Finite Group Databasebeta

← C2×#11754 #481671

Subgroups — C2×#11754 #481671

Order 512.

1390 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2598C211yesmaximal, characteristic
2C2×#1740C211yesmaximal, characteristic, complemented
3C2×#1464C211yesmaximal, characteristic, complemented
4#11754C218yesmaximal, direct product factor
5C2×#1352C212yesmaximal, complemented
6C2×#1343C212yesmaximal
7C22×C4×D4C411yescharacteristic, complemented
8C22×#117—21no—
9C22×#117C2211yescharacteristic
10C2×#432—21no—
11#2598—22no—
12#2598—22no—
13#2598C414yescomplemented
14#2598C2214yes—
15C2×#400C411yescharacteristic, complemented
16C2×#399C2211yescharacteristic
17C2×#394C411yescharacteristic, complemented
18C2×#392C2211yescharacteristic
19C2×#388—21no—
20C2×#386—21no—
21C2×#386—21no—
22C2×#385—21no—
23C2×#383C411yescharacteristic, complemented
24#1740—22no—
25#1740C2214yescomplemented
26C2×#342—21no—
27C2×#342C2212yes—
28#1464—22no—
29#1464C2214yescomplemented
30C2×#324—21no—
31C2×#323C2212yes—
32#1352C2218yescomplemented
33#1343C2218yes—
34C23×D4—21no—
35C23×D4C2×C411yescharacteristic
36C24×C4—21no—
37C24×C4D411yescharacteristic
38C2×C4×D4—22no—
39C2×C4×D4—42no—
40C2×C4×D4—41no—
41C2×C4×D4—41no—
42C2×C4×D4C2×C414yescomplemented
43C2×#117—28no—
44C2×#117—24no—
45C2×#117—41no—
46C2×#117—41no—
47C2×#117—42no—
48C2×#117—42no—
49C2×#117—44no—
50C2×#117C2×C412yes—
51C2×#117D412yes—
52C2×#117D412yes—
53C2×#117C2312yes—
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⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4D411yescharacteristic
66C22×C22⋊C4C2311yescharacteristic
67C22×C42—41no—
68C22×C42C2×C411yescharacteristic
69C22×C42D411yescharacteristic
70#432—24no—
71#432—42no—
72C2×C4⋊C8—41no—
73C2×C22⋊C8—22no—
74C2×#97—41no—
75#400—22no—
76#400C2×C414yescomplemented
77#399—21no—
78#399—21no—
79#399C2×C412yes—
80#399C2312yes—
81#394—22no—
82#394C2×C414yescomplemented
83#392—21no—
84#392—21no—
85#392C2×C412yes—
86#392C2312yes—
87#388—24no—
88#388—24no—
89#386—24no—
90#386—24no—
91#386—24no—
92#386—24no—
93#385—24no—
94#385—24no—
95#383—22no—
96#383C2×C414yescomplemented
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—
109C25—21no—
110C25C2×D411yescharacteristic
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—21no—
114C22×D4—41no—
115C22×D4—41no—
116C22×D4—44no—
117C22×D4—44no—
118C22×D4—42no—
119C22×D4—42no—
120C22×D4—82no—
121C22×D4C22×C412yes—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—24no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—24no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—82no—
136C23×C4—82no—
137C23×C4C22⋊C412yes—
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C22×C8—42no—
146C4×D4—44no—
147C4×D4—44no—
148C4×D4—44no—
149C4×D4—44no—
150#117—216no—
151#117—42no—
152#117—42no—
153#117—42no—
154#117—42no—
155#117—42no—
156#117—42no—
157#117—42no—
158#117—42no—
159#117—416no—
160#117—48no—
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—48no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—82no—
199C2×C22⋊C4—82no—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C22×C411yescharacteristic
202C2×C22⋊C4C2×D412yes—
203C2×C22⋊C4C2×D411yescharacteristic
204C2×C22⋊C4C2×D411yescharacteristic
205C2×C22⋊C4C2411yescharacteristic
206C2×C42—22no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—48no—
213C2×C42—44no—
214C2×C42C22⋊C412yes—
215C2×C42C22⋊C412yes—
216C2×C42C22×C412yes—
217C2×C42C2×D412yes—
218C4⋊C8—44no—
219C22⋊C8—24no—
220C22⋊C8—42no—
221#97—44no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—21no—
227#95—21no—
228#95C22×C412yes—
229#95C22×C412yes—
230C24—21no—
231C24—21no—
232C24—22no—
233C24—24no—
234C24—22no—
235C24—42no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24—41no—
241C24—41no—
242C24—42no—
243C24—42no—
244C24C2×C22⋊C411yescharacteristic
245C24C2×C22⋊C411yescharacteristic
246C24AGL(1,ℤ/8)12yes—
247C24#13712yes—
