Finite Group Databasebeta

← C2×#20248309 #26388049

Subgroups — C2×#20248309 #26388049

Order 12032.

3614 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094002C212yesmaximal, complemented
2C22×C4×Dic94C211yesmaximal, characteristic, complemented
3C22×#13093986C211yesmaximal, characteristic
4#20248309C2116yesmaximal, direct product factor
5C2×#15689047C212yesmaximal, complemented
6C2×#15689045C212yesmaximal, complemented
7C2×#15689025C211yesmaximal, characteristic
8C2×#15689024C211yesmaximal, characteristic
9C2×#15689019C211yesmaximal, characteristic, complemented
10C2×#15688992C211yesmaximal, characteristic, complemented
11C2×#15688982C212yesmaximal, complemented
12C2×C47×#389C211yesmaximal, characteristic, complemented
13C23×Dic94C2211yescharacteristic
14C23×D188C2211yescharacteristic
15C23×C4×D47C2212yes—
16C2×#13094002—216no—
17C2×#13094002C22116yescomplemented
18C2×C4×Dic94—28no—
19C2×C4×Dic94C2218yescomplemented
20C2×#13093986—28no—
21C2×#13093986C2218yes—
22C22×C4⋊Dic47C2211yescharacteristic
23C22×C4⋊Dic47C2211yescharacteristic
24C22×Dic47⋊C4C2211yescharacteristic
25C22×Dic47⋊C4C2212yes—
26C22×Dic47⋊C4C2211yescharacteristic
27C22×Dic47⋊C4C2212yes—
28C22×Dic47⋊C4C2212yes—
29C22×(C2×D47)⋊C4C2212yes—
30C22×(C2×D47)⋊C4C2212yes—
31C22×(C2×D47)⋊C4C2212yes—
32C22×(C2×D47)⋊C4C2212yes—
33C22×(C2×D47)⋊C4C2212yes—
34C22×C4×Dic47C2211yescharacteristic
35C22×C4×Dic47C2211yescharacteristic
36#15689047C22116yescomplemented
37#15689045C22116yescomplemented
38#15689025C2218yes—
39#15689024C2218yes—
40#15689019C2218yescomplemented
41#15688992C2218yescomplemented
42#15688982C22116yescomplemented
43C2×C47⋊#95C2212yes—
44C22×C4⋊C4×C47C2211yescharacteristic
45C22×C4⋊C4×C47C2211yescharacteristic
46C22×C4⋊C4×C47C2212yes—
47C22×C42×C47C2211yescharacteristic
48C47×#389C2218yescomplemented
49C2×#95×C47C2212yes—
50C24×D47C2312yes—
51C22×Dic94—24no—
52C22×Dic94—24no—
53C22×Dic94D418yescomplemented
54C22×Dic94C2314yes—
55C22×D188—24no—
56C22×D188—24no—
57C22×D188Q818yescomplemented
58C22×D188C2314yes—
59C23×Dic47C2311yescharacteristic
60C23×Dic47C2312yes—
61C23×Dic47C2311yescharacteristic
62C23×Dic47C2311yescharacteristic
63C23×Dic47C2311yescharacteristic
64C22×C4×D47—28no—
65C22×C4×D47—28no—
66C22×C4×D47—28no—
67C22×C4×D47C2318yes—
68#13094002—264no—
69C4×Dic94—232no—
70#13093986—232no—
71C2×C4⋊Dic47—24no—
72C2×C4⋊Dic47—24no—
73C2×C4⋊Dic47C2314yes—
74C2×C4⋊Dic47C2314yes—
75C2×Dic47⋊C4—24no—
76C2×Dic47⋊C4—24no—
77C2×Dic47⋊C4—28no—
78C2×Dic47⋊C4—28no—
79C2×Dic47⋊C4—44no—
80C2×Dic47⋊C4C2314yes—
81C2×Dic47⋊C4C2318yes—
82C2×Dic47⋊C4C2318yes—
83C2×Dic47⋊C4C2314yes—
84C2×Dic47⋊C4C2318yes—
85C2×(C2×D47)⋊C4—28no—
86C2×(C2×D47)⋊C4—28no—
87C2×(C2×D47)⋊C4—28no—
88C2×(C2×D47)⋊C4—44no—
89C2×(C2×D47)⋊C4—44no—
90C2×(C2×D47)⋊C4C2318yes—
91C2×(C2×D47)⋊C4C2318yes—
92C2×(C2×D47)⋊C4C2318yes—
