Finite Group Databasebeta

← C2×#20254416 #27087553

Subgroups — C2×#20254416 #27087553

Order 12032.

2146 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D47C211yesmaximal, characteristic, complemented
2C2×#15689658C211yesmaximal, characteristic
3#20254416C2116yesmaximal, direct product factor
4C2×#15689517C211yesmaximal, characteristic, complemented
5C2×#15689515C211yesmaximal, characteristic, complemented
6C2×#15689493C211yesmaximal, characteristic, complemented
7C2×#15689332C212yesmaximal, complemented
8C2×#15689327C212yesmaximal, complemented
9C2×#15689280C212yesmaximal, complemented
10C2×#15689263C212yesmaximal
11C2×#15689178C211yesmaximal, characteristic, complemented
12C2×C47×#491C211yesmaximal, characteristic, complemented
13C22×Q8×D47C2212yes—
14C22×Dic188C2212yes—
15C22×#10625337C2212yes—
16C22×#10625325C2212yes—
17C22×#10625336C2212yes—
18C2×#13094024C2211yescharacteristic
19C2×#13094023—21no—
20C2×#13094023—21no—
21C2×#13094022C2212yes—
22C4⋊Q8×D47—24no—
23C4⋊Q8×D47C2218yescomplemented
24#15689658—24no—
25#15689658C2218yes—
26C2×#13094019—21no—
27C2×#13094012C2211yescharacteristic
28C2×#13094009—21no—
29C2×#13094004C2211yescharacteristic
30C2×#13093999C2211yescharacteristic
31C2×#13093989C2211yescharacteristic
32C2×#13093986—21no—
33C2×#13093986C2211yescharacteristic
34C2×#13093982—21no—
35C2×C4⋊C4×D47—21no—
36C2×C4⋊C4×D47—21no—
37C2×C42×D47C2211yescharacteristic
38#15689517C2218yescomplemented
39#15689515C2218yescomplemented
40#15689493C2218yescomplemented
41#15689332C22116yescomplemented
42#15689327C22116yescomplemented
43#15689280C22116yescomplemented
44#15689263C22116yes—
45#15689178C2218yescomplemented
46C2×Dic47⋊C8C2212yes—
47C2×#13093947C2212yes—
48C2×#13093947C2212yes—
49C2×(C2×D47)⋊C8C2212yes—
50C2×#13093935C2211yescharacteristic
51C2×C4⋊Q8×C47C2211yescharacteristic
52C2×#125×C47C2211yescharacteristic
53C47×#491C2218yescomplemented
54C2×#103×C47C2212yes—
55C2×#103×C47C2212yes—
56C2×#97×C47C2211yescharacteristic
57C22×Dic94—22no—
58C22×Dic94C2311yescharacteristic
59C22×Dic94C2312yes—
60C2×Q8×D47—24no—
61C2×Q8×D47—44no—
62C2×Q8×D47—44no—
63C2×Q8×D47C2318yes—
64C22×D188—21no—
65C22×D188C2311yescharacteristic
66C22×C4×D47—22no—
67C22×C4×D47—21no—
68C22×C4×D47D411yescharacteristic
69C22×C4×D47D411yescharacteristic
70C22×C4×D47C2311yescharacteristic
71C2×Dic188—44no—
72C2×Dic188C2318yes—
73C2×#10625337—44no—
74C2×#10625337C2318yes—
75C2×#10625325—44no—
76C2×#10625325C2318yes—
77C2×#10625336—44no—
78C2×#10625336C2318yes—
79C22×C47⋊C8C2312yes—
80#13094024—22no—
81#13094024C2314yes—
82#13094023—28no—
83#13094023—28no—
84#13094022—24no—
85#13094022C2318yes—
86#13094019—28no—
87#13094012—22no—
88#13094012C2314yes—
89#13094009—28no—
