Finite Group Databasebeta

← C2×#5550 #85780

Subgroups — C2×#5550 #85780

Order 384.

1900 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#235C211yesmaximal, characteristic, complemented
2C23×D12C211yesmaximal, characteristic, complemented
3C2×C3×C4×D4C211yesmaximal, characteristic, complemented
4C2×#937C212yesmaximal, complemented
5C2×#930C211yesmaximal, characteristic, complemented
6C2×#924C211yesmaximal, characteristic, complemented
7#5550C2116yesmaximal, direct product factor
8C2×#894C212yesmaximal, complemented
9C2×#884C212yesmaximal, complemented
10C2×D3×C22⋊C4C212yesmaximal, complemented
11C2×#875C211yesmaximal, characteristic, complemented
12C2×C4×D12C211yesmaximal, characteristic, complemented
13C2×#546—31nomaximal, Sylow
14C24×D3C2212yes—
15C22×C3×D4C2211yescharacteristic
16C23×C3×C4C2211yescharacteristic
17C23×C3×C4C2211yescharacteristic, complemented
18C22×#67C2212yescomplemented
19C22×#67C2212yescomplemented
20C2×#235—28no—
21C2×#235C2218yescomplemented
22C22×D12—28no—
23C22×D12C2211yescharacteristic
24C22×D12C2212yes—
25C22×D12C2211yescharacteristic
26C22×D12C2218yescomplemented
27C22×C4×D3C2212yescomplemented
28C22×C4×D3C2212yescomplemented
29C22×Dic6C2211yescharacteristic, complemented
30C3×C4×D4C2218yescomplemented
31C2×C3×C4⋊C4C2211yescharacteristic, complemented
32C2×C3×C22⋊C4C2211yescharacteristic
33C2×C3×C22⋊C4C2211yescharacteristic, complemented
34C2×C3×C42C2211yescharacteristic, complemented
35C2×#217C2212yes—
36#937C22116yescomplemented
37#930C2218yescomplemented
38C2×(C2×D3)⋊C4C2212yescomplemented
39C2×(C2×D3)⋊C4C2212yescomplemented
40C2×(C2×D3)⋊C4C2212yescomplemented
41C2×(C2×D3)⋊C4C2212yescomplemented
42C2×C4⋊Dic3C2211yescharacteristic, complemented
43#924C2218yescomplemented
44C2×Dic3⋊C4C2212yescomplemented
45#894C22116yescomplemented
46#884C22116yescomplemented
47D3×C22⋊C4C22116yescomplemented
48#875C2218yescomplemented
49C4×D12C2218yescomplemented
50C22×#41—31no—
51C23×D4—31no—
52#546—316no—
53C2×#124—31no—
54C2×#123—32no—
55C2×C22⋊Q8—31no—
56C2×#121—31no—
57C2×#121—32no—
58C2×#120—32no—
59C2×C4×D4—31no—
60C2×C4×D4D311yescharacteristic, complemented
61C22×C22⋊C4—32no—
62C24×C3C2311yescharacteristic
63C24×C3C2311yescharacteristic
64C23×D3—28no—
65C23×D3—28no—
66C23×D3—44no—
67C23×D3C2312yes—
68C23×D3C2318yes—
69C23×D3C2312yes—
70C2×C3×D4—24no—
71C2×C3×D4C2314yes—
72C22×C3×C4—24no—
73C22×C3×C4—24no—
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2314yes—
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2314yescomplemented
81C2×#67—28no—
82C2×#67—44no—
83C2×#67C2318yescomplemented
84C2×#67C2318yescomplemented
85C22×Dic3C2312yes—
86C22×Dic3C2312yes—
87#235—232no—
88C2×D12—28no—
89C2×D12—24no—
90C2×D12—232no—
91C2×D12D418yescomplemented
92C2×D12C2314yes—
93C2×D12C2318yes—
94C2×D12C2314yes—
95C2×C4×D3—28no—
96C2×C4×D3—44no—
97C2×C4×D3C2318yescomplemented
98C2×C4×D3C2318yescomplemented
99C2×Dic6—24no—
100C2×Dic6C2314yescomplemented
101C3×C4⋊C4C2314yescomplemented
102C3×C22⋊C4C2314yes—
103C3×C22⋊C4C2314yescomplemented
104C3×C42C2314yescomplemented
105#217C2318yes—
106(C2×D3)⋊C4C2318yescomplemented
107(C2×D3)⋊C4C2318yescomplemented
108(C2×D3)⋊C4C2318yescomplemented
109(C2×D3)⋊C4C2318yescomplemented
110C4⋊Dic3C2314yescomplemented
111Dic3⋊C4C2318yescomplemented
112C25—32no—
113C2×#41—38no—
114C2×#41—68no—
115C22×Q8—31no—
116C22×D4—32no—
117C22×D4—32no—
118C22×D4—31no—
119C22×D4—32no—
