Finite Group Databasebeta

← C2×#20246990 #26384301

Subgroups — C2×#20246990 #26384301

Order 12032.

3246 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
1C22×#13094000C211yesmaximal, characteristic
2C22×#13094000C211yesmaximal, characteristic
3C22×Q8×Dic47C211yesmaximal, characteristic, complemented
4#20246990C2116yesmaximal, direct product factor
5C2×#15689056C212yesmaximal, complemented
6C2×#15689045C212yesmaximal
7C2×#15689030C211yesmaximal, characteristic, complemented
8C2×#15689021C211yesmaximal, characteristic, complemented
9C2×#15689018C212yesmaximal, complemented
10C2×#15689013C212yesmaximal
11C2×C4×(C2×D47)⋊C4C211yesmaximal, characteristic
12C22×C4⋊Q8×C47C211yesmaximal, characteristic, complemented
13C23×C4×D47C2211yescharacteristic
14C2×#13094000—28no—
15C2×#13094000—28no—
16C2×#13094000C2218yes—
17C2×#13094000C2218yes—
18C2×Q8×Dic47—28no—
19C2×Q8×Dic47C2218yescomplemented
20C22×C4⋊Dic47C2212yes—
21C22×C4⋊Dic47C2211yescharacteristic
22C22×C4⋊Dic47C2211yescharacteristic
23C22×Dic47⋊C4C2212yes—
24C22×Dic47⋊C4C2212yes—
25C22×Dic47⋊C4C2212yes—
26C22×(C2×D47)⋊C4C2212yes—
27C22×(C2×D47)⋊C4C2212yes—
28C22×(C2×D47)⋊C4C2211yescharacteristic
29C22×(C2×D47)⋊C4C2211yescharacteristic
30C22×C4×Dic47C2212yes—
31C22×C4×Dic47C2211yescharacteristic
32#15689056C22116yescomplemented
33#15689045C22116yes—
34#15689030C2218yescomplemented
35#15689021C2218yescomplemented
36#15689018C22116yescomplemented
37#15689013C22116yes—
38C4×(C2×D47)⋊C4C2218yes—
39C2×C47⋊#95C2212yes—
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2211yescharacteristic
42C2×C47⋊#95C2212yes—
43C2×C47⋊#95C2211yescharacteristic
44C23×Q8×C47C2211yescharacteristic
45C23×Q8×C47C2211yescharacteristic
46C2×C4⋊Q8×C47—28no—
47C2×C4⋊Q8×C47C2218yescomplemented
48C22×C4⋊C4×C47C2212yes—
49C22×C4⋊C4×C47C2212yes—
50C22×C42×C47C2211yescharacteristic
51C24×D47C2311yescharacteristic
52C23×Dic47C2311yescharacteristic
53C23×Dic47C2312yes—
54C23×Dic47C2311yescharacteristic
55C23×Dic47C2312yes—
56C23×Dic47C2311yescharacteristic
57C22×C4×D47—24no—
58C22×C4×D47—24no—
59C22×C4×D47—24no—
60C22×C4×D47C2314yes—
61#13094000—232no—
62#13094000—232no—
63Q8×Dic47—232no—
64C2×C4⋊Dic47—24no—
65C2×C4⋊Dic47—28no—
66C2×C4⋊Dic47D418yes—
67C2×C4⋊Dic47C2314yes—
68C2×C4⋊Dic47C2318yes—
69C2×C4⋊Dic47C2314yes—
70C2×Dic47⋊C4—28no—
71C2×Dic47⋊C4—28no—
72C2×Dic47⋊C4—28no—
73C2×Dic47⋊C4C2318yes—
74C2×Dic47⋊C4C2318yes—
75C2×Dic47⋊C4C2318yes—
76C2×(C2×D47)⋊C4—28no—
77C2×(C2×D47)⋊C4—24no—
78C2×(C2×D47)⋊C4—28no—
79C2×(C2×D47)⋊C4Q818yescomplemented
80C2×(C2×D47)⋊C4C2314yes—
81C2×(C2×D47)⋊C4C2318yes—
82C2×(C2×D47)⋊C4C2318yes—
83C2×(C2×D47)⋊C4C2314yes—
84C2×C4×Dic47—24no—
85C2×C4×Dic47—28no—
86C2×C4×Dic47C2314yes—
87C2×C4×Dic47C2318yes—
88C47⋊#95C2314yes—
89C47⋊#95C2314yes—
