Finite Group Databasebeta

← C22×#3205 #10575477

Subgroups — C22×#3205 #10575477

Order 512.

12058 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#532C211yesmaximal, characteristic, complemented
2C23×#127C211yesmaximal, characteristic, complemented
3C2×#3205C2148yesmaximal, direct product factor
4C22×#493C212yesmaximal, complemented
5C22×#486C212yesmaximal, complemented
6C22×#459C214yesmaximal, complemented
7C22×#447C214yesmaximal, complemented
8C22×#431C211yesmaximal, characteristic, complemented
9C23×#41—21no—
10C24×D4—21no—
11C24×D4—21no—
12C24×D4C2212yes—
13C23×D8C2218yescomplemented
14C2×#532—212no—
15C2×#532C22124yescomplemented
16C22×C4⋊Q8C2211yescharacteristic, complemented
17C22×#127—212no—
18C22×#127—21no—
19C22×#127—21no—
20C22×#127C2212yes—
21C22×#127C2212yes—
22C22×#127C22124yescomplemented
23C22×#127C2211yescharacteristic, complemented
24C22×#124—21no—
25C22×#121—21no—
26C22×#121C2212yes—
27C22×C4×D4—21no—
28C22×C4×D4C2212yescomplemented
29C23×C42C2211yescharacteristic
30#3205C221256yesdirect product factor
31C2×#493C22148yescomplemented
32C2×#486C22148yescomplemented
33C2×#459C22196yescomplemented
34C2×#447C22196yescomplemented
35C2×#431C22124yescomplemented
36C22×C4⋊C8C2212yescomplemented
37C22×#102C2218yescomplemented
38C22×C22⋊C8C2212yes—
39C22×C4×C8C2212yescomplemented
40C26—22no—
41C22×#41—224no—
42C22×#41—416no—
43C23×Q8—21no—
44C23×D4—22no—
45C23×D4—22no—
46C23×D4—22no—
47C23×D4—22no—
48C23×D4—22no—
49C23×D4—224no—
50C23×D4—224no—
51C23×D4—212no—
52C23×D4—22no—
53C23×D4—21no—
54C23×D4—21no—
55C23×D4—416no—
56C23×D4—416no—
57C23×D4—416no—
58C23×D4C2314yes—
59C23×D4C2312yes—
60C23×D4C23124yes—
61C24×C4—22no—
62C24×C4D411yescharacteristic
63C24×C4D411yescharacteristic
64C24×C4C2311yescharacteristic
65C22×D8—432no—
66C22×D8C23196yescomplemented
67C23×C8C2314yes—
68#532—296no—
69#532C23164yescomplemented
70C2×C4⋊Q8—26no—
71C2×C4⋊Q8C23112yescomplemented
72C2×#127—212no—
73C2×#127—212no—
74C2×#127—224no—
75C2×#127—224no—
76C2×#127—296no—
77C2×#127—26no—
78C2×#127C23124yes—
79C2×#127C23124yes—
80C2×#127C23164yescomplemented
81C2×#127C23112yescomplemented
82C2×#124—224no—
83C2×#121—224no—
84C2×#121—212no—
85C2×#121C23124yes—
86C2×C4×D4—224no—
87C2×C4×D4—212no—
88C2×C4×D4C23124yescomplemented
89C22×C4⋊C4—21no—
90C22×C4⋊C4C2312yes—
91C22×C22⋊C4—22no—
92C22×C22⋊C4—22no—
93C22×C42D411yescharacteristic
94C22×C42D411yescharacteristic
95C22×C42D4112yes—
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C23112yes—
99#493C231128yescomplemented
100#486C231128yescomplemented
101#459C231256yescomplemented
102#447C231256yescomplemented
103#431C23164yescomplemented
104C2×C4⋊C8C23124yescomplemented
105C2×#102C23196yescomplemented
106C2×C22⋊C8C23124yes—
107C2×C4×C8C23124yescomplemented
108C25—224no—
109C25—22no—
110C25—24no—
111C25—416no—
112C25—416no—
113C25—416no—
114C25C2×D411yescharacteristic
115C2×#41—264no—
116C2×#41—4192no—
117C22×Q8—212no—
118C22×Q8—48no—
119C22×D4—224no—
120C22×D4—26no—
121C22×D4—212no—
122C22×D4—212no—
123C22×D4—224no—
124C22×D4—248no—
125C22×D4—212no—