120C22×D4—31no—
121C22×D4—38no—
122C22×D4—68no—
123C22×D4C2×D311yescharacteristic
124C23×C4—32no—
125C23×C4—32no—
126C23×C4C2×D311yescharacteristic
127C23×C4C2×D311yescharacteristic, complemented
128#124—38no—
129#123—316no—
130C22⋊Q8—38no—
131#121—38no—
132#121—316no—
133#120—316no—
134C4×D4—38no—
135C4×D4C2×D318yescomplemented
136C2×C4⋊C4—31no—
137C2×C4⋊C4—32no—
138C2×C4⋊C4C2×D311yescharacteristic, complemented
139C2×C22⋊C4—32no—
140C2×C22⋊C4—32no—
141C2×C22⋊C4—32no—
142C2×C22⋊C4—32no—
143C2×C22⋊C4—32no—
144C2×C22⋊C4—316no—
145C2×C22⋊C4C2×D311yescharacteristic
146C2×C22⋊C4C2×D311yescharacteristic, complemented
147C2×C42C2×D311yescharacteristic, complemented
148C23×C3—22no—
149C23×C3—22no—
150C23×C3—22no—
151C23×C3—22no—
152C23×C3—22no—
153C23×C3#4114yescomplemented
154C23×C3C2412yes—
155C23×C3C2412yes—
156C23×C3C2411yescharacteristic
157C22×D3—24no—
158C22×D3—24no—
159C22×D3—24no—
160C22×D3—24no—
161C22×D3—216no—
162C22×D3—24no—
163C22×D3—216no—
164C22×D3—48no—
165C22×D3—432no—
166C22×D3C2×D418yes—
167C22×D3C2414yes—
168C22×D3C2414yes—
169C3×D4—28no—
170C2×C3×C4—22no—
171C2×C3×C4—22no—
172C2×C3×C4—22no—
173C2×C3×C4—22no—
174C2×C3×C4—28no—
175C2×C3×C4—28no—
176C2×C3×C4C2×D414yes—
177C2×C3×C4C2412yes—
178C2×C3×C4C2412yes—
179C2×C3×C4C2412yes—
180C2×C3×C4C2412yes—
181C2×C3×C4C2412yes—
182#67—216no—
183#67—48no—
184C2×Dic3—24no—
185C2×Dic3—24no—
186C2×Dic3C2414yes—
187C2×Dic3C2414yes—
188D12—216no—
189D12—28no—
190D12C2×D4116yescomplemented
191C4×D3—216no—
192C4×D3—48no—
193Dic6—28no—
194C24—32no—
195C24—32no—
196C24—38no—
197C24—68no—
198C24—68no—
199C24—124no—
200C24C22×D311yescharacteristic
201C24C22×D311yescharacteristic
202#41—632no—
203C2×Q8—34no—
204C2×Q8—64no—
205C2×D4—24no—
206C2×D4—38no—
207C2×D4—38no—
208C2×D4—38no—
209C2×D4—34no—
210C2×D4—38no—
211C2×D4—34no—
212C2×D4—632no—
213C2×D4—68no—
214C2×D4—68no—
215C2×D4—64no—
216C2×D4—124no—
217C2×D4C22×D314yes—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—32no—
221C22×C4—32no—
222C22×C4—38no—
223C22×C4—38no—
224C22×C4—68no—
225C22×C4—124no—
226C22×C4C22×D314yes—
227C22×C4C22×D311yescharacteristic
228C22×C4C22×D311yescharacteristic
229C22×C4C22×D311yescharacteristic
230C22×C4C22×D311yescharacteristic
231C22×C4C22×D311yescharacteristic
232C22×C4C22×D314yescomplemented
233C4⋊C4—34no—
234C4⋊C4—38no—
235C4⋊C4C22×D314yescomplemented
236C22⋊C4—38no—
237C22⋊C4—38no—
238C22⋊C4—38no—
239C22⋊C4—38no—
240C22⋊C4—38no—
241C22⋊C4C22×D314yes—
242C22⋊C4C22×D314yescomplemented
243C42C22×D314yescomplemented
244C22×C3—28no—
245C22×C3—22no—
246C22×C3—22no—
247C22×C3—22no—
248C22×C3—22no—
249C22×C3—22no—
250C22×C3—44no—
251C22×C3C22×D412yes—
252C22×C3C2×#4112yes—
253C22×C3C2×#4114yescomplemented
254C22×C3H(5,𝔽2)12yes—
255C22×C3C2511yescharacteristic
256C2×D3—216no—
257C2×D3—24no—
258C2×D3—24no—
259C2×D3—24no—
260C2×D3—24no—
261C2×D3—24no—
262C2×D3—432no—
263C2×D3—48no—
264C2×D3C22×D418yes—
265C3×C4—22no—
266C3×C4—22no—
267C3×C4—22no—
268C3×C4—22no—
269C3×C4C22×D414yes—
270Dic3—24no—
271Dic3—24no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—38no—
278C23—34no—
279C23—34no—
280C23—616no—
281C23—64no—
282C23—64no—
283C23—64no—
284C23—616no—
285C23—64no—
286C23—64no—
287C23—128no—
288C23—1232no—
289C23#23514yescomplemented
290C23C23×D312yes—
291C23C23×D312yes—
292C23C23×D311yescharacteristic
293Q8—68no—
294D4—28no—
295D4—316no—
296D4—616no—
297D4—616no—
298D4—68no—
