Finite Group Databasebeta

← C2×#5774 #87190

Subgroups — C2×#5774 #87190

Order 384.

1512 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1009C211yesmaximal, characteristic, complemented
2C2×#1004C211yesmaximal, characteristic, complemented
3C2×C3×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
4C2×#950C211yesmaximal, characteristic, complemented
5C22×#213C211yesmaximal, characteristic, complemented
6#5774C2116yesmaximal, direct product factor
7C2×#916C212yesmaximal, complemented
8C2×#914C212yesmaximal, complemented
9C2×#911C212yesmaximal, complemented
10C2×D3×D8C212yesmaximal, complemented
11C2×#908C211yesmaximal, characteristic, complemented
12C2×#907C211yesmaximal, characteristic, complemented
13C2×#576—31nomaximal, Sylow
14C2×C3×#41C2211yescharacteristic, complemented
15C22×C3×D4C2211yescharacteristic, complemented
16C22×#67—22no—
17#1009—24no—
18#1009C2218yescomplemented
19C2×#239C2212yescomplemented
20#1004—24no—
21#1004C2218yescomplemented
22C2×#237—21no—
23C2×#237C2212yescomplemented
24C2×D3×D4—22no—
25C2×D3×D4—21no—
26C2×D3×D4C2212yescomplemented
27C2×D3×D4C2212yescomplemented
28C2×#235—21no—
29C2×#235C2211yescharacteristic
30C2×#235C2211yescharacteristic, complemented
31C22×D12—21no—
32C22×D12C2211yescharacteristic, complemented
33C3×AGL(1,ℤ/8)C2218yescomplemented
34C2×C3×SD16C2212yescomplemented
35C2×C3×D8C2212yescomplemented
36C2×C3×M16C2211yescharacteristic, complemented
37#950C2218yescomplemented
38C2×#216C2211yescharacteristic, complemented
39C2×#215C2211yescharacteristic, complemented
40C2×#214C2211yescharacteristic, complemented
41C2×#213C2211yescharacteristic, complemented
42C2×#213C2212yescomplemented
43C2×#213C2212yescomplemented
44C2×#213C2218yescomplemented
45C22×C3⋊C8C2211yescharacteristic, complemented
46#916C22116yescomplemented
47#914C22116yescomplemented
48#911C22116yescomplemented
49D3×D8C22116yescomplemented
50#908C2218yescomplemented
51#907C2218yescomplemented
52C2×D24C2212yescomplemented
53C2×#204C2212yescomplemented
54C2×ΓL(1,𝔽25)C2212yescomplemented
55C2×D3×C8C2212yescomplemented
56C2×H(5,𝔽2)—31no—
57C2×H(5,𝔽2)—31no—
58#576—316no—
59C2×AGL(1,ℤ/8)—32no—
60C2×AGL(1,ℤ/8)—32no—
61C2×AGL(1,ℤ/8)—31no—
62C2×AGL(1,ℤ/8)D311yescharacteristic, complemented
63C2×#135—31no—
64C2×#135—32no—
65C22×D8—31no—
66C22×D8—32no—
67C2×#131—31no—
68C24×C3—21no—
69C23×D3—21no—
70C23×D3—22no—
71C23×D3—22no—
72C3×#41—22no—
73C3×#41C2314yescomplemented
74C2×C3×Q8C2311yescharacteristic
75C2×C3×D4—21no—
76C2×C3×D4—21no—
77C2×C3×D4—22no—
78C2×C3×D4C2312yes—
79C2×C3×D4C2311yescharacteristic
80C2×C3×D4C2314yescomplemented
81C22×C3×C4—21no—
82C22×C3×C4C2311yescharacteristic
83C2×#67—21no—
84C2×#67—22no—
85C2×#67—216no—
86C2×#67—22no—
87C2×#67—22no—
88C2×#67D412yes—
89C22×Dic3—22no—
90#239—24no—
