Finite Group Databasebeta

← C2×#5629 #85976

Subgroups — C2×#5629 #85976

Order 384.

1640 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#239C211yesmaximal, characteristic, complemented
2C22×#235C211yesmaximal, characteristic, complemented
3C2×C3×C22⋊Q8C211yesmaximal, characteristic, complemented
4C2×#946C211yesmaximal, characteristic, complemented
5C2×#930C211yesmaximal, characteristic, complemented
6#5629C2116yesmaximal, direct product factor
7C2×#896C212yesmaximal, complemented
8C2×#895C211yesmaximal, characteristic, complemented
9C2×#895C212yesmaximal, complemented
10C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
11C2×#884C212yesmaximal, complemented
12C2×#884C212yesmaximal, complemented
13C2×#537—31nomaximal, Sylow
14C22×C3×Q8C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C22×#67C2211yescharacteristic
17C22×#67C2211yescharacteristic
18C22×#67C2212yes—
19C2×#239—28no—
20C2×#239C2218yescomplemented
21C2×#235—28no—
22C2×#235C2218yescomplemented
23C22×D12C2211yescharacteristic
24C22×D12C2212yes—
25C22×D12C2211yescharacteristic
26C22×D12C2212yes—
27C22×C4×D3C2211yescharacteristic
28C22×C4×D3C2212yes—
29C22×C4×D3C2212yes—
30C22×C4×D3C2211yescharacteristic
31C22×Dic6C2211yescharacteristic
32C3×C22⋊Q8C2218yescomplemented
33C2×C3×C4⋊C4C2211yescharacteristic
34C2×C3×C4⋊C4C2212yes—
35C2×C3×C22⋊C4C2212yes—
36#946C2218yescomplemented
37#930C2218yescomplemented
38C2×(C2×D3)⋊C4C2212yes—
39C2×(C2×D3)⋊C4C2212yes—
40C2×(C2×D3)⋊C4C2212yes—
41C2×C4⋊Dic3C2211yescharacteristic
42C2×Dic3⋊C4C2212yes—
43C2×Dic3⋊C4C2212yes—
44#896C22116yescomplemented
45#895C2218yescomplemented
46#895C22116yescomplemented
47D3×C4⋊C4C2218yescomplemented
48#884C22116yescomplemented
49#884C22116yescomplemented
50C22×#41—31no—
51C22×#41—31no—
52#537—316no—
53C2×C22⋊Q8—31no—
54C2×C22⋊Q8—32no—
55C2×C22⋊Q8D311yescharacteristic, complemented
56C2×#121—31no—
57C2×#121—31no—
58C2×#121—32no—
59C2×#121—32no—
60C2×#121—32no—
61C22×C4⋊C4—31no—
62C24×C3C2311yescharacteristic
63C23×D3C2312yes—
64C23×D3C2311yescharacteristic
65C23×D3C2311yescharacteristic
66C2×C3×Q8—24no—
67C2×C3×Q8C2314yes—
68C22×C3×C4—24no—
69C22×C3×C4C2312yes—
70C22×C3×C4C2312yes—
71C22×C3×C4C2311yescharacteristic
72C22×C3×C4C2314yes—
73C22×C3×C4C2311yescharacteristic
74C2×#67—24no—
75C2×#67—24no—
76C2×#67—44no—
77C2×#67C2314yes—
78C2×#67C2314yes—
79C2×#67C2318yes—
80C22×Dic3C2312yes—
81C22×Dic3C2311yescharacteristic
82C22×Dic3C2311yescharacteristic
83#239—232no—
84#235—232no—
85C2×D12—24no—
86C2×D12—28no—
87C2×D12—24no—
88C2×D12—44no—
89C2×D12C2318yes—
90C2×D12C2314yes—
91C2×D12C2314yes—
92C2×D12C2318yes—
93C2×C4×D3—24no—
94C2×C4×D3—28no—
95C2×C4×D3—44no—
96C2×C4×D3D418yescomplemented
97C2×C4×D3C2318yes—
98C2×C4×D3C2314yes—
99C2×C4×D3C2318yes—
100C2×C4×D3C2314yes—
101C2×Dic6—24no—
102C2×Dic6C2314yes—
