Finite Group Databasebeta

← C2×#11505 #480255

Subgroups — C2×#11505 #480255

Order 512.

2694 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 2694 conjugacy classes of subgroups — too many to draw. Only its 250 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#390C211yesmaximal, characteristic
2C2×#1463C212yesmaximal, complemented
3#11505C218yesmaximal, direct product factor
4C2×#1318C214yesmaximal, complemented
5C24×D4C411yescharacteristic
6C23×C4⋊C4C2211yescharacteristic
7C23×C22⋊C4—22no—
8C23×C42C411yescharacteristic
9C2×#407—22no—
10C2×#390—22no—
11C2×#390—24no—
12C2×#390C414yes—
13C2×#390C412yes—
14C2×#390C2212yes—
15C2×#390C2214yes—
16#1463—24no—
17#1463C2218yescomplemented
18C2×#324—22no—
19C2×#324C2214yescomplemented
20#1318C22116yescomplemented
21C26—21no—
22C23×D4—28no—
23C23×D4—22no—
24C23×D4—21no—
25C23×D4—42no—
26C23×D4—41no—
27C23×D4—41no—
28C23×D4—44no—
29C23×D4C2×C414yes—
30C23×D4C2×C412yes—
31C24×C4—21no—
32C24×C4C2×C411yescharacteristic
33C24×C4D412yes—
34C22×M16—42no—
35C22×C4⋊C4—41no—
36C22×C4⋊C4—41no—
37C22×C4⋊C4—42no—
38C22×C4⋊C4C2×C412yes—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4D412yes—
41C22×C4⋊C4D414yes—
42C22×C4⋊C4C2311yescharacteristic
43C22×C4⋊C4C2312yes—
44C22×C22⋊C4—28no—
45C22×C22⋊C4—24no—
46C22×C22⋊C4—24no—
47C22×C22⋊C4—28no—
48C22×C22⋊C4—416no—
49C22×C42—22no—
50C22×C42—21no—
51C22×C42—44no—
52C22×C42C2×C412yes—
53C22×C42C2×C414yes—
54#407—28no—
55#407—44no—
56C2×C22⋊C8—24no—
57C2×C22⋊C8—42no—
58#390—24no—
59#390—28no—
60#390—24no—
61#390—28no—
62#390C2×C414yes—
63#390C2×C418yes—
64#390C2314yes—
65#324—28no—
66#324—24no—
67#324C2318yescomplemented
68C25—28no—
69C25—24no—
70C25—22no—
71C25—48no—
72C25—44no—
73C25—48no—
74C25—42no—
75C25—41no—
76C25—41no—
77C25C22⋊C411yescharacteristic
78C22×D4—28no—
79C22×D4—216no—
80C22×D4—216no—
81C22×D4—22no—
82C22×D4—24no—
83C22×D4—24no—
84C22×D4—42no—
85C22×D4—44no—
86C22×D4—44no—
87C22×D4—42no—
88C22×D4—44no—
89C22×D4—432no—
90C22×D4—416no—
91C22×D4—416no—
92C22×D4—44no—
93C22×D4—44no—
94C22×D4—42no—
95C22×D4—42no—
96C22×D4—48no—
97C22×D4—48no—
98C22×D4—44no—
99C22×D4M1618yescomplemented
100C22×D4C22×C414yes—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—24no—
105C23×C4—24no—
106C23×C4—28no—
107C23×C4—22no—
108C23×C4—24no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—24no—
112C23×C4—44no—
113C23×C4—44no—
114C23×C4—48no—
115C23×C4C22⋊C412yes—
116C23×C4C22⋊C411yescharacteristic
117C23×C4C22⋊C414yes—
118C23×C4C22⋊C412yes—
119C23×C4C22×C412yes—
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D412yes—
122C23×C4C2×D414yes—
123C2×M16—48no—
124C2×M16—88no—
125C22×C8—44no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4C22⋊C412yes—
139C2×C4⋊C4C22⋊C414yes—
140C2×C4⋊C4C22×C411yescharacteristic
141C2×C4⋊C4C22×C412yes—
142C2×C4⋊C4C2×D412yes—
143C2×C4⋊C4C2×D414yes—
144C2×C4⋊C4C2411yescharacteristic
145C2×C22⋊C4—28no—
146C2×C22⋊C4—216no—
147C2×C22⋊C4—28no—
148C2×C22⋊C4—432no—
149C2×C22⋊C4—416no—
150C2×C22⋊C4—416no—
151C2×C22⋊C4—432no—
152C2×C42—22no—
153C2×C42—24no—
154C2×C42—44no—
155C2×C42—48no—
156C2×C42—44no—
157C2×C42—48no—
158C2×C42C22×C414yes—
159C22⋊C8—28no—
160C22⋊C8—44no—
161C22⋊C8—48no—
162C24—24no—
163C24—24no—
164C24—22no—
165C24—22no—
166C24—216no—
167C24—24no—
168C24—28no—
169C24—416no—
170C24—48no—
171C24—48no—
172C24—416no—
173C24—48no—
174C24—48no—
175C24—464no—
176C24—416no—
177C24—48no—
178C24—44no—
179C24—44no—
180C24—42no—
181C24—42no—
182C24—44no—
183C24—832no—
184C24—88no—
185C24—84no—
186C24—88no—
187C24—84no—
188C24—84no—
189C24—84no—
190C24#9912yes—
191C24#9911yescharacteristic
