Finite Group Databasebeta

← C2×#20251739 #26422213

Subgroups — C2×#20251739 #26422213

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×#13094002C212yesmaximal
3C22×C42×D47C211yesmaximal, characteristic
4#20251739C2116yesmaximal, direct product factor
5C2×#15689027C212yesmaximal, complemented
6C2×#15689021C211yesmaximal, characteristic, complemented
7C2×#15689018C212yesmaximal, complemented
8C2×#15688992C211yesmaximal, characteristic, complemented
9C2×#15688982C212yesmaximal
10C2×#15688980C211yesmaximal, characteristic
11C2×#15688980C211yesmaximal, characteristic
12C2×C47×#389C211yesmaximal, characteristic, complemented
13C23×Dic94C2212yes—
14C23×C4×D47C2211yescharacteristic
15C23×C4×D47C2211yescharacteristic
16C23×C4×D47C2211yescharacteristic
17C2×#13094023—28no—
18C2×#13094023C2218yescomplemented
19C2×#13094002—216no—
20C2×#13094002C22116yes—
21C22×C4⋊Dic47C2211yescharacteristic
22C22×C4⋊Dic47C2211yescharacteristic
23C22×Dic47⋊C4C2212yes—
24C22×Dic47⋊C4C2212yes—
25C22×Dic47⋊C4C2212yes—
26C22×Dic47⋊C4C2212yes—
27C22×(C2×D47)⋊C4C2212yes—
28C22×(C2×D47)⋊C4C2212yes—
29C22×C4×Dic47C2211yescharacteristic
30C22×C4×Dic47C2211yescharacteristic
31C22×C4×Dic47C2211yescharacteristic
32C2×C42×D47—28no—
33C2×C42×D47C2218yes—
34#15689027C22116yescomplemented
35#15689021C2218yescomplemented
36#15689018C22116yescomplemented
37#15688992C2218yescomplemented
38#15688982C22116yes—
39#15688980C2218yes—
40#15688980C2218yes—
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2212yes—
43C2×C47⋊#95C2212yes—
44C22×C4⋊C4×C47C2211yescharacteristic
45C22×C4⋊C4×C47C2212yes—
46C22×C4⋊C4×C47C2211yescharacteristic
47C22×C42×C47C2211yescharacteristic
48C47×#389C2218yescomplemented
49C2×#95×C47C2212yes—
50C24×D47C2311yescharacteristic
51C22×Dic94—28no—
52C22×Dic94—28no—
53C22×Dic94—28no—
54C22×Dic94C2318yes—
55C23×Dic47C2312yes—
56C23×Dic47C2312yes—
57C23×Dic47C2311yescharacteristic
58C23×Dic47C2311yescharacteristic
59C23×Dic47C2311yescharacteristic
60C22×C4×D47—24no—
61C22×C4×D47—24no—
62C22×C4×D47—24no—
63C22×C4×D47—24no—
64C22×C4×D47—24no—
65C22×C4×D47—24no—
66C22×C4×D47D418yes—
67C22×C4×D47Q818yescomplemented
68C22×C4×D47Q818yescomplemented
69C22×C4×D47C2314yes—
70C22×C4×D47C2314yes—
71C22×C4×D47C2314yes—
72#13094023—232no—
73#13094002—264no—
74C2×C4⋊Dic47—24no—
75C2×C4⋊Dic47—24no—
76C2×C4⋊Dic47C2314yes—
77C2×C4⋊Dic47C2314yes—
78C2×Dic47⋊C4—28no—
79C2×Dic47⋊C4—28no—
80C2×Dic47⋊C4—28no—
81C2×Dic47⋊C4—28no—
82C2×Dic47⋊C4C2318yes—
83C2×Dic47⋊C4C2318yes—
84C2×Dic47⋊C4C2318yes—
85C2×Dic47⋊C4C2318yes—
86C2×(C2×D47)⋊C4—28no—
87C2×(C2×D47)⋊C4—28no—
88C2×(C2×D47)⋊C4C2318yes—
89C2×(C2×D47)⋊C4C2318yes—
90C2×C4×Dic47—24no—
91C2×C4×Dic47—24no—
92C2×C4×Dic47D418yes—
93C2×C4×Dic47C2314yes—
