Finite Group Databasebeta

← C2×#5778 #87195

Subgroups — C2×#5778 #87195

Order 384.

1460 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1009C211yesmaximal, characteristic, complemented
2C2×#1007C211yesmaximal, characteristic, complemented
3C2×C3×#137C211yesmaximal, characteristic, complemented
4C2×#950C211yesmaximal, characteristic, complemented
5C22×#215C211yesmaximal, characteristic, complemented
6#5778C2116yesmaximal, direct product factor
7C2×#919C212yesmaximal, complemented
8C2×#918C212yesmaximal, complemented
9C2×#914C212yesmaximal, complemented
10C2×D3×SD16C212yesmaximal, complemented
11C2×#908C211yesmaximal, characteristic, complemented
12C2×#907C211yesmaximal, characteristic, complemented
13C2×#577—31nomaximal, Sylow
14C2×C3×#41C2211yescharacteristic
15C22×C3×Q8C2211yescharacteristic, complemented
16#1009—24no—
17#1009C2218yescomplemented
18#1007—24no—
19#1007C2218yescomplemented
20C2×#239—21no—
21C2×#239C2212yes—
22C2×#239C2212yescomplemented
23C2×D3×Q8—21no—
24C2×D3×Q8C2212yescomplemented
25C2×D3×D4—22no—
26C2×D3×D4C2212yes—
27C2×#235—22no—
28C2×#235—21no—
29C2×#235C2211yescharacteristic
30C2×#235C2211yescharacteristic, complemented
31C22×D12—21no—
32C22×D12C2211yescharacteristic
33C3×#137C2218yescomplemented
34C2×C3×Q16C2212yescomplemented
35C2×C3×SD16C2212yescomplemented
36C2×C3×M16C2211yescharacteristic, complemented
37#950C2218yescomplemented
38C2×#216C2211yescharacteristic, complemented
39C2×#215C2212yescomplemented
40C2×#215C2212yescomplemented
41C2×#215C2211yescharacteristic, complemented
42C2×#215C2218yescomplemented
43C2×#214C2211yescharacteristic, complemented
44C2×#213C2211yescharacteristic, complemented
45C22×C3⋊C8C2211yescharacteristic, complemented
46#919C22116yescomplemented
47#918C22116yescomplemented
48#914C22116yescomplemented
49D3×SD16C22116yescomplemented
50#908C2218yescomplemented
51#907C2218yescomplemented
52C2×D24C2212yescomplemented
53C2×#204C2212yescomplemented
54C2×ΓL(1,𝔽25)C2212yescomplemented
55C2×D3×C8C2212yescomplemented
56C2×2−1+4—31no—
57C2×H(5,𝔽2)—31no—
58#577—316no—
59C2×#137—32no—
60C2×#137D311yescharacteristic, complemented
61C2×AGL(1,ℤ/8)—32no—
62C2×AGL(1,ℤ/8)—31no—
63C2×#135—31no—
64C2×#135—32no—
65C22×SD16—31no—
66C22×SD16—32no—
67C2×#131—31no—
68C23×D3—21no—
69C23×D3—22no—
70C3×#41—22no—
71C3×#41C2314yes—
72C2×C3×Q8—21no—
73C2×C3×Q8—22no—
74C2×C3×Q8C2312yes—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×Q8C2314yescomplemented
77C2×C3×D4—21no—
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4C2311yescharacteristic
82C2×#67—21no—
83C2×#67—22no—
84C2×#67D412yes—
85#239—28no—
86#239—24no—
87#239—24no—