299D4—128no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—28no—
305C2×C4—28no—
306C2×C4—34no—
307C2×C4—34no—
308C2×C4—616no—
309C2×C4—64no—
310C2×C4—64no—
311C2×C4—128no—
312C2×C4D3×D414yes—
313C2×C4C23×D312yes—
314C2×C4C23×D312yes—
315C2×C4C23×D312yes—
316C2×C4C23×D312yes—
317C2×C4C23×D312yes—
318C2×C3—24no—
319C2×C3—42no—
320C2×C3#54614yescomplemented
321C2×C3C23×D411yescharacteristic
322C2×C3C22×#4111yescharacteristic
323C2×C3C2×H(5,𝔽2)11yescharacteristic
324D3—28no—
325D3—44no—
326C22—22no—
327C22—22no—
328C22—28no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—38no—
333C22—44no—
334C22—64no—
335C22—64no—
336C22—616no—
337C22—64no—
338C22—64no—
339C22—64no—
340C22—1232no—
341C22—128no—
342C22C2×#23512yes—
343C22C2×#23514yescomplemented
344C22C2×D3×D412yes—
345C22#100912yes—
346C22C24×D311yescharacteristic
347C4—22no—
348C4—22no—
349C4—22no—
350C4—22no—
351C4—64no—
352C4—64no—
353C4C2×D3×D414yes—
354C3C2×#54611yescharacteristic, complemented, Sylow
355C2—24no—
356C2—42no—
357C2—68no—
358C2—124no—
359C2#555014yesdirect product factor
360C2C22×#23511yescharacteristic
361C2C22×D3×D411yescharacteristic
362C2C2×#100911yescharacteristic
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×C4×D4C2
socleC23×C3C24
hypercenterC2×C4×D4D3
hypocenterC3C2×#546
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×#235C2semidirect
2. C23×D12C2semidirect
3. C2×C3×C4×D4C2semidirect
4. C2×#937C2semidirect
5. C2×#930C2semidirect
6. C2×#924C2semidirect
7. #5550C2direct (N × H)
8. C2×#894C2semidirect
9. C2×#884C2semidirect
10. C2×D3×C22⋊C4C2semidirect
11. C2×#875C2semidirect
12. C2×C4×D12C2semidirect
17. C23×C3×C4C22semidirect
18. C22×#67C22semidirect
19. C22×#67C22semidirect
21. C2×#235C22semidirect
26. C22×D12C22semidirect
27. C22×C4×D3C22semidirect
28. C22×C4×D3C22semidirect
29. C22×Dic6C22semidirect
30. C3×C4×D4C22semidirect
31. C2×C3×C4⋊C4C22semidirect
33. C2×C3×C22⋊C4C22semidirect
34. C2×C3×C42C22semidirect
36. #937C22semidirect
37. #930C22semidirect
38. C2×(C2×D3)⋊C4C22semidirect
39. C2×(C2×D3)⋊C4C22semidirect
40. C2×(C2×D3)⋊C4C22semidirect
41. C2×(C2×D3)⋊C4C22semidirect
42. C2×C4⋊Dic3C22semidirect
43. #924C22semidirect
44. C2×Dic3⋊C4C22semidirect
45. #894C22semidirect
46. #884C22semidirect
47. D3×C22⋊C4C22semidirect
48. #875C22semidirect
49. C4×D12C22semidirect
60. C2×C4×D4D3semidirect
80. C22×C3×C4C23semidirect
83. C2×#67C23semidirect
84. C2×#67C23semidirect
91. C2×D12D4semidirect
97. C2×C4×D3C23semidirect
98. C2×C4×D3C23semidirect
100. C2×Dic6C23semidirect
101. C3×C4⋊C4C23semidirect
103. C3×C22⋊C4C23semidirect
104. C3×C42C23semidirect
106. (C2×D3)⋊C4C23semidirect
107. (C2×D3)⋊C4C23semidirect
108. (C2×D3)⋊C4C23semidirect
109. (C2×D3)⋊C4C23semidirect
110. C4⋊Dic3C23semidirect
111. Dic3⋊C4C23semidirect
127. C23×C4C2×D3semidirect
135. C4×D4C2×D3semidirect
138. C2×C4⋊C4C2×D3semidirect
146. C2×C22⋊C4C2×D3semidirect
147. C2×C42C2×D3semidirect
153. C23×C3#41semidirect
190. D12C2×D4semidirect
232. C22×C4C22×D3semidirect
235. C4⋊C4C22×D3semidirect
242. C22⋊C4C22×D3semidirect
243. C42C22×D3semidirect
253. C22×C3C2×#41semidirect
289. C23#235semidirect
320. C2×C3#546semidirect
343. C22C2×#235semidirect
354. C3C2×#546semidirect
359. C2#5550direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#2352
C23×D122
C2×C3×C4×D42
C2×#9372
C2×#9302
C2×#9242
#55502
C2×#8942
C2×#8842
C2×D3×C22⋊C42
C2×#8752
C2×C4×D122
C2×#5463