Finite Group Databasebeta

← C2×#20253201 #27083807

Subgroups — C2×#20253201 #27083807

Order 12032.

2358 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094023C211yesmaximal, characteristic, complemented
2C2×#15689591C211yesmaximal, characteristic
3#20253201C2116yesmaximal, direct product factor
4C2×#15689417C212yesmaximal, complemented
5C2×#15689407C212yesmaximal, complemented
6C2×#15689412C212yesmaximal, complemented
7C2×#15689402C212yesmaximal, complemented
8C2×#15689166C211yesmaximal, characteristic, complemented
9C2×#15689146C211yesmaximal, characteristic, complemented
10C2×#15689125C211yesmaximal, characteristic
11C2×#15689122C211yesmaximal, characteristic
12C2×C47×#428C211yesmaximal, characteristic, complemented
13C22×#10625350C2211yescharacteristic
14C23×Dic94—21no—
15C22×#10625329C2211yescharacteristic
16C2×#13094023—21no—
17C2×#13094023—21no—
18C2×#13094023—24no—
19C2×#13094023C2212yes—
20C2×#13094023C2212yes—
21C2×#13094023C2218yescomplemented
22#15689591—24no—
23#15689591C2218yes—
24C2×#13093989C2211yescharacteristic
25C2×#13093989C2211yescharacteristic
26C2×#13093986C2211yescharacteristic
27C2×#13093986C2211yescharacteristic
28C2×(C22⋊D47)⋊C4—21no—
29C2×(C22⋊D47)⋊C4—21no—
30C2×#13093981—21no—
31C22×C4⋊Dic47C2211yescharacteristic
32C22×Dic47⋊C4—21no—
33C22×C4×Dic47C2211yescharacteristic
34C2×C42×D47—21no—
35#15689417C22116yescomplemented
36#15689412C22116yescomplemented
37#15689407C22116yescomplemented
38#15689402C22116yescomplemented
39#15689166C2218yescomplemented
40C2×#13093957C2212yes—
41C2×#13093955C2212yes—
42#15689146C2218yescomplemented
43C2×Dic47⋊C8C2212yes—
44C2×Dic47⋊C8C2212yes—
45#15689125C2218yes—
46#15689122C2218yes—
47C2×#13093949C2212yes—
48C2×#13093947C2212yes—
49C2×#13093946C2212yes—
50C2×#13093944C2212yes—
51C22×M16×C47C2211yescharacteristic
52C2×#117×C47C2211yescharacteristic
53C22×C4⋊C4×C47C2211yescharacteristic
54C47×#428C2218yescomplemented
55C2×#107×C47C2212yes—
56C2×#106×C47C2212yes—
57C2×#10625350—22no—
58C2×#10625350—24no—
59C2×#10625350Q818yescomplemented
60C2×#10625350C2314yes—
61C22×Dic94—22no—
62C22×Dic94—22no—
63C22×Dic94—28no—
64C22×Dic94—48no—
65C22×Dic94C2311yescharacteristic
66C22×D188C2311yescharacteristic
67C22×C22⋊D47—21no—
68C23×Dic47—21no—
69C23×Dic47D411yescharacteristic
70C23×Dic47D411yescharacteristic
71C22×C4×D47—21no—
72C22×C4×D47—21no—
73C22×C4×D47—21no—
74C2×#10625329—24no—
75C2×#10625329C2314yes—
76C22×C47⋊C8C2312yes—
77#13094023—24no—
78#13094023—24no—
79#13094023—28no—
80#13094023—28no—
81#13094023C2318yes—
82#13094023C2318yes—
83#13093989—22no—
84#13093989—22no—
85#13093989C2314yes—
86#13093989C2314yes—
87#13093986—22no—
88#13093986—22no—
89#13093986C2314yes—
90#13093986C2314yes—
91(C22⋊D47)⋊C4—28no—
92(C22⋊D47)⋊C4—28no—
93#13093981—28no—
94C2×C4⋊Dic47—21no—
95C2×C4⋊Dic47—22no—
96C2×C4⋊Dic47C2312yes—
97C2×C4⋊Dic47C2314yes—
98C2×Dic47⋊C4—22no—
