Finite Group Databasebeta

← C2×#20246333 #26375623

Subgroups — C2×#20246333 #26375623

Order 12032.

3678 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093986C211yesmaximal, characteristic
2C22×#13093981C212yesmaximal, complemented
3#20246333C2116yesmaximal, direct product factor
4C2×#15689025C211yesmaximal, characteristic
5C2×#15689024C211yesmaximal, characteristic
6C2×#15689014C212yesmaximal, complemented
7C2×#15689012C212yesmaximal, complemented
8C2×#15688992C211yesmaximal, characteristic, complemented
9C2×#15688982C212yesmaximal, complemented
10C2×C4⋊C4×Dic47C211yesmaximal, characteristic, complemented
11C2×C4×Dic47⋊C4C211yesmaximal, characteristic, complemented
12C2×C4×C4⋊C4×C47C211yesmaximal, characteristic, complemented
13C23×D188C2211yescharacteristic
14C23×C4×D47C2212yes—
15C2×#13093986C4116yescomplemented
16C2×#13093986C2218yes—
17C2×#13093981—216no—
18C2×#13093981C22116yescomplemented
19C22×C4⋊Dic47C2211yescharacteristic
20C22×Dic47⋊C4C2211yescharacteristic
21C22×Dic47⋊C4C2212yes—
22C22×Dic47⋊C4C2212yes—
23C22×Dic47⋊C4C2211yescharacteristic
24C22×(C2×D47)⋊C4C2212yes—
25C22×(C2×D47)⋊C4C2212yes—
26C22×(C2×D47)⋊C4C2212yes—
27C22×(C2×D47)⋊C4C2212yes—
28C22×(C2×D47)⋊C4C2212yes—
29C22×C4×Dic47C2212yes—
30C22×C4×Dic47C2211yescharacteristic
31C22×C4×Dic47C2211yescharacteristic
32#15689025C2218yes—
33#15689024C2218yes—
34#15689014C22116yescomplemented
35#15689012C22116yescomplemented
36#15688992C2218yescomplemented
37#15688982C22116yescomplemented
38C4⋊C4×Dic47C2218yescomplemented
39C4×Dic47⋊C4C2218yescomplemented
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2211yescharacteristic
42C2×C47⋊#95C2211yescharacteristic
43C22×C4⋊C4×C47C2211yescharacteristic
44C22×C4⋊C4×C47C2211yescharacteristic
45C22×C42×C47C2212yes—
46C22×C42×C47C2211yescharacteristic
47C4×C4⋊C4×C47C2218yescomplemented
48C2×#95×C47C2212yes—
49C24×D47C2312yes—
50C22×D188—24no—
51C22×D188—24no—
52C22×D188C2×C418yes—
53C22×D188C2314yes—
54C23×Dic47C2311yescharacteristic
55C23×Dic47C2312yes—
56C23×Dic47C2311yescharacteristic
57C23×Dic47C2311yescharacteristic
58C23×Dic47C2311yescharacteristic
59C22×C4×D47—28no—
60C22×C4×D47—28no—
61C22×C4×D47C2×C4116yes—
62C22×C4×D47C2318yes—
63#13093986C2×C4164yescomplemented
64#13093981—264no—
65C2×C4⋊Dic47—24no—
66C2×C4⋊Dic47C2314yes—
67C2×Dic47⋊C4—28no—
68C2×Dic47⋊C4—24no—
69C2×Dic47⋊C4—44no—
70C2×Dic47⋊C4D418yescomplemented
71C2×Dic47⋊C4C2314yes—
72C2×Dic47⋊C4C2318yes—
73C2×Dic47⋊C4C2318yes—
74C2×Dic47⋊C4C2314yes—
75C2×(C2×D47)⋊C4—28no—
76C2×(C2×D47)⋊C4—28no—
77C2×(C2×D47)⋊C4—44no—
78C2×(C2×D47)⋊C4—44no—
79C2×(C2×D47)⋊C4C2×C4116yes—
80C2×(C2×D47)⋊C4C2318yes—
81C2×(C2×D47)⋊C4C2318yes—
82C2×(C2×D47)⋊C4C2318yes—
83C2×(C2×D47)⋊C4C2318yes—
84C2×(C2×D47)⋊C4C2318yes—
85C2×C4×Dic47—24no—
86C2×C4×Dic47—28no—
87C2×C4×Dic47C2×C418yes—
