Finite Group Databasebeta

← C2×#20254088 #27086191

Subgroups — C2×#20254088 #27086191

Order 12032.

2278 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689674C211yesmaximal, characteristic, complemented
2C2×#15689658C211yesmaximal, characteristic, complemented
3#20254088C2116yesmaximal, direct product factor
4C2×#15689512C211yesmaximal, characteristic, complemented
5C2×#15689489C211yesmaximal, characteristic, complemented
6C2×#15689485C211yesmaximal, characteristic, complemented
7C2×#15689360C212yesmaximal, complemented
8C2×#15689353C212yesmaximal, complemented
9C2×#15689234C212yesmaximal, complemented
10C2×#15689222C212yesmaximal, complemented
11C2×#97×D47C211yesmaximal, characteristic, complemented
12C2×C47×#501C211yesmaximal, characteristic, complemented
13C22×#10625354—21no—
14C22×#10625325C2212yes—
15C22×D376C2212yes—
16C22×C8×D47C2212yes—
17#15689674—24no—
18#15689674C2218yescomplemented
19C2×#13094021C2211yescharacteristic
20#15689658—24no—
21#15689658C2218yescomplemented
22C2×#13094019—21no—
23C2×#13094013—21no—
24C2×#13094012C2211yescharacteristic
25C2×#13094009—21no—
26C2×#13094004C2211yescharacteristic
27C2×#13093999—21no—
28C2×#13093999C2212yes—
29C2×#13093999C2211yescharacteristic
30C2×#13093989C2211yescharacteristic
31C2×#13093986—21no—
32C2×#13093986—21no—
33C2×#13093986C2212yes—
34C2×#13093986C2211yescharacteristic
35C2×#13093982—21no—
36C2×C42×D47C2211yescharacteristic
37#15689512C2218yescomplemented
38#15689489C2218yescomplemented
39#15689485C2218yescomplemented
40#15689360C22116yescomplemented
41#15689353C22116yescomplemented
42#15689234C22116yescomplemented
43#15689222C22116yescomplemented
44C2×#13093947C2212yes—
45C2×#13093944C2212yes—
46C2×#13093944C2212yes—
47C2×#13093944C2212yes—
48C2×#13093935C2211yescharacteristic
49C2×#13093934C2212yes—
50#97×D47C2218yescomplemented
51C2×C4⋊Q8×C47C2211yescharacteristic
52C2×#125×C47C2211yescharacteristic
53C47×#501C2218yescomplemented
54C2×#107×C47C2212yes—
55C2×#106×C47C2212yes—
56C2×#97×C47C2211yescharacteristic
57C2×#10625354—28no—
58C2×#10625354—48no—
59C22×Dic94C2311yescharacteristic
60C22×D188—21no—
61C22×D188—22no—
62C22×D188—21no—
63C22×D188—21no—
64C22×D188C2311yescharacteristic
65C22×D188C2312yes—
66C22×C4×D47—21no—
67C22×C4×D47—22no—
68C22×C4×D47D411yescharacteristic
69C22×C4×D47D411yescharacteristic
70C22×C4×D47C2311yescharacteristic
71C2×#10625325—28no—
72C2×#10625325C2318yes—
73C2×D376—28no—
74C2×D376C2318yes—
75C22×C47⋊C8C2312yes—
76C2×C8×D47—28no—
77C2×C8×D47C2318yes—
78#13094021—22no—
79#13094021C2314yes—
80#13094019—28no—
81#13094013—28no—
82#13094012—22no—
83#13094012C2314yes—
84#13094009—28no—
85#13094004—22no—
86#13094004C2314yes—
87#13093999—28no—
88#13093999—22no—
89#13093999—24no—
90#13093999C2318yes—
