Finite Group Databasebeta

← C2×#11531 #480335

Subgroups — C2×#11531 #480335

Order 512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#124C211yesmaximal, characteristic
2C2×#1465C212yesmaximal
3#11531C218yesmaximal, direct product factor
4C2×#1324C212yesmaximal, complemented
5C2×#1319C212yesmaximal, complemented
6C22×#124C411yescharacteristic
7C22×C4×Q8C411yescharacteristic
8C22×C4×D4C2211yescharacteristic
9C2×#407—22no—
10C4×#124—24no—
11C4×#124C414yes—
12C4×#124C2214yes—
13C2×#391C2211yescharacteristic
14C2×#390C411yescharacteristic
15C2×#388—22no—
16C2×#382C2211yescharacteristic
17C2×C4×C22⋊C4—22no—
18C2×C43C411yescharacteristic
19#1465—24no—
20#1465C2218yes—
21C2×#324C2214yes—
22C2×#323—22no—
23#1324C2218yescomplemented
24#1319C2218yescomplemented
25C23×Q8C2×C411yescharacteristic
26C23×D4C2×C411yescharacteristic
27C24×C4D412yes—
28C22×M16—42no—
29C2×#124—24no—
30C2×#124—22no—
31C2×#124—24no—
32C2×#124C2×C414yes—
33C2×C4×Q8—22no—
34C2×C4×Q8—42no—
35C2×C4×Q8—41no—
36C2×C4×Q8—41no—
37C2×C4×Q8C2×C414yes—
38C2×C4×D4—41no—
39C2×C4×D4—41no—
40C2×C4×D4—42no—
41C2×C4×D4C2×C412yes—
42C2×C4×D4D414yes—
43C2×C4×D4C2312yes—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2311yescharacteristic
47C22×C22⋊C4—22no—
48C22×C22⋊C4—22no—
49C22×C22⋊C4D412yes—
50C22×C42—21no—
51C22×C42—21no—
52C22×C42—21no—
53C22×C42C2×C411yescharacteristic
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C411yescharacteristic
56#407—28no—
57#407—44no—
58C2×C22⋊C8—24no—
59C2×C22⋊C8—42no—
60#391—22no—
61#391C2×C412yes—
62#391C2312yes—
63#390—22no—
64#390C2×C414yes—
65#388—28no—
66#388—28no—
67#382—22no—
68#382C2×C412yes—
69#382C2312yes—
70C4×C22⋊C4—28no—
71C4×C22⋊C4—28no—
72C2×#95—22no—
73C2×#95—22no—
74C43—22no—
75C43C2×C414yes—
76#324—24no—
77#324C2318yes—
78#323—28no—
79C25C22⋊C412yes—
80C22×Q8—21no—
81C22×Q8—22no—
82C22×Q8—41no—
83C22×Q8—41no—
84C22×Q8—42no—
85C22×Q8M1614yescomplemented
86C22×Q8C22×C412yes—
87C22×D4—22no—
88C22×D4—22no—
89C22×D4—41no—
90C22×D4—41no—
91C22×D4—42no—
92C22×D4C22⋊C412yes—
93C22×D4C22×C412yes—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—24no—
100C23×C4—21no—
101C23×C4—48no—
102C23×C4C22⋊C411yescharacteristic
103C23×C4C22⋊C414yes—
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22×C411yescharacteristic
106C23×C4C2×D412yes—
107C23×C4C2×D414yes—
108C2×M16—48no—
109C2×M16—88no—
110C22×C8—44no—
111#124—216no—
112#124—28no—
113#124—28no—
114C4×Q8—44no—
115C4×Q8—44no—
116C4×Q8—44no—
117C4×Q8—44no—
118C4×D4—42no—
119C4×D4—42no—
120C4×D4—44no—
121C4×D4—44no—
122C4×D4—42no—
123C4×D4—42no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4C22×C412yes—
133C2×C4⋊C4C22×C411yescharacteristic
134C2×C4⋊C4C2×D412yes—
135C2×C4⋊C4C2411yescharacteristic
136C2×C22⋊C4—216no—
137C2×C22⋊C4—28no—
138C2×C22⋊C4—28no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—48no—
142C2×C22⋊C4C22⋊C414yes—
143C2×C22⋊C4C2×D414yes—
144C2×C42—22no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—22no—
149C2×C42—24no—
150C2×C42—21no—
151C2×C42—24no—
152C2×C42—24no—
153C2×C42—42no—
154C2×C42—44no—
155C2×C42—44no—
156C2×C42—42no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42C22⋊C412yes—
160C2×C42M1614yescomplemented
161C2×C42C22×C412yes—
162C2×C42C22×C412yes—
163C2×C42C22×C412yes—
164C22⋊C8—28no—
