Finite Group Databasebeta

← C2×#18761 #6348675

Subgroups — C2×#18761 #6348675

Order 512.

2868 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2C2×#3229C211yesmaximal, characteristic
3C2×#3104C211yesmaximal, characteristic
4C2×#3078C211yesmaximal, characteristic
5#18761C2116yesmaximal, direct product factor
6C2×#1653C214yesmaximal, complemented
7C2×#1636C214yesmaximal, complemented
8C2×#1546C211yesmaximal, characteristic
9C2×#1487C211yesmaximal, characteristic
10C2×#1479C211yesmaximal, characteristic
11C2×#557—22no—
12C2×#557—24no—
13C2×#557—24no—
14C2×#557—22no—
15C2×#557—22no—
16C2×#557—24no—
17C2×#557C2214yes—
18C2×#557C2218yes—
19C2×#533C2211yescharacteristic
20C22×C4⋊Q8—21no—
21C22×C4⋊Q8C2211yescharacteristic
22C22×#125—21no—
23C22×#125—22no—
24C22×C4×Q8—21no—
25C22×C4×Q8—21no—
26C22×C4×Q8C2211yescharacteristic
27C22×C4×Q8C2211yescharacteristic
28#3229—24no—
29#3229C2218yes—
30C2×#498C2212yes—
31C2×#487C2212yes—
32#3104—24no—
33#3104C2218yes—
34C2×#484—21no—
35C2×#483—22no—
36#3078—24no—
37#3078C2218yes—
38C2×#479—22no—
39C2×#477C2212yes—
40C2×#477C2212yes—
41C2×#475—22no—
42C2×#475C2212yes—
43C2×#462C2212yes—
44C2×#451—21no—
45C2×C4×Q16—22no—
46C2×#431C2211yescharacteristic
47C2×#431C2211yescharacteristic
48C2×#428—21no—
49C2×#423C2211yescharacteristic
50C2×#419—21no—
51#1653C22132yescomplemented
52#1636C22132yescomplemented
53#1546C2218yes—
54C2×#332C2214yes—
55#1487C2218yes—
56#1479C2218yes—
57C2×Q8⋊C8C2214yes—
58C2×Q8⋊C8C2214yes—
59C23×Q8—21no—
60C23×Q8D411yescharacteristic
61C23×Q8D411yescharacteristic
62C22×Q16—22no—
63C22×M16—21no—
64#557—216no—
65#557—216no—
66#557—216no—
67#557—216no—
68#557—28no—
69#557—216no—
70#557—28no—
71#557—28no—
72#557—216no—
73#557C23116yes—
74#533—22no—
75#533—22no—
76#533—22no—
77#533C2314yes—
78C2×C4⋊Q8—28no—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—28no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8D412yes—
87C2×C4⋊Q8D414yes—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2314yes—
92C2×#125—216no—
93C2×#125—28no—
94C2×#125—28no—
95C2×#125—24no—
96C2×#125—28no—
97C2×#125—216no—
98C2×#125—24no—
99C2×#125—28no—
100C2×#125—21no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—21no—
103C2×C4×Q8—28no—
104C2×C4×Q8—24no—
105C2×C4×Q8—28no—
106C2×C4×Q8—24no—
107C2×C4×Q8—28no—
108C2×C4×Q8—24no—
109C2×C4×Q8—24no—
110C2×C4×Q8—28no—
111C2×C4×Q8—21no—
112C2×C4×Q8—22no—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8—21no—
116C2×C4×Q8—22no—
117C2×C4×Q8—21no—
118C2×C4×Q8—22no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8D412yes—
122C2×C4×Q8D414yes—
123C2×C4×Q8D414yes—
124C2×C4×Q8D412yes—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2314yes—
128C2×C4×Q8C2314yes—
129C2×#117—21no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4D412yes—
137C22×C4⋊C4D411yescharacteristic
138C22×C4⋊C4D411yescharacteristic
139C22×C42—22no—
140C22×C42—21no—
141C22×C42—21no—
142C22×C42C2311yescharacteristic
143#498—24no—
144#498C2318yes—
145#487—24no—
146#487C2318yes—
147#484—28no—
148#483—216no—
149#479—216no—
150#477—24no—
151#477—24no—
152#477C2318yes—
153#477C2318yes—
154#475—216no—
155#475—24no—
156#475C2318yes—
157#462—24no—
158#462C2318yes—
159#451—28no—
160C4×Q16—216no—
161#431—22no—
