Finite Group Databasebeta

← C2×#11519 #480296

Subgroups — C2×#11519 #480296

Order 512.

1606 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1606 conjugacy classes of subgroups — too many to draw. Only its 234 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
1C2×#2315C211yesmaximal, characteristic
2C2×#1462C212yesmaximal
3#11519C218yesmaximal, direct product factor
4C2×#1326C214yesmaximal, complemented
5C23×C22⋊C4C411yescharacteristic
6C23×C22⋊C4C411yescharacteristic
7C23×C22⋊C4C2211yescharacteristic
8C2×#407—22no—
9#2315—24no—
10#2315C414yes—
11#2315C2214yes—
12C2×#385—22no—
13C2×#385C412yes—
14C2×#385C2212yes—
15C2×#380—22no—
16#1462—24no—
17#1462C2218yes—
18C2×#323—22no—
19C2×#323C2214yes—
20#1326C22116yescomplemented
21C26C2×C411yescharacteristic
22C24×C4—21no—
23C24×C4C2×C412yes—
24C24×C4D412yes—
25C22×M16—42no—
26C22×C4⋊C4—22no—
27C22×C4⋊C4—22no—
28C22×C22⋊C4—22no—
29C22×C22⋊C4—21no—
30C22×C22⋊C4—28no—
31C22×C22⋊C4—22no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4—41no—
35C22×C22⋊C4—41no—
36C22×C22⋊C4—42no—
37C22×C22⋊C4—42no—
38C22×C22⋊C4—41no—
39C22×C22⋊C4—41no—
40C22×C22⋊C4C2×C414yes—
41C22×C22⋊C4C2×C412yes—
42C22×C22⋊C4C2×C414yes—
43C22×C22⋊C4C2×C412yes—
44C22×C22⋊C4C2×C411yescharacteristic
45C22×C22⋊C4D414yes—
46C22×C22⋊C4D412yes—
47C22×C22⋊C4C2312yes—
48C22×C22⋊C4C2311yescharacteristic
49#407—28no—
50#407—44no—
51C2×C22⋊C8—24no—
52C2×C22⋊C8—42no—
53#385—28no—
54#385—28no—
55#385—24no—
56#385—22no—
57#385—22no—
58#385C2×C414yes—
59#385C2×C418yes—
60#385C2314yes—
61#380—28no—
62#380—28no—
63C2×#95—22no—
64C2×#95—22no—
65#323—28no—
66#323—24no—
67#323C2318yes—
68C25—22no—
69C25—21no—
70C25—21no—
71C25—24no—
72C25—24no—
73C25—84no—
74C25C22⋊C412yes—
75C25C22⋊C411yescharacteristic
76C25C22⋊C411yescharacteristic
77C25C22×C412yes—
78C25C22×C411yescharacteristic
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—28no—
82C23×C4—22no—
83C23×C4—24no—
84C23×C4—24no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—42no—
88C23×C4—42no—
89C23×C4—84no—
90C23×C4—84no—
91C23×C4—84no—
92C23×C4C22⋊C412yes—
93C23×C4C22⋊C414yes—
94C23×C4C22×C414yes—
95C23×C4C2×D412yes—
96C23×C4C2×D414yes—
97C2×M16—48no—
98C2×M16—88no—
99C22×C8—44no—
100C2×C4⋊C4—28no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—84no—
104C2×C4⋊C4—84no—
105C2×C22⋊C4—28no—
106C2×C22⋊C4—216no—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—216no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—44no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—44no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—41no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—44no—
128C2×C22⋊C4—44no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—44no—
133C2×C22⋊C4—44no—
134C2×C22⋊C4C22⋊C414yes—
135C2×C22⋊C4C22⋊C412yes—
136C2×C22⋊C4M1618yescomplemented
137C2×C22⋊C4C22×C414yes—
138C2×C22⋊C4C22×C411yescharacteristic
139C2×C22⋊C4C22×C412yes—
140C2×C22⋊C4C2×D414yes—
141C2×C22⋊C4C2×D412yes—
142C2×C22⋊C4C2411yescharacteristic
143C22⋊C8—28no—
144C22⋊C8—44no—
145C22⋊C8—48no—
146#95—28no—
147#95—24no—
148#95—24no—
149C24—216no—
150C24—28no—
151C24—28no—
152C24—22no—
153C24—22no—
154C24—21no—
155C24—21no—
156C24—21no—
157C24—21no—
158C24—22no—
159C24—22no—
160C24—24no—
161C24—24no—
162C24—22no—
163C24—22no—
164C24—22no—
165C24—22no—
166C24—24no—
167C24—24no—
168C24—44no—
169C24—44no—
170C24—44no—
