Finite Group Databasebeta

← C2×#20246010 #26374897

Subgroups — C2×#20246010 #26374897

Order 12032.

3502 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093982C211yesmaximal, characteristic
2C22×#13093981C211yesmaximal, characteristic
3C22×C4⋊C4×D47C211yesmaximal, characteristic
4#20246010C2116yesmaximal, direct product factor
5C2×#15689022C211yesmaximal, characteristic, complemented
6C2×#15689021C211yesmaximal, characteristic, complemented
7C2×#15689019C211yesmaximal, characteristic, complemented
8C2×#15688997C212yesmaximal
9C2×#15688994C212yesmaximal
10C2×#15688992C212yesmaximal, complemented
11C2×#15688987C212yesmaximal, complemented
12C2×C47×#389C211yesmaximal, characteristic, complemented
13C23×C4×D47C2212yes—
14C23×C4×D47C2211yescharacteristic
15C2×#13093982C4116yescomplemented
16C2×#13093982C2218yes—
17C2×#13093981—28no—
18C2×#13093981C2218yes—
19C22×C4⋊Dic47C2211yescharacteristic
20C22×C4⋊Dic47C2212yes—
21C22×C4⋊Dic47C2211yescharacteristic
22C22×Dic47⋊C4C2211yescharacteristic
23C22×Dic47⋊C4C2212yes—
24C22×Dic47⋊C4C2211yescharacteristic
25C22×Dic47⋊C4C2212yes—
26C22×Dic47⋊C4C2212yes—
27C2×C4⋊C4×D47—28no—
28C2×C4⋊C4×D47C2218yes—
29C22×(C2×D47)⋊C4C2211yescharacteristic
30C22×(C2×D47)⋊C4C2211yescharacteristic
31C22×(C2×D47)⋊C4C2212yes—
32C22×C4×Dic47C2211yescharacteristic
33C22×C4×Dic47C2212yes—
34#15689022C2218yescomplemented
35#15689021C2218yescomplemented
36#15689019C2218yescomplemented
37#15688997C22116yes—
38#15688994C22116yes—
39#15688992C22116yescomplemented
40#15688987C22116yescomplemented
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
51C23×Dic47C2311yescharacteristic
52C23×Dic47C2312yes—
53C23×Dic47C2311yescharacteristic
54C23×Dic47C2312yes—
55C23×Dic47C2311yescharacteristic
56C22×C4×D47—28no—
57C22×C4×D47—44no—
58C22×C4×D47—44no—
59C22×C4×D47C2×C418yes—
60C22×C4×D47D418yes—
61C22×C4×D47Q818yes—
62C22×C4×D47C2318yes—
63C22×C4×D47C2314yes—
64#13093982C2×C4164yescomplemented
65#13093981—232no—
66C2×C4⋊Dic47—24no—
67C2×C4⋊Dic47—44no—
68C2×C4⋊Dic47C2×C418yescomplemented
69C2×C4⋊Dic47C2314yes—
70C2×C4⋊Dic47C2318yes—
71C2×C4⋊Dic47C2314yes—
72C2×Dic47⋊C4—28no—
73C2×Dic47⋊C4—24no—
74C2×Dic47⋊C4—24no—
75C2×Dic47⋊C4—44no—
76C2×Dic47⋊C4—44no—
77C2×Dic47⋊C4C2314yes—
78C2×Dic47⋊C4C2318yes—
79C2×Dic47⋊C4C2314yes—
80C2×Dic47⋊C4C2318yes—
81C2×Dic47⋊C4C2318yes—
82C4⋊C4×D47—232no—
83C2×(C2×D47)⋊C4—24no—
84C2×(C2×D47)⋊C4—24no—
85C2×(C2×D47)⋊C4C2×C4116yes—
86C2×(C2×D47)⋊C4C2314yes—
87C2×(C2×D47)⋊C4C2314yes—
88C2×(C2×D47)⋊C4C2318yes—
89C2×C4×Dic47—24no—
90C2×C4×Dic47C2×C4116yescomplemented
91C2×C4×Dic47C2314yes—
92C2×C4×Dic47C2318yes—
93C47⋊#95C2318yes—
94C47⋊#95C2318yes—
95C47⋊#95C2318yes—
96C23×C4×C47C2311yescharacteristic
97C23×C4×C47C2312yes—
98C23×C4×C47C2311yescharacteristic
