Finite Group Databasebeta

← C2×#20246391 #26375793

Subgroups — C2×#20246391 #26375793

Order 12032.

3614 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094003C211yesmaximal, characteristic
2C22×C4×Dic94C211yesmaximal, characteristic, complemented
3C23×C4⋊Dic47C211yesmaximal, characteristic
4#20246391C2116yesmaximal, direct product factor
5C2×#15689029C211yesmaximal, characteristic, complemented
6C2×#15689021C211yesmaximal, characteristic, complemented
7C2×#15689020C212yesmaximal, complemented
8C2×#15689010C212yesmaximal, complemented
9C2×#15688995C212yesmaximal
10C2×C4×C4⋊Dic47C211yesmaximal, characteristic, complemented
11C2×C22⋊C4×Dic47C212yesmaximal
12C2×C4×C22⋊C4×C47C211yesmaximal, characteristic
13C23×Dic94C2211yescharacteristic
14C24×Dic47C2212yes—
15C2×#13094003C4116yescomplemented
16C2×#13094003C2218yes—
17C2×C4×Dic94—28no—
18C2×C4×Dic94C2218yescomplemented
19C22×C4⋊Dic47—28no—
20C22×C4⋊Dic47C2212yes—
21C22×C4⋊Dic47C2211yescharacteristic
22C22×C4⋊Dic47C2218yes—
23C22×C4⋊Dic47C2211yescharacteristic
24C22×C4⋊Dic47C2211yescharacteristic
25C22×Dic47⋊C4C2212yes—
26C22×Dic47⋊C4C2212yes—
27C22×Dic47⋊C4C2212yes—
28C22×C22⋊Dic47C2212yes—
29C22×C4×Dic47C2212yes—
30C22×C4×Dic47C2212yes—
31#15689029C2218yescomplemented
32#15689021C2218yescomplemented
33#15689020C22116yescomplemented
34#15689010C22116yescomplemented
35#15688995C22116yes—
36C4×C4⋊Dic47C2218yescomplemented
37C22⋊C4×Dic47C22116yes—
38C2×C47⋊#95C2212yes—
39C2×C47⋊#95C2212yes—
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2212yes—
42C24×C4×C47C2211yescharacteristic
43C22×C22⋊C4×C47C2211yescharacteristic
44C22×C22⋊C4×C47C2211yescharacteristic
45C22×C42×C47C2211yescharacteristic
46C22×C42×C47C2211yescharacteristic
47C4×C22⋊C4×C47C2218yes—
48C2×#95×C47C2211yescharacteristic
49C2×#95×C47C2211yescharacteristic
50C22×Dic94—24no—
51C22×Dic94—24no—
52C22×Dic94C2×C418yescomplemented
53C22×Dic94C2314yes—
54C23×Dic47—28no—
55C23×Dic47—28no—
56C23×Dic47—44no—
57C23×Dic47C2312yes—
58C23×Dic47C2318yes—
59C23×Dic47C2312yes—
60C23×Dic47C2312yes—
61C23×Dic47C2312yes—
62#13094003C2×C4164yescomplemented
63C4×Dic94—232no—
64C2×C4⋊Dic47—24no—
65C2×C4⋊Dic47—28no—
66C2×C4⋊Dic47—232no—
67C2×C4⋊Dic47C2×C418yescomplemented
68C2×C4⋊Dic47D418yes—
69C2×C4⋊Dic47C2318yes—
70C2×C4⋊Dic47C2314yes—
71C2×C4⋊Dic47C2314yes—
72C2×C4⋊Dic47C2314yes—
73C2×Dic47⋊C4—28no—
74C2×Dic47⋊C4—44no—
75C2×Dic47⋊C4C2×C4116yescomplemented
76C2×Dic47⋊C4C2318yes—
77C2×Dic47⋊C4C2318yes—
78C2×Dic47⋊C4C2318yes—
79C2×C22⋊Dic47C2×C4116yes—
80C2×C22⋊Dic47C2318yes—
81C2×C4×Dic47—28no—
82C2×C4×Dic47—44no—
83C2×C4×Dic47C2318yes—
84C2×C4×Dic47C2318yes—
85C47⋊#95C2318yes—
86C47⋊#95C2318yes—
87C47⋊#95C2318yes—
88C47⋊#95C2318yes—
89C25×C47C2311yescharacteristic
90C23×C4×C47—24no—
91C23×C4×C47—24no—
92C23×C4×C47C2×C418yes—
93C23×C4×C47C2311yescharacteristic