162#431—22no—
163#431C2314yes—
164#431C2314yes—
165#428—28no—
166C2×#107—22no—
167C2×#107—22no—
168C2×#106—24no—
169C2×#106—22no—
170#423—22no—
171#423C2314yes—
172C2×C4⋊C8—21no—
173C2×C4⋊C8—21no—
174C2×C4⋊C8C2312yes—
175C2×C4⋊C8C2312yes—
176#419—28no—
177C2×#103—22no—
178C2×#103—22no—
179C2×#103—22no—
180C2×#103—24no—
181C2×#103—22no—
182C2×C22⋊C8—21no—
183C2×C22⋊C8—21no—
184C2×C4×C8—21no—
185C2×C4×C8C2312yes—
186#332C23116yes—
187Q8⋊C8C23116yes—
188Q8⋊C8C23116yes—
189C22×Q8—28no—
190C22×Q8—22no—
191C22×Q8—21no—
192C22×Q8—22no—
193C22×Q8—21no—
194C22×Q8—24no—
195C22×Q8—22no—
196C22×Q8—24no—
197C22×Q8—22no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—21no—
201C22×Q8—21no—
202C22×Q8—44no—
203C22×Q8—44no—
204C22×Q8C2×D414yes—
205C22×Q8C2×D412yes—
206C22×Q8C2×D414yes—
207C22×Q8C2×D412yes—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—22no—
212C23×C4—21no—
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C2×Q16—28no—
217C2×Q16—48no—
218C2×M16—24no—
219C2×M16—44no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C4⋊Q8—232no—
224C4⋊Q8—216no—
225C4⋊Q8—216no—
226C4⋊Q8—24no—
227C4⋊Q8—24no—
228C4⋊Q8—24no—
229C4⋊Q8—24no—
230C4⋊Q8—24no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8C2×D418yes—
240C4⋊Q8C2×D414yes—
241C4⋊Q8C2412yes—
242C4⋊Q8C2412yes—
243C4⋊Q8C2414yes—
244#125—264no—
245#125—232no—
246#125—232no—
247#125—216no—
248#125—232no—
249#125—216no—
250#125—24no—
251#125—42no—
252C22⋊Q8—24no—
253C22⋊Q8—24no—
254C22⋊Q8—42no—
255C22⋊Q8—42no—
256C4×Q8—232no—
257C4×Q8—216no—
258C4×Q8—232no—
259C4×Q8—216no—
260C4×Q8—216no—
261C4×Q8—216no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—22no—
269C4×Q8—24no—
270C4×Q8—22no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—22no—
274C4×Q8—24no—
275C4×Q8—22no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—42no—
279C4×Q8—42no—
280C4×Q8C2×D418yes—
281C4×Q8C2×D414yes—
282C4×Q8C2×D414yes—
283C4×Q8C2×D418yes—
284C4×Q8C2414yes—
285C4×Q8C2414yes—
286#117—24no—
287#117—42no—
288C2×C4⋊C4—216no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—216no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—216no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—41no—
317C2×C4⋊C4—41no—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D414yes—
322C2×C4⋊C4C2×D414yes—
323C2×C4⋊C4C2×D418yes—
324C2×C22⋊C4—41no—
325C2×C42—216no—
326C2×C42—28no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—41no—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2412yes—
344#107—28no—
345#107—28no—
346#106—28no—
347#106—216no—
348C4⋊C8—24no—
349C4⋊C8—24no—
350C4⋊C8—22no—
351C4⋊C8—22no—
352C4⋊C8C2414yes—
353C4⋊C8C2414yes—
354#103—28no—
355#103—28no—
356#103—28no—
357#103—28no—
358#103—216no—
359C22⋊C8—24no—
360C22⋊C8—24no—
361C4×C8—24no—
362C4×C8—22no—
363C4×C8C2414yes—
364C24#12011yescharacteristic
365C2×Q8—216no—
366C2×Q8—28no—
367C2×Q8—216no—
368C2×Q8—24no—
369C2×Q8—22no—
370C2×Q8—24no—
371C2×Q8—22no—
372C2×Q8—28no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—28no—
378C2×Q8—24no—
379C2×Q8—28no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—44no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—44no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×Q8—44no—
389C2×Q8—48no—