88C2×C4×Dic47C2314yes—
89C2×C4×Dic47C2318yes—
90C2×C4×Dic47C2314yes—
91C47⋊#95C2314yes—
92C47⋊#95C2314yes—
93C47⋊#95C2318yes—
94C23×C4×C47C2311yescharacteristic
95C23×C4×C47C2312yes—
96C23×C4×C47C2312yes—
97C23×C4×C47C2311yescharacteristic
98C23×C4×C47C2311yescharacteristic
99C2×C4⋊C4×C47—24no—
100C2×C4⋊C4×C47C2×C418yes—
101C2×C4⋊C4×C47C2314yes—
102C2×C4⋊C4×C47C2314yes—
103C2×C42×C47—28no—
104C2×C42×C47—24no—
105C2×C42×C47C2314yes—
106C2×C42×C47C2318yes—
107#95×C47C2318yes—
108C24×C47C2411yescharacteristic
109C23×D47—24no—
110C23×D47—24no—
111C23×D47—24no—
112C23×D47—24no—
113C23×D47—24no—
114C23×D47—24no—
115C23×D47C22×C418yes—
116C23×D47C2414yes—
117C22×C4×C47—24no—
118C22×C4×C47—22no—
119C22×C4×C47—24no—
120C22×C4×C47—22no—
121C22×C4×C47—24no—
122C22×C4×C47—22no—
123C22×C4×C47—24no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—24no—
127C22×C4×C47C22×C418yes—
128C22×C4×C47C22×C414yes—
129C22×C4×C47C2×D414yes—
130C22×C4×C47#4114yes—
131C22×C4×C47#4114yes—
132C22×C4×C47C2414yes—
133C22×C4×C47C2412yes—
134C22×C4×C47C2414yes—
135C22×C4×C47C2412yes—
136C22×C4×C47C2412yes—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—24no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—24no—
145C22×Dic47—22no—
146C22×Dic47—24no—
147C22×Dic47C22×C414yes—
148C22×Dic47C22×C414yes—
149C22×Dic47C2×D414yes—
150C22×Dic47C2×D414yes—
151C22×Dic47#4114yes—
152C22×Dic47C2412yes—
153C22×Dic47C2414yes—
154C22×Dic47C2412yes—
155C22×Dic47C2412yes—
156C22×Dic47C2412yes—
157C2×D188—232no—
158C2×D188—28no—
159C2×D188—28no—
160C2×D188C22×C4116yes—
161C2×C4×D47—264no—
162C2×C4×D47—216no—
163C2×C4×D47—216no—
164C2×C4×D47C22×C4132yes—
165C4⋊C4×C47—28no—
166C4⋊C4×C47C22×C4116yes—
167C42×C47—216no—
168C42×C47—28no—
169(C2×D47)⋊C4—216no—
170(C2×D47)⋊C4—216no—
171(C2×D47)⋊C4—48no—
172(C2×D47)⋊C4—48no—
173(C2×D47)⋊C4C22×C4132yes—
174C4⋊Dic47—28no—
175Dic47⋊C4—216no—
176Dic47⋊C4—28no—
177Dic47⋊C4—48no—
178Dic47⋊C4C2×D4116yescomplemented
179C4×Dic47—28no—
180C4×Dic47—216no—
181C4×Dic47C22×C4116yes—
182C23×C47C23×C412yes—
183C23×C47C22×D412yes—
184C23×C47C2×#4112yes—
185C23×C47C2×#4112yes—
186C23×C47C2×#4112yes—
187C23×C47H(5,𝔽2)12yes—
188C23×C472−1+412yes—
189C23×C47C2511yescharacteristic
190C22×D47—216no—
191C22×D47—216no—
192C22×D47—216no—
193C22×D47—24no—
194C22×D47—24no—
195C22×D47—24no—
196C22×D47—24no—
197C22×D47—24no—
198C22×D47—24no—
199C22×D47—48no—
200C22×D47—48no—
201C22×D47—48no—
202C22×D47—48no—
203C22×D47C23×C418yes—
204C2×C4×C47—216no—
205C2×C4×C47—28no—
206C2×C4×C47—24no—
207C2×C4×C47—22no—
208C2×C4×C47—24no—
209C2×C4×C47—22no—
210C2×C4×C47—24no—
211C2×C4×C47—22no—
212C2×C4×C47—22no—
213C2×C4×C47—24no—
214C2×C4×C47—24no—
215C2×C4×C47—22no—
216C2×C4×C47—48no—
217C2×C4×C47—44no—
218C2×C4×C47#117116yescomplemented
219C2×C4×C47C23×C418yes—
