Finite Group Databasebeta

← #26234643

Subgroups — #26234643

Order 12032.

982 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689224C211yesmaximal, characteristic
2C2×SD16×Dic47C211yesmaximal, characteristic
3Q8×C4⋊Dic47C211yesmaximal, characteristic, complemented
4#20246298C211yesmaximal, characteristic
5#20244127C211yesmaximal, characteristic, complemented
6#20244122C211yesmaximal, characteristic, complemented
7#20243658C211yesmaximal, characteristic, complemented
8#20243651C211yesmaximal, characteristic
9#20243644C211yesmaximal, characteristic, complemented
10#20243637C211yesmaximal, characteristic
11#20243556C211yesmaximal, characteristic, complemented
12#20243403C211yesmaximal, characteristic, complemented
13#20243388C211yesmaximal, characteristic
14#20243260C211yesmaximal, characteristic, complemented
15C2×C47×#440C211yesmaximal, characteristic
16C2×Q8×Dic47C2211yescharacteristic
17C2×Q8×Dic47C2211yescharacteristic
18C2×D4×Dic47C2211yescharacteristic
19C2×D4×Dic47C2211yescharacteristic
20#15689224—24no—
21SD16×Dic47—24no—
22C2×#13093958C2211yescharacteristic
23C2×#13093958C2211yescharacteristic
24C2×#13093956C2211yescharacteristic
25C2×#13093956C2211yescharacteristic
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×#13093948C2211yescharacteristic
28C2×#13093948C2211yescharacteristic
29C2×#13093945C2211yescharacteristic
30C2×#13093945C2211yescharacteristic
31C2×#13093934C2211yescharacteristic
32C2×C8×Dic47C2211yescharacteristic
33#15689022—21no—
34#15689022—21no—
35#15689022C2211yescharacteristic
36#15689022C2211yescharacteristic
37#15689022C2211yescharacteristic
38#15689022C2211yescharacteristic
39#15688998—21no—
40#15688998—21no—
41#15688993—21no—
42#15688993C2211yescharacteristic
43#15688989C2211yescharacteristic
44#15688983—21no—
45C4×C4⋊Dic47—21no—
46C4×C4⋊Dic47C2211yescharacteristic
47C2×C47⋊#95—21no—
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688877C2211yescharacteristic
53#15688877C2211yescharacteristic
54C22×SD16×C47C2211yescharacteristic
55C2×C4×Q8×C47C2211yescharacteristic
56C2×C4×D4×C47C2211yescharacteristic
57C47×#440C418yescomplemented
58C2×#107×C47C2211yescharacteristic
59C2×#103×C47C2211yescharacteristic
60C2×#102×C47C2211yescharacteristic
61C2×#97×C47C2211yescharacteristic
62C23×Dic47—21no—
63C23×Dic47—21no—
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66Q8×Dic47—22no—
67Q8×Dic47—22no—
68Q8×Dic47—41no—
69Q8×Dic47—41no—
70D4×Dic47—22no—
71D4×Dic47—22no—
72D4×Dic47—41no—
73D4×Dic47—41no—
74C2×C4⋊Dic47—21no—
75C2×C4⋊Dic47—21no—
76C2×C4⋊Dic47—21no—
77C2×C4⋊Dic47—21no—
78C2×C4⋊Dic47—21no—
79C2×C4⋊Dic47—21no—
80C2×C4⋊Dic47—21no—
81C2×C4⋊Dic47—21no—
82C2×C4⋊Dic47D411yescharacteristic
83C2×C4⋊Dic47D411yescharacteristic
84C2×C4⋊Dic47C2311yescharacteristic
85C2×C4⋊Dic47C2311yescharacteristic
86C2×C4⋊Dic47C2311yescharacteristic
87C2×C4⋊Dic47C2311yescharacteristic
88C2×C22⋊Dic47—21no—
89C2×C22⋊Dic47—21no—
90C2×C4×Dic47—21no—
91C2×C4×Dic47—21no—
92C2×C4×Dic47C2311yescharacteristic
