Finite Group Databasebeta

← #82578

Subgroups — #82578

Order 384.

808 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#120C211yesmaximal, characteristic
2C22×#217C211yesmaximal, characteristic, complemented
3C2×D4×Dic3C211yesmaximal, characteristic, complemented
4#5247C211yesmaximal, characteristic, complemented
5#5221C212yesmaximal, complemented
6#5220C212yesmaximal, complemented
7#5220C211yesmaximal, characteristic, complemented
8#4951C212yesmaximal, complemented
9#4951C212yesmaximal, complemented
10#4950C211yesmaximal, characteristic, complemented
11Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
12#2321—31nomaximal, Sylow
13C25×C3C2211yescharacteristic
14C22×C3×D4C2211yescharacteristic
15C22×C3×D4C2212yes—
16C23×Dic3C2211yescharacteristic
17C23×Dic3C2211yescharacteristic
18C23×Dic3C2211yescharacteristic, complemented
19C3×#120C418yescomplemented
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2212yes—
22C2×#217—24no—
23C2×#217C2211yescharacteristic
24C2×#217C2211yescharacteristic
25C2×#217C2212yes—
26C2×#217C2212yes—
27C2×#217C2212yes—
28C2×#217C2212yes—
29C2×#217C2211yescharacteristic, complemented
30C2×#217C2212yescomplemented
31D4×Dic3—24no—
32C2×C4⋊Dic3C2211yescharacteristic, complemented
33C2×C4⋊Dic3C2212yescomplemented
34C2×C4×Dic3C2211yescharacteristic, complemented
35C2×C4×Dic3C2212yescomplemented
36C3⋊#95C2212yescomplemented
37C3⋊#95C2212yescomplemented
38C3⋊#95C2212yescomplemented
39C2×#120D311yescharacteristic
40C2×C4×D4—31no—
41C22×C22⋊C4—31no—
42#390—32no—
43#388—32no—
44#388—32no—
45#386—32no—
46#386—31no—
47#385—31no—
48#379—31no—
49C4×C22⋊C4—31no—
50C24×C3—22no—
51C24×C3—22no—
52C24×C3—42no—
53C24×C3—42no—
54C24×C3C2×C414yescomplemented
55C24×C3C2312yes—
56C24×C3C2311yescharacteristic
57C24×C3C2311yescharacteristic
58C2×C3×D4—22no—
59C2×C3×D4—22no—
60C2×C3×D4—42no—
61C2×C3×D4—42no—
62C2×C3×D4C2×C414yes—
63C2×C3×D4C2×C418yescomplemented
64C22×C3×C4C2311yescharacteristic
65C22×C3×C4C2312yes—
66C22×Dic3—22no—
67C22×Dic3—22no—
68C22×Dic3—42no—
69C22×Dic3—42no—
70C22×Dic3—42no—
71C22×Dic3D414yescomplemented
72C22×Dic3C2312yes—
73C22×Dic3C2311yescharacteristic
74C22×Dic3C2312yes—
75C22×Dic3C2311yescharacteristic
76C22×Dic3C2311yescharacteristic
77C22×Dic3C2311yescharacteristic
78C3×C22⋊C4C2×C418yescomplemented
79C3×C22⋊C4C2×C414yescomplemented
80#217—22no—
81#217—22no—
82#217—22no—
83#217—24no—
84#217—24no—
85#217—42no—
86#217—42no—
87#217—42no—
88C4⋊Dic3—22no—
89C4⋊Dic3—42no—
90C4×Dic3—22no—
91C4×Dic3—42no—
92C25C2×D311yescharacteristic
93C22×D4C2×D311yescharacteristic
94C22×D4C2×D312yes—
95C23×C4—31no—
96C23×C4—31no—
97C23×C4—31no—
98#120Dic318yescomplemented
99C4×D4—64no—
100C2×C4⋊C4—31no—
101C2×C4⋊C4—32no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4—31no—
105C2×C22⋊C4—31no—
106C2×C22⋊C4—31no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—64no—
111C2×C22⋊C4C2×D311yescharacteristic
112C2×C22⋊C4C2×D312yes—
113C2×C42—31no—
114C2×C42—32no—
115#95—32no—
116#95—32no—
117#95—32no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—28no—
121C23×C3—22no—
122C23×C3—21no—
123C23×C3—21no—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3—22no—
127C23×C3—22no—
128C23×C3—22no—
129C23×C3—22no—
130C23×C3—22no—
131C23×C3—48no—
132C23×C3—44no—
133C23×C3—44no—
134C23×C3—48no—
135C23×C3C22×C414yes—
136C23×C3C22×C412yes—
137C23×C3C22×C412yes—
138C23×C3C2×D412yes—
139C23×C3#4112yes—
140C23×C3C2411yescharacteristic
141C3×D4—28no—
142C3×D4—48no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4C22×C412yes—
148C2×C3×C4C22×C414yes—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—21no—
152C2×Dic3—21no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—22no—
158C2×Dic3—22no—
159C2×Dic3—22no—
160C2×Dic3—22no—
161C2×Dic3—44no—
162C2×Dic3—44no—
163C2×Dic3—44no—
164C2×Dic3C2×D412yes—
165C2×Dic3C2×D412yes—
166C24—22no—
167C24—22no—
168C24—42no—
169C24—42no—
170C24C2×Dic314yescomplemented
171C24C22×D312yes—
172C24C22×D311yescharacteristic
