Finite Group Databasebeta

← C2×#28487 #6465694

Subgroups — C2×#28487 #6465694

Order 512.

3606 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#495C211yesmaximal, characteristic, complemented
2C22×#470C212yesmaximal, complemented
3C22×#469C212yesmaximal, complemented
4C22×#436C211yesmaximal, characteristic, complemented
5C2×#2540C212yesmaximal, complemented
6#28487C2116yesmaximal, direct product factor
7C2×#2000C214yesmaximal, complemented
8C2×#1983C212yesmaximal, complemented
9C2×C4×#106C211yesmaximal, characteristic, complemented
10C23×Q16C2211yescharacteristic
11C23×SD16C2212yes—
12C23×D8C2211yescharacteristic
13C23×M16C2212yes—
14C22×#124C2212yes—
15C22×C22⋊Q8C2214yes—
16C22×#121C2214yes—
17C22×C4×D4C2211yescharacteristic
18C2×#495—28no—
19C2×#495C2218yescomplemented
20C2×#470—216no—
21C2×#470C22116yescomplemented
22C2×#469—216no—
23C2×#469C22116yescomplemented
24C2×#436—28no—
25C2×#436C2218yescomplemented
26C22×#106C2212yes—
27C22×C4⋊C8C2211yescharacteristic
28C22×#103C2214yes—
29C22×#102C2214yes—
30C22×C22⋊C8C2212yes—
31C22×C4×C8C2211yescharacteristic
32#2540—28no—
33#2540C22116yescomplemented
34C2×#399—22no—
35C2×#392—22no—
36C2×#390—24no—
37C2×C4×C4⋊C4C2212yes—
38#2000C22132yescomplemented
39#1983C22116yescomplemented
40C4×#106C2218yescomplemented
41C2×#340C2212yes—
42C23×Q8C2312yes—
43C23×D4—24no—
44C23×D4D411yescharacteristic
45C23×D4C2312yes—
46C24×C4C2312yes—
47C22×Q16—24no—
48C22×Q16—28no—
49C22×Q16C2314yes—
50C22×SD16—28no—
51C22×SD16—48no—
52C22×SD16C2318yes—
53C22×D8—24no—
54C22×D8—28no—
55C22×D8C2314yes—
56C22×M16—28no—
57C22×M16—28no—
58C22×M16—28no—
59C22×M16C2318yes—
60C23×C8C2312yes—
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×#124—24no—
64C2×#124—28no—
65C2×#124—28no—
66C2×#124C2318yes—
67C2×C22⋊Q8—28no—
68C2×C22⋊Q8—216no—
69C2×C22⋊Q8—48no—
70C2×C22⋊Q8C23116yes—
71C2×#121—28no—
72C2×#121—216no—
73C2×#121—48no—
74C2×#121C23116yes—
75C2×C4×D4—24no—
76C2×C4×D4—24no—
77C2×C4×D4D414yes—
78C2×C4×D4C2314yes—
79C22×C4⋊C4—24no—
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4C2314yes—
82C22×C22⋊C4—24no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—24no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4D412yes—
87C22×C42—22no—
88C22×C42C2311yescharacteristic
89#495—232no—
90#470—264no—
91#469—264no—
92#436—232no—
93C2×#106D4116yescomplemented
94C2×#106C2318yes—
95C2×C4⋊C8—24no—
96C2×C4⋊C8C2314yes—
97C2×#103—216no—
98C2×#103C23116yes—
99C2×#102—216no—
100C2×#102C23116yes—
101C2×C22⋊C8—28no—
102C2×C22⋊C8C2318yes—
103C2×C4×C8D418yescomplemented
104C2×C4×C8C2314yes—
105#399—216no—
106#392—216no—
107#390—232no—
108C4×C4⋊C4—24no—
109C4×C4⋊C4C2318yes—
110C2×#95—22no—
111#340C2318yes—
112C25—22no—
113C25C2×D412yes—
114C22×Q8—22no—
115C22×Q8—28no—
116C22×Q8—24no—
117C22×Q8—24no—
118C22×Q8—28no—
119C22×Q8C2414yes—