93C2×(C2×D47)⋊C4C2318yes—
94C2×(C2×D47)⋊C4C2318yes—
95C2×C4×Dic47—24no—
96C2×C4×Dic47D418yes—
97C2×C4×Dic47C2314yes—
98C2×C4×Dic47C2314yes—
99C47⋊#95C2318yes—
100C23×C4×C47C2312yes—
101C23×C4×C47C2311yescharacteristic
102C23×C4×C47C2312yes—
103C23×C4×C47C2311yescharacteristic
104C23×C4×C47C2311yescharacteristic
105C2×C4⋊C4×C47—24no—
106C2×C4⋊C4×C47—28no—
107C2×C4⋊C4×C47—24no—
108C2×C4⋊C4×C47C2314yes—
109C2×C4⋊C4×C47C2314yes—
110C2×C4⋊C4×C47C2318yes—
111C2×C42×C47—24no—
112C2×C42×C47C2314yes—
113#95×C47C2318yes—
114C24×C47C2411yescharacteristic
115C23×D47—24no—
116C23×D47—24no—
117C23×D47—24no—
118C23×D47—24no—
119C23×D47—24no—
120C23×D47—24no—
121C23×D47C2×Q818yes—
122C23×D47C2414yes—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—24no—
127C22×C4×C47—24no—
128C22×C4×C47—24no—
129C22×C4×C47—24no—
130C22×C4×C47—24no—
131C22×C4×C47—22no—
132C22×C4×C47—24no—
133C22×C4×C47—22no—
134C22×C4×C47C2×D414yes—
135C22×C4×C47C2×D414yes—
136C22×C4×C47C2×Q814yes—
137C22×C4×C47#4114yes—
138C22×C4×C47C2412yes—
139C22×C4×C47C2414yes—
140C22×C4×C47C2414yes—
141C22×C4×C47C2412yes—
142C22×C4×C47C2412yes—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—24no—
149C22×Dic47—24no—
150C22×Dic47—22no—
151C22×Dic47—24no—
152C22×Dic47—22no—
153C22×Dic47C2×D414yes—
154C22×Dic47C2×D414yes—
155C22×Dic47C2×D414yes—
156C22×Dic47C2×D414yes—
157C22×Dic47#4114yes—
158C22×Dic47C2412yes—
159C22×Dic47C2414yes—
160C22×Dic47C2412yes—
161C22×Dic47C2412yes—
162C22×Dic47C2412yes—
163C2×D188—232no—
164C2×D188—28no—
165C2×D188—28no—
166C2×D188C2×Q8116yescomplemented
167C2×C4×D47—216no—
168C2×C4×D47—216no—
169C2×C4×D47—216no—
170C2×C4×D47—432no—
171C2×Dic94—232no—
172C2×Dic94—28no—
173C2×Dic94—28no—
174C2×Dic94C2×D4116yescomplemented
175C4⋊C4×C47—28no—
176C4⋊C4×C47—216no—
177C4⋊C4×C47—28no—
178C42×C47—28no—
179(C2×D47)⋊C4—216no—
180(C2×D47)⋊C4—216no—
181(C2×D47)⋊C4—216no—
182(C2×D47)⋊C4—48no—
183(C2×D47)⋊C4—48no—
184C4⋊Dic47—28no—
185C4⋊Dic47—28no—
186Dic47⋊C4—28no—
187Dic47⋊C4—28no—
188Dic47⋊C4—216no—
189Dic47⋊C4—216no—
190Dic47⋊C4—48no—
191C4×Dic47—28no—
192C4×Dic47C2×D4116yes—
193C23×C47C22×D412yes—
194C23×C47C22×D412yes—
195C23×C47C22×Q812yes—
196C23×C47C2×#4112yes—
197C23×C47C2×#4112yes—
198C23×C47H(5,𝔽2)12yes—
199C23×C472−1+412yes—
200C23×C47C2511yescharacteristic
201C22×D47—216no—
202C22×D47—216no—
203C22×D47—216no—
204C22×D47—24no—
205C22×D47—24no—
206C22×D47—24no—
207C22×D47—24no—
208C22×D47—24no—
209C22×D47—24no—
210C22×D47—48no—
211C22×D47—48no—
212C22×D47—48no—
213C22×D47—48no—
214C22×D47C22×Q818yes—
215C2×C4×C47—28no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—24no—
220C2×C4×C47—24no—
221C2×C4×C47—24no—
222C2×C4×C47—24no—
223C2×C4×C47—24no—
224C2×C4×C47—24no—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
227C2×C4×C47—48no—
228C2×C4×C47—44no—
229C2×C4×C47—48no—
