Finite Group Databasebeta

← C2×#3224 #60793

Subgroups — C2×#3224 #60793

Order 256.

1139 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#542C211yesmaximal, characteristic, complemented
2C2×#535C211yesmaximal, characteristic, complemented
3#3224C2116yesmaximal, direct product factor
4C2×#496C211yesmaximal, characteristic, complemented
5C2×#495C211yesmaximal, characteristic, complemented
6C2×#489C211yesmaximal, characteristic, complemented
7C2×#486C211yesmaximal, characteristic, complemented
8C2×#464C211yesmaximal, characteristic, complemented
9C2×#460C211yesmaximal, characteristic, complemented
10C2×#463C211yesmaximal, characteristic, complemented
11C2×#459C211yesmaximal, characteristic, complemented
12C2×#452C211yesmaximal, characteristic, complemented
13C2×#449C211yesmaximal, characteristic, complemented
14C2×#450C211yesmaximal, characteristic, complemented
15C2×#447C211yesmaximal, characteristic, complemented
16C2×#433C211yesmaximal, characteristic, complemented
17C22×#41C2211yescharacteristic
18C23×D4C2211yescharacteristic
19C22×Q16C2211yescharacteristic, complemented
20C22×SD16C2211yescharacteristic
21C22×SD16C2211yescharacteristic
22C22×SD16C2211yescharacteristic, complemented
23C22×SD16C2211yescharacteristic, complemented
24C22×D8C2211yescharacteristic
25C22×D8C2211yescharacteristic
26C22×D8C2211yescharacteristic, complemented
27#542—24no—
28#542C2218yescomplemented
29#535—24no—
30#535C2218yescomplemented
31C2×C4⋊Q8C2211yescharacteristic, complemented
32C2×#127C2211yescharacteristic
33C2×#127C2211yescharacteristic
34C2×#126—21no—
35C2×#124—21no—
36C2×#124C2211yescharacteristic
37C2×#124C2211yescharacteristic
38C2×#124C2211yescharacteristic, complemented
39C2×#123—21no—
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8C2211yescharacteristic
42C2×#121—21no—
43C2×#121—21no—
44C2×#121C2211yescharacteristic
45C2×#120—21no—
46C2×#120—21no—
47C2×C4×Q8C2211yescharacteristic, complemented
48C2×C4×D4C2211yescharacteristic, complemented
49C2×#117C2211yescharacteristic
50#496C2218yescomplemented
51#495C2218yescomplemented
52#489C2218yescomplemented
53#486C2218yescomplemented
54#464C2218yescomplemented
55#463C2218yescomplemented
56#460C2218yescomplemented
57#459C2218yescomplemented
58#452C2218yescomplemented
59#450C2218yescomplemented
60#449C2218yescomplemented
61#447C2218yescomplemented
62#433C2218yescomplemented
63C2×C4⋊C8C2211yescharacteristic, complemented
64C2×C4⋊C8C2211yescharacteristic, complemented
65C2×#103C2211yescharacteristic, complemented
66C2×#103C2211yescharacteristic, complemented
67C2×#102C2211yescharacteristic
68C2×#102C2211yescharacteristic
69C2×#102C2211yescharacteristic
70C2×#102C2211yescharacteristic
71C2×#102C2211yescharacteristic, complemented
72C2×#102C2211yescharacteristic, complemented
73C2×C22⋊C8C2211yescharacteristic
74C2×C22⋊C8C2211yescharacteristic
75C2×#97C2211yescharacteristic, complemented
76C2×C4×C8C2211yescharacteristic, complemented
77C25—21no—
78C2×#41—22no—
79C2×#41—42no—
80C2×#41—42no—
81C2×#41C2314yes—
82C22×Q8—21no—
83C22×Q8C2311yescharacteristic
84C22×Q8C2311yescharacteristic
85C22×D4—21no—
86C22×D4—21no—
87C22×D4—21no—
88C22×D4—22no—
89C22×D4—21no—
90C22×D4—21no—
91C22×D4—21no—
92C22×D4—21no—
93C22×D4—42no—
94C22×D4—42no—
95C22×D4C2311yescharacteristic
96C22×D4C2311yescharacteristic
