Finite Group Databasebeta

← C2×#20246297 #26375573

Subgroups — C2×#20246297 #26375573

Order 12032.

4198 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D188C211yesmaximal, characteristic
2C22×#13093986C211yesmaximal, characteristic
3C22×#13093986C211yesmaximal, characteristic
4#20246297C2116yesmaximal, direct product factor
5C2×#15689022C211yesmaximal, characteristic, complemented
6C2×#15688997C212yesmaximal, complemented
7C2×#15688994C212yesmaximal, complemented
8C2×#15688988C211yesmaximal, characteristic, complemented
9C2×#15688980C212yesmaximal, complemented
10C2×C4×Dic47⋊C4C211yesmaximal, characteristic, complemented
11C2×#95×D47C212yesmaximal, complemented
12C2×C47×#389C211yesmaximal, characteristic, complemented
13C23×D188C2211yescharacteristic
14C23×C4×D47C2212yes—
15C23×C4×D47C2212yes—
16C23×C4×D47C2212yes—
17C2×C4×D188—28no—
18C2×C4×D188C2218yes—
19C2×#13093986—28no—
20C2×#13093986C4116yescomplemented
21C2×#13093986C2218yes—
22C2×#13093986C2218yes—
23C22×C4⋊Dic47C2211yescharacteristic
24C22×C4⋊Dic47C2212yes—
25C22×Dic47⋊C4C2211yescharacteristic
26C22×Dic47⋊C4C2211yescharacteristic
27C22×(C2×D47)⋊C4C2212yes—
28C22×(C2×D47)⋊C4C2212yes—
29C22×(C2×D47)⋊C4C2212yes—
30C22×C4×Dic47C2211yescharacteristic
31C22×C4×Dic47C2211yescharacteristic
32#15689022C2218yescomplemented
33#15688997C22116yescomplemented
34#15688994C22116yescomplemented
35#15688988C2218yescomplemented
36#15688980C22116yescomplemented
37C4×Dic47⋊C4C2218yescomplemented
38C2×C47⋊#95C2212yes—
39C2×C47⋊#95C2212yes—
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2211yescharacteristic
42C2×C47⋊#95C2211yescharacteristic
43#95×D47C22116yescomplemented
44C22×C4⋊C4×C47C2211yescharacteristic
45C22×C4⋊C4×C47C2212yes—
46C22×C4⋊C4×C47C2211yescharacteristic
47C22×C42×C47C2211yescharacteristic
48C47×#389C2218yescomplemented
49C2×#95×C47C2212yes—
50C24×D47C2312yes—
51C22×D188C2×C418yes—
52C22×D188D418yes—
53C22×D188Q818yes—
54C22×D188C2314yes—
55C23×Dic47C2312yes—
56C23×Dic47C2311yescharacteristic
57C23×Dic47C2311yescharacteristic
58C23×Dic47C2311yescharacteristic
59C23×Dic47C2311yescharacteristic
60C22×C4×D47—28no—
61C22×C4×D47—28no—
62C22×C4×D47—28no—
63C22×C4×D47—28no—
64C22×C4×D47—44no—
65C22×C4×D47—44no—
66C22×C4×D47—44no—
67C22×C4×D47—44no—
68C22×C4×D47C2×C4116yescomplemented
69C22×C4×D47C2318yes—
70C22×C4×D47C2318yes—
71C22×C4×D47C2318yes—
72C4×D188—232no—
73#13093986—232no—
74#13093986C2×C4164yescomplemented
75C2×C4⋊Dic47—24no—
76C2×C4⋊Dic47—44no—
77C2×C4⋊Dic47C2314yes—
78C2×C4⋊Dic47C2318yes—
79C2×Dic47⋊C4—24no—
80C2×Dic47⋊C4—24no—
81C2×Dic47⋊C4C2314yes—
82C2×Dic47⋊C4C2314yes—
83C2×(C2×D47)⋊C4—28no—
84C2×(C2×D47)⋊C4—28no—
85C2×(C2×D47)⋊C4C2×C4116yescomplemented
86C2×(C2×D47)⋊C4C2318yes—
87C2×(C2×D47)⋊C4C2318yes—
88C2×(C2×D47)⋊C4C2318yes—
89C2×C4×Dic47—24no—
90C2×C4×Dic47C2×C418yescomplemented
91C2×C4×Dic47C2314yes—
92C2×C4×Dic47C2314yes—
93C47⋊#95C2314yes—
