Finite Group Databasebeta

← C2×#9462 #201281

Subgroups — C2×#9462 #201281

Order 512.

2134 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2434C211yesmaximal, characteristic
2C2×#2310C211yesmaximal, characteristic
3C2×#1784C211yesmaximal, characteristic
4C2×#1511C212yesmaximal
5#9462C218yesmaximal, direct product factor
6C2×#1284C212yesmaximal
7C24×Q8C2211yescharacteristic
8C22×C4⋊Q8C412yescomplemented
9C22×C22⋊Q8C2211yescharacteristic
10C22×#117—21no—
11C22×#101—22no—
12C2×#407C2212yes—
13#2434—22no—
14#2434—22no—
15#2434C414yescomplemented
16#2434C2214yes—
17C2×#395—22no—
18#2310—24no—
19#2310—22no—
20#2310C2214yes—
21C2×#391—24no—
22C2×#391C412yescomplemented
23C2×#385—21no—
24C2×#385—21no—
25C2×#381—21no—
26C2×#381C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#1784—22no—
29#1784C2214yes—
30#1511—24no—
31#1511C2218yes—
32C2×#328—24no—
33#1284C2218yes—
34C23×Q8—22no—
35C23×Q8—22no—
36C23×Q8—22no—
37C23×Q8—22no—
38C23×Q8—48no—
39C23×Q8C2×C411yescharacteristic
40C23×Q8C2×C412yes—
41C23×Q8C2×C412yes—
42C23×Q8D412yes—
43C23×Q8D412yes—
44C23×Q8D412yes—
45C23×Q8C2312yes—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—48no—
52C2×C4⋊Q8C2×C418yescomplemented
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8C2×C412yes—
60C2×C22⋊Q8C2312yes—
61C2×#117—24no—
62C2×#117—24no—
63C2×#117—48no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—21no—
72C22×C42—22no—
73C22×C42C2×C412yes—
74C2×#101—28no—
75C2×#101—48no—
76#407—22no—
77#407—22no—
78#407—22no—
79#407C2314yes—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82#395—28no—
83#395—28no—
84#391—216no—
85#391—24no—
86#391—216no—
87#391C2×C418yescomplemented
88#385—24no—
89#385—24no—
90#385—24no—
91#385—24no—
92#381—24no—
93#381—22no—
94#381—22no—
95#381—24no—
96#381D414yes—
97#381C2314yes—
98C4×C22⋊C4—21no—
99C4×C22⋊C4—21no—
100C4×C22⋊C4C2×C412yes—
101C4×C22⋊C4C2312yes—
102C2×#95—24no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95D414yes—
108#328—216no—
109C25C2×D411yescharacteristic
110C22×Q8—28no—
111C22×Q8—28no—
112C22×Q8—21no—
113C22×Q8—24no—
114C22×Q8—24no—
115C22×Q8—24no—
116C22×Q8—24no—
117C22×Q8—24no—
118C22×Q8—24no—
119C22×Q8—24no—
120C22×Q8—24no—
121C22×Q8—24no—
122C22×Q8—24no—
123C22×Q8—24no—
124C22×Q8—24no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—44no—
128C22×Q8—44no—
129C22×Q8—48no—
130C22×Q8—48no—
131C22×Q8—48no—
132C22×Q8—48no—
133C22×Q8—48no—
134C22×Q8—48no—
135C22×Q8—416no—
136C22×Q8—416no—
137C22×Q8—416no—
138C22×Q8—44no—
139C22×Q8C22⋊C414yes—
140C22×Q8C22⋊C414yes—
141C22×Q8C22×C412yes—
142C22×Q8C22×C414yes—
143C22×Q8C2×D414yes—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—28no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—24no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4C22⋊C412yes—
160C23×C4C22⋊C412yes—
161C23×C4C22⋊C412yes—
162C23×C4C22⋊C412yes—
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4#4112yes—
173C23×C4#4112yes—
174C23×C4C2411yescharacteristic
175C2×M16—24no—
176C2×M16—42no—
177C2×M16—48no—
178C22×C8—42no—
179C4⋊Q8—416no—
180C4⋊Q8—416no—
181C22⋊Q8—24no—
182C22⋊Q8—24no—
183C22⋊Q8—24no—
184C22⋊Q8—24no—
185C22⋊Q8—42no—
186C22⋊Q8—42no—
187C22⋊Q8—42no—
188C22⋊Q8—42no—
189#117—416no—
190#117—416no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—48no—
200C2×C4⋊C4—48no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4C22×C412yes—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—48no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C42—28no—
218C2×C42—22no—
219C2×C42—44no—
220C2×C42—48no—
221C2×C42C22×C414yes—
222#101—416no—
223C22⋊C8—24no—
224C22⋊C8—24no—
225C22⋊C8—42no—
226C22⋊C8—42no—
227#95—216no—
228#95—22no—
229#95—28no—
230#95—28no—
231#95—24no—
232#95—24no—
233#95—22no—
234#95—24no—
235#95C2×D418yes—
236C24—24no—
237C24—24no—
238C24—42no—
239C24C2×C22⋊C411yescharacteristic