97C22×D4C2311yescharacteristic
98C22×D4C2311yescharacteristic
99C22×D4C2314yes—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4C2311yescharacteristic
103C2×Q16—42no—
104C2×Q16C2314yescomplemented
105C2×SD16—42no—
106C2×SD16—42no—
107C2×SD16—42no—
108C2×SD16—42no—
109C2×SD16C2314yes—
110C2×SD16C2314yes—
111C2×SD16C2314yescomplemented
112C2×SD16C2314yescomplemented
113C2×D8—42no—
114C2×D8—42no—
115C2×D8—42no—
116C2×D8C2314yes—
117C2×D8C2314yes—
118C2×D8C2314yescomplemented
119C22×C8C2311yescharacteristic
120C22×C8C2311yescharacteristic
121C22×C8C2311yescharacteristic
122C22×C8C2311yescharacteristic
123C4⋊Q8—22no—
124C4⋊Q8C2314yescomplemented
125#127—22no—
126#127—22no—
127#127C2314yes—
128#127C2314yes—
129#126—28no—
130#124—28no—
131#124—22no—
132#124—22no—
133#124—22no—
134#124C2314yes—
135#124C2314yes—
136#124C2314yescomplemented
137#123—28no—
138C22⋊Q8—22no—
139C22⋊Q8—28no—
140C22⋊Q8C2314yes—
141#121—22no—
142#121—28no—
143#121—28no—
144#121C2314yes—
145#120—28no—
146#120—28no—
147C4×Q8—22no—
148C4×Q8C2314yescomplemented
149C4×D4—22no—
150C4×D4C2314yescomplemented
151#117D414yes—
152#117C2314yes—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4D411yescharacteristic
157C2×C4⋊C4D411yescharacteristic
158C2×C4⋊C4C2311yescharacteristic
159C2×C4⋊C4C2311yescharacteristic
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4D411yescharacteristic
170C2×C22⋊C4D411yescharacteristic
171C2×C42—21no—
172C2×C42C2311yescharacteristic
173C2×C42C2311yescharacteristic
174C4⋊C8C2314yescomplemented
175C4⋊C8C2314yescomplemented
176#103C2314yescomplemented
177#103C2314yescomplemented
178#102C2314yes—
179#102C2314yes—
180#102C2314yes—
181#102C2314yes—
182#102C2314yescomplemented
183#102C2314yescomplemented
184C22⋊C8C2314yes—
185C22⋊C8C2314yes—
186#97C2314yescomplemented
187C4×C8C2314yescomplemented
188C24—21no—
189C24—21no—
190C24—21no—
191C24—21no—
192C24—24no—
193C24—44no—
194C24—82no—
195C24—82no—
196C24C2×D411yescharacteristic
197#41—44no—
198#41—44no—
199#41—48no—
200C2×Q8—21no—
201C2×Q8—21no—
202C2×Q8—24no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—82no—
208C2×Q8C2412yes—
209C2×Q8C2412yes—
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—21no—
213C2×D4—24no—
214C2×D4—21no—
215C2×D4—21no—
216C2×D4—21no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—44no—
230C2×D4—44no—
231C2×D4—44no—
232C2×D4—48no—
233C2×D4—44no—
234C2×D4—82no—
235C2×D4—82no—
236C2×D4—82no—
237C2×D4—82no—
238C2×D4—82no—
239C2×D4C2412yes—
240C2×D4C2412yes—
241C2×D4C2412yes—
242C2×D4C2412yes—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—44no—
252C22×C4—82no—
253C22×C4C2×D412yes—
254C22×C4C2×D411yescharacteristic
255C22×C4C2×D411yescharacteristic
256C22×C4C2×D411yescharacteristic
257C22×C4C2×D411yescharacteristic
258C22×C4C2×D411yescharacteristic
259C22×C4C2412yes—
260C22×C4C2411yescharacteristic
261Q16—44no—
262SD16—44no—
263SD16—44no—
264SD16—44no—
265SD16—44no—
266D8—44no—
267D8—44no—
268D8—44no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8C2412yes—
274C2×C8C2412yes—
275C2×C8C2412yes—
276C2×C8C2412yes—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4C2×D414yes—
283C4⋊C4C2×D414yes—
284C4⋊C4C2412yes—
285C4⋊C4C2412yes—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4C2×D414yes—
296C22⋊C4C2×D414yes—
297C42—24no—
298C42C2×D412yes—
