Finite Group Databasebeta

← C2×#5552 #85754

Subgroups — C2×#5552 #85754

Order 384.

1620 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
1C22×#235C211yesmaximal, characteristic, complemented
2C23×Dic6C211yesmaximal, characteristic, complemented
3C2×C3×C4×D4C211yesmaximal, characteristic, complemented
4C2×#939C212yesmaximal, complemented
5C2×#930C211yesmaximal, characteristic, complemented
6C2×#924C211yesmaximal, characteristic, complemented
7#5552C2116yesmaximal, direct product factor
8C2×#896C212yesmaximal, complemented
9C2×#885C212yesmaximal, complemented
10C2×#67⋊C4C212yesmaximal, complemented
11C2×#875C211yesmaximal, characteristic, complemented
12C2×C4×Dic6C211yesmaximal, characteristic, complemented
13C2×#548—31nomaximal, Sylow
14C22×C3×D4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C23×C3×C4C2211yescharacteristic
17C22×#67C2212yes—
18C22×#67C2212yes—
19C23×Dic3C2212yes—
20C2×#235—28no—
21C2×#235C2218yescomplemented
22C22×D12C2211yescharacteristic
23C22×C4×D3C2212yes—
24C22×Dic6—28no—
25C22×Dic6C2211yescharacteristic
26C22×Dic6C2212yes—
27C22×Dic6C2211yescharacteristic
28C22×Dic6C2218yescomplemented
29C3×C4×D4C2218yescomplemented
30C2×C3×C4⋊C4C2211yescharacteristic
31C2×C3×C22⋊C4C2211yescharacteristic
32C2×C3×C22⋊C4C2211yescharacteristic
33C2×C3×C42C2211yescharacteristic
34C2×#217C2212yes—
35#939C22116yescomplemented
36#930C2218yescomplemented
37C2×(C2×D3)⋊C4C2212yes—
38C2×(C2×D3)⋊C4C2212yes—
39C2×(C2×D3)⋊C4C2212yes—
40C2×C4⋊Dic3C2211yescharacteristic
41#924C2218yescomplemented
42C2×Dic3⋊C4C2212yes—
43C2×Dic3⋊C4C2212yes—
44C2×C4×Dic3C2212yes—
45#896C22116yescomplemented
46#885C22116yescomplemented
47#67⋊C4C22116yescomplemented
48#875C2218yescomplemented
49C4×Dic6C2218yescomplemented
50C22×#41—31no—
51C23×Q8—31no—
52#548—316no—
53C2×#124—32no—
54C2×#124—31no—
55C2×C22⋊Q8—31no—
56C2×C22⋊Q8—32no—
57C2×#121—32no—
58C2×#121—31no—
59C2×C4×Q8—31no—
60C2×C4×D4—32no—
61C2×C4×D4D311yescharacteristic, complemented
62C24×C3C2311yescharacteristic
63C24×C3C2311yescharacteristic
64C23×D3C2312yes—
65C2×C3×D4—24no—
66C2×C3×D4C2314yes—
67C22×C3×C4—24no—
68C22×C3×C4—24no—
69C22×C3×C4C2311yescharacteristic
70C22×C3×C4C2311yescharacteristic
71C22×C3×C4C2311yescharacteristic
72C22×C3×C4C2314yes—
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2314yes—
75C22×C3×C4C2311yescharacteristic
76C2×#67—28no—
77C2×#67—28no—
78C2×#67C2318yes—
79C2×#67C2318yes—
80C22×Dic3—28no—
81C22×Dic3C2318yes—
82C22×Dic3C2312yes—
83C22×Dic3C2312yes—
84C22×Dic3C2312yes—
85#235—232no—
86C2×D12—24no—
87C2×D12C2314yes—
88C2×C4×D3—28no—
89C2×C4×D3C2318yes—
90C2×Dic6—24no—
91C2×Dic6—28no—
92C2×Dic6—232no—
93C2×Dic6D418yescomplemented
94C2×Dic6C2314yes—
95C2×Dic6C2318yes—
96C2×Dic6C2314yes—
97C3×C4⋊C4C2314yes—
98C3×C22⋊C4C2314yes—
99C3×C22⋊C4C2314yes—