103C3×C4⋊C4C2314yes—
104C3×C4⋊C4C2318yes—
105C3×C22⋊C4C2318yes—
106(C2×D3)⋊C4C2318yes—
107(C2×D3)⋊C4C2318yes—
108(C2×D3)⋊C4C2318yes—
109C4⋊Dic3C2314yes—
110Dic3⋊C4C2318yes—
111Dic3⋊C4C2318yes—
112C2×#41—38no—
113C2×#41—38no—
114C2×#41—68no—
115C2×#41—68no—
116C22×Q8—31no—
117C22×Q8C2×D311yescharacteristic
118C22×D4—31no—
119C22×D4—31no—
120C22×D4—32no—
121C22×D4—31no—
122C22×D4—32no—
123C22×D4—31no—
124C22×D4—32no—
125C23×C4—31no—
126C23×C4—32no—
127C23×C4—31no—
128C23×C4—32no—
129C23×C4C2×D311yescharacteristic
130C22⋊Q8—38no—
131C22⋊Q8—316no—
132C22⋊Q8C2×D318yescomplemented
133#121—38no—
134#121—38no—
135#121—316no—
136#121—316no—
137#121—316no—
138C2×C4⋊C4—32no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—32no—
141C2×C4⋊C4—38no—
142C2×C4⋊C4C2×D311yescharacteristic
143C2×C4⋊C4C2×D312yes—
144C2×C22⋊C4—32no—
145C2×C22⋊C4—32no—
146C2×C22⋊C4—32no—
147C2×C22⋊C4C2×D312yes—
148C23×C3—22no—
149C23×C3—22no—
150C23×C3—22no—
151C23×C3C2412yes—
152C23×C3C2411yescharacteristic
153C22×D3—22no—
154C22×D3—22no—
155C22×D3—24no—
156C22×D3—22no—
157C22×D3—22no—
158C22×D3—24no—
159C22×D3—24no—
160C22×D3—22no—
161C22×D3C2×D414yes—
162C22×D3C2414yes—
163C22×D3C2412yes—
164C22×D3C2412yes—
165C3×Q8—28no—
166C2×C3×C4—22no—
167C2×C3×C4—24no—
168C2×C3×C4—24no—
169C2×C3×C4—28no—
170C2×C3×C4C2×D414yes—
171C2×C3×C4C2414yes—
172C2×C3×C4C2414yes—
173C2×C3×C4C2412yes—
174C2×C3×C4C2412yes—
175#67—28no—
176#67—28no—
177#67—48no—
178C2×Dic3—22no—
179C2×Dic3—24no—
180C2×Dic3C2×D414yes—
181C2×Dic3C2414yes—
182C2×Dic3C2412yes—
183C2×Dic3C2412yes—
184D12—28no—
185D12—216no—
186D12—28no—
187D12—48no—
188C4×D3—28no—
189C4×D3—216no—
190C4×D3—48no—
191C4×D3C2×D4116yescomplemented
192Dic6—28no—
193C24—31no—
194C24—31no—
195C24—32no—
196C24C22×D311yescharacteristic
197#41—632no—
198#41—632no—
199C2×Q8—24no—
200C2×Q8—34no—
201C2×Q8—64no—
202C2×Q8C22×D314yes—
203C2×D4—34no—
204C2×D4—34no—
205C2×D4—38no—
206C2×D4—34no—
207C2×D4—38no—
208C2×D4—34no—
209C2×D4—38no—
210C2×D4—64no—
211C2×D4—68no—
212C2×D4—64no—
213C2×D4—64no—
214C2×D4—64no—
215C2×D4—124no—
216C2×D4—124no—
217C22×C4—24no—
218C22×C4—31no—
219C22×C4—38no—
220C22×C4—34no—
221C22×C4—32no—
222C22×C4—31no—
223C22×C4—38no—
224C22×C4—34no—
225C22×C4—38no—
226C22×C4—68no—
227C22×C4—64no—
228C22×C4—124no—
229C22×C4C22×D314yes—
230C22×C4C22×D311yescharacteristic
231C22×C4C22×D312yes—
232C22×C4C22×D312yes—
233C22×C4C22×D311yescharacteristic
234C4⋊C4—38no—
235C4⋊C4—34no—
236C4⋊C4—38no—
237C4⋊C4C22×D314yes—
238C4⋊C4C22×D318yes—
239C22⋊C4—38no—
240C22⋊C4—38no—
241C22⋊C4—38no—
242C22⋊C4C22×D318yes—
243C22×C3—22no—
244C22×C3—22no—
245C22×C3—22no—
246C22×C3—44no—
247C22×C3C22×D412yes—
248C22×C3H(5,𝔽2)12yes—
249C22×C32−1+412yes—
250C22×C3C2511yescharacteristic
251C2×D3—28no—
252C2×D3—22no—
253C2×D3—22no—
254C2×D3—24no—
