Finite Group Databasebeta

← C2×#11781 #481611

Subgroups — C2×#11781 #481611

Order 512.

1328 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2548C211yesmaximal, characteristic
2C2×C4×#101C211yesmaximal, characteristic, complemented
3C2×#1467C211yesmaximal, characteristic, complemented
4#11781C218yesmaximal, direct product factor
5C2×#1345C214yesmaximal, complemented
6C22×#127C411yescharacteristic
7C22×#124—21no—
8C22×C4×Q8C2211yescharacteristic
9C2×#432—21no—
10C22×#101—21no—
11C2×C4×M16—21no—
12#2548—22no—
13#2548—22no—
14#2548C414yes—
15#2548C2214yes—
16C2×#394C412yes—
17C2×#394C2212yes—
18C2×#390—21no—
19C2×#390—21no—
20C2×#390—21no—
21C2×C4×C4⋊C4C411yescharacteristic
22C4×#101—22no—
23C4×#101C2214yescomplemented
24C2×#343C2212yes—
25#1467—22no—
26#1467C2214yescomplemented
27C2×#324—21no—
28C2×#324C2212yes—
29#1345C22116yescomplemented
30C23×Q8D411yescharacteristic
31C23×D4—21no—
32C23×D4—21no—
33C23×D4C2×C412yes—
34C22×M16—22no—
35C2×#127—22no—
36C2×#127—42no—
37C2×#127—41no—
38C2×#127—41no—
39C2×#127C2×C414yes—
40C2×#124—24no—
41C2×#124—24no—
42C2×#124—48no—
43C2×C4×Q8—41no—
44C2×C4×Q8—41no—
45C2×C4×Q8—42no—
46C2×C4×Q8C2×C412yes—
47C2×C4×Q8D412yes—
48C2×C4×Q8D412yes—
49C2×C4×Q8C2312yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C42—21no—
59C22×C42C2×C411yescharacteristic
60C22×C42D411yescharacteristic
61C22×C42D411yescharacteristic
62#432—24no—
63#432—42no—
64C2×C4⋊C8—41no—
65C2×#101—28no—
66C2×#101—24no—
67C2×C22⋊C8—22no—
68C4×M16—24no—
69C4×M16—42no—
70C2×#97—41no—
71C2×#97—41no—
72C2×C4×C8—41no—
73#394—24no—
74#394—22no—
75#394—22no—
76#394C2×C418yes—
77#394C2×C414yes—
78#394C2314yes—
79#390—24no—
80#390—24no—
81#390—24no—
82#390—24no—
83#390—24no—
84#390—24no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4C2×C414yes—
87C2×#95C2×C412yes—
88#343—22no—
89#343C2314yes—
90#324—24no—
91#324—22no—
92#324C2314yes—
93C25—22no—
94C22×Q8—24no—
95C22×Q8—24no—
96C22×Q8—41no—
97C22×Q8—41no—
98C22×Q8C22⋊C412yes—
99C22×Q8C2×D412yes—
100C22×D4—24no—
101C22×D4—24no—
102C22×D4—22no—
103C22×D4—42no—
104C22×D4—42no—
105C22×D4—44no—
106C22×D4—42no—
107C22×D4—42no—
108C22×D4—44no—
109C22×D4—44no—
110C22×D4—42no—
111C22×D4—41no—
112C22×D4—41no—
113C22×D4—44no—
114C22×D4—84no—
115C22×D4C22×C414yes—
116C23×C4—22no—
117C23×C4C22⋊C411yescharacteristic
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22×C411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C2×M16—28no—
125C2×M16—24no—
126C2×M16—42no—
127C2×M16—44no—
128C22×C8—42no—
129C22×C8—42no—
130#127—44no—
131#127—44no—
132#127—44no—
133#127—44no—
134#124—416no—
135#124—416no—
136C4×Q8—42no—
137C4×Q8—42no—
138C4×Q8—42no—
139C4×Q8—42no—
140C4×Q8—42no—
141C4×Q8—42no—
142C4×Q8—42no—
143C4×Q8—42no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22×C412yes—
155C2×C4⋊C4C22×C411yescharacteristic
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×D411yescharacteristic
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C4⋊C4C2411yescharacteristic
160C2×C22⋊C4—28no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—48no—
167C2×C22⋊C4—48no—
168C2×C22⋊C4—84no—
169C2×C22⋊C4—84no—
170C2×C42—22no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—24no—
180C2×C42—44no—
181C2×C42C22⋊C412yes—
182C2×C42C22⋊C412yes—
183C2×C42C22⋊C412yes—
