Finite Group Databasebeta

← #26144702

Subgroups — #26144702

Order 12032.

1390 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689022C212yesmaximal
2C2×#15689009C212yesmaximal
3#20241653C212yesmaximal
4#20241641C212yesmaximal
5#20241641C212yesmaximal
6#20241622C212yesmaximal
7#20241618C211yesmaximal, characteristic
8#20241618C211yesmaximal, characteristic
9C2×C47×#389C211yesmaximal, characteristic
10C22×C4⋊Dic47C2212yes—
11C22×C4⋊Dic47C2212yes—
12C22×C4⋊Dic47C2211yescharacteristic
13C22×C4⋊Dic47C2211yescharacteristic
14C22×C4⋊Dic47C2211yescharacteristic
15C22×C4⋊Dic47C2211yescharacteristic
16C22×C4×Dic47C2212yes—
17#15689022—28no—
18#15689009—28no—
19C2×C47⋊#95C2212yes—
20C2×C47⋊#95C2212yes—
21C2×C47⋊#95C2212yes—
22C2×C47⋊#95C2212yes—
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C22×C4⋊C4×C47C2212yes—
29C22×C4⋊C4×C47C2211yescharacteristic
30C22×C4⋊C4×C47C2211yescharacteristic
31C22×C42×C47C2211yescharacteristic
32C47×#389C418yescomplemented
33C2×#95×C47C2212yes—
34C23×Dic47C2312yes—
35C23×Dic47C2312yes—
36C23×Dic47C2312yes—
37C23×Dic47C2312yes—
38C2×C4⋊Dic47—24no—
39C2×C4⋊Dic47—22no—
40C2×C4⋊Dic47—22no—
41C2×C4⋊Dic47—24no—
42C2×C4⋊Dic47—24no—
43C2×C4⋊Dic47—41no—
44C2×C4⋊Dic47—41no—
45C2×C4⋊Dic47—41no—
46C2×C4⋊Dic47—41no—
47C2×C4⋊Dic47—42no—
48C2×C4⋊Dic47—42no—
49C2×C4⋊Dic47—42no—
50C2×C4⋊Dic47—41no—
51C2×C4⋊Dic47—41no—
52C2×C4⋊Dic47—41no—
53C2×C4⋊Dic47—41no—
54C2×C4⋊Dic47D414yes—
55C2×C4⋊Dic47Q814yes—
56C2×C4×Dic47—24no—
57C2×C4×Dic47—24no—
58C2×C4×Dic47—42no—
59C47⋊#95—24no—
60C47⋊#95—24no—
61C47⋊#95—24no—
62C47⋊#95—24no—
63C47⋊#95—24no—
64C47⋊#95—24no—
65C47⋊#95—24no—
66C47⋊#95—42no—
67C47⋊#95—42no—
68C23×C4×C47C2312yes—
69C23×C4×C47C2311yescharacteristic
70C23×C4×C47C2312yes—
71C23×C4×C47C2311yescharacteristic
72C23×C4×C47C2311yescharacteristic
73C2×C4⋊C4×C47—24no—
74C2×C4⋊C4×C47—24no—
75C2×C4⋊C4×C47—41no—
76C2×C4⋊C4×C47—41no—
77C2×C4⋊C4×C47C2×C418yes—
78C2×C4⋊C4×C47C2×C414yes—
79C2×C4⋊C4×C47C2×C414yes—
80C2×C4⋊C4×C47D414yes—
81C2×C4⋊C4×C47Q814yes—
82C2×C42×C47—41no—
83C2×C42×C47—41no—
84C2×C42×C47C2×C414yes—
85#95×C47C2×C418yes—
86C24×C47C2411yescharacteristic
87C22×C4×C47—22no—
88C22×C4×C47—22no—
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47—21no—
93C22×C4×C47—21no—
94C22×C4×C47—21no—
95C22×C4×C47—22no—
96C22×C4×C47—21no—
97C22×C4×C47—21no—
98C22×C4×C47—21no—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—21no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47—22no—
105C22×C4×C47—22no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—22no—
109C22×C4×C47C22×C414yes—
110C22×C4×C47C22×C412yes—
111C22×C4×C47C22×C414yes—
112C22×C4×C47C22×C412yes—
