Finite Group Databasebeta

← C2×#20247084 #26384105

Subgroups — C2×#20247084 #26384105

Order 12032.

3550 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094023C211yesmaximal, characteristic, complemented
2C22×#13094003C212yesmaximal
3#20247084C2116yesmaximal, direct product factor
4C2×#15689056C212yesmaximal, complemented
5C2×#15689043C212yesmaximal, complemented
6C2×#15689022C211yesmaximal, characteristic, complemented
7C2×#15689019C211yesmaximal, characteristic, complemented
8C2×#15688996C212yesmaximal
9C2×#15688992C211yesmaximal, characteristic, complemented
10C2×#15688983C211yesmaximal, characteristic
11C2×C4×C22⋊Dic47C211yesmaximal, characteristic
12C23×C4⋊C4×C47C211yesmaximal, characteristic
13C23×Dic94C2212yes—
14C2×#13094023—28no—
15C2×#13094023C2218yescomplemented
16C2×#13094003—216no—
17C2×#13094003C22116yes—
18C22×C4⋊Dic47C2211yescharacteristic
19C22×C4⋊Dic47C2211yescharacteristic
20C22×C4⋊Dic47C2212yes—
21C22×Dic47⋊C4C2212yes—
22C22×Dic47⋊C4C2212yes—
23C22×Dic47⋊C4C2212yes—
24C22×Dic47⋊C4C2212yes—
25C22×C22⋊Dic47C2212yes—
26C22×C22⋊Dic47C2211yescharacteristic
27C22×C22⋊Dic47C2211yescharacteristic
28C22×C4×Dic47C2211yescharacteristic
29C22×C4×Dic47C2211yescharacteristic
30#15689056C22116yescomplemented
31#15689043C22116yescomplemented
32#15689022C2218yescomplemented
33#15689019C2218yescomplemented
34#15688996C22116yes—
35#15688992C2218yescomplemented
36#15688983C2218yes—
37C4×C22⋊Dic47C2218yes—
38C2×C47⋊#95C2212yes—
39C2×C47⋊#95C2212yes—
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2211yescharacteristic
42C2×C47⋊#95C2211yescharacteristic
43C24×C4×C47C2212yes—
44C24×C4×C47C2211yescharacteristic
45C22×C4⋊C4×C47—28no—
46C22×C4⋊C4×C47C2211yescharacteristic
47C22×C4⋊C4×C47C2218yes—
48C22×C4⋊C4×C47C2212yes—
49C22×C4⋊C4×C47C2211yescharacteristic
50C22×Dic94—28no—
51C22×Dic94—28no—
52C22×Dic94—44no—
53C22×Dic94C2318yes—
54C23×Dic47C2312yes—
55C23×Dic47C2312yes—
56C23×Dic47C2311yescharacteristic
57C23×Dic47C2311yescharacteristic
58C23×Dic47C2311yescharacteristic
59C23×Dic47C2311yescharacteristic
60#13094023—232no—
61#13094003—264no—
62C2×C4⋊Dic47—24no—
63C2×C4⋊Dic47—24no—
64C2×C4⋊Dic47—28no—
65C2×C4⋊Dic47C2314yes—
66C2×C4⋊Dic47C2314yes—
67C2×C4⋊Dic47C2318yes—
68C2×Dic47⋊C4—28no—
69C2×Dic47⋊C4—28no—
70C2×Dic47⋊C4—28no—
71C2×Dic47⋊C4—44no—
72C2×Dic47⋊C4C2318yes—
73C2×Dic47⋊C4C2318yes—
74C2×Dic47⋊C4C2318yes—
75C2×Dic47⋊C4C2318yes—
76C2×C22⋊Dic47—28no—
77C2×C22⋊Dic47D418yes—
78C2×C22⋊Dic47Q818yescomplemented
79C2×C22⋊Dic47C2318yes—
80C2×C22⋊Dic47C2314yes—
81C2×C22⋊Dic47C2314yes—
82C2×C4×Dic47—24no—
83C2×C4×Dic47—24no—
84C2×C4×Dic47C2314yes—
85C2×C4×Dic47C2314yes—
86C47⋊#95C2314yes—
87C47⋊#95C2314yes—
88C47⋊#95C2318yes—
89C47⋊#95C2318yes—
90C47⋊#95C2318yes—
91C25×C47C2311yescharacteristic
92C23×C4×C47—24no—
93C23×C4×C47—24no—
