Finite Group Databasebeta

← C2×#3319 #61136

Subgroups — C2×#3319 #61136

Order 256.

1159 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#555C211yesmaximal, characteristic, complemented
2C2×#546C211yesmaximal, characteristic, complemented
3#3319C2116yesmaximal, direct product factor
4C2×#489C211yesmaximal, characteristic, complemented
5C2×#485C211yesmaximal, characteristic, complemented
6C2×#482C211yesmaximal, characteristic, complemented
7C2×#474C211yesmaximal, characteristic, complemented
8C2×#468C211yesmaximal, characteristic, complemented
9C2×#466C211yesmaximal, characteristic, complemented
10C2×#463C211yesmaximal, characteristic, complemented
11C2×#449C211yesmaximal, characteristic, complemented
12C2×#447C211yesmaximal, characteristic, complemented
13C2×#442C211yesmaximal, characteristic, complemented
14C2×#435C211yesmaximal, characteristic, complemented
15C2×#418C211yesmaximal, characteristic, complemented
16C2×#416C211yesmaximal, characteristic, complemented
17C22×#41C2211yescharacteristic
18C23×D4C2211yescharacteristic
19C22×SD16C2211yescharacteristic, complemented
20C22×D8C2211yescharacteristic
21C22×D8C2211yescharacteristic
22C22×M16C2211yescharacteristic
23C23×C8C2211yescharacteristic, complemented
24#555—24no—
25#555C2218yescomplemented
26#546—24no—
27#546C2218yescomplemented
28C2×C4⋊Q8C2211yescharacteristic, complemented
29C2×#124—21no—
30C2×#124—21no—
31C2×#124C2211yescharacteristic
32C2×#123—21no—
33C2×C22⋊Q8C2211yescharacteristic
34C2×#121—21no—
35C2×#121—21no—
36C2×#121C2211yescharacteristic
37C2×#121C2211yescharacteristic
38C2×#121C2211yescharacteristic, complemented
39C2×#120—21no—
40C2×C4×D4C2211yescharacteristic
41C2×C4×D4C2211yescharacteristic
42C2×C4×D4C2211yescharacteristic, complemented
43C2×#117—21no—
44C2×#117C2211yescharacteristic
45C2×#117C2211yescharacteristic, complemented
46C22×C22⋊C4—21no—
47#489C2218yescomplemented
48#485C2218yescomplemented
49#482C2218yescomplemented
50#474C2218yescomplemented
51#468C2218yescomplemented
52#466C2218yescomplemented
53#463C2218yescomplemented
54#449C2218yescomplemented
55#447C2218yescomplemented
56#442C2218yescomplemented
57#435C2218yescomplemented
58C2×#107C2211yescharacteristic, complemented
59C2×#107C2211yescharacteristic, complemented
60C2×#106C2211yescharacteristic, complemented
61C2×C4⋊C8C2211yescharacteristic, complemented
62#418C2218yescomplemented
63#416C2218yescomplemented
64C2×#103C2211yescharacteristic, complemented
65C2×#103C2211yescharacteristic, complemented
66C2×#103C2211yescharacteristic, complemented
67C2×#102C2211yescharacteristic
68C2×#102C2211yescharacteristic
69C2×#102C2211yescharacteristic
70C2×#102C2211yescharacteristic
71C2×#102C2211yescharacteristic
72C2×#102C2211yescharacteristic
73C2×#102C2211yescharacteristic, complemented
74C2×C22⋊C8C2211yescharacteristic
75C2×C22⋊C8C2211yescharacteristic, complemented
76C2×#97C2211yescharacteristic, complemented
77C25—21no—
78C2×#41—22no—
79C2×#41—24no—
80C2×#41—42no—
81C2×#41C2314yes—
82C22×Q8—21no—
83C22×Q8C2311yescharacteristic
84C22×D4—24no—
85C22×D4—21no—
86C22×D4—21no—
87C22×D4—21no—
88C22×D4—21no—
89C22×D4—21no—
90C22×D4—22no—
91C22×D4—42no—
92C22×D4D411yescharacteristic
93C22×D4C2311yescharacteristic
94C22×D4C2311yescharacteristic
95C22×D4C2314yes—
