Finite Group Databasebeta

← C2×#5586 #85909

Subgroups — C2×#5586 #85909

Order 384.

1880 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×Dic3C211yesmaximal, characteristic, complemented
2C22×#237C211yesmaximal, characteristic, complemented
3C2×C3×#120C211yesmaximal, characteristic, complemented
4C2×#953C211yesmaximal, characteristic, complemented
5C2×#939C212yesmaximal, complemented
6C2×#935C211yesmaximal, characteristic, complemented
7C2×D4×Dic3C212yesmaximal, complemented
8#5586C2116yesmaximal, direct product factor
9C2×#886C211yesmaximal, characteristic, complemented
10C2×#67⋊C4C212yesmaximal, complemented
11C2×#878C212yesmaximal, complemented
12C2×#877C211yesmaximal, characteristic, complemented
13C2×#525—31nomaximal, Sylow
14C25×C3C2211yescharacteristic
15C22×C3×D4C2211yescharacteristic
16C22×C3×D4C2212yes—
17C22×#67C2211yescharacteristic
18C22×#67C2212yes—
19C22×#67C2211yescharacteristic, complemented
20C23×Dic3—28no—
21C23×Dic3C2212yes—
22C23×Dic3C2212yes—
23C23×Dic3C2211yescharacteristic
24C23×Dic3C2211yescharacteristic
25C23×Dic3C2218yescomplemented
26C23×Dic3C2211yescharacteristic, complemented
27C2×#237—28no—
28C2×#237C2218yescomplemented
29C22×C4×D3C2211yescharacteristic, complemented
30C22×Dic6C2211yescharacteristic, complemented
31C3×#120C2218yescomplemented
32C2×C3×C22⋊C4C2211yescharacteristic
33C2×C3×C22⋊C4C2212yes—
34#953C2218yescomplemented
35C2×#217C2211yescharacteristic
36C2×#217C2212yes—
37C2×#217C2212yescomplemented
38#939C22116yescomplemented
39#935C2218yescomplemented
40D4×Dic3C22116yescomplemented
41C2×(C2×D3)⋊C4C2212yescomplemented
42C2×C4⋊Dic3C2212yescomplemented
43C2×Dic3⋊C4C2212yescomplemented
44C2×Dic3⋊C4C2212yescomplemented
45C2×C4×Dic3C2212yescomplemented
46#886C2218yescomplemented
47#67⋊C4C22116yescomplemented
48#878C22116yescomplemented
49#877C2218yescomplemented
50C22×#41—31no—
51C24×C4—31no—
52#525—316no—
53C2×#123—31no—
54C2×#123—31no—
55C2×C22⋊Q8—32no—
56C2×#121—32no—
57C2×#120—31no—
58C2×#120D311yescharacteristic, complemented
59C2×C4×D4—32no—
60C2×C4×D4—32no—
61C2×#117—31no—
62C24×C3—24no—
63C24×C3—28no—
64C24×C3C2314yes—
65C24×C3C2311yescharacteristic
66C24×C3C2311yescharacteristic
67C24×C3C2312yes—
68C23×D3C2311yescharacteristic
69C2×C3×D4—24no—
70C2×C3×D4—28no—
71C2×C3×D4C2318yes—
72C2×C3×D4C2314yes—
73C22×C3×C4C2312yes—
74C22×C3×C4C2311yescharacteristic
75C2×#67—24no—
76C2×#67—28no—
77C2×#67—24no—
78C2×#67C2318yes—
79C2×#67C2314yes—
80C2×#67C2314yescomplemented
81C22×Dic3—24no—
82C22×Dic3—24no—
83C22×Dic3—28no—
84C22×Dic3—28no—
85C22×Dic3—232no—
86C22×Dic3D418yescomplemented
87C22×Dic3C2318yes—
88C22×Dic3C2318yes—
89C22×Dic3C2314yes—
90C22×Dic3C2312yes—
91C22×Dic3C2312yes—
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2314yes—
96C22×Dic3C2314yescomplemented
97#237—232no—
98C2×C4×D3—24no—
99C2×C4×D3C2314yescomplemented
100C2×Dic6—24no—
101C2×Dic6C2314yescomplemented
102C3×C22⋊C4C2314yes—
103C3×C22⋊C4C2318yes—
104#217C2314yes—
105#217C2318yes—
106#217C2318yescomplemented
107(C2×D3)⋊C4C2318yescomplemented