220C2×C4×C47C23×C414yes—
221C2×C4×C47C22×D414yes—
222C2×C4×C47C2×#4114yes—
223C2×C4×C47C2×#4114yes—
224C2×Dic47—28no—
225C2×Dic47—28no—
226C2×Dic47—22no—
227C2×Dic47—22no—
228C2×Dic47—22no—
229C2×Dic47—22no—
230C2×Dic47—24no—
231C2×Dic47—22no—
232C2×Dic47—22no—
233C2×Dic47—24no—
234C2×Dic47—24no—
235C2×Dic47—22no—
236C2×Dic47—48no—
237C2×Dic47—44no—
238C2×Dic47C4×D4116yes—
239C2×Dic47C23×C414yes—
240C2×Dic47C23×C414yes—
241C2×Dic47C22×D414yes—
242C2×Dic47C22×D414yes—
243C2×Dic47C2×#4114yes—
244D188—232no—
245C4×D47—264no—
246C2×#2354—471nomaximal, Sylow
247C22×C47C2×#11714yes—
248C22×C47C2×C4×D414yes—
249C22×C47#52014yes—
250C22×C47#54714yes—
251C22×C47#55014yes—
252C22×C47#55514yes—
253C22×C47#55614yes—
254C22×C47C24×C411yescharacteristic
255C22×C47C23×D411yescharacteristic
256C22×C47C22×#4111yescharacteristic
257C22×C47C22×#4111yescharacteristic
258C22×C47C22×#4111yescharacteristic
259C22×C47C2×H(5,𝔽2)11yescharacteristic
260C22×C47C2×2−1+411yescharacteristic
261C2×D47—28no—
262C2×D47—28no—
263C2×D47—28no—
264C2×D47—432no—
265C2×D47—44no—
266C2×D47—44no—
267C2×D47—44no—
268C2×D47—44no—
269C4×C47—28no—
270C4×C47—24no—
271C4×C47—44no—
272C4×C47—42no—
273C4×C47C2×#11718yescomplemented
274Dic47—24no—
275Dic47—24no—
276Dic47—44no—
277Dic47—42no—
278Dic47C2×C4×D418yes—
279C22×C4×D4—471no—
280C22×#117—472no—
281#2354—4716no—
282C2×#390—471no—
283C2×#390—471no—
284C2×#388—472no—
285C2×#388—472no—
286C2×#384—471no—
287C2×#380—472no—
288C2×C4×C4⋊C4—471no—
289C2×C4×C4⋊C4—471no—
290C2×C4×C4⋊C4D4711yescharacteristic, complemented
291C2×C47#235418yescomplemented
292C2×C47C22×#11711yescharacteristic
293C2×C47C22×C4×D411yescharacteristic
294C2×C47C2×#52011yescharacteristic
295C2×C47C2×#54711yescharacteristic
296C2×C47C2×#55011yescharacteristic
297C2×C47C2×#55511yescharacteristic
298C2×C47C2×#55611yescharacteristic
299D47—48no—
300C23×D4—471no—
301C24×C4—472no—
302C2×C4×D4—4716no—
303C2×C4×D4—478no—
304C2×#117—4716no—
305C2×#117—9416no—
306C22×C4⋊C4—471no—
307C22×C4⋊C4—472no—
308C22×C4⋊C4—471no—
309C22×C4⋊C4—471no—
310C22×C4⋊C4—472no—
311C22×C4⋊C4C2×D4711yescharacteristic
312C22×C4⋊C4C2×D4711yescharacteristic
313C22×C22⋊C4—472no—
314C22×C22⋊C4—472no—
315C22×C22⋊C4—472no—
316C22×C22⋊C4—472no—
317C22×C22⋊C4—472no—
318C22×C42—471no—
319C22×C42—472no—
320C22×C42—471no—
321C22×C42C2×D4712yes—
322C22×C42C2×D4711yescharacteristic
323#390—478no—
324#390—478no—
325#388—4716no—
326#388—4716no—
327#384—478no—
328#380—4716no—
329C4×C4⋊C4—478no—
330C4×C4⋊C4—478no—
331C4×C4⋊C4C2×D4718yescomplemented
332C2×#95—472no—
333C2×#95—471no—
334C2×#95—471no—
335C2×#95C2×D4712yes—
336C47C2×#235411yescharacteristic, complemented, Sylow
337C25—472no—
338C22×D4—478no—
339C22×D4—474no—
340C22×D4—944no—
341C22×D4—944no—
342C23×C4—471no—
343C23×C4—471no—
344C23×C4—471no—
345C23×C4—4716no—