99C2×Dic47⋊C4—22no—
100C2×Dic47⋊C4—28no—
101C2×Dic47⋊C4—21no—
102C2×Dic47⋊C4—21no—
103C2×(C2×D47)⋊C4—21no—
104C2×(C2×D47)⋊C4—21no—
105C2×C4×Dic47—21no—
106C2×C4×Dic47—21no—
107C2×C4×Dic47D411yescharacteristic
108C2×C4×Dic47D411yescharacteristic
109C2×C4×Dic47D414yes—
110C2×C4×Dic47C2311yescharacteristic
111C2×C4×Dic47C2311yescharacteristic
112C2×C4×Dic47C2314yes—
113C42×D47—28no—
114#13093957C2318yes—
115#13093955C2318yes—
116Dic47⋊C8C2318yes—
117Dic47⋊C8C2318yes—
118#13093949C2318yes—
119#13093947C2318yes—
120#13093946C2318yes—
121#13093944C2318yes—
122C23×C4×C47—21no—
123C23×C4×C47C2311yescharacteristic
124C2×M16×C47—24no—
125C2×M16×C47C2314yes—
126C22×C8×C47C2312yes—
127#117×C47—22no—
128#117×C47C2314yes—
129C2×C4⋊C4×C47—21no—
130C2×C4⋊C4×C47—22no—
131C2×C4⋊C4×C47C2312yes—
132C2×C4⋊C4×C47C2314yes—
133C2×C4⋊C4×C47C2311yescharacteristic
134C2×C4⋊C4×C47C2311yescharacteristic
135C2×C22⋊C4×C47—21no—
136C2×C42×C47—21no—
137#107×C47C2318yes—
138#106×C47C2318yes—
139C24×C47C2×D411yescharacteristic
140C23×D47—21no—
141C22×C4×C47—21no—
142C22×C4×C47—21no—
143C22×C4×C47—24no—
144C22×C4×C47—22no—
145C22×C4×C47—44no—
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D412yes—
148C22×C4×C47C2×Q814yes—
149C22×C4×C47#4114yes—
150C22×C4×C47C2411yescharacteristic
151C22×C4×C47C2412yes—
152C2×C22⋊D47—28no—
153C2×C22⋊D47—24no—
154C22×Dic47—21no—
155C22×Dic47—24no—
156C22×Dic47—24no—
157C22×Dic47—24no—
158C22×Dic47—22no—
159C22×Dic47—44no—
160C22×Dic47C2×D411yescharacteristic
161C22×Dic47C2×D411yescharacteristic
162C22×Dic47C2×D414yes—
163C22×Dic47C2×D414yes—
164C22×Dic47C2×D411yescharacteristic
165C22×Dic47C2×D411yescharacteristic
166#10625350—216no—
167#10625350—28no—
168#10625350C2×Q8116yescomplemented
169C2×D188—21no—
170C2×D188—22no—
171C2×D188C2×Q814yescomplemented
172C2×D188C2412yes—
173C2×C4×D47—28no—
174C2×C4×D47—24no—
175C2×C4×D47—24no—
176C2×C4×D47—24no—
177C2×C4×D47—44no—
178C2×C4×D47—44no—
179C2×Dic94—21no—
180C2×Dic94—22no—
181C2×Dic94—28no—
182C2×Dic94—28no—
183C2×Dic94—48no—
184C2×Dic94—432no—
185C2×Dic94—48no—
186C2×Dic94C2×Q814yescomplemented
187C2×Dic94C2412yes—
188M16×C47—28no—
189C2×C8×C47—24no—
190C2×C8×C47C2414yes—
191C4⋊C4×C47—24no—
192C4⋊C4×C47—22no—
193C4⋊C4×C47—21no—
194C4⋊C4×C47—21no—
195C4⋊C4×C47C2412yes—
196C4⋊C4×C47C2414yes—
197C4⋊C4×C47C2412yes—
198C22⋊C4×C47—24no—
199C42×C47—24no—
200(C2×D47)⋊C4—24no—
201(C2×D47)⋊C4—24no—
202C4⋊Dic47—24no—
203C4⋊Dic47—22no—
204C4⋊Dic47C2414yes—
205Dic47⋊C4—28no—
206Dic47⋊C4—28no—
207Dic47⋊C4—24no—
208Dic47⋊C4—24no—
209C4×Dic47—24no—
210C4×Dic47—24no—
211C4×Dic47C2×D414yes—
212C4×Dic47C2×D414yes—
213C4×Dic47C2×D412yes—
214C4×Dic47C2×D412yes—
215C4×Dic47C2412yes—
216C4×Dic47C2412yes—
217#10625329—28no—
218C2×C47⋊C8—24no—
219C2×C47⋊C8C2414yes—
220C23×C47—22no—
221C23×C47—22no—
222C23×C47C22⋊Q814yes—
223C23×C47C22×D412yes—
224C23×C47C22×D411yescharacteristic