94C2×C4×Dic47C2314yes—
95C2×C4×Dic47C2314yes—
96C42×D47—232no—
97C47⋊#95C2318yes—
98C47⋊#95C2318yes—
99C47⋊#95C2318yes—
100C23×C4×C47C2312yes—
101C23×C4×C47C2312yes—
102C23×C4×C47C2311yescharacteristic
103C23×C4×C47C2311yescharacteristic
104C23×C4×C47C2311yescharacteristic
105C2×C4⋊C4×C47—28no—
106C2×C4⋊C4×C47—24no—
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—22no—
116C23×D47—22no—
117C23×D47—22no—
118C23×D47—22no—
119C23×D47C2×D414yes—
120C23×D47C2×Q814yes—
121C23×D47C2×Q814yes—
122C23×D47C2412yes—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—24no—
128C22×C4×C47—24no—
129C22×C4×C47—24no—
130C22×C4×C47—24no—
131C22×C4×C47—24no—
132C22×C4×C47—24no—
133C22×C4×C47C2×D414yes—
134C22×C4×C47C2×D414yes—
135C22×C4×C47C2×Q814yes—
136C22×C4×C47C2×Q814yes—
137C22×C4×C47#4114yes—
138C22×C4×C47C2414yes—
139C22×C4×C47C2414yes—
140C22×C4×C47C2412yes—
141C22×C4×C47C2412yes—
142C22×C4×C47C2412yes—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—24no—
146C22×Dic47—24no—
147C22×Dic47—24no—
148C22×Dic47—24no—
149C22×Dic47—24no—
150C22×Dic47—24no—
151C22×Dic47C2×D414yes—
152C22×Dic47C2×D414yes—
153C22×Dic47C2×D414yes—
154C22×Dic47C2×Q814yes—
155C22×Dic47C2×Q814yes—
156C22×Dic47#4114yes—
157C22×Dic47#4114yes—
158C22×Dic47C2414yes—
159C22×Dic47C2414yes—
160C22×Dic47C2412yes—
161C22×Dic47C2412yes—
162C22×Dic47C2412yes—
163C2×C4×D47—232no—
164C2×C4×D47—28no—
165C2×C4×D47—28no—
166C2×C4×D47—232no—
167C2×C4×D47—232no—
168C2×C4×D47—28no—
169C2×C4×D47—28no—
170C2×C4×D47—28no—
171C2×C4×D47—28no—
172C2×C4×D47C2×D4116yes—
173C2×C4×D47C2×Q8116yescomplemented
174C2×C4×D47C2×Q8116yescomplemented
175C2×Dic94—216no—
176C2×Dic94—216no—
177C2×Dic94—216no—
178C2×Dic94—432no—
179C4⋊C4×C47—216no—
180C4⋊C4×C47—28no—
181C4⋊C4×C47—28no—
182C42×C47—28no—
183(C2×D47)⋊C4—216no—
184(C2×D47)⋊C4—216no—
185C4⋊Dic47—28no—
186C4⋊Dic47—28no—
187Dic47⋊C4—216no—
188Dic47⋊C4—216no—
189Dic47⋊C4—216no—
190Dic47⋊C4—216no—
191C4×Dic47—28no—
192C4×Dic47—28no—
193C4×Dic47C2×D4116yes—
194C23×C47C22×D412yes—
195C23×C47C22×D412yes—
196C23×C47C22×Q812yes—
197C23×C47C22×Q812yes—
198C23×C47C2×#4112yes—
199C23×C47C2×#4112yes—
200C23×C47C2×#4112yes—
201C23×C47C2511yescharacteristic
202C22×D47—28no—
203C22×D47—28no—
204C22×D47—28no—
205C22×D47—28no—
206C22×D47—28no—
207C22×D47—28no—
208C22×D47—22no—
209C22×D47—22no—
210C22×D47—22no—
211C22×D47—22no—
212C22×D47C4⋊Q8116yescomplemented
213C22×D47C22×D414yes—
214C22×D47C22×Q814yes—
215C22×D47C22×Q814yes—
216C2×C4×C47—28no—
217C2×C4×C47—28no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—24no—
223C2×C4×C47—24no—
224C2×C4×C47—24no—
225C2×C4×C47—24no—
226C2×C4×C47—24no—
227C2×C4×C47—24no—
228C2×C4×C47—48no—
229C2×C4×C47—48no—
230C2×C4×C47C22⋊Q8116yescomplemented