120C22×D4—216no—
121C22×D4—24no—
122C22×D4—28no—
123C22×D4—24no—
124C22×D4—22no—
125C22×D4—24no—
126C22×D4—28no—
127C22×D4—44no—
128C22×D4—416no—
129C22×D4—416no—
130C22×D4—88no—
131C22×D4C2×D414yes—
132C22×D4C2414yes—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4C2×D414yes—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2414yes—
146C23×C4C2411yescharacteristic
147C2×Q16—216no—
148C2×Q16—28no—
149C2×Q16—416no—
150C2×SD16—216no—
151C2×SD16—416no—
152C2×SD16—432no—
153C2×D8—216no—
154C2×D8—28no—
155C2×D8—416no—
156C2×M16—216no—
157C2×M16—216no—
158C2×M16—216no—
159C2×M16—432no—
160C22×C8—24no—
161C22×C8—24no—
162C22×C8—24no—
163C22×C8—24no—
164C22×C8C2×D414yes—
165C22×C8C2×D414yes—
166C22×C8C2×D418yes—
167C22×C8C2414yes—
168C22×C8C2412yes—
169C22×C8C2412yes—
170#124—216no—
171#124—216no—
172#124—416no—
173C22⋊Q8—232no—
174C22⋊Q8—416no—
175C22⋊Q8—432no—
176#121—232no—
177#121—416no—
178#121—432no—
179C4×D4—216no—
180C4×D4—28no—
181C4×D4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—216no—
186C2×C4⋊C4—416no—
187C2×C4⋊C4C2×D414yes—
188C2×C4⋊C4C2×D4116yes—
189C2×C4⋊C4C2418yes—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—216no—
192C2×C22⋊C4—216no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—416no—
196C2×C22⋊C4—416no—
197C2×C22⋊C4—48no—
198C2×C22⋊C4—48no—
199C2×C22⋊C4C2×D418yes—
200C2×C42—28no—
201C2×C42—84no—
202C2×C42C2×D412yes—
203C2×C42C2×D414yes—
204C2×C42#4114yes—
205C2×C42C2412yes—
206#106C2×D4132yescomplemented
207C4⋊C8—28no—
208#103—232no—
209#102—232no—
210C22⋊C8—216no—
211C4×C8C2×D4116yescomplemented
212#95—28no—
213C24—24no—
214C24—24no—
215C24—24no—
216C24—28no—
217C24—28no—
218C24—24no—
219C24—44no—
220C24—48no—
221C24—48no—
222C24—44no—
223C24—44no—
224C24C22×D414yes—
225C24C22×D411yescharacteristic
226C2×Q8—28no—
227C2×Q8—24no—
228C2×Q8—24no—
229C2×Q8—28no—
230C2×Q8—44no—
231C2×Q8—48no—
232C2×Q8—48no—
233C2×Q8—48no—
234C2×Q8—416no—
235C2×D4—28no—
236C2×D4—28no—
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—28no—
240C2×D4—48no—
241C2×D4—416no—
242C2×D4—432no—
243C2×D4—432no—
244C2×D4—44no—
245C2×D4—48no—
246C2×D4—48no—
247C2×D4—48no—
248C2×D4—416no—
249C2×D4—816no—
250C2×D4—864no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—216no—
256C22×C4—24no—
257C22×C4—28no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—28no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4—48no—
271C22×C4—48no—
272C22×C4—48no—
273C22×C4—48no—
274C22×C4—44no—
275C22×C4—48no—
276C22×C4#12114yes—
277C22×C4#12114yes—
278C22×C4C22×D414yes—
279C22×C4C22×D412yes—
280C22×C4C22×D412yes—
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D412yes—
283C22×C4C22×D414yes—
284C22×C4C2×#4112yes—
285C22×C4H(5,𝔽2)14yes—
286C22×C4C2511yescharacteristic
287Q16—416no—
288SD16—432no—
289D8—416no—
290M16—432no—