94C47⋊#95C2318yes—
95C47⋊#95C2318yes—
96C47⋊#95C2314yes—
97C47⋊#95C2318yes—
98C23×C4×C47C2312yes—
99C23×C4×C47C2311yescharacteristic
100C23×C4×C47C2312yes—
101C23×C4×C47C2311yescharacteristic
102C23×C4×C47C2311yescharacteristic
103C2×C4⋊C4×C47—24no—
104C2×C4⋊C4×C47—44no—
105C2×C4⋊C4×C47C2×C418yescomplemented
106C2×C4⋊C4×C47C2314yes—
107C2×C4⋊C4×C47C2318yes—
108C2×C4⋊C4×C47C2314yes—
109C2×C42×C47—24no—
110C2×C42×C47C2314yes—
111#95×C47C2318yes—
112C24×C47C2411yescharacteristic
113C23×D47—24no—
114C23×D47—24no—
115C23×D47—24no—
116C23×D47—24no—
117C23×D47C22×C418yes—
118C23×D47C2×D418yes—
119C23×D47C2×Q818yes—
120C23×D47C2414yes—
121C22×C4×C47—22no—
122C22×C4×C47—24no—
123C22×C4×C47—24no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—24no—
127C22×C4×C47—22no—
128C22×C4×C47—24no—
129C22×C4×C47—22no—
130C22×C4×C47—24no—
131C22×C4×C47C22×C418yes—
132C22×C4×C47C22×C414yes—
133C22×C4×C47C2×D414yes—
134C22×C4×C47C2×Q814yes—
135C22×C4×C47#4114yes—
136C22×C4×C47C2412yes—
137C22×C4×C47C2414yes—
138C22×C4×C47C2414yes—
139C22×C4×C47C2412yes—
140C22×C4×C47C2412yes—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—24no—
149C22×Dic47—22no—
150C22×Dic47—24no—
151C22×Dic47—22no—
152C22×Dic47—24no—
153C22×Dic47C22×C414yes—
154C22×Dic47C22×C414yes—
155C22×Dic47#4114yes—
156C22×Dic47C2414yes—
157C22×Dic47C2412yes—
158C22×Dic47C2412yes—
159C22×Dic47C2412yes—
160C22×Dic47C2412yes—
161C2×D188C4⋊C4164yescomplemented
162C2×D188C22×C4116yes—
163C2×D188C2×D4116yes—
164C2×D188C2×Q8116yes—
165C2×C4×D47—216no—
166C2×C4×D47—216no—
167C2×C4×D47—216no—
168C2×C4×D47—216no—
169C2×C4×D47—432no—
170C2×C4×D47—48no—
171C2×C4×D47—48no—
172C2×C4×D47—48no—
173C2×C4×D47—48no—
174C2×C4×D47—432no—
175C2×C4×D47—432no—
176C2×C4×D47C22×C4132yescomplemented
177C4⋊C4×C47—28no—
178C4⋊C4×C47—48no—
179C4⋊C4×C47C22×C4116yescomplemented
180C42×C47—28no—
181(C2×D47)⋊C4—216no—
182(C2×D47)⋊C4—216no—
183(C2×D47)⋊C4C22×C4132yescomplemented
184C4⋊Dic47—28no—
185C4⋊Dic47—48no—
186Dic47⋊C4—28no—
187Dic47⋊C4—28no—
188C4×Dic47—28no—
189C4×Dic47C22×C4116yescomplemented
190C23×C47C23×C412yes—
191C23×C47C22×D412yes—
192C23×C47C22×Q812yes—
193C23×C47C2×#4112yes—
194C23×C47C2×#4112yes—
195C23×C47H(5,𝔽2)12yes—
196C23×C47H(5,𝔽2)12yes—
197C23×C47C2511yescharacteristic
198C22×D47—216no—
199C22×D47—216no—
200C22×D47—216no—
201C22×D47—216no—
202C22×D47—216no—
203C22×D47—216no—
204C22×D47—24no—
205C22×D47—24no—
206C22×D47—24no—
207C22×D47—24no—
208C22×D47C2×C4⋊C4132yes—
209C22×D47C23×C418yes—
210C22×D47C22×D418yes—
211C22×D47C22×Q818yes—
212C2×C4×C47—216no—
213C2×C4×C47—28no—
214C2×C4×C47—22no—
215C2×C4×C47—24no—
216C2×C4×C47—24no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—24no—
220C2×C4×C47—22no—
221C2×C4×C47—24no—
222C2×C4×C47—22no—
223C2×C4×C47—24no—
224C2×C4×C47—48no—
225C2×C4×C47—44no—
226C2×C4×C47C2×C4⋊C4116yes—
227C2×C4×C47C23×C418yes—
228C2×C4×C47C23×C414yes—