93C2×C4×Dic47C2311yescharacteristic
94#13093958—22no—
95#13093958—41no—
96#13093956—22no—
97#13093956—41no—
98C4⋊(C47⋊C8)—22no—
99#13093948—22no—
100#13093948—22no—
101#13093945—22no—
102#13093945—22no—
103#13093934—22no—
104C8×Dic47—22no—
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×D4×C47C2311yescharacteristic
115C23×C4×C47—21no—
116C2×SD16×C47C2×C414yes—
117C2×SD16×C47D414yes—
118C2×SD16×C47Q814yes—
119C22×C8×C47C2311yescharacteristic
120C22×C8×C47C2311yescharacteristic
121C4×Q8×C47—22no—
122C4×Q8×C47C2×C414yescomplemented
123C4×D4×C47—22no—
124C4×D4×C47C2×C414yes—
125C2×C4⋊C4×C47—21no—
126C2×C4⋊C4×C47C2311yescharacteristic
127C2×C4⋊C4×C47C2311yescharacteristic
128C2×C22⋊C4×C47—21no—
129C2×C42×C47—21no—
130C2×C42×C47C2311yescharacteristic
131#107×C47C2×C414yescomplemented
132#103×C47C2×C414yescomplemented
133#102×C47C2×C414yes—
134#97×C47C2×C414yescomplemented
135C24×C47—21no—
136C2×Q8×C47—21no—
137C2×Q8×C47—21no—
138C2×Q8×C47—21no—
139C2×Q8×C47C22×C412yes—
140C2×Q8×C47C2×D412yes—
141C2×Q8×C47C2×Q812yes—
142C2×D4×C47—21no—
143C2×D4×C47—21no—
144C2×D4×C47—21no—
145C2×D4×C47C22×C412yes—
146C2×D4×C47C2×D412yes—
147C2×D4×C47C2×Q812yes—
148C22×C4×C47—21no—
149C22×C4×C47—21no—
150C22×C4×C47—21no—
151C22×C4×C47—24no—
152C22×C4×C47—81no—
153C22×C4×C47—81no—
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2411yescharacteristic
157C22×Dic47—21no—
158C22×Dic47—21no—
159C22×Dic47—21no—
160C22×Dic47—21no—
161C22×Dic47—42no—
162C22×Dic47—42no—
163C22×Dic47—42no—
164C22×Dic47—42no—
165C22×Dic47—81no—
166C22×Dic47—81no—
167C22×Dic47C2×D411yescharacteristic
168C22×Dic47C2×D411yescharacteristic
169C22×Dic47C2×D411yescharacteristic
170C22×Dic47C2×D411yescharacteristic
171SD16×C47C4⋊C4116yescomplemented
172C2×C8×C47—21no—
173C2×C8×C47—21no—
174C2×C8×C47C22×C412yes—
175C2×C8×C47C22×C412yes—
176C2×C8×C47C2×D412yes—
177C2×C8×C47C2×Q812yes—
178C4⋊C4×C47—24no—
179C4⋊C4×C47—21no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47C22×C412yes—
182C4⋊C4×C47C22×C412yes—
183C22⋊C4×C47—24no—
184C42×C47—24no—
185C42×C47C22×C412yes—
186C42×C47#4112yes—
187C22⋊Dic47—42no—
188C22⋊Dic47—42no—
189C4⋊Dic47—22no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47—21no—
193C4⋊Dic47—21no—
194C4⋊Dic47—21no—
195C4⋊Dic47—21no—
196C4⋊Dic47—21no—
197C4⋊Dic47—21no—
198C4⋊Dic47—41no—
199C4⋊Dic47—42no—
200C4⋊Dic47—42no—
201C4⋊Dic47—42no—
202C4⋊Dic47—81no—
203C4⋊Dic47—81no—
204C4⋊Dic47—81no—
205C4⋊Dic47—81no—
206C4⋊Dic47—81no—
207C4×Dic47—21no—
208C4×Dic47—21no—
209C4×Dic47—21no—
210C4×Dic47—21no—
211C4×Dic47—42no—
212C4×Dic47—42no—
213C2×C47⋊C8—21no—
214C2×C47⋊C8—21no—
215C2×C47⋊C8—21no—
216C2×C47⋊C8—21no—
217C2×C47⋊C8—21no—
218C2×C47⋊C8—21no—
219C23×C47—22no—
220C23×C47—22no—
221C23×C47—22no—
222C23×C47—41no—
223C23×C47—41no—
224C23×C47—41no—
225C23×C47—41no—
226C23×C47C22×D411yescharacteristic
227Q8×C47—22no—
228Q8×C47—22no—
229Q8×C47—22no—
230Q8×C47C2×C4⋊C414yes—
231D4×C47—22no—
232D4×C47—22no—
233D4×C47—22no—
234D4×C47C2×C4⋊C414yes—
