Finite Group Databasebeta

← C2×#11789 #481631

Subgroups — C2×#11789 #481631

Order 512.

1250 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2564C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3C2×#1466C211yesmaximal, characteristic
4#11789C218yesmaximal, direct product factor
5C2×#1352C212yesmaximal, complemented
6C2×#1351C212yesmaximal, complemented
7C22×C22⋊Q8C411yescharacteristic, complemented
8C22×C4×D4C2211yescharacteristic
9C2×#432—21no—
10C22×#99—21no—
11#2564—22no—
12#2564—22no—
13#2564C414yescomplemented
14#2564C2214yes—
15C2×#400C411yescharacteristic, complemented
16C2×#399C2211yescharacteristic
17C2×#395C411yescharacteristic, complemented
18C2×#393C2211yescharacteristic
19C2×#387—21no—
20C2×#386—21no—
21C2×#385—21no—
22C2×#380—21no—
23C2×C4×C22⋊C4—21no—
24C22×#95C411yescharacteristic, complemented
25C4×#99—22no—
26C4×#99C2214yes—
27C2×#342C2212yes—
28#1466—22no—
29#1466C2214yes—
30C2×#323—21no—
31C2×#323C2212yes—
32#1352C2218yescomplemented
33#1351C2218yescomplemented
34C23×Q8C2×C411yescharacteristic
35C23×D4D411yescharacteristic
36C24×C4—21no—
37C24×C4C2×C411yescharacteristic
38C24×C4D411yescharacteristic
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—42no—
41C2×C22⋊Q8—41no—
42C2×C22⋊Q8—41no—
43C2×C22⋊Q8C2×C414yescomplemented
44C2×C4×D4—41no—
45C2×C4×D4—41no—
46C2×C4×D4—42no—
47C2×C4×D4C2×C412yes—
48C2×C4×D4D412yes—
49C2×C4×D4D412yes—
50C2×C4×D4C2312yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C22⋊C4—22no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4D411yescharacteristic
61C22×C22⋊C4C2311yescharacteristic
62C22×C42—41no—
63C22×C42D411yescharacteristic
64#432—24no—
65#432—42no—
66C2×C4⋊C8—41no—
67C2×#99—28no—
68C2×#99—24no—
69C2×C22⋊C8—22no—
70C2×#97—41no—
71#400—22no—
72#400C2×C414yescomplemented
73#399—21no—
74#399—21no—
75#399C2×C412yes—
76#399C2312yes—
77#395—22no—
78#395C2×C414yescomplemented
79#393—21no—
80#393—21no—
81#393C2×C412yes—
82#393C2312yes—
83#387—24no—
84#387—24no—
85#386—24no—
86#386—24no—
87#385—24no—
88#385—24no—
89#380—24no—
90#380—24no—
91C4×C22⋊C4—24no—
92C4×C22⋊C4—42no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—41no—
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C414yescomplemented
103#342—22no—
104#342C2314yes—
105#323—24no—
106#323—22no—
107#323C2314yes—
108C25C22⋊C411yescharacteristic
109C25C2×D411yescharacteristic
110C22×Q8—21no—
111C22×Q8—41no—
112C22×Q8—41no—
113C22×Q8—82no—
114C22×Q8C22×C412yes—
115C22×D4—24no—
116C22×D4—24no—
117C22×D4—41no—
118C22×D4—41no—
119C22×D4C22⋊C412yes—
120C22×D4C2×D412yes—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—24no—
132C23×C4—44no—
133C23×C4—42no—
134C23×C4—44no—
135C23×C4—82no—
136C23×C4—82no—
137C23×C4—82no—
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C412yes—
141C23×C4C22⋊C412yes—
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C22×C8—42no—
148C22⋊Q8—44no—
149C22⋊Q8—44no—
150C22⋊Q8—44no—
151C22⋊Q8—44no—
152C4×D4—42no—
153C4×D4—42no—
154C4×D4—42no—
155C4×D4—42no—
156C4×D4—42no—
157C4×D4—42no—
158C4×D4—42no—
159C4×D4—42no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—82no—
194C2×C22⋊C4C22⋊C412yes—
195C2×C22⋊C4C22×C411yescharacteristic
196C2×C22⋊C4C2×D411yescharacteristic
197C2×C22⋊C4C2×D412yes—
198C2×C22⋊C4C2×D411yescharacteristic
199C2×C22⋊C4C2411yescharacteristic
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—44no—
204C2×C42—84no—
205C2×C42C22⋊C412yes—
206C2×C42C2×D412yes—
207C4⋊C8—44no—
208#99—216no—
209C22⋊C8—24no—
210C22⋊C8—42no—
211#97—44no—
212#95—21no—
213#95—21no—
214#95—24no—
215#95—24no—
216#95—22no—
217#95—24no—
218#95—24no—
219#95—22no—
220#95—44no—
221#95C22×C412yes—
222#95C22×C412yes—
223C24—21no—
224C24—21no—
225C24—21no—