229C2×C4×C47C22×D414yes—
230C2×C4×C47C22×Q814yes—
231C2×C4×C47C2×#4114yes—
232C2×Dic47—28no—
233C2×Dic47—28no—
234C2×Dic47—28no—
235C2×Dic47—22no—
236C2×Dic47—22no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—22no—
240C2×Dic47—22no—
241C2×Dic47—22no—
242C2×Dic47—24no—
243C2×Dic47—22no—
244C2×Dic47—24no—
245C2×Dic47—22no—
246C2×Dic47—24no—
247C2×Dic47—48no—
248C2×Dic47—44no—
249C2×Dic47C23×C414yes—
250C2×Dic47C23×C414yes—
251C2×Dic47C2×#4114yes—
252D188C2×C4⋊C4164yescomplemented
253C4×D47—432no—
254C4×D47—432no—
255C4×D47—432no—
256C2×#2349—471nomaximal, Sylow
257C22×C47C22×C4⋊C414yes—
258C22×C47C4×#4114yes—
259C22×C47#51814yes—
260C22×C47#54614yes—
261C22×C47#54614yes—
262C22×C47#55414yes—
263C22×C47#55414yes—
264C22×C47C24×C411yescharacteristic
265C22×C47C23×D411yescharacteristic
266C22×C47C23×Q811yescharacteristic
267C22×C47C22×#4111yescharacteristic
268C22×C47C22×#4111yescharacteristic
269C22×C47C2×H(5,𝔽2)11yescharacteristic
270C22×C47C2×H(5,𝔽2)11yescharacteristic
271C2×D47—264no—
272C2×D47—28no—
273C2×D47—28no—
274C2×D47—28no—
275C2×D47—28no—
276C2×D47—28no—
277C2×D47—28no—
278C2×D47C22×C4⋊C4116yes—
279C4×C47—28no—
280C4×C47—24no—
281C4×C47—44no—
282C4×C47—42no—
283C4×C47C22×C4⋊C418yes—
284Dic47—24no—
285Dic47—24no—
286Dic47—24no—
287Dic47—44no—
288Dic47—42no—
289C22×C4×D4—471no—
290C22×C4×D4—471no—
291C22×C4×D4—471no—
292#2349—4716no—
293C2×#389—471no—
294C2×#389D4711yescharacteristic, complemented
295C2×#385—472no—
296C2×#385—472no—
297C2×#382—471no—
298C2×#380—472no—
299C2×C4×C4⋊C4—471no—
300C22×#95—472no—
301C2×C47#234918yescomplemented
302C2×C47C23×C4⋊C411yescharacteristic
303C2×C47C2×C4×#4111yescharacteristic
304C2×C47C2×#51811yescharacteristic
305C2×C47C2×#54611yescharacteristic
306C2×C47C2×#54611yescharacteristic
307C2×C47C2×#55411yescharacteristic
308C2×C47C2×#55411yescharacteristic
309D47—216no—
310C23×D4—471no—
311C24×C4—472no—
312C24×C4—472no—
313C24×C4—472no—
314C2×C4×D4—4716no—
315C2×C4×D4—478no—
316C2×C4×D4—478no—
317C2×C4×D4—478no—
318C2×C4×D4—948no—
319C2×C4×D4—948no—
320C22×C4⋊C4—471no—
321C22×C4⋊C4—471no—
322C22×C4⋊C4—471no—
323C22×C4⋊C4—472no—
324C22×C4⋊C4C2×D4711yescharacteristic
325C22×C4⋊C4C2×D4712yes—
326C22×C4⋊C4C2×D4711yescharacteristic
327C22×C22⋊C4—472no—
328C22×C22⋊C4—472no—
329C22×C22⋊C4—472no—
330C22×C42—471no—
331C22×C42—471no—
332C22×C42C2×D4711yescharacteristic
333#389—478no—
334#389C2×D4718yescomplemented
335#385—4716no—
336#385—4716no—
337#382—478no—
338#380—4716no—
339C4×C4⋊C4—478no—
340C2×#95—471no—
341C2×#95—472no—
342C2×#95—472no—
343C2×#95—471no—
344C2×#95—472no—
345C2×#95—4716no—
346C2×#95C2×D4712yes—
347C47C2×#234911yescharacteristic, complemented, Sylow
348C25—472no—
349C22×D4—478no—
350C22×D4—478no—
351C22×D4—478no—
352C22×D4—474no—
353C23×C4—471no—
354C23×C4—471no—
355C23×C4—471no—
356C23×C4—4716no—
357C23×C4—478no—
358C23×C4—471no—
359C23×C4—472no—
360C23×C4—478no—