225C22×D47—24no—
226C22×D47—22no—
227C22×D47—42no—
228C22×D47—42no—
229C2×C4×C47—22no—
230C2×C4×C47—22no—
231C2×C4×C47—24no—
232C2×C4×C47—48no—
233C2×C4×C47—44no—
234C2×C4×C47—42no—
235C2×C4×C47—42no—
236C2×C4×C47C22⋊Q814yes—
237C2×C4×C47C22⋊Q818yes—
238C2×C4×C47C22×D412yes—
239C2×C4×C47C22×D411yescharacteristic
240C2×C4×C47C22×Q812yes—
241C2×C4×C47C22×Q814yes—
242C2×C4×C47C2×#4112yes—
243C2×C4×C47C2×#4114yes—
244C2×C4×C47C2511yescharacteristic
245C22⋊D47—216no—
246C2×Dic47—24no—
247C2×Dic47—22no—
248C2×Dic47—22no—
249C2×Dic47—22no—
250C2×Dic47—28no—
251C2×Dic47—28no—
252C2×Dic47—24no—
253C2×Dic47—48no—
254C2×Dic47—44no—
255C2×Dic47AGL(1,ℤ/8)14yes—
256C2×Dic47#13714yes—
257C2×Dic47C22×D412yes—
258C2×Dic47C22×D412yes—
259C2×Dic47C22×D412yes—
260C2×Dic47C22×D412yes—
261D188—24no—
262D188—22no—
263D188C22×Q814yescomplemented
264C4×D47—216no—
265C4×D47—48no—
266C4×D47—48no—
267Dic94—24no—
268Dic94—22no—
269Dic94—416no—
270Dic94—416no—
271Dic94C22×Q814yescomplemented
272C8×C47—24no—
273C47⋊C8—24no—
274C2×#3079—471nomaximal, Sylow
275C22×C47—28no—
276C22×C47—22no—
277C22×C47—22no—
278C22×C47C2×C22⋊Q814yes—
279C22×C47C2×C22⋊Q812yes—
280C22×C47C2×AGL(1,ℤ/8)12yes—
281C22×C47C2×#13712yes—
282C22×C47C23×D411yescharacteristic
283C2×D47—24no—
284C2×D47—48no—
285C2×D47—42no—
286C2×D47—42no—
287C4×C47—44no—
288C4×C47—42no—
289C4×C47—42no—
290C4×C47C2×C22⋊Q812yes—
291C4×C47C2×C22⋊Q814yes—
292C4×C47C23×Q811yescharacteristic
293C4×C47C22×#4111yescharacteristic
294Dic47—24no—
295Dic47—22no—
296Dic47—22no—
297Dic47—44no—
298Dic47C2×AGL(1,ℤ/8)14yes—
299Dic47C2×#13714yes—
300C22×C4⋊Q8—471no—
301C2×C4×#41—471no—
302#3079—4716no—
303C2×#477—472no—
304C2×#475—472no—
305C2×#476—472no—
306C2×#474—472no—
307C2×#428—471no—
308C2×#428D4711yescharacteristic, complemented
309C2×#423—471no—
310C2×#417—471no—
311C2×#417—471no—
312C2×C47—24no—
313C2×C47#307914yescomplemented
314C2×C47C22×C22⋊Q811yescharacteristic
315C2×C47C22×AGL(1,ℤ/8)11yescharacteristic
316C2×C47C22×#13711yescharacteristic
317D47—44no—
318C22×#41—471no—
319C23×Q8—941no—
320C22×M16—471no—
321C22×M16C2×D4711yescharacteristic
322C2×C4⋊Q8—472no—
323C2×C4⋊Q8—472no—
324C2×C4⋊Q8—478no—
325C2×C4⋊Q8—941no—
326C2×C4⋊Q8—944no—
327C2×C4⋊Q8—941no—
328C4×#41—478no—
329C4×#41—944no—
330C2×C4×Q8—471no—
331C2×C4×Q8—471no—
332C2×C4×D4—471no—
333C2×C4×D4—471no—
334C2×C4×D4—941no—
335C2×C4×D4—941no—
336C2×#117—941no—
337C2×#117C2×D4711yescharacteristic
338C22×C4⋊C4—471no—
339C22×C4⋊C4—941no—
340C22×C4⋊C4C2×D4711yescharacteristic
341C22×C42—471no—
342C22×C42—941no—
343#477—4716no—
344#476—4716no—
345#475—4716no—
346#474—4716no—
347#428—478no—
348#428C2×D4718yescomplemented
349C2×#107—472no—
350C2×#107C2×D4712yes—
351C2×#106—472no—
352C2×#106C2×D4712yes—
353#423—478no—
354C2×C4⋊C8—472no—
355C2×C4⋊C8—472no—
356#417—478no—
357#417—478no—
358C2×#103—472no—
359C2×#103—472no—
360C2×#102—472no—