94C23×C4×C47—28no—
95C23×C4×C47—28no—
96C23×C4×C47—44no—
97C23×C4×C47Q818yescomplemented
98C23×C4×C47C2312yes—
99C23×C4×C47C2318yes—
100C23×C4×C47C2312yes—
101C23×C4×C47C2311yescharacteristic
102C23×C4×C47C2314yes—
103C23×C4×C47C2311yescharacteristic
104C2×C4⋊C4×C47—28no—
105C2×C4⋊C4×C47—24no—
106C2×C4⋊C4×C47—232no—
107C2×C4⋊C4×C47D418yes—
108C2×C4⋊C4×C47C2314yes—
109C2×C4⋊C4×C47C2318yes—
110C2×C4⋊C4×C47C2314yes—
111C24×C47—22no—
112C24×C47—22no—
113C24×C47—22no—
114C24×C47—22no—
115C24×C47C2×D414yes—
116C24×C47C2×Q814yes—
117C24×C47C2×Q814yes—
118C24×C47C2412yes—
119C24×C47C2411yescharacteristic
120C22×C4×C47—22no—
121C22×C4×C47—24no—
122C22×C4×C47—22no—
123C22×C4×C47—24no—
124C22×C4×C47—232no—
125C22×C4×C47—28no—
126C22×C4×C47—28no—
127C22×C4×C47—24no—
128C22×C4×C47—24no—
129C22×C4×C47—216no—
130C22×C4×C47—24no—
131C22×C4×C47—216no—
132C22×C4×C47—48no—
133C22×C4×C47—432no—
134C22×C4×C47C2×D418yes—
135C22×C4×C47C2×D414yes—
136C22×C4×C47C2×Q814yes—
137C22×C4×C47C2×Q814yes—
138C22×C4×C47C2×Q8116yescomplemented
139C22×C4×C47#4114yes—
140C22×C4×C47C2414yes—
141C22×C4×C47C2414yes—
142C22×C4×C47C2412yes—
143C22×C4×C47C2412yes—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—24no—
153C22×Dic47—24no—
154C22×Dic47—24no—
155C22×Dic47—24no—
156C22×Dic47—24no—
157C22×Dic47—24no—
158C22×Dic47C2×D414yes—
159C22×Dic47C2×D414yes—
160C22×Dic47C2×Q814yes—
161C22×Dic47C2×Q814yes—
162C22×Dic47C2414yes—
163C22×Dic47C2414yes—
164C22×Dic47C2412yes—
165C22×Dic47C2412yes—
166C22×Dic47C2412yes—
167C22×Dic47C2412yes—
168C2×Dic94—216no—
169C2×Dic94—216no—
170C2×Dic94—48no—
171C2×Dic94—432no—
172C4⋊C4×C47—216no—
173C4⋊C4×C47—28no—
174C4⋊C4×C47C2×D4116yes—
175C22⋊Dic47—216no—
176C22⋊Dic47C2×D4116yes—
177C22⋊Dic47C2×Q8116yescomplemented
178C4⋊Dic47—28no—
179C4⋊Dic47—28no—
180C4⋊Dic47—216no—
181Dic47⋊C4—216no—
182Dic47⋊C4—216no—
183Dic47⋊C4—216no—
184Dic47⋊C4—48no—
185C4×Dic47—28no—
186C4×Dic47—28no—
187C23×C47—28no—
188C23×C47—28no—
189C23×C47—28no—
190C23×C47—28no—
191C23×C47—28no—
192C23×C47—28no—
193C23×C47—22no—
194C23×C47—22no—
195C23×C47—22no—
196C23×C47—22no—
197C23×C47C4⋊Q8116yescomplemented
198C23×C47C22×D414yes—
199C23×C47C22×D412yes—
200C23×C47C22×D412yes—
201C23×C47C22×Q814yes—
202C23×C47C22×Q812yes—
203C23×C47C22×Q814yes—
204C23×C47C22×Q812yes—
205C23×C47C2×#4112yes—
206C23×C47H(5,𝔽2)12yes—
207C23×C472−1+412yes—
208C23×C47C2511yescharacteristic
209C2×C4×C47—28no—
210C2×C4×C47—216no—
211C2×C4×C47—24no—
212C2×C4×C47—22no—
213C2×C4×C47—22no—
214C2×C4×C47—24no—
215C2×C4×C47—232no—
216C2×C4×C47—24no—
217C2×C4×C47—24no—
218C2×C4×C47—24no—
219C2×C4×C47—432no—
220C2×C4×C47—48no—
221C2×C4×C47C22⋊Q8116yescomplemented
222C2×C4×C47C22×D418yes—
223C2×C4×C47C22×D414yes—
224C2×C4×C47C22×Q814yes—
225C2×C4×C47C22×Q814yes—
226C2×C4×C47C2×#4114yes—
227C2×Dic47—28no—
