Finite Group Databasebeta

← C22×#15689806 #27123266

Subgroups — C22×#15689806 #27123266

Order 12032.

7934 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
1C2×#15689806C2148yesmaximal, direct product factor
2C22×#13094019C211yesmaximal, characteristic, complemented
3C22×#13094018C211yesmaximal, characteristic, complemented
4C22×#13094016C211yesmaximal, characteristic, complemented
5C22×#13094014C211yesmaximal, characteristic, complemented
6C22×#13094014C211yesmaximal, characteristic, complemented
7C22×#13094008C211yesmaximal, characteristic
8C22×#13094007C211yesmaximal, characteristic
9C22×#13094002C211yesmaximal, characteristic, complemented
10C22×#13094001C211yesmaximal, characteristic, complemented
11C22×C4×Dic94C211yesmaximal, characteristic, complemented
12C22×(C22⋊D47)⋊C4C211yesmaximal, characteristic
13C22×#13093981C211yesmaximal, characteristic, complemented
14C22×#13093980C211yesmaximal, characteristic, complemented
15C23×Dic47⋊C4C211yesmaximal, characteristic, complemented
16C22×#117×C47C211yesmaximal, characteristic, complemented
17C23×Dic94C2211yescharacteristic
18C23×C22⋊D47C2211yescharacteristic
19C24×Dic47C2211yescharacteristic
20C24×Dic47C2211yescharacteristic
21C23×C4×D47C2211yescharacteristic
22#15689806C221256yesdirect product factor
23C2×#13094019C22124yescomplemented
24C2×#13094018C22124yescomplemented
25C2×#13094016C22124yescomplemented
26C2×#13094014C22124yescomplemented
27C2×#13094014C22124yescomplemented
28C2×#13094008C22124yes—
29C2×#13094007C22124yes—
30C2×#13094002C22124yescomplemented
31C2×#13094001C22124yescomplemented
32C2×C4×Dic94C22124yescomplemented
33C2×(C22⋊D47)⋊C4C22124yes—
34C2×#13093981C22124yescomplemented
35C2×#13093980C22124yescomplemented
36C22×C4⋊Dic47C2211yescharacteristic
37C22×C4⋊Dic47C2211yescharacteristic
38C22×C4⋊Dic47C2211yescharacteristic
39C22×Dic47⋊C4C2211yescharacteristic
40C22×Dic47⋊C4C2211yescharacteristic
41C22×Dic47⋊C4C2211yescharacteristic
42C22×Dic47⋊C4C2211yescharacteristic
43C22×Dic47⋊C4C2211yescharacteristic
44C22×Dic47⋊C4C2211yescharacteristic
45C22×Dic47⋊C4C2211yescharacteristic
46C22×Dic47⋊C4C2211yescharacteristic
47C22×Dic47⋊C4C2211yescharacteristic
48C22×Dic47⋊C4C2211yescharacteristic
49C22×Dic47⋊C4C2211yescharacteristic
50C22×Dic47⋊C4C22124yescomplemented
51C22×C22⋊Dic47C2211yescharacteristic
52C22×(C2×D47)⋊C4C2211yescharacteristic
53C22×(C2×D47)⋊C4C2211yescharacteristic
54C22×(C2×D47)⋊C4C2211yescharacteristic
55C22×(C2×D47)⋊C4C2211yescharacteristic
56C22×(C2×D47)⋊C4C2211yescharacteristic
57C22×C4×Dic47C2211yescharacteristic
58C22×C4×Dic47C2211yescharacteristic
59C22×C4×Dic47C2211yescharacteristic
