Finite Group Databasebeta

← C2×#20251738 #26422212

Subgroups — C2×#20251738 #26422212

Order 12032.

3790 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
1C22×#13094023C211yesmaximal, characteristic, complemented
2C22×#13094001C212yesmaximal
3C23×C4×Dic47C211yesmaximal, characteristic
4#20251738C2116yesmaximal, direct product factor
5C2×#15689027C212yesmaximal, complemented
6C2×#15689022C211yesmaximal, characteristic, complemented
7C2×#15689020C212yesmaximal, complemented
8C2×#15689019C211yesmaximal, characteristic, complemented
9C2×#15688992C211yesmaximal, characteristic, complemented
10C2×#15688983C211yesmaximal, characteristic
11C2×#15688981C212yesmaximal
12C2×C47×#380C211yesmaximal, characteristic
13C23×Dic94C2212yes—
14C24×Dic47C2211yescharacteristic
15C24×Dic47C2211yescharacteristic
16C2×#13094023—28no—
17C2×#13094023C2218yescomplemented
18C2×#13094001—216no—
19C2×#13094001C22116yes—
20C22×C4⋊Dic47C2211yescharacteristic
21C22×C4⋊Dic47C2211yescharacteristic
22C22×C4⋊Dic47C2212yes—
23C22×Dic47⋊C4C2212yes—
24C22×Dic47⋊C4C2212yes—
25C22×Dic47⋊C4C2212yes—
26C22×Dic47⋊C4C2212yes—
27C22×C22⋊Dic47C2212yes—
28C22×C4×Dic47—28no—
29C22×C4×Dic47C2211yescharacteristic
30C22×C4×Dic47C2211yescharacteristic
31C22×C4×Dic47C2218yes—
32C22×C4×Dic47C2211yescharacteristic
33C22×C4×Dic47C2211yescharacteristic
34#15689027C22116yescomplemented
35#15689022C2218yescomplemented
36#15689020C22116yescomplemented
37#15689019C2218yescomplemented
38#15688992C2218yescomplemented
39#15688983C2218yes—
40#15688981C22116yes—
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2212yes—
43C2×C47⋊#95C2212yes—
44C24×C4×C47C2211yescharacteristic
45C22×C4⋊C4×C47C2211yescharacteristic
46C22×C4⋊C4×C47C2211yescharacteristic
47C22×C22⋊C4×C47C2212yes—
48C47×#380C2218yes—
49C2×#95×C47C2212yes—
50C22×Dic94—28no—
51C22×Dic94—28no—
52C22×Dic94—28no—
53C22×Dic94C2318yes—
54C23×Dic47—24no—
55C23×Dic47—24no—
56C23×Dic47—24no—
57C23×Dic47—24no—
58C23×Dic47D418yes—
59C23×Dic47Q818yescomplemented
60C23×Dic47C2312yes—
61C23×Dic47C2312yes—
62C23×Dic47C2311yescharacteristic
63C23×Dic47C2314yes—
64C23×Dic47C2311yescharacteristic
65C23×Dic47C2314yes—
66C23×Dic47C2311yescharacteristic
67C23×Dic47C2311yescharacteristic
68#13094023—232no—
69#13094001—264no—
70C2×C4⋊Dic47—24no—
71C2×C4⋊Dic47—24no—
72C2×C4⋊Dic47—28no—
73C2×C4⋊Dic47C2314yes—
74C2×C4⋊Dic47C2314yes—
75C2×C4⋊Dic47C2318yes—
76C2×Dic47⋊C4—28no—
77C2×Dic47⋊C4—28no—
78C2×Dic47⋊C4—28no—
79C2×Dic47⋊C4—28no—
80C2×Dic47⋊C4C2318yes—
81C2×Dic47⋊C4C2318yes—
82C2×Dic47⋊C4C2318yes—
83C2×Dic47⋊C4C2318yes—
84C2×C22⋊Dic47—28no—
85C2×C22⋊Dic47C2318yes—
86C2×C4×Dic47—24no—
87C2×C4×Dic47—24no—
88C2×C4×Dic47—24no—
89C2×C4×Dic47—232no—
90C2×C4×Dic47D418yes—
91C2×C4×Dic47C2314yes—
92C2×C4×Dic47C2314yes—
93C2×C4×Dic47C2314yes—
94C2×C4×Dic47C2314yes—
95C47⋊#95C2318yes—
96C47⋊#95C2318yes—
97C47⋊#95C2318yes—