113C22×C4×C47C22×C412yes—
114C22×C4×C47C2×D412yes—
115C22×C4×C47C2×D414yes—
116C22×C4×C47C2×D412yes—
117C22×C4×C47C2×Q812yes—
118C22×C4×C47C2×Q814yes—
119C22×C4×C47C2×Q812yes—
120C22×C4×C47#4112yes—
121C22×C4×C47#4112yes—
122C22×Dic47—22no—
123C22×Dic47—22no—
124C22×Dic47—22no—
125C22×Dic47—22no—
126C22×Dic47—22no—
127C22×Dic47—22no—
128C22×Dic47—22no—
129C22×Dic47—22no—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47C2×D414yes—
149C22×Dic47C2×Q814yes—
150C4⋊C4×C47—216no—
151C4⋊C4×C47—44no—
152C4⋊C4×C47C4⋊C4116yescomplemented
153C42×C47—44no—
154C4⋊Dic47—44no—
155C4⋊Dic47—44no—
156C4⋊Dic47—48no—
157C4⋊Dic47—48no—
158C4⋊Dic47—44no—
159C4⋊Dic47—44no—
160C4×Dic47—48no—
161C23×C47C23×C411yescharacteristic
162C23×C47C22×D411yescharacteristic
163C23×C47C22×D411yescharacteristic
164C23×C47C22×Q811yescharacteristic
165C23×C47C22×Q811yescharacteristic
166C23×C47C2×#4111yescharacteristic
167C23×C47C2×#4111yescharacteristic
168C23×C47H(5,𝔽2)11yescharacteristic
169C23×C47H(5,𝔽2)11yescharacteristic
170C23×C47H(5,𝔽2)11yescharacteristic
171C23×C47H(5,𝔽2)11yescharacteristic
172C23×C472−1+411yescharacteristic
173C23×C472−1+411yescharacteristic
174C23×C472−1+411yescharacteristic
175C23×C472−1+411yescharacteristic
176C2×C4×C47—24no—
177C2×C4×C47—22no—
178C2×C4×C47—22no—
179C2×C4×C47—22no—
180C2×C4×C47—22no—
181C2×C4×C47—22no—
182C2×C4×C47—22no—
183C2×C4×C47—24no—
184C2×C4×C47—22no—
185C2×C4×C47—22no—
186C2×C4×C47—22no—
187C2×C4×C47—22no—
188C2×C4×C47—22no—
189C2×C4×C47—22no—
190C2×C4×C47—22no—
191C2×C4×C47—22no—
192C2×C4×C47—22no—
193C2×C4×C47—22no—
194C2×C4×C47—22no—
195C2×C4×C47—22no—
196C2×C4×C47—24no—
197C2×C4×C47—24no—
198C2×C4×C47—24no—
199C2×C4×C47—24no—
200C2×C4×C47—24no—
201C2×C4×C47—24no—
202C2×C4×C47—24no—
203C2×C4×C47—42no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47C2×C4⋊C414yes—
208C2×C4×C47C2×C4⋊C418yes—
209C2×C4×C47C4×D414yes—
210C2×C4×C47C4×Q814yes—
211C2×Dic47—24no—
212C2×Dic47—24no—
213C2×Dic47—24no—
214C2×Dic47—24no—
215C2×Dic47—24no—
216C2×Dic47—24no—
217C2×Dic47—42no—
218C2×Dic47—42no—
219C2×Dic47—42no—
220C2×Dic47—42no—
221C2×Dic47—42no—
222C2×Dic47—42no—
223C2×Dic47—42no—
224C2×Dic47—42no—
225C2×Dic47—42no—
226C2×Dic47—42no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47—42no—
235C2×Dic47—42no—
236C2×Dic47—42no—
237C2×Dic47—42no—
238C2×Dic47—42no—
239#14441—471nomaximal, Sylow
240C22×C47C22×C4⋊C411yescharacteristic
241C22×C47C2×C4×D411yescharacteristic
242C22×C47C2×C4×Q811yescharacteristic
243C22×C47#51811yescharacteristic
244C22×C47#51911yescharacteristic
245C22×C47#52011yescharacteristic
246C22×C47#52011yescharacteristic
