Finite Group Databasebeta

← C2×#20254755 #27089128

Subgroups — C2×#20254755 #27089128

Order 12032.

1898 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689657C211yesmaximal, characteristic
2#20254755C2116yesmaximal, direct product factor
3C2×#15689430C212yesmaximal, complemented
4C2×#15689415C211yesmaximal, characteristic, complemented
5C2×#15689410C211yesmaximal, characteristic, complemented
6C2×#15689379C212yesmaximal, complemented
7C2×#15689371C212yesmaximal
8C2×#15689345C212yesmaximal, complemented
9C2×#15689330C211yesmaximal, characteristic, complemented
10C2×#15689325C211yesmaximal, characteristic, complemented
11C2×#15689147C211yesmaximal, characteristic, complemented
12C2×C47×#533C211yesmaximal, characteristic, complemented
13C22×#10625337C2212yes—
14C22×#10625336C2212yes—
15C22×#10625329C2212yes—
16#15689657—24no—
17#15689657C2218yes—
18C2×#13094018—21no—
19C2×#13094016C2211yescharacteristic
20C2×#13094012C2211yescharacteristic
21C2×#13094007—21no—
22C2×#13094003C2211yescharacteristic
23C2×#13093998C2211yescharacteristic
24C2×C4×Dic94C2211yescharacteristic
25C2×C4×D188C2211yescharacteristic
26C2×C4×C22⋊D47—21no—
27C2×#13093981—21no—
28#15689430C22116yescomplemented
29#15689415C2218yescomplemented
30#15689410C2218yescomplemented
31#15689379C22116yescomplemented
32#15689371C22116yes—
33#15689345C22116yescomplemented
34#15689330C2218yescomplemented
35#15689325C2218yescomplemented
36C2×#13093957C2212yes—
37C2×#13093955C2212yes—
38#15689147C2218yescomplemented
39C2×C4⋊(C47⋊C8)C2211yescharacteristic
40C2×C4⋊(C47⋊C8)C2211yescharacteristic
41C2×C4⋊(C47⋊C8)C2212yes—
42C2×#13093948C2212yes—
43C2×#13093948C2212yes—
44C2×#13093947C2212yes—
45C2×#13093944C2212yes—
46C47×#533—24no—
47C47×#533C2218yescomplemented
48C2×C4⋊Q8×C47C2211yescharacteristic
49C2×C4⋊Q8×C47C2211yescharacteristic
50C2×#125×C47—21no—
51C2×C22⋊Q8×C47—21no—
52C2×C22⋊Q8×C47C2212yes—
53C2×C4×Q8×C47—21no—
54C2×C4×Q8×C47C2212yes—
55C2×#117×C47—21no—
56C22×C42×C47C2211yescharacteristic
57C22×Dic94C2311yescharacteristic
58C22×D188C2311yescharacteristic
59C22×C22⋊D47—21no—
60C22×C4×D47—21no—
61C2×#10625337—44no—
62C2×#10625337C2318yes—
63C2×#10625336—44no—
64C2×#10625336C2318yes—
65C2×#10625329—44no—
66C2×#10625329C2318yes—
67C22×C47⋊C8C2312yes—
68C22×C47⋊C8C2312yes—
69#13094018—28no—
70#13094016—22no—
71#13094016C2314yes—
72#13094012—22no—
73#13094012C2314yes—
74#13094007—28no—
75#13094003—22no—
76#13094003C2314yes—
77#13093998—22no—
78#13093998C2314yes—
79C4×Dic94—22no—
80C4×Dic94C2314yes—
81C4×D188—22no—
82C4×D188C2314yes—
83C4×C22⋊D47—28no—
84#13093981—28no—
85C2×C4⋊Dic47C2311yescharacteristic
86C2×C4⋊Dic47C2311yescharacteristic
87C2×Dic47⋊C4—21no—
88C2×Dic47⋊C4—21no—
89C2×Dic47⋊C4—21no—