226C24—22no—
227C24—21no—
228C24—22no—
229C24—42no—
230C24—42no—
231C24—42no—
232C24—42no—
233C24#10414yes—
234C24C2×C22⋊C411yescharacteristic
235C24C2×C22⋊C411yescharacteristic
236C24C2×C22⋊C412yes—
237C24#13714yes—
238C24C22×D411yescharacteristic
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—82no—
242C2×Q8—82no—
243C2×Q8—84no—
244C2×Q8—84no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—48no—
252C2×D4—48no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—48no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—48no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—48no—
297C22×C4—42no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—84no—
304C22×C4—84no—
305C22×C4—84no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—816no—
309C22×C4—84no—
310C22×C4—84no—
311C22×C4#9911yescharacteristic
312C22×C4#9911yescharacteristic
313C22×C4#9912yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C412yes—
319C22×C4C23×C411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C2×C8—44no—
322C2×C8—84no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—84no—
332C22⋊C4—28no—
333C22⋊C4—42no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—42no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—48no—
342C22⋊C4—48no—
343C22⋊C4—44no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—88no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—21no—
355C23—21no—
356C23—22no—
357C23—24no—
358C23—24no—
359C23—24no—
360C23—21no—
361C23—21no—
362C23—24no—
363C23—42no—
364C23—42no—
365C23—48no—
366C23—42no—
367C23—48no—
368C23—42no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—48no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—81no—
379C23—84no—
380C23—81no—
381C23C2×#9911yescharacteristic
382C23C2×#9911yescharacteristic
383C23#41912yes—
384C23#41912yes—
385C23C2×#10412yes—
386C23C2×#10414yes—
387C23C22×C22⋊C411yescharacteristic
388C23C2×#13712yes—
389Q8—84no—
390Q8—84no—
391D4—48no—
392D4—48no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—48no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—48no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—816no—
441C2×C4—82no—
442C2×C4—84no—
443C2×C4—82no—
444C2×C4—84no—
445C2×C4—816no—
446C2×C4—84no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4C2×#9912yes—
450C2×C4C2×#9912yes—
451C8—84no—
452C22—24no—
453C22—24no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—22no—
459C22—22no—
460C22—24no—
461C22—44no—
462C22—42no—
463C22—42no—
464C22—44no—
465C22—48no—
466C22—48no—
467C22—48no—
468C22—41no—
469C22—41no—
470C22—42no—
471C22—41no—
472C22—42no—
473C22—41no—
474C22—81no—
475C22—81no—
476C22—84no—
477C22#150812yes—
478C22#150812yes—
479C22#212112yes—
480C22#212612yes—
481C22C22×#9911yescharacteristic
482C22C2×#41911yescharacteristic
483C22C22×#10411yescharacteristic
484C4—22no—
485C4—22no—
486C4—44no—
487C4—42no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—41no—
492C4—42no—
493C4—82no—
494C4—88no—
495C4—82no—
496C4—82no—
497C2—21no—
498C2—21no—
499C2—22no—
500C2—42no—
501C2—44no—
502C2—42no—
503C2#1178914yesdirect product factor
504C2C2×#150811yescharacteristic
505C2C2×#212111yescharacteristic
506C2C2×#212611yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11789C2direct (N × H)
5. C2×#1352C2semidirect
6. C2×#1351C2semidirect
7. C22×C22⋊Q8C4semidirect
13. #2564C4semidirect
15. C2×#400C4semidirect
17. C2×#395C4semidirect
24. C22×#95C4semidirect
32. #1352C22semidirect
33. #1351C22semidirect
43. C2×C22⋊Q8C2×C4semidirect
72. #400C2×C4semidirect
78. #395C2×C4semidirect
102. C2×#95C2×C4semidirect
503. C2#11789direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25642
C2×C4×#992
C2×#14662
#117892
C2×#13522
C2×#13512