230C2×C4×C47C22⋊Q8116yescomplemented
231C2×C4×C47C22×D414yes—
232C2×C4×C47C22×D414yes—
233C2×C4×C47C22×Q814yes—
234C2×C4×C47C2×#4114yes—
235C2×Dic47—28no—
236C2×Dic47—28no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—22no—
240C2×Dic47—22no—
241C2×Dic47—22no—
242C2×Dic47—24no—
243C2×Dic47—24no—
244C2×Dic47—24no—
245C2×Dic47—22no—
246C2×Dic47—22no—
247C2×Dic47—44no—
248C2×Dic47—48no—
249C2×Dic47#121116yes—
250C2×Dic47C22×D414yes—
251C2×Dic47C22×D414yes—
252C2×Dic47C22×D414yes—
253C2×Dic47C22×D414yes—
254C2×Dic47C2×#4114yes—
255D188—232no—
256C4×D47—432no—
257Dic94—232no—
258C2×#2538—471nomaximal, Sylow
259C22×C47C2×#12114yes—
260C22×C47C2×C22⋊Q814yes—
261C22×C47#53714yes—
262C22×C47D4×Q814yes—
263C22×C47#55014yes—
264C22×C47#55414yes—
265C22×C47#55614yes—
266C22×C47C23×D411yescharacteristic
267C22×C47C23×D411yescharacteristic
268C22×C47C23×Q811yescharacteristic
269C22×C47C22×#4111yescharacteristic
270C22×C47C22×#4111yescharacteristic
271C22×C47C2×H(5,𝔽2)11yescharacteristic
272C22×C47C2×2−1+411yescharacteristic
273C2×D47—28no—
274C2×D47—28no—
275C2×D47—28no—
276C2×D47—44no—
277C2×D47—44no—
278C2×D47—44no—
279C2×D47—44no—
280C2×D47—432no—
281C4×C47—24no—
282C4×C47—44no—
283C4×C47—42no—
284C4×C47—44no—
285C4×C47C2×C22⋊Q818yescomplemented
286Dic47—24no—
287Dic47—24no—
288Dic47—42no—
289Dic47—44no—
290Dic47C2×#12118yes—
291C22×C22⋊Q8—472no—
292C22×C4×Q8—471no—
293C22×C4×D4—471no—
294#2538—4716no—
295C2×#396—472no—
296C2×#396—472no—
297C2×#390—471no—
298C2×#390—471no—
299C2×#389—471no—
300C2×#389D4711yescharacteristic, complemented
301C2×#384—471no—
302C2×#380—472no—
303C2×C47#253818yescomplemented
304C2×C47C22×#12111yescharacteristic
305C2×C47C22×C22⋊Q811yescharacteristic
306C2×C47C2×#53711yescharacteristic
307C2×C47C2×D4×Q811yescharacteristic
308C2×C47C2×#55011yescharacteristic
309C2×C47C2×#55411yescharacteristic
310C2×C47C2×#55611yescharacteristic
311D47—48no—
312C23×Q8—471no—
313C23×D4—471no—
314C24×C4—472no—
315C2×C22⋊Q8—4716no—
316C2×C22⋊Q8—9416no—
317C2×C4×Q8—478no—
318C2×C4×Q8—948no—
319C2×C4×D4—478no—
320C2×C4×D4—948no—
321C22×C4⋊C4—471no—
322C22×C4⋊C4—471no—
323C22×C4⋊C4—472no—
324C22×C4⋊C4—471no—
325C22×C4⋊C4—472no—
326C22×C4⋊C4—471no—
327C22×C4⋊C4—472no—
328C22×C4⋊C4C2×D4712yes—
329C22×C4⋊C4C2×D4711yescharacteristic
330C22×C4⋊C4C2×D4711yescharacteristic
331C22×C22⋊C4—472no—
332C22×C22⋊C4—472no—
333C22×C22⋊C4—472no—
334C22×C22⋊C4—472no—
335C22×C22⋊C4—472no—
336C22×C42—471no—
337C22×C42—471no—
338C22×C42C2×D4711yescharacteristic
339#396—4716no—
340#396—4716no—
341#390—478no—
342#390—478no—
343#389—478no—
344#389C2×D4718yescomplemented
345#384—478no—
346#380—4716no—
347C2×#95—472no—
348C2×#95C2×D4712yes—
349C47C2×#253811yescharacteristic, complemented, Sylow
350C25—472no—
351C22×Q8—478no—
352C22×Q8—474no—
353C22×Q8—944no—
354C22×Q8—944no—