184C2×C42C22×C412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C4⋊C8—44no—
188#101—216no—
189C22⋊C8—24no—
190C22⋊C8—42no—
191#97—44no—
192#97—44no—
193C4×C8—44no—
194#95—22no—
195#95C22×C414yes—
196C24—24no—
197C24—44no—
198C24—44no—
199C24—44no—
200C24—44no—
201C24—42no—
202C24—42no—
203C24—44no—
204C24—44no—
205C24C2×C22⋊C411yescharacteristic
206C2×Q8—24no—
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—24no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—48no—
213C2×Q8—48no—
214C2×D4—44no—
215C2×D4—48no—
216C2×D4—48no—
217C2×D4—44no—
218C2×D4—48no—
219C2×D4—48no—
220C2×D4—44no—
221C2×D4—44no—
222C2×D4—48no—
223C2×D4—84no—
224C2×D4—88no—
225C2×D4—88no—
226C2×D4—88no—
227C2×D4—84no—
228C2×D4—88no—
229C2×D4—88no—
230C2×D4—84no—
231C2×D4—84no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—24no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4#9912yes—
257C22×C4#9911yescharacteristic
258C22×C4#9911yescharacteristic
259C22×C4#10414yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C412yes—
267C22×C4AGL(1,ℤ/8)14yes—
268C22×C4C23×C411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271M16—28no—
272M16—48no—
273M16—44no—
274C2×C8—44no—
275C2×C8—48no—
276C2×C8—44no—
277C2×C8—84no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—42no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—48no—
287C22⋊C4—416no—
288C22⋊C4—48no—
289C22⋊C4—48no—
290C22⋊C4—88no—
291C22⋊C4—84no—
292C22⋊C4—84no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—48no—
306C23—22no—
307C23—22no—
308C23—44no—
309C23—44no—
310C23—44no—
311C23—44no—
312C23—44no—
313C23—44no—
314C23—44no—
315C23—84no—
316C23—84no—
317C23—88no—
318C23—88no—
319C23—88no—
320C23—88no—
321C23—84no—
322C23—84no—
323C23—84no—
324C23—84no—
325C23C2×#9911yescharacteristic
326C23C2×#9911yescharacteristic
327C23#41812yes—
328C23C2×#10412yes—
329C23C22×C22⋊C411yescharacteristic
330Q8—48no—
331Q8—48no—
332D4—88no—
333D4—816no—
334D4—88no—
335D4—88no—
336D4—88no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—88no—
375C2×C4#35514yes—
376C2×C4C2×#9912yes—
377C2×C4C2×#9912yes—
378C2×C4#41812yes—
379C2×C4C2×#10414yes—
380C2×C4C2×AGL(1,ℤ/8)12yes—
381C8—48no—
382C8—84no—
383C22—24no—
384C22—24no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—84no—
390C22—84no—
391C22—84no—
392C22—84no—
393C22—84no—
394C22—84no—
395C22—84no—
396C22—168no—
397C22—168no—
398C22#150712yes—
399C22#150712yes—
400C22#212212yes—
401C22C2×#35512yes—
402C22C22×#9911yescharacteristic
403C22C2×#41811yescharacteristic
404C22C22×#10411yescharacteristic
405C4—22no—
406C4—22no—
407C4—22no—
408C4—24no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—42no—
414C4—42no—
415C4—84no—
416C4C2×#35514yes—
417C2—21no—
418C2—21no—
419C2—22no—
420C2—164no—
421C2#1178114yesdirect product factor
422C2C2×#150711yescharacteristic
423C2C2×#212211yescharacteristic
424C2C22×#35511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#127C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×#101C2semidirect
3. C2×#1467C2semidirect
4. #11781C2direct (N × H)
5. C2×#1345C2semidirect
23. C4×#101C22semidirect
26. #1467C22semidirect
29. #1345C22semidirect
421. C2#11781direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25482
C2×C4×#1012
C2×#14672
#117812
C2×#13452