100C3×C42C2314yes—
101#217C2318yes—
102(C2×D3)⋊C4C2318yes—
103(C2×D3)⋊C4C2318yes—
104(C2×D3)⋊C4C2318yes—
105C4⋊Dic3C2314yes—
106Dic3⋊C4C2318yes—
107Dic3⋊C4C2318yes—
108C4×Dic3C2318yes—
109C2×#41—38no—
110C2×#41—68no—
111C22×Q8—31no—
112C22×Q8—32no—
113C22×Q8—31no—
114C22×Q8—38no—
115C22×Q8—68no—
116C22×D4—32no—
117C22×D4—32no—
118C22×D4—31no—
119C22×D4C2×D311yescharacteristic
120C23×C4—32no—
121C23×C4—32no—
122C23×C4C2×D311yescharacteristic
123C23×C4C2×D311yescharacteristic
124#124—316no—
125#124—38no—
126C22⋊Q8—38no—
127C22⋊Q8—316no—
128#121—316no—
129#121—38no—
130C4×Q8—38no—
131C4×D4—316no—
132C4×D4C2×D318yescomplemented
133C2×C4⋊C4—32no—
134C2×C4⋊C4—32no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4C2×D311yescharacteristic
137C2×C22⋊C4—32no—
138C2×C22⋊C4—32no—
139C2×C22⋊C4—32no—
140C2×C22⋊C4—32no—
141C2×C22⋊C4C2×D311yescharacteristic
142C2×C22⋊C4C2×D311yescharacteristic
143C2×C42—32no—
144C2×C42C2×D311yescharacteristic
145C23×C3—22no—
146C23×C3—22no—
147C23×C3—22no—
148C23×C3—22no—
149C23×C3—22no—
150C23×C3#4114yescomplemented
151C23×C3C2412yes—
152C23×C3C2412yes—
153C23×C3C2411yescharacteristic
154C22×D3—24no—
155C22×D3—24no—
156C22×D3—24no—
157C22×D3C2414yes—
158C3×D4—28no—
159C2×C3×C4—22no—
160C2×C3×C4—22no—
161C2×C3×C4—22no—
162C2×C3×C4—22no—
163C2×C3×C4—28no—
164C2×C3×C4—28no—
165C2×C3×C4C2×D414yes—
166C2×C3×C4C2412yes—
167C2×C3×C4C2412yes—
168C2×C3×C4C2412yes—
169C2×C3×C4C2412yes—
170C2×C3×C4C2412yes—
171#67—216no—
172#67—216no—
173C2×Dic3—24no—
174C2×Dic3—24no—
175C2×Dic3—216no—
176C2×Dic3C2×D418yes—
177C2×Dic3C2414yes—
178C2×Dic3C2414yes—
179C2×Dic3C2414yes—
180D12—28no—
181C4×D3—216no—
182Dic6—28no—
183Dic6—216no—
184Dic6C2×D4116yescomplemented
185C24—32no—
186C24C22×D311yescharacteristic
187C24C22×D311yescharacteristic
188#41—632no—
189C2×Q8—38no—
190C2×Q8—34no—
191C2×Q8—38no—
192C2×Q8—34no—
193C2×Q8—68no—
194C2×Q8—632no—
195C2×Q8—64no—
196C2×D4—24no—
197C2×D4—38no—
198C2×D4—38no—
199C2×D4—34no—
200C2×D4—68no—
201C2×D4—68no—
202C2×D4—64no—
203C2×D4C22×D314yes—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—32no—
207C22×C4—38no—
208C22×C4—32no—
209C22×C4—32no—
210C22×C4—38no—
211C22×C4—68no—
212C22×C4—68no—
213C22×C4C22×D314yes—
214C22×C4C22×D314yes—
215C22×C4C22×D311yescharacteristic
216C22×C4C22×D311yescharacteristic
217C22×C4C22×D311yescharacteristic
218C22×C4C22×D311yescharacteristic
219C22×C4C22×D311yescharacteristic
220C4⋊C4—38no—
221C4⋊C4—38no—
222C4⋊C4—34no—
223C4⋊C4C22×D314yes—
224C22⋊C4—38no—
225C22⋊C4—38no—
226C22⋊C4—38no—
227C22⋊C4—38no—
228C22⋊C4C22×D314yes—
229C22⋊C4C22×D314yes—
230C42—38no—
231C42C22×D314yes—
232C22×C3—28no—
233C22×C3—22no—
234C22×C3—22no—
235C22×C3—22no—
236C22×C3—22no—
237C22×C3—22no—
238C22×C3—44no—
239C22×C3C22×D412yes—
240C22×C3C2×#4112yes—
