Finite Group Databasebeta

← #26238571

Subgroups — #26238571

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×#15689342C211yesmaximal, characteristic
2C2×C4×#10625337C211yesmaximal, characteristic, complemented
3C2×C4×#10625335C211yesmaximal, characteristic
4#20246308C211yesmaximal, characteristic, complemented
5#20244200C211yesmaximal, characteristic
6#20244156C211yesmaximal, characteristic, complemented
7#20244074C211yesmaximal, characteristic
8#20244067C211yesmaximal, characteristic, complemented
9#20243746C211yesmaximal, characteristic
10#20243732C211yesmaximal, characteristic, complemented
11#20243656C211yesmaximal, characteristic, complemented
12#20243635C211yesmaximal, characteristic
13C4⋊C4×C47⋊C8C211yesmaximal, characteristic, complemented
14C4×#13093947C211yesmaximal, characteristic, complemented
15C47×#2367C211yesmaximal, characteristic
16C22×#10625337C2211yescharacteristic
17C22×#10625335C2211yescharacteristic
18C2×C4×Dic94C2211yescharacteristic
19C2×C4×Dic94C2211yescharacteristic
20C2×C4×Dic94C2211yescharacteristic
21#15689342C418yescomplemented
22C4×#10625337—24no—
23C4×#10625335—24no—
24C2×#13093957C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×#13093948C2211yescharacteristic
28C2×#13093948C2211yescharacteristic
29C2×#13093948C2211yescharacteristic
30C2×#13093947C2211yescharacteristic
31C2×#13093947C2211yescharacteristic
32C2×#13093945C2211yescharacteristic
33C2×#13093945C2211yescharacteristic
34C2×#13093945C2211yescharacteristic
35C2×C4×C47⋊C8C2211yescharacteristic
36C2×C4×C47⋊C8C2211yescharacteristic
37C2×C4×C47⋊C8C2211yescharacteristic
38#15689021C2211yescharacteristic
39#15689021C2211yescharacteristic
40#15689018—21no—
41#15689018—21no—
42#15688989C2211yescharacteristic
43#15688987—21no—
44#15688974—21no—
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688878C2211yescharacteristic
50#15688878C2211yescharacteristic
51C2×#124×C47C2211yescharacteristic
52C2×C4×Q8×C47C2211yescharacteristic
53C2×C4×D4×C47C2211yescharacteristic
54C47×#391C2211yescharacteristic
55C47×#390C2211yescharacteristic
56C47×#388—21no—
57C47×#388—21no—
58C47×#385—21no—
59C4×C4⋊C4×C47C2211yescharacteristic
60C4×C4⋊C4×C47C2211yescharacteristic
61C4×C22⋊C4×C47—21no—
62C22×Dic94C2311yescharacteristic
63C2×#10625337—22no—
64C2×#10625337—22no—
65C2×#10625337C2×C414yescomplemented
66C2×#10625335—22no—
67C2×#10625335—22no—
68C2×#10625335C2×C414yes—
69C22×C47⋊C8C2311yescharacteristic
70C22×C47⋊C8C2311yescharacteristic
71C22×C47⋊C8C2311yescharacteristic
72C22×C47⋊C8C2311yescharacteristic
73C4×Dic94—22no—
74C4×Dic94—22no—
75C4×Dic94—22no—
76C4×Dic94—41no—
77C4×Dic94—41no—
78C4×Dic94C2×C414yescomplemented
79C2×C4⋊Dic47C2311yescharacteristic
80C2×C4⋊Dic47C2311yescharacteristic
81C2×C4⋊Dic47C2311yescharacteristic
82C2×Dic47⋊C4—21no—
83C2×Dic47⋊C4—21no—
84C2×Dic47⋊C4—21no—
85C2×Dic47⋊C4—21no—
86C2×Dic47⋊C4—21no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47—21no—
90#13093957—22no—
91#13093955—22no—
92C4⋊(C47⋊C8)C2×C414yescomplemented
93#13093948—22no—
94#13093948—41no—
95#13093948C2×C414yescomplemented
96#13093947—22no—
97#13093947D414yes—
98#13093945—22no—
99#13093945—41no—
100#13093945C2×C414yes—
101C4×C47⋊C8—22no—
102C4×C47⋊C8—22no—
103C4×C47⋊C8—22no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C22×Q8×C47C2311yescharacteristic
109C22×D4×C47C2311yescharacteristic