94C23×C4×C47C2311yescharacteristic
95C23×C4×C47C2311yescharacteristic
96C23×C4×C47C2314yes—
97C23×C4×C47C2311yescharacteristic
98C23×C4×C47C2311yescharacteristic
99C23×C4×C47C2311yescharacteristic
100C2×C22⋊C4×C47—24no—
101C2×C22⋊C4×C47Q818yescomplemented
102C2×C22⋊C4×C47C2314yes—
103C2×C22⋊C4×C47C2314yes—
104C2×C42×C47—24no—
105C2×C42×C47—24no—
106C2×C42×C47C2314yes—
107C2×C42×C47C2314yes—
108#95×C47C2314yes—
109#95×C47C2314yes—
110C24×C47—22no—
111C24×C47—22no—
112C24×C47—22no—
113C24×C47—22no—
114C24×C47C22×C414yes—
115C24×C47C2×Q814yes—
116C24×C47#4114yes—
117C24×C47C2412yes—
118C24×C47C2411yescharacteristic
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—232no—
126C22×C4×C47—28no—
127C22×C4×C47—28no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47C22×C414yes—
135C22×C4×C47C22×C4116yes—
136C22×C4×C47C22×C414yes—
137C22×C4×C47C2×D414yes—
138C22×C4×C47C2×Q814yes—
139C22×C4×C47C2×Q814yes—
140C22×C4×C47#4114yes—
141C22×C4×C47C2412yes—
142C22×C4×C47C2412yes—
143C22×C4×C47C2412yes—
144C22×C4×C47C2412yes—
145C22×C4×C47C2412yes—
146C22×C4×C47C2412yes—
147C22×Dic47—24no—
148C22×Dic47—24no—
149C22×Dic47—24no—
150C22×Dic47—24no—
151C22×Dic47—24no—
152C22×Dic47—24no—
153C22×Dic47—24no—
154C22×Dic47—216no—
155C22×Dic47—24no—
156C22×Dic47—24no—
157C22×Dic47—216no—
158C22×Dic47—48no—
159C22×Dic47—432no—
160C22×Dic47C22×C418yes—
161C22×Dic47C22×C418yes—
162C22×Dic47C2×D418yes—
163C22×Dic47C2414yes—
164C22×Dic47C2414yes—
165C22×Dic47C2414yes—
166C22×Dic47C2414yes—
167C2×Dic94—232no—
168C2×Dic94—28no—
169C2×Dic94—28no—
170C2×Dic94C22×C4116yescomplemented
171C22⋊C4×C47—28no—
172C22⋊C4×C47C2×Q8116yescomplemented
173C42×C47—28no—
174C42×C47—28no—
175C22⋊Dic47C22×C4132yes—
176C4⋊Dic47—28no—
177C4⋊Dic47—216no—
178C4⋊Dic47C22×C4116yescomplemented
179C4⋊Dic47C2×D4116yes—
180Dic47⋊C4—216no—
181Dic47⋊C4—48no—
182Dic47⋊C4C22×C4132yescomplemented
183C4×Dic47—216no—
184C4×Dic47—48no—
185C23×C47—28no—
186C23×C47—28no—
187C23×C47—28no—
188C23×C47—28no—
189C23×C47—28no—
190C23×C47—28no—
191C23×C47—22no—
192C23×C47—22no—
193C23×C47—22no—
194C23×C47—22no—
195C23×C47C4×Q8116yescomplemented
196C23×C47C23×C414yes—
197C23×C47C23×C412yes—
198C23×C47C22×D412yes—
199C23×C47C22×Q814yes—
200C23×C47C22×Q812yes—
201C23×C47C2×#4112yes—
202C23×C47C2×#4114yes—
203C23×C47C2×#4112yes—
204C23×C47H(5,𝔽2)12yes—
205C23×C472−1+412yes—
206C23×C47C2511yescharacteristic
207C2×C4×C47—28no—
208C2×C4×C47—28no—
209C2×C4×C47—28no—
210C2×C4×C47—22no—
211C2×C4×C47—22no—
212C2×C4×C47—22no—
213C2×C4×C47—22no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—232no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—44no—
224C2×C4×C47—44no—
225C2×C4×C47C4×D4116yes—
226C2×C4×C47C23×C414yes—
227C2×C4×C47C23×C414yes—
228C2×C4×C47C22×D414yes—
229C2×C4×C47C22×Q814yes—
230C2×C4×C47C22×Q814yes—
