Finite Group Databasebeta

← C2×#3310 #61127

Subgroups — C2×#3310 #61127

Order 256.

1289 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#552C211yesmaximal, characteristic, complemented
2C2×D42C211yesmaximal, characteristic
3#3310C2116yesmaximal, direct product factor
4C2×#490C211yesmaximal, characteristic, complemented
5C2×#482C211yesmaximal, characteristic, complemented
6C2×#480C211yesmaximal, characteristic, complemented
7C2×#478C211yesmaximal, characteristic, complemented
8C2×#468C211yesmaximal, characteristic, complemented
9C2×#465C211yesmaximal, characteristic, complemented
10C2×#460C211yesmaximal, characteristic, complemented
11C2×#450C211yesmaximal, characteristic, complemented
12C2×#450C211yesmaximal, characteristic, complemented
13C2×#440C211yesmaximal, characteristic, complemented
14C2×#435C211yesmaximal, characteristic, complemented
15C2×#418C211yesmaximal, characteristic, complemented
16C22×#102C211yesmaximal, characteristic, complemented
17C23×D4—21no—
18C23×D4C2211yescharacteristic
19C23×D4C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×SD16C2211yescharacteristic, complemented
22C22×SD16C2211yescharacteristic, complemented
23C22×M16C2211yescharacteristic, complemented
24C23×C8C2211yescharacteristic, complemented
25#552—24no—
26#552C2218yescomplemented
27D42—24no—
28D42C2218yes—
29C2×#127C2211yescharacteristic
30C2×#126—21no—
31C2×#125—21no—
32C2×#125C2211yescharacteristic, complemented
33C2×#123—21no—
34C2×C22⋊Q8C2211yescharacteristic, complemented
35C2×C22⋊Q8C2211yescharacteristic, complemented
36C2×#121—21no—
37C2×#121C2211yescharacteristic
38C2×#121C2211yescharacteristic
39C2×#120—21no—
40C2×#120—21no—
41C2×C4×Q8C2211yescharacteristic, complemented
42C2×C4×D4C2211yescharacteristic
43C2×C4×D4C2211yescharacteristic
44C2×#117—21no—
45C2×#117C2211yescharacteristic, complemented
46C22×C4⋊C4C2211yescharacteristic, complemented
47#490C2218yescomplemented
48#482C2218yescomplemented
49#480C2218yescomplemented
50#478C2218yescomplemented
51#468C2218yescomplemented
52#465C2218yescomplemented
53#460C2218yescomplemented
54#450C2218yescomplemented
55#450C2218yescomplemented
56#440C2218yescomplemented
57#435C2218yescomplemented
58C2×#107C2211yescharacteristic, complemented
59C2×#107C2211yescharacteristic, complemented
60C2×#106C2211yescharacteristic, complemented
61C2×C4⋊C8C2211yescharacteristic, complemented
62#418C2218yescomplemented
63C2×#103C2211yescharacteristic, complemented
64C2×#102C2211yescharacteristic
65C2×#102C2211yescharacteristic
66C2×#102C2211yescharacteristic
67C2×#102C2218yescomplemented
68C2×#102C2211yescharacteristic, complemented
69C2×#102C2211yescharacteristic, complemented
70C2×#102C2211yescharacteristic, complemented
71C2×#102C2211yescharacteristic, complemented
72C2×#102C2211yescharacteristic, complemented
73C2×#102C2211yescharacteristic, complemented
74C2×C22⋊C8C2211yescharacteristic, complemented
75C2×C22⋊C8C2211yescharacteristic, complemented
76C2×#97C2211yescharacteristic, complemented
77C25—21no—
78C25—21no—
79C22×Q8C2311yescharacteristic
80C22×D4—24no—
81C22×D4—24no—
82C22×D4—21no—
83C22×D4—21no—
84C22×D4—28no—
85C22×D4—21no—
86C22×D4—21no—
87C22×D4—21no—
88C22×D4—22no—
89C22×D4—22no—
90C22×D4—21no—
91C22×D4—42no—
92C22×D4—42no—
93C22×D4—48no—
94C22×D4D411yescharacteristic
95C22×D4C2311yescharacteristic
96C22×D4C2311yescharacteristic
97C22×D4C2314yes—
98C22×D4C2314yes—
99C22×D4C2311yescharacteristic
100C23×C4—21no—
101C23×C4—21no—
102C23×C4C2311yescharacteristic
103C23×C4C2311yescharacteristic
104C2×SD16—24no—
105C2×SD16—42no—