231C2×C4×C47C22×D414yes—
232C2×C4×C47C22×D414yes—
233C2×C4×C47C22×Q814yes—
234C2×C4×C47C22×Q814yes—
235C2×C4×C47C2×#4114yes—
236C2×Dic47—28no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—24no—
240C2×Dic47—24no—
241C2×Dic47—24no—
242C2×Dic47—24no—
243C2×Dic47—24no—
244C2×Dic47—24no—
245C2×Dic47—48no—
246C2×Dic47—48no—
247C2×Dic47#121116yes—
248C2×Dic47C22⋊Q8116yescomplemented
249C2×Dic47C22×D414yes—
250C2×Dic47C22×D414yes—
251C2×Dic47C22×D414yes—
252C2×Dic47C22×Q814yes—
253C2×Dic47C22×Q814yes—
254C2×Dic47C2×#4114yes—
255C2×Dic47C2×#4114yes—
256C4×D47—232no—
257C4×D47—232no—
258C4×D47—232no—
259Dic94—432no—
260C2×#2857—471nomaximal, Sylow
261C22×C47C2×#12114yes—
262C22×C47C2×C22⋊Q814yes—
263C22×C47C2×C22⋊Q814yes—
264C22×C47C2×C4⋊Q814yes—
265C22×C47#53214yes—
266C22×C47#53314yes—
267C22×C47#53314yes—
268C22×C47C23×D411yescharacteristic
269C22×C47C23×D411yescharacteristic
270C22×C47C23×Q811yescharacteristic
271C22×C47C23×Q811yescharacteristic
272C22×C47C22×#4111yescharacteristic
273C22×C47C22×#4111yescharacteristic
274C22×C47C22×#4111yescharacteristic
275C2×D47—232no—
276C2×D47—24no—
277C2×D47—24no—
278C2×D47—24no—
279C2×D47—24no—
280C2×D47—24no—
281C2×D47—24no—
282C2×D47C2×C4⋊Q818yescomplemented
283C4×C47—24no—
284C4×C47—24no—
285C4×C47—44no—
286C4×C47—44no—
287C4×C47C2×C22⋊Q818yescomplemented
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—471no—
299C2×#389—472no—
300C2×#389D4711yescharacteristic, complemented
301C2×#384—471no—
302C2×#380—472no—
303C2×#380—471no—
304C2×#380—471no—
305C2×C47#285718yescomplemented
306C2×C47C22×#12111yescharacteristic
307C2×C47C22×C22⋊Q811yescharacteristic
308C2×C47C22×C22⋊Q811yescharacteristic
309C2×C47C22×C4⋊Q811yescharacteristic
310C2×C47C2×#53211yescharacteristic
311C2×C47C2×#53311yescharacteristic
312C2×C47C2×#53311yescharacteristic
313D47—28no—
314C23×Q8—472no—
315C24×C4—471no—
316C24×C4—471no—
317C24×C4—471no—
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—471no—
327C22×C4⋊C4—471no—
328C22×C4⋊C4C2×D4712yes—
329C22×C4⋊C4C2×D4711yescharacteristic
330C22×C4⋊C4C2×D4711yescharacteristic
331C22×C22⋊C4—472no—
332C22×C22⋊C4—472no—
333C22×C42—471no—
334C22×C42—471no—
335C22×C42—471no—
336C22×C42—478no—
337C22×C42—948no—
338C22×C42C2×D4711yescharacteristic
339#391—4716no—
340#389—4716no—
341#389—478no—
342#389C2×D4718yescomplemented
343#384—478no—
344#380—478no—
345#380—478no—
346#380—4716no—
347C2×#95—472no—
348C2×#95—472no—
349C2×#95—472no—
350C2×#95C2×D4712yes—
351C47C2×#285711yescharacteristic, complemented, Sylow
352C25—471no—
353C22×Q8—478no—
354C22×Q8—948no—
355C22×Q8—948no—
356C22×Q8—948no—
357C23×C4—471no—
358C23×C4—471no—
359C23×C4—471no—
360C23×C4—478no—
361C23×C4—478no—
362C23×C4—478no—
363C23×C4—474no—