231C2×C4×C47C2×#4114yes—
232C2×Dic47—216no—
233C2×Dic47—216no—
234C2×Dic47—216no—
235C2×Dic47—24no—
236C2×Dic47—24no—
237C2×Dic47—24no—
238C2×Dic47—24no—
239C2×Dic47—24no—
240C2×Dic47—24no—
241C2×Dic47—24no—
242C2×Dic47—24no—
243C2×Dic47—24no—
244C2×Dic47—432no—
245C2×Dic47—48no—
246C2×Dic47C23×C418yes—
247C2×Dic47C23×C418yes—
248C2×Dic47C22×D418yes—
249Dic94—232no—
250C2×#2360—471nomaximal, Sylow
251C22×C47—232no—
252C22×C47—24no—
253C22×C47—24no—
254C22×C47—24no—
255C22×C47—24no—
256C22×C47—24no—
257C22×C47—24no—
258C22×C47C2×C4×D414yes—
259C22×C47C2×C4×Q814yes—
260C22×C47C2×C4×Q818yescomplemented
261C22×C47#52014yes—
262C22×C47#54614yes—
263C22×C47D4×Q814yes—
264C22×C47#55214yes—
265C22×C47#55414yes—
266C22×C47C24×C411yescharacteristic
267C22×C47C23×D411yescharacteristic
268C22×C47C23×Q811yescharacteristic
269C22×C47C22×#4111yescharacteristic
270C22×C47C22×#4111yescharacteristic
271C22×C47C2×H(5,𝔽2)11yescharacteristic
272C22×C47C2×2−1+411yescharacteristic
273C4×C47—24no—
274C4×C47—24no—
275C4×C47—24no—
276C4×C47—42no—
277C4×C47—42no—
278C4×C47C2×C4×D418yes—
279Dic47—28no—
280Dic47—28no—
281Dic47—28no—
282Dic47—44no—
283C22×C22⋊Q8—471no—
284C22×C4×Q8—471no—
285C23×C4⋊C4—471no—
286#2360—4716no—
287C2×#391—471no—
288C2×#389—471no—
289C2×#389—472no—
290C2×#387—472no—
291C2×#385—472no—
292C2×C4×C4⋊C4—471no—
293C2×C4×C22⋊C4—472no—
294C2×C4×C22⋊C4D4711yescharacteristic
295C2×C47—28no—
296C2×C47#236018yescomplemented
297C2×C47C22×C4×D411yescharacteristic
298C2×C47C22×C4×Q811yescharacteristic
299C2×C47C2×#52011yescharacteristic
300C2×C47C2×#54611yescharacteristic
301C2×C47C2×D4×Q811yescharacteristic
302C2×C47C2×#55211yescharacteristic
303C2×C47C2×#55411yescharacteristic
304C23×Q8—471no—
305C24×C4—472no—
306C24×C4C2×D4711yescharacteristic
307C2×C22⋊Q8—4716no—
308C2×C22⋊Q8—478no—
309C2×C4×Q8—478no—
310C2×C4×Q8—948no—
311C22×C4⋊C4—471no—
312C22×C4⋊C4—472no—
313C22×C4⋊C4—471no—
314C22×C4⋊C4—478no—
315C22×C4⋊C4—472no—
316C22×C4⋊C4—471no—
317C22×C4⋊C4—472no—
318C22×C4⋊C4—472no—
319C22×C4⋊C4—948no—
320C22×C22⋊C4—472no—
321C22×C22⋊C4C2×D4711yescharacteristic
322C22×C22⋊C4C2×D4711yescharacteristic
323C22×C42—472no—
324C22×C42—472no—
325C22×C42C2×D4711yescharacteristic
326C22×C42C2×D4711yescharacteristic
327#391—478no—
328#389—478no—
329#389—4716no—
330#387—4716no—
331#385—4716no—
332C4×C4⋊C4—478no—
333C4×C22⋊C4—4716no—
334C4×C22⋊C4C2×D4718yes—
335C2×#95—472no—
336C2×#95—472no—
337C2×#95—472no—
338C2×#95—472no—
339C2×#95C2×D4711yescharacteristic
340C2×#95C2×D4711yescharacteristic
341C47C2×#236011yescharacteristic, complemented, Sylow
342C25C22×D4711yescharacteristic
343C22×Q8—478no—
344C22×Q8—474no—
345C22×Q8—944no—
346C22×Q8—944no—
347C23×C4—24no—
348C23×C4—24no—
349C23×C4—472no—
350C23×C4—472no—
351C23×C4—472no—
352C23×C4—472no—
353C23×C4—478no—
