Finite Group Databasebeta

← C2×#20243376 #26215673

Subgroups — C2×#20243376 #26215673

Order 12032.

2066 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×#13093939C212yesmaximal
2C2×#15688983C212yesmaximal, complemented
3#20243376C218yesmaximal, direct product factor
4C2×#15688890C212yesmaximal, complemented
5C22×C4×D4×C47C211yesmaximal, characteristic
6C22×C4⋊Dic47—22no—
7C22×C22⋊Dic47C2214yes—
8C2×#13093939—28no—
9C2×#13093939C2218yes—
10#15688983—24no—
11#15688983C2218yescomplemented
12C2×C47⋊#95—22no—
13#15688890C2218yescomplemented
14C23×D4×C47C2211yescharacteristic
15C24×C4×C47C411yescharacteristic, complemented
16C24×C4×C47C2211yescharacteristic
17C2×C4×D4×C47—22no—
18C2×C4×D4×C47—22no—
19C2×C4×D4×C47—24no—
20C2×C4×D4×C47C414yescomplemented
21C2×C4×D4×C47C2214yes—
22C22×C4⋊C4×C47C411yescharacteristic, complemented
23C22×C22⋊C4×C47C411yescharacteristic, complemented
24C22×C22⋊C4×C47C2211yescharacteristic
25C22×C42×C47C411yescharacteristic, complemented
26C23×Dic47—24no—
27C2×C4⋊Dic47—28no—
28C2×C4⋊Dic47—48no—
29C2×C22⋊Dic47—28no—
30C2×C22⋊Dic47—28no—
31C2×C22⋊Dic47—24no—
32C2×C22⋊Dic47C2318yes—
33#13093939—216no—
34C47⋊#95—28no—
35C25×C47C2×C411yescharacteristic
36C25×C47C2311yescharacteristic
37C22×D4×C47—22no—
38C22×D4×C47—22no—
39C22×D4×C47—24no—
40C22×D4×C47C2×C412yes—
41C22×D4×C47D414yes—
42C22×D4×C47Q814yes—
43C22×D4×C47C2312yes—
44C23×C4×C47—22no—
45C23×C4×C47—21no—
46C23×C4×C47—21no—
47C23×C4×C47—44no—
48C23×C4×C47—44no—
49C23×C4×C47C2×C411yescharacteristic
50C23×C4×C47C2×C411yescharacteristic
51C23×C4×C47C2×C411yescharacteristic
52C23×C4×C47C2×C411yescharacteristic
53C23×C4×C47C2×C411yescharacteristic
54C23×C4×C47C2×C412yes—
55C23×C4×C47C2×C414yescomplemented
56C23×C4×C47D412yes—
57C23×C4×C47D412yes—
58C23×C4×C47D414yes—
59C23×C4×C47C2312yes—
60C4×D4×C47—28no—
61C4×D4×C47—28no—
62C4×D4×C47—28no—
63C4×D4×C47—28no—
64C2×C4⋊C4×C47—22no—
65C2×C4⋊C4×C47—22no—
66C2×C4⋊C4×C47C2×C414yescomplemented
67C2×C22⋊C4×C47—22no—
68C2×C22⋊C4×C47—22no—
69C2×C22⋊C4×C47—22no—
70C2×C22⋊C4×C47—22no—
71C2×C22⋊C4×C47C2×C412yes—
72C2×C22⋊C4×C47C2×C414yescomplemented
73C2×C22⋊C4×C47C2312yes—
74C2×C42×C47—21no—
75C2×C42×C47—22no—
76C2×C42×C47—21no—
77C2×C42×C47C2×C414yescomplemented
78C24×C47—21no—
79C24×C47—21no—
80C24×C47—22no—
81C24×C47—42no—
82C24×C47—42no—
83C24×C47—42no—
84C24×C47—42no—
85C24×C47C4⋊C414yescomplemented
86C24×C47C22×C411yescharacteristic
87C24×C47C22×C412yes—
88C24×C47C22×C411yescharacteristic
89C24×C47C2×D412yes—
90C24×C47C2×D412yes—
91C24×C47C2×D412yes—
92C24×C47C2×Q812yes—
93C24×C47#4114yes—
94C24×C47C2411yescharacteristic
95C2×D4×C47—24no—
96C2×D4×C47—24no—
97C2×D4×C47—28no—
98C2×D4×C47—28no—
99C2×D4×C47—28no—
100C2×D4×C47—28no—
101C2×D4×C47—24no—
102C2×D4×C47—24no—