91#13093999C2314yes—
92#13093989—22no—
93#13093989C2314yes—
94#13093986—28no—
95#13093986—28no—
96#13093986—22no—
97#13093986—24no—
98#13093986C2318yes—
99#13093986C2314yes—
100#13093982—28no—
101C2×C4⋊Dic47—21no—
102C2×Dic47⋊C4—21no—
103C2×(C2×D47)⋊C4—21no—
104C2×(C2×D47)⋊C4—21no—
105C2×(C2×D47)⋊C4—21no—
106C2×(C2×D47)⋊C4—21no—
107C2×(C2×D47)⋊C4—22no—
108C2×(C2×D47)⋊C4—22no—
109C2×(C2×D47)⋊C4—21no—
110C2×C4×Dic47—21no—
111C2×C4×Dic47D411yescharacteristic
112C2×C4×Dic47D411yescharacteristic
113C2×C4×Dic47C2311yescharacteristic
114C42×D47D414yes—
115C42×D47C2314yes—
116#13093947C2318yes—
117#13093944C2318yes—
118#13093944C2318yes—
119#13093944C2318yes—
120#13093935C2314yes—
121#13093934C2318yes—
122C22×Q8×C47—21no—
123C22×C8×C47C2312yes—
124C4⋊Q8×C47—22no—
125C4⋊Q8×C47C2314yes—
126#125×C47—22no—
127#125×C47C2314yes—
128C2×C4⋊C4×C47—21no—
129C2×C4⋊C4×C47—21no—
130C2×C4⋊C4×C47—21no—
131C2×C4⋊C4×C47C2311yescharacteristic
132C2×C4⋊C4×C47C2311yescharacteristic
133C2×C4⋊C4×C47C2312yes—
134C2×C42×C47C2311yescharacteristic
135#107×C47C2318yes—
136#106×C47C2318yes—
137#97×C47C2314yes—
138C23×D47—21no—
139C23×D47—22no—
140C23×D47C2×D411yescharacteristic
141C2×Q8×C47—24no—
142C2×Q8×C47—44no—
143C22×C4×C47—21no—
144C22×C4×C47—21no—
145C22×C4×C47—22no—
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2411yescharacteristic
149C22×Dic47—21no—
150C22×Dic47C2×D411yescharacteristic
151C22×Dic47C2×D411yescharacteristic
152C22×Dic47C2×D411yescharacteristic
153#10625354—432no—
154C2×D188—24no—
155C2×D188—24no—
156C2×D188—24no—
157C2×D188—24no—
158C2×D188—28no—
159C2×D188—21no—
160C2×D188—22no—
161C2×D188—24no—
162C2×D188—24no—
163C2×D188—44no—
164C2×D188—44no—
165C2×D188—44no—
166C2×D188—48no—
167C2×D188C2412yes—
168C2×D188C2414yes—
169C2×C4×D47—24no—
170C2×C4×D47—24no—
171C2×C4×D47—24no—
172C2×C4×D47—28no—
173C2×C4×D47—42no—
174C2×C4×D47—44no—
175C2×C4×D47—48no—
176C2×C4×D47C2×D412yes—
177C2×C4×D47C2×D414yes—
178C2×C4×D47C2×D414yes—
179C2×C4×D47C2412yes—
180C2×Dic94—24no—
181C2×Dic94—21no—
182C2×Dic94C2412yes—
183C2×C8×C47#4118yescomplemented
184C2×C8×C47C2414yes—
185C4⋊C4×C47—24no—
186C4⋊C4×C47—24no—
187C4⋊C4×C47—24no—
188C4⋊C4×C47—21no—
189C4⋊C4×C47—21no—
190C4⋊C4×C47—22no—
191C4⋊C4×C47C2412yes—
192C4⋊C4×C47C2414yes—
193C4⋊C4×C47C2412yes—
194C42×C47C2×D412yes—
195C42×C47C2412yes—
196(C2×D47)⋊C4—24no—
197(C2×D47)⋊C4—24no—
198(C2×D47)⋊C4—24no—
199(C2×D47)⋊C4—24no—
200(C2×D47)⋊C4—28no—
201(C2×D47)⋊C4—28no—
202(C2×D47)⋊C4—24no—
203C4⋊Dic47—24no—
204Dic47⋊C4—24no—
205C4×Dic47—24no—
206C4×Dic47C2×D412yes—
207C4×Dic47C2×D414yes—
208C4×Dic47C2×D414yes—
209C4×Dic47C2412yes—
210C2×C47⋊C8—24no—
211C2×C47⋊C8C2414yes—
212D376—216no—
213#10625325—216no—
214C8×D47—216no—
215C23×C47C22×D411yescharacteristic
216C22×D47—22no—
217C22×D47—22no—