361C23×C4—478no—
362C23×C4—948no—
363C23×C4—948no—
364C23×C4—948no—
365C23×C4—948no—
366C23×C4—1884no—
367C23×C4—1884no—
368C23×C4—1884no—
369C23×C4—1884no—
370C23×C4C22×D4711yescharacteristic
371C23×C4C22×D4712yes—
372C23×C4C22×D4712yes—
373C23×C4C22×D4711yescharacteristic
374C23×C4C22×D4711yescharacteristic
375C4×D4—4764no—
376C4×D4—9432no—
377C4×D4—9432no—
378C2×C4⋊C4—24no—
379C2×C4⋊C4—44no—
380C2×C4⋊C4—474no—
381C2×C4⋊C4—474no—
382C2×C4⋊C4—474no—
383C2×C4⋊C4—478no—
384C2×C4⋊C4—944no—
385C2×C4⋊C4—944no—
386C2×C4⋊C4—944no—
387C2×C4⋊C4—1884no—
388C2×C4⋊C4C4×D4718yescomplemented
389C2×C4⋊C4C22×D4714yes—
390C2×C4⋊C4C22×D4718yes—
391C2×C4⋊C4C22×D4714yes—
392C2×C22⋊C4—4716no—
393C2×C22⋊C4—478no—
394C2×C22⋊C4—478no—
395C2×C22⋊C4—478no—
396C2×C22⋊C4—948no—
397C2×C22⋊C4—948no—
398C2×C42—24no—
399C2×C42—478no—
400C2×C42—474no—
401C2×C42—474no—
402C2×C42—944no—
403C2×C42C22×D4714yes—
404#95—474no—
405#95—478no—
406#95—478no—
407#95—474no—
408#95—478no—
409#95C22×D4718yes—
410C24—478no—
411C24—478no—
412C24—478no—
413C24—474no—
414C24—944no—
415C24—944no—
416C24—944no—
417C24—944no—
418C24C23×D4711yescharacteristic
419C2×D4—4764no—
420C2×D4—4716no—
421C2×D4—4716no—
422C2×D4—4716no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—474no—
434C22×C4—474no—
435C22×C4—474no—
436C22×C4—472no—
437C22×C4—472no—
438C22×C4—472no—
439C22×C4—4732no—
440C22×C4—474no—
441C22×C4—472no—
442C22×C4—9416no—
443C22×C4—9416no—
444C22×C4—9416no—
445C22×C4—942no—
446C22×C4—942no—
447C22×C4—942no—
448C22×C4—942no—
449C22×C4—942no—
450C22×C4—942no—
451C22×C4—942no—
452C22×C4—944no—
453C22×C4—942no—
454C22×C4—944no—
455C22×C4—942no—
456C22×C4—944no—
457C22×C4—9416no—
458C22×C4—18832no—
459C22×C4—1888no—
460C22×C4—1888no—
461C22×C4—1888no—
462C22×C4—1888no—
463C22×C4—18832no—
464C22×C4—18832no—
465C22×C4C2×C4×D4718yes—
466C22×C4C2×C4×D4714yes—
467C22×C4D4×D4714yes—
468C22×C4Q8×D4714yes—
469C22×C4#1062535414yes—
470C22×C4C23×D4714yes—
471C22×C4C23×D4712yes—
472C22×C4C23×D4714yes—
473C22×C4C23×D4712yes—
474C22×C4C23×D4712yes—
475C4⋊C4—28no—
476C4⋊C4—48no—
477C4⋊C4—948no—
478C4⋊C4—948no—
479C4⋊C4—948no—
480C4⋊C4—1888no—
481C4⋊C4C2×C4×D47116yescomplemented
482C22⋊C4—4732no—
483C22⋊C4—9416no—
484C22⋊C4—9416no—
485C42—28no—
486C42—4716no—
487C42—948no—
488C23—4732no—
489C23—478no—
490C23—478no—
491C23—478no—
492C23—9416no—
493C23—9416no—
494C23—9416no—
495C23—9416no—
496C23—9416no—
497C23—9416no—
498C23—944no—
499C23—944no—
500C23—944no—
501C23—944no—
502C23C22×C4×D4712yes—
503C23C2×D4×D4712yes—
504C23C2×Q8×D4712yes—
505C23#41×D4712yes—
506C23C2×#1062535412yes—
507C23#1309406912yes—
508C23#1309407012yes—
509C23C24×D4711yescharacteristic
510D4—4764no—
511C2×C4—28no—
512C2×C4—216no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—44no—
524C2×C4—48no—
525C2×C4—474no—
526C2×C4—474no—
527C2×C4—474no—
528C2×C4—948no—
529C2×C4—948no—
530C2×C4—948no—
531C2×C4—942no—