90C2×Dic47⋊C4—21no—
91C2×C22⋊Dic47—21no—
92C2×(C2×D47)⋊C4—21no—
93C2×(C2×D47)⋊C4—21no—
94C2×(C2×D47)⋊C4—21no—
95C2×(C2×D47)⋊C4—21no—
96C2×C4×Dic47—21no—
97#13093957C2318yes—
98#13093955C2318yes—
99C4⋊(C47⋊C8)C2314yes—
100C4⋊(C47⋊C8)C2314yes—
101C4⋊(C47⋊C8)C2318yes—
102#13093948C2318yes—
103#13093948C2318yes—
104#13093947C2318yes—
105#13093944C2318yes—
106C22×Q8×C47—21no—
107C22×Q8×C47C2312yes—
108C23×C4×C47D411yescharacteristic
109C23×C4×C47D411yescharacteristic
110C23×C4×C47C2311yescharacteristic
111C4⋊Q8×C47—22no—
112C4⋊Q8×C47—22no—
113C4⋊Q8×C47C2314yes—
114C4⋊Q8×C47C2314yes—
115#125×C47—28no—
116C22⋊Q8×C47—28no—
117C22⋊Q8×C47—24no—
118C22⋊Q8×C47C2318yes—
119C4×Q8×C47—28no—
120C4×Q8×C47—24no—
121C4×Q8×C47C2318yes—
122#117×C47—28no—
123C2×C4⋊C4×C47—21no—
124C2×C4⋊C4×C47—21no—
125C2×C4⋊C4×C47—21no—
126C2×C4⋊C4×C47—22no—
127C2×C4⋊C4×C47—22no—
128C2×C4⋊C4×C47C2312yes—
129C2×C22⋊C4×C47—22no—
130C2×C42×C47—22no—
131C2×C42×C47D411yescharacteristic
132C2×C42×C47D411yescharacteristic
133C2×C42×C47D414yes—
134C2×C42×C47C2311yescharacteristic
135C2×C42×C47C2311yescharacteristic
136C2×C42×C47C2314yes—
137C24×C47C2×D411yescharacteristic
138C23×D47—21no—
139C2×Q8×C47—22no—
140C2×Q8×C47—24no—
141C2×Q8×C47—24no—
142C2×Q8×C47—24no—
143C2×Q8×C47—44no—
144C2×Q8×C47C2414yes—
145C22×C4×C47—24no—
146C22×C4×C47—22no—
147C22×C4×C47—22no—
148C22×C4×C47—22no—
149C22×C4×C47—22no—
150C22×C4×C47—42no—
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D412yes—
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D414yes—
157C22×C4×C47C2×D414yes—
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2412yes—
160C22×C4×C47C2411yescharacteristic
161C2×C22⋊D47—24no—
162C2×C22⋊D47—44no—
163C22×Dic47—21no—
164C22×Dic47—21no—
165C22×Dic47—21no—
166C2×D188—22no—
167C2×D188—22no—
168C2×D188—21no—
169C2×D188C2412yes—
170C2×C4×D47—24no—
171C2×C4×D47—44no—
172C2×Dic94—22no—
173C2×Dic94—22no—
174C2×Dic94—21no—
175C2×Dic94C2412yes—
176C4⋊C4×C47—24no—
177C4⋊C4×C47—24no—
178C4⋊C4×C47—24no—
179C4⋊C4×C47—28no—
180C4⋊C4×C47—28no—
181C4⋊C4×C47—22no—
182C4⋊C4×C47C2414yes—
183C22⋊C4×C47—28no—
184C42×C47—28no—
185C42×C47C2×D412yes—
186C42×C47C2×D414yes—
187C42×C47C2×D414yes—
188C42×C47C2×D412yes—
189C42×C47C2412yes—
190C42×C47C2412yes—
191C22⋊Dic47—24no—
192#10625337—48no—
193#10625336—48no—
194(C2×D47)⋊C4—24no—
195(C2×D47)⋊C4—24no—
196(C2×D47)⋊C4—24no—
197(C2×D47)⋊C4—24no—
198C4⋊Dic47—21no—
199C4⋊Dic47—21no—
200C4⋊Dic47C2412yes—
201C4⋊Dic47C2412yes—
202Dic47⋊C4—24no—
203Dic47⋊C4—24no—
204Dic47⋊C4—24no—
205Dic47⋊C4—24no—
206C4×Dic47—24no—
207#10625329—48no—
208C2×C47⋊C8—24no—
209C2×C47⋊C8—24no—
210C2×C47⋊C8C2414yes—
211C2×C47⋊C8C2414yes—
212C23×C47—22no—
213C23×C47—22no—
214C23×C47—22no—
215C23×C47C22×D412yes—