126C22×D4—224no—
127C22×D4—224no—
128C22×D4—224no—
129C22×D4—224no—
130C22×D4—264no—
131C22×D4—264no—
132C22×D4—216no—
133C22×D4—232no—
134C22×D4—496no—
135C22×D4—416no—
136C22×D4—416no—
137C22×D4—496no—
138C22×D4—4192no—
139C22×D4—416no—
140C22×D4—416no—
141C22×D4—416no—
142C22×D4—48no—
143C22×D4—4192no—
144C22×D4—48no—
145C22×D4—416no—
146C22×D4C24112yes—
147C22×D4C24124yes—
148C22×D4C24132yes—
149C23×C4—28no—
150C23×C4—28no—
151C23×C4—22no—
152C23×C4—224no—
153C23×C4—416no—
154C23×C4D8116yescomplemented
155C23×C4C2×D4112yes—
156C23×C4C2×D416yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D4112yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2416yes—
164C23×C4C2411yescharacteristic
165C2×D8—4192no—
166C2×D8C241128yescomplemented
167C22×C8—216no—
168C22×C8C24124yes—
169C4⋊Q8—224no—
170C4⋊Q8C24116yescomplemented
171#127—248no—
172#127—248no—
173#127—264no—
174#127—264no—
175#127—224no—
176#127C24132yes—
177#127C24132yes—
178#127C24116yescomplemented
179#124—264no—
180#121—264no—
181#121—248no—
182#121C24132yes—
183C4×D4—264no—
184C4×D4—248no—
185C4×D4C24132yescomplemented
186C2×C4⋊C4—212no—
187C2×C4⋊C4—26no—
188C2×C4⋊C4C24112yes—
189C2×C22⋊C4—224no—
190C2×C22⋊C4—224no—
191C2×C42C2×D4112yes—
192C2×C42C2×D416yes—
193C2×C42C2×D4112yes—
194C2×C42C2×D416yes—
195C2×C42C2×D4148yes—
196C2×C42C2416yes—
197C2×C42C2416yes—
198C2×C42C24116yes—
199C4⋊C8C24132yescomplemented
200#102C241128yescomplemented
201C22⋊C8C24132yes—
202C4×C8C24132yescomplemented
203C24—24no—
204C24—24no—
205C24—212no—
206C24—232no—
207C24—224no—
208C24—496no—
209C24—496no—
210C24—416no—
211C24—48no—
212C24—48no—
213C24—496no—
214C24—416no—
215C24—416no—
216C24—48no—
217C24—8128no—
218C24C2×D818yes—
219C24C22×D416yes—
220C24C22×D411yescharacteristic
221#41—4256no—
222C2×Q8—216no—
223C2×Q8—448no—
224C2×D4—232no—
225C2×D4—212no—
226C2×D4—216no—
227C2×D4—216no—
228C2×D4—232no—
229C2×D4—224no—
230C2×D4—232no—
231C2×D4—232no—
232C2×D4—232no—
233C2×D4—232no—
234C2×D4—248no—
235C2×D4—296no—
236C2×D4—464no—
237C2×D4—496no—
238C2×D4—496no—
239C2×D4—4192no—
240C2×D4—4256no—
241C2×D4—496no—
242C2×D4—424no—
243C2×D4—448no—
244C2×D4—496no—
245C2×D4—496no—
246C2×D4—448no—
247C2×D4—4256no—
248C2×D4—448no—
249C2×D4—496no—
250C2×D4C25116yes—
251C2×D4C2518yes—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—224no—
255C22×C4—248no—
256C22×C4—224no—
257C22×C4—212no—
258C22×C4—232no—
259C22×C4—48no—
260C22×C4—496no—
261C22×C4C2×D818yes—
262C22×C4C2×D818yes—
263C22×C4C2×D8196yescomplemented
264C22×C4AGL(1,ℤ/8)18yes—
265C22×C4C22×D416yes—
266C22×C4C22×D4112yes—
267C22×C4C22×D416yes—
268C22×C4C22×D4116yes—
269C22×C4C22×D413yes—
270C22×C4C22×D416yes—
271C22×C4C22×D416yes—
272C22×C4C22×D4116yes—
273C22×C4C22×D416yes—
274C22×C4H(5,𝔽2)18yes—
275C22×C4C2514yes—
276C22×C4C2513yes—
277D8—4128no—
278C2×C8—248no—
279C2×C8C25116yes—
280C4⋊C4—216no—
281C4⋊C4—212no—
282C4⋊C4C2518yes—
283C22⋊C4—232no—
284C22⋊C4—232no—
285C42C22×D4116yes—
286C42C22×D4112yes—
287C42C22×D4116yes—
288C42C22×D4112yes—
289C42C2514yes—
290C42C2514yes—
