Finite Group Databasebeta

← C2×#28445 #6465529

Subgroups — C2×#28445 #6465529

Order 512.

4018 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#496C212yesmaximal, complemented
2C22×#463C214yesmaximal, complemented
3C2×#2858C211yesmaximal, characteristic, complemented
4C2×C4×#127C211yesmaximal, characteristic, complemented
5#28445C2116yesmaximal, direct product factor
6C2×#2013C212yesmaximal, complemented
7C2×#2011C212yesmaximal, complemented
8C2×#1959C212yesmaximal, complemented
9C2×#1767C211yesmaximal, characteristic, complemented
10C23×SD16C2214yes—
11C23×D8C2214yes—
12C22×C4⋊Q8C2211yescharacteristic
13C22×#127C2211yescharacteristic
14C22×#127C2211yescharacteristic
15C22×#124C2212yes—
16C22×#124C2212yes—
17C22×C4×D4—22no—
18C22×C4×D4C2212yes—
19C2×#496—216no—
20C2×#496C22116yescomplemented
21C2×#463—232no—
22C2×#463C22132yescomplemented
23C22×C4⋊C8C2212yes—
24C22×#103C2214yes—
25C22×#102C2214yes—
26C22×#97C2212yes—
27#2858—24no—
28#2858C2218yescomplemented
29C4×#127—24no—
30C4×#127C2218yescomplemented
31C2×#390—22no—
32C2×#390—22no—
33C2×#390—22no—
34C2×#390C2212yes—
35C2×#384C2211yescharacteristic
36C2×C43C2211yescharacteristic
37#2013C22116yescomplemented
38#2011C22116yescomplemented
39#1959C22116yescomplemented
40#1767C2218yescomplemented
41C2×#336C2212yes—
42C23×Q8C2312yes—
43C23×D4—22no—
44C23×D4—22no—
45C23×D4C2312yes—
46C23×D4C2312yes—
47C24×C4—22no—
48C22×SD16—216no—
49C22×SD16—216no—
50C22×SD16—48no—
51C22×SD16C23116yes—
52C22×D8—216no—
53C22×D8—216no—
54C22×D8—48no—
55C22×D8C23116yes—
56C23×C8C2314yes—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—44no—
59C2×C4⋊Q8C2314yes—
60C2×#127—22no—
61C2×#127—22no—
62C2×#127—24no—
63C2×#127—44no—
64C2×#127D418yescomplemented
65C2×#127C2314yes—
66C2×#127C2314yes—
67C2×#124—24no—
68C2×#124—24no—
69C2×#124—28no—
70C2×#124—216no—
71C2×#124—28no—
72C2×#124C2318yes—
73C2×#124C2318yes—
74C2×C4×D4—24no—
75C2×C4×D4—216no—
76C2×C4×D4—216no—
77C2×C4×D4—416no—
78C2×C4×D4C2318yes—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4C2312yes—
82C22×C22⋊C4—24no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—24no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—22no—
87C22×C42D412yes—
88C22×C42D412yes—
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#496—264no—
93#463—2128no—
94C2×C4⋊C8D4116yescomplemented
95C2×C4⋊C8C2318yes—
96C2×#103—216no—
97C2×#103C23116yes—
98C2×#102—216no—
99C2×#102C23116yes—
100C2×#97—28no—
101C2×#97C2318yes—
102#390—24no—
103#390—216no—
104#390—216no—
105#390—216no—
106#390C2318yes—
107#384—22no—
108#384C2314yes—
109C43D414yes—
110C43C2314yes—
111#336C2318yes—
112C25—22no—
113C25—22no—
114C22×Q8—22no—
115C22×Q8—24no—
116C22×Q8—28no—
117C22×Q8—24no—
118C22×Q8—28no—
119C22×Q8C2414yes—
120C22×D4—216no—
121C22×D4—28no—
122C22×D4—22no—
123C22×D4—28no—