355C22×D4—478no—
356C22×D4—474no—
357C22×D4—944no—
358C22×D4—944no—
359C23×C4—471no—
360C23×C4—471no—
361C23×C4—471no—
362C23×C4—472no—
363C23×C4—471no—
364C23×C4—478no—
365C23×C4—948no—
366C23×C4—948no—
367C23×C4—948no—
368C23×C4C22×D4712yes—
369C23×C4C22×D4711yescharacteristic
370C23×C4C22×D4712yes—
371C23×C4C22×D4711yescharacteristic
372C23×C4C22×D4711yescharacteristic
373C22⋊Q8—9464no—
374C4×Q8—9432no—
375C4×D4—9432no—
376C2×C4⋊C4—24no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4—28no—
379C2×C4⋊C4—474no—
380C2×C4⋊C4—474no—
381C2×C4⋊C4—474no—
382C2×C4⋊C4—478no—
383C2×C4⋊C4—478no—
384C2×C4⋊C4—474no—
385C2×C4⋊C4—478no—
386C2×C4⋊C4—944no—
387C2×C4⋊C4—944no—
388C2×C4⋊C4—944no—
389C2×C4⋊C4—948no—
390C2×C4⋊C4—948no—
391C2×C4⋊C4—944no—
392C2×C4⋊C4—1884no—
393C2×C4⋊C4C22×D4714yes—
394C2×C4⋊C4C22×D4718yes—
395C2×C4⋊C4C22×D4714yes—
396C2×C22⋊C4—478no—
397C2×C22⋊C4—478no—
398C2×C22⋊C4—478no—
399C2×C22⋊C4—478no—
400C2×C22⋊C4—478no—
401C2×C22⋊C4—948no—
402C2×C22⋊C4—948no—
403C2×C22⋊C4—948no—
404C2×C22⋊C4—1884no—
405C2×C22⋊C4—1884no—
406C2×C42—24no—
407C2×C42—478no—
408C2×C42—474no—
409C2×C42—474no—
410C2×C42—944no—
411C2×C42C22×D4714yes—
412#95—478no—
413#95C22×D4718yes—
414C24—478no—
415C24—474no—
416C24—944no—
417C24—944no—
418C24—944no—
419C24—944no—
420C24—944no—
421C24—944no—
422C24C23×D4711yescharacteristic
423C2×Q8—4716no—
424C2×Q8—948no—
425C2×Q8—9432no—
426C2×Q8—948no—
427C2×D4—4716no—
428C2×D4—9432no—
429C2×D4—948no—
430C2×D4—948no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—474no—
443C22×C4—474no—
444C22×C4—474no—
445C22×C4—472no—
446C22×C4—474no—
447C22×C4—474no—
448C22×C4—472no—
449C22×C4—472no—
450C22×C4—472no—
451C22×C4—474no—
452C22×C4—9416no—
453C22×C4—9416no—
454C22×C4—9416no—
455C22×C4—942no—
456C22×C4—942no—
457C22×C4—942no—
458C22×C4—942no—
459C22×C4—942no—
460C22×C4—944no—
461C22×C4—944no—
462C22×C4—942no—
463C22×C4—944no—
464C22×C4—942no—
465C22×C4—18832no—
466C22×C4#1062535014yes—
467C22×C4D4×D4714yes—
468C22×C4D4×D4714yes—
469C22×C4Q8×D4714yes—
470C22×C4C23×D4714yes—
471C22×C4C23×D4712yes—
472C22×C4C23×D4714yes—
473C22×C4C23×D4712yes—
474C22×C4C23×D4712yes—
475C4⋊C4—28no—
476C4⋊C4—216no—
477C4⋊C4—28no—
478C4⋊C4—948no—
479C4⋊C4—948no—
480C4⋊C4—9416no—
481C4⋊C4—9416no—
482C4⋊C4—948no—
483C4⋊C4—948no—
484C4⋊C4—1888no—
485C22⋊C4—9416no—
486C22⋊C4—9416no—
487C22⋊C4—9416no—
488C22⋊C4—1888no—
489C22⋊C4—1888no—
490C42—28no—
491C42—4716no—
492C42—948no—
493C23—478no—
494C23—9416no—
495C23—9416no—
496C23—9416no—
497C23—944no—
498C23—944no—
499C23—944no—
500C23—944no—
501C23—944no—
502C23—944no—
503C23—1888no—
504C23—1888no—
505C23—1888no—
506C23—1888no—
507C23C2×D4×D4712yes—
508C23C2×D4×D4712yes—
509C23C2×Q8×D4712yes—
510C23#41×D4712yes—
511C23C2×#1062535012yes—
512C23#1309407012yes—
513C23#1309407212yes—
514C23C24×D4711yescharacteristic