218C22×D47—22no—
219C22×D47—24no—
220C22×D47—42no—
221C22×D47—44no—
222C22×D47—44no—
223C22×D47—42no—
224C22×D47—42no—
225C22×D47—44no—
226C22×D47AGL(1,ℤ/8)14yes—
227C22×D47C22×D412yes—
228Q8×C47—48no—
229C2×C4×C47—22no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—24no—
234C2×C4×C47—44no—
235C2×C4×C47—42no—
236C2×C4×C47—42no—
237C2×C4×C47C22×D412yes—
238C2×C4×C47C22×D412yes—
239C2×C4×C47C22×D411yescharacteristic
240C2×C4×C47C2×#4114yes—
241C2×C4×C47C2511yescharacteristic
242C2×Dic47—22no—
243C2×Dic47—22no—
244C2×Dic47—22no—
245C2×Dic47—22no—
246C2×Dic47—42no—
247C2×Dic47C22×D412yes—
248C2×Dic47C22×D412yes—
249C2×Dic47C22×D412yes—
250D188—24no—
251D188—24no—
252D188—24no—
253D188—48no—
254D188—48no—
255D188—48no—
256D188—416no—
257D188—42no—
258D188—44no—
259C4×D47—28no—
260C4×D47—24no—
261C4×D47—24no—
262C4×D47—44no—
263C4×D47—48no—
264C4×D47—416no—
265Dic94—24no—
266Dic94—42no—
267C8×C47C2×#4118yescomplemented
268C47⋊C8—24no—
269C2×#3180—471nomaximal, Sylow
270C22×C47#53212yes—
271C22×C47C2×AGL(1,ℤ/8)12yes—
272C22×C47#57512yes—
273C22×C47C23×D411yescharacteristic
274C2×D47—28no—
275C2×D47—22no—
276C2×D47—22no—
277C2×D47—44no—
278C2×D47—42no—
279C2×D47—44no—
280C2×D47—42no—
281C2×D47—42no—
282C2×D47—44no—
283C2×D47—88no—
284C2×D47—84no—
285C2×D47C2×AGL(1,ℤ/8)14yes—
286C4×C47—22no—
287C4×C47—22no—
288C4×C47—44no—
289C4×C47—42no—
290C4×C47—42no—
291C4×C47#53212yes—
292C4×C47C22×#4112yes—
293Dic47—22no—
294Dic47—22no—
295Dic47—22no—
296Dic47—42no—
297C2×#532—471no—
298C2×#529—471no—
299#3180—4716no—
300C2×#501D4711yescharacteristic, complemented
301C2×#496—471no—
302C2×#490—471no—
303C2×#489—471no—
304C2×#466—472no—
305C2×#465—472no—
306C2×#442—472no—
307C2×#440—472no—
308C22×#97—471no—
309C2×C47#318014yescomplemented
310C2×C47C2×#53211yescharacteristic
311C2×C47C22×AGL(1,ℤ/8)11yescharacteristic
312C2×C47C2×#57511yescharacteristic
313D47—24no—
314D47—84no—
315D47—82no—
316C22×#41—941no—
317C22×SD16—472no—
318C22×D8—472no—
319C23×C8—472no—
320#532—478no—
321#532—944no—
322C2×C4⋊Q8C2×D4711yescharacteristic
323C2×#127—471no—
324#529—478no—
325#529—944no—
326C2×#126—941no—
327C2×#125C2×D4711yescharacteristic
328C2×#124—471no—
329C2×#124—941no—
330C2×#123—941no—
331C2×C22⋊Q8—471no—
332C2×#121—471no—
333C2×#121—472no—
334C2×#121—941no—
335C2×C4×Q8—471no—
336C2×C4×D4—471no—
337C2×C4×D4—472no—
338C2×C4×D4—941no—
339C2×C4×D4—941no—
340C2×#117—941no—
341C22×C42—471no—
342#501C2×D4718yescomplemented
343#496—478no—
344#490—478no—
345#489—478no—
346#466—4716no—
347#465—4716no—
348#442—4716no—
349#440—4716no—
350C2×#107C2×D4712yes—
351C2×#106C2×D4712yes—
352C2×#103—472no—
353C2×#102—472no—
354C2×#102—472no—
355C2×#102—472no—
356C2×#97—472no—
357C2×#97—471no—
358C2×#97—478no—
359C2×#97C2×D4711yescharacteristic