88#239C2318yes—
89#239C2318yescomplemented
90D3×Q8—24no—
91D3×Q8—28no—
92D3×Q8C2318yescomplemented
93D3×D4—216no—
94D3×D4—24no—
95D3×D4C2318yes—
96#235—22no—
97#235—216no—
98#235—28no—
99#235D414yes—
100#235C2314yes—
101#235C2314yescomplemented
102C2×D12—22no—
103C2×D12—22no—
104C2×D12—21no—
105C2×D12—22no—
106C2×D12—28no—
107C2×D12D411yescharacteristic
108C2×D12C2312yes—
109C2×D12C2311yescharacteristic
110C2×D12C2314yes—
111C2×C4×D3—22no—
112C2×C4×D3—21no—
113C2×C4×D3—22no—
114C2×C4×D3—22no—
115C2×C4×D3C2312yes—
116C2×Dic6—22no—
117C2×Dic6D411yescharacteristic
118C2×Dic6C2311yescharacteristic
119C3×Q16C2318yescomplemented
120C3×SD16C2318yescomplemented
121C3×M16C2314yescomplemented
122C2×C3×C8C2312yes—
123#216C2314yescomplemented
124#215C2318yescomplemented
125#215C2318yescomplemented
126#215C2314yescomplemented
127#214C2314yescomplemented
128#213C2314yescomplemented
129C2×C3⋊C8C2311yescharacteristic
130C2×C3⋊C8C2311yescharacteristic
131C2×C3⋊C8C2314yescomplemented
132D24C2318yescomplemented
133#204C2318yescomplemented
134ΓL(1,𝔽25)C2318yescomplemented
135D3×C8C2318yescomplemented
1362−1+4—38no—
1372−1+4—64no—
138H(5,𝔽2)—38no—
139H(5,𝔽2)—64no—
140C2×#41—31no—
141C2×#41—32no—
142C2×#41—32no—
143C2×#41—31no—
144C2×#41—62no—
145C2×#41—61no—
146C2×#41—61no—
147C2×#41C2×D311yescharacteristic
148C22×Q8—32no—
149C22×Q8—61no—
150C22×Q8C2×D311yescharacteristic, complemented
151C22×D4—32no—
152C22×D4—31no—
153C22×D4—61no—
154C22×D4—62no—
155#137—316no—
156#137C2×D318yescomplemented
157AGL(1,ℤ/8)—316no—
158AGL(1,ℤ/8)—38no—
159#135—38no—
160#135—316no—
161C2×Q16—31no—
162C2×Q16C2×D312yescomplemented
163C2×SD16—32no—
164C2×SD16—32no—
165C2×SD16—31no—
166C2×SD16—38no—
167C2×SD16—31no—
168C2×SD16—316no—
169C2×SD16—32no—
170C2×SD16C2×D312yescomplemented
171C2×D8—31no—
172C2×D8—32no—
173#131—38no—
174C2×M16—32no—
175C2×M16C2×D311yescharacteristic, complemented
176C22×C8—31no—
177C22×C8—32no—
178C23×C3—21no—
179C23×C3C2×D411yescharacteristic
180C22×D3—21no—
181C22×D3—22no—
182C22×D3—24no—
183C22×D3—28no—
184C22×D3—84no—
185C22×D3—82no—
186C22×D3C2×D412yes—
187C3×Q8—22no—
188C3×Q8—24no—
189C3×Q8—21no—
190C3×Q8C2414yes—
191C3×Q8C2412yes—
192C3×D4—24no—
193C3×D4—21no—
194C3×D4C2412yes—
195C2×C3×C4—22no—
196C2×C3×C4—21no—
197C2×C3×C4—24no—
198C2×C3×C4—24no—
199C2×C3×C4C2×D411yescharacteristic
200C2×C3×C4C2×D412yes—
201C2×C3×C4C2411yescharacteristic
202C2×C3×C4C2412yes—
203#67—24no—
204#67—28no—
205#67C2×D418yes—
206C2×Dic3—21no—
207C2×Dic3C2×D412yes—
208D12—28no—
209D12—28no—
210D12—24no—
211D12—22no—
212D12—21no—
213D12C2×D414yes—
214D12C2414yes—
215D12C2412yes—
216C4×D3—28no—
217C4×D3—24no—
218C4×D3—28no—