90#13094004—22no—
91#13094004C2314yes—
92#13093999—22no—
93#13093999C2314yes—
94#13093989—22no—
95#13093989C2314yes—
96#13093986—28no—
97#13093986—22no—
98#13093986C2314yes—
99#13093982—28no—
100C2×C4⋊Dic47—21no—
101C2×C4⋊Dic47—21no—
102C2×C4⋊Dic47—21no—
103C2×Dic47⋊C4—22no—
104C2×Dic47⋊C4—22no—
105C2×Dic47⋊C4—21no—
106C4⋊C4×D47—28no—
107C4⋊C4×D47—28no—
108C2×(C2×D47)⋊C4—21no—
109C2×(C2×D47)⋊C4—21no—
110C2×(C2×D47)⋊C4—21no—
111C2×(C2×D47)⋊C4—21no—
112C2×(C2×D47)⋊C4—21no—
113C2×C4×Dic47—21no—
114C2×C4×Dic47D411yescharacteristic
115C2×C4×Dic47D411yescharacteristic
116C2×C4×Dic47C2311yescharacteristic
117C42×D47D414yes—
118C42×D47C2314yes—
119Dic47⋊C8C2318yes—
120#13093947C2318yes—
121#13093947C2318yes—
122(C2×D47)⋊C8C2318yes—
123#13093935C2314yes—
124C22×Q8×C47C2312yes—
125C22×C8×C47C2312yes—
126C4⋊Q8×C47—22no—
127C4⋊Q8×C47C2314yes—
128#125×C47—22no—
129#125×C47C2314yes—
130C2×C4⋊C4×C47—21no—
131C2×C4⋊C4×C47—21no—
132C2×C4⋊C4×C47—21no—
133C2×C4⋊C4×C47—21no—
134C2×C4⋊C4×C47C2311yescharacteristic
135C2×C4⋊C4×C47C2311yescharacteristic
136C2×C42×C47C2311yescharacteristic
137#103×C47C2318yes—
138#103×C47C2318yes—
139#97×C47C2314yes—
140C23×D47—21no—
141C23×D47C2×D411yescharacteristic
142C2×Q8×C47—22no—
143C2×Q8×C47—24no—
144C2×Q8×C47—24no—
145C2×Q8×C47C2414yes—
146C22×C4×C47—21no—
147C22×C4×C47—21no—
148C22×C4×C47—22no—
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2411yescharacteristic
152C22×Dic47—21no—
153C22×Dic47—22no—
154C22×Dic47C2×D411yescharacteristic
155C22×Dic47C2×D411yescharacteristic
156C22×Dic47C2×D411yescharacteristic
157Q8×D47—48no—
158Q8×D47—48no—
159Q8×D47—416no—
160C2×D188—24no—
161C2×D188—24no—
162C2×D188—21no—
163C2×D188—44no—
164C2×D188C2412yes—
165C2×C4×D47—24no—
166C2×C4×D47—24no—
167C2×C4×D47—24no—
168C2×C4×D47—28no—
169C2×C4×D47—24no—
170C2×C4×D47—48no—
171C2×C4×D47—44no—
172C2×C4×D47C2×D412yes—
173C2×C4×D47C2×D414yes—
174C2×C4×D47C2×D414yes—
175C2×C4×D47C2412yes—
176C2×Dic94—24no—
177C2×Dic94—24no—
178C2×Dic94—24no—
179C2×Dic94—21no—
180C2×Dic94—22no—
181C2×Dic94—28no—
182C2×Dic94—48no—
183C2×Dic94C2414yes—
184C2×Dic94C2412yes—
185C2×C8×C47—24no—
186C2×C8×C47C2414yes—
187C4⋊C4×C47—24no—
188C4⋊C4×C47—24no—
189C4⋊C4×C47—24no—
190C4⋊C4×C47—24no—
191C4⋊C4×C47—21no—
192C4⋊C4×C47—21no—
193C4⋊C4×C47C2412yes—
194C4⋊C4×C47C2412yes—
195C42×C47C2×D412yes—
196C42×C47C2412yes—
197#10625337—48no—
198#10625336—48no—
199(C2×D47)⋊C4—24no—
200(C2×D47)⋊C4—24no—
201(C2×D47)⋊C4—24no—
202(C2×D47)⋊C4—24no—
203(C2×D47)⋊C4—24no—
204C4⋊Dic47—24no—
205C4⋊Dic47—24no—
206C4⋊Dic47—24no—
207Dic47⋊C4—28no—
208Dic47⋊C4—28no—
209Dic47⋊C4—24no—
210C4×Dic47—24no—
211C4×Dic47C2×D412yes—
212C4×Dic47C2×D414yes—
213C4×Dic47C2×D414yes—
214C4×Dic47C2412yes—