216C23×C47C22×D411yescharacteristic
217C22×D47—22no—
218C22×D47—42no—
219C22×D47—42no—
220C22×D47—42no—
221Q8×C47—24no—
222Q8×C47—24no—
223Q8×C47—48no—
224Q8×C47—44no—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
227C2×C4×C47—22no—
228C2×C4×C47—22no—
229C2×C4×C47—24no—
230C2×C4×C47—28no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—24no—
234C2×C4×C47—24no—
235C2×C4×C47—44no—
236C2×C4×C47—44no—
237C2×C4×C47—44no—
238C2×C4×C47#13714yes—
239C2×C4×C47C22×D412yes—
240C2×C4×C47C22×D412yes—
241C2×C4×C47C22×D412yes—
242C2×C4×C47C22×D412yes—
243C2×C4×C47C22×D411yescharacteristic
244C2×C4×C47C22×D412yes—
245C2×C4×C47H(5,𝔽2)12yes—
246C2×C4×C472−1+412yes—
247C2×C4×C47C2511yescharacteristic
248C22⋊D47—48no—
249C2×Dic47—22no—
250C2×Dic47—22no—
251C2×Dic47—22no—
252C2×Dic47—42no—
253C2×Dic47—42no—
254C2×Dic47—42no—
255D188—22no—
256D188—22no—
257D188—42no—
258C4×D47—48no—
259Dic94—22no—
260Dic94—22no—
261Dic94—42no—
262C47⋊C8—24no—
263C47⋊C8—24no—
264C2×#3252—471nomaximal, Sylow
265C22×C47—22no—
266C22×C47—22no—
267C22×C47—22no—
268C22×C47—44no—
269C22×C47#53712yes—
270C22×C47C2×#13712yes—
271C22×C47#57512yes—
272C22×C47C23×D411yescharacteristic
273C2×D47—42no—
274C2×D47—42no—
275C2×D47—42no—
276C2×D47—84no—
277C4×C47—22no—
278C4×C47—22no—
279C4×C47—22no—
280C4×C47—22no—
281C4×C47—44no—
282C4×C47—44no—
283C4×C47#53712yes—
284C4×C47C2×#13714yes—
285C4×C47C2×H(5,𝔽2)11yescharacteristic
286C4×C47C2×2−1+411yescharacteristic
287Dic47—42no—
288Dic47—42no—
289Dic47—42no—
290C2×#533D4711yescharacteristic, complemented
291C2×#529—471no—
292#3252—4716no—
293C2×#479—472no—
294C2×#477—471no—
295C2×#476—471no—
296C2×#470—472no—
297C2×#468—472no—
298C2×#464—472no—
299C2×#462—471no—
300C2×#461—471no—
301C2×#423—471no—
302C2×C47—42no—
303C2×C47#325214yescomplemented
304C2×C47C2×#53711yescharacteristic
305C2×C47C22×#13711yescharacteristic
306C2×C47C2×#57511yescharacteristic
307D47—82no—
308C22×Q16—472no—
309C22×SD16—472no—
310C22×M16—472no—
311#533—24no—
312#533C2×D4718yescomplemented
313C2×C4⋊Q8C2×D4711yescharacteristic
314C2×C4⋊Q8C2×D4711yescharacteristic
315#529—478no—
316#529—944no—
317C2×#126—941no—
318C2×#125—21no—
319C2×#125—471no—
320C2×#124—471no—
321C2×#123—941no—
322C2×C22⋊Q8—21no—
323C2×C22⋊Q8—471no—
324C2×C22⋊Q8C2×D4712yes—
325C2×#121—471no—
326C2×C4×Q8—21no—
327C2×C4×Q8—471no—
328C2×C4×Q8C2×D4712yes—
329C2×C4×D4—471no—
330C2×C4×D4—941no—
331C2×#117—21no—
332C2×#117—941no—
333C22×C42C2×D4711yescharacteristic
334#479—4716no—
335#477—478no—
336#476—478no—
337#470—4716no—
338#468—4716no—
339#464—4716no—
340#462—478no—
341#461—478no—
342C2×#107—472no—
343C2×#106—472no—
344#423—478no—
345C2×C4⋊C8—471no—
346C2×C4⋊C8—472no—
347C2×C4⋊C8—471no—
348C2×#103—472no—
349C2×#103—472no—
350C2×#103—472no—
