Finite Group Databasebeta

← C2×#28606 #6462926

Subgroups — C2×#28606 #6462926

Order 512.

2642 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#487C211yesmaximal, characteristic, complemented
2C22×#436C211yesmaximal, characteristic
3C2×#2547C211yesmaximal, characteristic
4C2×#2542C211yesmaximal, characteristic
5#28606C2116yesmaximal, direct product factor
6C2×#2036C214yesmaximal, complemented
7C2×#1988C211yesmaximal, characteristic, complemented
8C2×#1981C211yesmaximal, characteristic, complemented
9C2×#1894C214yesmaximal, complemented
10C2×C4×#107C211yesmaximal, characteristic, complemented
11C23×M16C2212yes—
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×#124—21no—
15C22×C22⋊Q8C2214yes—
16C22×C4×D4C2211yescharacteristic
17C2×#487—28no—
18C2×#487C2218yescomplemented
19C2×#436—28no—
20C2×#436C2218yes—
21C22×#107C2212yes—
22C22×C4⋊C8C2211yescharacteristic
23C22×#103C2212yes—
24C22×#103C2212yes—
25C22×#103C2214yes—
26C22×C22⋊C8C2212yes—
27C22×C4×C8C2211yescharacteristic
28#2547—24no—
29#2547C2218yes—
30#2542—24no—
31#2542C2218yes—
32C2×#399—22no—
33C2×#397—22no—
34C2×#391—21no—
35C2×#390—21no—
36C2×#384—21no—
37C2×#380C2214yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40#2036C22132yescomplemented
41#1988C2218yescomplemented
42#1981C2218yescomplemented
43#1894C22132yescomplemented
44C4×#107C2218yescomplemented
45C2×#340C2214yes—
46C2×#340C2212yes—
47C23×Q8—21no—
48C23×Q8C2312yes—
49C23×D4D411yescharacteristic
50C24×C4C2312yes—
51C22×M16—28no—
52C22×M16—48no—
53C22×M16C2318yes—
54C23×C8C2312yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—24no—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8—24no—
62C2×C4⋊Q8—24no—
63C2×C4⋊Q8C2314yes—
64C2×C4⋊Q8C2314yes—
65C2×#124—28no—
66C2×#124—48no—
67C2×C22⋊Q8—28no—
68C2×C22⋊Q8—216no—
69C2×C22⋊Q8—48no—
70C2×C22⋊Q8C23116yes—
71C2×C4×D4—24no—
72C2×C4×D4—24no—
73C2×C4×D4D414yes—
74C2×C4×D4C2314yes—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—24no—
84C22×C22⋊C4—24no—
85C22×C22⋊C4D412yes—
86C22×C42—21no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89#487—232no—
90#436—232no—
91C2×#107—28no—
92C2×#107C2318yes—
93C2×C4⋊C8—24no—
94C2×C4⋊C8C2314yes—
95C2×#103—28no—
96C2×#103—28no—
97C2×#103—48no—
98C2×#103C2318yes—
99C2×#103C2318yes—
100C2×#103C23116yes—
101C2×C22⋊C8—28no—
102C2×C22⋊C8C2318yes—
103C2×C4×C8D418yes—
104C2×C4×C8C2314yes—
105#399—216no—
106#397—216no—
107#391—28no—
108#390—28no—
109#384—28no—
110#380—28no—
111#380C23116yes—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4C2314yes—
115C4×C4⋊C4C2314yes—
116C2×#95—21no—
117C2×#95—24no—
118C2×#95—21no—
119#340C2318yes—
120#340C23116yes—
121C25C2×D412yes—
122C22×Q8—22no—
123C22×Q8—24no—
124C22×Q8—24no—
125C22×Q8—28no—
126C22×Q8—24no—
127C22×Q8—28no—
128C22×Q8—44no—
129C22×Q8—44no—
130C22×Q8—44no—
131C22×Q8C2414yes—
132C22×D4—24no—
133C22×D4—24no—
134C22×D4—24no—
135C22×D4C2×D414yes—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—28no—
140C23×C4—28no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4C2×D414yes—
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2414yes—
149C23×C4C2411yescharacteristic
150C2×M16—216no—
151C2×M16—416no—
152C2×M16—432no—
153C22×C8—24no—
154C22×C8—24no—
155C22×C8—24no—
156C22×C8—28no—
157C22×C8C2×D414yes—
158C22×C8C2×D414yes—
159C22×C8C2414yes—
160C22×C8C2412yes—
161C22×C8C2412yes—
162C4⋊Q8—28no—
163C4⋊Q8—28no—
164C4⋊Q8—28no—
165C4⋊Q8—28no—
166C4⋊Q8—48no—
167C4⋊Q8—48no—
168#124—432no—
169C22⋊Q8—232no—
170C22⋊Q8—416no—
171C22⋊Q8—432no—
172C4×D4—216no—
173C4×D4—28no—
174C4×D4—28no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—216no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—416no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4C2×D414yes—
193C2×C4⋊C4#4118yes—
194C2×C4⋊C4C2414yes—
195C2×C4⋊C4C2414yes—
196C2×C4⋊C4C2414yes—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—216no—
199C2×C22⋊C4—216no—
200C2×C22⋊C4—416no—
201C2×C22⋊C4—416no—
202C2×C22⋊C4C2×D418yes—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42—44no—
206C2×C42—44no—
207C2×C42C2×D412yes—