291C23—212no—
292C23—232no—
293C23—212no—
294C23—28no—
295C23—216no—
296C23—464no—
297C23—464no—
298C23—448no—
299C23—424no—
300C23—424no—
301C23—464no—
302C23—448no—
303C23—448no—
304C23—424no—
305C23—8384no—
306C23—832no—
307C23—864no—
308C23#53514yes—
309C23C22×D8124yes—
310C23C22×D814yes—
311C23C2×AGL(1,ℤ/8)14yes—
312C23C23×D414yes—
313C23C23×D413yes—
314Q8—432no—
315D4—216no—
316D4—232no—
317D4—464no—
318D4—464no—
319D4—464no—
320D4—424no—
321D4—448no—
322D4—464no—
323D4—464no—
324D4—432no—
325D4—432no—
326D4—464no—
327C2×C4—212no—
328C2×C4—212no—
329C2×C4—28no—
330C2×C4—224no—
331C2×C4—232no—
332C2×C4—28no—
333C2×C4—424no—
334C2×C4—464no—
335C2×C4#535112yes—
336C2×C4C22×D8124yes—
337C2×C4C22×D8124yes—
338C2×C4C22×D8164yescomplemented
339C2×C4C2×AGL(1,ℤ/8)124yes—
340C2×C4C23×D414yes—
341C2×C4C23×D414yes—
342C2×C4C23×D413yes—
343C2×C4C23×D414yes—
344C2×C4C23×D414yes—
345C2×C4C23×D414yes—
346C2×C4C2×H(5,𝔽2)112yes—
347C2×C4C2611yescharacteristic
348C8—216no—
349C22—248no—
350C22—24no—
351C22—24no—
352C22—48no—
353C22—416no—
354C22—416no—
355C22—48no—
356C22—416no—
357C22—48no—
358C22—8128no—
359C22—848no—
360C22—896no—
361C22#3205116yesdirect product factor
362C22C2×#53516yes—
363C22C23×D818yes—
364C22C23×D816yes—
365C22C22×AGL(1,ℤ/8)16yes—
366C22C24×D411yescharacteristic
367C4—24no—
368C4—24no—
369C4—48no—
370C4C2×#53516yes—
371C4C23×D818yes—
372C4C23×D818yes—
373C4C22×AGL(1,ℤ/8)18yes—
374C4C22×H(5,𝔽2)12yes—
375C2—28no—
376C2—88no—
377C2—816no—
378C2C2×#3205112yesdirect product factor
379C2C22×#53511yescharacteristic
380C2C24×D811yescharacteristic
381C2C23×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC2×C4C26
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#532C2semidirect
2. C23×#127C2semidirect
3. C2×#3205C2direct (N × H)
4. C22×#493C2semidirect
5. C22×#486C2semidirect
6. C22×#459C2semidirect
7. C22×#447C2semidirect
8. C22×#431C2semidirect
13. C23×D8C22semidirect
15. C2×#532C22semidirect
16. C22×C4⋊Q8C22semidirect
22. C22×#127C22semidirect
23. C22×#127C22semidirect
28. C22×C4×D4C22semidirect
30. #3205C22direct (N × H)
31. C2×#493C22semidirect
32. C2×#486C22semidirect
33. C2×#459C22semidirect
34. C2×#447C22semidirect
35. C2×#431C22semidirect
36. C22×C4⋊C8C22semidirect
37. C22×#102C22semidirect
39. C22×C4×C8C22semidirect
66. C22×D8C23semidirect
69. #532C23semidirect
71. C2×C4⋊Q8C23semidirect
80. C2×#127C23semidirect
81. C2×#127C23semidirect
88. C2×C4×D4C23semidirect
99. #493C23semidirect
100. #486C23semidirect
101. #459C23semidirect
102. #447C23semidirect
103. #431C23semidirect
104. C2×C4⋊C8C23semidirect
105. C2×#102C23semidirect
107. C2×C4×C8C23semidirect
154. C23×C4D8semidirect
166. C2×D8C24semidirect
170. C4⋊Q8C24semidirect
178. #127C24semidirect
185. C4×D4C24semidirect
199. C4⋊C8C24semidirect
200. #102C24semidirect
202. C4×C8C24semidirect
263. C22×C4C2×D8semidirect
338. C2×C4C22×D8semidirect
361. C22#3205direct (N × H)
378. C2C2×#3205direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5322
C23×#1272
C2×#32052
C22×#4932
C22×#4862
C22×#4592
C22×#4472
C22×#4312