215C2×C47⋊C8—24no—
216C2×C47⋊C8C2414yes—
217Dic188—48no—
218#10625325—48no—
219C23×C47C22×D411yescharacteristic
220C22×D47—22no—
221C22×D47—22no—
222C22×D47—22no—
223C22×D47—42no—
224C22×D47—42no—
225C22×D47—42no—
226C22×D47#13714yes—
227C22×D47C22×D412yes—
228Q8×C47—24no—
229Q8×C47—24no—
230Q8×C47—44no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—22no—
234C2×C4×C47—24no—
235C2×C4×C47—22no—
236C2×C4×C47—44no—
237C2×C4×C47—42no—
238C2×C4×C47—42no—
239C2×C4×C47C22×D412yes—
240C2×C4×C47C22×D412yes—
241C2×C4×C47C22×D411yescharacteristic
242C2×C4×C47H(5,𝔽2)14yes—
243C2×C4×C47C2511yescharacteristic
244C2×Dic47—22no—
245C2×Dic47—22no—
246C2×Dic47—22no—
247C2×Dic47—24no—
248C2×Dic47—44no—
249C2×Dic47—42no—
250C2×Dic47#13714yes—
251C2×Dic47C22×D412yes—
252C2×Dic47C22×D412yes—
253C2×Dic47C22×D412yes—
254D188—24no—
255D188—48no—
256D188—42no—
257C4×D47—28no—
258C4×D47—24no—
259C4×D47—24no—
260C4×D47—28no—
261C4×D47—416no—
262C4×D47—48no—
263Dic94—24no—
264Dic94—24no—
265Dic94—24no—
266Dic94—416no—
267Dic94—42no—
268Dic94—44no—
269C8×C47—24no—
270C47⋊C8—24no—
271C2×#3218—471nomaximal, Sylow
272C22×C47#53512yes—
273C22×C47C2×#13712yes—
274C22×C47C2×#13712yes—
275C22×C47C23×D411yescharacteristic
276C2×D47—22no—
277C2×D47—28no—
278C2×D47—22no—
279C2×D47—42no—
280C2×D47—42no—
281C2×D47—42no—
282C2×D47—84no—
283C2×D47C2×#13714yes—
284C4×C47—22no—
285C4×C47—22no—
286C4×C47—44no—
287C4×C47—42no—
288C4×C47—42no—
289C4×C47#53512yes—
290C4×C47C2×H(5,𝔽2)12yes—
291Dic47—22no—
292Dic47—22no—
293Dic47—44no—
294Dic47—42no—
295Dic47C2×#13714yes—
296C22×C4⋊Q8—471no—
297C2×#529—471no—
298#3218—4716no—
299C2×#497—471no—
300C2×#497—471no—
301C2×#491—471no—
302C2×#491D4711yescharacteristic, complemented
303C2×#462—472no—
304C2×#461—472no—
305C2×#451—472no—
306C2×#448—472no—
307C2×#432—471no—
308C2×C47#321814yescomplemented
309C2×C47C2×#53511yescharacteristic
310C2×C47C22×#13711yescharacteristic
311C2×C47C22×#13711yescharacteristic
312D47—24no—
313D47—82no—
314C23×Q8—472no—
315C22×Q16—472no—
316C22×Q16—472no—
317C22×SD16—472no—
318C22×SD16—472no—
319C2×C4⋊Q8—472no—
320C2×C4⋊Q8—478no—
321C2×C4⋊Q8—471no—
322C2×C4⋊Q8—941no—
323C2×C4⋊Q8—941no—
324C2×C4⋊Q8—944no—
325C2×C4⋊Q8C2×D4711yescharacteristic
326#529—478no—
327#529—944no—
328C2×#126—941no—
329C2×#125C2×D4711yescharacteristic
330C2×#124—471no—
331C2×#123—941no—
332C2×C22⋊Q8—471no—
333C2×#121—471no—
334C2×C4×Q8—471no—
335C2×C4×D4—471no—
336C2×C4×D4—941no—
337C2×#117—941no—
338C22×C4⋊C4—941no—
339C22×C4⋊C4—941no—
340C22×C42—471no—
341#497—478no—
342#497—478no—
343#491—478no—
344#491C2×D4718yescomplemented
345#462—4716no—
346#461—4716no—
347#451—4716no—
348#448—4716no—
349#432—478no—
350C2×C4⋊C8—472no—