108C4⋊Dic3C2318yescomplemented
109Dic3⋊C4C2318yescomplemented
110Dic3⋊C4C2318yescomplemented
111C4×Dic3C2318yescomplemented
112C25C2×D311yescharacteristic
113C2×#41—38no—
114C2×#41—68no—
115C22×Q8—31no—
116C22×D4—32no—
117C22×D4—31no—
118C22×D4—31no—
119C22×D4C2×D312yes—
120C22×D4C2×D311yescharacteristic
121C23×C4—31no—
122C23×C4—32no—
123C23×C4—31no—
124C23×C4—32no—
125C23×C4—31no—
126C23×C4—38no—
127C23×C4—31no—
128C23×C4—68no—
129#123—38no—
130#123—38no—
131C22⋊Q8—316no—
132#121—316no—
133#120—38no—
134#120C2×D318yescomplemented
135C4×D4—316no—
136C4×D4—316no—
137#117—38no—
138C2×C4⋊C4—32no—
139C2×C4⋊C4—32no—
140C2×C4⋊C4—32no—
141C2×C22⋊C4—31no—
142C2×C22⋊C4—32no—
143C2×C22⋊C4—32no—
144C2×C22⋊C4—32no—
145C2×C22⋊C4C2×D311yescharacteristic
146C2×C22⋊C4C2×D312yes—
147C2×C42—32no—
148C23×C3—232no—
149C23×C3—28no—
150C23×C3—24no—
151C23×C3—24no—
152C23×C3—24no—
153C23×C3—216no—
154C23×C3—22no—
155C23×C3—24no—
156C23×C3C2×D414yes—
157C23×C3#4118yescomplemented
158C23×C3C2412yes—
159C23×C3C2412yes—
160C23×C3C2414yes—
161C23×C3C2411yescharacteristic
162C22×D3—22no—
163C22×D3—24no—
164C22×D3C2412yes—
165C3×D4—28no—
166C3×D4—216no—
167C2×C3×C4—22no—
168C2×C3×C4—24no—
169C2×C3×C4C2414yes—
170C2×C3×C4C2412yes—
171#67—28no—
172#67—216no—
173#67—28no—
174C2×Dic3—22no—
175C2×Dic3—28no—
176C2×Dic3—24no—
177C2×Dic3—24no—
178C2×Dic3—28no—
179C2×Dic3—216no—
180C2×Dic3—216no—
181C2×Dic3C2×D414yes—
182C2×Dic3C2×D414yes—
183C2×Dic3C2×D4116yescomplemented
184C2×Dic3C2414yes—
185C2×Dic3C2414yes—
186C2×Dic3C2412yes—
187C2×Dic3C2412yes—
188C2×Dic3C2412yes—
189C4×D3—28no—
190Dic6—28no—
191C24—24no—
192C24—28no—
193C24—31no—
194C24C22×D314yes—
195C24C22×D311yescharacteristic
196C24C22×D311yescharacteristic
197C24C22×D312yes—
198#41—632no—
199C2×Q8—34no—
200C2×Q8—64no—
201C2×D4—24no—
202C2×D4—28no—
203C2×D4—38no—
204C2×D4—34no—
205C2×D4—34no—
206C2×D4—64no—
207C2×D4—64no—
208C2×D4—68no—
209C2×D4C22×D318yes—
210C2×D4C22×D314yes—
211C22×C4—38no—
212C22×C4—34no—
213C22×C4—31no—
214C22×C4—31no—
215C22×C4—38no—
216C22×C4—34no—
217C22×C4—38no—
218C22×C4—34no—
219C22×C4—32no—
220C22×C4—32no—
221C22×C4—31no—
222C22×C4—34no—
223C22×C4—64no—
224C22×C4—68no—
225C22×C4—68no—
226C22×C4—632no—
227C22×C4—64no—
228C22×C4—64no—
229C22×C4C22×D312yes—
230C22×C4C22×D311yescharacteristic
231C4⋊C4—38no—
232C4⋊C4—38no—
233C4⋊C4—38no—
234C22⋊C4—34no—
235C22⋊C4—38no—
236C22⋊C4—38no—
237C22⋊C4—38no—
238C22⋊C4C22×D314yes—
239C22⋊C4C22×D318yes—
240C42—38no—
241C22×C3—232no—
242C22×C3—28no—
243C22×C3—216no—
244C22×C3—24no—
245C22×C3—24no—
246C22×C3—24no—
247C22×C3—22no—
248C22×C3—24no—
249C22×C3—44no—
250C22×C3C22×D414yes—
251C22×C3C22×D412yes—
252C22×C3C2×#4114yes—
253C22×C3C2×#4118yescomplemented
254C22×C3C2511yescharacteristic
255C2×D3—22no—
256C2×D3—24no—
257C2×D3—44no—
258C3×C4—22no—
259C3×C4—24no—
260Dic3—22no—
261Dic3—24no—
262Dic3—24no—
263Dic3C22×D414yes—
264Dic3C22×D414yes—
265C23—232no—
266C23—28no—
267C23—24no—
268C23—24no—
269C23—24no—
270C23—216no—
271C23—22no—
272C23—24no—
273C23—32no—