98C25×C47C2311yescharacteristic
99C23×C4×C47—24no—
100C23×C4×C47—24no—
101C23×C4×C47Q818yescomplemented
102C23×C4×C47C2312yes—
103C23×C4×C47C2312yes—
104C23×C4×C47C2311yescharacteristic
105C23×C4×C47C2314yes—
106C23×C4×C47C2311yescharacteristic
107C2×C4⋊C4×C47—24no—
108C2×C4⋊C4×C47—24no—
109C2×C4⋊C4×C47C2314yes—
110C2×C4⋊C4×C47C2314yes—
111C2×C22⋊C4×C47—28no—
112C2×C22⋊C4×C47C2318yes—
113#95×C47C2318yes—
114C24×C47—22no—
115C24×C47—22no—
116C24×C47—22no—
117C24×C47—22no—
118C24×C47C2×D414yes—
119C24×C47C2×Q814yes—
120C24×C47C2×Q814yes—
121C24×C47C2412yes—
122C24×C47C2411yescharacteristic
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—232no—
126C22×C4×C47—28no—
127C22×C4×C47—28no—
128C22×C4×C47—24no—
129C22×C4×C47—24no—
130C22×C4×C47—24no—
131C22×C4×C47—24no—
132C22×C4×C47—24no—
133C22×C4×C47—24no—
134C22×C4×C47C2×D414yes—
135C22×C4×C47C2×Q814yes—
136C22×C4×C47C2×Q814yes—
137C22×C4×C47C2×Q8116yescomplemented
138C22×C4×C47#4114yes—
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—232no—
148C22×Dic47—232no—
149C22×Dic47—28no—
150C22×Dic47—28no—
151C22×Dic47—28no—
152C22×Dic47—28no—
153C22×Dic47—24no—
154C22×Dic47—24no—
155C22×Dic47—24no—
156C22×Dic47—24no—
157C22×Dic47—24no—
158C22×Dic47—24no—
159C22×Dic47C2×D414yes—
160C22×Dic47C2×D4116yes—
161C22×Dic47C2×D414yes—
162C22×Dic47C2×D414yes—
163C22×Dic47C2×D414yes—
164C22×Dic47C2×Q814yes—
165C22×Dic47C2×Q814yes—
166C22×Dic47C2×Q8116yescomplemented
167C22×Dic47#4114yes—
168C22×Dic47#4114yes—
169C22×Dic47C2414yes—
170C22×Dic47C2414yes—
171C22×Dic47C2412yes—
172C22×Dic47C2412yes—
173C22×Dic47C2412yes—
174C22×Dic47C2412yes—
175C2×Dic94—216no—
176C2×Dic94—216no—
177C2×Dic94—216no—
178C2×Dic94—432no—
179C4⋊C4×C47—28no—
180C4⋊C4×C47—28no—
181C22⋊C4×C47—216no—
182C22⋊Dic47—216no—
183C4⋊Dic47—28no—
184C4⋊Dic47—28no—
185C4⋊Dic47—216no—
186Dic47⋊C4—216no—
187Dic47⋊C4—216no—
188Dic47⋊C4—216no—
189Dic47⋊C4—216no—
190C4×Dic47—28no—
191C4×Dic47—28no—
192C4×Dic47—28no—
193C4×Dic47C2×D4116yes—
194C23×C47—28no—
195C23×C47—28no—
196C23×C47—28no—
197C23×C47—28no—
198C23×C47—28no—
199C23×C47—28no—
200C23×C47—22no—
201C23×C47—22no—
202C23×C47—22no—
203C23×C47—22no—
204C23×C47C4⋊Q8116yescomplemented
205C23×C47C22×D414yes—
206C23×C47C22×D412yes—
207C23×C47C22×D412yes—
208C23×C47C22×Q814yes—
209C23×C47C22×Q812yes—
210C23×C47C22×Q814yes—
211C23×C47C22×Q812yes—
212C23×C47C2×#4112yes—
213C23×C47C2×#4112yes—
214C23×C47C2×#4112yes—
215C23×C47C2511yescharacteristic
216C2×C4×C47—28no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—232no—
220C2×C4×C47—24no—
221C2×C4×C47—24no—
222C2×C4×C47—24no—
223C2×C4×C47—24no—
224C2×C4×C47—24no—
225C2×C4×C47—24no—
226C2×C4×C47—48no—
227C2×C4×C47—48no—
228C2×C4×C47C22⋊Q8116yescomplemented
229C2×C4×C47C22×D414yes—
230C2×C4×C47C22×Q814yes—
231C2×C4×C47C22×Q814yes—
232C2×C4×C47C2×#4114yes—
233C2×Dic47—28no—
234C2×Dic47—28no—
235C2×Dic47—22no—