91#239C2318yescomplemented
92#237—24no—
93#237—28no—
94#237C2318yescomplemented
95D3×D4—216no—
96D3×D4—28no—
97D3×D4—24no—
98D3×D4—24no—
99D3×D4C2318yescomplemented
100D3×D4C2318yescomplemented
101#235—22no—
102#235—28no—
103#235D414yes—
104#235C2314yes—
105#235C2314yescomplemented
106C2×D12—22no—
107C2×D12—21no—
108C2×D12—22no—
109C2×D12—28no—
110C2×D12D411yescharacteristic
111C2×D12C2312yes—
112C2×D12C2311yescharacteristic
113C2×D12C2314yescomplemented
114C2×C4×D3—21no—
115C2×C4×D3—22no—
116C2×C4×D3C2312yes—
117C2×Dic6D411yescharacteristic
118C2×Dic6C2311yescharacteristic
119C3×SD16C2318yescomplemented
120C3×D8C2318yescomplemented
121C3×M16C2314yescomplemented
122C2×C3×C8C2312yes—
123#216C2314yescomplemented
124#215C2314yescomplemented
125#214C2314yescomplemented
126#213C2314yescomplemented
127#213C2318yescomplemented
128#213C2318yescomplemented
129C2×C3⋊C8C2311yescharacteristic
130C2×C3⋊C8C2311yescharacteristic
131C2×C3⋊C8C2314yescomplemented
132D24C2318yescomplemented
133#204C2318yescomplemented
134ΓL(1,𝔽25)C2318yescomplemented
135D3×C8C2318yescomplemented
136H(5,𝔽2)—38no—
137H(5,𝔽2)—38no—
138H(5,𝔽2)—64no—
139H(5,𝔽2)—64no—
140C2×#41—31no—
141C2×#41—32no—
142C2×#41—32no—
143C2×#41—31no—
144C2×#41—61no—
145C2×#41—61no—
146C2×#41C2×D311yescharacteristic, complemented
147C22×D4—32no—
148C22×D4—32no—
149C22×D4—31no—
150C22×D4—61no—
151C22×D4—62no—
152C22×D4—61no—
153C22×D4—62no—
154C22×D4C2×D311yescharacteristic, complemented
155AGL(1,ℤ/8)—316no—
156AGL(1,ℤ/8)—316no—
157AGL(1,ℤ/8)—38no—
158AGL(1,ℤ/8)C2×D318yescomplemented
159#135—38no—
160#135—316no—
161C2×Q16—31no—
162C2×SD16—31no—
163C2×SD16—31no—
164C2×SD16—32no—
165C2×SD16C2×D312yescomplemented
166C2×D8—32no—
167C2×D8—31no—
168C2×D8—32no—
169C2×D8—38no—
170C2×D8—316no—
171C2×D8—32no—
172C2×D8C2×D312yescomplemented
173#131—38no—
174C2×M16—32no—
175C2×M16C2×D311yescharacteristic, complemented
176C22×C8—31no—
177C22×C8—32no—
178C23×C3—22no—
179C23×C3—21no—
180C23×C3—24no—
181C23×C3—82no—
182C23×C3C2×D411yescharacteristic
183C22×D3—22no—
184C22×D3—21no—
185C22×D3—24no—
186C22×D3—28no—
187C22×D3—28no—
188C22×D3—84no—
189C22×D3—84no—
190C22×D3—82no—
191C22×D3C2×D412yes—
192C3×Q8—21no—
193C3×Q8C2412yes—
194C3×D4—24no—
195C3×D4—21no—
196C3×D4—22no—
197C3×D4—24no—
198C3×D4C2414yes—
199C3×D4C2412yes—
200C2×C3×C4—21no—
201C2×C3×C4—24no—
202C2×C3×C4C2×D411yescharacteristic
203C2×C3×C4C2×D412yes—
204C2×C3×C4C2411yescharacteristic
205C2×C3×C4C2412yes—
206#67—24no—
207#67—28no—
208#67—28no—
209#67—28no—
210#67C2×D418yes—
211C2×Dic3—21no—
212C2×Dic3—28no—
213C2×Dic3C2×D412yes—
214D12—28no—
215D12—24no—
216D12—22no—
217D12—21no—
218D12C2×D414yes—
219D12C2414yes—
220D12C2412yes—
221C4×D3—24no—
222C4×D3—28no—
223C4×D3C2×D414yes—
224C4×D3C2414yes—
225Dic6—21no—
226Dic6C2×D414yes—
227Dic6C2412yes—