90C47⋊#95C2318yes—
91C47⋊#95C2318yes—
92C47⋊#95C2318yes—
93C22×Q8×C47—24no—
94C22×Q8×C47—24no—
95C22×Q8×C47—24no—
96C22×Q8×C47—24no—
97C22×Q8×C47—24no—
98C22×Q8×C47D418yes—
99C22×Q8×C47C2314yes—
100C22×Q8×C47C2314yes—
101C23×C4×C47C2312yes—
102C23×C4×C47C2312yes—
103C23×C4×C47C2311yescharacteristic
104C23×C4×C47C2311yescharacteristic
105C23×C4×C47C2311yescharacteristic
106C4⋊Q8×C47—232no—
107C2×C4⋊C4×C47—28no—
108C2×C4⋊C4×C47—28no—
109C2×C4⋊C4×C47C2318yes—
110C2×C4⋊C4×C47C2318yes—
111C2×C42×C47—24no—
112C2×C42×C47C2314yes—
113C24×C47C2411yescharacteristic
114C23×D47—22no—
115C23×D47—22no—
116C23×D47—22no—
117C23×D47—22no—
118C23×D47—22no—
119C23×D47—22no—
120C23×D47C2×Q814yes—
121C23×D47C2412yes—
122C2×Q8×C47—232no—
123C2×Q8×C47—28no—
124C2×Q8×C47—28no—
125C2×Q8×C47—28no—
126C2×Q8×C47—28no—
127C2×Q8×C47—28no—
128C2×Q8×C47—416no—
129C2×Q8×C47C2×D4116yes—
130C22×C4×C47—22no—
131C22×C4×C47—22no—
132C22×C4×C47—24no—
133C22×C4×C47—24no—
134C22×C4×C47—22no—
135C22×C4×C47—22no—
136C22×C4×C47—24no—
137C22×C4×C47—24no—
138C22×C4×C47—24no—
139C22×C4×C47C2×D418yes—
140C22×C4×C47C2×D414yes—
141C22×C4×C47C2×D414yes—
142C22×C4×C47C2×Q814yes—
143C22×C4×C47#4114yes—
144C22×C4×C47#4114yes—
145C22×C4×C47C2414yes—
146C22×C4×C47C2414yes—
147C22×C4×C47C2412yes—
148C22×C4×C47C2412yes—
149C22×C4×C47C2412yes—
150C22×Dic47—22no—
151C22×Dic47—24no—
152C22×Dic47—24no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—24no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47—24no—
160C22×Dic47—22no—
161C22×Dic47—22no—
162C22×Dic47—24no—
163C22×Dic47C2×D418yes—
164C22×Dic47C2×Q814yes—
165C22×Dic47C2412yes—
166C22×Dic47C2414yes—
167C22×Dic47C2412yes—
168C22×Dic47C2414yes—
169C22×Dic47C2412yes—
170C2×C4×D47—28no—
171C2×C4×D47—28no—
172C2×C4×D47—28no—
173C2×C4×D47—416no—
174C4⋊C4×C47—216no—
175C4⋊C4×C47—216no—
176C42×C47—28no—
177(C2×D47)⋊C4—216no—
178(C2×D47)⋊C4—28no—
179(C2×D47)⋊C4—216no—
180(C2×D47)⋊C4C2×Q8116yescomplemented
181C4⋊Dic47—28no—
182C4⋊Dic47—216no—
183C4⋊Dic47C2×D4116yes—
184Dic47⋊C4—216no—
185Dic47⋊C4—216no—
186Dic47⋊C4—216no—
187C4×Dic47—28no—
188C4×Dic47—216no—
189C23×C47C22×D412yes—
190C23×C47C22×D412yes—
191C23×C47C22×Q812yes—
192C23×C47C2×#4112yes—
193C23×C47C2×#4112yes—
194C23×C472−1+412yes—
195C23×C472−1+412yes—
196C23×C47C2511yescharacteristic
197C22×D47—28no—
198C22×D47—28no—
199C22×D47—28no—
200C22×D47—22no—
201C22×D47—22no—
202C22×D47—22no—
203C22×D47—22no—
204C22×D47—22no—
205C22×D47—22no—
206C22×D47—44no—
207C22×D47—44no—
208C22×D47—44no—
209C22×D47—44no—
210C22×D47C22×Q814yes—
211Q8×C47—232no—
212Q8×C47—416no—
213C2×C4×C47—216no—
214C2×C4×C47—28no—
215C2×C4×C47—28no—
216C2×C4×C47—24no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—24no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—24no—