106C2×SD16—42no—
107C2×SD16C2314yes—
108C2×SD16C2314yescomplemented
109C2×SD16C2314yescomplemented
110C2×M16—42no—
111C2×M16C2314yescomplemented
112C22×C8—42no—
113C22×C8C2311yescharacteristic
114C22×C8C2311yescharacteristic
115C22×C8C2311yescharacteristic
116C22×C8C2311yescharacteristic
117C22×C8C2314yescomplemented
118#127—22no—
119#127C2314yes—
120#126—28no—
121#125—28no—
122#125—22no—
123#125C2314yescomplemented
124#123—28no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8C2314yescomplemented
128C22⋊Q8C2314yescomplemented
129#121—22no—
130#121—22no—
131#121—28no—
132#121C2314yes—
133#121C2314yes—
134#120—28no—
135#120—28no—
136C4×Q8—22no—
137C4×Q8C2314yescomplemented
138C4×D4—22no—
139C4×D4D414yes—
140C4×D4C2314yes—
141C4×D4C2314yes—
142#117—22no—
143#117—28no—
144#117C2314yescomplemented
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4D411yescharacteristic
153C2×C4⋊C4C2311yescharacteristic
154C2×C4⋊C4C2311yescharacteristic
155C2×C4⋊C4C2311yescharacteristic
156C2×C4⋊C4C2311yescharacteristic
157C2×C4⋊C4C2314yescomplemented
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4D411yescharacteristic
165C2×C22⋊C4D411yescharacteristic
166C2×C42—21no—
167C2×C42—21no—
168C2×C42C2311yescharacteristic
169#107C2314yescomplemented
170#107C2314yescomplemented
171#106C2314yescomplemented
172C4⋊C8C2314yescomplemented
173#103C2314yescomplemented
174#102C2314yes—
175#102C2314yes—
176#102C2314yes—
177#102C2314yescomplemented
178#102C2314yescomplemented
179#102C2314yescomplemented
180#102C2314yescomplemented
181#102C2314yescomplemented
182#102C2314yescomplemented
183C22⋊C8C2314yescomplemented
184C22⋊C8C2314yescomplemented
185#97C2314yescomplemented
186C24—21no—
187C24—21no—
188C24—21no—
189C24—24no—
190C24—24no—
191C24—44no—
192C24—44no—
193C24—44no—
194C24—44no—
195C24—82no—
196C24—82no—
197C24C2×D411yescharacteristic
198C24C2×D411yescharacteristic
199C2×Q8—21no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8C2412yes—
203C2×D4—24no—
204C2×D4—28no—
205C2×D4—28no—
206C2×D4—24no—
207C2×D4—24no—
208C2×D4—21no—
209C2×D4—24no—
210C2×D4—24no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—44no—
221C2×D4—44no—
222C2×D4—48no—
223C2×D4—48no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—432no—
228C2×D4—44no—
229C2×D4—44no—
230C2×D4—82no—
231C2×D4C2×D414yes—
232C2×D4#4114yes—
233C2×D4C2412yes—
234C2×D4C2412yes—
235C2×D4C2412yes—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—24no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4C2×D412yes—
250C22×C4C2×D412yes—
251C22×C4C2×D411yescharacteristic
252C22×C4C2×D411yescharacteristic
253C22×C4C2×D411yescharacteristic
254C22×C4C2×D411yescharacteristic
255C22×C4C2412yes—
256C22×C4C2412yes—
257C22×C4C2411yescharacteristic
258SD16—28no—
259SD16—44no—
260SD16—44no—
261M16—44no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—44no—
267C2×C8C2412yes—
268C2×C8C2412yes—
269C2×C8C2412yes—
270C2×C8C2412yes—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4C2×D414yes—
282C4⋊C4C2412yes—
283C4⋊C4C2412yes—
284C4⋊C4C2412yes—
285C4⋊C4C2412yes—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4C2×D414yes—
293C22⋊C4C2×D414yes—
294C42—24no—
295C42—24no—
296C42C2×D412yes—
297C42C2412yes—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—24no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—48no—
308C23—48no—
309C23—48no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—48no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—84no—
320C23—84no—
321C23—816no—
322C23—816no—
323C23AGL(1,ℤ/8)14yes—
324C23C22×D412yes—