99C23×C4×C47C2312yes—
100C23×C4×C47C2311yescharacteristic
101C2×C4⋊C4×C47—24no—
102C2×C4⋊C4×C47—44no—
103C2×C4⋊C4×C47C2×C418yescomplemented
104C2×C4⋊C4×C47C2314yes—
105C2×C4⋊C4×C47C2318yes—
106C2×C4⋊C4×C47C2314yes—
107C2×C42×C47—24no—
108C2×C42×C47C2314yes—
109#95×C47C2318yes—
110C24×C47C2411yescharacteristic
111C23×D47—22no—
112C23×D47—22no—
113C23×D47—22no—
114C23×D47—22no—
115C23×D47C22×C414yes—
116C23×D47C2×D414yes—
117C23×D47C2×Q814yes—
118C23×D47C2412yes—
119C22×C4×C47—24no—
120C22×C4×C47—24no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—24no—
124C22×C4×C47—22no—
125C22×C4×C47—24no—
126C22×C4×C47—22no—
127C22×C4×C47—24no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47C22×C418yes—
131C22×C4×C47C22×C414yes—
132C22×C4×C47C2×D414yes—
133C22×C4×C47C2×Q814yes—
134C22×C4×C47C2412yes—
135C22×C4×C47C2412yes—
136C22×C4×C47C2414yes—
137C22×C4×C47C2414yes—
138C22×C4×C47C2412yes—
139C22×Dic47—24no—
140C22×Dic47—24no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—24no—
144C22×Dic47—22no—
145C22×Dic47—24no—
146C22×Dic47—22no—
147C22×Dic47—24no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47C22×C418yes—
151C22×Dic47C22×C414yes—
152C22×Dic47C2×D414yes—
153C22×Dic47C2×Q814yes—
154C22×Dic47C2412yes—
155C22×Dic47C2414yes—
156C22×Dic47C2412yes—
157C22×Dic47C2414yes—
158C22×Dic47C2412yes—
159C2×C4×D47—216no—
160C2×C4×D47—48no—
161C2×C4×D47—48no—
162C2×C4×D47—432no—
163C2×C4×D47C4⋊C4164yescomplemented
164C2×C4×D47C22×C4116yes—
165C2×C4×D47C2×D4116yes—
166C2×C4×D47C2×Q8116yes—
167C4⋊C4×C47—28no—
168C4⋊C4×C47—48no—
169C4⋊C4×C47C22×C4116yescomplemented
170C42×C47—28no—
171(C2×D47)⋊C4—28no—
172(C2×D47)⋊C4—28no—
173(C2×D47)⋊C4C22×C4132yes—
174C4⋊Dic47—28no—
175C4⋊Dic47—48no—
176C4⋊Dic47C22×C4116yescomplemented
177Dic47⋊C4—216no—
178Dic47⋊C4—28no—
179Dic47⋊C4—28no—
180Dic47⋊C4—48no—
181Dic47⋊C4—48no—
182C4×Dic47—28no—
183C4×Dic47C22×C4132yescomplemented
184C23×C47C23×C412yes—
185C23×C47C22×D412yes—
186C23×C47C22×Q812yes—
187C23×C47H(5,𝔽2)12yes—
188C23×C47H(5,𝔽2)12yes—
189C23×C472−1+412yes—
190C23×C472−1+412yes—
191C23×C47C2511yescharacteristic
192C22×D47—28no—
193C22×D47—28no—
194C22×D47—28no—
195C22×D47—28no—
196C22×D47—28no—
197C22×D47—28no—
198C22×D47—22no—
199C22×D47—22no—
200C22×D47—22no—
201C22×D47—22no—
202C22×D47C2×C4⋊C4116yes—
203C22×D47C23×C414yes—
204C22×D47C22×D414yes—
205C22×D47C22×Q814yes—
206C2×C4×C47—216no—
207C2×C4×C47—24no—
208C2×C4×C47—24no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—24no—
212C2×C4×C47—22no—
213C2×C4×C47—24no—
214C2×C4×C47—22no—
215C2×C4×C47—24no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—48no—
219C2×C4×C47—44no—
220C2×C4×C47—44no—
221C2×C4×C47C2×C4⋊C4116yes—
222C2×C4×C47C23×C418yes—
223C2×C4×C47C23×C414yes—
224C2×C4×C47C22×D414yes—
225C2×C4×C47C22×Q814yes—