236C2×Dic47—22no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—232no—
240C2×Dic47—232no—
241C2×Dic47—24no—
242C2×Dic47—24no—
243C2×Dic47—24no—
244C2×Dic47—24no—
245C2×Dic47—24no—
246C2×Dic47—24no—
247C2×Dic47—48no—
248C2×Dic47—48no—
249C2×Dic47#121116yes—
250C2×Dic47C22⋊Q8116yescomplemented
251C2×Dic47C22×D414yes—
252C2×Dic47C22×D414yes—
253C2×Dic47C22×D414yes—
254C2×Dic47C22×D414yes—
255C2×Dic47C22×Q814yes—
256C2×Dic47C22×Q814yes—
257C2×Dic47C2×#4114yes—
258C2×Dic47C2×#4114yes—
259Dic94—432no—
260C2×#2857—471nomaximal, Sylow
261C22×C47—232no—
262C22×C47—24no—
263C22×C47—24no—
264C22×C47—24no—
265C22×C47—24no—
266C22×C47—24no—
267C22×C47—24no—
268C22×C47C2×#12114yes—
269C22×C47C2×C22⋊Q814yes—
270C22×C47C2×C22⋊Q814yes—
271C22×C47C2×C4⋊Q814yes—
272C22×C47C2×C4⋊Q818yescomplemented
273C22×C47#53214yes—
274C22×C47#53314yes—
275C22×C47#53314yes—
276C22×C47C23×D411yescharacteristic
277C22×C47C23×D411yescharacteristic
278C22×C47C23×Q811yescharacteristic
279C22×C47C23×Q811yescharacteristic
280C22×C47C22×#4111yescharacteristic
281C22×C47C22×#4111yescharacteristic
282C22×C47C22×#4111yescharacteristic
283C4×C47—24no—
284C4×C47—44no—
285C4×C47—44no—
286C4×C47C2×C22⋊Q818yescomplemented
287Dic47—24no—
288Dic47—24no—
289Dic47—44no—
290Dic47—44no—
291Dic47C2×#12118yes—
292Dic47C2×C22⋊Q818yescomplemented
293C22×C4⋊Q8—471no—
294C22×C22⋊Q8—472no—
295C23×C42—471no—
296#2857—4716no—
297C2×#391—472no—
298C2×#389—472no—
299C2×#389—471no—
300C2×#389—471no—
301C2×#384—471no—
302C2×#380—471no—
303C2×#380—472no—
304C2×#380D4711yescharacteristic
305C2×C47—28no—
306C2×C47#285718yescomplemented
307C2×C47C22×#12111yescharacteristic
308C2×C47C22×C22⋊Q811yescharacteristic
309C2×C47C22×C22⋊Q811yescharacteristic
310C2×C47C22×C4⋊Q811yescharacteristic
311C2×C47C2×#53211yescharacteristic
312C2×C47C2×#53311yescharacteristic
313C2×C47C2×#53311yescharacteristic
314C23×Q8—472no—
315C24×C4—471no—
316C24×C4—471no—
317C24×C4C2×D4711yescharacteristic
318C2×C4⋊Q8—478no—
319C2×C4⋊Q8—948no—
320C2×C22⋊Q8—4716no—
321C2×C22⋊Q8—9416no—
322C22×C4⋊C4—472no—
323C22×C4⋊C4—472no—
324C22×C4⋊C4—472no—
325C22×C4⋊C4—472no—
326C22×C4⋊C4—472no—
327C22×C4⋊C4—471no—
328C22×C4⋊C4—471no—
329C22×C4⋊C4C2×D4711yescharacteristic
330C22×C4⋊C4C2×D4711yescharacteristic
331C22×C22⋊C4—472no—
332C22×C22⋊C4C2×D4712yes—
333C22×C42—471no—
334C22×C42—478no—
335C22×C42—471no—
336C22×C42—471no—
337C22×C42—471no—
338C22×C42—948no—
339#391—4716no—
340#389—478no—
341#389—478no—
342#389—4716no—
343#384—478no—
344#380—478no—
345#380—4716no—
346#380C2×D4718yes—
347C2×#95—472no—
348C2×#95—472no—
349C2×#95—472no—
350C2×#95C2×D4712yes—
351C47C2×#285711yescharacteristic, complemented, Sylow
352C25C22×D4711yescharacteristic
353C22×Q8—478no—
354C22×Q8—948no—
355C22×Q8—948no—
356C22×Q8—948no—
357C23×C4—24no—
358C23×C4—24no—
359C23×C4—471no—
360C23×C4—471no—
361C23×C4—478no—
362C23×C4—478no—