219C4×D3—28no—
220C4×D3C2×D414yes—
221C4×D3C2414yes—
222Dic6—21no—
223Dic6—28no—
224Dic6C2×D414yes—
225Dic6C2412yes—
226C3×C8C2414yes—
227C3⋊C8C2412yes—
228C3⋊C8C2412yes—
229C24—62no—
230C24—61no—
231#41—22no—
232#41—34no—
233#41—34no—
234#41—38no—
235#41—38no—
236#41—34no—
237#41—616no—
238#41—68no—
239#41—64no—
240#41—64no—
241#41—68no—
242#41—62no—
243#41C22×D314yes—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—31no—
247C2×Q8—38no—
248C2×Q8—31no—
249C2×Q8—62no—
250C2×Q8—64no—
251C2×Q8—68no—
252C2×Q8C22×D312yes—
253C2×Q8C22×D311yescharacteristic
254C2×Q8C22×D314yescomplemented
255C2×D4—21no—
256C2×D4—32no—
257C2×D4—38no—
258C2×D4—31no—
259C2×D4—31no—
260C2×D4—32no—
261C2×D4—34no—
262C2×D4—68no—
263C2×D4—62no—
264C2×D4—62no—
265C2×D4—64no—
266C2×D4—62no—
267C2×D4—616no—
268C2×D4—61no—
269C2×D4—61no—
270C2×D4—62no—
271C2×D4C22×D311yescharacteristic
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—32no—
275C22×C4—62no—
276C22×C4—62no—
277C22×C4—62no—
278C22×C4—61no—
279C22×C4C22×D311yescharacteristic
280Q16—34no—
281Q16C22×D318yescomplemented
282SD16—38no—
283SD16—38no—
284SD16—34no—
285SD16—34no—
286SD16—38no—
287SD16C22×D318yescomplemented
288D8—34no—
289D8—38no—
290M16—38no—
291M16C22×D314yescomplemented
292C2×C8—31no—
293C2×C8—31no—
294C2×C8—34no—
295C2×C8—38no—
296C2×C8C22×D312yes—
297C22×C3—22no—
298C22×C3—22no—
299C22×C3—42no—
300C22×C3C22×D412yes—
301C22×C3C22×D411yescharacteristic
302C2×D3—24no—
303C2×D3—22no—
304C2×D3—24no—
305C2×D3—42no—
306C2×D3—44no—
307C2×D3—88no—
308C2×D3—84no—
309C2×D3C22×D414yes—
310C3×C4—24no—
311C3×C4—22no—
312C3×C4C22×D411yescharacteristic
313C3×C4C22×D412yes—
314C3×C4C2511yescharacteristic
315Dic3—22no—
316Dic3C22×D414yes—
317C23—21no—
318C23—32no—
319C23—68no—
320C23—61no—
321C23—62no—
322C23—64no—
323C23—242no—
324C23—244no—
325C23D3×D411yescharacteristic
326Q8—22no—
327Q8—24no—
328Q8—21no—
329Q8—34no—
330Q8—32no—
331Q8—68no—
332Q8—61no—
333Q8C23×D314yes—
334Q8C23×D312yes—
335D4—24no—
336D4—21no—
337D4—38no—
338D4—34no—
339D4—32no—
340D4—34no—
341D4—68no—
342D4—68no—
343D4—68no—
344D4—64no—
345D4—64no—
346D4—62no—
347D4—61no—
348D4C23×D312yes—
349C2×C4—24no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—32no—
354C2×C4—34no—
355C2×C4—34no—
356C2×C4—68no—
357C2×C4—68no—
358C2×C4—68no—
359C2×C4—61no—
360C2×C4—64no—
361C2×C4D3×D412yes—
362C2×C4D3×D411yescharacteristic
363C2×C4C23×D312yes—
364C2×C4C23×D311yescharacteristic
365C8—32no—
366C8—32no—
367C8C23×D314yes—
368C2×C3—22no—
369C2×C3—42no—
370C2×C3#57712yescomplemented
371C2×C3C23×D411yescharacteristic
372D3—24no—
373D3—42no—
374D3—44no—
375C22—22no—
376C22—22no—