240C24C2×C22⋊C411yescharacteristic
241C24#12011yescharacteristic
242C24#12011yescharacteristic
243C24#12311yescharacteristic
244C24#12311yescharacteristic
245C24C22×D411yescharacteristic
246C2×Q8—28no—
247C2×Q8—28no—
248C2×Q8—28no—
249C2×Q8—28no—
250C2×Q8—28no—
251C2×Q8—28no—
252C2×Q8—28no—
253C2×Q8—28no—
254C2×Q8—24no—
255C2×Q8—44no—
256C2×Q8—44no—
257C2×Q8—44no—
258C2×Q8—44no—
259C2×Q8—416no—
260C2×Q8—416no—
261C2×Q8—416no—
262C2×Q8—416no—
263C2×Q8—48no—
264C2×Q8—48no—
265C2×Q8—48no—
266C2×Q8—48no—
267C2×Q8—416no—
268C2×Q8—416no—
269C2×Q8—416no—
270C2×Q8—416no—
271C2×Q8—416no—
272C2×Q8—416no—
273C2×Q8—44no—
274C2×Q8—44no—
275C2×Q8—42no—
276C2×Q8—84no—
277C2×Q8—84no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—28no—
281C22×C4—28no—
282C22×C4—28no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—28no—
293C22×C4—28no—
294C22×C4—28no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—28no—
307C22×C4—44no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—42no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—48no—
324C22×C4—48no—
325C22×C4—44no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—44no—
331C22×C4—48no—
332C22×C4#9918yes—
333C22×C4C2×C22⋊C414yes—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C414yes—
337C22×C4C4×D412yes—
338C22×C4C4×D412yes—
339C22×C4C4×D412yes—
340C22×C4C4×D412yes—
341C22×C4#12012yes—
342C22×C4#12112yes—
343C22×C4#12112yes—
344C22×C4#12112yes—
345C22×C4#12112yes—
346C22×C4#12312yes—
347C22×C4C23×C411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D412yes—
350C22×C4C2×#4112yes—
351M16—48no—
352M16—84no—
353C2×C8—44no—
354C2×C8—84no—
355C4⋊C4—24no—
356C4⋊C4—416no—
357C4⋊C4—416no—
358C4⋊C4—44no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—416no—
362C22⋊C4—28no—
363C22⋊C4—28no—
364C22⋊C4—48no—
365C22⋊C4—48no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C42—44no—
371C42—416no—
372C42—44no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—416no—
393C23#38611yescharacteristic
394C23#38611yescharacteristic
395C23#38611yescharacteristic
396C23#38611yescharacteristic
397C23C2×#9914yes—
398C23#45614yes—
399C23C22×C22⋊C411yescharacteristic
400C23C2×#12011yescharacteristic
401C23C2×#12311yescharacteristic
402Q8—416no—
403Q8—416no—
404Q8—416no—
405Q8—416no—
406Q8—84no—
407Q8—84no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—28no—
411C2×C4—28no—
412C2×C4—28no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—28no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—416no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—48no—
447C2×C4—48no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—48no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—816no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4#38612yes—
459C2×C4#38612yes—
460C2×C4C2×#9918yes—
461C2×C4C2×C4×D412yes—
462C2×C4C2×#12112yes—
463C8—84no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—28no—
473C22—28no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—44no—
481C22—44no—
482C22—44no—
483C22—44no—
484C22—44no—
485C22#188612yes—
486C22#188612yes—
487C22#189812yes—
488C22#189812yes—
489C22#191212yes—
490C22C2×#38611yescharacteristic
491C22C22×#9912yes—
492C22C2×#45612yes—
493C4—22no—
494C4—28no—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—48no—
499C4—44no—
500C4—88no—
501C4—82no—
502C4—82no—
503C2—22no—
504C2—22no—
505C2—22no—
506C2—22no—
507C2—41no—
508C2—41no—
509C2#946214yesdirect product factor
510C2C2×#188611yescharacteristic
511C2C2×#189811yescharacteristic
512C2C2×#191211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9462C2direct (N × H)
8. C22×C4⋊Q8C4semidirect
15. #2434C4semidirect
22. C2×#391C4semidirect
52. C2×C4⋊Q8C2×C4semidirect
87. #391C2×C4semidirect
509. C2#9462direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24342
C2×#23102
C2×#17842
C2×#15112
#94622
C2×#12842