Finite Group Databasebeta

← C2×#11763 #481694

Subgroups — C2×#11763 #481694

Order 512.

1166 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2599C211yesmaximal, characteristic
2C2×#1742C211yesmaximal, characteristic
3C2×#1464C211yesmaximal, characteristic
4#11763C218yesmaximal, direct product factor
5C2×#1354C212yesmaximal, complemented
6C2×#1346C212yesmaximal, complemented
7C23×M16—21no—
8C22×C4×Q8C411yescharacteristic
9C22×#117C2211yescharacteristic
10C2×#432—21no—
11#2599—22no—
12#2599—22no—
13#2599C414yes—
14#2599C2214yes—
15C2×#398C2211yescharacteristic
16C2×#397C411yescharacteristic
17C2×#395C2211yescharacteristic
18C2×#393C411yescharacteristic
19C2×#388—21no—
20C2×#387—21no—
21C2×#387—21no—
22C2×#385—21no—
23C2×#383C411yescharacteristic
24#1742—22no—
25#1742C2214yes—
26C2×#343—21no—
27C2×#343C2212yes—
28#1464—22no—
29#1464C2214yes—
30C2×#324C2212yes—
31C2×#323—21no—
32#1354C2218yescomplemented
33#1346C2218yescomplemented
34C23×Q8C2×C411yescharacteristic
35C24×C4D411yescharacteristic
36C22×M16—28no—
37C22×M16—22no—
38C22×M16—24no—
39C22×M16—41no—
40C22×M16—42no—
41C22×M16—44no—
42C23×C8—41no—
43C2×C4×Q8—22no—
44C2×C4×Q8—42no—
45C2×C4×Q8—41no—
46C2×C4×Q8—41no—
47C2×C4×Q8C2×C414yes—
48C2×#117—41no—
49C2×#117—41no—
50C2×#117—42no—
51C2×#117C2×C412yes—
52C2×#117D412yes—
53C2×#117D412yes—
54C2×#117C2312yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4D411yescharacteristic
65C22×C42—21no—
66C22×C42C2×C411yescharacteristic
67C22×C42D411yescharacteristic
68#432—24no—
69#432—42no—
70C2×C4⋊C8—41no—
71C2×C22⋊C8—22no—
72C2×#97—41no—
73#398—21no—
74#398—21no—
75#398C2×C412yes—
76#398C2312yes—
77#397—22no—
78#397C2×C414yes—
79#395—21no—
80#395—21no—
81#395C2×C412yes—
82#395C2312yes—
83#393—22no—
84#393C2×C414yes—
85#388—24no—
86#388—24no—
87#387—24no—
88#387—24no—
89#387—24no—
90#387—24no—
91#385—24no—
92#385—24no—
93#383—22no—
94#383C2×C414yes—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103#343—24no—
104#343—22no—
105#343C2314yes—
106#324—22no—
107#324C2314yes—
108#323—24no—
109C25C22⋊C411yescharacteristic
110C22×Q8—21no—
111C22×Q8—44no—
112C22×Q8—41no—
113C22×Q8—41no—
114C22×Q8C22×C412yes—
115C23×C4—24no—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4C22⋊C412yes—
126C23×C4C22⋊C411yescharacteristic
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C2×M16—28no—
134C2×M16—216no—
135C2×M16—24no—
136C2×M16—416no—
137C2×M16—48no—
138C2×M16—42no—
139C2×M16—44no—
140C2×M16—48no—
141C2×M16—44no—
142C22×C8—48no—
143C22×C8—42no—
144C22×C8—44no—
145C22×C8—42no—
146C4×Q8—44no—
147C4×Q8—44no—
148C4×Q8—44no—
149C4×Q8—44no—
150#117—42no—
151#117—42no—
152#117—42no—
153#117—42no—
154#117—42no—
155#117—42no—
156#117—42no—
157#117—42no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C4⋊C4C2×D412yes—
180C2×C4⋊C4C2×D411yescharacteristic
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C4⋊C4C2411yescharacteristic
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4C22⋊C412yes—
192C2×C22⋊C4C22⋊C412yes—
193C2×C22⋊C4C22×C412yes—
194C2×C22⋊C4C2×D412yes—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—24no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42C22⋊C412yes—
205C2×C42C22⋊C412yes—
206C2×C42C22×C412yes—
207C2×C42C2×D412yes—
208C4⋊C8—44no—
209C22⋊C8—24no—
210C22⋊C8—42no—
211#97—44no—
212#95—24no—
213#95—24no—
214#95—22no—
215#95—24no—
216#95—24no—
217#95—22no—
218#95—21no—
219#95—21no—
220#95—22no—
221#95—22no—
222#95C22×C412yes—
223#95C22×C412yes—
224C24—21no—
225C24—21no—
226C24—22no—
227C24—42no—
228C24—42no—
229C24#10414yes—
230C24C2×C22⋊C411yescharacteristic
231C24C2×C22⋊C412yes—
232C2×Q8—44no—
233C2×Q8—42no—
234C2×Q8—44no—
235C2×Q8—42no—
236C2×Q8—48no—
237C2×Q8—44no—
238C2×Q8—44no—
239C22×C4—24no—
240C22×C4—28no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—28no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4#9912yes—
290C22×C4#9911yescharacteristic
291C22×C4#9911yescharacteristic
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4AGL(1,ℤ/8)12yes—
299C22×C4#13712yes—
300C22×C4C23×C411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303M16—28no—
304M16—416no—
305M16—48no—
306M16—44no—
307C2×C8—416no—
308C2×C8—44no—
309C2×C8—48no—
310C2×C8—44no—
311C2×C8—84no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—48no—
332C22⋊C4—48no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—48no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—24no—
346C23—22no—
347C23—24no—
348C23—42no—
349C23—48no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—81no—
354C23—81no—
355C23—84no—
356C23C2×#9911yescharacteristic
357C23C2×#9911yescharacteristic
358C23#41712yes—
359C23C2×#10412yes—
360C23C2×#10414yes—
361C23C22×C22⋊C411yescharacteristic
362Q8—44no—
363Q8—44no—
364Q8—48no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—28no—
370C2×C4—28no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—84no—
419C2×C4—84no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4C2×#9912yes—
423C2×C4C2×#9912yes—
424C2×C4#41712yes—
425C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
426C2×C4C2×#13711yescharacteristic
427C8—48no—
428C8—84no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—24no—
434C22—24no—
435C22—24no—
436C22—44no—
437C22—42no—
438C22—42no—
439C22—48no—
440C22—82no—
441C22—84no—
442C22#150412yes—
443C22#150412yes—
444C22#212012yes—
445C22#213512yes—
446C22C22×#9911yescharacteristic
447C22C2×#41711yescharacteristic
448C22C22×#10411yescharacteristic
449C4—21no—
450C4—21no—
451C4—22no—
452C4—22no—
453C4—22no—
454C4—41no—
455C4—44no—
456C4—41no—
457C4—41no—
458C4—41no—
459C4—42no—
460C4—42no—
461C4—82no—
462C4—82no—
463C4—82no—
464C2—21no—
465C2—21no—
466C2—22no—
467C2—44no—
468C2#1176314yesdirect product factor
469C2C2×#150411yescharacteristic
470C2C2×#212011yescharacteristic
471C2C2×#213511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ 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
4. #11763C2direct (N × H)
5. C2×#1354C2semidirect
6. C2×#1346C2semidirect
32. #1354C22semidirect
33. #1346C22semidirect
468. C2#11763direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25992
C2×#17422
C2×#14642
#117632
C2×#13542
C2×#13462