223C2×C4×C47—24no—
224C2×C4×C47—24no—
225C2×C4×C47—48no—
226C2×C4×C47#121116yes—
227C2×C4×C47C22×D414yes—
228C2×C4×C47C22×D418yes—
229C2×C4×C47C22×D414yes—
230C2×C4×C47C22×Q814yes—
231C2×C4×C47C2×#4114yes—
232C2×C4×C47C2×#4114yes—
233C2×Dic47—216no—
234C2×Dic47—28no—
235C2×Dic47—22no—
236C2×Dic47—24no—
237C2×Dic47—24no—
238C2×Dic47—22no—
239C2×Dic47—22no—
240C2×Dic47—22no—
241C2×Dic47—22no—
242C2×Dic47—24no—
243C2×Dic47—24no—
244C2×Dic47—22no—
245C2×Dic47—22no—
246C2×Dic47—22no—
247C2×Dic47—24no—
248C2×Dic47—44no—
249C2×Dic47—44no—
250C2×Dic47—48no—
251C2×Dic47C22×D418yes—
252C2×Dic47C22×Q814yes—
253C4×D47—416no—
254C2×#2429—471nomaximal, Sylow
255C22×C47C2×#12114yes—
256C22×C47#53314yes—
257C22×C47#54114yes—
258C22×C47#54714yes—
259C22×C47#54814yes—
260C22×C47D4×Q814yes—
261C22×C47D4×Q814yes—
262C22×C47C23×D411yescharacteristic
263C22×C47C23×D411yescharacteristic
264C22×C47C23×Q811yescharacteristic
265C22×C47C22×#4111yescharacteristic
266C22×C47C22×#4111yescharacteristic
267C22×C47C2×2−1+411yescharacteristic
268C22×C47C2×2−1+411yescharacteristic
269C2×D47—24no—
270C2×D47—24no—
271C2×D47—24no—
272C2×D47—42no—
273C2×D47—416no—
274C2×D47—42no—
275C2×D47—42no—
276C2×D47—42no—
277C4×C47—28no—
278C4×C47—24no—
279C4×C47—24no—
280C4×C47—44no—
281C4×C47C2×#12118yes—
282Dic47—28no—
283Dic47—24no—
284Dic47—42no—
285Dic47—42no—
286Dic47—44no—
287C22×C4⋊Q8D4711yescharacteristic, complemented
288C22×C22⋊Q8—471no—
289C22×C22⋊Q8—471no—
290C22×C4×Q8—471no—
291#2429—4716no—
292C2×#398—472no—
293C2×#396—472no—
294C2×#391—471no—
295C2×#389—471no—
296C2×#389—472no—
297C2×#388—472no—
298C2×C4×C22⋊C4—471no—
299C2×C47#242918yescomplemented
300C2×C47C22×#12111yescharacteristic
301C2×C47C2×#53311yescharacteristic
302C2×C47C2×#54111yescharacteristic
303C2×C47C2×#54711yescharacteristic
304C2×C47C2×#54811yescharacteristic
305C2×C47C2×D4×Q811yescharacteristic
306C2×C47C2×D4×Q811yescharacteristic
307D47—44no—
308C23×Q8C2×D4711yescharacteristic
309C23×Q8C2×D4711yescharacteristic
310C24×C4—471no—
311C2×C4⋊Q8—28no—
312C2×C4⋊Q8C2×D4718yescomplemented
313C2×C22⋊Q8—478no—
314C2×C22⋊Q8—478no—
315C2×C22⋊Q8—948no—
316C2×C22⋊Q8—948no—
317C2×C4×Q8—478no—
318C2×C4×Q8—948no—
319C22×C4⋊C4—471no—
320C22×C4⋊C4—472no—
321C22×C4⋊C4—472no—
322C22×C4⋊C4—472no—
323C22×C4⋊C4—472no—
324C22×C4⋊C4—471no—
325C22×C4⋊C4C2×D4712yes—
326C22×C4⋊C4C2×D4712yes—
327C22×C22⋊C4—471no—
328C22×C22⋊C4—471no—
329C22×C22⋊C4—472no—
330C22×C22⋊C4—472no—
331C22×C42—471no—
332C22×C42—472no—
333C22×C42C2×D4711yescharacteristic
334#398—4716no—
335#396—4716no—
336#391—478no—
337#389—4716no—
338#389—478no—
339#388—4716no—
340C4×C22⋊C4—478no—
341C2×#95—472no—
342C2×#95—471no—
343C2×#95—472no—
344C2×#95—471no—
345C2×#95—472no—