390C2×Q8—44no—
391C2×Q8—416no—
392C2×Q8—48no—
393C2×Q8—44no—
394C2×Q8#13718yes—
395C2×Q8C22×D414yes—
396C2×Q8C22×D414yes—
397C22×C4—28no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—28no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—42no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—48no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4#12012yes—
417C22×C4#12011yescharacteristic
418C22×C4#12011yescharacteristic
419C22×C4#12012yes—
420C22×C4#12012yes—
421C22×C4#12011yescharacteristic
422C22×C4#13514yes—
423C22×C4#13714yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430Q16—416no—
431M16—48no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—44no—
436C2×C8—44no—
437C2×C8—44no—
438C4⋊C4—232no—
439C4⋊C4—232no—
440C4⋊C4—232no—
441C4⋊C4—28no—
442C4⋊C4—24no—
443C4⋊C4—28no—
444C4⋊C4—24no—
445C4⋊C4—28no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—28no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4C22×D418yes—
467C4⋊C4C22×D414yes—
468C4⋊C4C22×D414yes—
469C22⋊C4—44no—
470C42—232no—
471C42—28no—
472C42—24no—
473C42—28no—
474C42—24no—
475C42—24no—
476C42—24no—
477C42—44no—
478C42#12012yes—
479C42#12012yes—
480C42#12012yes—
481C42C22×D412yes—
482C42C22×D411yescharacteristic
483C42C22×D412yes—
484C42C22×D412yes—
485C42C22×D411yescharacteristic
486C42C22×D411yescharacteristic
487C42C2511yescharacteristic
488C23—22no—
489C23—22no—
490C23#45214yes—
491C23C2×#12012yes—
492C23C2×#12011yescharacteristic
493Q8—216no—
494Q8—28no—
495Q8—24no—
496Q8—24no—
497Q8—24no—
498Q8—44no—
499Q8—48no—
500Q8—44no—
501Q8—416no—
502Q8—48no—
503Q8—44no—
504Q8—44no—
505Q8—44no—
506Q8—82no—
507Q8—82no—
508Q8C2×#13718yes—
509C2×C4—216no—
510C2×C4—24no—
511C2×C4—28no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—48no—
524C2×C4—48no—
525C2×C4—48no—
526C2×C4—44no—
527C2×C4—416no—
528C2×C4—48no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4#45218yes—
536C2×C4#45214yes—
537C2×C4C2×#12012yes—
538C2×C4C2×#12012yes—
539C2×C4C2×#12011yescharacteristic
540C2×C4C2×#12012yes—
541C2×C4C2×#12011yescharacteristic
542C2×C4C2×#12011yescharacteristic
543C2×C4C2×#13512yes—
544C2×C4C2×#13514yes—
545C2×C4C2×#13714yes—
546C2×C4C2×#13712yes—
547C2×C4C2×#13714yes—
548C2×C4C23×D411yescharacteristic
549C2×C4C23×D411yescharacteristic
550C2×C4C23×D411yescharacteristic
551C8—44no—
552C8—44no—
553C8—44no—
554C22—28no—
555C22—22no—
556C22—22no—
557C22C2×#45214yes—
558C22C2×#45212yes—
559C22#300412yes—
560C22#302012yes—
561C22C22×#12011yescharacteristic
562C4—28no—
563C4—22no—
564C4—21no—
565C4—22no—
566C4—21no—
567C4—44no—
568C4—48no—
569C4—44no—
570C4—44no—
571C4—82no—
572C4—82no—
573C4C2×#45214yes—
574C4C2×#45212yes—
575C4#300412yes—
576C4C22×#13511yescharacteristic
577C4C22×#13711yescharacteristic
578C4C22×#13712yes—
579C2—24no—
580C2#1876114yesdirect product factor
581C2C22×#45211yescharacteristic
582C2C2×#300411yescharacteristic
583C2C2×#302011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #18761C2direct (N × H)
6. C2×#1653C2semidirect
7. C2×#1636C2semidirect
51. #1653C22semidirect
52. #1636C22semidirect
580. C2#18761direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#32292
C2×#31042
C2×#30782
#187612
C2×#16532
C2×#16362
C2×#15462
C2×#14872
C2×#14792