364C23×C4—474no—
365C23×C4—474no—
366C23×C4—472no—
367C23×C4—472no—
368C23×C4—944no—
369C23×C4—944no—
370C23×C4—944no—
371C23×C4—944no—
372C23×C4—944no—
373C23×C4—944no—
374C23×C4C22×D4712yes—
375C23×C4C22×D4712yes—
376C23×C4C22×D4711yescharacteristic
377C23×C4C22×D4711yescharacteristic
378C23×C4C22×D4711yescharacteristic
379C4⋊Q8—9432no—
380C22⋊Q8—9464no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—28no—
383C2×C4⋊C4—24no—
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—944no—
394C2×C4⋊C4—948no—
395C2×C4⋊C4—944no—
396C2×C4⋊C4C22×D4718yes—
397C2×C4⋊C4C22×D4714yes—
398C2×C4⋊C4C22×D4714yes—
399C2×C22⋊C4—478no—
400C2×C22⋊C4—478no—
401C2×C22⋊C4—948no—
402C2×C22⋊C4—948no—
403C2×C42—24no—
404C2×C42—478no—
405C2×C42—474no—
406C2×C42—474no—
407C2×C42—474no—
408C2×C42—9432no—
409C2×C42—944no—
410C2×C42—944no—
411C2×C42C22×D4714yes—
412#95—478no—
413#95—478no—
414#95—478no—
415#95C22×D4718yes—
416C24—474no—
417C24—474no—
418C24—474no—
419C24—472no—
420C24—942no—
421C24—942no—
422C24—942no—
423C24—942no—
424C24C23×D4711yescharacteristic
425C2×Q8—9416no—
426C2×Q8—9416no—
427C2×Q8—9416no—
428C2×Q8—18832no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—474no—
440C22×C4—474no—
441C22×C4—474no—
442C22×C4—474no—
443C22×C4—474no—
444C22×C4—474no—
445C22×C4—472no—
446C22×C4—472no—
447C22×C4—474no—
448C22×C4—472no—
449C22×C4—4716no—
450C22×C4—4716no—
451C22×C4—4716no—
452C22×C4—474no—
453C22×C4—474no—
454C22×C4—948no—
455C22×C4—948no—
456C22×C4—9432no—
457C22×C4—9432no—
458C22×C4—948no—
459C22×C4—948no—
460C22×C4—942no—
461C22×C4—942no—
462C22×C4—9432no—
463C22×C4—948no—
464C22×C4—948no—
465C22×C4—944no—
466C22×C4—944no—
467C22×C4—944no—
468C22×C4—944no—
469C22×C4—944no—
470C22×C4—944no—
471C22×C4#1062535014yes—
472C22×C4D4×D4714yes—
473C22×C4D4×D4714yes—
474C22×C4Q8×D4714yes—
475C22×C4Q8×D4714yes—
476C22×C4C23×D4714yes—
477C22×C4C23×D4714yes—
478C22×C4C23×D4712yes—
479C22×C4C23×D4712yes—
480C22×C4C23×D4712yes—
481C4⋊C4—216no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—9416no—
485C4⋊C4—9416no—
486C4⋊C4—9416no—
487C4⋊C4—9416no—
488C4⋊C4—948no—
489C4⋊C4—948no—
490C22⋊C4—9416no—
491C22⋊C4—9416no—
492C42—28no—
493C42—4716no—
494C42—948no—
495C42—948no—
496C23—4716no—
497C23—474no—
498C23—474no—
499C23—474no—
500C23—948no—
501C23—948no—
502C23—948no—
503C23—948no—
504C23—948no—
505C23—948no—
506C23—942no—
507C23—942no—
508C23—942no—
509C23—942no—
510C23C2×D4×D4712yes—
511C23C2×D4×D4712yes—
512C23C2×Q8×D4712yes—
513C23C2×Q8×D4712yes—
514C23#41×D4712yes—
515C23#41×D4712yes—
516C23C2×#1062535012yes—
517C23C24×D4711yescharacteristic
518Q8—18832no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—48no—
532C2×C4—48no—
533C2×C4—4716no—
534C2×C4—4716no—