124C22×D4—24no—
125C22×D4—28no—
126C22×D4—22no—
127C22×D4—28no—
128C22×D4—24no—
129C22×D4—24no—
130C22×D4—28no—
131C22×D4—28no—
132C22×D4—48no—
133C22×D4—48no—
134C22×D4—48no—
135C22×D4—48no—
136C22×D4—48no—
137C22×D4C2×D418yes—
138C22×D4C2414yes—
139C22×D4C2414yes—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—28no—
143C23×C4—416no—
144C23×C4—48no—
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2411yescharacteristic
150C2×SD16—232no—
151C2×SD16—232no—
152C2×SD16—416no—
153C2×SD16—464no—
154C2×D8—232no—
155C2×D8—232no—
156C2×D8—416no—
157C2×D8—464no—
158C22×C8—28no—
159C22×C8—28no—
160C22×C8C2×D4116yes—
161C22×C8C2418yes—
162C4⋊Q8—48no—
163C4⋊Q8—48no—
164#127—216no—
165#127—28no—
166#127—48no—
167#127—48no—
168#127C2×D4116yescomplemented
169#124—232no—
170#124—216no—
171#124—216no—
172#124—416no—
173#124—416no—
174C4×D4—232no—
175C4×D4—464no—
176C4×D4—416no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—48no—
183C2×C4⋊C4C2414yes—
184C2×C22⋊C4—216no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—216no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—416no—
190C2×C22⋊C4—48no—
191C2×C22⋊C4—48no—
192C2×C22⋊C4—48no—
193C2×C22⋊C4—416no—
194C2×C42—28no—
195C2×C42—24no—
196C2×C42—44no—
197C2×C42C2×D418yes—
198C2×C42C2×D412yes—
199C2×C42C2×D418yes—
200C2×C42C2×D412yes—
201C2×C42C2×D414yes—
202C2×C42C2×D412yes—
203C2×C42C2×D418yes—
204C2×C42#4114yes—
205C2×C42C2412yes—
206C2×C42C2412yes—
207C2×C42C2412yes—
208C4⋊C8C2×D4132yescomplemented
209#103—232no—
210#102—232no—
211#97—216no—
212C24—28no—
213C24—24no—
214C24—24no—
215C24—44no—
216C24—44no—
217C24—48no—
218C24—48no—
219C24—44no—
220C24—44no—
221C24—44no—
222C24—48no—
223C24—48no—
224C24C22×D411yescharacteristic
225C2×Q8—28no—
226C2×Q8—24no—
227C2×Q8—24no—
228C2×Q8—28no—
229C2×Q8—44no—
230C2×Q8—44no—
231C2×Q8—44no—
232C2×Q8—416no—
233C2×Q8—416no—
234C2×D4—28no—
235C2×D4—232no—
236C2×D4—232no—
237C2×D4—28no—
238C2×D4—28no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—28no—
243C2×D4—28no—
244C2×D4—416no—
245C2×D4—464no—
246C2×D4—416no—
247C2×D4—416no—
248C2×D4—416no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4—44no—
254C2×D4—416no—
255C2×D4—416no—
256C2×D4—416no—
257C2×D4—416no—
258C2×D4—832no—
259C2×D4C22×D418yes—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—432no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—48no—
274C22×C4—416no—
275C22×C4—832no—
276C22×C4#12118yes—
277C22×C4#12714yes—
278C22×C4#12714yes—
279C22×C4AGL(1,ℤ/8)18yes—
280C22×C4C22×D414yes—
281C22×C4C22×D412yes—
282C22×C4C22×D412yes—
283C22×C4C22×D414yes—
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D412yes—
286C22×C4C22×D414yes—
287C22×C4C2×#4112yes—
288C22×C4H(5,𝔽2)14yes—
289C22×C4C2511yescharacteristic
290SD16—464no—
291D8—464no—
292C2×C8—232no—
293C2×C8—28no—