354C23×C4—948no—
355C23×C4—948no—
356C23×C4—1884no—
357C23×C4C4×D4718yes—
358C23×C4C22×D4711yescharacteristic
359C23×C4C22×D4711yescharacteristic
360C23×C4C22×D4711yescharacteristic
361C23×C4C22×D4711yescharacteristic
362C23×C4C22×D4714yes—
363C23×C4C22×D4711yescharacteristic
364C23×C4C22×D4711yescharacteristic
365C22⋊Q8—4764no—
366C4×Q8—9432no—
367C2×C4⋊C4—4716no—
368C2×C4⋊C4—478no—
369C2×C4⋊C4—478no—
370C2×C4⋊C4—474no—
371C2×C4⋊C4—478no—
372C2×C4⋊C4—474no—
373C2×C4⋊C4—478no—
374C2×C4⋊C4—474no—
375C2×C4⋊C4—478no—
376C2×C4⋊C4—478no—
377C2×C4⋊C4—9432no—
378C2×C4⋊C4—948no—
379C2×C4⋊C4—948no—
380C2×C4⋊C4—944no—
381C2×C4⋊C4—1884no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—4716no—
384C2×C22⋊C4—478no—
385C2×C22⋊C4Dic9418yescomplemented
386C2×C22⋊C4C22×D4714yes—
387C2×C22⋊C4C22×D4714yes—
388C2×C42—24no—
389C2×C42—24no—
390C2×C42—478no—
391C2×C42—478no—
392C2×C42—948no—
393C2×C42—1884no—
394C2×C42C22×D4714yes—
395C2×C42C22×D4714yes—
396#95—478no—
397#95—478no—
398#95—478no—
399#95—478no—
400#95C22×D4714yes—
401#95C22×D4714yes—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24C2×Dic9414yes—
407C24C2×C4×D4714yes—
408C24#1062535014yes—
409C24C23×D4712yes—
410C24C23×D4711yescharacteristic
411C2×Q8—4716no—
412C2×Q8—9432no—
413C2×Q8—948no—
414C2×Q8—948no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—232no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—478no—
431C22×C4—478no—
432C22×C4—478no—
433C22×C4—474no—
434C22×C4—474no—
435C22×C4—474no—
436C22×C4—474no—
437C22×C4—9416no—
438C22×C4—944no—
439C22×C4—944no—
440C22×C4—944no—
441C22×C4—944no—
442C22×C4—944no—
443C22×C4—944no—
444C22×C4—944no—
445C22×C4—9416no—
446C22×C4—944no—
447C22×C4—944no—
448C22×C4—1888no—
449C22×C4—18832no—
450C22×C4C2×Dic9414yes—
451C22×C4C2×Dic9414yes—
452C22×C4C2×C4×D4714yes—
453C22×C4C2×C4×D47116yes—
454C22×C4C2×C4×D4714yes—
455C22×C4D4×D4714yes—
456C22×C4#1062535214yes—
457C22×C4C23×D4712yes—
458C22×C4C23×D4712yes—
459C22×C4C23×D4712yes—
460C22×C4C23×D4712yes—
461C22×C4C23×D4712yes—
462C22×C4C23×D4712yes—
463C4⋊C4—4732no—
464C4⋊C4—4716no—
465C4⋊C4—4716no—
466C4⋊C4—9416no—
467C4⋊C4—9416no—
468C4⋊C4—948no—
469C4⋊C4—1888no—
470C22⋊C4—28no—
471C22⋊C4—4732no—
472C22⋊C4C2×Dic94116yescomplemented
473C42—28no—
474C42—28no—
475C42—9416no—
476C42—1888no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23C4×Dic94116yescomplemented
488C23C22×C4×D4714yes—
489C23C22×C4×D4712yes—
490C23C2×D4×D4712yes—
491C23C22×Dic9414yes—
492C23C22×Dic9412yes—
493C23C2×#1062535212yes—
494C23C2×#1062535014yes—
495C23C2×#1062535012yes—
496C23#1309407012yes—
497C23#1309407112yes—
498C23C24×D4711yescharacteristic
499Q8—9432no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—232no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—478no—
519C2×C4—478no—
520C2×C4—478no—
521C2×C4—9416no—