165C22⋊C8—44no—
166C22⋊C8—48no—
167#95—24no—
168#95—24no—
169#95—24no—
170#95—24no—
171C24—24no—
172C24—22no—
173C24—22no—
174C24—24no—
175C24—44no—
176C24—44no—
177C24C2×C22⋊C411yescharacteristic
178C24C2×C22⋊C414yes—
179C2×Q8—22no—
180C2×Q8—24no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—44no—
184C2×Q8—44no—
185C2×Q8—42no—
186C2×Q8—44no—
187C2×Q8—44no—
188C2×Q8—42no—
189C2×Q8C2×M1614yescomplemented
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—24no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—44no—
198C2×D4—44no—
199C2×D4—44no—
200C2×D4—44no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—48no—
221C22×C4—48no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—48no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—88no—
236C22×C4—84no—
237C22×C4#9912yes—
238C22×C4#9911yescharacteristic
239C22×C4#9911yescharacteristic
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C412yes—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×M1612yes—
245C22×C4C2×M1612yes—
246C22×C4C23×C411yescharacteristic
247C22×C4C22×D412yes—
248M16—816no—
249C2×C8—48no—
250C2×C8—88no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—48no—
257C22⋊C4—232no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—416no—
261C42—22no—
262C42—24no—
263C42—22no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—42no—
268C42—42no—
269C42—48no—
270C42—48no—
271C42—44no—
272C42—44no—
273C42C2×M1614yescomplemented
274C23—22no—
275C23—22no—
276C23—24no—
277C23—24no—
278C23—24no—
279C23—44no—
280C23—44no—
281C23—416no—
282C23—44no—
283C23—44no—
284C23—44no—
285C23—44no—
286C23—82no—
287C23—82no—
288C23—88no—
289C23#40712yes—
290C23C2×#9911yescharacteristic
291C23C2×#9911yescharacteristic
292C23#41312yes—
293C23C22×C22⋊C411yescharacteristic
294Q8—44no—
295Q8—44no—
296Q8—48no—
297D4—44no—
298D4—44no—
299D4—48no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—48no—
321C2×C4—48no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—48no—
331C2×C4—48no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4#35414yes—
343C2×C4#40712yes—
344C2×C4C2×#9912yes—
345C2×C4C2×#9912yes—
346C2×C4C22×M1611yescharacteristic
347C2×C4C22×M1611yescharacteristic
348C8—88no—
349C22—24no—
350C22—24no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—48no—
356C22—44no—
357C22—44no—
358C22—44no—
359C22—84no—
360C22—88no—
361C22—88no—
362C22#146512yes—
363C22#146512yes—
364C22C2×#35412yes—
365C22#213212yes—
366C22C2×#40711yescharacteristic
367C22C22×#9911yescharacteristic
368C22C2×#41311yescharacteristic
369C4—22no—
370C4—21no—
371C4—21no—
372C4—22no—
373C4—44no—
374C4—44no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4—44no—
379C4—82no—
380C4C2×#35414yes—
381C2—21no—
382C2—21no—
383C2—22no—
384C2—84no—
385C2#1153114yesdirect product factor
386C2C2×#146511yescharacteristic
387C2C22×#35411yescharacteristic
388C2C2×#213211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11531C2direct (N × H)
4. C2×#1324C2semidirect
5. C2×#1319C2semidirect
23. #1324C22semidirect
24. #1319C22semidirect
85. C22×Q8M16semidirect
160. C2×C42M16semidirect
189. C2×Q8C2×M16semidirect
273. C42C2×M16semidirect
385. C2#11531direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#1242
C2×#14652
#115312
C2×#13242
C2×#13192