294C2×C8—28no—
295C2×C8C22×D4116yes—
296C4⋊C4—28no—
297C4⋊C4—416no—
298C4⋊C4—416no—
299C4⋊C4—44no—
300C22⋊C4—432no—
301C22⋊C4—416no—
302C22⋊C4—416no—
303C22⋊C4—416no—
304C22⋊C4—432no—
305C42—24no—
306C42—216no—
307C42—24no—
308C42—48no—
309C42#12118yes—
310C42#12718yes—
311C42C22×D414yes—
312C42C22×D418yes—
313C42C2×#4114yes—
314C23—28no—
315C23—44no—
316C23—44no—
317C23—432no—
318C23—416no—
319C23—48no—
320C23—48no—
321C23—44no—
322C23—44no—
323C23—44no—
324C23—48no—
325C23—48no—
326C23—88no—
327C23—88no—
328C23—816no—
329C23—88no—
330C23—88no—
331C23—816no—
332C23—816no—
333C23—88no—
334C23C2×#12112yes—
335C23C2×#12712yes—
336C23#53512yes—
337C23C2×AGL(1,ℤ/8)14yes—
338C23#57514yes—
339C23C23×D411yescharacteristic
340Q8—48no—
341Q8—48no—
342Q8—88no—
343D4—216no—
344D4—464no—
345D4—48no—
346D4—48no—
347D4—48no—
348D4—416no—
349D4—832no—
350D4—88no—
351C2×C4—28no—
352C2×C4—28no—
353C2×C4—24no—
354C2×C4—216no—
355C2×C4—216no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—48no—
366C2×C4—832no—
367C2×C4—88no—
368C2×C4—88no—
369C2×C4C2×#12118yes—
370C2×C4C2×#12112yes—
371C2×C4C2×#12714yes—
372C2×C4C2×#12714yes—
373C2×C4C2×#12712yes—
374C2×C4#53512yes—
375C2×C4D4218yes—
376C2×C4#55018yes—
377C2×C4C2×AGL(1,ℤ/8)18yes—
378C2×C4C23×D411yescharacteristic
379C2×C4C23×D412yes—
380C2×C4C22×#4111yescharacteristic
381C2×C4C2×H(5,𝔽2)12yes—
382C8—216no—
383C22—48no—
384C22—416no—
385C22—832no—
386C22—88no—
387C22—84no—
388C22—84no—
389C22—84no—
390C22—88no—
391C22—84no—
392C22—84no—
393C22—88no—
394C22—1616no—
395C22#254114yes—
396C22#303618yes—
397C22#315014yes—
398C22#315114yes—
399C22#321318yes—
400C22C22×#12111yescharacteristic
401C22C22×#12711yescharacteristic
402C22C2×#53511yescharacteristic
403C22C22×AGL(1,ℤ/8)12yes—
404C22C2×#57512yes—
405C4—28no—
406C4—24no—
407C4—24no—
408C4—28no—
409C4—84no—
410C4—84no—
411C4#254114yes—
412C4C2×D4212yes—
413C4C2×#55012yes—
414C2—88no—
415C2—164no—
416C2#2844518yesdirect product factor
417C2C2×#254111yescharacteristic
418C2C2×#303612yes—
419C2C2×#315011yescharacteristic
420C2C2×#315111yescharacteristic
421C2C2×#321312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#496C2semidirect
2. C22×#463C2semidirect
3. C2×#2858C2semidirect
4. C2×C4×#127C2semidirect
5. #28445C2direct (N × H)
6. C2×#2013C2semidirect
7. C2×#2011C2semidirect
8. C2×#1959C2semidirect
9. C2×#1767C2semidirect
20. C2×#496C22semidirect
22. C2×#463C22semidirect
28. #2858C22semidirect
30. C4×#127C22semidirect
37. #2013C22semidirect
38. #2011C22semidirect
39. #1959C22semidirect
40. #1767C22semidirect
64. C2×#127D4semidirect
94. C2×C4⋊C8D4semidirect
168. #127C2×D4semidirect
208. C4⋊C8C2×D4semidirect
416. C2#28445direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4962
C22×#4632
C2×#28582
C2×C4×#1272
#284452
C2×#20132
C2×#20112
C2×#19592
C2×#17672