Finite Group Databasebeta

← C2×#11500 #480243

Subgroups — C2×#11500 #480243

Order 512.

2326 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 2326 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×#381C211yesmaximal, characteristic
2C2×#1462C212yesmaximal
3#11500C218yesmaximal, direct product factor
4C2×#1317C214yesmaximal, complemented
5C25×C4C411yescharacteristic
6C23×C22⋊C4C411yescharacteristic
7C23×C22⋊C4C2211yescharacteristic
8C2×#407—22no—
9C2×#381—22no—
10C2×#381—24no—
11C2×#381C414yes—
12C2×#381C412yes—
13C2×#381C2212yes—
14C2×#381C2214yes—
15C22×#95—22no—
16#1462—24no—
17#1462C2218yes—
18C2×#323—22no—
19C2×#323C2214yes—
20#1317C22116yescomplemented
21C26C2×C411yescharacteristic
22C24×C4—22no—
23C24×C4—21no—
24C24×C4—21no—
25C24×C4—28no—
26C24×C4—21no—
27C24×C4—42no—
28C24×C4—42no—
29C24×C4—44no—
30C24×C4C2×C414yes—
31C24×C4C2×C412yes—
32C24×C4D412yes—
33C22×M16—42no—
34C22×C22⋊C4—22no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—42no—
37C22×C22⋊C4—42no—
38C22×C22⋊C4—44no—
39C22×C22⋊C4C2×C412yes—
40C22×C22⋊C4C2×C414yes—
41C22×C22⋊C4C2×C412yes—
42C22×C22⋊C4C2×C411yescharacteristic
43C22×C22⋊C4D412yes—
44C22×C22⋊C4D414yes—
45C22×C22⋊C4C2311yescharacteristic
46C22×C22⋊C4C2312yes—
47#407—28no—
48#407—44no—
49C2×C22⋊C8—24no—
50C2×C22⋊C8—42no—
51#381—24no—
52#381—28no—
53#381—24no—
54#381—28no—
55#381C2×C414yes—
56#381C2×C418yes—
57#381C2314yes—
58C2×#95—28no—
59C2×#95—24no—
60C2×#95—24no—
61C2×#95—28no—
62#323—28no—
63#323—24no—
64#323C2318yes—
65C25—22no—
66C25—22no—
67C25—24no—
68C25—24no—
69C25—48no—
70C25C22⋊C412yes—
71C25C22⋊C411yescharacteristic
72C25C22⋊C411yescharacteristic
73C25C22×C412yes—
74C25C22×C411yescharacteristic
75C23×C4—28no—
76C23×C4—216no—
77C23×C4—28no—
78C23×C4—22no—
79C23×C4—24no—
80C23×C4—22no—
81C23×C4—24no—
82C23×C4—22no—
83C23×C4—24no—
84C23×C4—22no—
85C23×C4—24no—
86C23×C4—216no—
87C23×C4—24no—
88C23×C4—22no—
89C23×C4—432no—
90C23×C4—432no—
91C23×C4—48no—
92C23×C4—42no—
93C23×C4—42no—
94C23×C4—44no—
95C23×C4—48no—
96C23×C4—42no—
97C23×C4—42no—
98C23×C4—44no—
99C23×C4—48no—
100C23×C4—48no—
101C23×C4—42no—
102C23×C4—42no—
103C23×C4—44no—
104C23×C4—48no—
105C23×C4—44no—
106C23×C4—48no—
107C23×C4C22⋊C414yes—
108C23×C4C22⋊C412yes—
109C23×C4M1618yescomplemented
110C23×C4C22×C414yes—
111C23×C4C2×D412yes—
112C23×C4C2×D414yes—
113C2×M16—48no—
114C2×M16—88no—
115C22×C8—44no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—44no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—44no—
122C2×C22⋊C4—44no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—44no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—48no—
129C2×C22⋊C4—44no—
130C2×C22⋊C4—44no—
131C2×C22⋊C4—48no—
132C2×C22⋊C4C22⋊C412yes—
133C2×C22⋊C4C22⋊C414yes—
134C2×C22⋊C4C22×C414yes—
135C2×C22⋊C4C22×C411yescharacteristic
136C2×C22⋊C4C22×C412yes—
137C2×C22⋊C4C2×D412yes—
138C2×C22⋊C4C2×D414yes—
139C2×C22⋊C4C2411yescharacteristic
140C22⋊C8—28no—
141C22⋊C8—44no—
142C22⋊C8—48no—
143#95—28no—
144#95—216no—
145#95—28no—
146C24—28no—
147C24—22no—
148C24—22no—
149C24—216no—
150C24—216no—
151C24—24no—
152C24—24no—
153C24—22no—
154C24—24no—
155C24—21no—
156C24—24no—
157C24—21no—
158C24—21no—
159C24—21no—
160C24—24no—
161C24—24no—
162C24—22no—
163C24—44no—
164C24—48no—
165C24—44no—
166C24—48no—
167C24—432no—
168C24—432no—
169C24—416no—
170C24—48no—
171C24—48no—
172C24#9912yes—
173C24#9912yes—
174C24#10014yes—
175C24C2×C22⋊C412yes—
176C24C2×C22⋊C411yescharacteristic
177C24C2×C22⋊C411yescharacteristic
178C24C2×C22⋊C412yes—