171C24—84no—
172C24—84no—
173C24—84no—
174C24—84no—
175C24—88no—
176C24—816no—
177C24—816no—
178C24#9911yescharacteristic
179C24#9911yescharacteristic
180C24#9912yes—
181C24#10114yes—
182C24C2×C22⋊C411yescharacteristic
183C24C2×C22⋊C411yescharacteristic
184C24C2×C22⋊C412yes—
185C24C2×C22⋊C412yes—
186C24C2×M1614yes—
187C24C23×C411yescharacteristic
188C22×C4—24no—
189C22×C4—28no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—216no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—48no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—44no—
211C22×C4—84no—
212C22×C4—84no—
213C22×C4—84no—
214C22×C4—88no—
215C22×C4—88no—
216C22×C4—816no—
217C22×C4—84no—
218C22×C4—84no—
219C22×C4—88no—
220C22×C4—88no—
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C22⋊C414yes—
223C22×C4C22×D412yes—
224M16—816no—
225C2×C8—48no—
226C2×C8—88no—
227C4⋊C4—88no—
228C4⋊C4—84no—
229C4⋊C4—84no—
230C22⋊C4—216no—
231C22⋊C4—24no—
232C22⋊C4—28no—
233C22⋊C4—44no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—41no—
237C22⋊C4—44no—
238C22⋊C4—42no—
239C22⋊C4—41no—
240C22⋊C4—48no—
241C22⋊C4—44no—
242C22⋊C4—44no—
243C22⋊C4C2×M1618yescomplemented
244C23—24no—
245C23—28no—
246C23—28no—
247C23—28no—
248C23—28no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—22no—
256C23—21no—
257C23—24no—
258C23—24no—
259C23—24no—
260C23—22no—
261C23—22no—
262C23—24no—
263C23—24no—
264C23—24no—
265C23—24no—
266C23—44no—
267C23—44no—
268C23—42no—
269C23—44no—
270C23—42no—
271C23—48no—
272C23—48no—
273C23—44no—
274C23—48no—
275C23—44no—
276C23—48no—
277C23—48no—
278C23—48no—
279C23—44no—
280C23—44no—
281C23—84no—
282C23—84no—
283C23—84no—
284C23—84no—
285C23—88no—
286C23—88no—
287C23—88no—
288C23—88no—
289C23—832no—
290C23—816no—
291C23—816no—
292C23#40712yes—
293C23#40712yes—
294C23C2×#9912yes—
295C23C2×#9912yes—
296C23C2×#9911yescharacteristic
297C23C2×#9911yescharacteristic
298C23C2×#10112yes—
299C23C2×#10114yes—
300C23C22×C22⋊C411yescharacteristic
301C23C22×M1612yes—
302C2×C4—28no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—48no—
309C2×C4—416no—
310C2×C4—48no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—84no—
322C2×C4—84no—
323C2×C4—816no—
324C2×C4—84no—
325C2×C4—88no—
326C2×C4—84no—
327C2×C4—88no—
328C8—88no—
329C22—22no—
330C22—22no—
331C22—24no—
332C22—24no—
333C22—24no—
334C22—24no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—24no—
340C22—42no—
341C22—42no—
342C22—44no—
343C22—44no—
344C22—44no—
345C22—44no—
346C22—42no—
347C22—42no—
348C22—48no—
349C22—48no—
350C22—48no—
351C22—42no—
352C22—42no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—48no—
357C22—48no—
358C22—88no—
359C22—88no—
360C22—88no—
361C22—88no—
362C22#146312yes—
363C22#146312yes—
364C22#212114yes—
365C22C2×#40711yescharacteristic
366C22C22×#9911yescharacteristic
367C22C22×#10111yescharacteristic
368C4—48no—
369C4—42no—
370C4—44no—
371C4—42no—
372C4—84no—
373C2—21no—
374C2—21no—
375C2—22no—
376C2—42no—
377C2—42no—
378C2—42no—
379C2—42no—
380C2—44no—
381C2#1151914yesdirect product factor
382C2C2×#146311yescharacteristic
383C2C2×#212112yes—
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. #11519C2direct (N × H)
4. C2×#1326C2semidirect
20. #1326C22semidirect
136. C2×C22⋊C4M16semidirect
243. C22⋊C4C2×M16semidirect
381. C2#11519direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#23152
C2×#14622
#115192
C2×#13262