103C2×D4×C47C4⋊C418yescomplemented
104C2×D4×C47C2×D414yes—
105C2×D4×C47C2×Q814yes—
106C22×C4×C47—21no—
107C22×C4×C47—21no—
108C22×C4×C47—22no—
109C22×C4×C47—24no—
110C22×C4×C47—28no—
111C22×C4×C47—21no—
112C22×C4×C47—21no—
113C22×C4×C47—21no—
114C22×C4×C47—21no—
115C22×C4×C47—24no—
116C22×C4×C47—24no—
117C22×C4×C47—44no—
118C22×C4×C47—48no—
119C22×C4×C47—44no—
120C22×C4×C47—42no—
121C22×C4×C47—44no—
122C22×C4×C47—44no—
123C22×C4×C47—48no—
124C22×C4×C47—48no—
125C22×C4×C47—44no—
126C22×C4×C47—42no—
127C22×C4×C47—42no—
128C22×C4×C47C22⋊C411yescharacteristic
129C22×C4×C47C22⋊C412yes—
130C22×C4×C47C22⋊C412yes—
131C22×C4×C47C22⋊C414yes—
132C22×C4×C47C22⋊C412yes—
133C22×C4×C47C22⋊C414yes—
134C22×C4×C47C22⋊C411yescharacteristic
135C22×C4×C47C4⋊C414yescomplemented
136C22×C4×C47C22×C412yes—
137C22×C4×C47C22×C412yes—
138C22×C4×C47C22×C412yes—
139C22×C4×C47C22×C412yes—
140C22×C4×C47C22×C412yes—
141C22×C4×C47C2×D414yes—
142C22×C4×C47C2×D414yes—
143C22×Dic47—28no—
144C22×Dic47—48no—
145C22×Dic47—48no—
146C22×Dic47—48no—
147C4⋊C4×C47—28no—
148C22⋊C4×C47—24no—
149C22⋊C4×C47—24no—
150C22⋊C4×C47—28no—
151C42×C47—24no—
152C42×C47—24no—
153C22⋊Dic47—28no—
154C22⋊Dic47—28no—
155C22⋊Dic47—48no—
156C4⋊Dic47—416no—
157C23×C47—24no—
158C23×C47—22no—
159C23×C47—24no—
160C23×C47—22no—
161C23×C47—22no—
162C23×C47—22no—
163C23×C47—24no—
164C23×C47—24no—
165C23×C47—41no—
166C23×C47—42no—
167C23×C47—41no—
168C23×C47—41no—
169C23×C47—44no—
170C23×C47—42no—
171C23×C47—44no—
172C23×C47—48no—
173C23×C47—41no—
174C23×C47—42no—
175C23×C47—48no—
176C23×C47—42no—
177C23×C47—42no—
178C23×C47—48no—
179C23×C47—42no—
180C23×C47—42no—
181C23×C47—42no—
182C23×C47C2×C22⋊C412yes—
183C23×C47C2×C22⋊C411yescharacteristic
184C23×C47C2×C22⋊C411yescharacteristic
185C23×C47C2×C4⋊C412yes—
186C23×C47C2×C4⋊C412yes—
187C23×C47C2×C4⋊C414yescomplemented
188C23×C47#11712yes—
189C23×C47#11712yes—
190C23×C47#12114yes—
191C23×C47#12114yes—
192C23×C47C22⋊Q814yes—
193C23×C47C22⋊Q814yes—
194C23×C47C23×C411yescharacteristic
195C23×C47C22×D412yes—
196C23×C47C22×D411yescharacteristic
197C23×C47C22×Q811yescharacteristic
198C23×C47C2×#4112yes—
199D4×C47—28no—
200D4×C47—28no—
201D4×C47—216no—
202C2×C4×C47—22no—
203C2×C4×C47—22no—
204C2×C4×C47—24no—
205C2×C4×C47—24no—
206C2×C4×C47—22no—
207C2×C4×C47—24no—
208C2×C4×C47—28no—
209C2×C4×C47—24no—
210C2×C4×C47—24no—
211C2×C4×C47—28no—
212C2×C4×C47—22no—
213C2×C4×C47—22no—
214C2×C4×C47—44no—
215C2×C4×C47—42no—
216C2×C4×C47—42no—
217C2×C4×C47—44no—
218C2×C4×C47—48no—
219C2×C4×C47—42no—
220C2×C4×C47—42no—
221C2×C4×C47—42no—
222C2×C4×C47—42no—
223C2×C4×C47—48no—
224C2×C4×C47—42no—
225C2×C4×C47—42no—
226C2×C4×C47#9914yescomplemented
227C2×C4×C47C2×C22⋊C412yes—
228C2×C4×C47C2×C22⋊C414yes—
229C2×C4×C47C2×C22⋊C412yes—