235C2×C4×C47—22no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—22no—
243C2×C4×C47—22no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—84no—
249C2×C4×C47C4×D412yes—
250C2×C4×C47C4×D412yes—
251C2×C4×C47C23×C411yescharacteristic
252C2×C4×C47C22×D411yescharacteristic
253C2×C4×C47C22×Q811yescharacteristic
254C2×C4×C47C2×#4111yescharacteristic
255C2×C4×C47H(5,𝔽2)11yescharacteristic
256C2×C4×C472−1+411yescharacteristic
257C2×Dic47—21no—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—21no—
261C2×Dic47—21no—
262C2×Dic47—21no—
263C2×Dic47—21no—
264C2×Dic47—21no—
265C2×Dic47—21no—
266C2×Dic47—21no—
267C2×Dic47—21no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278C2×Dic47—41no—
279C2×Dic47—41no—
280C2×Dic47—84no—
281C2×Dic47—84no—
282C2×Dic47#13712yes—
283C8×C47—22no—
284C8×C47C2×C4⋊C414yes—
285C47⋊C8—41no—
286C47⋊C8—41no—
287#22678—471nomaximal, Sylow
288C22×C47—24no—
289C22×C47—42no—
290C22×C47—42no—
291C22×C47—42no—
292C22×C47—42no—
293C22×C47—42no—
294C22×C47—42no—
295C22×C47C2×C4×D411yescharacteristic
296C22×C47D4211yescharacteristic
297C22×C47D4×Q811yescharacteristic
298C22×C47C2×#13711yescharacteristic
299C22×C47#57511yescharacteristic
300C22×C47#57611yescharacteristic
301C22×C47#57711yescharacteristic
302C4×C47—24no—
303C4×C47—22no—
304C4×C47—22no—
305C4×C47—42no—
306C4×C47—42no—
307C4×C47C22×C4⋊C411yescharacteristic
308C4×C47#52011yescharacteristic
309C4×C47#54711yescharacteristic
310C4×C47#55411yescharacteristic
311Dic47—22no—
312Dic47—41no—
313Dic47—41no—
314Dic47—41no—
315Dic47—81no—
316Dic47—81no—
317Dic47—81no—
318Dic47—81no—
319C2×#440—471no—
320C2×#440D4711yescharacteristic
321C2×C4×SD16—471no—
322Q8×C4⋊C4—471no—
323#2349—471no—
324#1943—471no—
325#1942—471no—
326#1865—471no—
327#1864—471no—
328#1863—471no—
329#1862—471no—
330#1847—471no—
331#1805—471no—
332#1802—471no—
333#1771—471no—
334C2×C47—44no—
335C2×C47D4×C4⋊C411yescharacteristic
336C2×C47C4×#13711yescharacteristic
337C2×C47#294611yescharacteristic
338C2×C47#326611yescharacteristic
339C2×C47#326811yescharacteristic
340C2×C47#338511yescharacteristic
341C2×C47#338811yescharacteristic
342C22×SD16C2×D4711yescharacteristic
343C2×C4×Q8—471no—
344C2×C4×Q8—471no—
345C2×C4×Q8C2×D4711yescharacteristic
346C2×C4×D4—471no—
347C2×C4×D4—471no—
348C2×C4×D4C2×D4711yescharacteristic
349#440—944no—
350#440Dic4718yescomplemented
351C4×SD16—944no—
352C2×#107—471no—
353C2×#107—471no—
354C2×#107C2×D4711yescharacteristic
355C2×#106—471no—
356C2×#106—471no—
357C2×C4⋊C8—471no—
358C2×#103—471no—
359C2×#103—471no—
360C2×#103C2×D4711yescharacteristic
361C2×#102—471no—
362C2×#102—471no—
363C2×#102C2×D4711yescharacteristic
364C2×#97—471no—
365C2×#97C2×D4711yescharacteristic
366C2×C4×C8—471no—
367#389—471no—
368#389—471no—
369#389—471no—
370#389—471no—
371#389—941no—
372#389—941no—
373#385—941no—
374#385—941no—
375#384—471no—
376#384—941no—
377#382—471no—
378#380—941no—
379C4×C4⋊C4—471no—
380C4×C4⋊C4—941no—
381C2×#95—941no—
382#340—471no—
383#340—471no—
384#340—471no—
385#340—471no—