377C22—34no—
378C22—42no—
379C22—64no—
380C22—62no—
381C22—64no—
382C22—124no—
383C22—122no—
384C22—244no—
385C22—248no—
386C22C2×D3×D412yes—
387C22C2×D3×D411yescharacteristic
388C4—22no—
389C4—24no—
390C4—34no—
391C4—62no—
392C4C2×D3×D412yes—
393C4C2×D3×D411yescharacteristic
394C4C24×D311yescharacteristic
395C3C2×#57711yescharacteristic, complemented, Sylow
396C2—22no—
397C2—42no—
398C2—64no—
399C2—124no—
400C2—122no—
401C2#577812yesdirect product factor
402C2C22×D3×D411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC3×C4C25
FrattiniC4C24×D3
FittingC2×C3×#137C2
socleC22×C3C22×D4
hypercenterC2×#137D3
hypocenterC3C2×#577
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#1009C2semidirect
2. C2×#1007C2semidirect
3. C2×C3×#137C2semidirect
4. C2×#950C2semidirect
5. C22×#215C2semidirect
6. #5778C2direct (N × H)
7. C2×#919C2semidirect
8. C2×#918C2semidirect
9. C2×#914C2semidirect
10. C2×D3×SD16C2semidirect
11. C2×#908C2semidirect
12. C2×#907C2semidirect
15. C22×C3×Q8C22semidirect
17. #1009C22semidirect
19. #1007C22semidirect
22. C2×#239C22semidirect
24. C2×D3×Q8C22semidirect
30. C2×#235C22semidirect
33. C3×#137C22semidirect
34. C2×C3×Q16C22semidirect
35. C2×C3×SD16C22semidirect
36. C2×C3×M16C22semidirect
37. #950C22semidirect
38. C2×#216C22semidirect
39. C2×#215C22semidirect
40. C2×#215C22semidirect
41. C2×#215C22semidirect
42. C2×#215C22semidirect
43. C2×#214C22semidirect
44. C2×#213C22semidirect
45. C22×C3⋊C8C22semidirect
46. #919C22semidirect
47. #918C22semidirect
48. #914C22semidirect
49. D3×SD16C22semidirect
50. #908C22semidirect
51. #907C22semidirect
52. C2×D24C22semidirect
53. C2×#204C22semidirect
54. C2×ΓL(1,𝔽25)C22semidirect
55. C2×D3×C8C22semidirect
60. C2×#137D3semidirect
76. C2×C3×Q8C23semidirect
89. #239C23semidirect
92. D3×Q8C23semidirect
101. #235C23semidirect
119. C3×Q16C23semidirect
120. C3×SD16C23semidirect
121. C3×M16C23semidirect
123. #216C23semidirect
124. #215C23semidirect
125. #215C23semidirect
126. #215C23semidirect
127. #214C23semidirect
128. #213C23semidirect
131. C2×C3⋊C8C23semidirect
132. D24C23semidirect
133. #204C23semidirect
134. ΓL(1,𝔽25)C23semidirect
135. D3×C8C23semidirect
150. C22×Q8C2×D3semidirect
156. #137C2×D3semidirect
162. C2×Q16C2×D3semidirect
170. C2×SD16C2×D3semidirect
175. C2×M16C2×D3semidirect
254. C2×Q8C22×D3semidirect
281. Q16C22×D3semidirect
287. SD16C22×D3semidirect
291. M16C22×D3semidirect
370. C2×C3#577semidirect
395. C3C2×#577semidirect
401. C2#5778direct (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×#10072
C2×C3×#1372
C2×#9502
C22×#2152
#57782
C2×#9192
C2×#9182
C2×#9142
C2×D3×SD162
C2×#9082
C2×#9072
C2×#5773