291C2×C8—28no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—48no—
297C2×C8#127116yes—
298C2×C8C22×D414yes—
299C2×C8C22×D414yes—
300C2×C8C22×D418yes—
301C4⋊C4—28no—
302C4⋊C4—216no—
303C4⋊C4—28no—
304C4⋊C4—432no—
305C4⋊C4C22×D4116yes—
306C22⋊C4—216no—
307C22⋊C4—432no—
308C22⋊C4—432no—
309C22⋊C4—416no—
310C22⋊C4—416no—
311C42—88no—
312C42#12118yes—
313C42C22×D414yes—
314C42C2×#4114yes—
315C23—24no—
316C23—24no—
317C23—24no—
318C23—28no—
319C23—28no—
320C23—416no—
321C23—416no—
322C23—48no—
323C23—48no—
324C23—48no—
325C23—44no—
326C23—48no—
327C23—48no—
328C23—44no—
329C23—44no—
330C23—816no—
331C23—816no—
332C23—88no—
333C23—88no—
334C23—88no—
335C23C2×#12112yes—
336C23D4214yes—
337C23#57514yes—
338C23#57714yes—
339C23C23×D411yescharacteristic
340Q8—44no—
341Q8—44no—
342Q8—48no—
343Q8—88no—
344D4—416no—
345D4—44no—
346D4—44no—
347D4—48no—
348D4—864no—
349D4—88no—
350C2×C4—28no—
351C2×C4—28no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—216no—
357C2×C4—24no—
358C2×C4—432no—
359C2×C4—48no—
360C2×C4—48no—
361C2×C4—48no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—416no—
367C2×C4—44no—
368C2×C4—48no—
369C2×C4—88no—
370C2×C4C2×#12114yes—
371C2×C4C2×#12114yes—
372C2×C4C2×#12112yes—
373C2×C4C2×#12714yes—
374C2×C4#53514yes—
375C2×C4D4214yes—
376C2×C4#55018yes—
377C2×C4C23×D411yescharacteristic
378C2×C4C23×D412yes—
379C2×C4C22×#4111yescharacteristic
380C2×C4C2×H(5,𝔽2)12yes—
381C8—24no—
382C8—44no—
383C8C2×#12718yes—
384C22—44no—
385C22—44no—
386C22—48no—
387C22—48no—
388C22—44no—
389C22—816no—
390C22—84no—
391C22—84no—
392C22—88no—
393C22—88no—
394C22—84no—
395C22—1616no—
396C22#254114yes—
397C22#305414yes—
398C22#306014yes—
399C22#3275116yes—
400C22C22×#12111yescharacteristic
401C22C2×D4212yes—
402C22C2×#57512yes—
403C22C2×#57712yes—
404C4—24no—
405C4—24no—
406C4—416no—
407C4—44no—
408C4—84no—
409C4#254114yes—
410C4C22×#12711yescharacteristic
411C4C2×#53511yescharacteristic
412C4C2×#55012yes—
413C2—84no—
414C2—164no—
415C2#2848718yesdirect product factor
416C2C2×#254111yescharacteristic
417C2C2×#305411yescharacteristic
418C2C2×#306011yescharacteristic
419C2C2×#327514yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#495C2semidirect
2. C22×#470C2semidirect
3. C22×#469C2semidirect
4. C22×#436C2semidirect
5. C2×#2540C2semidirect
6. #28487C2direct (N × H)
7. C2×#2000C2semidirect
8. C2×#1983C2semidirect
9. C2×C4×#106C2semidirect
19. C2×#495C22semidirect
21. C2×#470C22semidirect
23. C2×#469C22semidirect
25. C2×#436C22semidirect
33. #2540C22semidirect
38. #2000C22semidirect
39. #1983C22semidirect
40. C4×#106C22semidirect
93. C2×#106D4semidirect
103. C2×C4×C8D4semidirect
206. #106C2×D4semidirect
211. C4×C8C2×D4semidirect
415. C2#28487direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4952
C22×#4702
C22×#4692
C22×#4362
C2×#25402
#284872
C2×#20002
C2×#19832
C2×C4×#1062