241C22×C3C2×#4114yescomplemented
242C22×C32−1+412yes—
243C22×C3C2511yescharacteristic
244C2×D3—24no—
245C2×D3—24no—
246C2×D3—24no—
247C2×D3—48no—
248C3×C4—22no—
249C3×C4—22no—
250C3×C4—22no—
251C3×C4—22no—
252C3×C4C22×D414yes—
253Dic3—24no—
254Dic3—24no—
255Dic3C22×D418yes—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—34no—
262C23—64no—
263C23—64no—
264C23—64no—
265C23#23514yescomplemented
266C23C23×D312yes—
267C23C23×D312yes—
268C23C23×D311yescharacteristic
269Q8—316no—
270Q8—616no—
271Q8—68no—
272D4—28no—
273D4—616no—
274D4—616no—
275D4—68no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—28no—
281C2×C4—28no—
282C2×C4—38no—
283C2×C4—34no—
284C2×C4—34no—
285C2×C4—34no—
286C2×C4—616no—
287C2×C4—616no—
288C2×C4—64no—
289C2×C4—64no—
290C2×C4D3×D414yes—
291C2×C4C23×D312yes—
292C2×C4C23×D312yes—
293C2×C4C23×D312yes—
294C2×C4C23×D312yes—
295C2×C4C23×D312yes—
296C2×C3—24no—
297C2×C3—42no—
298C2×C3#54814yescomplemented
299C2×C3C23×D411yescharacteristic
300C2×C3C22×#4111yescharacteristic
301C2×C3C2×2−1+411yescharacteristic
302D3—44no—
303C22—22no—
304C22—22no—
305C22—28no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—44no—
310C22—64no—
311C22—64no—
312C22—64no—
313C22—128no—
314C22C2×#23512yes—
315C22C2×#23514yescomplemented
316C22C2×D3×D412yes—
317C22#101012yes—
318C22C24×D311yescharacteristic
319C4—22no—
320C4—22no—
321C4—22no—
322C4—22no—
323C4—38no—
324C4—64no—
325C4—64no—
326C4C2×D3×D414yes—
327C3C2×#54811yescharacteristic, complemented, Sylow
328C2—24no—
329C2—42no—
330C2—124no—
331C2#555214yesdirect product factor
332C2C22×#23511yescharacteristic
333C2C22×D3×D411yescharacteristic
334C2C2×#101011yescharacteristic
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×C4×D4C2
socleC23×C3C24
hypercenterC2×C4×D4D3
hypocenterC3C2×#548
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×Dic6C2semidirect
3. C2×C3×C4×D4C2semidirect
4. C2×#939C2semidirect
5. C2×#930C2semidirect
6. C2×#924C2semidirect
7. #5552C2direct (N × H)
8. C2×#896C2semidirect
9. C2×#885C2semidirect
10. C2×#67⋊C4C2semidirect
11. C2×#875C2semidirect
12. C2×C4×Dic6C2semidirect
21. C2×#235C22semidirect
28. C22×Dic6C22semidirect
29. C3×C4×D4C22semidirect
35. #939C22semidirect
36. #930C22semidirect
41. #924C22semidirect
45. #896C22semidirect
46. #885C22semidirect
47. #67⋊C4C22semidirect
48. #875C22semidirect
49. C4×Dic6C22semidirect
61. C2×C4×D4D3semidirect
93. C2×Dic6D4semidirect
132. C4×D4C2×D3semidirect
150. C23×C3#41semidirect
184. Dic6C2×D4semidirect
241. C22×C3C2×#41semidirect
265. C23#235semidirect
298. C2×C3#548semidirect
315. C22C2×#235semidirect
327. C3C2×#548semidirect
331. C2#5552direct (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×Dic62
C2×C3×C4×D42
C2×#9392
C2×#9302
C2×#9242
#55522
C2×#8962
C2×#8852
C2×#67⋊C42
C2×#8752
C2×C4×Dic62
C2×#5483