226C2×Dic47—216no—
227C2×Dic47—24no—
228C2×Dic47—24no—
229C2×Dic47—22no—
230C2×Dic47—22no—
231C2×Dic47—24no—
232C2×Dic47—22no—
233C2×Dic47—24no—
234C2×Dic47—22no—
235C2×Dic47—24no—
236C2×Dic47—22no—
237C2×Dic47—22no—
238C2×Dic47—48no—
239C2×Dic47—44no—
240C2×Dic47—44no—
241C2×Dic47C2×C4⋊C4116yes—
242C2×Dic47C23×C418yes—
243C2×Dic47C23×C414yes—
244C2×Dic47C22×D414yes—
245C2×Dic47C22×Q814yes—
246C4×D47—432no—
247C4×D47C2×C4⋊C4164yescomplemented
248C2×#2317—471nomaximal, Sylow
249C22×C47C22×C4⋊C414yes—
250C22×C47#52014yes—
251C22×C47#52014yes—
252C22×C47#53514yes—
253C22×C47#53614yes—
254C22×C47#54414yes—
255C22×C47#54414yes—
256C22×C47C24×C411yescharacteristic
257C22×C47C23×D411yescharacteristic
258C22×C47C23×Q811yescharacteristic
259C22×C47C2×H(5,𝔽2)11yescharacteristic
260C22×C47C2×H(5,𝔽2)11yescharacteristic
261C22×C47C2×2−1+411yescharacteristic
262C22×C47C2×2−1+411yescharacteristic
263C2×D47—232no—
264C2×D47—24no—
265C2×D47—24no—
266C2×D47—24no—
267C2×D47—24no—
268C2×D47—24no—
269C2×D47—24no—
270C2×D47C22×C4⋊C418yes—
271C4×C47—28no—
272C4×C47—44no—
273C4×C47—42no—
274C4×C47—42no—
275C4×C47C22×C4⋊C418yes—
276Dic47—28no—
277Dic47—44no—
278Dic47—42no—
279Dic47—42no—
280Dic47C22×C4⋊C418yes—
281C22×#117—471no—
282C22×#117—471no—
283C23×C4⋊C4—471no—
284#2317—4716no—
285C2×#389—471no—
286C2×#389—471no—
287C2×#389—471no—
288C2×#389D4711yescharacteristic, complemented
289C2×#385—472no—
290C2×#385—472no—
291C2×#384—472no—
292C2×#382—472no—
293C2×C47#231718yescomplemented
294C2×C47C23×C4⋊C411yescharacteristic
295C2×C47C2×#52011yescharacteristic
296C2×C47C2×#52011yescharacteristic
297C2×C47C2×#53511yescharacteristic
298C2×C47C2×#53611yescharacteristic
299C2×C47C2×#54411yescharacteristic
300C2×C47C2×#54411yescharacteristic
301D47—28no—
302C24×C4—471no—
303C24×C4—472no—
304C2×#117—4716no—
305C2×#117—478no—
306C2×#117—478no—
307C2×#117—948no—
308C22×C4⋊C4—471no—
309C22×C4⋊C4—472no—
310C22×C4⋊C4—471no—
311C22×C4⋊C4—471no—
312C22×C4⋊C4—471no—
313C22×C4⋊C4—472no—
314C22×C4⋊C4—472no—
315C22×C4⋊C4—472no—
316C22×C4⋊C4—478no—
317C22×C4⋊C4—948no—
318C22×C4⋊C4C2×D4711yescharacteristic
319C22×C4⋊C4C2×D4712yes—
320C22×C4⋊C4C2×D4711yescharacteristic
321C22×C22⋊C4—472no—
322C22×C22⋊C4—471no—
323C22×C22⋊C4—471no—
324C22×C42—472no—
325C22×C42—471no—
326C22×C42C2×D4711yescharacteristic
327#389—478no—
328#389—478no—
329#389—478no—
330#389C2×D4718yescomplemented
331#385—4716no—
332#385—4716no—
333#384—4716no—
334#382—4716no—
335C2×#95—472no—
336C2×#95—472no—
337C2×#95—472no—
338C2×#95C2×D4712yes—
339C47C2×#231711yescharacteristic, complemented, Sylow
340C25—471no—
341C23×C4—471no—
342C23×C4—478no—
343C23×C4—478no—
344C23×C4—478no—
345C23×C4—474no—
346C23×C4—472no—
347C23×C4—471no—
348C23×C4—471no—
349C23×C4—472no—