351C2×#103—472no—
352C2×#103—472no—
353C2×#103C2×D4712yes—
354C2×#103C2×D4712yes—
355C2×C22⋊C8—472no—
356C2×#97—471no—
357C2×#97C2×D4711yescharacteristic
358C47C2×#321811yescharacteristic, complemented, Sylow
359C22×Q8—478no—
360C22×Q8—471no—
361C22×Q8—472no—
362C22×Q8—942no—
363C22×Q8—944no—
364C22×Q8—1884no—
365C22×Q8—1884no—
366C22×Q8C22×D4712yes—
367C22×D4—471no—
368C22×D4—941no—
369C23×C4—471no—
370C23×C4—471no—
371C23×C4—471no—
372C23×C4—941no—
373C23×C4—942no—
374C2×Q16—478no—
375C2×Q16—478no—
376C2×Q16—1884no—
377C2×Q16—1884no—
378C2×SD16—478no—
379C2×SD16—478no—
380C2×SD16—1884no—
381C2×SD16—1884no—
382C22×C8—472no—
383C22×C8C22×D4712yes—
384C4⋊Q8—22no—
385C4⋊Q8—478no—
386C4⋊Q8—474no—
387C4⋊Q8—948no—
388C4⋊Q8—948no—
389C4⋊Q8—944no—
390C4⋊Q8—942no—
391C4⋊Q8C22×D4714yes—
392#126—948no—
393#125—22no—
394#125C22×D4714yes—
395#124—474no—
396#124—942no—
397#123—948no—
398C22⋊Q8—474no—
399C22⋊Q8—942no—
400#121—474no—
401#121—942no—
402C4×Q8—474no—
403C4×Q8—942no—
404C4×D4—474no—
405C4×D4—942no—
406C4×D4—948no—
407#117—948no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—948no—
413C2×C4⋊C4—948no—
414C2×C4⋊C4—942no—
415C2×C4⋊C4—942no—
416C2×C4⋊C4—941no—
417C2×C4⋊C4—941no—
418C2×C4⋊C4—941no—
419C2×C4⋊C4—941no—
420C2×C4⋊C4C22×D4711yescharacteristic
421C2×C4⋊C4C22×D4711yescharacteristic
422C2×C22⋊C4—941no—
423C2×C22⋊C4—941no—
424C2×C22⋊C4—941no—
425C2×C22⋊C4—941no—
426C2×C22⋊C4—941no—
427C2×C42—471no—
428C2×C42—471no—
429C2×C42—471no—
430C2×C42—474no—
431C2×C42—474no—
432C2×C42—941no—
433C2×C42C22×D4711yescharacteristic
434C4⋊C8—478no—
435#103—478no—
436#103—478no—
437#103C22×D4718yes—
438#103C22×D4718yes—
439C22⋊C8—478no—
440#97—474no—
441#97C22×D4714yes—
442C24—471no—
443C24—941no—
444C2×Q8—22no—
445C2×Q8—24no—
446C2×Q8—24no—
447C2×Q8—474no—
448C2×Q8—472no—
449C2×Q8—948no—
450C2×Q8—941no—
451C2×Q8—942no—
452C2×Q8—944no—
453C2×Q8—944no—
454C2×Q8—944no—
455C2×Q8—1888no—
456C2×Q8—1888no—
457C2×Q8—1888no—
458C2×Q8—18816no—
459C2×Q8C23×D4714yes—
460C2×D4—472no—
461C2×D4—944no—
462C2×D4—941no—
463C2×D4—944no—
464C2×D4—1884no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—471no—
469C22×C4—471no—
470C22×C4—471no—
471C22×C4—474no—
472C22×C4—474no—
473C22×C4—472no—
474C22×C4—472no—
475C22×C4—944no—
476C22×C4—944no—
477C22×C4—944no—
478C22×C4—948no—
479C22×C4—944no—
480C22×C4—941no—
481C22×C4—942no—
482C22×C4—1884no—
483C22×C4—1888no—
484C22×C4D4×D4711yescharacteristic
485C22×C4D4×D4711yescharacteristic
486C22×C4C23×D4711yescharacteristic
487Q16—1888no—
488Q16—1888no—
489SD16—1888no—
490SD16—1888no—
491C2×C8—24no—
492C2×C8—474no—
493C2×C8—944no—
494C2×C8C23×D4714yes—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—948no—
502C4⋊C4—948no—