60C24×C4×C47C2211yescharacteristic
61C2×#117×C47C22124yescomplemented
62C22×C4⋊C4×C47C2211yescharacteristic
63C22×C4⋊C4×C47C2211yescharacteristic
64C22×C22⋊C4×C47C2211yescharacteristic
65C22×C22⋊C4×C47C2211yescharacteristic
66C22×C42×C47C2211yescharacteristic
67C22×C42×C47C2211yescharacteristic
68C24×D47C2311yescharacteristic
69C22×Dic94—28no—
70C22×Dic94C23112yes—
71C22×C22⋊D47—28no—
72C22×C22⋊D47C23112yes—
73C23×Dic47—28no—
74C23×Dic47—28no—
75C23×Dic47C2311yescharacteristic
76C23×Dic47C2311yescharacteristic
77C23×Dic47C2311yescharacteristic
78C23×Dic47C2311yescharacteristic
79C23×Dic47C2311yescharacteristic
80C23×Dic47C2311yescharacteristic
81C23×Dic47C23112yes—
82C23×Dic47C23112yes—
83C23×Dic47C2311yescharacteristic
84C22×C4×D47—28no—
85C22×C4×D47C23112yes—
86#13094019C23164yescomplemented
87#13094018C23164yescomplemented
88#13094016C23164yescomplemented
89#13094014C23164yescomplemented
90#13094014C23164yescomplemented
91#13094008C23164yes—
92#13094007C23164yes—
93#13094002C23164yescomplemented
94#13094001C23164yescomplemented
95C4×Dic94C23164yescomplemented
96(C22⋊D47)⋊C4C23164yes—
97#13093981C23164yescomplemented
98#13093980C23164yescomplemented
99C2×C4⋊Dic47C23112yes—
100C2×C4⋊Dic47C23112yes—
101C2×C4⋊Dic47C23112yes—
102C2×Dic47⋊C4C23112yes—
103C2×Dic47⋊C4C23112yes—
104C2×Dic47⋊C4C23112yes—
105C2×Dic47⋊C4C23112yes—
106C2×Dic47⋊C4C23112yes—
107C2×Dic47⋊C4C23112yes—
108C2×Dic47⋊C4C23112yes—
109C2×Dic47⋊C4C23112yes—
110C2×Dic47⋊C4C23112yes—
111C2×Dic47⋊C4C23112yes—
112C2×Dic47⋊C4C23112yes—
113C2×Dic47⋊C4C23164yescomplemented
114C2×C22⋊Dic47C23112yes—
115C2×(C2×D47)⋊C4C23112yes—
116C2×(C2×D47)⋊C4C23112yes—
117C2×(C2×D47)⋊C4C23112yes—
118C2×(C2×D47)⋊C4C23112yes—
119C2×(C2×D47)⋊C4C23112yes—
120C2×C4×Dic47C23112yes—
121C2×C4×Dic47C23112yes—
122C2×C4×Dic47C23112yes—
123C25×C47C2311yescharacteristic
124C23×C4×C47—28no—
125C23×C4×C47C2311yescharacteristic
126C23×C4×C47C2311yescharacteristic
127C23×C4×C47C2311yescharacteristic
128C23×C4×C47C2311yescharacteristic
129C23×C4×C47C2311yescharacteristic
130C23×C4×C47C2311yescharacteristic
131C23×C4×C47C23112yes—
132#117×C47C23164yescomplemented
133C2×C4⋊C4×C47C23112yes—
134C2×C4⋊C4×C47C23112yes—
135C2×C22⋊C4×C47C23112yes—
136C2×C22⋊C4×C47C23112yes—
137C2×C42×C47C23112yes—
138C2×C42×C47C23112yes—
139C24×C47—24no—
140C24×C47—24no—
141C24×C47#4118yes—
142C24×C47C2416yes—
143C24×C47C2411yescharacteristic
144C23×D47—24no—
145C23×D47—24no—
146C23×D47—24no—
147C23×D47C2416yes—
148C22×C4×C47—248no—
149C22×C4×C47—24no—
150C22×C4×C47—24no—
151C22×C4×C47—24no—
152C22×C4×C47—24no—
153C22×C4×C47—24no—
154C22×C4×C47—24no—
155C22×C4×C47C2416yes—