361C2×#102—472no—
362C47C2×#307911yescharacteristic, complemented, Sylow
363C2×#41—478no—
364C2×#41—474no—
365C2×#41—944no—
366C2×#41—942no—
367C22×Q8—471no—
368C22×Q8—948no—
369C22×Q8—942no—
370C22×Q8—942no—
371C22×Q8—1888no—
372C22×D4—471no—
373C22×D4—941no—
374C23×C4—21no—
375C23×C4—471no—
376C23×C4—471no—
377C23×C4—941no—
378C23×C4—941no—
379C23×C4—941no—
380C23×C4—941no—
381C23×C4C22×D4711yescharacteristic
382C2×M16—24no—
383C2×M16—474no—
384C2×M16—944no—
385C2×M16C22×D4714yes—
386C22×C8—472no—
387C22×C8C22×D4712yes—
388C4⋊Q8—478no—
389C4⋊Q8—478no—
390C4⋊Q8—948no—
391C4⋊Q8—948no—
392C4⋊Q8—944no—
393C4⋊Q8—944no—
394C4×Q8—474no—
395C4×Q8—474no—
396C4×Q8—942no—
397C4×Q8—942no—
398C4×D4—474no—
399C4×D4—474no—
400C4×D4—942no—
401C4×D4—942no—
402C4×D4—948no—
403C4×D4—948no—
404#117—22no—
405#117—948no—
406#117C22×D4714yes—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—472no—
410C2×C4⋊C4—474no—
411C2×C4⋊C4—948no—
412C2×C4⋊C4—942no—
413C2×C4⋊C4—942no—
414C2×C4⋊C4—941no—
415C2×C4⋊C4—941no—
416C2×C4⋊C4—942no—
417C2×C4⋊C4—941no—
418C2×C4⋊C4C22×D4712yes—
419C2×C4⋊C4C22×D4714yes—
420C2×C4⋊C4C22×D4711yescharacteristic
421C2×C4⋊C4C22×D4711yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—941no—
424C2×C22⋊C4—941no—
425C2×C42—21no—
426C2×C42—471no—
427C2×C42—474no—
428C2×C42—471no—
429C2×C42—471no—
430C2×C42—471no—
431C2×C42—474no—
432C2×C42—948no—
433C2×C42—941no—
434C2×C42—941no—
435#107—478no—
436#107C22×D4718yes—
437#106—478no—
438#106C22×D4718yes—
439C4⋊C8—478no—
440C4⋊C8—478no—
441#103—478no—
442#103—478no—
443#102—478no—
444#102—478no—
445C24—941no—
446C24D4×D4711yescharacteristic
447#41—4716no—
448#41—9416no—
449#41—948no—
450C2×Q8—474no—
451C2×Q8—472no—
452C2×Q8—948no—
453C2×Q8—948no—
454C2×Q8—941no—
455C2×Q8—942no—
456C2×Q8—1888no—
457C2×Q8—18832no—
458C2×Q8—1888no—
459C2×D4—474no—
460C2×D4—472no—
461C2×D4—948no—
462C2×D4—944no—
463C2×D4—941no—
464C2×D4—942no—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—44no—
470C22×C4—471no—
471C22×C4—471no—
472C22×C4—474no—
473C22×C4—474no—
474C22×C4—471no—
475C22×C4—471no—
476C22×C4—944no—
477C22×C4—944no—
478C22×C4—944no—
479C22×C4—944no—
480C22×C4—941no—
481C22×C4—948no—
482C22×C4—944no—
483C22×C4—944no—
484C22×C4—942no—
485C22×C4—1884no—
486C22×C4—1884no—
487C22×C4—1884no—
488C22×C4#1062535014yes—
489C22×C4D4×D4712yes—
490C22×C4D4×D4711yescharacteristic
491C22×C4Q8×D4714yes—
492C22×C4C23×D4712yes—
493C22×C4C23×D4711yescharacteristic
494M16—28no—
495M16—948no—
496C2×C8—24no—
497C2×C8—474no—
498C2×C8—944no—
499C2×C8C23×D4714yes—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—474no—
505C4⋊C4—948no—
506C4⋊C4—948no—
507C4⋊C4—944no—
508C4⋊C4—944no—
509C4⋊C4—944no—
510C4⋊C4—942no—
511C4⋊C4C23×D4714yes—
512C4⋊C4C23×D4712yes—
513C4⋊C4C23×D4712yes—
514C22⋊C4—24no—
515C22⋊C4—944no—
516C22⋊C4—944no—
517C42—24no—
518C42—474no—
519C42—474no—
