Finite Group Databasebeta

← #26238581

Subgroups — #26238581

Order 12032.

958 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q16×Dic47C211yesmaximal, characteristic, complemented
2C2×SD16×Dic47C211yesmaximal, characteristic
3#20246463C211yesmaximal, characteristic
4#20246309C211yesmaximal, characteristic, complemented
5#20244200C211yesmaximal, characteristic
6#20244158C211yesmaximal, characteristic, complemented
7#20244073C211yesmaximal, characteristic
8#20244066C211yesmaximal, characteristic, complemented
9#20243745C211yesmaximal, characteristic
10#20243731C211yesmaximal, characteristic, complemented
11#20243658C211yesmaximal, characteristic, complemented
12#20243637C211yesmaximal, characteristic
13#20243253C211yesmaximal, characteristic, complemented
14C4×#13093948C211yesmaximal, characteristic, complemented
15C2×C47×#464C211yesmaximal, characteristic
16C2×Q8×Dic47C2211yescharacteristic
17C2×Q8×Dic47C2211yescharacteristic
18C2×Q8×Dic47C2211yescharacteristic
19C2×D4×Dic47C2211yescharacteristic
20Q16×Dic47—24no—
21SD16×Dic47—24no—
22C2×#13093958C2211yescharacteristic
23C2×#13093956C2211yescharacteristic
24C2×#13093948C2211yescharacteristic
25C2×#13093948C2211yescharacteristic
26C2×#13093948C2211yescharacteristic
27C2×#13093948C2211yescharacteristic
28C2×#13093945C2211yescharacteristic
29C2×#13093945C2211yescharacteristic
30C2×C4×C47⋊C8C2211yescharacteristic
31C2×C8×Dic47C2211yescharacteristic
32C2×C8×Dic47C2211yescharacteristic
33#15689030C2211yescharacteristic
34#15689026C2211yescharacteristic
35#15689022—21no—
36#15689022—21no—
37#15689022C2211yescharacteristic
38#15689022C2211yescharacteristic
39#15689016—21no—
40#15689016—21no—
41#15688998—21no—
42#15688989—21no—
43#15688989C2211yescharacteristic
44C4×C4⋊Dic47—21no—
45C4×C4⋊Dic47C2211yescharacteristic
46C4×C4⋊Dic47C2211yescharacteristic
47C22⋊C4×Dic47—21no—
48#15688886C2211yescharacteristic
49#15688886C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688877C2211yescharacteristic
53#15688877C2211yescharacteristic
54C22×Q16×C47C2211yescharacteristic
55C22×SD16×C47C2211yescharacteristic
56C2×#124×C47C2211yescharacteristic
57C2×C4×Q8×C47C2211yescharacteristic
58C47×#464C418yescomplemented
59C2×C4⋊C8×C47C2211yescharacteristic
60C2×#103×C47C2211yescharacteristic
61C2×#102×C47C2211yescharacteristic
62C23×Dic47—21no—
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65Q8×Dic47—22no—
66Q8×Dic47—22no—
67Q8×Dic47—22no—
68Q8×Dic47—41no—
69Q8×Dic47—41no—
70Q8×Dic47—41no—
71D4×Dic47—22no—
72D4×Dic47—41no—
73C2×C4⋊Dic47—21no—
74C2×C4⋊Dic47—21no—
75C2×C4⋊Dic47—21no—
76C2×C4⋊Dic47—21no—
77C2×C4⋊Dic47—21no—
78C2×C4⋊Dic47D411yescharacteristic
79C2×C4⋊Dic47D411yescharacteristic
80C2×C4⋊Dic47C2311yescharacteristic
81C2×C4⋊Dic47C2311yescharacteristic
82C2×C4⋊Dic47C2311yescharacteristic
83C2×C4⋊Dic47C2311yescharacteristic
84C2×C22⋊Dic47—21no—
85C2×C22⋊Dic47—21no—
86C2×C22⋊Dic47—21no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47—21no—
90C2×C4×Dic47—21no—
91C2×C4×Dic47C2311yescharacteristic
92C2×C4×Dic47C2311yescharacteristic