299C42C2×D412yes—
300C42C2412yes—
301C42C2412yes—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—48no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—84no—
323C23—84no—
324C23—816no—
325C23C22×D412yes—
326C23C22×D411yescharacteristic
327Q8—22no—
328Q8—22no—
329Q8—22no—
330Q8—22no—
331Q8—42no—
332Q8—42no—
333Q8—84no—
334D4—22no—
335D4—22no—
336D4—22no—
337D4—22no—
338D4—22no—
339D4—22no—
340D4—22no—
341D4—22no—
342D4—48no—
343D4—48no—
344D4—42no—
345D4—42no—
346D4—42no—
347D4—42no—
348D4—84no—
349D4—84no—
350D4—84no—
351D4—84no—
352D4—84no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—48no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—84no—
371C2×C4C22×D412yes—
372C2×C4C22×D412yes—
373C2×C4C22×D412yes—
374C2×C4C22×D412yes—
375C2×C4C22×D412yes—
376C2×C4C22×D411yescharacteristic
377C2×C4H(5,𝔽2)12yes—
378C2×C4H(5,𝔽2)12yes—
379C2×C4C2511yescharacteristic
380C8—22no—
381C8—22no—
382C8—22no—
383C8—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—44no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—816no—
401C22—84no—
402C22—84no—
403C22—84no—
404C22—84no—
405C22#53512yes—
406C22#57612yes—
407C22#57712yes—
408C22C23×D411yescharacteristic
409C4—22no—
410C4—22no—
411C4—22no—
412C4—22no—
413C4—22no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4#53512yes—
419C4C2×H(5,𝔽2)11yescharacteristic
420C4C2×H(5,𝔽2)11yescharacteristic
421C2—42no—
422C2—82no—
423C2—82no—
424C2—82no—
425C2—82no—
426C2#322414yesdirect product factor
427C2C2×#53511yescharacteristic
428C2C2×#57611yescharacteristic
429C2C2×#57711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#542C2semidirect
2. C2×#535C2semidirect
3. #3224C2direct (N × H)
4. C2×#496C2semidirect
5. C2×#495C2semidirect
6. C2×#489C2semidirect
7. C2×#486C2semidirect
8. C2×#464C2semidirect
9. C2×#460C2semidirect
10. C2×#463C2semidirect
11. C2×#459C2semidirect
12. C2×#452C2semidirect
13. C2×#449C2semidirect
14. C2×#450C2semidirect
15. C2×#447C2semidirect
16. C2×#433C2semidirect
19. C22×Q16C22semidirect
22. C22×SD16C22semidirect
23. C22×SD16C22semidirect
26. C22×D8C22semidirect
28. #542C22semidirect
30. #535C22semidirect
31. C2×C4⋊Q8C22semidirect
38. C2×#124C22semidirect
47. C2×C4×Q8C22semidirect
48. C2×C4×D4C22semidirect
50. #496C22semidirect
51. #495C22semidirect
52. #489C22semidirect
53. #486C22semidirect
54. #464C22semidirect
55. #463C22semidirect
56. #460C22semidirect
57. #459C22semidirect
58. #452C22semidirect
59. #450C22semidirect
60. #449C22semidirect
61. #447C22semidirect
62. #433C22semidirect
63. C2×C4⋊C8C22semidirect
64. C2×C4⋊C8C22semidirect
65. C2×#103C22semidirect
66. C2×#103C22semidirect
71. C2×#102C22semidirect
72. C2×#102C22semidirect
75. C2×#97C22semidirect
76. C2×C4×C8C22semidirect
104. C2×Q16C23semidirect
111. C2×SD16C23semidirect
112. C2×SD16C23semidirect
118. C2×D8C23semidirect
124. C4⋊Q8C23semidirect
136. #124C23semidirect
148. C4×Q8C23semidirect
150. C4×D4C23semidirect
174. C4⋊C8C23semidirect
175. C4⋊C8C23semidirect
176. #103C23semidirect
177. #103C23semidirect
182. #102C23semidirect
183. #102C23semidirect
186. #97C23semidirect
187. C4×C8C23semidirect
426. C2#3224direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5422
C2×#5352
#32242
C2×#4962
C2×#4952
C2×#4892
C2×#4862
C2×#4642
C2×#4602
C2×#4632
C2×#4592
C2×#4522
C2×#4492
C2×#4502
C2×#4472
C2×#4332