350C23×C4—478no—
351C23×C4—948no—
352C23×C4—1884no—
353C23×C4—1884no—
354C23×C4C22×D4711yescharacteristic
355C23×C4C22×D4711yescharacteristic
356C23×C4C22×D4712yes—
357C23×C4C22×D4712yes—
358C23×C4C22×D4711yescharacteristic
359#117—4764no—
360#117—9432no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4—44no—
363C2×C4⋊C4—478no—
364C2×C4⋊C4—474no—
365C2×C4⋊C4—478no—
366C2×C4⋊C4—474no—
367C2×C4⋊C4—474no—
368C2×C4⋊C4—474no—
369C2×C4⋊C4—478no—
370C2×C4⋊C4—478no—
371C2×C4⋊C4—478no—
372C2×C4⋊C4—9432no—
373C2×C4⋊C4—948no—
374C2×C4⋊C4—944no—
375C2×C4⋊C4—944no—
376C2×C4⋊C4—944no—
377C2×C4⋊C4—1884no—
378C2×C4⋊C4—1884no—
379C2×C4⋊C4—1884no—
380C2×C4⋊C4C4×D4718yescomplemented
381C2×C4⋊C4C22×D4714yes—
382C2×C4⋊C4C22×D4718yes—
383C2×C4⋊C4C22×D4714yes—
384C2×C22⋊C4—4716no—
385C2×C22⋊C4—478no—
386C2×C22⋊C4—474no—
387C2×C22⋊C4—474no—
388C2×C22⋊C4—944no—
389C2×C22⋊C4—944no—
390C2×C42—24no—
391C2×C42—4716no—
392C2×C42—478no—
393C2×C42—474no—
394C2×C42—944no—
395C2×C42C22×D4714yes—
396#95—478no—
397#95—478no—
398#95—478no—
399#95C22×D4718yes—
400C24—474no—
401C24—474no—
402C24—474no—
403C24—472no—
404C24—942no—
405C24—942no—
406C24—942no—
407C24—942no—
408C24C23×D4711yescharacteristic
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—474no—
421C22×C4—474no—
422C22×C4—474no—
423C22×C4—472no—
424C22×C4—4764no—
425C22×C4—4716no—
426C22×C4—4716no—
427C22×C4—4716no—
428C22×C4—478no—
429C22×C4—474no—
430C22×C4—472no—
431C22×C4—474no—
432C22×C4—472no—
433C22×C4—9416no—
434C22×C4—944no—
435C22×C4—944no—
436C22×C4—942no—
437C22×C4—942no—
438C22×C4—944no—
439C22×C4—942no—
440C22×C4—944no—
441C22×C4—942no—
442C22×C4—944no—
443C22×C4—942no—
444C22×C4—942no—
445C22×C4—1888no—
446C22×C4—1888no—
447C22×C4—18832no—
448C22×C4C2×C4×D4718yes—
449C22×C4C2×C4×D4714yes—
450C22×C4D4×D4714yes—
451C22×C4Q8×D4714yes—
452C22×C4C23×D4712yes—
453C22×C4C23×D4714yes—
454C22×C4C23×D4712yes—
455C22×C4C23×D4714yes—
456C22×C4C23×D4712yes—
457C4⋊C4—28no—
458C4⋊C4—48no—
459C4⋊C4—4716no—
460C4⋊C4—9416no—
461C4⋊C4—948no—
462C4⋊C4—948no—
463C4⋊C4—948no—
464C4⋊C4—1888no—
465C4⋊C4—1888no—
466C4⋊C4—1888no—
467C4⋊C4C2×C4×D47116yescomplemented
468C22⋊C4—4732no—
469C22⋊C4—948no—
470C22⋊C4—948no—
471C42—28no—
472C42—4732no—
473C42—948no—
474C23—4716no—
475C23—474no—
476C23—474no—
477C23—474no—
478C23—948no—
479C23—948no—
480C23—948no—
481C23—948no—
482C23—948no—
483C23—948no—
484C23—942no—
485C23—942no—
486C23—942no—
487C23—942no—
488C23C22×C4×D4712yes—
489C23C2×D4×D4712yes—
490C23C2×Q8×D4712yes—
491C23#1309406912yes—
492C23#1309407012yes—
493C23#1309407112yes—
494C23#1309407212yes—
495C23C24×D4711yescharacteristic
496C2×C4—216no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—44no—
509C2×C4—48no—
510C2×C4—44no—
511C2×C4—4716no—
512C2×C4—474no—
513C2×C4—474no—