93#13093958—22no—
94#13093956—22no—
95#13093948—22no—
96#13093948—22no—
97#13093948—41no—
98#13093948D414yes—
99#13093945—22no—
100#13093945—41no—
101C4×C47⋊C8—22no—
102C8×Dic47—22no—
103C8×Dic47—22no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C47⋊#95—21no—
109C47⋊#95—21no—
110C47⋊#95—21no—
111C47⋊#95D411yescharacteristic
112C47⋊#95D411yescharacteristic
113C22×Q8×C47C2311yescharacteristic
114C22×Q8×C47C2311yescharacteristic
115C22×D4×C47C2311yescharacteristic
116C2×Q16×C47—22no—
117C2×Q16×C47—22no—
118C2×Q16×C47C2×C414yescomplemented
119C2×SD16×C47—22no—
120C2×SD16×C47—22no—
121C2×SD16×C47C2×C414yes—
122C22×C8×C47C2311yescharacteristic
123C22×C8×C47C2311yescharacteristic
124#124×C47—22no—
125#124×C47C2×C414yes—
126C4×Q8×C47—22no—
127C4×Q8×C47C2×C414yescomplemented
128C2×C4⋊C4×C47—21no—
129C2×C4⋊C4×C47C2311yescharacteristic
130C2×C22⋊C4×C47—21no—
131C2×C22⋊C4×C47—21no—
132C2×C42×C47—21no—
133C2×C42×C47C2311yescharacteristic
134C4⋊C8×C47C2×C414yescomplemented
135#103×C47C2×C414yescomplemented
136#102×C47C2×C414yes—
137C24×C47—21no—
138C2×Q8×C47—21no—
139C2×Q8×C47—21no—
140C2×Q8×C47—21no—
141C2×Q8×C47—21no—
142C2×Q8×C47—21no—
143C2×Q8×C47—21no—
144C2×Q8×C47—21no—
145C2×Q8×C47—21no—
146C2×Q8×C47C22×C412yes—
147C2×Q8×C47C22×C412yes—
148C2×Q8×C47C2×D412yes—
149C2×Q8×C47#4112yes—
150C2×D4×C47—21no—
151C2×D4×C47—21no—
152C2×D4×C47—21no—
153C2×D4×C47—21no—
154C2×D4×C47—21no—
155C2×D4×C47C22×C412yes—
156C22×C4×C47—21no—
157C22×C4×C47—21no—
158C22×C4×C47—21no—
159C22×C4×C47C2×D411yescharacteristic
160C22×C4×C47C2×D411yescharacteristic
161C22×C4×C47C2411yescharacteristic
162C22×Dic47—21no—
163C22×Dic47—21no—
164C22×Dic47—21no—
165C22×Dic47—21no—
166C22×Dic47—42no—
167C22×Dic47—42no—
168C22×Dic47—81no—
169C22×Dic47C2×D411yescharacteristic
170C22×Dic47C2×D411yescharacteristic
171C22×Dic47C2×D411yescharacteristic
172C22×Dic47C2×D411yescharacteristic
173Q16×C47—28no—
174SD16×C47—28no—
175C2×C8×C47—21no—
176C2×C8×C47—21no—
177C2×C8×C47—21no—
178C2×C8×C47—21no—
179C2×C8×C47C22×C412yes—
180C2×C8×C47C22×C412yes—
181C4⋊C4×C47—24no—
182C4⋊C4×C47—21no—
183C4⋊C4×C47C22×C412yes—
184C22⋊C4×C47—24no—
185C22⋊C4×C47—24no—
186C42×C47—24no—
187C42×C47C22×C412yes—
188C42×C47#4112yes—
189C22⋊Dic47—42no—
190C22⋊Dic47—42no—
191C22⋊Dic47—81no—
192C4⋊Dic47—21no—
193C4⋊Dic47—22no—
194C4⋊Dic47—22no—
195C4⋊Dic47—21no—
196C4⋊Dic47—21no—
197C4⋊Dic47—21no—
198C4⋊Dic47—21no—
199C4⋊Dic47—21no—
200C4⋊Dic47—21no—
201C4⋊Dic47—42no—
202C4⋊Dic47—42no—
203C4⋊Dic47—42no—
204C4⋊Dic47—42no—
205C4⋊Dic47—42no—
206C4⋊Dic47C2×D412yes—
207C4×Dic47—21no—
208C4×Dic47—21no—
209C4×Dic47—21no—
210C4×Dic47—21no—
211C4×Dic47—42no—
212C4×Dic47—42no—
213C4×Dic47—42no—
214C4×Dic47—81no—
215C2×C47⋊C8—21no—
216C2×C47⋊C8—21no—
217C2×C47⋊C8—21no—
218C2×C47⋊C8—21no—
219C2×C47⋊C8—21no—
220C2×C47⋊C8C2×D412yes—
221C23×C47—22no—
222C23×C47—41no—
223C23×C47—41no—
224C23×C47—41no—
225C23×C47—41no—
226C23×C47—41no—
227C23×C47—41no—
228C23×C47C22×D411yescharacteristic
229Q8×C47—22no—
230Q8×C47—22no—
231Q8×C47—22no—
232Q8×C47—22no—