515Q8—9432no—
516D4—9432no—
517C2×C4—28no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—44no—
532C2×C4—4716no—
533C2×C4—474no—
534C2×C4—474no—
535C2×C4—474no—
536C2×C4—474no—
537C2×C4—474no—
538C2×C4—948no—
539C2×C4—942no—
540C2×C4—942no—
541C2×C4—942no—
542C2×C4—942no—
543C2×C4—948no—
544C2×C4—942no—
545C2×C4—944no—
546C2×C4—944no—
547C2×C4—942no—
548C2×C4—944no—
549C2×C4—942no—
550C2×C4—18832no—
551C2×C4—1888no—
552C2×C4—1884no—
553C2×C4#13094002116yescomplemented
554C2×C4C2×D4×D4714yes—
555C2×C4C2×D4×D4714yes—
556C2×C4C2×Q8×D4714yes—
557C2×C4C2×#1062535014yes—
558C22—948no—
559C22—948no—
560C22—948no—
561C22—1884no—
562C22—1884no—
563C22—1884no—
564C22—1884no—
565C22—18832no—
566C22#121×D4714yes—
567C22C2×#1309400214yes—
568C22#1568970314yes—
569C22#1568978814yes—
570C22#1568979414yes—
571C22#1568982914yes—
572C22#1568984514yes—
573C22C22×D4×D4711yescharacteristic
574C22C22×D4×D4711yescharacteristic
575C22C22×Q8×D4711yescharacteristic
576C22C2×#41×D4711yescharacteristic
577C22C22×#1062535011yescharacteristic
578C22C2×#1309407011yescharacteristic
579C22C2×#1309407211yescharacteristic
580C4—24no—
581C4—44no—
582C4—44no—
583C4—42no—
584C4—478no—
585C4—944no—
586C4—944no—
587C4—1884no—
588C4—1882no—
589C4C2×#1309400218yescomplemented
590C2—1888no—
591C2#2024830918yesdirect product factor
592C2C2×#121×D4711yescharacteristic
593C2C22×#1309400211yescharacteristic
594C2C2×#1568970311yescharacteristic
595C2C2×#1568978811yescharacteristic
596C2C2×#1568979411yescharacteristic
597C2C2×#1568982911yescharacteristic
598C2C2×#1568984511yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#389C2
socleC24×C47C24
prime radicalC23×D188C22
hypercenterC2×#389D47
hypocenterC47C2×#2538
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×#13094002C2semidirect
2. C22×C4×Dic94C2semidirect
4. #20248309C2direct (N × H)
5. C2×#15689047C2semidirect
6. C2×#15689045C2semidirect
9. C2×#15689019C2semidirect
10. C2×#15688992C2semidirect
11. C2×#15688982C2semidirect
12. C2×C47×#389C2semidirect
17. C2×#13094002C22semidirect
19. C2×C4×Dic94C22semidirect
36. #15689047C22semidirect
37. #15689045C22semidirect
40. #15689019C22semidirect
41. #15688992C22semidirect
42. #15688982C22semidirect
48. C47×#389C22semidirect
53. C22×Dic94D4semidirect
57. C22×D188Q8semidirect
166. C2×D188C2×Q8semidirect
174. C2×Dic94C2×D4semidirect
230. C2×C4×C47C22⋊Q8semidirect
285. C4×C47C2×C22⋊Q8semidirect
300. C2×#389D47semidirect
303. C2×C47#2538semidirect
344. #389C2×D47semidirect
349. C47C2×#2538semidirect
553. C2×C4#13094002semidirect
589. C4C2×#13094002semidirect
591. C2#20248309direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940022
C22×C4×Dic942
C22×#130939862
#202483092
C2×#156890472
C2×#156890452
C2×#156890252
C2×#156890242
C2×#156890192
C2×#156889922
C2×#156889822
C2×C47×#3892
C2×#253847