325C23C22×D412yes—
326C23C22×D411yescharacteristic
327Q8—22no—
328Q8—22no—
329Q8—42no—
330D4—28no—
331D4—24no—
332D4—22no—
333D4—22no—
334D4—22no—
335D4—22no—
336D4—22no—
337D4—48no—
338D4—48no—
339D4—48no—
340D4—48no—
341D4—48no—
342D4—42no—
343D4—42no—
344D4—84no—
345D4C2×#4114yes—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—48no—
362C2×C4—42no—
363C2×C4—48no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4C22×D412yes—
369C2×C4C22×D412yes—
370C2×C4C22×D412yes—
371C2×C4C22×D412yes—
372C2×C4C22×D411yescharacteristic
373C2×C4C2×#4112yes—
374C2×C4H(5,𝔽2)12yes—
375C2×C4C2511yescharacteristic
376C8—22no—
377C8—22no—
378C8—22no—
379C8—22no—
380C22—22no—
381C22—28no—
382C22—22no—
383C22—22no—
384C22—24no—
385C22—22no—
386C22—22no—
387C22—44no—
388C22—48no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—816no—
398C22—816no—
399C22—84no—
400C22—84no—
401C22#54612yes—
402C22C2×AGL(1,ℤ/8)14yes—
403C22C2×AGL(1,ℤ/8)12yes—
404C22#57712yes—
405C22C23×D411yescharacteristic
406C4—22no—
407C4—22no—
408C4—22no—
409C4—22no—
410C4—42no—
411C4—42no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4#54612yes—
416C4C22×#4111yescharacteristic
417C4C2×H(5,𝔽2)11yescharacteristic
418C2—24no—
419C2—42no—
420C2—44no—
421C2—82no—
422C2—82no—
423C2#331014yesdirect product factor
424C2C2×#54611yescharacteristic
425C2C22×AGL(1,ℤ/8)11yescharacteristic
426C2C2×#57711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC2×D42C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#552C2semidirect
3. #3310C2direct (N × H)
4. C2×#490C2semidirect
5. C2×#482C2semidirect
6. C2×#480C2semidirect
7. C2×#478C2semidirect
8. C2×#468C2semidirect
9. C2×#465C2semidirect
10. C2×#460C2semidirect
11. C2×#450C2semidirect
12. C2×#450C2semidirect
13. C2×#440C2semidirect
14. C2×#435C2semidirect
15. C2×#418C2semidirect
16. C22×#102C2semidirect
21. C22×SD16C22semidirect
22. C22×SD16C22semidirect
23. C22×M16C22semidirect
24. C23×C8C22semidirect
26. #552C22semidirect
32. C2×#125C22semidirect
34. C2×C22⋊Q8C22semidirect
35. C2×C22⋊Q8C22semidirect
41. C2×C4×Q8C22semidirect
45. C2×#117C22semidirect
46. C22×C4⋊C4C22semidirect
47. #490C22semidirect
48. #482C22semidirect
49. #480C22semidirect
50. #478C22semidirect
51. #468C22semidirect
52. #465C22semidirect
53. #460C22semidirect
54. #450C22semidirect
55. #450C22semidirect
56. #440C22semidirect
57. #435C22semidirect
58. C2×#107C22semidirect
59. C2×#107C22semidirect
60. C2×#106C22semidirect
61. C2×C4⋊C8C22semidirect
62. #418C22semidirect
63. C2×#103C22semidirect
67. C2×#102C22semidirect
68. C2×#102C22semidirect
69. C2×#102C22semidirect
70. C2×#102C22semidirect
71. C2×#102C22semidirect
72. C2×#102C22semidirect
73. C2×#102C22semidirect
74. C2×C22⋊C8C22semidirect
75. C2×C22⋊C8C22semidirect
76. C2×#97C22semidirect
108. C2×SD16C23semidirect
109. C2×SD16C23semidirect
111. C2×M16C23semidirect
117. C22×C8C23semidirect
123. #125C23semidirect
127. C22⋊Q8C23semidirect
128. C22⋊Q8C23semidirect
137. C4×Q8C23semidirect
144. #117C23semidirect
157. C2×C4⋊C4C23semidirect
169. #107C23semidirect
170. #107C23semidirect
171. #106C23semidirect
172. C4⋊C8C23semidirect
173. #103C23semidirect
177. #102C23semidirect
178. #102C23semidirect
179. #102C23semidirect
180. #102C23semidirect
181. #102C23semidirect
182. #102C23semidirect
183. C22⋊C8C23semidirect
184. C22⋊C8C23semidirect
185. #97C23semidirect
423. C2#3310direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5522
C2×D422
#33102
C2×#4902
C2×#4822
C2×#4802
C2×#4782
C2×#4682
C2×#4652
C2×#4602
C2×#4502
C2×#4402
C2×#4352
C2×#4182
C22×#1022