Finite Group Databasebeta

← C2×#5548 #85772

Subgroups — C2×#5548 #85772

Order 384.

1728 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×#235C212yesmaximal, complemented
2C2×C3×C4×D4C211yesmaximal, characteristic, complemented
3C2×#940C211yesmaximal, characteristic, complemented
4C2×#938C211yesmaximal, characteristic, complemented
5C2×C4×#67C212yesmaximal, complemented
6#5548C2116yesmaximal, direct product factor
7C2×#895C211yesmaximal, characteristic, complemented
8C2×#888C211yesmaximal, characteristic, complemented
9C2×#885C212yesmaximal, complemented
10C2×#884C212yesmaximal, complemented
11C2×C4×D12C211yesmaximal, characteristic, complemented
12C2×C42×D3C211yesmaximal, characteristic, complemented
13C2×#532—31nomaximal, Sylow
14C22×C3×D4C2211yescharacteristic
15C23×C3×C4C2212yes—
16C22×#67C2212yes—
17C22×#67C2212yes—
18C22×#67C2212yes—
19C2×#235—216no—
20C2×#235C22116yescomplemented
21C22×D12C2212yes—
22C22×D12C2211yescharacteristic
23C22×C4×D3C2212yes—
24C22×C4×D3C2211yescharacteristic
25C22×C4×D3C2211yescharacteristic
26C22×C4×D3C2211yescharacteristic
27C22×Dic6C2212yes—
28C3×C4×D4C2218yescomplemented
29C2×C3×C4⋊C4C2211yescharacteristic
30C2×C3×C22⋊C4C2212yes—
31C2×C3×C42C2211yescharacteristic
32C2×#217C2212yes—
33#940C2218yescomplemented
34#938C2218yescomplemented
35C4×#67C22116yescomplemented
36C2×(C2×D3)⋊C4C2212yes—
37C2×(C2×D3)⋊C4C2212yes—
38C2×C4⋊Dic3C2211yescharacteristic
39C2×Dic3⋊C4C2212yes—
40C2×C4×Dic3C2211yescharacteristic
41C2×C4×Dic3C2211yescharacteristic
42C2×C4×Dic3C2211yescharacteristic
43#895C2218yescomplemented
44#888C2218yescomplemented
45#885C22116yescomplemented
46#884C22116yescomplemented
47C4×D12C2218yescomplemented
48C42×D3C2218yescomplemented
49C22×#41—32no—
50#532—316no—
51C2×C4⋊Q8—31no—
52C2×#127—31no—
53C2×#124—32no—
54C2×#121—31no—
55C2×#121—31no—
56C2×#121—32no—
57C2×C4×D4—32no—
58C2×C4×D4—31no—
59C2×C4×D4D311yescharacteristic, complemented
60C22×C42—31no—
61C24×C3C2312yes—
62C23×D3C2312yes—
63C23×D3C2311yescharacteristic
64C2×C3×D4—24no—
65C2×C3×D4C2314yes—
66C22×C3×C4—28no—
67C22×C3×C4C2311yescharacteristic
68C22×C3×C4C2312yes—
69C22×C3×C4C2318yes—
70C22×C3×C4C2311yescharacteristic
71C22×C3×C4C2311yescharacteristic
72C2×#67—28no—
73C2×#67—28no—
74C2×#67—28no—
75C2×#67C2318yes—
76C2×#67C2318yes—
77C2×#67C2318yes—
78C22×Dic3C2311yescharacteristic
79C22×Dic3C2312yes—
80C22×Dic3C2311yescharacteristic
81C22×Dic3C2311yescharacteristic
82#235—264no—
83C2×D12—28no—
84C2×D12—24no—
85C2×D12C2314yes—
86C2×D12C2318yes—
87C2×C4×D3—28no—
88C2×C4×D3—24no—
89C2×C4×D3—24no—
90C2×C4×D3D418yescomplemented
91C2×C4×D3C2314yes—
92C2×C4×D3C2314yes—
93C2×C4×D3C2318yes—
94C2×C4×D3C2314yes—
95C2×Dic6—28no—
96C2×Dic6C2318yes—
97C3×C4⋊C4C2314yes—
98C3×C22⋊C4C2318yes—
99C3×C42C2314yes—