230C2×C4×C47C2×C4⋊C414yescomplemented
231C2×Dic47—48no—
232C2×Dic47—48no—
233C2×Dic47—48no—
234C2×Dic47—816no—
235C2×#1799—471nomaximal, Sylow
236C22×C47—28no—
237C22×C47—24no—
238C22×C47—24no—
239C22×C47—28no—
240C22×C47—22no—
241C22×C47—21no—
242C22×C47—21no—
243C22×C47—22no—
244C22×C47—22no—
245C22×C47—21no—
246C22×C47—21no—
247C22×C47—22no—
248C22×C47—24no—
249C22×C47—42no—
250C22×C47—42no—
251C22×C47—44no—
252C22×C47—44no—
253C22×C47—41no—
254C22×C47—41no—
255C22×C47—42no—
256C22×C47—44no—
257C22×C47—44no—
258C22×C47—48no—
259C22×C47—48no—
260C22×C47—42no—
261C22×C47—42no—
262C22×C47#38012yes—
263C22×C47#38014yes—
264C22×C47#38012yes—
265C22×C47C2×#9912yes—
266C22×C47#41012yes—
267C22×C47C22×C22⋊C411yescharacteristic
268C22×C47C22×C4⋊C411yescharacteristic
269C22×C47C2×#11711yescharacteristic
270C22×C47C2×#12112yes—
271C22×C47C2×C22⋊Q812yes—
272C4×C47—22no—
273C4×C47—24no—
274C4×C47—24no—
275C4×C47—42no—
276C4×C47—42no—
277C4×C47C2×#9914yescomplemented
278Dic47—88no—
279C22×C4×D4D4711yescharacteristic
280C22×#99—472no—
281C2×#380—472no—
282#1799—478no—
283C2×#342—472no—
284C2×C47—22no—
285C2×C47—21no—
286C2×C47—21no—
287C2×C47—22no—
288C2×C47—22no—
289C2×C47—24no—
290C2×C47—44no—
291C2×C47#179914yescomplemented
292C2×C47C2×#38011yescharacteristic
293C2×C47C22×#9911yescharacteristic
294C2×C47C2×#41011yescharacteristic
295C23×D4C2×D4711yescharacteristic
296C24×C4Dic4711yescharacteristic, complemented
297C24×C4C2×D4711yescharacteristic
298C2×C4×D4—22no—
299C2×C4×D4—22no—
300C2×C4×D4—24no—
301C2×C4×D4Dic4714yescomplemented
302C2×C4×D4C2×D4714yes—
303C22×C4⋊C4—942no—
304C22×C4⋊C4Dic4711yescharacteristic, complemented
305C22×C22⋊C4—474no—
306C22×C22⋊C4Dic4711yescharacteristic, complemented
307C22×C22⋊C4C2×D4711yescharacteristic
308C22×C42Dic4711yescharacteristic, complemented
309C2×#99—478no—
310C2×#99—948no—
311#380—478no—
312#380—944no—
313C2×#95—942no—
314#342—478no—
315C47C2×#179911yescharacteristic, complemented, Sylow
316C25C2×Dic4711yescharacteristic
317C25C22×D4711yescharacteristic
318C22×D4—24no—
319C22×D4—22no—
320C22×D4—22no—
321C22×D4Dic9414yes—
322C22×D4D18814yes—
323C22×D4C2×Dic4712yes—
324C22×D4C22×D4712yes—
325C23×C4—22no—
326C23×C4—21no—
327C23×C4—21no—
328C23×C4—44no—
329C23×C4—44no—
330C23×C4—944no—
331C23×C4C2×Dic4711yescharacteristic
332C23×C4C2×Dic4711yescharacteristic
333C23×C4C2×Dic4711yescharacteristic
334C23×C4C2×Dic4711yescharacteristic
335C23×C4C2×Dic4711yescharacteristic
336C23×C4C2×Dic4712yes—
337C23×C4C2×Dic4714yescomplemented
338C23×C4C22⋊D4712yes—
339C23×C4C22⋊D4712yes—
340C23×C4C22⋊D4714yes—
341C23×C4C22×D4712yes—
342C4×D4—28no—
343C4×D4—28no—
344C4×D4—28no—
345C4×D4—28no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—948no—
349C2×C4⋊C4—1888no—
350C2×C4⋊C4C2×Dic4714yescomplemented