363C23×C4—474no—
364C23×C4—474no—
365C23×C4—471no—
366C23×C4—471no—
367C23×C4—472no—
368C23×C4—472no—
369C23×C4—944no—
370C23×C4—944no—
371C23×C4—944no—
372C23×C4—944no—
373C23×C4Dic9418yescomplemented
374C23×C4C22×D4712yes—
375C23×C4C22×D4712yes—
376C23×C4C22×D4711yescharacteristic
377C23×C4C22×D4714yes—
378C23×C4C22×D4711yescharacteristic
379C4⋊Q8—9432no—
380C22⋊Q8—9464no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4—478no—
384C2×C4⋊C4—478no—
385C2×C4⋊C4—478no—
386C2×C4⋊C4—478no—
387C2×C4⋊C4—474no—
388C2×C4⋊C4—478no—
389C2×C4⋊C4—474no—
390C2×C4⋊C4—948no—
391C2×C4⋊C4—948no—
392C2×C4⋊C4—948no—
393C2×C4⋊C4—948no—
394C2×C4⋊C4—948no—
395C2×C4⋊C4—944no—
396C2×C4⋊C4—944no—
397C2×C4⋊C4C22×D4714yes—
398C2×C4⋊C4C22×D4714yes—
399C2×C22⋊C4—28no—
400C2×C22⋊C4—478no—
401C2×C22⋊C4—948no—
402C2×C22⋊C4C22×D4718yes—
403C2×C42—478no—
404C2×C42—474no—
405C2×C42—474no—
406C2×C42—474no—
407C2×C42—474no—
408C2×C42—9432no—
409C2×C42—944no—
410C2×C42—944no—
411C2×C42—944no—
412#95—478no—
413#95—478no—
414#95—478no—
415#95C22×D4718yes—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24C2×Dic9414yes—
421C24D4×D4714yes—
422C24Q8×D4714yes—
423C24C23×D4712yes—
424C24C23×D4711yescharacteristic
425C2×Q8—9416no—
426C2×Q8—9416no—
427C2×Q8—9416no—
428C2×Q8—18832no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—232no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—474no—
441C22×C4—474no—
442C22×C4—474no—
443C22×C4—474no—
444C22×C4—474no—
445C22×C4—474no—
446C22×C4—472no—
447C22×C4—472no—
448C22×C4—474no—
449C22×C4—4716no—
450C22×C4—474no—
451C22×C4—4716no—
452C22×C4—472no—
453C22×C4—472no—
454C22×C4—474no—
455C22×C4—474no—
456C22×C4—948no—
457C22×C4—948no—
458C22×C4—9432no—
459C22×C4—9432no—
460C22×C4—948no—
461C22×C4—948no—
462C22×C4—942no—
463C22×C4—942no—
464C22×C4—942no—
465C22×C4—942no—
466C22×C4—944no—
467C22×C4—944no—
468C22×C4—944no—
469C22×C4—944no—
470C22×C4—944no—
471C22×C4—944no—
472C22×C4C2×Dic9414yes—
473C22×C4C2×Dic9414yes—
474C22×C4C2×Dic94116yescomplemented
475C22×C4D4×D4714yes—
476C22×C4#1062535214yes—
477C22×C4C23×D4714yes—
478C22×C4C23×D4714yes—
479C22×C4C23×D4712yes—
480C22×C4C23×D4712yes—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—9416no—
484C4⋊C4—9416no—
485C4⋊C4—9416no—
486C4⋊C4—9416no—
487C4⋊C4—9416no—
488C4⋊C4—948no—
489C4⋊C4—948no—
490C22⋊C4—216no—
491C22⋊C4—9416no—
492C42—4716no—
493C42—948no—
494C42—948no—
495C42—948no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23#13094023116yescomplemented
507C23C2×D4×D4714yes—
508C23C2×D4×D4712yes—
509C23C2×D4×D4712yes—
510C23C2×Q8×D4714yes—
511C23C2×Q8×D4712yes—
512C23C22×Dic9414yes—
513C23C22×Dic9412yes—
514C23#41×D4712yes—
515C23#41×D4712yes—
516C23C2×#1062535212yes—
517C23C24×D4711yescharacteristic
518Q8—18832no—
519C2×C4—28no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—232no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—4716no—