520C42—472no—
521C42—472no—
522C42—472no—
523C42—472no—
524C42—944no—
525C42—944no—
526C23—22no—
527C23—22no—
528C23—944no—
529C23—942no—
530C23—1882no—
531C23—1882no—
532C23#1309400214yes—
533C23C2×D4×D4712yes—
534C23C2×D4×D4711yescharacteristic
535Q8—474no—
536Q8—944no—
537Q8—942no—
538Q8—18816no—
539Q8—18816no—
540D4—474no—
541D4—9416no—
542D4—944no—
543D4—942no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—48no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—474no—
552C2×C4—474no—
553C2×C4—472no—
554C2×C4—472no—
555C2×C4—472no—
556C2×C4—472no—
557C2×C4—948no—
558C2×C4—948no—
559C2×C4—944no—
560C2×C4—942no—
561C2×C4—942no—
562C2×C4—942no—
563C2×C4—9416no—
564C2×C4—944no—
565C2×C4—1888no—
566C2×C4—1888no—
567C2×C4—1888no—
568C2×C4—1884no—
569C2×C4#1309400214yes—
570C2×C4#1309400218yes—
571C2×C4C2×D4×D4712yes—
572C2×C4C2×D4×D4711yescharacteristic
573C2×C4C2×Q8×D4712yes—
574C2×C4C2×Q8×D4714yes—
575C2×C4C2×#1062535012yes—
576C2×C4C2×#1062535014yes—
577C2×C4C24×D4711yescharacteristic
578C8—24no—
579C8—944no—
580C22—28no—
581C22—22no—
582C22—22no—
583C22—944no—
584C22—1888no—
585C22—1882no—
586C22—1882no—
587C22C2×#1309400214yes—
588C22C2×#1309400212yes—
589C22AGL(1,ℤ/8)×D4712yes—
590C22#137×D4712yes—
591C22C22×D4×D4711yescharacteristic
592C4—44no—
593C4—42no—
594C4—42no—
595C4—474no—
596C4—474no—
597C4—942no—
598C4—942no—
599C4—944no—
600C4—1884no—
601C4C2×#1309400214yes—
602C4C2×#1309400212yes—
603C4C22×Q8×D4711yescharacteristic
604C4C22×#1062535011yescharacteristic
605C2—24no—
606C2—1884no—
607C2#2025320114yesdirect product factor
608C2C22×#1309400211yescharacteristic
609C2C2×AGL(1,ℤ/8)×D4711yescharacteristic
610C2C2×#137×D4711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#428C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
hypercenterC2×#428D47
hypocenterC47C2×#3079
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C22×#13094023C2semidirect
3. #20253201C2direct (N × H)
4. C2×#15689417C2semidirect
5. C2×#15689407C2semidirect
6. C2×#15689412C2semidirect
7. C2×#15689402C2semidirect
8. C2×#15689166C2semidirect
9. C2×#15689146C2semidirect
12. C2×C47×#428C2semidirect
21. C2×#13094023C22semidirect
35. #15689417C22semidirect
36. #15689412C22semidirect
37. #15689407C22semidirect
38. #15689402C22semidirect
39. #15689166C22semidirect
42. #15689146C22semidirect
54. C47×#428C22semidirect
59. C2×#10625350Q8semidirect
168. #10625350C2×Q8semidirect
171. C2×D188C2×Q8semidirect
186. C2×Dic94C2×Q8semidirect
263. D188C22×Q8semidirect
271. Dic94C22×Q8semidirect
308. C2×#428D47semidirect
313. C2×C47#3079semidirect
348. #428C2×D47semidirect
362. C47C2×#3079semidirect
607. C2#20253201direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940232
C2×#156895912
#202532012
C2×#156894172
C2×#156894072
C2×#156894122
C2×#156894022
C2×#156891662
C2×#156891462
C2×#156891252
C2×#156891222
C2×C47×#4282
C2×#307947