351C2×#102—472no—
352C47C2×#325211yescharacteristic, complemented, Sylow
353C22×Q8—21no—
354C22×Q8—471no—
355C22×Q8C22×D4712yes—
356C22×D4—471no—
357C22×D4—941no—
358C23×C4—941no—
359C23×C4C22⋊D4711yescharacteristic
360C23×C4C22⋊D4711yescharacteristic
361C23×C4C22×D4711yescharacteristic
362C2×Q16—478no—
363C2×Q16—1884no—
364C2×SD16—478no—
365C2×SD16—1884no—
366C2×M16—478no—
367C2×M16—1884no—
368C22×C8—472no—
369C22×C8—472no—
370C4⋊Q8—22no—
371C4⋊Q8—22no—
372C4⋊Q8C22×D4714yes—
373C4⋊Q8C22×D4714yes—
374#126—948no—
375#125—28no—
376#125—474no—
377#125—942no—
378#124—474no—
379#124—942no—
380#123—948no—
381C22⋊Q8—28no—
382C22⋊Q8—24no—
383C22⋊Q8—474no—
384C22⋊Q8—942no—
385C22⋊Q8C22×D4718yes—
386#121—474no—
387#121—942no—
388C4×Q8—28no—
389C4×Q8—24no—
390C4×Q8—474no—
391C4×Q8—942no—
392C4×Q8C22×D4718yes—
393C4×D4—474no—
394C4×D4—948no—
395C4×D4—942no—
396#117—28no—
397#117—948no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—471no—
404C2×C4⋊C4—471no—
405C2×C4⋊C4—941no—
406C2×C4⋊C4—941no—
407C2×C4⋊C4—941no—
408C2×C4⋊C4—941no—
409C2×C4⋊C4C22×D4712yes—
410C2×C22⋊C4—22no—
411C2×C22⋊C4—941no—
412C2×C22⋊C4—941no—
413C2×C22⋊C4—941no—
414C2×C22⋊C4—941no—
415C2×C22⋊C4—941no—
416C2×C42—22no—
417C2×C42—941no—
418C2×C42C22⋊D4711yescharacteristic
419C2×C42C22⋊D4711yescharacteristic
420C2×C42C22⋊D4714yes—
421C2×C42C22×D4711yescharacteristic
422C2×C42C22×D4711yescharacteristic
423C2×C42C22×D4714yes—
424#107—478no—
425#106—478no—
426C4⋊C8—474no—
427C4⋊C8—474no—
428C4⋊C8—478no—
429#103—478no—
430#103—478no—
431#103—478no—
432#102—478no—
433C24—941no—
434C24C2×C22⋊D4711yescharacteristic
435C2×Q8—22no—
436C2×Q8—24no—
437C2×Q8—24no—
438C2×Q8—24no—
439C2×Q8—44no—
440C2×Q8—472no—
441C2×Q8—941no—
442C2×Q8—942no—
443C2×Q8—942no—
444C2×Q8C23×D4714yes—
445C2×D4—472no—
446C2×D4—944no—
447C2×D4—941no—
448C2×D4—942no—
449C2×D4—942no—
450C2×D4—1884no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—42no—
457C22×C4—941no—
458C22×C4—941no—
459C22×C4—941no—
460C22×C4—944no—
461C22×C4—1884no—
462C22×C4C2×C22⋊D4712yes—
463C22×C4C2×C22⋊D4714yes—
464C22×C4C2×C22⋊D4711yescharacteristic
465C22×C4C2×C22⋊D4711yescharacteristic
466C22×C4C2×C22⋊D4711yescharacteristic
467C22×C4C2×C22⋊D4711yescharacteristic
468C22×C4C2×C22⋊D4714yes—
469C22×C4C2×C22⋊D4711yescharacteristic
470C22×C4C23×D4712yes—
471C22×C4C23×D4711yescharacteristic
472Q16—1888no—
473SD16—1888no—
474M16—1888no—
475C2×C8—474no—
476C2×C8—474no—
477C2×C8—944no—
478C2×C8—944no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—22no—
485C4⋊C4—472no—
486C4⋊C4—472no—
487C4⋊C4—944no—
488C4⋊C4—944no—
489C4⋊C4—944no—
490C4⋊C4—944no—
491C4⋊C4—941no—
492C4⋊C4—941no—
493C4⋊C4C23×D4714yes—
494C22⋊C4—28no—
495C22⋊C4—944no—
496C22⋊C4—944no—
497C22⋊C4—944no—