156C22×C4×C47C2416yes—
157C22×C4×C47C2416yes—
158C22×C4×C47C2416yes—
159C22×C4×C47C2416yes—
160C22×C4×C47C2416yes—
161C22×C4×C47C24116yes—
162C2×C22⋊D47—248no—
163C2×C22⋊D47C24116yes—
164C22×Dic47—24no—
165C22×Dic47—248no—
166C22×Dic47—24no—
167C22×Dic47—24no—
168C22×Dic47—24no—
169C22×Dic47—24no—
170C22×Dic47—24no—
171C22×Dic47—248no—
172C22×Dic47#4118yes—
173C22×Dic47C2416yes—
174C22×Dic47C2416yes—
175C22×Dic47C2416yes—
176C22×Dic47C2416yes—
177C22×Dic47C2416yes—
178C22×Dic47C2416yes—
179C22×Dic47C24116yes—
180C22×Dic47C24116yes—
181C22×Dic47C2416yes—
182C2×C4×D47—248no—
183C2×C4×D47C24116yes—
184C2×Dic94—248no—
185C2×Dic94C24116yes—
186C4⋊C4×C47C24116yes—
187C4⋊C4×C47C24116yes—
188C22⋊C4×C47C24116yes—
189C22⋊C4×C47C24116yes—
190C42×C47C24116yes—
191C42×C47C24116yes—
192C22⋊Dic47C24116yes—
193(C2×D47)⋊C4C24116yes—
194(C2×D47)⋊C4C24116yes—
195(C2×D47)⋊C4C24116yes—
196(C2×D47)⋊C4C24116yes—
197(C2×D47)⋊C4C24116yes—
198C4⋊Dic47C24116yes—
199C4⋊Dic47C24116yes—
200C4⋊Dic47C24116yes—
201Dic47⋊C4C24116yes—
202Dic47⋊C4C24116yes—
203Dic47⋊C4C24116yes—
204Dic47⋊C4C24116yes—
205Dic47⋊C4C24116yes—
206Dic47⋊C4C24116yes—
207Dic47⋊C4C24116yes—
208Dic47⋊C4C24116yes—
209Dic47⋊C4C24116yes—
210Dic47⋊C4C24116yes—
211Dic47⋊C4C24116yes—
212C4×Dic47C24116yes—
213C4×Dic47C24116yes—
214C4×Dic47C24116yes—
215C23×C47—232no—
216C23×C47—212no—
217C23×C47—212no—
218C23×C47C2×#4114yes—
219C23×C47C2×#41124yes—
220C23×C47C2×#4114yes—
221C23×C472−1+414yes—
222C23×C47C2514yes—
223C23×C47C2513yes—
224C22×D47—212no—
225C22×D47—212no—
226C22×D47—212no—
227C22×D47—416no—
228C22×D47C2514yes—
229C2×C4×C47—212no—
230C2×C4×C47—212no—
231C2×C4×C47—232no—
232C2×C4×C47—212no—
233C2×C4×C47—212no—
234C2×C4×C47—212no—
235C2×C4×C47—212no—
236C2×C4×C47C2514yes—
237C2×C4×C47C2514yes—
238C2×C4×C47C2514yes—
239C2×C4×C47C2514yes—
240C2×C4×C47C2514yes—
241C2×C4×C47C2514yes—
242C22⋊D47—232no—
243C2×Dic47—212no—
244C2×Dic47—212no—
245C2×Dic47—232no—
246C2×Dic47—212no—
247C2×Dic47—212no—
248C2×Dic47—212no—
249C2×Dic47—212no—
250C2×Dic47—232no—
251C2×Dic47C2×#41124yes—
252C2×Dic47C2514yes—
253C2×Dic47C2514yes—
254C2×Dic47C2514yes—
255C2×Dic47C2514yes—
256C2×Dic47C2514yes—
257C2×Dic47C2514yes—
258C2×Dic47C2514yes—
259C4×D47—232no—
260Dic94—232no—
261C22×#552—471nomaximal, Sylow
262C22×C47—24no—
263C22×C47—24no—
264C22×C47—248no—
265C22×C47#552116yescomplemented
266C22×C47C22×#4116yes—
267C22×C47C22×#4118yes—
268C22×C47C22×#4116yes—
269C22×C47C2×2−1+416yes—
270C22×C47C2611yescharacteristic
271C2×D47—24no—
272C2×D47—24no—
273C2×D47—24no—
274C2×D47—424no—
275C4×C47—24no—
276C4×C47—24no—
277C4×C47—24no—
278C4×C47—24no—
279C4×C47—24no—
280C4×C47—24no—