274C23—62no—
275C23—64no—
276C23D3×D414yes—
277C23#23718yescomplemented
278C23C23×D312yes—
279C23C23×D312yes—
280C23C23×D314yes—
281C23C23×D311yescharacteristic
282Q8—68no—
283D4—28no—
284D4—216no—
285D4—68no—
286D4—68no—
287D4—616no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—34no—
291C2×C4—34no—
292C2×C4—316no—
293C2×C4—32no—
294C2×C4—32no—
295C2×C4—34no—
296C2×C4—34no—
297C2×C4—32no—
298C2×C4—68no—
299C2×C4—616no—
300C2×C4—616no—
301C2×C4—68no—
302C2×C4—68no—
303C2×C4—64no—
304C2×C4—62no—
305C2×C4—64no—
306C2×C4C23×D314yes—
307C2×C4C23×D312yes—
308C2×C3—24no—
309C2×C3—28no—
310C2×C3—42no—
311C2×C3#52514yescomplemented
312C2×C3C23×D411yescharacteristic
313C2×C3C22×#4112yes—
314D3—42no—
315C22—232no—
316C22—24no—
317C22—24no—
318C22—24no—
319C22—28no—
320C22—216no—
321C22—22no—
322C22—24no—
323C22—44no—
324C22—62no—
325C22—64no—
326C22—124no—
327C22C2×D3×D414yes—
328C22C2×D3×D412yes—
329C22C2×#23714yes—
330C22C2×#23718yescomplemented
331C22C24×D311yescharacteristic
332C4—22no—
333C4—24no—
334C4—34no—
335C4—34no—
336C4—64no—
337C4—62no—
338C4—64no—
339C3C2×#52511yescharacteristic, complemented, Sylow
340C2—24no—
341C2—28no—
342C2—42no—
343C2—122no—
344C2#558614yesdirect product factor
345C2C22×D3×D411yescharacteristic
346C2C22×#23712yes—
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×#120C2
socleC23×C3C24
hypercenterC2×#120D3
hypocenterC3C2×#525
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C24×Dic3C2semidirect
2. C22×#237C2semidirect
3. C2×C3×#120C2semidirect
4. C2×#953C2semidirect
5. C2×#939C2semidirect
6. C2×#935C2semidirect
7. C2×D4×Dic3C2semidirect
8. #5586C2direct (N × H)
9. C2×#886C2semidirect
10. C2×#67⋊C4C2semidirect
11. C2×#878C2semidirect
12. C2×#877C2semidirect
19. C22×#67C22semidirect
25. C23×Dic3C22semidirect
26. C23×Dic3C22semidirect
28. C2×#237C22semidirect
29. C22×C4×D3C22semidirect
30. C22×Dic6C22semidirect
31. C3×#120C22semidirect
34. #953C22semidirect
37. C2×#217C22semidirect
38. #939C22semidirect
39. #935C22semidirect
40. D4×Dic3C22semidirect
41. C2×(C2×D3)⋊C4C22semidirect
42. C2×C4⋊Dic3C22semidirect
43. C2×Dic3⋊C4C22semidirect
44. C2×Dic3⋊C4C22semidirect
45. C2×C4×Dic3C22semidirect
46. #886C22semidirect
47. #67⋊C4C22semidirect
48. #878C22semidirect
49. #877C22semidirect
58. C2×#120D3semidirect
80. C2×#67C23semidirect
86. C22×Dic3D4semidirect
96. C22×Dic3C23semidirect
99. C2×C4×D3C23semidirect
101. C2×Dic6C23semidirect
106. #217C23semidirect
107. (C2×D3)⋊C4C23semidirect
108. C4⋊Dic3C23semidirect
109. Dic3⋊C4C23semidirect
110. Dic3⋊C4C23semidirect
111. C4×Dic3C23semidirect
134. #120C2×D3semidirect
157. C23×C3#41semidirect
183. C2×Dic3C2×D4semidirect
253. C22×C3C2×#41semidirect
277. C23#237semidirect
311. C2×C3#525semidirect
330. C22C2×#237semidirect
339. C3C2×#525semidirect
344. C2#5586direct (N × H)

Maximal subgroups

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

subgroupindex
C24×Dic32
C22×#2372
C2×C3×#1202
C2×#9532
C2×#9392
C2×#9352
C2×D4×Dic32
#55862
C2×#8862
C2×#67⋊C42
C2×#8782
C2×#8772
C2×#5253