192C24#9911yescharacteristic
193C24C2×C22⋊C412yes—
194C24C2×C22⋊C411yescharacteristic
195C2×D4—216no—
196C2×D4—24no—
197C2×D4—28no—
198C2×D4—48no—
199C2×D4—48no—
200C2×D4—44no—
201C2×D4—44no—
202C2×D4—432no—
203C2×D4—432no—
204C2×D4—432no—
205C2×D4—432no—
206C2×D4—44no—
207C2×D4—48no—
208C2×D4—44no—
209C2×D4—416no—
210C2×D4—416no—
211C2×D4—48no—
212C2×D4—48no—
213C2×D4—48no—
214C2×D4C2×M1618yescomplemented
215C22×C4—216no—
216C22×C4—28no—
217C22×C4—28no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—216no—
233C22×C4—28no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—48no—
244C22×C4—432no—
245C22×C4—48no—
246C22×C4—44no—
247C22×C4—48no—
248C22×C4—44no—
249C22×C4—48no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—48no—
253C22×C4—84no—
254C22×C4—88no—
255C22×C4#10014yes—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C414yes—
260C22×C4C2×M1614yes—
261C22×C4C23×C411yescharacteristic
262C22×C4C22×D412yes—
263M16—816no—
264C2×C8—48no—
265C2×C8—88no—
266C4⋊C4—42no—
267C4⋊C4—44no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—44no—
272C4⋊C4—41no—
273C22⋊C4—416no—
274C22⋊C4—432no—
275C22⋊C4—416no—
276C42—44no—
277C42—44no—
278C42—48no—
279C23—216no—
280C23—216no—
281C23—22no—
282C23—22no—
283C23—24no—
284C23—22no—
285C23—24no—
286C23—22no—
287C23—24no—
288C23—22no—
289C23—24no—
290C23—22no—
291C23—24no—
292C23—22no—
293C23—22no—
294C23—28no—
295C23—48no—
296C23—416no—
297C23—416no—
298C23—464no—
299C23—48no—
300C23—432no—
301C23—44no—
302C23—44no—
303C23—48no—
304C23—48no—
305C23—832no—
306C23—864no—
307C23—88no—
308C23—84no—
309C23—88no—
310C23—84no—
311C23—816no—
312C23—832no—
313C23—832no—
314C23—88no—
315C23—84no—
316C23—88no—
317C23—84no—
318C23#35314yes—
319C23#35514yes—
320C23#40712yes—
321C23C2×#9912yes—
322C23C2×#9912yes—
323C23C2×#9911yescharacteristic
324C23C2×#9911yescharacteristic
325C23C2×#10012yes—
326C23C22×C22⋊C411yescharacteristic
327D4—48no—
328D4—48no—
329D4—416no—
330D4—432no—
331C2×C4—28no—
332C2×C4—28no—
333C2×C4—28no—
334C2×C4—28no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—28no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—432no—
350C2×C4—44no—
351C2×C4—416no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—48no—
355C2×C4—48no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4—82no—
364C2×C4#40712yes—
365C2×C4C2×#10014yes—
366C2×C4C22×M1612yes—
367C8—88no—
368C22—28no—
369C22—28no—
370C22—216no—
371C22—21no—
372C22—22no—
373C22—21no—
374C22—21no—
375C22—22no—
376C22—21no—
377C22—22no—
378C22—22no—
379C22—24no—
380C22—24no—
381C22—28no—
382C22—28no—
383C22—416no—
384C22—48no—
385C22—88no—
386C22—832no—
387C22—832no—
388C22—816no—
389C22—84no—
390C22—84no—
391C22—816no—
392C22#146212yes—
393C22#146212yes—
394C22C2×#35314yes—
395C22C2×#35312yes—
396C22C2×#35514yes—
397C22C2×#35512yes—
398C22C2×#40711yescharacteristic
399C22C22×#9911yescharacteristic
400C22C22×#10011yescharacteristic
401C4—22no—
402C4—22no—
403C4—24no—
404C4—48no—
405C4—42no—
406C4—42no—
407C4—44no—
408C2—21no—
409C2—21no—
410C2—24no—
411C2—24no—
412C2—22no—
413C2—88no—
414C2#1150514yesdirect product factor
415C2C2×#146211yescharacteristic
416C2C22×#35311yescharacteristic
417C2C22×#35511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC24×D4C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1463C2semidirect
3. #11505C2direct (N × H)
4. C2×#1318C2semidirect
17. #1463C22semidirect
19. C2×#324C22semidirect
20. #1318C22semidirect
67. #324C23semidirect
99. C22×D4M16semidirect
214. C2×D4C2×M16semidirect
414. C2#11505direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3902
C2×#14632
#115052
C2×#13182