173C24C22×D311yescharacteristic
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4C2×Dic314yes—
179C2×D4C2×Dic318yescomplemented
180C22×C4—31no—
181C22×C4—31no—
182C22×C4—34no—
183C22×C4—32no—
184C22×C4—32no—
185C22×C4—31no—
186C22×C4—31no—
187C22×C4—62no—
188C22×C4—62no—
189C22×C4—122no—
190C22×C4—122no—
191C22×C4—122no—
192C22×C4C22×D311yescharacteristic
193C22×C4C22×D312yes—
194C4⋊C4—62no—
195C4⋊C4—122no—
196C22⋊C4—64no—
197C22⋊C4—64no—
198C22⋊C4—62no—
199C22⋊C4—62no—
200C22⋊C4—62no—
201C22⋊C4—122no—
202C22⋊C4—122no—
203C22⋊C4—122no—
204C22⋊C4C2×Dic318yescomplemented
205C22⋊C4C2×Dic314yescomplemented
206C42—62no—
207C42—122no—
208C22×C3—24no—
209C22×C3—22no—
210C22×C3—22no—
211C22×C3—24no—
212C22×C3—22no—
213C22×C3—24no—
214C22×C3—24no—
215C22×C3—24no—
216C22×C3—24no—
217C22×C3—416no—
218C22×C3—41no—
219C22×C3—42no—
220C22×C3—41no—
221C22×C3—42no—
222C22×C3—42no—
223C22×C3—42no—
224C22×C3—42no—
225C22×C3C4×D414yes—
226C22×C3C23×C411yescharacteristic
227C22×C3C22×D411yescharacteristic
228C22×C3C2×#4111yescharacteristic
229C22×C3H(5,𝔽2)12yes—
230C22×C3H(5,𝔽2)12yes—
231C3×C4—22no—
232C3×C4—24no—
233Dic3—22no—
234Dic3—22no—
235Dic3—41no—
236Dic3—41no—
237Dic3—42no—
238Dic3—42no—
239C23—22no—
240C23—22no—
241C23—28no—
242C23—22no—
243C23—21no—
244C23—21no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—48no—
253C23—44no—
254C23—44no—
255C23—48no—
256C23D3×D412yes—
257C23#23712yes—
258C23C22×Dic314yes—
259C23C22×Dic312yes—
260C23C22×Dic312yes—
261C23C23×D311yescharacteristic
262D4—28no—
263D4—48no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—32no—
269C2×C4—32no—
270C2×C4—62no—
271C2×C4—62no—
272C2×C4—61no—
273C2×C4—62no—
274C2×C4—61no—
275C2×C4—62no—
276C2×C4—62no—
277C2×C4—62no—
278C2×C4—62no—
279C2×C4—62no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4—124no—
283C2×C4—124no—
284C2×C4—124no—
285C2×C4C22×Dic312yes—
286C2×C4C22×Dic314yes—
287C2×C3—24no—
288C2×C3—42no—
289C2×C3—44no—
290C2×C3C2×C4×D411yescharacteristic
291C2×C3#51812yes—
292C2×C3#53411yescharacteristic
293C2×C3#53511yescharacteristic
294C2×C3#53812yes—
295C22—24no—
296C22—22no—
297C22—22no—
298C22—24no—
299C22—22no—
300C22—24no—
301C22—24no—
302C22—24no—
303C22—24no—
304C22—416no—
305C22—41no—
306C22—42no—
307C22—42no—
308C22—41no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22D4×Dic314yes—
313C22C2×D3×D411yescharacteristic
314C22C2×#23711yescharacteristic
315C22#100412yes—
316C22#100412yes—
317C22C23×Dic311yescharacteristic
318C4—22no—
319C4—24no—
320C4—62no—
321C4—62no—
322C4—121no—
323C4—122no—
324C4—121no—
325C4—122no—
326C3#232111yescharacteristic, complemented, Sylow
327C2—24no—
328C2—44no—
329C2—42no—
330C2#558911yescharacteristic
331C2#559211yescharacteristic
332C2#559312yes—
333C2C2×D4×Dic311yescharacteristic
334C2#579812yes—
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
Fitting, prime radicalC2×C3×#120C2
socle, cosocleC23×C3C24
hypercenterC2×#120D3
hypocenterC3#2321
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
2. C22×#217C2semidirect
3. C2×D4×Dic3C2semidirect
4. #5247C2semidirect
5. #5221C2semidirect
6. #5220C2semidirect
7. #5220C2semidirect
8. #4951C2semidirect
9. #4951C2semidirect
10. #4950C2semidirect
11. Dic3×C22⋊C4C2semidirect
18. C23×Dic3C22semidirect
19. C3×#120C4semidirect
29. C2×#217C22semidirect
30. C2×#217C22semidirect
32. C2×C4⋊Dic3C22semidirect
33. C2×C4⋊Dic3C22semidirect
34. C2×C4×Dic3C22semidirect
35. C2×C4×Dic3C22semidirect
36. C3⋊#95C22semidirect
37. C3⋊#95C22semidirect
38. C3⋊#95C22semidirect
54. C24×C3C2×C4semidirect
63. C2×C3×D4C2×C4semidirect
71. C22×Dic3D4semidirect
78. C3×C22⋊C4C2×C4semidirect
79. C3×C22⋊C4C2×C4semidirect
98. #120Dic3semidirect
170. C24C2×Dic3semidirect
179. C2×D4C2×Dic3semidirect
204. C22⋊C4C2×Dic3semidirect
205. C22⋊C4C2×Dic3semidirect
326. C3#2321semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1202
C22×#2172
C2×D4×Dic32
#52472
#52212
#52202
#49512
#49502
Dic3×C22⋊C42
#23213