255C2×D3—22no—
256C2×D3—22no—
257C2×D3—24no—
258C2×D3—24no—
259C2×D3—22no—
260C2×D3—48no—
261C2×D3—44no—
262C2×D3C22×D414yes—
263C3×C4—22no—
264C3×C4—24no—
265C3×C4—24no—
266C3×C4C22×D414yes—
267Dic3—22no—
268Dic3—24no—
269Dic3C22×D414yes—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—34no—
274C23—32no—
275C23—32no—
276C23—34no—
277C23—62no—
278C23—62no—
279C23—64no—
280C23—62no—
281C23—62no—
282C23—64no—
283C23—62no—
284C23—64no—
285C23C23×D312yes—
286C23C23×D311yescharacteristic
287Q8—28no—
288Q8—68no—
289D4—68no—
290D4—616no—
291D4—68no—
292D4—68no—
293D4—68no—
294D4—128no—
295D4—128no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—28no—
300C2×C4—34no—
301C2×C4—32no—
302C2×C4—316no—
303C2×C4—34no—
304C2×C4—32no—
305C2×C4—616no—
306C2×C4—62no—
307C2×C4—64no—
308C2×C4—68no—
309C2×C4—128no—
310C2×C4D3×D414yes—
311C2×C4C23×D312yes—
312C2×C4C23×D314yes—
313C2×C4C23×D314yes—
314C2×C4C23×D312yes—
315C2×C3—42no—
316C2×C3#53714yescomplemented
317C2×C3C23×D411yescharacteristic
318C2×C3C2×H(5,𝔽2)11yescharacteristic
319C2×C3C2×2−1+411yescharacteristic
320D3—24no—
321D3—44no—
322D3—42no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—34no—
327C22—44no—
328C22—62no—
329C22—62no—
330C22—68no—
331C22—64no—
332C22—62no—
333C22—62no—
334C22—64no—
335C22—62no—
336C22—64no—
337C22—128no—
338C22—124no—
339C22C2×D3×D412yes—
340C22#100712yes—
341C22#100912yes—
342C22C24×D311yescharacteristic
343C4—22no—
344C4—24no—
345C4—24no—
346C4—34no—
347C4—62no—
348C4—64no—
349C4C2×D3×D414yes—
350C3C2×#53711yescharacteristic, complemented, Sylow
351C2—42no—
352C2—64no—
353C2—124no—
354C2—122no—
355C2#562914yesdirect product factor
356C2C22×D3×D411yescharacteristic
357C2C2×#100711yescharacteristic
358C2C2×#100911yescharacteristic
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×C22⋊Q8C2
socleC23×C3C24
hypercenterC2×C22⋊Q8D3
hypocenterC3C2×#537
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#239C2semidirect
2. C22×#235C2semidirect
3. C2×C3×C22⋊Q8C2semidirect
4. C2×#946C2semidirect
5. C2×#930C2semidirect
6. #5629C2direct (N × H)
7. C2×#896C2semidirect
8. C2×#895C2semidirect
9. C2×#895C2semidirect
10. C2×D3×C4⋊C4C2semidirect
11. C2×#884C2semidirect
12. C2×#884C2semidirect
20. C2×#239C22semidirect
22. C2×#235C22semidirect
32. C3×C22⋊Q8C22semidirect
36. #946C22semidirect
37. #930C22semidirect
44. #896C22semidirect
45. #895C22semidirect
46. #895C22semidirect
47. D3×C4⋊C4C22semidirect
48. #884C22semidirect
49. #884C22semidirect
55. C2×C22⋊Q8D3semidirect
96. C2×C4×D3D4semidirect
132. C22⋊Q8C2×D3semidirect
191. C4×D3C2×D4semidirect
316. C2×C3#537semidirect
350. C3C2×#537semidirect
355. C2#5629direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#2392
C22×#2352
C2×C3×C22⋊Q82
C2×#9462
C2×#9302
#56292
C2×#8962
C2×#8952
C2×D3×C4⋊C42
C2×#8842
C2×#5373