Finite Group Databasebeta

← C2×#11513 #480282

Subgroups — C2×#11513 #480282

Order 512.

1230 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic
2C2×#1464C212yesmaximal
3#11513C218yesmaximal, direct product factor
4C2×#1325C212yesmaximal, complemented
5C2×#1324C212yesmaximal, complemented
6C22×#117C411yescharacteristic
7C22×#117C2211yescharacteristic
8C23×C42C411yescharacteristic
9C2×#432—22no—
10#2285—24no—
11#2285C414yes—
12#2285C2214yes—
13C2×#384C411yescharacteristic
14C2×#384C2211yescharacteristic
15C2×#382C411yescharacteristic
16C2×#382C2211yescharacteristic
17C2×#380—22no—
18C2×C4×C4⋊C4—22no—
19#1464—24no—
20#1464C2218yes—
21C2×#324C2214yes—
22C2×#323—22no—
23#1325C2218yescomplemented
24#1324C2218yescomplemented
25C24×C4—21no—
26C24×C4C2×C411yescharacteristic
27C2×#117—22no—
28C2×#117—41no—
29C2×#117—41no—
30C2×#117—42no—
31C2×#117—42no—
32C2×#117—41no—
33C2×#117—41no—
34C2×#117C2×C414yes—
35C2×#117C2×C412yes—
36C2×#117D414yes—
37C2×#117C2312yes—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2311yescharacteristic
44C22×C22⋊C4—21no—
45C22×C22⋊C4D412yes—
46C22×C42—22no—
47C22×C42—21no—
48C22×C42—24no—
49C22×C42—24no—
50C22×C42—21no—
51C22×C42C2×C414yes—
52C22×C42C2×C411yescharacteristic
53C22×C42C2×C411yescharacteristic
54C22×C42D412yes—
55#432—28no—
56#432—44no—
57C2×C4⋊C8—42no—
58C2×C22⋊C8—24no—
59C2×#97—42no—
60#384—22no—
61#384—22no—
62#384C2×C414yes—
63#384C2×C412yes—
64#384C2312yes—
65#382—22no—
66#382—22no—
67#382C2×C414yes—
68#382C2×C412yes—
69#382C2312yes—
70#380—28no—
71#380—28no—
72C4×C4⋊C4—28no—
73C4×C4⋊C4—28no—
74C2×#95—22no—
75C2×#95—22no—
76#324—24no—
77#324C2318yes—
78#323—28no—
79C25C22⋊C411yescharacteristic
80C23×C4—24no—
81C23×C4—21no—
82C23×C4—22no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—28no—
88C23×C4—21no—
89C23×C4—24no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—44no—
93C23×C4—44no—
94C23×C4C22⋊C412yes—
95C23×C4C22⋊C412yes—
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22×C412yes—
98C23×C4C22×C411yescharacteristic
99C23×C4C2×D412yes—
100C22×C8—44no—
101#117—42no—
102#117—42no—
103#117—44no—
104#117—44no—
105#117—42no—
106#117—42no—
107#117—44no—
108#117—44no—
109#117—44no—
110#117—44no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—28no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—48no—
125C2×C4⋊C4—48no—
126C2×C4⋊C4C22⋊C412yes—
127C2×C4⋊C4C22×C412yes—
128C2×C4⋊C4C22×C412yes—
129C2×C4⋊C4C22×C412yes—
130C2×C4⋊C4C22×C411yescharacteristic
131C2×C4⋊C4C2×D412yes—
132C2×C4⋊C4C2411yescharacteristic
133C2×C22⋊C4—24no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4C22⋊C414yes—
139C2×C22⋊C4C2×D414yes—
140C2×C42—28no—
141C2×C42—216no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—24no—
145C2×C42—24no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—28no—
151C2×C42—28no—
152C2×C42—24no—
153C2×C42—42no—
154C2×C42—41no—
155C2×C42—41no—
156C2×C42C22⋊C414yes—
157C2×C42M1614yes—
158C2×C42M1614yes—
159C2×C42C22×C412yes—
160C2×C42C22×C412yes—
161C2×C42C2×D414yes—
162C4⋊C8—48no—
163C22⋊C8—28no—
164C22⋊C8—44no—
165#97—48no—
166#95—24no—
167#95—28no—
168#95—24no—
169C24—22no—
170C24—21no—
171C24—21no—
172C24—42no—
173C24—42no—
174C24#10114yes—
175C24C2×C22⋊C411yescharacteristic
176C24C2×C22⋊C412yes—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—28no—
180C22×C4—28no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—24no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—216no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—48no—
211C22×C4—48no—
212C22×C4—432no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4#9912yes—
224C22×C4#9911yescharacteristic
225C22×C4#9911yescharacteristic
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C414yes—
230C22×C4C2×M1612yes—
231C22×C4C2×M1612yes—
232C22×C4C23×C411yescharacteristic
233C22×C4C22×D412yes—
234C2×C8—48no—
235C2×C8—88no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—48no—
244C4⋊C4—416no—
245C4⋊C4—48no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C22⋊C4—44no—
249C22⋊C4—44no—
250C22⋊C4—48no—
251C42—216no—
252C42—22no—
253C42—22no—
254C42—24no—
255C42—24no—
256C42—24no—
257C42—24no—
258C42—24no—
259C42—24no—
260C42—44no—
261C42—44no—
262C42C2×M1614yes—
263C42C2×M1614yes—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—24no—
269C23—24no—
270C23—42no—
271C23—42no—
272C23—48no—
273C23—48no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23#40712yes—
279C23C2×#9911yescharacteristic
280C23C2×#9911yescharacteristic
281C23C2×#10112yes—
282C23C2×#10114yes—
283C23C22×C22⋊C411yescharacteristic
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—28no—
291C2×C4—28no—
292C2×C4—216no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—432no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—48no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4#40712yes—
331C2×C4C2×#9912yes—
332C2×C4C2×#9912yes—
333C2×C4C22×M1611yescharacteristic
334C2×C4C22×M1611yescharacteristic
335C8—88no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—24no—
341C22—24no—
342C22—24no—
343C22—44no—
344C22—44no—
345C22—48no—
346C22—42no—
347C22—42no—
348C22—48no—
349C22#146312yes—
350C22#146312yes—
351C22#212612yes—
352C22#213212yes—
353C22C2×#40711yescharacteristic
354C22C22×#9911yescharacteristic
355C22C22×#10111yescharacteristic
356C4—28no—
357C4—21no—
358C4—21no—
359C4—22no—
360C4—22no—
361C4—22no—
362C4—44no—
363C4—42no—
364C4—42no—
365C4—44no—
366C4—82no—
367C4—82no—
368C2—21no—
369C2—21no—
370C2—22no—
371C2—44no—
372C2#1151314yesdirect product factor
373C2C2×#146311yescharacteristic
374C2C2×#212611yescharacteristic
375C2C2×#213211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11513C2direct (N × H)
4. C2×#1325C2semidirect
5. C2×#1324C2semidirect
23. #1325C22semidirect
24. #1324C22semidirect
372. C2#11513direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#22852
C2×#14642
#115132
C2×#13252
C2×#13242