281Dic47—24no—
282Dic47—24no—
283Dic47—24no—
284Dic47—24no—
285Dic47—24no—
286Dic47—24no—
287Dic47C22×#4118yes—
288C2×#552—4748no—
289C22×#126—471no—
290C22×#126—471no—
291C22×#125—471no—
292C22×#125—471no—
293C22×#125—471no—
294C22×#123—471no—
295C22×#123—471no—
296C22×C22⋊Q8—471no—
297C22×C22⋊Q8—471no—
298C22×C4×Q8—471no—
299C22×C4×D4—471no—
300C22×#117—471no—
301C22×#117—471no—
302C22×#117D4711yescharacteristic, complemented
303C23×C4⋊C4—471no—
304C2×C47—28no—
305C2×C47C2×#552112yescomplemented
306C2×C47C23×#4111yescharacteristic
307C2×C47C23×#4111yescharacteristic
308C2×C47C22×2−1+411yescharacteristic
309D47—44no—
310C23×Q8—471no—
311C23×D4—471no—
312C24×C4—471no—
313C24×C4—471no—
314C24×C4—471no—
315C24×C4C2×D4711yescharacteristic
316#552—47256no—
317C2×#126—4724no—
318C2×#126—4724no—
319C2×#125—4724no—
320C2×#125—4724no—
321C2×#125—4724no—
322C2×#123—4724no—
323C2×#123—4724no—
324C2×C22⋊Q8—4724no—
325C2×C22⋊Q8—4724no—
326C2×C4×Q8—4724no—
327C2×C4×D4—4724no—
328C2×#117—4724no—
329C2×#117—4724no—
330C2×#117C2×D47124yescomplemented
331C22×C4⋊C4—471no—
332C22×C4⋊C4—471no—
333C22×C4⋊C4—471no—
334C22×C4⋊C4—471no—
335C22×C4⋊C4—4724no—
336C22×C4⋊C4—471no—
337C22×C4⋊C4—471no—
338C22×C4⋊C4—471no—
339C22×C4⋊C4—471no—
340C22×C4⋊C4—471no—
341C22×C4⋊C4—471no—
342C22×C4⋊C4—471no—
343C22×C4⋊C4—471no—
344C22×C4⋊C4—471no—
345C22×C4⋊C4—471no—
346C22×C4⋊C4C2×D4711yescharacteristic
347C22×C4⋊C4C2×D4711yescharacteristic
348C22×C22⋊C4—471no—
349C22×C22⋊C4—471no—
350C22×C22⋊C4—471no—
351C22×C22⋊C4—471no—
352C22×C22⋊C4—471no—
353C22×C22⋊C4—471no—
354C22×C22⋊C4C2×D4711yescharacteristic
355C22×C22⋊C4C2×D4711yescharacteristic
356C22×C42—471no—
357C22×C42—471no—
358C22×C42—471no—
359C22×C42C2×D4711yescharacteristic
360C22×C42C2×D4711yescharacteristic
361C47C22×#55211yescharacteristic, complemented, Sylow
362C25—471no—
363C25C22×D4711yescharacteristic
364C22×Q8—4712no—
365C22×Q8—948no—
366C22×D4—4712no—
367C22×D4—948no—
368C23×C4—28no—
369C23×C4—471no—
370C23×C4—4712no—
371C23×C4—4712no—
372C23×C4—471no—
373C23×C4—471no—
374C23×C4—471no—
375C23×C4—471no—
376C23×C4—471no—
377C23×C4—471no—
378C23×C4—4712no—
379C23×C4—948no—
380C23×C4—948no—
381C23×C4—948no—
382C23×C4C22×D47112yes—
383C23×C4C22×D4711yescharacteristic
384C23×C4C22×D4711yescharacteristic
385C23×C4C22×D4711yescharacteristic
386C23×C4C22×D4711yescharacteristic
387C23×C4C22×D4711yescharacteristic
388C23×C4C22×D4711yescharacteristic
389#126—4764no—
390#126—4764no—
391#125—4764no—
392#125—4764no—
393#125—4764no—
394#123—4764no—
395#123—4764no—
396C22⋊Q8—4764no—
397C22⋊Q8—4764no—
398C4×Q8—4764no—
399C4×D4—4764no—
400#117—4764no—
401#117—4764no—
