Finite Group Databasebeta

← C2×#11756 #481672

Subgroups — C2×#11756 #481672

Order 512.

1206 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2599C211yesmaximal, characteristic
2C2×#1740C211yesmaximal, characteristic
3C2×#1464C211yesmaximal, characteristic
4#11756C218yesmaximal, direct product factor
5C2×#1351C212yesmaximal
6C2×#1344C212yesmaximal
7C22×C4×Q8C411yescharacteristic, complemented
8C22×#117—21no—
9C22×#117C2211yescharacteristic
10C2×#432—21no—
11#2599—22no—
12#2599—22no—
13#2599C414yescomplemented
14#2599C2214yes—
15C2×#398C411yescharacteristic, complemented
16C2×#397C2211yescharacteristic
17C2×#395C411yescharacteristic, complemented
18C2×#393C2211yescharacteristic
19C2×#388—21no—
20C2×#387—21no—
21C2×#387—21no—
22C2×#385—21no—
23C2×#383C411yescharacteristic, complemented
24#1740—22no—
25#1740C2214yes—
26C2×#342—21no—
27C2×#342C2212yes—
28#1464—22no—
29#1464C2214yes—
30C2×#324—21no—
31C2×#323C2212yes—
32#1351C2218yes—
33#1344C2218yes—
34C23×Q8C2×C411yescharacteristic
35C24×C4D411yescharacteristic
36C2×C4×Q8—22no—
37C2×C4×Q8—42no—
38C2×C4×Q8—41no—
39C2×C4×Q8—41no—
40C2×C4×Q8C2×C414yescomplemented
41C2×#117—28no—
42C2×#117—24no—
43C2×#117—41no—
44C2×#117—41no—
45C2×#117—42no—
46C2×#117—42no—
47C2×#117—44no—
48C2×#117C2×C412yes—
49C2×#117D412yes—
50C2×#117D412yes—
51C2×#117C2312yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—41no—
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C22⋊C4—22no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4D411yescharacteristic
63C22×C22⋊C4C2311yescharacteristic
64C22×C42—21no—
65C22×C42—41no—
66C22×C42C2×C411yescharacteristic
67C22×C42D411yescharacteristic
68#432—24no—
69#432—42no—
70C2×C4⋊C8—41no—
71C2×C22⋊C8—22no—
72C2×#97—41no—
73#398—22no—
74#398C2×C414yescomplemented
75#397—21no—
76#397—21no—
77#397C2×C412yes—
78#397C2312yes—
79#395—22no—
80#395C2×C414yescomplemented
81#393—21no—
82#393—21no—
83#393C2×C412yes—
84#393C2312yes—
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×C414yescomplemented
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#342—24no—
104#342—22no—
105#342C2314yes—
106#324—24no—
107#323—22no—
108#323C2314yes—
109C25C2×D411yescharacteristic
110C22×Q8—21no—
111C22×Q8—44no—
112C22×Q8—41no—
113C22×Q8—41no—
114C22×Q8C22×C412yes—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—24no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—42no—
126C23×C4C22⋊C412yes—
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C22×C8—42no—
135C4×Q8—44no—
136C4×Q8—44no—
137C4×Q8—44no—
138C4×Q8—44no—
139#117—216no—
140#117—42no—
141#117—42no—
142#117—42no—
143#117—42no—
144#117—42no—
145#117—42no—
146#117—42no—
147#117—42no—
148#117—416no—
149#117—48no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—48no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C2×D412yes—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4C22⋊C412yes—
188C2×C22⋊C4C22×C411yescharacteristic
189C2×C22⋊C4C2×D412yes—
190C2×C22⋊C4C2×D411yescharacteristic
191C2×C22⋊C4C2×D411yescharacteristic
192C2×C22⋊C4C2411yescharacteristic
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—24no—
200C2×C42—48no—
201C2×C42—44no—
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—24no—
215#95—22no—
216#95—22no—
217#95—24no—
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—
229C24C2×C22⋊C411yescharacteristic
230C24C2×C22⋊C411yescharacteristic
231C24AGL(1,ℤ/8)12yes—
232C24#13712yes—
233C24C22×D411yescharacteristic
234C2×Q8—44no—
235C2×Q8—42no—
236C2×Q8—44no—
237C2×Q8—42no—
238C2×Q8—48no—
239C2×Q8—44no—
240C2×Q8—44no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—28no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—28no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—48no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—84no—
294C22×C4—84no—
295C22×C4#9912yes—
296C22×C4#9911yescharacteristic
297C22×C4#9911yescharacteristic
298C22×C4#10414yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C412yes—
305C22×C4C23×C411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C2×C8—44no—
308C2×C8—84no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—48no—
315C4⋊C4—42no—
316C4⋊C4—416no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—48no—
320C22⋊C4—28no—
321C22⋊C4—42no—
322C22⋊C4—41no—
323C22⋊C4—42no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—48no—
330C22⋊C4—42no—
331C22⋊C4—44no—
332C22⋊C4—42no—
333C22⋊C4—44no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—48no—
343C42—416no—
344C23—22no—
345C23—22no—
346C23—21no—
347C23—21no—
348C23—22no—
349C23—22no—
350C23—21no—
351C23—21no—
352C23—22no—
353C23—22no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—44no—
358C23—44no—
359C23—41no—
360C23—42no—
361C23—42no—
362C23—81no—
363C23—81no—
364C23—84no—
365C23C2×#9911yescharacteristic
366C23C2×#9911yescharacteristic
367C23#41712yes—
368C23#41712yes—
369C23C2×#10412yes—
370C23C22×C22⋊C411yescharacteristic
371C23C2×AGL(1,ℤ/8)11yescharacteristic
372C23C2×#13711yescharacteristic
373Q8—44no—
374Q8—44no—
375Q8—48no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—28no—
381C2×C4—28no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—48no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—84no—
427C2×C4—84no—
428C2×C4—84no—
429C2×C4—84no—
430C2×C4—816no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4C2×#9912yes—
434C2×C4C2×#9912yes—
435C2×C4C2×#10414yes—
436C8—84no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—24no—
442C22—24no—
443C22—21no—
444C22—21no—
445C22—22no—
446C22—41no—
447C22—41no—
448C22—42no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—44no—
454C22—44no—
455C22—81no—
456C22—81no—
457C22—82no—
458C22—82no—
459C22#150412yes—
460C22#150412yes—
461C22#212012yes—
462C22#212912yes—
463C22C22×#9911yescharacteristic
464C22C2×#41711yescharacteristic
465C22C22×#10411yescharacteristic
466C4—22no—
467C4—22no—
468C4—24no—
469C4—44no—
470C4—41no—
471C4—41no—
472C4—42no—
473C4—41no—
474C4—42no—
475C4—41no—
476C4—82no—
477C4—82no—
478C4—88no—
479C4—82no—
480C2—21no—
481C2—21no—
482C2—22no—
483C2—41no—
484C2—41no—
485C2—42no—
486C2#1175614yesdirect product factor
487C2C2×#150411yescharacteristic
488C2C2×#212011yescharacteristic
489C2C2×#212911yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11756C2direct (N × H)
7. C22×C4×Q8C4semidirect
13. #2599C4semidirect
15. C2×#398C4semidirect
17. C2×#395C4semidirect
23. C2×#383C4semidirect
40. C2×C4×Q8C2×C4semidirect
74. #398C2×C4semidirect
80. #395C2×C4semidirect
94. #383C2×C4semidirect
486. C2#11756direct (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×#17402
C2×#14642
#117562
C2×#13512
C2×#13442