248C24C22×D411yescharacteristic
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4—48no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—44no—
261C2×D4—82no—
262C2×D4—82no—
263C2×D4—88no—
264C2×D4—88no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—28no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—28no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—48no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—48no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—84no—
314C22×C4—84no—
315C22×C4—88no—
316C22×C4—88no—
317C22×C4—84no—
318C22×C4—84no—
319C22×C4#9912yes—
320C22×C4#9911yescharacteristic
321C22×C4#9911yescharacteristic
322C22×C4#10414yes—
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C2×C8—44no—
332C2×C8—84no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—416no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C22⋊C4—28no—
343C22⋊C4—42no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—42no—
348C22⋊C4—41no—
349C22⋊C4—48no—
350C22⋊C4—48no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—48no—
355C22⋊C4—82no—
356C22⋊C4—82no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—416no—
368C23—22no—
369C23—22no—
370C23—21no—
371C23—21no—
372C23—22no—
373C23—22no—
374C23—21no—
375C23—21no—
376C23—22no—
377C23—22no—
378C23—24no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—44no—
384C23—44no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—44no—
389C23—48no—
390C23—48no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—44no—
395C23—42no—
396C23—42no—
397C23—81no—
398C23—81no—
399C23—84no—
400C23—82no—
401C23—84no—
402C23—84no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23C2×#9911yescharacteristic
407C23C2×#9911yescharacteristic
408C23#41712yes—
409C23#41712yes—
410C23C2×#10412yes—
411C23C22×C22⋊C411yescharacteristic
412C23C2×AGL(1,ℤ/8)11yescharacteristic
413C23C2×#13711yescharacteristic
414D4—44no—
415D4—44no—
416D4—48no—
417D4—84no—
418D4—84no—
419D4—88no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—28no—
425C2×C4—28no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—48no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—816no—
466C2×C4—84no—
467C2×C4—82no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—84no—
471C2×C4—816no—
472C2×C4—82no—
473C2×C4C2×#9912yes—
474C2×C4C2×#9912yes—
475C2×C4C2×#10414yes—
476C8—84no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—24no—
482C22—24no—
483C22—21no—
484C22—21no—
485C22—22no—
486C22—41no—
487C22—41no—
488C22—42no—
489C22—44no—
490C22—44no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—44no—
496C22—44no—
497C22—44no—
498C22—81no—
499C22—81no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22—88no—
505C22—82no—
506C22—82no—
507C22—88no—
508C22#150412yes—
509C22#150412yes—
510C22#212012yes—
511C22#212812yes—
512C22C22×#9911yescharacteristic
513C22C2×#41711yescharacteristic
514C22C22×#10411yescharacteristic
515C4—22no—
516C4—22no—
517C4—24no—
518C4—41no—
519C4—41no—
520C4—42no—
521C4—41no—
522C4—42no—
523C4—41no—
524C4—82no—
525C4—82no—
526C4—88no—
527C4—82no—
528C2—21no—
529C2—21no—
530C2—22no—
531C2—41no—
532C2—41no—
533C2—42no—
534C2—84no—
535C2#1175414yesdirect product factor
536C2C2×#150411yescharacteristic
537C2C2×#212011yescharacteristic
538C2C2×#212811yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC2×#392C22
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
2. C2×#1740C2semidirect
3. C2×#1464C2semidirect
4. #11754C2direct (N × H)
5. C2×#1352C2semidirect
7. C22×C4×D4C4semidirect
13. #2598C4semidirect
15. C2×#400C4semidirect
17. C2×#394C4semidirect
23. C2×#383C4semidirect
25. #1740C22semidirect
29. #1464C22semidirect
32. #1352C22semidirect
42. C2×C4×D4C2×C4semidirect
76. #400C2×C4semidirect
82. #394C2×C4semidirect
96. #383C2×C4semidirect
535. C2#11754direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25982
C2×#17402
C2×#14642
#117542
C2×#13522
C2×#13432