386#336—471no—
387#336—471no—
388C47#2267811yescharacteristic, complemented, Sylow
389C22×Q8C22×D4711yescharacteristic
390C22×D4C22×D4711yescharacteristic
391C23×C4—21no—
392C23×C4—941no—
393C23×C4—941no—
394C2×SD16Dic9414yes—
395C2×SD16D18814yes—
396C2×SD16C2×Dic4714yes—
397C22×C8—471no—
398C22×C8—471no—
399C22×C8C22×D4711yescharacteristic
400C22×C8C22×D4711yescharacteristic
401C4×Q8—22no—
402C4×Q8—942no—
403C4×Q8—942no—
404C4×Q8—1881no—
405C4×Q8—1881no—
406C4×Q8C2×Dic4714yescomplemented
407C4×D4—22no—
408C4×D4—942no—
409C4×D4—942no—
410C4×D4—1881no—
411C4×D4—1881no—
412C4×D4C2×Dic4714yes—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—471no—
415C2×C4⋊C4—471no—
416C2×C4⋊C4—471no—
417C2×C4⋊C4—471no—
418C2×C4⋊C4—471no—
419C2×C4⋊C4—471no—
420C2×C4⋊C4—941no—
421C2×C4⋊C4—941no—
422C2×C4⋊C4—941no—
423C2×C4⋊C4—941no—
424C2×C4⋊C4—941no—
425C2×C4⋊C4—941no—
426C2×C4⋊C4—941no—
427C2×C4⋊C4—941no—
428C2×C4⋊C4C22×D4711yescharacteristic
429C2×C4⋊C4C22×D4711yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—941no—
432C2×C22⋊C4—941no—
433C2×C42—21no—
434C2×C42—471no—
435C2×C42—471no—
436C2×C42—941no—
437C2×C42—941no—
438C2×C42C22×D4711yescharacteristic
439#107—942no—
440#107—1881no—
441#107C2×Dic4714yescomplemented
442#106—942no—
443#106—1881no—
444C4⋊C8—942no—
445#103—942no—
446#103—942no—
447#103C2×Dic4714yescomplemented
448#102—942no—
449#102—942no—
450#102C2×Dic4714yes—
451#97—942no—
452#97C2×Dic4714yescomplemented
453C4×C8—942no—
454#95—471no—
455#95—471no—
456#95—941no—
457#95—941no—
458#95—941no—
459#95—941no—
460#95—941no—
461#95—941no—
462C24—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8C2×Dic9412yes—
467C2×Q8C2×D18812yes—
468C2×Q8C22×Dic4712yes—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C2×Dic9412yes—
473C2×D4C2×D18812yes—
474C2×D4C22×Dic4712yes—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—24no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4—471no—
482C22×C4—471no—
483C22×C4—471no—
484C22×C4—471no—
485C22×C4—941no—
486C22×C4—941no—
487C22×C4—941no—
488C22×C4—941no—
489C22×C4—3761no—
490C22×C4—3761no—
491C22×C4—1882no—
492C22×C4—1882no—
493C22×C4—1882no—
494C22×C4—1882no—
495C22×C4D4×D4711yescharacteristic
496C22×C4D4×D4711yescharacteristic
497C22×C4C23×D4711yescharacteristic
498SD16C4⋊Dic47116yescomplemented
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—941no—
502C2×C8—941no—
503C2×C8—941no—
504C2×C8—941no—
505C2×C8—941no—
506C2×C8—941no—
507C2×C8C2×Dic9412yes—
508C2×C8C2×D18812yes—
509C2×C8C22×Dic4712yes—
510C2×C8C22×Dic4712yes—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—942no—
515C4⋊C4—941no—
516C4⋊C4—941no—
517C4⋊C4—941no—
518C4⋊C4—941no—
519C4⋊C4—941no—
520C4⋊C4—941no—
521C4⋊C4—941no—
522C4⋊C4—941no—
523C4⋊C4—3761no—
524C4⋊C4—3761no—
525C4⋊C4—3761no—
526C4⋊C4—3761no—
527C4⋊C4—3761no—
528C4⋊C4—1881no—
529C4⋊C4—1882no—
530C4⋊C4—1882no—
531C4⋊C4—1882no—
532C4⋊C4C22×Dic4712yes—
533C4⋊C4C22×Dic4712yes—
534C22⋊C4—24no—
535C22⋊C4—1882no—
536C22⋊C4—1882no—
537C42—24no—