402#117C22×D47164yescomplemented
403C2×C4⋊C4—4712no—
404C2×C4⋊C4—4712no—
405C2×C4⋊C4—4712no—
406C2×C4⋊C4—4712no—
407C2×C4⋊C4—4764no—
408C2×C4⋊C4—4712no—
409C2×C4⋊C4—4712no—
410C2×C4⋊C4—4712no—
411C2×C4⋊C4—4712no—
412C2×C4⋊C4—4712no—
413C2×C4⋊C4—4712no—
414C2×C4⋊C4—4712no—
415C2×C4⋊C4—4712no—
416C2×C4⋊C4—4712no—
417C2×C4⋊C4—4712no—
418C2×C4⋊C4C22×D47112yes—
419C2×C4⋊C4C22×D47112yes—
420C2×C22⋊C4—4712no—
421C2×C22⋊C4—4712no—
422C2×C22⋊C4—4712no—
423C2×C22⋊C4—4712no—
424C2×C22⋊C4—4712no—
425C2×C22⋊C4—4712no—
426C2×C22⋊C4C22×D47112yes—
427C2×C22⋊C4C22×D47112yes—
428C2×C42—4712no—
429C2×C42—4712no—
430C2×C42—4712no—
431C2×C42C22×D47112yes—
432C2×C42C22×D47112yes—
433C24—24no—
434C24—24no—
435C24—476no—
436C24—944no—
437C24—944no—
438C24—944no—
439C24#1062535018yes—
440C24C23×D4716yes—
441C24C23×D4711yescharacteristic
442C2×Q8—4716no—
443C2×Q8—9448no—
444C2×D4—4716no—
445C2×D4—9448no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—248no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—478no—
454C22×C4—476no—
455C22×C4—476no—
456C22×C4—476no—
457C22×C4—476no—
458C22×C4—476no—
459C22×C4—476no—
460C22×C4—476no—
461C22×C4—4716no—
462C22×C4—4716no—
463C22×C4—4716no—
464C22×C4—9448no—
465C22×C4—9448no—
466C22×C4—944no—
467C22×C4—944no—
468C22×C4—9448no—
469C22×C4—944no—
470C22×C4—944no—
471C22×C4—944no—
472C22×C4—944no—
473C22×C4C23×D4716yes—
474C22×C4C23×D4716yes—
475C22×C4C23×D4716yes—
476C22×C4C23×D4716yes—
477C22×C4C23×D4716yes—
478C22×C4C23×D4716yes—
479C22×C4C23×D47116yes—
480C4⋊C4—4716no—
481C4⋊C4—4716no—
482C4⋊C4—4716no—
483C4⋊C4—4716no—
484C4⋊C4—4716no—
485C4⋊C4—4716no—
486C4⋊C4—4716no—
487C4⋊C4—4716no—
488C4⋊C4—4716no—
489C4⋊C4—4716no—
490C4⋊C4—4716no—
491C4⋊C4—4716no—
492C4⋊C4—4716no—
493C4⋊C4—4716no—
494C4⋊C4C23×D47116yes—
495C4⋊C4C23×D47116yes—
496C22⋊C4—4716no—
497C22⋊C4—4716no—
498C22⋊C4—4716no—
499C22⋊C4—4716no—
500C22⋊C4—4716no—
501C22⋊C4—4716no—
502C22⋊C4C23×D47116yes—
503C22⋊C4C23×D47116yes—
504C42—4716no—
505C42—4716no—
506C42—4716no—
507C42C23×D47116yes—
508C42C23×D47116yes—
509C23—232no—
510C23—212no—
511C23—212no—
512C23—474no—
513C23—9412no—
514C23—9412no—
515C23—9412no—
516C23—18816no—
517C23#41×D4714yes—
518C23C2×#10625350124yes—
519C23C2×#1062535014yes—
520C23#1309407114yes—
521C23C24×D4714yes—
522C23C24×D4713yes—
523Q8—9432no—
524D4—9432no—
525C2×C4—212no—
526C2×C4—212no—
527C2×C4—232no—
528C2×C4—212no—
529C2×C4—212no—
530C2×C4—212no—
531C2×C4—212no—
532C2×C4—4724no—
533C2×C4—474no—
534C2×C4—474no—
535C2×C4—474no—
536C2×C4—474no—
537C2×C4—474no—
538C2×C4—474no—
539C2×C4—474no—
540C2×C4—9432no—
541C2×C4—9432no—
542C2×C4—9412no—
543C2×C4—9412no—