360C47C2×#318011yescharacteristic, complemented, Sylow
361C2×#41—948no—
362C2×#41—1888no—
363C22×Q8—21no—
364C22×Q8—471no—
365C22×D4—471no—
366C22×D4—472no—
367C22×D4—941no—
368C22×D4—942no—
369C22×D4—941no—
370C22×D4—941no—
371C23×C4—471no—
372C23×C4—471no—
373C23×C4—471no—
374C23×C4—941no—
375C23×C4—942no—
376C2×SD16—478no—
377C2×SD16—948no—
378C2×D8—478no—
379C2×D8—948no—
380C22×C8—472no—
381C22×C8—478no—
382C22×C8—948no—
383C22×C8C22×D4712yes—
384C4⋊Q8—22no—
385C4⋊Q8C22×D4714yes—
386#127—474no—
387#127—942no—
388#126—948no—
389#125—22no—
390#125C22×D4714yes—
391#124—474no—
392#124—948no—
393#124—942no—
394#123—948no—
395C22⋊Q8—474no—
396C22⋊Q8—942no—
397#121—474no—
398#121—478no—
399#121—942no—
400#121—944no—
401#121—948no—
402C4×Q8—474no—
403C4×Q8—942no—
404C4×D4—474no—
405C4×D4—478no—
406C4×D4—942no—
407C4×D4—944no—
408C4×D4—948no—
409C4×D4—948no—
410#117—948no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—941no—
415C2×C4⋊C4—941no—
416C2×C4⋊C4C22×D4712yes—
417C2×C4⋊C4C22×D4711yescharacteristic
418C2×C4⋊C4C22×D4711yescharacteristic
419C2×C22⋊C4—941no—
420C2×C22⋊C4—942no—
421C2×C22⋊C4—941no—
422C2×C22⋊C4—942no—
423C2×C22⋊C4—941no—
424C2×C22⋊C4—941no—
425C2×C22⋊C4—941no—
426C2×C42—471no—
427C2×C42—471no—
428C2×C42—471no—
429C2×C42—474no—
430C2×C42—474no—
431C2×C42—941no—
432C2×C42C22×D4711yescharacteristic
433#107C22×D4718yes—
434#106C22×D4718yes—
435#103—478no—
436#102—478no—
437#102—478no—
438#102—478no—
439#97—478no—
440#97—474no—
441#97C22×D4714yes—
442C24—471no—
443C24—941no—
444C24—942no—
445#41—18832no—
446C2×Q8—24no—
447C2×Q8—44no—
448C2×Q8—472no—
449C2×Q8—941no—
450C2×Q8—944no—
451C2×D4—472no—
452C2×D4—474no—
453C2×D4—944no—
454C2×D4—948no—
455C2×D4—941no—
456C2×D4—944no—
457C2×D4—942no—
458C2×D4—944no—
459C2×D4—944no—
460C2×D4—944no—
461C2×D4—944no—
462C2×D4—1884no—
463C2×D4—1884no—
464C2×D4—1884no—
465C2×D4—1888no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—471no—
470C22×C4—471no—
471C22×C4—471no—
472C22×C4—472no—
473C22×C4—474no—
474C22×C4—472no—
475C22×C4—474no—
476C22×C4—944no—
477C22×C4—948no—
478C22×C4—944no—
479C22×C4—944no—
480C22×C4—941no—
481C22×C4—1882no—
482C22×C4—1884no—
483C22×C4—1888no—
484C22×C4D4×D4711yescharacteristic
485C22×C4D4×D4711yescharacteristic
486C22×C4C23×D4711yescharacteristic
487SD16—9416no—
488D8—9416no—
489C2×C8—474no—
490C2×C8—944no—
491C2×C8—9416no—
492C2×C8#1062535418yescomplemented
493C2×C8C23×D4714yes—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—944no—
501C4⋊C4—944no—
502C4⋊C4C23×D4712yes—
503C4⋊C4C23×D4712yes—
504C4⋊C4C23×D4714yes—
505C22⋊C4—944no—
506C22⋊C4—948no—
507C22⋊C4—944no—
508C22⋊C4—948no—
509C22⋊C4—944no—
510C22⋊C4—944no—
511C22⋊C4—944no—
512C42—474no—
513C42—472no—
514C42—474no—
515C42—472no—
516C42—944no—
517C42D4×D4712yes—