100#217C2318yes—
101(C2×D3)⋊C4C2318yes—
102(C2×D3)⋊C4C2318yes—
103C4⋊Dic3C2314yes—
104Dic3⋊C4C2318yes—
105C4×Dic3C2314yes—
106C4×Dic3C2314yes—
107C4×Dic3C2314yes—
108C2×#41—316no—
109C2×#41—616no—
110C22×Q8—32no—
111C22×D4—32no—
112C22×D4—32no—
113C22×D4—32no—
114C22×D4—31no—
115C22×D4—32no—
116C22×D4C2×D311yescharacteristic
117C23×C4—31no—
118C23×C4—31no—
119C23×C4—31no—
120C23×C4—32no—
121C23×C4C2×D312yes—
122C4⋊Q8—38no—
123#127—38no—
124#124—316no—
125#121—38no—
126#121—38no—
127#121—316no—
128C4×D4—316no—
129C4×D4—38no—
130C4×D4C2×D318yescomplemented
131C2×C4⋊C4—32no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4C2×D311yescharacteristic
134C2×C22⋊C4—32no—
135C2×C22⋊C4—32no—
136C2×C22⋊C4—32no—
137C2×C22⋊C4C2×D312yes—
138C2×C42—31no—
139C2×C42—31no—
140C2×C42—31no—
141C2×C42—38no—
142C2×C42C2×D311yescharacteristic
143C23×C3—24no—
144C23×C3—24no—
145C23×C3—24no—
146C23×C3C2414yes—
147C23×C3C2411yescharacteristic
148C22×D3—22no—
149C22×D3—22no—
150C22×D3—24no—
151C22×D3—24no—
152C22×D3—24no—
153C22×D3C2×D414yes—
154C22×D3C2414yes—
155C22×D3C2412yes—
156C3×D4—28no—
157C2×C3×C4—24no—
158C2×C3×C4—22no—
159C2×C3×C4—216no—
160C2×C3×C4C2×D414yes—
161C2×C3×C4#4114yescomplemented
162C2×C3×C4C2412yes—
163C2×C3×C4C2414yes—
164C2×C3×C4C2412yes—
165C2×C3×C4C2412yes—
166#67—216no—
167#67—216no—
168#67—216no—
169C2×Dic3—24no—
170C2×Dic3—22no—
171C2×Dic3C2×D414yes—
172C2×Dic3#4114yescomplemented
173C2×Dic3C2412yes—
174C2×Dic3C2414yes—
175C2×Dic3C2412yes—
176C2×Dic3C2412yes—
177D12—216no—
178D12—28no—
179C4×D3—216no—
180C4×D3—28no—
181C4×D3—28no—
182C4×D3C2×D4116yescomplemented
183Dic6—216no—
184C24—31no—
185C24—32no—
186C24C22×D312yes—
187#41—664no—
188C2×Q8—38no—
189C2×Q8—68no—
190C2×D4—24no—
191C2×D4—38no—
192C2×D4—38no—
193C2×D4—38no—
194C2×D4—34no—
195C2×D4—38no—
196C2×D4—68no—
197C2×D4—68no—
198C2×D4—68no—
199C2×D4—68no—
200C2×D4—64no—
201C2×D4C22×D314yes—
202C22×C4—28no—
203C22×C4—31no—
204C22×C4—31no—
205C22×C4—38no—
206C22×C4—34no—
207C22×C4—31no—
208C22×C4—32no—
209C22×C4—34no—
210C22×C4—34no—
211C22×C4—38no—
212C22×C4—64no—
213C22×C4—64no—
214C22×C4—68no—
215C22×C4C22×D311yescharacteristic
216C22×C4C22×D312yes—
217C22×C4C22×D318yes—
218C22×C4C22×D311yescharacteristic
219C22×C4C22×D311yescharacteristic
220C4⋊C4—38no—
221C4⋊C4—34no—
222C4⋊C4C22×D314yes—
223C22⋊C4—38no—
224C22⋊C4—38no—
225C22⋊C4—38no—
226C22⋊C4C22×D318yes—
227C42—34no—
228C42—34no—
229C42—34no—
230C42C22×D314yes—
231C22×C3—24no—
232C22×C3—24no—
233C22×C3—24no—
234C22×C3—48no—
235C22×C3C22×D412yes—
236C22×C3C2×#4112yes—
237C22×C3C2×#4112yes—
238C22×C3C2511yescharacteristic
239C2×D3—28no—
240C2×D3—22no—
241C2×D3—22no—
242C2×D3—24no—
243C2×D3—24no—
244C2×D3—24no—
245C2×D3—48no—
246C2×D3C22×D414yes—
247C3×C4—24no—
248C3×C4—22no—
249C3×C4C22×D414yes—