544C2×C4—9432no—
545C2×C4—9412no—
546C2×C4—9412no—
547C2×C4—9412no—
548C2×C4—9412no—
549C2×C4C24×D4714yes—
550C2×C4C24×D4714yes—
551C2×C4C24×D4714yes—
552C2×C4C24×D4714yes—
553C2×C4C24×D4714yes—
554C2×C4C24×D4714yes—
555C22—248no—
556C22—24no—
557C22—24no—
558C22—944no—
559C22—944no—
560C22—944no—
561C22—18824no—
562C22#15689806116yesdirect product factor
563C22C2×#41×D4716yes—
564C22C22×#1062535018yes—
565C22C22×#1062535016yes—
566C22C2×#1309407116yes—
567C22C25×D4711yescharacteristic
568C4—24no—
569C4—24no—
570C4—24no—
571C4—24no—
572C4—24no—
573C4—24no—
574C4—478no—
575C4—944no—
576C4—944no—
577C4—944no—
578C4—944no—
579C4—944no—
580C4—944no—
581C2—28no—
582C2—1884no—
583C2C2×#15689806112yesdirect product factor
584C2C22×#41×D4711yescharacteristic
585C2C23×#1062535011yescharacteristic
586C2C22×#1309407111yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC22×#117×C47C2
socleC24×C47C24
prime radicalC23×C22⋊D47C22
hypercenterC22×#117D47
hypocenterC47C22×#552
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
1. C2×#15689806C2direct (N × H)
2. C22×#13094019C2semidirect
3. C22×#13094018C2semidirect
4. C22×#13094016C2semidirect
5. C22×#13094014C2semidirect
6. C22×#13094014C2semidirect
9. C22×#13094002C2semidirect
10. C22×#13094001C2semidirect
11. C22×C4×Dic94C2semidirect
13. C22×#13093981C2semidirect
14. C22×#13093980C2semidirect
15. C23×Dic47⋊C4C2semidirect
16. C22×#117×C47C2semidirect
22. #15689806C22direct (N × H)
23. C2×#13094019C22semidirect
24. C2×#13094018C22semidirect
25. C2×#13094016C22semidirect
26. C2×#13094014C22semidirect
27. C2×#13094014C22semidirect
30. C2×#13094002C22semidirect
31. C2×#13094001C22semidirect
32. C2×C4×Dic94C22semidirect
34. C2×#13093981C22semidirect
35. C2×#13093980C22semidirect
50. C22×Dic47⋊C4C22semidirect
61. C2×#117×C47C22semidirect
86. #13094019C23semidirect
87. #13094018C23semidirect
88. #13094016C23semidirect
89. #13094014C23semidirect
90. #13094014C23semidirect
93. #13094002C23semidirect
94. #13094001C23semidirect
95. C4×Dic94C23semidirect
97. #13093981C23semidirect
98. #13093980C23semidirect
113. C2×Dic47⋊C4C23semidirect
132. #117×C47C23semidirect
265. C22×C47#552semidirect
302. C22×#117D47semidirect
305. C2×C47C2×#552semidirect
330. C2×#117C2×D47semidirect
361. C47C22×#552semidirect
402. #117C22×D47semidirect
562. C22#15689806direct (N × H)
583. C2C2×#15689806direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156898062
C22×#130940192
C22×#130940182
C22×#130940162
C22×#130940142
C22×#130940082
C22×#130940072
C22×#130940022
C22×#130940012
C22×C4×Dic942
C22×(C22⋊D47)⋊C42
C22×#130939812
C22×#130939802
C23×Dic47⋊C42
C22×#117×C472
C22×#55247