346C23×C4—478no—
347C23×C4—472no—
348C23×C4—471no—
349C23×C4—948no—
350C23×C4—948no—
351C23×C4C22×D4712yes—
352C23×C4C22×D4711yescharacteristic
353C23×C4C22×D4712yes—
354C23×C4C22×D4711yescharacteristic
355C23×C4C22×D4711yescharacteristic
356C4×D4—4764no—
357#117—9464no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—478no—
360C2×C4⋊C4—474no—
361C2×C4⋊C4—478no—
362C2×C4⋊C4—474no—
363C2×C4⋊C4—474no—
364C2×C4⋊C4—478no—
365C2×C4⋊C4—948no—
366C2×C4⋊C4—944no—
367C2×C4⋊C4—944no—
368C2×C4⋊C4—1884no—
369C2×C4⋊C4C4×D4718yes—
370C2×C4⋊C4C22×D4714yes—
371C2×C4⋊C4C22×D4714yes—
372C2×C22⋊C4—4716no—
373C2×C22⋊C4—478no—
374C2×C22⋊C4—478no—
375C2×C22⋊C4—478no—
376C2×C22⋊C4—478no—
377C2×C22⋊C4—478no—
378C2×C22⋊C4—948no—
379C2×C22⋊C4—948no—
380C2×C22⋊C4—1884no—
381C2×C22⋊C4—1884no—
382C2×C42—24no—
383C2×C42—28no—
384C2×C42—478no—
385C2×C42—474no—
386C2×C42—478no—
387C2×C42—474no—
388C2×C42—948no—
389C2×C42—944no—
390C2×C42C22×D4718yes—
391C2×C42C22×D4714yes—
392#95—478no—
393#95—474no—
394#95—474no—
395#95C22×D4718yes—
396C24—478no—
397C24—474no—
398C24—944no—
399C24—944no—
400C24—944no—
401C24—944no—
402C24—944no—
403C24—944no—
404C24C23×D4711yescharacteristic
405C2×D4—4716no—
406C2×D4—9432no—
407C2×D4—948no—
408C2×D4—948no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—474no—
420C22×C4—474no—
421C22×C4—474no—
422C22×C4—472no—
423C22×C4—474no—
424C22×C4—474no—
425C22×C4—472no—
426C22×C4—472no—
427C22×C4—4732no—
428C22×C4—472no—
429C22×C4—474no—
430C22×C4—9416no—
431C22×C4—9464no—
432C22×C4—9416no—
433C22×C4—942no—
434C22×C4—942no—
435C22×C4—942no—
436C22×C4—942no—
437C22×C4—944no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—944no—
441C22×C4—942no—
442C22×C4—944no—
443C22×C4C2×C4×D4718yes—
444C22×C4C2×C4×D4714yes—
445C22×C4#1062535014yes—
446C22×C4#1062535014yes—
447C22×C4D4×D4714yes—
448C22×C4C23×D4712yes—
449C22×C4C23×D4714yes—
450C22×C4C23×D4714yes—
451C22×C4C23×D4712yes—
452C22×C4C23×D4712yes—
453C4⋊C4—28no—
454C4⋊C4—4716no—
455C4⋊C4—948no—
456C4⋊C4—9416no—
457C4⋊C4—948no—
458C4⋊C4—1888no—
459C4⋊C4C2×C4×D47116yes—
460C22⋊C4—4732no—
461C22⋊C4—9416no—
462C22⋊C4—9416no—
463C22⋊C4—1888no—
464C22⋊C4—1888no—
465C42—216no—
466C42—28no—
467C42—4716no—
468C42—9416no—
469C42—948no—
470C23—478no—
471C23—9416no—
472C23—9416no—
473C23—9416no—
474C23—944no—
475C23—944no—
476C23—944no—
477C23—944no—
478C23—944no—
479C23—944no—
480C23—1888no—
481C23—1888no—
482C23—1888no—
483C23—1888no—
484C23C22×C4×D4712yes—
485C23C2×D4×D4712yes—
486C23#41×D4712yes—
487C23C2×#1062535012yes—
488C23C2×#1062535012yes—
489C23#1309407012yes—
490C23#1309407212yes—
491C23C24×D4711yescharacteristic
492D4—9432no—
493C2×C4—28no—
494C2×C4—216no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—48no—
506C2×C4—44no—
507C2×C4—4716no—
508C2×C4—474no—
509C2×C4—474no—
510C2×C4—474no—