498C22⋊C4—944no—
499C22⋊C4—944no—
500C42—28no—
501C42—944no—
502C42C2×C22⋊D4714yes—
503C42C2×C22⋊D4712yes—
504C42C2×C22⋊D4712yes—
505C42C2×C22⋊D4714yes—
506C42C23×D4712yes—
507C42C23×D4712yes—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—942no—
512C23—1882no—
513C23—1882no—
514C23—1882no—
515C23C22×C22⋊D4712yes—
516C23C22×C22⋊D4711yescharacteristic
517Q8—24no—
518Q8—24no—
519Q8—48no—
520Q8—44no—
521Q8—942no—
522Q8—942no—
523Q8—1882no—
524D4—942no—
525D4—942no—
526D4—1888no—
527D4—1882no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—28no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—942no—
542C2×C4—942no—
543C2×C4—942no—
544C2×C4—1882no—
545C2×C4—1882no—
546C2×C4—1882no—
547C2×C4—1888no—
548C2×C4#1309405514yes—
549C2×C4C22×C22⋊D4712yes—
550C2×C4C22×C22⋊D4712yes—
551C2×C4C22×C22⋊D4712yes—
552C2×C4C22×C22⋊D4712yes—
553C2×C4C22×C22⋊D4712yes—
554C2×C4C22×C22⋊D4711yescharacteristic
555C2×C4#1309407012yes—
556C2×C4#1309407112yes—
557C2×C4C24×D4711yescharacteristic
558C8—944no—
559C8—944no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—44no—
564C22—3764no—
565C22—1882no—
566C22—1882no—
567C22—1882no—
568C22#1568970112yes—
569C22C2×#1309405512yes—
570C22#1568992112yes—
571C22C23×C22⋊D4711yescharacteristic
572C4—22no—
573C4—22no—
574C4—22no—
575C4—22no—
576C4—44no—
577C4—44no—
578C4—1882no—
579C4—1882no—
580C4—1882no—
581C4#1568970112yes—
582C4C2×#1309405514yes—
583C4C2×#1309407011yescharacteristic
584C4C2×#1309407111yescharacteristic
585C2—42no—
586C2—3762no—
587C2#2025475514yesdirect product factor
588C2C2×#1568970111yescharacteristic
589C2C22×#1309405511yescharacteristic
590C2C2×#1568992111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#533C2
socleC23×C47C22×D4
prime radicalC2×#13093998C22
hypercenterC2×#533D47
hypocenterC47C2×#3252
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
2. #20254755C2direct (N × H)
3. C2×#15689430C2semidirect
4. C2×#15689415C2semidirect
5. C2×#15689410C2semidirect
6. C2×#15689379C2semidirect
8. C2×#15689345C2semidirect
9. C2×#15689330C2semidirect
10. C2×#15689325C2semidirect
11. C2×#15689147C2semidirect
12. C2×C47×#533C2semidirect
28. #15689430C22semidirect
29. #15689415C22semidirect
30. #15689410C22semidirect
31. #15689379C22semidirect
33. #15689345C22semidirect
34. #15689330C22semidirect
35. #15689325C22semidirect
38. #15689147C22semidirect
47. C47×#533C22semidirect
290. C2×#533D47semidirect
303. C2×C47#3252semidirect
312. #533C2×D47semidirect
352. C47C2×#3252semidirect
587. C2#20254755direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896572
#202547552
C2×#156894302
C2×#156894152
C2×#156894102
C2×#156893792
C2×#156893712
C2×#156893452
C2×#156893302
C2×#156893252
C2×#156891472
C2×C47×#5332
C2×#325247