96C22×D4C2311yescharacteristic
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4C2311yescharacteristic
101C23×C4C2311yescharacteristic
102C2×SD16—42no—
103C2×SD16C2314yescomplemented
104C2×D8—24no—
105C2×D8—42no—
106C2×D8C2314yes—
107C2×D8C2314yes—
108C2×M16—42no—
109C2×M16C2314yes—
110C22×C8—42no—
111C22×C8C2311yescharacteristic
112C22×C8C2311yescharacteristic
113C22×C8C2311yescharacteristic
114C22×C8C2311yescharacteristic
115C22×C8C2314yescomplemented
116C4⋊Q8—22no—
117C4⋊Q8C2314yescomplemented
118#124—28no—
119#124—28no—
120#124—22no—
121#124C2314yes—
122#123—28no—
123C22⋊Q8—22no—
124C22⋊Q8C2314yes—
125#121—28no—
126#121—22no—
127#121—22no—
128#121—22no—
129#121—28no—
130#121C2314yes—
131#121C2314yes—
132#121C2314yescomplemented
133#120—28no—
134C4×D4—22no—
135C4×D4—22no—
136C4×D4D414yes—
137C4×D4C2314yes—
138C4×D4C2314yes—
139C4×D4C2314yescomplemented
140#117—22no—
141#117—22no—
142#117—28no—
143#117C2314yes—
144#117C2314yescomplemented
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4D411yescharacteristic
148C2×C4⋊C4C2311yescharacteristic
149C2×C4⋊C4C2311yescharacteristic
150C2×C4⋊C4C2311yescharacteristic
151C2×C4⋊C4C2311yescharacteristic
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—28no—
163C2×C22⋊C4D411yescharacteristic
164C2×C22⋊C4D411yescharacteristic
165C2×C42—21no—
166C2×C42—21no—
167C2×C42C2311yescharacteristic
168#107C2314yescomplemented
169#107C2314yescomplemented
170#106C2314yescomplemented
171C4⋊C8C2314yescomplemented
172#103C2314yescomplemented
173#103C2314yescomplemented
174#103C2314yescomplemented
175#102C2314yes—
176#102C2314yes—
177#102C2314yes—
178#102C2314yes—
179#102C2314yes—
180#102C2314yes—
181#102C2314yescomplemented
182C22⋊C8C2314yes—
183C22⋊C8C2314yescomplemented
184#97C2314yescomplemented
185C24—21no—
186C24—21no—
187C24—21no—
188C24—24no—
189C24—44no—
190C24—44no—
191C24—82no—
192C24C2×D411yescharacteristic
193C24C2×D411yescharacteristic
194#41—28no—
195#41—44no—
196#41—48no—
197C2×Q8—21no—
198C2×Q8—24no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—82no—
202C2×Q8C2412yes—
203C2×D4—28no—
204C2×D4—24no—
205C2×D4—24no—
206C2×D4—24no—
207C2×D4—24no—
208C2×D4—21no—
209C2×D4—21no—
210C2×D4—24no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—44no—
218C2×D4—48no—
219C2×D4—42no—
220C2×D4—44no—
221C2×D4—44no—
222C2×D4—82no—
223C2×D4—82no—
224C2×D4—82no—
225C2×D4C2×D414yes—
226C2×D4#4114yes—
227C2×D4C2412yes—
228C2×D4C2412yes—
229C2×D4C2412yes—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—82no—
245C22×C4C2×D412yes—
246C22×C4C2×D412yes—
247C22×C4C2×D411yescharacteristic
248C22×C4C2×D411yescharacteristic
249C22×C4C2×D411yescharacteristic
250C22×C4C2×D411yescharacteristic
251C22×C4C2412yes—
252C22×C4C2412yes—
253C22×C4C2411yescharacteristic
254SD16—44no—
255D8—28no—
256D8—44no—
257M16—44no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—44no—
263C2×C8C2412yes—
264C2×C8C2412yes—
265C2×C8C2412yes—
266C2×C8C2412yes—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4C2×D414yes—
274C4⋊C4C2412yes—
275C4⋊C4C2412yes—
276C4⋊C4C2412yes—
277C4⋊C4C2412yes—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4C2×D414yes—
289C22⋊C4C2×D414yes—
290C42—24no—
291C42—24no—
292C42C2×D412yes—
293C42C2412yes—
294C23—22no—
295C23—22no—