346C47C2×#242911yescharacteristic, complemented, Sylow
347C25—471no—
348C22×Q8—24no—
349C22×Q8—24no—
350C22×Q8—24no—
351C22×Q8—24no—
352C22×Q8—24no—
353C22×Q8C22⋊D4718yes—
354C22×Q8C22×D4714yes—
355C22×Q8C22×D4714yes—
356C23×C4—472no—
357C23×C4—471no—
358C23×C4—472no—
359C23×C4—471no—
360C23×C4—471no—
361C23×C4—474no—
362C23×C4—944no—
363C23×C4—944no—
364C23×C4—944no—
365C23×C4C22×D4712yes—
366C23×C4C22×D4711yescharacteristic
367C23×C4C22×D4711yescharacteristic
368C23×C4C22×D4712yes—
369C23×C4C22×D4711yescharacteristic
370C4⋊Q8—232no—
371C22⋊Q8—9432no—
372C22⋊Q8—9432no—
373C4×Q8—9432no—
374C2×C4⋊C4—28no—
375C2×C4⋊C4—28no—
376C2×C4⋊C4—478no—
377C2×C4⋊C4—474no—
378C2×C4⋊C4—478no—
379C2×C4⋊C4—478no—
380C2×C4⋊C4—478no—
381C2×C4⋊C4—474no—
382C2×C4⋊C4—478no—
383C2×C4⋊C4—948no—
384C2×C4⋊C4—948no—
385C2×C4⋊C4—948no—
386C2×C4⋊C4—948no—
387C2×C4⋊C4—944no—
388C2×C4⋊C4C22×D4718yes—
389C2×C4⋊C4C22×D4718yes—
390C2×C22⋊C4—478no—
391C2×C22⋊C4—474no—
392C2×C22⋊C4—478no—
393C2×C22⋊C4—478no—
394C2×C22⋊C4—474no—
395C2×C22⋊C4—948no—
396C2×C22⋊C4—944no—
397C2×C22⋊C4—948no—
398C2×C42—24no—
399C2×C42—474no—
400C2×C42—478no—
401C2×C42—948no—
402C2×C42—944no—
403C2×C42C22×D4714yes—
404#95—474no—
405#95—474no—
406#95—478no—
407#95—478no—
408#95—478no—
409C24—474no—
410C24—472no—
411C24—942no—
412C24—942no—
413C24—942no—
414C24—942no—
415C24—942no—
416C24—942no—
417C24C23×D4711yescharacteristic
418C2×Q8—232no—
419C2×Q8—28no—
420C2×Q8—28no—
421C2×Q8—28no—
422C2×Q8—28no—
423C2×Q8—28no—
424C2×Q8—416no—
425C2×Q8C2×C22⋊D47116yes—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—478no—
436C22×C4—474no—
437C22×C4—472no—
438C22×C4—474no—
439C22×C4—472no—
440C22×C4—474no—
441C22×C4—472no—
442C22×C4—942no—
443C22×C4—944no—
444C22×C4—944no—
445C22×C4—942no—
446C22×C4—942no—
447C22×C4—942no—
448C22×C4—944no—
449C22×C4—942no—
450C22×C4—942no—
451C22×C4—942no—
452C22×C4—944no—
453C22×C4—942no—
454C22×C4—944no—
455C22×C4—948no—
456C22×C4—948no—
457C22×C4—948no—
458C22×C4—18816no—
459C22×C4D4×D4714yes—
460C22×C4#1062535214yes—
461C22×C4Q8×D4714yes—
462C22×C4#1062535414yes—
463C22×C4C2×C22⋊D4718yes—
464C22×C4C2×C22⋊D4714yes—
465C22×C4C23×D4714yes—
466C22×C4C23×D4714yes—
467C22×C4C23×D4712yes—
468C22×C4C23×D4712yes—
469C22×C4C23×D4712yes—
470C4⋊C4—216no—
471C4⋊C4—216no—
472C4⋊C4—4716no—
473C4⋊C4—9416no—
474C4⋊C4—9416no—
475C4⋊C4—9416no—
476C4⋊C4—948no—
477C4⋊C4—9416no—
478C22⋊C4—4716no—
479C22⋊C4—9416no—
480C22⋊C4—9416no—
481C22⋊C4—948no—
482C42—28no—
483C42—9416no—
484C42—948no—
485C23—474no—
486C23—948no—
487C23—948no—
488C23—948no—
489C23—942no—
490C23—942no—
491C23—942no—
492C23—942no—
493C23—942no—
494C23—942no—
495C23—1884no—
496C23—1884no—
497C23—1884no—
498C23—1884no—
499C23C2×D4×D4712yes—
500C23C22×C22⋊D4712yes—
501C23C2×Q8×D4712yes—