538C42—941no—
539C42—941no—
540C42—941no—
541C42—941no—
542C42—1882no—
543C42—1882no—
544C42#1062535212yes—
545C42C22×Dic4712yes—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23C2×D4×D4711yescharacteristic
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8C2×C4⋊Dic4714yes—
558D4—22no—
559D4—22no—
560D4—22no—
561D4C2×C4⋊Dic4714yes—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—472no—
577C2×C4—941no—
578C2×C4—941no—
579C2×C4—941no—
580C2×C4—941no—
581C2×C4—941no—
582C2×C4—941no—
583C2×C4—941no—
584C2×C4—941no—
585C2×C4—941no—
586C2×C4—941no—
587C2×C4—941no—
588C2×C4—3764no—
589C2×C4—3764no—
590C2×C4—1881no—
591C2×C4—1881no—
592C2×C4—1881no—
593C2×C4—1881no—
594C2×C4—1881no—
595C2×C4—1881no—
596C2×C4—1881no—
597C2×C4—1881no—
598C2×C4—1881no—
599C2×C4—1881no—
600C2×C4—1881no—
601C2×C4—1881no—
602C2×C4D4×Dic4712yes—
603C2×C4D4×Dic4712yes—
604C2×C4C23×Dic4711yescharacteristic
605C2×C4C22×D18811yescharacteristic
606C2×C4C22×Dic9411yescharacteristic
607C2×C4C2×#1062535211yescharacteristic
608C2×C4#1309407011yescharacteristic
609C2×C4#1309407111yescharacteristic
610C8—22no—
611C8—1881no—
612C8—1881no—
613C8C2×C4⋊Dic4714yes—
614C22—24no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22C2×D4×Dic4711yescharacteristic
622C22D4×D18811yescharacteristic
623C22D4×Dic9411yescharacteristic
624C22#137×D4711yescharacteristic
625C22#1568992211yescharacteristic
626C22#1568992711yescharacteristic
627C22#1568993311yescharacteristic
628C4—24no—
629C4—22no—
630C4—22no—
631C4—42no—
632C4—42no—
633C4—942no—
634C4—3761no—
635C4—3761no—
636C4—3761no—
637C4—3761no—
638C4—1881no—
639C4—1881no—
640C4—1881no—
641C4C22×C4⋊Dic4711yescharacteristic
642C4#1568960611yescharacteristic
643C4#1568977711yescharacteristic
644C4#1568983011yescharacteristic
645C2—44no—
646C2D4×C4⋊Dic4711yescharacteristic
647C2#137×Dic4711yescharacteristic
648C2#2025222111yescharacteristic
649C2#2025490011yescharacteristic
650C2#2025493011yescharacteristic
651C2#2025629911yescharacteristic
652C2#2025633411yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#440C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenterC2×#440D47
hypocenterC47#22678
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
3. Q8×C4⋊Dic47C2semidirect
5. #20244127C2semidirect
6. #20244122C2semidirect
7. #20243658C2semidirect
9. #20243644C2semidirect
11. #20243556C2semidirect
12. #20243403C2semidirect
14. #20243260C2semidirect
57. C47×#440C4semidirect
122. C4×Q8×C47C2×C4semidirect
131. #107×C47C2×C4semidirect
132. #103×C47C2×C4semidirect
134. #97×C47C2×C4semidirect
171. SD16×C47C4⋊C4semidirect
350. #440Dic47semidirect
388. C47#22678semidirect
406. C4×Q8C2×Dic47semidirect
441. #107C2×Dic47semidirect
447. #103C2×Dic47semidirect
452. #97C2×Dic47semidirect
498. SD16C4⋊Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156892242
C2×SD16×Dic472
Q8×C4⋊Dic472
#202462982
#202441272
#202441222
#202436582
#202436512
#202436442
#202436372
#202435562
#202434032
#202433882
#202432602
C2×C47×#4402
#2267847