532C2×C4—942no—
533C2×C4—942no—
534C2×C4—942no—
535C2×C4—942no—
536C2×C4—942no—
537C2×C4—942no—
538C2×C4—944no—
539C2×C4—942no—
540C2×C4—944no—
541C2×C4—942no—
542C2×C4—944no—
543C2×C4—18832no—
544C2×C4—18832no—
545C2×C4—18832no—
546C2×C4—1884no—
547C2×C4—1888no—
548C2×C4C4⋊C4×D47116yes—
549C2×C4C22×C4×D4718yes—
550C2×C4C22×C4×D4714yes—
551C2×C4C2×D4×D4714yes—
552C2×C4C2×Q8×D4714yes—
553C2×C4C2×#1062535414yes—
554C22—4716no—
555C22—9464no—
556C22—948no—
557C22—948no—
558C22—948no—
559C22—948no—
560C22—948no—
561C22—948no—
562C22C2×C4⋊C4×D4714yes—
563C22C4×#1062535414yes—
564C22#1568959514yes—
565C22#1568976414yes—
566C22#1568977014yes—
567C22#1568982914yes—
568C22#1568983414yes—
569C22C23×C4×D4711yescharacteristic
570C22C22×D4×D4711yescharacteristic
571C22C22×Q8×D4711yescharacteristic
572C22C2×#41×D4711yescharacteristic
573C22C22×#1062535411yescharacteristic
574C22C2×#1309406911yescharacteristic
575C22C2×#1309407011yescharacteristic
576C4—28no—
577C4—24no—
578C4—44no—
579C4—42no—
580C4—944no—
581C4—944no—
582C4—944no—
583C4—1884no—
584C4—1882no—
585C4C2×C4⋊C4×D4718yes—
586C2—9416no—
587C2#2024629718yesdirect product factor
588C2C22×C4⋊C4×D4711yescharacteristic
589C2C2×C4×#1062535411yescharacteristic
590C2C2×#1568959511yescharacteristic
591C2C2×#1568976411yescharacteristic
592C2C2×#1568977011yescharacteristic
593C2C2×#1568982911yescharacteristic
594C2C2×#1568983411yescharacteristic
subgroupG/H
centerC24C23×D47
commutatorC22×C47C24×C4
FrattiniC23C24×D47
FittingC2×C47×#389C2
socleC24×C47C24
prime radicalC23×D188C22
cosocleC23×C47C25
hypercenterC2×#389D47
hypocenterC47C2×#2349
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
4. #20246297C2direct (N × H)
5. C2×#15689022C2semidirect
6. C2×#15688997C2semidirect
7. C2×#15688994C2semidirect
8. C2×#15688988C2semidirect
9. C2×#15688980C2semidirect
10. C2×C4×Dic47⋊C4C2semidirect
11. C2×#95×D47C2semidirect
12. C2×C47×#389C2semidirect
20. C2×#13093986C4semidirect
32. #15689022C22semidirect
33. #15688997C22semidirect
34. #15688994C22semidirect
35. #15688988C22semidirect
36. #15688980C22semidirect
37. C4×Dic47⋊C4C22semidirect
43. #95×D47C22semidirect
48. C47×#389C22semidirect
68. C22×C4×D47C2×C4semidirect
74. #13093986C2×C4semidirect
85. C2×(C2×D47)⋊C4C2×C4semidirect
90. C2×C4×Dic47C2×C4semidirect
105. C2×C4⋊C4×C47C2×C4semidirect
161. C2×D188C4⋊C4semidirect
176. C2×C4×D47C22×C4semidirect
179. C4⋊C4×C47C22×C4semidirect
183. (C2×D47)⋊C4C22×C4semidirect
189. C4×Dic47C22×C4semidirect
252. D188C2×C4⋊C4semidirect
294. C2×#389D47semidirect
301. C2×C47#2349semidirect
334. #389C2×D47semidirect
347. C47C2×#2349semidirect
388. C2×C4⋊C4C4×D47semidirect
481. C4⋊C4C2×C4×D47semidirect
587. C2#20246297direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D1882
C22×#130939862
#202462972
C2×#156890222
C2×#156889972
C2×#156889942
C2×#156889882
C2×#156889802
C2×C4×Dic47⋊C42
C2×#95×D472
C2×C47×#3892
C2×#234947