522C2×C4—9416no—
523C2×C4—9416no—
524C2×C4—944no—
525C2×C4—944no—
526C2×C4—944no—
527C2×C4—944no—
528C2×C4—944no—
529C2×C4—944no—
530C2×C4—944no—
531C2×C4—944no—
532C2×C4—944no—
533C2×C4—18832no—
534C2×C4—1888no—
535C2×C4(C22⋊D47)⋊C4116yes—
536C2×C4C22×C4×D4714yes—
537C2×C4C22×C4×D4714yes—
538C2×C4C2×D4×D4714yes—
539C2×C4C22×Dic9414yes—
540C2×C4C22×Dic9414yes—
541C2×C4C2×#1062535214yes—
542C22—24no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—232no—
547C22—24no—
548C22—24no—
549C22C2×(C22⋊D47)⋊C414yes—
550C22C2×C4×Dic9414yes—
551C22C2×C4×Dic9418yescomplemented
552C22#1568960314yes—
553C22#1568976914yes—
554C22D4×Dic9414yes—
555C22#1568981314yes—
556C22#1568983014yes—
557C22C23×C4×D4711yescharacteristic
558C22C22×D4×D4711yescharacteristic
559C22C23×Dic9411yescharacteristic
560C22C22×#1062535211yescharacteristic
561C22C22×#1062535011yescharacteristic
562C22C2×#1309407011yescharacteristic
563C22C2×#1309407111yescharacteristic
564C4—24no—
565C4—24no—
566C4—24no—
567C4—42no—
568C4—42no—
569C4—948no—
570C4—948no—
571C4—948no—
572C4—1884no—
573C4C2×(C22⋊D47)⋊C418yes—
574C2—28no—
575C2#2024639118yesdirect product factor
576C2C22×(C22⋊D47)⋊C411yescharacteristic
577C2C22×C4×Dic9411yescharacteristic
578C2C2×#1568960311yescharacteristic
579C2C2×#1568976911yescharacteristic
580C2C2×D4×Dic9411yescharacteristic
581C2C2×#1568981311yescharacteristic
582C2C2×#1568983011yescharacteristic
subgroupG/H
centerC24C23×D47
commutatorC22×C47C24×C4
FrattiniC23C24×D47
FittingC2×C4×C22⋊C4×C47C2
socleC24×C47C24
prime radicalC25×C47C23
cosocleC23×C47C25
hypercenterC2×C4×C22⋊C4D47
hypocenterC47C2×#2360
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic94C2semidirect
4. #20246391C2direct (N × H)
5. C2×#15689029C2semidirect
6. C2×#15689021C2semidirect
7. C2×#15689020C2semidirect
8. C2×#15689010C2semidirect
10. C2×C4×C4⋊Dic47C2semidirect
15. C2×#13094003C4semidirect
18. C2×C4×Dic94C22semidirect
31. #15689029C22semidirect
32. #15689021C22semidirect
33. #15689020C22semidirect
34. #15689010C22semidirect
36. C4×C4⋊Dic47C22semidirect
52. C22×Dic94C2×C4semidirect
62. #13094003C2×C4semidirect
67. C2×C4⋊Dic47C2×C4semidirect
75. C2×Dic47⋊C4C2×C4semidirect
101. C2×C22⋊C4×C47Q8semidirect
170. C2×Dic94C22×C4semidirect
172. C22⋊C4×C47C2×Q8semidirect
178. C4⋊Dic47C22×C4semidirect
182. Dic47⋊C4C22×C4semidirect
195. C23×C47C4×Q8semidirect
260. C22×C47C2×C4×Q8semidirect
296. C2×C47#2360semidirect
341. C47C2×#2360semidirect
385. C2×C22⋊C4Dic94semidirect
472. C22⋊C4C2×Dic94semidirect
487. C23C4×Dic94semidirect
551. C22C2×C4×Dic94semidirect
575. C2#20246391direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940032
C22×C4×Dic942
C23×C4⋊Dic472
#202463912
C2×#156890292
C2×#156890212
C2×#156890202
C2×#156890102
C2×#156889952
C2×C4×C4⋊Dic472
C2×C22⋊C4×Dic472
C2×C4×C22⋊C4×C472
C2×#236047