228C2×Dic47—28no—
229C2×Dic47—28no—
230C2×Dic47—28no—
231C2×Dic47—22no—
232C2×Dic47—22no—
233C2×Dic47—22no—
234C2×Dic47—22no—
235C2×Dic47—22no—
236C2×Dic47—22no—
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×Dic47C22×D414yes—
248C2×Dic47C22×D414yes—
249C2×Dic47C22×Q814yes—
250C2×Dic47C22×Q814yes—
251Dic94—432no—
252C2×#2440—471nomaximal, Sylow
253C22×C47—24no—
254C22×C47—24no—
255C22×C47—24no—
256C22×C47—232no—
257C22×C47—24no—
258C22×C47—24no—
259C22×C47—24no—
260C22×C47C2×C22⋊Q814yes—
261C22×C47C2×C4⋊Q814yes—
262C22×C47C2×C4⋊Q818yescomplemented
263C22×C47#54414yes—
264C22×C47D4214yes—
265C22×C47#54714yes—
266C22×C47D4×Q814yes—
267C22×C47#55414yes—
268C22×C47C23×D411yescharacteristic
269C22×C47C23×D411yescharacteristic
270C22×C47C23×Q811yescharacteristic
271C22×C47C23×Q811yescharacteristic
272C22×C47C22×#4111yescharacteristic
273C22×C47C2×H(5,𝔽2)11yescharacteristic
274C22×C47C2×2−1+411yescharacteristic
275C4×C47—24no—
276C4×C47—28no—
277C4×C47—44no—
278C4×C47C2×C22⋊Q818yescomplemented
279Dic47—24no—
280Dic47—24no—
281Dic47—24no—
282Dic47—24no—
283Dic47—44no—
284Dic47—44no—
285C22×C4⋊Q8—471no—
286C22×C22⋊Q8—472no—
287C23×C4⋊C4D4711yescharacteristic
288#2440—4716no—
289C2×#398—472no—
290C2×#395—472no—
291C2×#389—471no—
292C2×#389—471no—
293C2×#385—472no—
294C2×#384—471no—
295C2×#380—471no—
296C2×C4×C22⋊C4—471no—
297C2×C47—28no—
298C2×C47#244018yescomplemented
299C2×C47C22×C22⋊Q811yescharacteristic
300C2×C47C22×C4⋊Q811yescharacteristic
301C2×C47C2×#54411yescharacteristic
302C2×C47C2×D4211yescharacteristic
303C2×C47C2×#54711yescharacteristic
304C2×C47C2×D4×Q811yescharacteristic
305C2×C47C2×#55411yescharacteristic
306C23×Q8—472no—
307C24×C4C2×D4712yes—
308C24×C4C2×D4711yescharacteristic
309C2×C4⋊Q8—478no—
310C2×C4⋊Q8—948no—
311C2×C22⋊Q8—4716no—
312C2×C22⋊Q8—9416no—
313C22×C4⋊C4—28no—
314C22×C4⋊C4—472no—
315C22×C4⋊C4—472no—
316C22×C4⋊C4—472no—
317C22×C4⋊C4—472no—
318C22×C4⋊C4—471no—
319C22×C4⋊C4—471no—
320C22×C4⋊C4—472no—
321C22×C4⋊C4C2×D4712yes—
322C22×C4⋊C4C2×D4718yes—
323C22×C4⋊C4C2×D4711yescharacteristic
324C22×C4⋊C4C2×D4711yescharacteristic
325C22×C22⋊C4—471no—
326C22×C22⋊C4—471no—
327C22×C22⋊C4—472no—
328C22×C42—471no—
329C22×C42—471no—
330#398—4716no—
331#395—4716no—
332#389—478no—
333#389—478no—
334#385—4716no—
335#384—478no—
336#380—478no—
337C4×C22⋊C4—478no—
338C2×#95—471no—
339C2×#95—472no—
340C2×#95—472no—
341C2×#95—471no—
342C2×#95—472no—
343C47C2×#244011yescharacteristic, complemented, Sylow
344C25C22×D4711yescharacteristic
345C22×Q8—478no—
346C22×Q8—948no—
347C22×Q8—948no—
348C22×Q8—1884no—
349C23×C4—24no—
350C23×C4—24no—
351C23×C4—28no—
352C23×C4—28no—
353C23×C4—44no—
354C23×C4—471no—
355C23×C4—471no—
356C23×C4—471no—
357C23×C4—471no—
358C23×C4—472no—
359C23×C4—472no—
360C23×C4Dic9418yescomplemented
361C23×C4C22×D4712yes—