110C23×C4×C47—21no—
111#124×C47—22no—
112#124×C47C2×C414yes—
113C4×Q8×C47—22no—
114C4×Q8×C47—41no—
115C4×D4×C47—22no—
116C4×D4×C47—41no—
117C2×C4⋊C4×C47—21no—
118C2×C4⋊C4×C47D411yescharacteristic
119C2×C4⋊C4×C47D411yescharacteristic
120C2×C4⋊C4×C47C2311yescharacteristic
121C2×C4⋊C4×C47C2311yescharacteristic
122C2×C22⋊C4×C47—21no—
123C2×C22⋊C4×C47—21no—
124C2×C22⋊C4×C47—21no—
125C2×C22⋊C4×C47—21no—
126C2×C22⋊C4×C47—21no—
127C2×C42×C47—21no—
128C2×C42×C47—21no—
129C2×C42×C47C2311yescharacteristic
130C2×C42×C47C2311yescharacteristic
131C2×C42×C47C2311yescharacteristic
132#95×C47—21no—
133#95×C47—21no—
134#95×C47—21no—
135#95×C47D411yescharacteristic
136#95×C47D411yescharacteristic
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×C47C22×C412yes—
144C2×D4×C47—21no—
145C2×D4×C47—21no—
146C2×D4×C47—21no—
147C2×D4×C47—21no—
148C2×D4×C47—21no—
149C2×D4×C47C22×C412yes—
150C22×C4×C47—21no—
151C22×C4×C47—21no—
152C22×C4×C47—21no—
153C22×C4×C47—42no—
154C22×C4×C47—42no—
155C22×C4×C47—81no—
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×D411yescharacteristic
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2×D411yescharacteristic
160C22×C4×C47C2×D411yescharacteristic
161C22×C4×C47C2×D411yescharacteristic
162C22×C4×C47C2411yescharacteristic
163C22×Dic47—21no—
164C22×Dic47—21no—
165C22×Dic47—21no—
166C22×Dic47—21no—
167C2×Dic94—21no—
168C2×Dic94—21no—
169C2×Dic94—21no—
170C2×Dic94C22×C412yes—
171C2×Dic94C2×D412yes—
172C2×Dic94#4112yes—
173C4⋊C4×C47—22no—
174C4⋊C4×C47—22no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—42no—
179C4⋊C4×C47C2×D412yes—
180C22⋊C4×C47—24no—
181C22⋊C4×C47—24no—
182C22⋊C4×C47—42no—
183C22⋊C4×C47—42no—
184C22⋊C4×C47—81no—
185C42×C47—21no—
186C42×C47—21no—
187C42×C47—21no—
188C42×C47—21no—
189C42×C47—42no—
190C42×C47—81no—
191C42×C47C22×C412yes—
192C42×C47#4112yes—
193#10625337—28no—
194#10625335—28no—
195C4⋊Dic47—21no—
196C4⋊Dic47—21no—
197C4⋊Dic47—21no—
198C4⋊Dic47—21no—
199C4⋊Dic47—21no—
200C4⋊Dic47C22×C412yes—
201Dic47⋊C4—24no—
202Dic47⋊C4—42no—
203Dic47⋊C4—42no—
204Dic47⋊C4—42no—
205Dic47⋊C4—42no—
206C4×Dic47—24no—
207C4×Dic47—42no—
208C4×Dic47—42no—
209C2×C47⋊C8—21no—
210C2×C47⋊C8—21no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8—21no—
214C2×C47⋊C8—21no—
215C2×C47⋊C8—21no—
216C2×C47⋊C8—21no—
217C2×C47⋊C8—21no—
218C2×C47⋊C8C22×C412yes—
219C2×C47⋊C8C22×C412yes—
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—41no—
232Q8×C47—41no—
233D4×C47—22no—
234D4×C47—22no—
235D4×C47—41no—
236D4×C47—41no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—22no—
246C2×C4×C47—21no—
247C2×C4×C47—21no—
248C2×C4×C47—22no—
249C2×C4×C47—21no—
250C2×C4×C47—21no—
251C2×C4×C47—21no—
252C2×C4×C47—21no—
253C2×C4×C47—41no—
254C2×C4×C47—41no—
255C2×C4×C47—41no—
256C2×C4×C47—41no—
257C2×C4×C47—41no—
258C2×C4×C47—41no—
259C2×C4×C47—41no—
260C2×C4×C47—84no—
261C2×C4×C47C4×D412yes—
262C2×C4×C47C4×D412yes—
263C2×C4×C47#13512yes—
264C2×C4×C47#13512yes—
265C2×C4×C47C23×C411yescharacteristic
266C2×C4×C47C22×D411yescharacteristic
267C2×C4×C47C2×#4111yescharacteristic
268C2×C4×C47C2×#4111yescharacteristic
269C2×C4×C47H(5,𝔽2)11yescharacteristic
270C2×C4×C472−1+411yescharacteristic
271C2×Dic47—22no—
272C2×Dic47—22no—
273C2×Dic47—22no—
274C2×Dic47—22no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278C2×Dic47—41no—
279C2×Dic47—41no—
280C2×Dic47—41no—
281C2×Dic47—41no—