228C3×C8C2414yes—
229C3⋊C8C2412yes—
230C3⋊C8C2412yes—
231C24—21no—
232C24—62no—
233C24—62no—
234C24—61no—
235#41—22no—
236#41—34no—
237#41—34no—
238#41—38no—
239#41—38no—
240#41—34no—
241#41—68no—
242#41—64no—
243#41—64no—
244#41—68no—
245#41—62no—
246#41C22×D314yescomplemented
247C2×Q8—31no—
248C2×Q8—31no—
249C2×Q8C22×D311yescharacteristic
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—22no—
253C2×D4—32no—
254C2×D4—38no—
255C2×D4—38no—
256C2×D4—31no—
257C2×D4—32no—
258C2×D4—31no—
259C2×D4—34no—
260C2×D4—68no—
261C2×D4—62no—
262C2×D4—64no—
263C2×D4—64no—
264C2×D4—62no—
265C2×D4—62no—
266C2×D4—616no—
267C2×D4—68no—
268C2×D4—616no—
269C2×D4—62no—
270C2×D4—61no—
271C2×D4—61no—
272C2×D4—62no—
273C2×D4C22×D312yes—
274C2×D4C22×D311yescharacteristic
275C2×D4C22×D314yescomplemented
276C22×C4—21no—
277C22×C4—32no—
278C22×C4—62no—
279C22×C4—62no—
280C22×C4—61no—
281C22×C4C22×D311yescharacteristic
282Q16—34no—
283SD16—34no—
284SD16—34no—
285SD16—38no—
286SD16C22×D318yescomplemented
287D8—38no—
288D8—34no—
289D8—38no—
290D8—38no—
291D8C22×D318yescomplemented
292M16—38no—
293M16C22×D314yescomplemented
294C2×C8—31no—
295C2×C8—31no—
296C2×C8—34no—
297C2×C8—38no—
298C2×C8C22×D312yes—
299C22×C3—22no—
300C22×C3—24no—
301C22×C3—22no—
302C22×C3—42no—
303C22×C3—44no—
304C22×C3—84no—
305C22×C3C22×D412yes—
306C22×C3C22×D411yescharacteristic
307C2×D3—24no—
308C2×D3—24no—
309C2×D3—22no—
310C2×D3—44no—
311C2×D3—42no—
312C2×D3—88no—
313C2×D3—88no—
314C2×D3—84no—
315C2×D3C22×D414yes—
316C3×C4—22no—
317C3×C4C22×D411yescharacteristic
318C3×C4C22×D412yes—
319C3×C4C2511yescharacteristic
320Dic3—22no—
321Dic3C22×D414yes—
322C23—21no—
323C23—22no—
324C23—24no—
325C23—32no—
326C23—68no—
327C23—68no—
328C23—62no—
329C23—61no—
330C23—64no—
331C23—82no—
332C23—242no—
333C23—244no—
334C23—244no—
335C23D3×D411yescharacteristic
336Q8—21no—
337Q8—34no—
338Q8—32no—
339Q8—61no—
340Q8C23×D312yes—
341D4—24no—
342D4—22no—
343D4—24no—
344D4—21no—
345D4—38no—
346D4—34no—
347D4—34no—
348D4—32no—
349D4—68no—
350D4—68no—
351D4—68no—
352D4—68no—
353D4—64no—
354D4—64no—
355D4—61no—
356D4—62no—
357D4C23×D314yes—
358D4C23×D312yes—
359C2×C4—21no—
360C2×C4—24no—
361C2×C4—32no—
362C2×C4—34no—
363C2×C4—34no—
364C2×C4—68no—
365C2×C4—68no—
366C2×C4—61no—
367C2×C4—64no—
368C2×C4D3×D412yes—
369C2×C4D3×D411yescharacteristic
370C2×C4C23×D312yes—
371C2×C4C23×D311yescharacteristic
372C8—32no—
373C8—32no—
374C8C23×D314yes—
375C2×C3—22no—
376C2×C3—42no—
377C2×C3—44no—
378C2×C3#57612yescomplemented
379C2×C3C23×D411yescharacteristic
380D3—24no—
381D3—44no—
382D3—42no—
383C22—22no—
384C22—22no—
385C22—24no—
386C22—34no—
387C22—44no—
388C22—42no—
389C22—64no—
390C22—64no—
391C22—62no—
392C22—84no—
393C22—124no—
394C22—122no—
395C22—244no—
396C22—248no—
397C22—248no—
398C22C2×D3×D412yes—