362C23×C4C22×D4718yes—
363C23×C4C22×D4711yescharacteristic
364C23×C4C22×D4714yes—
365C23×C4C22×D4712yes—
366C23×C4C22×D4711yescharacteristic
367C4⋊Q8—9432no—
368C22⋊Q8—9464no—
369C2×C4⋊C4—28no—
370C2×C4⋊C4—24no—
371C2×C4⋊C4—232no—
372C2×C4⋊C4—478no—
373C2×C4⋊C4—478no—
374C2×C4⋊C4—478no—
375C2×C4⋊C4—478no—
376C2×C4⋊C4—474no—
377C2×C4⋊C4—474no—
378C2×C4⋊C4—478no—
379C2×C4⋊C4—948no—
380C2×C4⋊C4—948no—
381C2×C4⋊C4—948no—
382C2×C4⋊C4—944no—
383C2×C4⋊C4—948no—
384C2×C4⋊C4—944no—
385C2×C4⋊C4—1884no—
386C2×C4⋊C4C22⋊D4718yes—
387C2×C4⋊C4C22×D4714yes—
388C2×C4⋊C4C22×D4718yes—
389C2×C4⋊C4C22×D4714yes—
390C2×C22⋊C4—478no—
391C2×C22⋊C4—478no—
392C2×C22⋊C4—474no—
393C2×C22⋊C4—474no—
394C2×C22⋊C4—478no—
395C2×C22⋊C4—948no—
396C2×C42—474no—
397C2×C42—474no—
398C2×C42—944no—
399C2×C42—944no—
400#95—474no—
401#95—474no—
402#95—478no—
403#95—478no—
404#95—478no—
405C24—22no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24C2×Dic9414yes—
410C24D4×D4714yes—
411C24Q8×D4714yes—
412C24C23×D4712yes—
413C24C23×D4711yescharacteristic
414C2×Q8—9416no—
415C2×Q8—9416no—
416C2×Q8—1888no—
417C2×Q8—18832no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—232no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—24no—
426C22×C4—216no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—216no—
430C22×C4—48no—
431C22×C4—432no—
432C22×C4—474no—
433C22×C4—474no—
434C22×C4—474no—
435C22×C4—474no—
436C22×C4—472no—
437C22×C4—472no—
438C22×C4—472no—
439C22×C4—472no—
440C22×C4—474no—
441C22×C4—474no—
442C22×C4—942no—
443C22×C4—942no—
444C22×C4—942no—
445C22×C4—942no—
446C22×C4—942no—
447C22×C4—942no—
448C22×C4—942no—
449C22×C4—942no—
450C22×C4—944no—
451C22×C4—944no—
452C22×C4—944no—
453C22×C4—944no—
454C22×C4—944no—
455C22×C4—944no—
456C22×C4C2×Dic9414yes—
457C22×C4C2×Dic9414yes—
458C22×C4C2×Dic94116yescomplemented
459C22×C4#1062535014yes—
460C22×C4C2×C22⋊D4718yes—
461C22×C4C2×C22⋊D4714yes—
462C22×C4C23×D4714yes—
463C22×C4C23×D4714yes—
464C22×C4C23×D4712yes—
465C22×C4C23×D4712yes—
466C4⋊C4—216no—
467C4⋊C4—28no—
468C4⋊C4—9416no—
469C4⋊C4—9416no—
470C4⋊C4—9416no—
471C4⋊C4—9416no—
472C4⋊C4—948no—
473C4⋊C4—948no—
474C4⋊C4—1888no—
475C4⋊C4C2×C22⋊D47116yes—
476C22⋊C4—4716no—
477C22⋊C4—4716no—
478C22⋊C4—9416no—
479C42—948no—
480C42—948no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23#13094023116yescomplemented
492C23C2×D4×D4714yes—
493C23C2×D4×D4712yes—
494C23C22×C22⋊D4712yes—
495C23C2×Q8×D4714yes—
496C23C2×Q8×D4712yes—
497C23C22×Dic9414yes—
498C23C22×Dic9412yes—
499C23C2×#1062535012yes—
500C23#1309406912yes—
501C23#1309407212yes—
502C23C24×D4711yescharacteristic
503Q8—18832no—
504C2×C4—216no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—232no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—432no—
515C2×C4—48no—
516C2×C4—474no—
517C2×C4—474no—