532C2×C4—4716no—
533C2×C4—474no—
534C2×C4—474no—
535C2×C4—474no—
536C2×C4—474no—
537C2×C4—474no—
538C2×C4—474no—
539C2×C4—474no—
540C2×C4—474no—
541C2×C4—9432no—
542C2×C4—9432no—
543C2×C4—948no—
544C2×C4—948no—
545C2×C4—942no—
546C2×C4—942no—
547C2×C4—942no—
548C2×C4—942no—
549C2×C4—944no—
550C2×C4—944no—
551C2×C4—944no—
552C2×C4—944no—
553C2×C4—944no—
554C2×C4—944no—
555C2×C4—1888no—
556C2×C4—1888no—
557C2×C4#13094001116yescomplemented
558C2×C4C2×D4×D4714yes—
559C2×C4C22×Dic9414yes—
560C2×C4C22×Dic9414yes—
561C2×C4C2×#1062535214yes—
562C22—232no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22#121×D4714yes—
570C22C22⋊Q8×D4714yes—
571C22C2×#1309400114yes—
572C22C2×#1309402314yes—
573C22C2×#1309402318yescomplemented
574C22#1568966814yes—
575C22#1568967514yes—
576C22#1568967614yes—
577C22C22×D4×D4711yescharacteristic
578C22C22×D4×D4711yescharacteristic
579C22C22×Q8×D4711yescharacteristic
580C22C23×Dic9411yescharacteristic
581C22C2×#41×D4711yescharacteristic
582C22C2×#41×D4711yescharacteristic
583C22C22×#1062535211yescharacteristic
584C4—24no—
585C4—44no—
586C4—44no—
587C4—478no—
588C4—478no—
589C4—944no—
590C4—944no—
591C4—1884no—
592C4—1884no—
593C4C2×#1309400118yescomplemented
594C2—28no—
595C2#2025173818yesdirect product factor
596C2C2×#121×D4711yescharacteristic
597C2C2×C22⋊Q8×D4711yescharacteristic
598C2C22×#1309400111yescharacteristic
599C2C22×#1309402311yescharacteristic
600C2C2×#1568966811yescharacteristic
601C2C2×#1568967511yescharacteristic
602C2C2×#1568967611yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#380C2
socleC24×C47C24
prime radicalC25×C47C23
hypercenterC2×#380D47
hypocenterC47C2×#2857
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. C22×#13094023C2semidirect
4. #20251738C2direct (N × H)
5. C2×#15689027C2semidirect
6. C2×#15689022C2semidirect
7. C2×#15689020C2semidirect
8. C2×#15689019C2semidirect
9. C2×#15688992C2semidirect
17. C2×#13094023C22semidirect
34. #15689027C22semidirect
35. #15689022C22semidirect
36. #15689020C22semidirect
37. #15689019C22semidirect
38. #15688992C22semidirect
59. C23×Dic47Q8semidirect
101. C23×C4×C47Q8semidirect
137. C22×C4×C47C2×Q8semidirect
166. C22×Dic47C2×Q8semidirect
204. C23×C47C4⋊Q8semidirect
228. C2×C4×C47C22⋊Q8semidirect
250. C2×Dic47C22⋊Q8semidirect
272. C22×C47C2×C4⋊Q8semidirect
286. C4×C47C2×C22⋊Q8semidirect
292. Dic47C2×C22⋊Q8semidirect
306. C2×C47#2857semidirect
351. C47C2×#2857semidirect
373. C23×C4Dic94semidirect
474. C22×C4C2×Dic94semidirect
506. C23#13094023semidirect
557. C2×C4#13094001semidirect
573. C22C2×#13094023semidirect
593. C4C2×#13094001semidirect
595. C2#20251738direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940232
C22×#130940012
C23×C4×Dic472
#202517382
C2×#156890272
C2×#156890222
C2×#156890202
C2×#156890192
C2×#156889922
C2×#156889832
C2×#156889812
C2×C47×#3802
C2×#285747