503C4⋊C4—944no—
504C4⋊C4—944no—
505C4⋊C4—944no—
506C4⋊C4—944no—
507C4⋊C4C23×D4712yes—
508C4⋊C4C23×D4712yes—
509C22⋊C4—944no—
510C22⋊C4—944no—
511C22⋊C4—944no—
512C22⋊C4—944no—
513C22⋊C4—944no—
514C42—474no—
515C42—472no—
516C42—474no—
517C42—472no—
518C42—944no—
519C42D4×D4712yes—
520C42C23×D4712yes—
521C23—474no—
522C23—472no—
523C23—942no—
524C23—942no—
525C23—942no—
526C23—1882no—
527C23—1882no—
528C23—1882no—
529C23C2×D4×D4711yescharacteristic
530Q8—24no—
531Q8—24no—
532Q8—44no—
533Q8—944no—
534Q8—944no—
535Q8—944no—
536Q8—18816no—
537Q8—1882no—
538Q8—1884no—
539D4—944no—
540D4—1888no—
541D4—1882no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—474no—
551C2×C4—472no—
552C2×C4—472no—
553C2×C4—472no—
554C2×C4—948no—
555C2×C4—948no—
556C2×C4—942no—
557C2×C4—942no—
558C2×C4—944no—
559C2×C4—944no—
560C2×C4—942no—
561C2×C4—944no—
562C2×C4—1888no—
563C2×C4—18816no—
564C2×C4—1884no—
565C2×C4—1882no—
566C2×C4C2×D4×D4712yes—
567C2×C4C2×D4×D4712yes—
568C2×C4C2×D4×D4711yescharacteristic
569C2×C4#1309407014yes—
570C2×C4C24×D4711yescharacteristic
571C8—24no—
572C8—944no—
573C22—474no—
574C22—942no—
575C22—942no—
576C22—948no—
577C22—3764no—
578C22—1882no—
579C22—1882no—
580C22—1882no—
581C22#1568969112yes—
582C22#137×D4712yes—
583C22#137×D4712yes—
584C22C22×D4×D4711yescharacteristic
585C4—22no—
586C4—22no—
587C4—42no—
588C4—42no—
589C4—44no—
590C4—474no—
591C4—942no—
592C4—942no—
593C4—1884no—
594C4—1882no—
595C4#1568969112yes—
596C4C2×#1309407012yes—
597C2—944no—
598C2—3762no—
599C2#2025441614yesdirect product factor
600C2C2×#1568969111yescharacteristic
601C2C2×#137×D4711yescharacteristic
602C2C2×#137×D4711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#491C2
socleC23×C47C22×D4
prime radicalC2×#13093999C22
hypercenterC2×#491D47
hypocenterC47C2×#3218
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊Q8×D47C2semidirect
3. #20254416C2direct (N × H)
4. C2×#15689517C2semidirect
5. C2×#15689515C2semidirect
6. C2×#15689493C2semidirect
7. C2×#15689332C2semidirect
8. C2×#15689327C2semidirect
9. C2×#15689280C2semidirect
11. C2×#15689178C2semidirect
12. C2×C47×#491C2semidirect
23. C4⋊Q8×D47C22semidirect
38. #15689517C22semidirect
39. #15689515C22semidirect
40. #15689493C22semidirect
41. #15689332C22semidirect
42. #15689327C22semidirect
43. #15689280C22semidirect
45. #15689178C22semidirect
53. C47×#491C22semidirect
302. C2×#491D47semidirect
308. C2×C47#3218semidirect
344. #491C2×D47semidirect
358. C47C2×#3218semidirect
599. C2#20254416direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D472
C2×#156896582
#202544162
C2×#156895172
C2×#156895152
C2×#156894932
C2×#156893322
C2×#156893272
C2×#156892802
C2×#156892632
C2×#156891782
C2×C47×#4912
C2×#321847