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—478no—
356C2×C22⋊C4—948no—
357C2×C22⋊C4—948no—
358C2×C22⋊C4—944no—
359C2×C22⋊C4C2×Dic4712yes—
360C2×C22⋊C4C2×Dic4714yescomplemented
361C2×C22⋊C4C22×D4712yes—
362C2×C42—22no—
363C2×C42—21no—
364C2×C42—21no—
365C2×C42C2×Dic4714yescomplemented
366#99—9416no—
367#95—948no—
368C24—21no—
369C24—21no—
370C24—22no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24C4⋊Dic4714yescomplemented
376C24C2×Dic9412yes—
377C24C2×D18812yes—
378C24#1062535014yes—
379C24C22×Dic4711yescharacteristic
380C24C22×Dic4712yes—
381C24C22×Dic4711yescharacteristic
382C24C2×C22⋊D4712yes—
383C24C2×C22⋊D4712yes—
384C24C23×D4711yescharacteristic
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—28no—
388C2×D4—28no—
389C2×D4—28no—
390C2×D4—28no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4C4⋊Dic4718yescomplemented
394C2×D4C2×Dic9414yes—
395C2×D4C2×D18814yes—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—44no—
408C22×C4—48no—
409C22×C4—44no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—44no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—948no—
419C22×C4—1888no—
420C22×C4—1888no—
421C22×C4—1888no—
422C22×C4C4⋊Dic4714yescomplemented
423C22×C4C22⋊Dic4712yes—
424C22×C4C22⋊Dic4714yes—
425C22×C4C22⋊Dic4711yescharacteristic
426C22×C4C22⋊Dic4712yes—
427C22×C4C22⋊Dic4712yes—
428C22×C4C22⋊Dic4714yes—
429C22×C4C22⋊Dic4711yescharacteristic
430C22×C4C22×Dic4712yes—
431C22×C4C22×Dic4712yes—
432C22×C4C22×Dic4712yes—
433C22×C4C22×Dic4712yes—
434C22×C4C22×Dic4712yes—
435C22×C4C2×C22⋊D4714yes—
436C22×C4C2×C22⋊D4714yes—
437C4⋊C4—28no—
438C4⋊C4—18816no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—28no—
442C22⋊C4—948no—
443C22⋊C4—948no—
444C22⋊C4—1888no—
445C42—24no—
446C42—24no—
447C23—22no—
448C23—24no—
449C23—22no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23—41no—
456C23—42no—
457C23—41no—
458C23—44no—
459C23—44no—
460C23—48no—
461C23—41no—
462C23—42no—
463C23—41no—
464C23—42no—
465C23—48no—
466C23—42no—
467C23—42no—
468C23—48no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23C2×C22⋊Dic4712yes—
473C23C2×C22⋊Dic4711yescharacteristic
474C23C2×C22⋊Dic4711yescharacteristic
475C23C2×C4⋊Dic4712yes—
476C23C2×C4⋊Dic4712yes—
477C23C2×C4⋊Dic4714yescomplemented
478C23#1309398312yes—
479C23#1309398312yes—
480C23#1309399814yes—
481C23#1309399814yes—
482C23#1309400314yes—
483C23#1309400314yes—
484C23C23×Dic4711yescharacteristic
485C23C22×C22⋊D4712yes—
486C23C22×D18811yescharacteristic
487C23C22×Dic9411yescharacteristic
488C23C2×#1062535012yes—
489D4—216no—
490D4—28no—
491D4—28no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—28no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—48no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—48no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—37616no—