399C22C2×D3×D411yescharacteristic
400C4—22no—
401C4—34no—
402C4—62no—
403C4C2×D3×D412yes—
404C4C2×D3×D411yescharacteristic
405C4C24×D311yescharacteristic
406C3C2×#57611yescharacteristic, complemented, Sylow
407C2—22no—
408C2—44no—
409C2—42no—
410C2—64no—
411C2—124no—
412C2—122no—
413C2#577412yesdirect product factor
414C2C22×D3×D411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC3×C4C25
FrattiniC4C24×D3
FittingC2×C3×AGL(1,ℤ/8)C2
socleC22×C3C22×D4
hypercenterC2×AGL(1,ℤ/8)D3
hypocenterC3C2×#576
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#1009C2semidirect
2. C2×#1004C2semidirect
3. C2×C3×AGL(1,ℤ/8)C2semidirect
4. C2×#950C2semidirect
5. C22×#213C2semidirect
6. #5774C2direct (N × H)
7. C2×#916C2semidirect
8. C2×#914C2semidirect
9. C2×#911C2semidirect
10. C2×D3×D8C2semidirect
11. C2×#908C2semidirect
12. C2×#907C2semidirect
14. C2×C3×#41C22semidirect
15. C22×C3×D4C22semidirect
18. #1009C22semidirect
19. C2×#239C22semidirect
21. #1004C22semidirect
23. C2×#237C22semidirect
26. C2×D3×D4C22semidirect
27. C2×D3×D4C22semidirect
30. C2×#235C22semidirect
32. C22×D12C22semidirect
33. C3×AGL(1,ℤ/8)C22semidirect
34. C2×C3×SD16C22semidirect
35. C2×C3×D8C22semidirect
36. C2×C3×M16C22semidirect
37. #950C22semidirect
38. C2×#216C22semidirect
39. C2×#215C22semidirect
40. C2×#214C22semidirect
41. C2×#213C22semidirect
42. C2×#213C22semidirect
43. C2×#213C22semidirect
44. C2×#213C22semidirect
45. C22×C3⋊C8C22semidirect
46. #916C22semidirect
47. #914C22semidirect
48. #911C22semidirect
49. D3×D8C22semidirect
50. #908C22semidirect
51. #907C22semidirect
52. C2×D24C22semidirect
53. C2×#204C22semidirect
54. C2×ΓL(1,𝔽25)C22semidirect
55. C2×D3×C8C22semidirect
62. C2×AGL(1,ℤ/8)D3semidirect
73. C3×#41C23semidirect
80. C2×C3×D4C23semidirect
91. #239C23semidirect
94. #237C23semidirect
99. D3×D4C23semidirect
100. D3×D4C23semidirect
105. #235C23semidirect
113. C2×D12C23semidirect
119. C3×SD16C23semidirect
120. C3×D8C23semidirect
121. C3×M16C23semidirect
123. #216C23semidirect
124. #215C23semidirect
125. #214C23semidirect
126. #213C23semidirect
127. #213C23semidirect
128. #213C23semidirect
131. C2×C3⋊C8C23semidirect
132. D24C23semidirect
133. #204C23semidirect
134. ΓL(1,𝔽25)C23semidirect
135. D3×C8C23semidirect
146. C2×#41C2×D3semidirect
154. C22×D4C2×D3semidirect
158. AGL(1,ℤ/8)C2×D3semidirect
165. C2×SD16C2×D3semidirect
172. C2×D8C2×D3semidirect
175. C2×M16C2×D3semidirect
246. #41C22×D3semidirect
275. C2×D4C22×D3semidirect
286. SD16C22×D3semidirect
291. D8C22×D3semidirect
293. M16C22×D3semidirect
378. C2×C3#576semidirect
406. C3C2×#576semidirect
413. C2#5774direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#10092
C2×#10042
C2×C3×AGL(1,ℤ/8)2
C2×#9502
C22×#2132
#57742
C2×#9162
C2×#9142
C2×#9112
C2×D3×D82
C2×#9082
C2×#9072
C2×#5763