2×#939C2
hypercenterC2×#123D3
hypocenterC3#2501
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×C3×#123C2semidirect
3. C2×#935C2semidirect
4. C2×#929C2semidirect
5. C22×Dic3⋊C4C2semidirect
6. C2×#886C2semidirect
7. #4993C2semidirect
8. #4975C2semidirect
9. #4957C2semidirect
10. #4949C2semidirect
11. Dic3×C22⋊C4C2semidirect
45. C2×#123D3semidirect
82. C3×C22⋊C4D4semidirect
85. #217D4semidirect
210. C22⋊C4#67semidirect
239. Dic3#123semidirect
320. C3#2501semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1232
C2×#9392
C2×#9352
C2×#9292
C22×Dic3⋊C42
C2×#8862
#49932
#49752
#49572
#49492
Dic3×C22⋊C42
#25013