517C2×C4—1888no—
518C2×C4—1888no—
519C2×C4—1888no—
520C2×C4#1309393914yescomplemented
521C2×C4C2×C22⋊Dic4712yes—
522C2×C4C2×C22⋊Dic4714yes—
523C2×C4C2×C22⋊Dic4712yes—
524C2×C4C2×C4⋊Dic4714yescomplemented
525C22—28no—
526C22—24no—
527C22—24no—
528C22—28no—
529C22—21no—
530C22—21no—
531C22—22no—
532C22—22no—
533C22—21no—
534C22—21no—
535C22—22no—
536C22—22no—
537C22—24no—
538C22—42no—
539C22—42no—
540C22—44no—
541C22—44no—
542C22—41no—
543C22—41no—
544C22—42no—
545C22—44no—
546C22—48no—
547C22—44no—
548C22—48no—
549C22—42no—
550C22—42no—
551C22#1568898312yes—
552C22#1568898314yes—
553C22#1568898312yes—
554C22C2×#1309393912yes—
555C22#1568909712yes—
556C22C22×C22⋊Dic4711yescharacteristic
557C22C22×C4⋊Dic4711yescharacteristic
558C22C2×#1309398311yescharacteristic
559C22C2×#1309399812yes—
560C22C2×#1309400312yes—
561C4—24no—
562C4—22no—
563C4—24no—
564C4—42no—
565C4—42no—
566C4—3768no—
567C4C2×#1309393914yescomplemented
568C2—22no—
569C2—22no—
570C2—21no—
571C2—21no—
572C2—24no—
573C2—22no—
574C2—44no—
575C2#2024337614yesdirect product factor
576C2C2×#1568898311yescharacteristic
577C2C22×#1309393911yescharacteristic
578C2C2×#1568909711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC22×C4×D4×C47C2
socleC23×C47C2×C22⋊C4
prime radicalC23×D4×C47C22
cosocleC24×C47C24
hypercenterC22×C4×D4D47
hypocenterC47C2×#1799
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#15688983C2semidirect
3. #20243376C2direct (N × H)
4. C2×#15688890C2semidirect
11. #15688983C22semidirect
13. #15688890C22semidirect
15. C24×C4×C47C4semidirect
20. C2×C4×D4×C47C4semidirect
22. C22×C4⋊C4×C47C4semidirect
23. C22×C22⋊C4×C47C4semidirect
25. C22×C42×C47C4semidirect
55. C23×C4×C47C2×C4semidirect
66. C2×C4⋊C4×C47C2×C4semidirect
72. C2×C22⋊C4×C47C2×C4semidirect
77. C2×C42×C47C2×C4semidirect
85. C24×C47C4⋊C4semidirect
103. C2×D4×C47C4⋊C4semidirect
135. C22×C4×C47C4⋊C4semidirect
187. C23×C47C2×C4⋊C4semidirect
226. C2×C4×C47#99semidirect
230. C2×C4×C47C2×C4⋊C4semidirect
277. C4×C47C2×#99semidirect
291. C2×C47#1799semidirect
296. C24×C4Dic47semidirect
301. C2×C4×D4Dic47semidirect
304. C22×C4⋊C4Dic47semidirect
306. C22×C22⋊C4Dic47semidirect
308. C22×C42Dic47semidirect
315. C47C2×#1799semidirect
337. C23×C4C2×Dic47semidirect
350. C2×C4⋊C4C2×Dic47semidirect
360. C2×C22⋊C4C2×Dic47semidirect
365. C2×C42C2×Dic47semidirect
375. C24C4⋊Dic47semidirect
393. C2×D4C4⋊Dic47semidirect
422. C22×C4C4⋊Dic47semidirect
477. C23C2×C4⋊Dic47semidirect
520. C2×C4#13093939semidirect
524. C2×C4C2×C4⋊Dic47semidirect
567. C4C2×#13093939semidirect
575. C2#20243376direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939392
C2×#156889832
#202433762
C2×#156888902
C22×C4×D4×C472
C2×#179947