250C3×C4C2×#4114yescomplemented
251Dic3—24no—
252Dic3—22no—
253Dic3C22×D414yes—
254Dic3C2×#4114yescomplemented
255C23—24no—
256C23—24no—
257C23—24no—
258C23—34no—
259C23—32no—
260C23—34no—
261C23—62no—
262C23—62no—
263C23—64no—
264C23—64no—
265C23—64no—
266C23C23×D314yes—
267C23C23×D311yescharacteristic
268Q8—616no—
269D4—28no—
270D4—616no—
271D4—616no—
272D4—616no—
273D4—616no—
274D4—68no—
275C2×C4—24no—
276C2×C4—22no—
277C2×C4—216no—
278C2×C4—34no—
279C2×C4—34no—
280C2×C4—32no—
281C2×C4—32no—
282C2×C4—316no—
283C2×C4—32no—
284C2×C4—34no—
285C2×C4—68no—
286C2×C4—62no—
287C2×C4—64no—
288C2×C4—68no—
289C2×C4—616no—
290C2×C4#23514yescomplemented
291C2×C4D3×D414yes—
292C2×C4C23×D312yes—
293C2×C4C23×D314yes—
294C2×C4C23×D312yes—
295C2×C4C23×D312yes—
296C2×C3—44no—
297C2×C3#53214yescomplemented
298C2×C3C23×D411yescharacteristic
299C2×C3C22×#4111yescharacteristic
300C2×C3C22×#4111yescharacteristic
301D3—24no—
302D3—44no—
303C22—24no—
304C22—24no—
305C22—24no—
306C22—34no—
307C22—48no—
308C22—62no—
309C22—62no—
310C22—68no—
311C22—64no—
312C22—64no—
313C22—64no—
314C22—128no—
315C22C2×#23512yes—
316C22C2×D3×D412yes—
317C22D3×#4112yes—
318C22C24×D311yescharacteristic
319C4—24no—
320C4—22no—
321C4—34no—
322C4—34no—
323C4—62no—
324C4—64no—
325C4C2×#23514yescomplemented
326C4C2×D3×D414yes—
327C3C2×#53211yescharacteristic, complemented, Sylow
328C2—44no—
329C2—64no—
330C2—124no—
331C2#554814yesdirect product factor
332C2C22×#23511yescharacteristic
333C2C22×D3×D411yescharacteristic
334C2C2×D3×#4111yescharacteristic
subgroupG/H
centerC22×C4C22×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×C4×D4C2
socleC23×C3C24
hypercenterC2×C4×D4D3
hypocenterC3C2×#532
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×#235C2semidirect
2. C2×C3×C4×D4C2semidirect
3. C2×#940C2semidirect
4. C2×#938C2semidirect
5. C2×C4×#67C2semidirect
6. #5548C2direct (N × H)
7. C2×#895C2semidirect
8. C2×#888C2semidirect
9. C2×#885C2semidirect
10. C2×#884C2semidirect
11. C2×C4×D12C2semidirect
12. C2×C42×D3C2semidirect
20. C2×#235C22semidirect
28. C3×C4×D4C22semidirect
33. #940C22semidirect
34. #938C22semidirect
35. C4×#67C22semidirect
43. #895C22semidirect
44. #888C22semidirect
45. #885C22semidirect
46. #884C22semidirect
47. C4×D12C22semidirect
48. C42×D3C22semidirect
59. C2×C4×D4D3semidirect
90. C2×C4×D3D4semidirect
130. C4×D4C2×D3semidirect
161. C2×C3×C4#41semidirect
172. C2×Dic3#41semidirect
182. C4×D3C2×D4semidirect
250. C3×C4C2×#41semidirect
254. Dic3C2×#41semidirect
290. C2×C4#235semidirect
297. C2×C3#532semidirect
325. C4C2×#235semidirect
327. C3C2×#532semidirect
331. C2#5548direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#2352
C2×C3×C4×D42
C2×#9402
C2×#9382
C2×C4×#672
#55482
C2×#8952
C2×#8882
C2×#8852
C2×#8842
C2×C4×D122
C2×C42×D32
C2×#5323