179C24C2×M1614yes—
180C24C23×C411yescharacteristic
181C22×C4—216no—
182C22×C4—216no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—28no—
186C22×C4—28no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—464no—
197C22×C4—464no—
198C22×C4—44no—
199C22×C4—48no—
200C22×C4—48no—
201C22×C4—44no—
202C22×C4—416no—
203C22×C4—416no—
204C22×C4—416no—
205C22×C4—416no—
206C22×C4—48no—
207C22×C4—48no—
208C22×C4—48no—
209C22×C4—48no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—48no—
213C22×C4—44no—
214C22×C4—48no—
215C22×C4—48no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—48no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—48no—
222C22×C4—48no—
223C22×C4—416no—
224C22×C4—48no—
225C22×C4—44no—
226C22×C4—84no—
227C22×C4—88no—
228C22×C4—816no—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C414yes—
231C22×C4C2×M1618yescomplemented
232C22×C4C22×D412yes—
233M16—816no—
234C2×C8—48no—
235C2×C8—88no—
236C22⋊C4—42no—
237C22⋊C4—44no—
238C22⋊C4—42no—
239C22⋊C4—41no—
240C22⋊C4—44no—
241C22⋊C4—42no—
242C22⋊C4—41no—
243C22⋊C4—44no—
244C22⋊C4—48no—
245C22⋊C4—44no—
246C23—216no—
247C23—216no—
248C23—22no—
249C23—22no—
250C23—28no—
251C23—24no—
252C23—28no—
253C23—24no—
254C23—24no—
255C23—24no—
256C23—216no—
257C23—216no—
258C23—22no—
259C23—21no—
260C23—22no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—24no—
265C23—24no—
266C23—24no—
267C23—24no—
268C23—48no—
269C23—42no—
270C23—44no—
271C23—48no—
272C23—42no—
273C23—416no—
274C23—416no—
275C23—416no—
276C23—416no—
277C23—464no—
278C23—48no—
279C23—44no—
280C23—48no—
281C23—44no—
282C23—48no—
283C23—464no—
284C23—48no—
285C23—48no—
286C23—48no—
287C23—44no—
288C23—44no—
289C23#35218yes—
290C23#40712yes—
291C23#40712yes—
292C23C2×#9914yes—
293C23C2×#9911yescharacteristic
294C23C2×#9911yescharacteristic
295C23C2×#10012yes—
296C23C2×#10014yes—
297C23C22×C22⋊C411yescharacteristic
298C23C22×M1612yes—
299C2×C4—28no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—432no—
304C2×C4—48no—
305C2×C4—416no—
306C2×C4—432no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—48no—
311C2×C4—48no—
312C2×C4—44no—
313C2×C4—48no—
314C2×C4—48no—
315C2×C4—48no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—48no—
320C2×C4—44no—
321C2×C4—82no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—82no—
325C2×C4—816no—
326C2×C4—88no—
327C8—88no—
328C22—28no—
329C22—28no—
330C22—216no—
331C22—22no—
332C22—24no—
333C22—22no—
334C22—24no—
335C22—24no—
336C22—216no—
337C22—24no—
338C22—22no—
339C22—22no—
340C22—24no—
341C22—42no—
342C22—42no—
343C22—44no—
344C22—42no—
345C22—42no—
346C22—44no—
347C22—432no—
348C22—432no—
349C22—48no—
350C22—48no—
351C22—42no—
352C22—44no—
353C22—42no—
354C22—48no—
355C22—48no—
356C22#146212yes—
357C22#146212yes—
358C22C2×#35218yes—
359C22C2×#35214yes—
360C22C2×#40711yescharacteristic
361C22C22×#9911yescharacteristic
362C22C22×#10011yescharacteristic
363C4—42no—
364C4—42no—
365C4—44no—
366C4—48no—
367C4—84no—
368C2—21no—
369C2—21no—
370C2—28no—
371C2—22no—
372C2—42no—
373C2—42no—
374C2—44no—
375C2#1150014yesdirect product factor
376C2C2×#146211yescharacteristic
377C2C22×#35212yes—
subgroupG/H
center, socleC23C2×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC26C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11500C2direct (N × H)
4. C2×#1317C2semidirect
20. #1317C22semidirect
109. C23×C4M16semidirect
231. C22×C4C2×M16semidirect
375. C2#11500direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3812
C2×#14622
#115002
C2×#13172