535C2×C4—474no—
536C2×C4—474no—
537C2×C4—474no—
538C2×C4—474no—
539C2×C4—474no—
540C2×C4—474no—
541C2×C4—474no—
542C2×C4—9432no—
543C2×C4—9432no—
544C2×C4—948no—
545C2×C4—942no—
546C2×C4—942no—
547C2×C4—9432no—
548C2×C4—944no—
549C2×C4—944no—
550C2×C4—944no—
551C2×C4—944no—
552C2×C4—944no—
553C2×C4—944no—
554C2×C4—1888no—
555C2×C4—1888no—
556C2×C4#13094002116yescomplemented
557C2×C4C2×D4×D4714yes—
558C2×C4C2×D4×D4714yes—
559C2×C4C2×Q8×D4714yes—
560C2×C4C2×Q8×D4714yes—
561C2×C4C2×#1062535014yes—
562C22—478no—
563C22—9432no—
564C22—944no—
565C22—944no—
566C22—944no—
567C22—944no—
568C22—944no—
569C22—944no—
570C22#121×D4714yes—
571C22C22⋊Q8×D4714yes—
572C22C2×#1309400214yes—
573C22C4⋊Q8×D4714yes—
574C22#1568967014yes—
575C22#1568967714yes—
576C22#1568967914yes—
577C22C22×D4×D4711yescharacteristic
578C22C22×D4×D4711yescharacteristic
579C22C22×Q8×D4711yescharacteristic
580C22C22×Q8×D4711yescharacteristic
581C22C2×#41×D4711yescharacteristic
582C22C2×#41×D4711yescharacteristic
583C22C22×#1062535011yescharacteristic
584C4—24no—
585C4—24no—
586C4—44no—
587C4—44no—
588C4—478no—
589C4—478no—
590C4—944no—
591C4—1884no—
592C4—1884no—
593C4C2×#1309400218yescomplemented
594C2—948no—
595C2#2025173918yesdirect product factor
596C2C2×#121×D4711yescharacteristic
597C2C2×C22⋊Q8×D4711yescharacteristic
598C2C22×#1309400211yescharacteristic
599C2C2×C4⋊Q8×D4711yescharacteristic
600C2C2×#1568967011yescharacteristic
601C2C2×#1568967711yescharacteristic
602C2C2×#1568967911yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#389C2
socleC24×C47C24
prime radicalC24×D47C23
hypercenterC2×#389D47
hypocenterC47C2×#2857
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×#13094023C2semidirect
4. #20251739C2direct (N × H)
5. C2×#15689027C2semidirect
6. C2×#15689021C2semidirect
7. C2×#15689018C2semidirect
8. C2×#15688992C2semidirect
12. C2×C47×#389C2semidirect
18. C2×#13094023C22semidirect
34. #15689027C22semidirect
35. #15689021C22semidirect
36. #15689018C22semidirect
37. #15688992C22semidirect
48. C47×#389C22semidirect
67. C22×C4×D47Q8semidirect
68. C22×C4×D47Q8semidirect
173. C2×C4×D47C2×Q8semidirect
174. C2×C4×D47C2×Q8semidirect
212. C22×D47C4⋊Q8semidirect
230. C2×C4×C47C22⋊Q8semidirect
248. C2×Dic47C22⋊Q8semidirect
282. C2×D47C2×C4⋊Q8semidirect
287. C4×C47C2×C22⋊Q8semidirect
292. Dic47C2×C22⋊Q8semidirect
300. C2×#389D47semidirect
305. C2×C47#2857semidirect
342. #389C2×D47semidirect
351. C47C2×#2857semidirect
556. C2×C4#13094002semidirect
593. C4C2×#13094002semidirect
595. C2#20251739direct (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×#130940022
C22×C42×D472
#202517392
C2×#156890272
C2×#156890212
C2×#156890182
C2×#156889922
C2×#156889822
C2×#156889802
C2×C47×#3892
C2×#285747