511C2×C4—474no—
512C2×C4—474no—
513C2×C4—9464no—
514C2×C4—948no—
515C2×C4—948no—
516C2×C4—942no—
517C2×C4—942no—
518C2×C4—942no—
519C2×C4—942no—
520C2×C4—944no—
521C2×C4—942no—
522C2×C4—942no—
523C2×C4—944no—
524C2×C4—942no—
525C2×C4—944no—
526C2×C4—1888no—
527C2×C4—1884no—
528C2×C4#13093981116yescomplemented
529C2×C4C22×C4×D4718yes—
530C2×C4C22×C4×D4714yes—
531C2×C4C2×D4×D4714yes—
532C2×C4C2×#1062535014yes—
533C2×C4C2×#1062535014yes—
534C22—948no—
535C22—948no—
536C22—948no—
537C22—18832no—
538C22—1884no—
539C22—1884no—
540C22—1884no—
541C22—1884no—
542C22C2×#1309398114yes—
543C22C4×D4×D4714yes—
544C22#1568960514yes—
545C22#1568977414yes—
546C22#1568979414yes—
547C22#1568983614yes—
548C22#1568984514yes—
549C22C23×C4×D4711yescharacteristic
550C22C22×D4×D4711yescharacteristic
551C22C2×#41×D4711yescharacteristic
552C22C22×#1062535011yescharacteristic
553C22C22×#1062535011yescharacteristic
554C22C2×#1309407011yescharacteristic
555C22C2×#1309407211yescharacteristic
556C4—28no—
557C4—24no—
558C4—44no—
559C4—42no—
560C4—478no—
561C4—944no—
562C4—944no—
563C4—1884no—
564C4—1882no—
565C4C2×#1309398118yescomplemented
566C2—1888no—
567C2#2024633318yesdirect product factor
568C2C22×#1309398111yescharacteristic
569C2C2×C4×D4×D4711yescharacteristic
570C2C2×#1568960511yescharacteristic
571C2C2×#1568977411yescharacteristic
572C2C2×#1568979411yescharacteristic
573C2C2×#1568983611yescharacteristic
574C2C2×#1568984511yescharacteristic
subgroupG/H
centerC24C23×D47
commutatorC22×C47C24×C4
FrattiniC23C24×D47
FittingC2×C4×C4⋊C4×C47C2
socleC24×C47C24
prime radicalC23×D188C22
cosocleC23×C47C25
hypercenterC2×C4×C4⋊C4D47
hypocenterC47C2×#2354
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×#13093981C2semidirect
3. #20246333C2direct (N × H)
6. C2×#15689014C2semidirect
7. C2×#15689012C2semidirect
8. C2×#15688992C2semidirect
9. C2×#15688982C2semidirect
10. C2×C4⋊C4×Dic47C2semidirect
11. C2×C4×Dic47⋊C4C2semidirect
12. C2×C4×C4⋊C4×C47C2semidirect
15. C2×#13093986C4semidirect
18. C2×#13093981C22semidirect
34. #15689014C22semidirect
35. #15689012C22semidirect
36. #15688992C22semidirect
37. #15688982C22semidirect
38. C4⋊C4×Dic47C22semidirect
39. C4×Dic47⋊C4C22semidirect
47. C4×C4⋊C4×C47C22semidirect
63. #13093986C2×C4semidirect
70. C2×Dic47⋊C4D4semidirect
178. Dic47⋊C4C2×D4semidirect
218. C2×C4×C47#117semidirect
273. C4×C47C2×#117semidirect
290. C2×C4×C4⋊C4D47semidirect
291. C2×C47#2354semidirect
331. C4×C4⋊C4C2×D47semidirect
336. C47C2×#2354semidirect
528. C2×C4#13093981semidirect
565. C4C2×#13093981semidirect
567. C2#20246333direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939862
C22×#130939812
#202463332
C2×#156890252
C2×#156890242
C2×#156890142
C2×#156890122
C2×#156889922
C2×#156889822
C2×C4⋊C4×Dic472
C2×C4×Dic47⋊C42
C2×C4×C4⋊C4×C472
C2×#235447