518C2×C4—474no—
519C2×C4—474no—
520C2×C4—948no—
521C2×C4—948no—
522C2×C4—948no—
523C2×C4—948no—
524C2×C4—942no—
525C2×C4—942no—
526C2×C4—942no—
527C2×C4—942no—
528C2×C4—942no—
529C2×C4—942no—
530C2×C4—942no—
531C2×C4—942no—
532C2×C4—944no—
533C2×C4—944no—
534C2×C4—944no—
535C2×C4—944no—
536C2×C4—944no—
537C2×C4—944no—
538C2×C4—1888no—
539C2×C4—1888no—
540C2×C4#13094003116yescomplemented
541C2×C4C22×C22⋊D4718yes—
542C2×C4C22×C22⋊D4714yes—
543C2×C4C22×Dic9414yes—
544C2×C4C22×Dic9414yes—
545C2×C4C2×#1062535014yes—
546C22—24no—
547C22—232no—
548C22—24no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—24no—
553C22C2×#1309400314yes—
554C22C2×#1309402314yes—
555C22C2×#1309402318yescomplemented
556C22#1568975414yes—
557C22D4×C22⋊D4714yes—
558C22#1568977414yes—
559C22Q8×C22⋊D4714yes—
560C22#1568982814yes—
561C22C22×D4×D4711yescharacteristic
562C22C23×C22⋊D4711yescharacteristic
563C22C22×Q8×D4711yescharacteristic
564C22C23×Dic9411yescharacteristic
565C22C22×#1062535011yescharacteristic
566C22C2×#1309406911yescharacteristic
567C22C2×#1309407211yescharacteristic
568C4—28no—
569C4—24no—
570C4—44no—
571C4—944no—
572C4—944no—
573C4—944no—
574C4—944no—
575C4—1884no—
576C4—1884no—
577C4C2×#1309400318yescomplemented
578C2—28no—
579C2#2024708418yesdirect product factor
580C2C22×#1309400311yescharacteristic
581C2C22×#1309402311yescharacteristic
582C2C2×#1568975411yescharacteristic
583C2C2×D4×C22⋊D4711yescharacteristic
584C2C2×#1568977411yescharacteristic
585C2C2×Q8×C22⋊D4711yescharacteristic
586C2C2×#1568982811yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC23×C4⋊C4×C47C2
socleC24×C47C24
prime radicalC25×C47C23
hypercenterC23×C4⋊C4D47
hypocenterC47C2×#2440
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#13094023C2semidirect
3. #20247084C2direct (N × H)
4. C2×#15689056C2semidirect
5. C2×#15689043C2semidirect
6. C2×#15689022C2semidirect
7. C2×#15689019C2semidirect
9. C2×#15688992C2semidirect
15. C2×#13094023C22semidirect
30. #15689056C22semidirect
31. #15689043C22semidirect
32. #15689022C22semidirect
33. #15689019C22semidirect
35. #15688992C22semidirect
78. C2×C22⋊Dic47Q8semidirect
97. C23×C4×C47Q8semidirect
138. C22×C4×C47C2×Q8semidirect
177. C22⋊Dic47C2×Q8semidirect
197. C23×C47C4⋊Q8semidirect
221. C2×C4×C47C22⋊Q8semidirect
262. C22×C47C2×C4⋊Q8semidirect
278. C4×C47C2×C22⋊Q8semidirect
298. C2×C47#2440semidirect
343. C47C2×#2440semidirect
360. C23×C4Dic94semidirect
458. C22×C4C2×Dic94semidirect
491. C23#13094023semidirect
540. C2×C4#13094003semidirect
555. C22C2×#13094023semidirect
577. C4C2×#13094003semidirect
579. C2#20247084direct (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×#130940032
#202470842
C2×#156890562
C2×#156890432
C2×#156890222
C2×#156890192
C2×#156889962
C2×#156889922
C2×#156889832
C2×C4×C22⋊Dic472
C23×C4⋊C4×C472
C2×#244047