518C42C23×D4712yes—
519C23—474no—
520C23—472no—
521C23—942no—
522C23—942no—
523C23—942no—
524C23—944no—
525C23—1884no—
526C23—1882no—
527C23—1882no—
528C23—1884no—
529C23—1882no—
530C23—1884no—
531C23C2×D4×D4711yescharacteristic
532Q8—48no—
533Q8—944no—
534Q8—1882no—
535D4—944no—
536D4—944no—
537D4—944no—
538D4—1888no—
539D4—1888no—
540D4—1888no—
541D4—18816no—
542D4—1882no—
543D4—1884no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—472no—
553C2×C4—472no—
554C2×C4—472no—
555C2×C4—948no—
556C2×C4—942no—
557C2×C4—942no—
558C2×C4—942no—
559C2×C4—944no—
560C2×C4—944no—
561C2×C4—942no—
562C2×C4—1884no—
563C2×C4—1888no—
564C2×C4—18816no—
565C2×C4—1882no—
566C2×C4C2×D4×D4712yes—
567C2×C4C2×D4×D4712yes—
568C2×C4C2×D4×D4711yescharacteristic
569C2×C4C2×#1062535414yes—
570C2×C4C24×D4711yescharacteristic
571C8—944no—
572C8C2×#1062535418yescomplemented
573C22—474no—
574C22—942no—
575C22—948no—
576C22—942no—
577C22—3768no—
578C22—3764no—
579C22—1884no—
580C22—1882no—
581C22—1882no—
582C22—1884no—
583C22—1882no—
584C22—1884no—
585C22#1568967412yes—
586C22AGL(1,ℤ/8)×D4712yes—
587C22#1568992212yes—
588C22C22×D4×D4711yescharacteristic
589C4—22no—
590C4—22no—
591C4—44no—
592C4—42no—
593C4—42no—
594C4—942no—
595C4—942no—
596C4—942no—
597C4—1882no—
598C4#1568967412yes—
599C4C22×#1062535412yes—
600C2—944no—
601C2—3764no—
602C2—3762no—
603C2#2025408814yesdirect product factor
604C2C2×#1568967411yescharacteristic
605C2C2×AGL(1,ℤ/8)×D4711yescharacteristic
606C2C2×#1568992211yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#501C2
socleC23×C47C22×D4
hypercenterC2×#501D47
hypocenterC47C2×#3180
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
1. C2×#15689674C2semidirect
2. C2×#15689658C2semidirect
3. #20254088C2direct (N × H)
4. C2×#15689512C2semidirect
5. C2×#15689489C2semidirect
6. C2×#15689485C2semidirect
7. C2×#15689360C2semidirect
8. C2×#15689353C2semidirect
9. C2×#15689234C2semidirect
10. C2×#15689222C2semidirect
11. C2×#97×D47C2semidirect
12. C2×C47×#501C2semidirect
18. #15689674C22semidirect
21. #15689658C22semidirect
37. #15689512C22semidirect
38. #15689489C22semidirect
39. #15689485C22semidirect
40. #15689360C22semidirect
41. #15689353C22semidirect
42. #15689234C22semidirect
43. #15689222C22semidirect
50. #97×D47C22semidirect
53. C47×#501C22semidirect
183. C2×C8×C47#41semidirect
267. C8×C47C2×#41semidirect
300. C2×#501D47semidirect
309. C2×C47#3180semidirect
342. #501C2×D47semidirect
360. C47C2×#3180semidirect
492. C2×C8#10625354semidirect
572. C8C2×#10625354semidirect
603. C2#20254088direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896742
C2×#156896582
#202540882
C2×#156895122
C2×#156894892
C2×#156894852
C2×#156893602
C2×#156893532
C2×#156892342
C2×#156892222
C2×#97×D472
C2×C47×#5012
C2×#318047