Finite Group Databasebeta

← C2×#11768 #481697

Subgroups — C2×#11768 #481697

Order 512.

1270 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2583C211yesmaximal, characteristic
2C22×#343C211yesmaximal, characteristic
3C2×#1463C211yesmaximal, characteristic
4#11768C218yesmaximal, direct product factor
5C2×#1354C212yesmaximal, complemented
6C2×#1346C212yesmaximal, complemented
7C23×M16—21no—
8C22×C22⋊Q8C411yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10C2×#407—21no—
11#2583—22no—
12#2583—22no—
13#2583C414yes—
14#2583C2214yes—
15C2×#398C411yescharacteristic
16C2×#398C2211yescharacteristic
17C2×#395C411yescharacteristic
18C2×#395C2211yescharacteristic
19C2×#389—21no—
20C2×#389—21no—
21C2×#385—21no—
22C2×#385—21no—
23C2×C4×C22⋊C4C411yescharacteristic
24C2×#343—22no—
25C2×#343—21no—
26C2×#343C2212yes—
27C2×#343C2214yes—
28#1463—22no—
29#1463C2214yes—
30C2×#324—21no—
31C2×#324C2212yes—
32#1354C2218yescomplemented
33#1346C2218yescomplemented
34C23×Q8C2×C411yescharacteristic
35C24×C4C2×C411yescharacteristic
36C24×C4D411yescharacteristic
37C24×C4D411yescharacteristic
38C22×M16—28no—
39C22×M16—24no—
40C22×M16—22no—
41C22×M16—42no—
42C22×M16—41no—
43C22×M16—44no—
44C22×M16—41no—
45C23×C8—41no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—42no—
48C2×C22⋊Q8—41no—
49C2×C22⋊Q8—41no—
50C2×C22⋊Q8C2×C414yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—41no—
57C22×C4⋊C4—41no—
58C22×C4⋊C4—42no—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4D412yes—
63C22×C4⋊C4D412yes—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C42—21no—
71#407—24no—
72#407—42no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—41no—
75#398—22no—
76#398—21no—
77#398—21no—
78#398C2×C412yes—
79#398C2×C414yes—
80#398C2312yes—
81#395—22no—
82#395—21no—
83#395—21no—
84#395C2×C412yes—
85#395C2×C414yes—
86#395C2312yes—
87#389—24no—
88#389—24no—
89#389—24no—
90#389—24no—
91#385—24no—
92#385—24no—
93#385—24no—
94#385—24no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4C2×C414yes—
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—24no—
107#324—22no—
108#324C2314yes—
109C25C22⋊C411yescharacteristic
110C22×Q8—21no—
111C22×Q8—41no—
112C22×Q8—41no—
113C22×Q8—82no—
114C22×Q8C22×C412yes—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—24no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—82no—
132C23×C4—82no—
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C412yes—
138C23×C4C22⋊C412yes—
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C2×M16—28no—
146C2×M16—216no—
147C2×M16—24no—
148C2×M16—416no—
149C2×M16—48no—
150C2×M16—44no—
151C2×M16—42no—
152C2×M16—48no—
153C2×M16—44no—
154C2×M16—44no—
155C2×M16—84no—
156C22×C8—48no—
157C22×C8—44no—
158C22×C8—42no—
159C22×C8—42no—
160C22⋊Q8—44no—
161C22⋊Q8—44no—
162C22⋊Q8—44no—
163C22⋊Q8—44no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2411yescharacteristic
208C2×C22⋊C4—28no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C42—24no—
215C2×C42—82no—
216C22⋊C8—24no—
217C22⋊C8—42no—
218C22⋊C8—44no—
219#95—22no—
220#95—24no—
221#95—24no—
222#95—22no—
223#95—21no—
224#95—21no—
225#95—22no—
226#95—22no—
227#95C22×C412yes—
228#95C22×C412yes—
229C24—24no—
230C24—22no—
231C24—22no—
232C24—42no—
233C24#9911yescharacteristic
234C24#9912yes—
235C24#9911yescharacteristic
236C24C2×C22⋊C412yes—
237C24C2×C22⋊C411yescharacteristic
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—82no—
241C2×Q8—82no—
242C2×Q8—84no—
243C2×Q8—84no—
244C22×C4—24no—
245C22×C4—28no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—28no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—84no—
310C22×C4—84no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4#10414yes—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C412yes—
320C22×C4AGL(1,ℤ/8)12yes—
321C22×C4#13712yes—
322C22×C4C23×C411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325M16—28no—
326M16—416no—
327M16—48no—
328M16—44no—
329M16—88no—
330C2×C8—416no—
331C2×C8—44no—
332C2×C8—48no—
333C2×C8—44no—
334C2×C8—84no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—42no—
346C4⋊C4—48no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—44no—
351C4⋊C4—48no—
352C4⋊C4—84no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C22⋊C4—216no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C42—84no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—24no—
366C23—24no—
367C23—22no—
368C23—22no—
369C23—24no—
370C23—22no—
371C23—42no—
372C23—42no—
373C23—48no—
374C23—48no—
375C23#35614yes—
376C23C2×#9912yes—
377C23C2×#9912yes—
378C23C2×#9911yescharacteristic
379C23C2×#9911yescharacteristic
380C23#41712yes—
381C23C2×#10412yes—
382C23C22×C22⋊C411yescharacteristic
383Q8—84no—
384Q8—84no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—28no—
390C2×C4—28no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—48no—
415C2×C4—48no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4#41712yes—
450C2×C4C2×#10414yes—
451C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
452C2×C4C2×#13711yescharacteristic
453C8—48no—
454C8—84no—
455C22—21no—
456C22—21no—
457C22—22no—
458C22—21no—
459C22—21no—
460C22—22no—
461C22—24no—
462C22—24no—
463C22—28no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—44no—
468C22—44no—
469C22—48no—
470C22—44no—
471C22#150412yes—
472C22#150412yes—
473C22#212112yes—
474C22C2×#35614yes—
475C22C2×#35612yes—
476C22C22×#9911yescharacteristic
477C22C2×#41711yescharacteristic
478C22C22×#10411yescharacteristic
479C4—21no—
480C4—21no—
481C4—22no—
482C4—24no—
483C4—44no—
484C4—44no—
485C4—41no—
486C4—42no—
487C4—41no—
488C4—42no—
489C4—41no—
490C4—41no—
491C4—82no—
492C4—82no—
493C2—21no—
494C2—21no—
495C2—24no—
496C2—22no—
497C2—42no—
498C2#1176814yesdirect product factor
499C2C2×#150411yescharacteristic
500C2C2×#212111yescharacteristic
501C2C22×#35611yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11768C2direct (N × H)
5. C2×#1354C2semidirect
6. C2×#1346C2semidirect
32. #1354C22semidirect
33. #1346C22semidirect
498. C2#11768direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25832
C22×#3432
C2×#14632
#117682
C2×#13542
C2×#13462