247C22×C47#53411yescharacteristic
248C22×C47#53711yescharacteristic
249C22×C47#53711yescharacteristic
250C22×C47#53711yescharacteristic
251C22×C47#53811yescharacteristic
252C22×C47#53811yescharacteristic
253C22×C47#53911yescharacteristic
254C22×C47#53911yescharacteristic
255C22×C47#54311yescharacteristic
256C22×C47#54411yescharacteristic
257C22×C47#54411yescharacteristic
258C22×C47#54411yescharacteristic
259C22×C47D4211yescharacteristic
260C22×C47#54711yescharacteristic
261C22×C47#54711yescharacteristic
262C22×C47D4×Q811yescharacteristic
263C22×C47D4×Q811yescharacteristic
264C22×C47D4×Q811yescharacteristic
265C22×C47#55011yescharacteristic
266C22×C47#55411yescharacteristic
267C22×C47#55611yescharacteristic
268C22×C47#55711yescharacteristic
269C22×C47#55711yescharacteristic
270C22×C47Q8211yescharacteristic
271C22×C47#56211yescharacteristic
272C22×C47#56311yescharacteristic
273C22×C47#56311yescharacteristic
274C22×C47#56411yescharacteristic
275C4×C47—24no—
276C4×C47—24no—
277C4×C47—24no—
278C4×C47—28no—
279C4×C47—44no—
280Dic47—44no—
281Dic47—44no—
282Dic47—82no—
283Dic47—82no—
284C2×#389—472no—
285C2×#389D4711yescharacteristic
286C2×#387—472no—
287#1444—472no—
288#1441—472no—
289#1441—472no—
290#1436—472no—
291C4⋊#95—471no—
292C4⋊#95—471no—
293C2×C47#231611yescharacteristic
294C2×C47#232511yescharacteristic
295C2×C47#232911yescharacteristic
296C2×C47D4×C4⋊C411yescharacteristic
297C2×C47Q8×C4⋊C411yescharacteristic
298C2×C47#236511yescharacteristic
299C2×C47#238211yescharacteristic
300C2×C47#268211yescharacteristic
301C2×C47#268311yescharacteristic
302C2×C47#272611yescharacteristic
303C2×C47#272611yescharacteristic
304C2×C47#272711yescharacteristic
305C2×C47#273411yescharacteristic
306C2×C47#280511yescharacteristic
307C2×C47#280611yescharacteristic
308C22×C4⋊C4—471no—
309C22×C4⋊C4—471no—
310C22×C4⋊C4—472no—
311C22×C4⋊C4—471no—
312C22×C4⋊C4—471no—
313C22×C4⋊C4—472no—
314C22×C4⋊C4C2×D4712yes—
315C22×C4⋊C4C2×D4711yescharacteristic
316C22×C4⋊C4C2×D4711yescharacteristic
317C22×C42—472no—
318C22×C42C2×D4711yescharacteristic
319#389—948no—
320#389Dic4718yescomplemented
321#387—948no—
322C2×#95—472no—
323C2×#95—472no—
324C2×#95—472no—
325C2×#95—472no—
326C2×#95—472no—
327C2×#95—472no—
328C2×#95—472no—
329C2×#95—472no—
330C2×#95—472no—
331C2×#95C2×D4712yes—
332C47#1444111yescharacteristic, complemented, Sylow
333C23×C4—472no—
334C23×C4—472no—
335C23×C4—472no—
336C23×C4—472no—
337C23×C4C22×D4712yes—
338C23×C4C22×D4712yes—
339C23×C4C22×D4711yescharacteristic
340C23×C4C22×D4711yescharacteristic
341C23×C4C22×D4711yescharacteristic
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—41no—
345C2×C4⋊C4—41no—