233Q8×C47—22no—
234Q8×C47—41no—
235Q8×C47—41no—
236Q8×C47C4×D414yes—
237D4×C47—22no—
238D4×C47—22no—
239D4×C47—41no—
240D4×C47—41no—
241C2×C4×C47—22no—
242C2×C4×C47—22no—
243C2×C4×C47—22no—
244C2×C4×C47—21no—
245C2×C4×C47—21no—
246C2×C4×C47—21no—
247C2×C4×C47—21no—
248C2×C4×C47—22no—
249C2×C4×C47—22no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47—41no—
254C2×C4×C47C4×D412yes—
255C2×C4×C47C4×D412yes—
256C2×C4×C47C23×C411yescharacteristic
257C2×C4×C47C22×D411yescharacteristic
258C2×C4×C47C2×#4111yescharacteristic
259C2×C4×C47C2×#4111yescharacteristic
260C2×C4×C47H(5,𝔽2)11yescharacteristic
261C2×C4×C472−1+411yescharacteristic
262C2×Dic47—21no—
263C2×Dic47—21no—
264C2×Dic47—21no—
265C2×Dic47—21no—
266C2×Dic47—21no—
267C2×Dic47—21no—
268C2×Dic47—22no—
269C2×Dic47—21no—
270C2×Dic47—21no—
271C2×Dic47—21no—
272C2×Dic47—21no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278C2×Dic47—41no—
279C2×Dic47—41no—
280C2×Dic47—41no—
281C2×Dic47—41no—
282C2×Dic47—41no—
283C2×Dic47—41no—
284C2×Dic47—84no—
285C2×Dic47#13512yes—
286C2×Dic47#13512yes—
287C8×C47—22no—
288C8×C47—22no—
289C47⋊C8—22no—
290C47⋊C8—41no—
291#23000—471nomaximal, Sylow
292C22×C47—42no—
293C22×C47—42no—
294C22×C47—42no—
295C22×C47—81no—
296C22×C47—81no—
297C22×C47—81no—
298C22×C47—81no—
299C22×C47C2×C4×D411yescharacteristic
300C22×C47D4211yescharacteristic
301C22×C47#54811yescharacteristic
302C22×C47C2×#13511yescharacteristic
303C22×C47C2×#13511yescharacteristic
304C22×C47#57711yescharacteristic
305C22×C47#57811yescharacteristic
306C4×C47—24no—
307C4×C47—22no—
308C4×C47—22no—
309C4×C47—42no—
310C4×C47—42no—
311C4×C47C2×C4×D411yescharacteristic
312C4×C47#52011yescharacteristic
313C4×C47#54711yescharacteristic
314C4×C47#55311yescharacteristic
315Dic47—22no—
316Dic47—22no—
317Dic47—41no—
318Dic47—42no—
319Dic47—41no—
320Dic47—81no—
321Dic47—81no—
322Dic47—81no—
323C2×#464D4711yescharacteristic
324C2×C4×Q16—471no—
325C2×C4×SD16—471no—
326#2367—471no—
327#2351—471no—
328#1958—471no—
329#1949—471no—
330#1934—471no—
331#1933—471no—
332#1880—471no—
333#1878—471no—
334#1865—471no—
335#1862—471no—
336C8×C4⋊C4—471no—
337C4×#103—471no—
338C2×C47—82no—
339C2×C47#236111yescharacteristic
340C2×C47C4×#13511yescharacteristic
341C2×C47#294711yescharacteristic
342C2×C47#328211yescharacteristic
343C2×C47#328511yescharacteristic
344C2×C47#336411yescharacteristic
345C2×C47#336511yescharacteristic
346C22×Q16C2×D4711yescharacteristic
347C22×SD16C2×D4711yescharacteristic
348C2×#124C2×D4711yescharacteristic
349C2×C4×Q8—471no—
350C2×C4×Q8—471no—
351C2×C4×Q8—471no—
352C2×C4×Q8C2×D4711yescharacteristic
353C2×C4×D4—471no—
354#464Dic4718yescomplemented
355C4×Q16—944no—
356C4×SD16—944no—
357C2×#107—471no—
358C2×#106—471no—
359C2×C4⋊C8C2×D4711yescharacteristic
360C2×#103—471no—
361C2×#103—471no—
362C2×#103—471no—
363C2×#103—471no—
364C2×#103C2×D4711yescharacteristic
365C2×#102—471no—
366C2×#102—471no—
367C2×#102C2×D4711yescharacteristic
368C2×C4×C8—471no—
369C2×C4×C8—471no—
370C2×C4×C8—471no—
371#391—471no—
372#390—471no—
373#389—471no—