296C23—24no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—48no—
305C23—48no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—84no—
315C23—816no—
316C23C22×D412yes—
317C23C22×D412yes—
318C23C22×D411yescharacteristic
319Q8—22no—
320Q8—22no—
321Q8—42no—
322Q8—84no—
323D4—24no—
324D4—22no—
325D4—22no—
326D4—22no—
327D4—22no—
328D4—22no—
329D4—44no—
330D4—48no—
331D4—48no—
332D4—42no—
333D4—42no—
334D4—84no—
335D4—84no—
336D4—84no—
337D4C2×#4114yes—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—48no—
354C2×C4—48no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—84no—
361C2×C4C22×D412yes—
362C2×C4C22×D412yes—
363C2×C4C22×D412yes—
364C2×C4C22×D412yes—
365C2×C4C22×D411yescharacteristic
366C2×C4C2×#4112yes—
367C2×C4H(5,𝔽2)12yes—
368C2×C4C2511yescharacteristic
369C8—22no—
370C8—22no—
371C8—22no—
372C8—22no—
373C22—22no—
374C22—22no—
375C22—24no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—44no—
381C22—44no—
382C22—48no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—816no—
392C22—84no—
393C22—84no—
394C22#54612yes—
395C22#57512yes—
396C22#57612yes—
397C22C23×D411yescharacteristic
398C4—22no—
399C4—22no—
400C4—22no—
401C4—22no—
402C4—42no—
403C4—42no—
404C4—42no—
405C4—42no—
406C4—42no—
407C4#54612yes—
408C4C22×#4111yescharacteristic
409C4C2×H(5,𝔽2)11yescharacteristic
410C2—42no—
411C2—42no—
412C2—44no—
413C2—82no—
414C2—82no—
415C2#331914yesdirect product factor
416C2C2×#54611yescharacteristic
417C2C2×#57511yescharacteristic
418C2C2×#57611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
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×#555C2semidirect
2. C2×#546C2semidirect
3. #3319C2direct (N × H)
4. C2×#489C2semidirect
5. C2×#485C2semidirect
6. C2×#482C2semidirect
7. C2×#474C2semidirect
8. C2×#468C2semidirect
9. C2×#466C2semidirect
10. C2×#463C2semidirect
11. C2×#449C2semidirect
12. C2×#447C2semidirect
13. C2×#442C2semidirect
14. C2×#435C2semidirect
15. C2×#418C2semidirect
16. C2×#416C2semidirect
19. C22×SD16C22semidirect
23. C23×C8C22semidirect
25. #555C22semidirect
27. #546C22semidirect
28. C2×C4⋊Q8C22semidirect
38. C2×#121C22semidirect
42. C2×C4×D4C22semidirect
45. C2×#117C22semidirect
47. #489C22semidirect
48. #485C22semidirect
49. #482C22semidirect
50. #474C22semidirect
51. #468C22semidirect
52. #466C22semidirect
53. #463C22semidirect
54. #449C22semidirect
55. #447C22semidirect
56. #442C22semidirect
57. #435C22semidirect
58. C2×#107C22semidirect
59. C2×#107C22semidirect
60. C2×#106C22semidirect
61. C2×C4⋊C8C22semidirect
62. #418C22semidirect
63. #416C22semidirect
64. C2×#103C22semidirect
65. C2×#103C22semidirect
66. C2×#103C22semidirect
73. C2×#102C22semidirect
75. C2×C22⋊C8C22semidirect
76. C2×#97C22semidirect
103. C2×SD16C23semidirect
115. C22×C8C23semidirect
117. C4⋊Q8C23semidirect
132. #121C23semidirect
139. C4×D4C23semidirect
144. #117C23semidirect
168. #107C23semidirect
169. #107C23semidirect
170. #106C23semidirect
171. C4⋊C8C23semidirect
172. #103C23semidirect
173. #103C23semidirect
174. #103C23semidirect
181. #102C23semidirect
183. C22⋊C8C23semidirect
184. #97C23semidirect
415. C2#3319direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5552
C2×#5462
#33192
C2×#4892
C2×#4852
C2×#4822
C2×#4742
C2×#4682
C2×#4662
C2×#4632
C2×#4492
C2×#4472
C2×#4422
C2×#4352
C2×#4182
C2×#4162