502C23C2×#1062535212yes—
503C23C2×#1062535412yes—
504C23#1309407212yes—
505C23#1309407212yes—
506C23C24×D4711yescharacteristic
507Q8—232no—
508Q8—416no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—216no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—48no—
522C2×C4—478no—
523C2×C4—474no—
524C2×C4—9416no—
525C2×C4—948no—
526C2×C4—942no—
527C2×C4—944no—
528C2×C4—944no—
529C2×C4—942no—
530C2×C4—942no—
531C2×C4—942no—
532C2×C4—944no—
533C2×C4—942no—
534C2×C4—942no—
535C2×C4—942no—
536C2×C4—944no—
537C2×C4—942no—
538C2×C4—944no—
539C2×C4—1884no—
540C2×C4—1888no—
541C2×C4—1884no—
542C2×C4—18816no—
543C2×C4#13093996116yes—
544C2×C4C2×D4×D4714yes—
545C2×C4C22×C22⋊D4718yes—
546C2×C4C22×C22⋊D4714yes—
547C2×C4C2×Q8×D4714yes—
548C2×C4C2×#1062535214yes—
549C2×C4C2×#1062535414yes—
550C22—944no—
551C22—944no—
552C22—944no—
553C22—1882no—
554C22—1882no—
555C22—1882no—
556C22—18816no—
557C22—1882no—
558C22C2×#1309399614yes—
559C22#1568967814yes—
560C22#1568973414yes—
561C22#1568977614yes—
562C22#1568978414yes—
563C22Q8×C22⋊D4714yes—
564C22#1568978814yes—
565C22C22×D4×D4711yescharacteristic
566C22C23×C22⋊D4711yescharacteristic
567C22C22×Q8×D4711yescharacteristic
568C22C22×#1062535211yescharacteristic
569C22C22×#1062535411yescharacteristic
570C22C2×#1309407211yescharacteristic
571C22C2×#1309407211yescharacteristic
572C4—24no—
573C4—24no—
574C4—28no—
575C4—44no—
576C4—948no—
577C4—944no—
578C4—1882no—
579C4—1884no—
580C4—1882no—
581C4C2×#1309399618yes—
582C2—1884no—
583C2#2024699018yesdirect product factor
584C2C22×#1309399611yescharacteristic
585C2C2×#1568967811yescharacteristic
586C2C2×#1568973411yescharacteristic
587C2C2×#1568977611yescharacteristic
588C2C2×#1568978411yescharacteristic
589C2C2×Q8×C22⋊D4711yescharacteristic
590C2C2×#1568978811yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC22×C4⋊Q8×C47C2
socleC24×C47C24
prime radicalC24×D47C23
hypercenterC22×C4⋊Q8D47
hypocenterC47C2×#2429
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. C22×Q8×Dic47C2semidirect
4. #20246990C2direct (N × H)
5. C2×#15689056C2semidirect
7. C2×#15689030C2semidirect
8. C2×#15689021C2semidirect
9. C2×#15689018C2semidirect
12. C22×C4⋊Q8×C47C2semidirect
19. C2×Q8×Dic47C22semidirect
32. #15689056C22semidirect
34. #15689030C22semidirect
35. #15689021C22semidirect
36. #15689018C22semidirect
47. C2×C4⋊Q8×C47C22semidirect
79. C2×(C2×D47)⋊C4Q8semidirect
180. (C2×D47)⋊C4C2×Q8semidirect
287. C22×C4⋊Q8D47semidirect
299. C2×C47#2429semidirect
312. C2×C4⋊Q8C2×D47semidirect
346. C47C2×#2429semidirect
583. C2#20246990direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940002
C22×Q8×Dic472
#202469902
C2×#156890562
C2×#156890452
C2×#156890302
C2×#156890212
C2×#156890182
C2×#156890132
C2×C4×(C2×D47)⋊C42
C22×C4⋊Q8×C472
C2×#242947