374#389—471no—
375#389—941no—
376#389—941no—
377#388—941no—
378#388—941no—
379#385—941no—
380#382—471no—
381#382—941no—
382C4×C4⋊C4—471no—
383C4×C4⋊C4—471no—
384C4×C4⋊C4—941no—
385C4×C22⋊C4—941no—
386#340—471no—
387#340—471no—
388#340—471no—
389#340—471no—
390#336—471no—
391#336—471no—
392C47#2300011yescharacteristic, complemented, Sylow
393C22×Q8C22×D4711yescharacteristic
394C22×Q8C22×D4711yescharacteristic
395C22×D4C22×D4711yescharacteristic
396C23×C4—941no—
397C2×Q16—22no—
398C2×Q16—22no—
399C2×Q16C2×Dic4714yescomplemented
400C2×SD16—22no—
401C2×SD16—22no—
402C2×SD16C2×Dic4714yes—
403C22×C8—471no—
404C22×C8—471no—
405C22×C8C22×D4711yescharacteristic
406C22×C8C22×D4711yescharacteristic
407#124—22no—
408#124C2×Dic4714yes—
409C4×Q8—22no—
410C4×Q8—942no—
411C4×Q8—942no—
412C4×Q8—942no—
413C4×Q8—1881no—
414C4×Q8—1881no—
415C4×Q8—1881no—
416C4×Q8C2×Dic4714yescomplemented
417C4×D4—942no—
418C4×D4—1881no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—471no—
421C2×C4⋊C4—471no—
422C2×C4⋊C4—471no—
423C2×C4⋊C4—471no—
424C2×C4⋊C4—471no—
425C2×C4⋊C4—471no—
426C2×C4⋊C4—941no—
427C2×C4⋊C4—941no—
428C2×C4⋊C4—941no—
429C2×C4⋊C4—941no—
430C2×C4⋊C4—941no—
431C2×C4⋊C4C22×D4711yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—941no—
435C2×C22⋊C4—941no—
436C2×C22⋊C4—941no—
437C2×C42—21no—
438C2×C42—471no—
439C2×C42—471no—
440C2×C42—941no—
441C2×C42—941no—
442C2×C42—941no—
443C2×C42—941no—
444C2×C42C22×D4711yescharacteristic
445#107—942no—
446#106—942no—
447C4⋊C8C2×Dic4714yescomplemented
448#103—474no—
449#103—942no—
450#103—942no—
451#103—1881no—
452#103C2×Dic4714yescomplemented
453#102—942no—
454#102—1881no—
455#102C2×Dic4714yes—
456C4×C8—942no—
457C4×C8—942no—
458C4×C8—942no—
459#95—471no—
460#95—471no—
461#95—941no—
462#95—941no—
463#95—941no—
464#95—941no—
465#95—941no—
466#95—941no—
467#95—941no—
468C24—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8D4×D4712yes—
478C2×Q8#1062535212yes—
479C2×Q8C22×Dic4712yes—
480C2×Q8C22×Dic4712yes—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C22×Dic4712yes—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—471no—
491C22×C4—471no—
492C22×C4—471no—
493C22×C4—471no—
494C22×C4—941no—
495C22×C4—941no—
496C22×C4—941no—
497C22×C4—941no—
498C22×C4—3761no—
499C22×C4—1882no—
500C22×C4—1882no—
501C22×C4D4×D4711yescharacteristic
502C22×C4D4×D4711yescharacteristic
503C22×C4C23×D4711yescharacteristic
504Q16—28no—
505SD16—28no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—472no—
511C2×C8—941no—
512C2×C8—941no—
513C2×C8—941no—
514C2×C8—941no—
515C2×C8—941no—
516C2×C8C22×Dic4712yes—
517C2×C8C22×Dic4712yes—
518C4⋊C4—24no—
519C4⋊C4—21no—
520C4⋊C4—472no—
521C4⋊C4—942no—
522C4⋊C4—942no—
523C4⋊C4—941no—
524C4⋊C4—941no—
525C4⋊C4—941no—
526C4⋊C4—941no—
527C4⋊C4—941no—
528C4⋊C4—941no—
529C4⋊C4—941no—
530C4⋊C4—1882no—
531C4⋊C4—1882no—
532C4⋊C4—1882no—
533C4⋊C4—1882no—
534C4⋊C4—1882no—
535C4⋊C4C22×Dic4712yes—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—3761no—
539C22⋊C4—1882no—
540C22⋊C4—1882no—
541C42—24no—
542C42—941no—