282C2×Dic47—41no—
283Dic94—22no—
284Dic94—22no—
285Dic94—22no—
286Dic94C4×D414yes—
287C47⋊C8—22no—
288C47⋊C8—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—22no—
307C4×C47—22no—
308C4×C47—22no—
309C4×C47—22no—
310C4×C47—42no—
311C4×C47—41no—
312C4×C47—42no—
313C4×C47—41no—
314C4×C47—81no—
315C4×C47C2×C4×D411yescharacteristic
316C4×C47#52011yescharacteristic
317C4×C47#54711yescharacteristic
318C4×C47#55311yescharacteristic
319Dic47—24no—
320Dic47—42no—
321Dic47—81no—
322Dic47—81no—
323C2×#464—471no—
324C2×C4×Q16—471no—
325C2×C4×SD16—471no—
326#2367D4711yescharacteristic
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×Q16—471no—
347C22×SD16—471no—
348C2×#124C2×D4711yescharacteristic
349C2×C4×Q8—471no—
350C2×C4×Q8—471no—
351C2×C4×Q8—471no—
352C2×C4×Q8C2×D4711yescharacteristic
353C2×C4×D4C2×D4711yescharacteristic
354#464—478no—
355C4×Q16—944no—
356C4×SD16—944no—
357C2×#107—471no—
358C2×#106—471no—
359C2×C4⋊C8—471no—
360C2×#103—471no—
361C2×#103—471no—
362C2×#103—471no—
363C2×#103—471no—
364C2×#103—471no—
365C2×#102—471no—
366C2×#102—471no—
367C2×#102—471no—
368C2×C4×C8—471no—
369C2×C4×C8—471no—
370C2×C4×C8—471no—
371#391C2×D4711yescharacteristic
372#390C2×D4711yescharacteristic
373#389—471no—
374#389—471no—
375#389—941no—
376#389—941no—
377#388—21no—
378#388—21no—
379#385—21no—
380#382—471no—
381#382—941no—
382C4×C4⋊C4—941no—
383C4×C4⋊C4C2×D4711yescharacteristic
384C4×C4⋊C4C2×D4711yescharacteristic
385C4×C22⋊C4—21no—
386#340—471no—
387#340—471no—
388#340—471no—
389#340—471no—
390#336—471no—
391#336—471no—
392C47#2300011yescharacteristic, complemented, Sylow
393C22×Q8—471no—
394C22×Q8C22×D4711yescharacteristic
395C22×D4C22×D4711yescharacteristic
396C23×C4—21no—
397C2×Q16—474no—
398C2×Q16—942no—
399C2×Q16—942no—
400C2×SD16—474no—
401C2×SD16—942no—
402C2×SD16—942no—
403C22×C8—471no—
404C22×C8—471no—
405C22×C8—471no—
406C22×C8—471no—
407#124—22no—
408#124C4×D4714yes—
409C4×Q8—22no—
410C4×Q8—41no—
411C4×Q8—474no—
412C4×Q8—942no—
413C4×Q8—942no—
414C4×Q8—942no—
415C4×Q8—1881no—
416C4×Q8—1881no—
417C4×D4—22no—
418C4×D4—41no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—471no—
421C2×C4⋊C4—471no—
422C2×C4⋊C4—471no—
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×C4⋊C4C22×D4711yescharacteristic
431C2×C4⋊C4C22×D4711yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C42—21no—
438C2×C42—21no—
439C2×C42—941no—
440C2×C42—941no—
441C2×C42—941no—
442C2×C42C22×D4711yescharacteristic
443C2×C42C22×D4711yescharacteristic
444C2×C42C22×D4711yescharacteristic
445#107—942no—
446#106—942no—
447C4⋊C8—474no—
448#103—474no—
449#103—474no—
450#103—942no—
451#103—942no—
452#103—1881no—
453#102—474no—
454#102—942no—
455#102—1881no—
456C4×C8—942no—
457C4×C8—942no—
458C4×C8—942no—
459#95—21no—
460#95—21no—
461#95—21no—
462#95—941no—
463#95—941no—
464#95—941no—
465#95—941no—
466#95C22⋊D4711yescharacteristic
467#95C22⋊D4711yescharacteristic
468C24—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—472no—
475C2×Q8—472no—
476C2×Q8—472no—
477C2×Q8—941no—
478C2×Q8—941no—
479C2×Q8—941no—
480C2×Q8C2×C4×D4712yes—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C2×C4×D4712yes—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—81no—
493C22×C4—941no—
494C22×C4—941no—
495C22×C4—941no—
496C22×C4—941no—
497C22×C4C2×C22⋊D4711yescharacteristic
498C22×C4C2×C22⋊D4711yescharacteristic