346C2×C4⋊C4—474no—
347C2×C4⋊C4—474no—
348C2×C4⋊C4—944no—
349C2×C4⋊C4—944no—
350C2×C4⋊C4—944no—
351C2×C4⋊C4—942no—
352C2×C4⋊C4—942no—
353C2×C4⋊C4—1881no—
354C2×C4⋊C4—1881no—
355C2×C4⋊C4—1881no—
356C2×C4⋊C4—1881no—
357C2×C4⋊C4—1882no—
358C2×C4⋊C4—1881no—
359C2×C4⋊C4—1881no—
360C2×C4⋊C4—1881no—
361C2×C4⋊C4—1881no—
362C2×C4⋊C4—1882no—
363C2×C4⋊C4—1882no—
364C2×C4⋊C4Dic9414yes—
365C2×C4⋊C4D18814yes—
366C2×C4⋊C4C2×Dic4718yes—
367C2×C4⋊C4C2×Dic4714yes—
368C2×C4⋊C4C2×Dic4714yes—
369C2×C42—41no—
370C2×C42—41no—
371C2×C42—944no—
372C2×C42—944no—
373C2×C42—1882no—
374C2×C42C2×Dic4714yes—
375#95—944no—
376#95—944no—
377#95—944no—
378#95—944no—
379#95—944no—
380#95—944no—
381#95—944no—
382#95—1882no—
383#95—1882no—
384#95C2×Dic4718yes—
385C24C23×D4711yescharacteristic
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—474no—
409C22×C4—474no—
410C22×C4—942no—
411C22×C4—942no—
412C22×C4—942no—
413C22×C4—942no—
414C22×C4—942no—
415C22×C4—942no—
416C22×C4—942no—
417C22×C4—942no—
418C22×C4—942no—
419C22×C4—942no—
420C22×C4—942no—
421C22×C4—942no—
422C22×C4—942no—
423C22×C4—942no—
424C22×C4—942no—
425C22×C4—942no—
426C22×C4—942no—
427C22×C4—942no—
428C22×C4—942no—
429C22×C4—942no—
430C22×C4—942no—
431C22×C4—942no—
432C22×C4—942no—
433C22×C4—942no—
434C22×C4—942no—
435C22×C4—942no—
436C22×C4C2×Dic9412yes—
437C22×C4C2×Dic9414yes—
438C22×C4C2×D18812yes—
439C22×C4C2×D18814yes—
440C22×C4D4×D4712yes—
441C22×C4#1062535212yes—
442C22×C4Q8×D4712yes—
443C22×C4#1062535412yes—
444C22×C4C22×Dic4714yes—
445C22×C4C22×Dic4714yes—
446C22×C4C22×Dic4712yes—
447C22×C4C22×Dic4712yes—
448C22×C4C22×Dic4712yes—
449C4⋊C4—216no—
450C4⋊C4—44no—
451C4⋊C4—1884no—
452C4⋊C4—1884no—
453C4⋊C4—1888no—
454C4⋊C4—1884no—
455C4⋊C4—1884no—
456C4⋊C4—1888no—
457C4⋊C4C4⋊Dic47116yescomplemented
458C42—44no—
459C42—1888no—
460C23C23×Dic4711yescharacteristic
461C23C2×D4×D4711yescharacteristic
462C23C22×D18811yescharacteristic
463C23C2×Q8×D4711yescharacteristic
464C23C22×Dic9411yescharacteristic
465C23C2×#1062535211yescharacteristic
466C23C2×#1062535411yescharacteristic
467C23#1309406911yescharacteristic
468C23#1309406911yescharacteristic
469C23#1309407011yescharacteristic
470C23#1309407011yescharacteristic
471C23#1309407111yescharacteristic
472C23#1309407111yescharacteristic
473C23#1309407211yescharacteristic
474C23#1309407211yescharacteristic
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—944no—
507C2×C4—944no—
508C2×C4—944no—
509C2×C4—944no—
510C2×C4—944no—
511C2×C4—944no—
512C2×C4—1882no—
513C2×C4—1882no—
514C2×C4—1882no—
515C2×C4—1882no—
516C2×C4—1882no—
517C2×C4—1882no—
518C2×C4—1882no—
519C2×C4—1882no—
520C2×C4—1882no—