208C2×C42C2×D414yes—
209C2×C42#4114yes—
210C2×C42C2412yes—
211#107—216no—
212C4⋊C8—28no—
213#103—216no—
214#103—216no—
215#103—416no—
216C22⋊C8—216no—
217C4×C8C2×D4116yes—
218#95—24no—
219#95—24no—
220#95—216no—
221C24—24no—
222C24—28no—
223C24—24no—
224C24—24no—
225C24—28no—
226C24C22×D414yes—
227C24C22×D411yescharacteristic
228C2×Q8—24no—
229C2×Q8—28no—
230C2×Q8—24no—
231C2×Q8—28no—
232C2×Q8—48no—
233C2×Q8—44no—
234C2×Q8—48no—
235C2×Q8—48no—
236C2×Q8—48no—
237C2×Q8—48no—
238C2×Q8—48no—
239C2×Q8—416no—
240C2×Q8—816no—
241C2×D4—28no—
242C2×D4—28no—
243C2×D4—28no—
244C2×D4—416no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—28no—
248C22×C4—28no—
249C22×C4—24no—
250C22×C4—28no—
251C22×C4—216no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—28no—
255C22×C4—28no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—48no—
262C22×C4—48no—
263C22×C4—416no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—48no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4#12114yes—
273C22×C4#12114yes—
274C22×C4AGL(1,ℤ/8)14yes—
275C22×C4#13714yes—
276C22×C4C22×D414yes—
277C22×C4C22×D412yes—
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D412yes—
280C22×C4C22×D412yes—
281C22×C4C2×#4112yes—
282C22×C4C2×#4114yes—
283C22×C4H(5,𝔽2)14yes—
284C22×C4C2511yescharacteristic
285M16—432no—
286C2×C8—28no—
287C2×C8—28no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—28no—
292C2×C8—48no—
293C2×C8C22×D414yes—
294C2×C8C22×D414yes—
295C4⋊C4—28no—
296C4⋊C4—28no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—432no—
303C4⋊C4—44no—
304C4⋊C4—48no—
305C4⋊C4—48no—
306C4⋊C4—48no—
307C4⋊C4—44no—
308C4⋊C4C2×#4118yes—
309C22⋊C4—216no—
310C22⋊C4—432no—
311C22⋊C4—432no—
312C42—48no—
313C42—48no—
314C42#12118yes—
315C42C22×D414yes—
316C42C2×#4114yes—
317C23—24no—
318C23—28no—
319C23—24no—
320C23—24no—
321C23—28no—
322C23—416no—
323C23—416no—
324C23—48no—
325C23—48no—
326C23—48no—
327C23C2×#12112yes—
328C23#54614yes—
329C23C2×AGL(1,ℤ/8)12yes—
330C23C2×#13712yes—
331C23#57814yes—
332C23C23×D411yescharacteristic
333Q8—44no—
334Q8—44no—
335Q8—48no—
336Q8—816no—
337Q8—88no—
338D4—416no—
339C2×C4—28no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—416no—
345C2×C4—48no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—48no—
351C2×C4—416no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—48no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—88no—
361C2×C4—88no—
362C2×C4—88no—
363C2×C4#468116yes—
364C2×C4C2×#12114yes—
365C2×C4C2×#12114yes—
366C2×C4C2×#12112yes—
367C2×C4#53214yes—
368C2×C4#53514yes—
369C2×C4#54614yes—
370C2×C4#55918yes—
371C2×C4C2×AGL(1,ℤ/8)14yes—
372C2×C4C2×#13714yes—
373C2×C4C23×D411yescharacteristic
374C2×C4C22×#4112yes—
375C2×C4C22×#4111yescharacteristic
376C2×C4C2×H(5,𝔽2)12yes—
377C8—24no—
378C8—24no—
379C8—44no—
380C22—44no—
381C22—44no—
382C22—44no—
383C22—48no—
384C22—48no—
385C22—816no—
386C22#254414yes—
387C22C2×#46814yes—
388C22#305914yes—
389C22#332818yes—
390C22#333018yes—
391C22C22×#12111yescharacteristic
392C22C2×#54612yes—
393C22C22×AGL(1,ℤ/8)11yescharacteristic
394C22C22×#13711yescharacteristic
395C22C2×#57812yes—
396C4—24no—
397C4—44no—
398C4—48no—
399C4—84no—
400C4—84no—
401C4—84no—
402C4#254414yes—
403C4C2×#46818yes—
404C4C2×#53211yescharacteristic
405C4C2×#53511yescharacteristic
406C4C2×#55912yes—
407C2—84no—
408C2#2860618yesdirect product factor
409C2C2×#254411yescharacteristic
410C2C22×#46811yescharacteristic
411C2C2×#305911yescharacteristic
412C2C2×#332812yes—
413C2C2×#333012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
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×#487C2semidirect
5. #28606C2direct (N × H)
6. C2×#2036C2semidirect
7. C2×#1988C2semidirect
8. C2×#1981C2semidirect
9. C2×#1894C2semidirect
10. C2×C4×#107C2semidirect
18. C2×#487C22semidirect
40. #2036C22semidirect
41. #1988C22semidirect
42. #1981C22semidirect
43. #1894C22semidirect
44. C4×#107C22semidirect
408. C2#28606direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4872
C22×#4362
C2×#25472
C2×#25422
#286062
C2×#20362
C2×#19882
C2×#19812
C2×#18942
C2×C4×#1072