514C2×C4—474no—
515C2×C4—4764no—
516C2×C4—478no—
517C2×C4—9416no—
518C2×C4—944no—
519C2×C4—944no—
520C2×C4—942no—
521C2×C4—942no—
522C2×C4—944no—
523C2×C4—942no—
524C2×C4—944no—
525C2×C4—942no—
526C2×C4—944no—
527C2×C4—942no—
528C2×C4—942no—
529C2×C4—18832no—
530C2×C4—1884no—
531C2×C4—1888no—
532C2×C4—1884no—
533C2×C4C4⋊C4×D47116yes—
534C2×C4C22×C4×D4718yes—
535C2×C4C22×C4×D4714yes—
536C2×C4C2×D4×D4714yes—
537C2×C4C2×Q8×D4714yes—
538C22—478no—
539C22—944no—
540C22—944no—
541C22—944no—
542C22—944no—
543C22—944no—
544C22—944no—
545C22—9432no—
546C22C2×C4⋊C4×D4714yes—
547C22#1568960214yes—
548C22#1568960514yes—
549C22#1568968914yes—
550C22#1568969614yes—
551C22#1568975014yes—
552C22#1568975314yes—
553C22C23×C4×D4711yescharacteristic
554C22C22×D4×D4711yescharacteristic
555C22C22×Q8×D4711yescharacteristic
556C22C2×#1309406911yescharacteristic
557C22C2×#1309407011yescharacteristic
558C22C2×#1309407111yescharacteristic
559C22C2×#1309407211yescharacteristic
560C4—28no—
561C4—44no—
562C4—42no—
563C4—42no—
564C4—478no—
565C4—948no—
566C4—1884no—
567C4—1882no—
568C4—1882no—
569C4C2×C4⋊C4×D4718yes—
570C2—948no—
571C2#2024601018yesdirect product factor
572C2C22×C4⋊C4×D4711yescharacteristic
573C2C2×#1568960211yescharacteristic
574C2C2×#1568960511yescharacteristic
575C2C2×#1568968911yescharacteristic
576C2C2×#1568969611yescharacteristic
577C2C2×#1568975011yescharacteristic
578C2C2×#1568975311yescharacteristic
subgroupG/H
centerC24C23×D47
commutatorC22×C47C24×C4
FrattiniC23C24×D47
FittingC2×C47×#389C2
socleC24×C47C24
prime radicalC24×D47C23
cosocleC23×C47C25
hypercenterC2×#389D47
hypocenterC47C2×#2317
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
4. #20246010C2direct (N × H)
5. C2×#15689022C2semidirect
6. C2×#15689021C2semidirect
7. C2×#15689019C2semidirect
10. C2×#15688992C2semidirect
11. C2×#15688987C2semidirect
12. C2×C47×#389C2semidirect
15. C2×#13093982C4semidirect
34. #15689022C22semidirect
35. #15689021C22semidirect
36. #15689019C22semidirect
39. #15688992C22semidirect
40. #15688987C22semidirect
48. C47×#389C22semidirect
64. #13093982C2×C4semidirect
68. C2×C4⋊Dic47C2×C4semidirect
90. C2×C4×Dic47C2×C4semidirect
103. C2×C4⋊C4×C47C2×C4semidirect
163. C2×C4×D47C4⋊C4semidirect
169. C4⋊C4×C47C22×C4semidirect
176. C4⋊Dic47C22×C4semidirect
183. C4×Dic47C22×C4semidirect
247. C4×D47C2×C4⋊C4semidirect
288. C2×#389D47semidirect
293. C2×C47#2317semidirect
330. #389C2×D47semidirect
339. C47C2×#2317semidirect
380. C2×C4⋊C4C4×D47semidirect
467. C4⋊C4C2×C4×D47semidirect
571. C2#20246010direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939822
C22×#130939812
C22×C4⋊C4×D472
#202460102
C2×#156890222
C2×#156890212
C2×#156890192
C2×#156889972
C2×#156889942
C2×#156889922
C2×#156889872
C2×C47×#3892
C2×#231747