521C2×C4—1882no—
522C2×C4—1882no—
523C2×C4—1882no—
524C2×C4—1882no—
525C2×C4—1882no—
526C2×C4—1882no—
527C2×C4—1882no—
528C2×C4—1882no—
529C2×C4—1882no—
530C2×C4—1882no—
531C2×C4—1882no—
532C2×C4—1882no—
533C2×C4—1882no—
534C2×C4C2×C4⋊Dic4718yes—
535C2×C4C2×C4⋊Dic4714yes—
536C2×C4D4×Dic4714yes—
537C2×C4Q8×Dic4714yes—
538C22C22×C4⋊Dic4711yescharacteristic
539C22C2×D4×Dic4711yescharacteristic
540C22C2×Q8×Dic4711yescharacteristic
541C22#1568959811yescharacteristic
542C22#1568960111yescharacteristic
543C22#1568960611yescharacteristic
544C22#1568960611yescharacteristic
545C22#1568968111yescharacteristic
546C22#1568970011yescharacteristic
547C22#1568970411yescharacteristic
548C22#1568970411yescharacteristic
549C22#1568971011yescharacteristic
550C22#1568971311yescharacteristic
551C22#1568971811yescharacteristic
552C22#1568972211yescharacteristic
553C22#1568974811yescharacteristic
554C22#1568975211yescharacteristic
555C22#1568975411yescharacteristic
556C22#1568975411yescharacteristic
557C22D4×D18811yescharacteristic
558C22#1568977611yescharacteristic
559C22#1568977711yescharacteristic
560C22#1568978811yescharacteristic
561C22D4×Dic9411yescharacteristic
562C22Q8×D18811yescharacteristic
563C22#1568979511yescharacteristic
564C22#1568983011yescharacteristic
565C22#1568984711yescharacteristic
566C22#1568984811yescharacteristic
567C22#1568985211yescharacteristic
568C22Q8×Dic9411yescharacteristic
569C22#1568987111yescharacteristic
570C22#1568987711yescharacteristic
571C22#1568987711yescharacteristic
572C22C47⋊#56411yescharacteristic
573C4—28no—
574C4—24no—
575C4—24no—
576C4—24no—
577C4—44no—
578C4—3762no—
579C4—3762no—
580C4—1884no—
581C4—1884no—
582C2#2024600611yescharacteristic
583C2#2024609311yescharacteristic
584C2#2024612311yescharacteristic
585C2D4×C4⋊Dic4711yescharacteristic
586C2Q8×C4⋊Dic4711yescharacteristic
587C2#2024644811yescharacteristic
588C2#2024657411yescharacteristic
589C2#2024975011yescharacteristic
590C2#2024975911yescharacteristic
591C2#2025033911yescharacteristic
592C2#2025034011yescharacteristic
593C2#2025034911yescharacteristic
594C2#2025041911yescharacteristic
595C2#2025134111yescharacteristic
596C2#2025135011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C47×#389C2
socle, prime radical, cosocleC24×C47C24
hypercenterC2×#389D47
hypocenterC47#14441
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
32. C47×#389C4semidirect
152. C4⋊C4×C47C4⋊C4semidirect
320. #389Dic47semidirect
332. C47#14441semidirect
457. C4⋊C4C4⋊Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156890222
C2×#156890092
#202416532
#202416412
#202416222
#202416182
C2×C47×#3892
#1444147