499C22×C4C2×C22⋊D4711yescharacteristic
500C22×C4C2×C22⋊D4711yescharacteristic
501C22×C4C2×C22⋊D4711yescharacteristic
502C22×C4C2×C22⋊D4711yescharacteristic
503C22×C4C23×D4711yescharacteristic
504Q16—948no—
505SD16—948no—
506C2×C8—472no—
507C2×C8—472no—
508C2×C8—472no—
509C2×C8—941no—
510C2×C8—941no—
511C2×C8—941no—
512C2×C8—941no—
513C2×C8—941no—
514C2×C8—941no—
515C2×C8—941no—
516C2×C8—941no—
517C2×C8—941no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—42no—
524C4⋊C4—472no—
525C4⋊C4—944no—
526C4⋊C4—941no—
527C4⋊C4—941no—
528C4⋊C4—941no—
529C4⋊C4—941no—
530C4⋊C4—941no—
531C4⋊C4—1882no—
532C4⋊C4—1882no—
533C4⋊C4—1882no—
534C4⋊C4—1882no—
535C4⋊C4D4×D4712yes—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—81no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—42no—
546C42—81no—
547C42—944no—
548C42—1882no—
549C42—1882no—
550C42C2×C4×D4712yes—
551C42#1062535012yes—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C22×C22⋊D4711yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—41no—
563Q8—41no—
564Q8—474no—
565Q8—942no—
566Q8—942no—
567Q8—942no—
568D4—22no—
569D4—22no—
570D4—41no—
571D4—41no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—942no—
597C2×C4—942no—
598C2×C4—942no—
599C2×C4—942no—
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×C4C4×C22⋊D4712yes—
609C2×C4C4×C22⋊D4712yes—
610C2×C4#1309404412yes—
611C2×C4#1309404412yes—
612C2×C4C22×C4×D4711yescharacteristic
613C2×C4C2×D4×D4711yescharacteristic
614C2×C4#41×D4711yescharacteristic
615C2×C4C2×#1062535011yescharacteristic
616C2×C4#1309406911yescharacteristic
617C2×C4#1309407111yescharacteristic
618C8—942no—
619C8—942no—
620C8—942no—
621C8—1881no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22C2×C4×C22⋊D4711yescharacteristic
630C22D4×C22⋊D4711yescharacteristic
631C22#1568978011yescharacteristic
632C22C2×#1309404411yescharacteristic
633C22C2×#1309404411yescharacteristic
634C22#1568993411yescharacteristic
635C22#1568993811yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—41no—
641C4—42no—
642C4—42no—
643C4—41no—
644C4—81no—
645C4—944no—
646C4—3761no—
647C4—3761no—
648C4—1882no—
649C4C4×D4×D4711yescharacteristic
650C4#1568960211yescharacteristic
651C4#1568977811yescharacteristic
652C4#1568982311yescharacteristic
653C2—82no—
654C2#2024640011yescharacteristic
655C2C4×#1309404411yescharacteristic
656C2#2025222711yescharacteristic
657C2#2025509111yescharacteristic
658C2#2025512511yescharacteristic
659C2#2025606911yescharacteristic
660C2#2025608011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2367C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#2367D47
hypocenterC47#23000
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
2. C2×C4×#10625337C2semidirect
4. #20246308C2semidirect
6. #20244156C2semidirect
8. #20244067C2semidirect
10. #20243732C2semidirect
11. #20243656C2semidirect
13. C4⋊C4×C47⋊C8C2semidirect
14. C4×#13093947C2semidirect
21. #15689342C4semidirect
65. C2×#10625337C2×C4semidirect
78. C4×Dic94C2×C4semidirect
92. C4⋊(C47⋊C8)C2×C4semidirect
95. #13093948C2×C4semidirect
392. C47#23000semidirect

Maximal subgroups

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

subgroupindex
C2×#156893422
C2×C4×#106253372
C2×C4×#106253352
#202463082
#202442002
#202441562
#202440742
#202440672
#202437462
#202437322
#202436562
#202436352
C4⋊C4×C47⋊C82
C4×#130939472
C47×#23672
#2300047