543C42—941no—
544C42—941no—
545C42—941no—
546C42—3761no—
547C42—1882no—
548C42—1882no—
549C42—1882no—
550C42#1062535212yes—
551C42C22×Dic4712yes—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C2×D4×D4711yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—22no—
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567Q8D4×Dic4714yes—
568D4—22no—
569D4—22no—
570D4—41no—
571D4—41no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—472no—
586C2×C4—472no—
587C2×C4—941no—
588C2×C4—941no—
589C2×C4—941no—
590C2×C4—941no—
591C2×C4—941no—
592C2×C4—941no—
593C2×C4—942no—
594C2×C4—941no—
595C2×C4—941no—
596C2×C4—941no—
597C2×C4—941no—
598C2×C4—3764no—
599C2×C4—1881no—
600C2×C4—1881no—
601C2×C4—1881no—
602C2×C4—1881no—
603C2×C4—1881no—
604C2×C4—1881no—
605C2×C4—1881no—
606C2×C4—1881no—
607C2×C4—1881no—
608C2×C4—1881no—
609C2×C4—1881no—
610C2×C4D4×Dic4712yes—
611C2×C4D4×Dic4712yes—
612C2×C4C23×Dic4711yescharacteristic
613C2×C4C2×D4×D4711yescharacteristic
614C2×C4C2×#1062535211yescharacteristic
615C2×C4C2×#1062535211yescharacteristic
616C2×C4#1309407011yescharacteristic
617C2×C4#1309407111yescharacteristic
618C8—22no—
619C8—22no—
620C8—942no—
621C8—1881no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22C2×D4×Dic4711yescharacteristic
630C22#1568975711yescharacteristic
631C22#1568977911yescharacteristic
632C22#135×D4711yescharacteristic
633C22#135×D4711yescharacteristic
634C22#1568993211yescharacteristic
635C22#1568993911yescharacteristic
636C4—22no—
637C4—22no—
638C4—24no—
639C4—42no—
640C4—42no—
641C4—942no—
642C4—942no—
643C4—3761no—
644C4—3761no—
645C4—3761no—
646C4—1881no—
647C4—1882no—
648C4—1881no—
649C4C2×D4×Dic4711yescharacteristic
650C4#1568960611yescharacteristic
651C4#1568977311yescharacteristic
652C4#1568982511yescharacteristic
653C2—82no—
654C2#2024640811yescharacteristic
655C2#135×Dic4711yescharacteristic
656C2#2025223211yescharacteristic
657C2#2025509011yescharacteristic
658C2#2025512411yescharacteristic
659C2#2025606811yescharacteristic
660C2#2025607911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#464C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenterC2×#464D47
hypocenterC47#23000
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#464

Complements

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

NH ≅ G/Ntype
1. C2×Q16×Dic47C2semidirect
4. #20246309C2semidirect
6. #20244158C2semidirect
8. #20244066C2semidirect
10. #20243731C2semidirect
11. #20243658C2semidirect
13. #20243253C2semidirect
14. C4×#13093948C2semidirect
58. C47×#464C4semidirect
118. C2×Q16×C47C2×C4semidirect
127. C4×Q8×C47C2×C4semidirect
134. C4⋊C8×C47C2×C4semidirect
135. #103×C47C2×C4semidirect
354. #464Dic47semidirect
392. C47#23000semidirect
399. C2×Q16C2×Dic47semidirect
416. C4×Q8C2×Dic47semidirect
447. C4⋊C8C2×Dic47semidirect
452. #103C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×Dic472
C2×SD16×Dic472
#202464632
#202463092
#202442002
#202441582
#202440732
#202440662
#202437452
#202437312
#202436582
#202436372
#202432532
C4×#130939482
C2×C47×#4642
#2300047