Finite Group Databasebeta

← #26139764

Subgroups — #26139764

Order 12032.

2358 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094013C211yesmaximal, characteristic
2C22×Q8×Dic47C211yesmaximal, characteristic, complemented
3C2×#15689025C211yesmaximal, characteristic
4C2×#15689024C211yesmaximal, characteristic
5#20241634C214yesmaximal, complemented
6#20241624C214yesmaximal, complemented
7#20241617C211yesmaximal, characteristic, complemented
8C4×C47⋊#95C211yesmaximal, characteristic, complemented
9C2×C47×#391C211yesmaximal, characteristic, complemented
10C23×D188C2211yescharacteristic
11C2×#13094013—24no—
12C2×#13094013—24no—
13C2×#13094013C418yescomplemented
14C2×Q8×Dic47—24no—
15C2×Q8×Dic47—24no—
16C2×Q8×Dic47—24no—
17C22×C4⋊Dic47C2212yes—
18C22×C4⋊Dic47C2211yescharacteristic
19C22×(C2×D47)⋊C4C2212yes—
20C22×(C2×D47)⋊C4C2214yes—
21C22×(C2×D47)⋊C4C2214yes—
22C22×(C2×D47)⋊C4C2212yes—
23C22×C4×Dic47C2212yes—
24C22×C4×Dic47C2211yescharacteristic
25C22×C4×Dic47C2211yescharacteristic
26#15689025—24no—
27#15689024—24no—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2214yes—
31C23×Q8×C47C2211yescharacteristic
32C22×C4⋊C4×C47C2211yescharacteristic
33C22×C42×C47C2211yescharacteristic
34C47×#391—24no—
35C2×#95×C47C2214yes—
36C24×D47C2312yes—
37C22×D188—22no—
38C22×D188—22no—
39C22×D188—22no—
40C22×D188—22no—
41C22×D188—22no—
42C22×D188—22no—
43C22×D188C2×C414yes—
44C23×Dic47C2312yes—
45C23×Dic47C2311yescharacteristic
46C23×Dic47C2312yes—
47C23×Dic47C2311yescharacteristic
48#13094013—232no—
49Q8×Dic47—232no—
50C2×C4⋊Dic47—24no—
51C2×C4⋊Dic47—22no—
52C2×C4⋊Dic47—22no—
53C2×C4⋊Dic47—24no—
54C2×C4⋊Dic47—24no—
55C2×C4⋊Dic47D414yes—
56C2×(C2×D47)⋊C4—28no—
57C2×(C2×D47)⋊C4—28no—
58C2×(C2×D47)⋊C4—28no—
59C2×(C2×D47)⋊C4—24no—
60C2×(C2×D47)⋊C4—24no—
61C2×(C2×D47)⋊C4—28no—
62C2×(C2×D47)⋊C4—28no—
63C2×(C2×D47)⋊C4—44no—
64C2×(C2×D47)⋊C4—44no—
65C2×(C2×D47)⋊C4C2×C4116yes—
66C2×C4×Dic47—24no—
67C2×C4×Dic47—22no—
68C2×C4×Dic47—22no—
69C2×C4×Dic47—22no—
70C2×C4×Dic47—22no—
71C2×C4×Dic47—24no—
72C2×C4×Dic47—24no—
73C2×C4×Dic47C2×C414yes—
74C2×C4×Dic47D414yes—
75C47⋊#95—24no—
76C47⋊#95—24no—
77C47⋊#95—44no—
78C22×Q8×C47—22no—
79C22×Q8×C47—22no—
80C22×Q8×C47—22no—
81C22×Q8×C47—22no—
82C22×Q8×C47C2×C414yes—
83C22×Q8×C47D414yes—
84C22×Q8×C47D414yes—
85C23×C4×C47C2312yes—
86C23×C4×C47C2312yes—
87C23×C4×C47C2312yes—
88C23×C4×C47C2311yescharacteristic
89C2×C4⋊C4×C47—22no—
90C2×C4⋊C4×C47—22no—
91C2×C4⋊C4×C47—22no—
92C2×C42×C47—22no—
93C2×C42×C47—22no—
94C2×C42×C47—22no—
95#95×C47—28no—
96C24×C47C2411yescharacteristic
97C23×D47—22no—
98C23×D47—22no—
99C23×D47—22no—
100C23×D47—22no—
101C23×D47—24no—
102C23×D47—24no—
103C23×D47—22no—
104C23×D47—22no—
105C23×D47—24no—
106C23×D47—24no—
107C23×D47C22×C414yes—
108C2×Q8×C47—28no—
109C2×Q8×C47—28no—
110C2×Q8×C47—28no—
111C2×Q8×C47—28no—
112C2×Q8×C47—28no—
113C2×Q8×C47—28no—
114C2×Q8×C47C22⋊C4116yescomplemented
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—24no—
129C22×C4×C47—24no—
130C22×C4×C47C22×C414yes—
131C22×C4×C47C22×C412yes—
132C22×C4×C47C2×D414yes—
133C22×C4×C47C2×D414yes—
134C22×C4×C47C2×D412yes—
135C22×C4×C47C2×D412yes—
136C22×C4×C47C2×D412yes—
137C22×C4×C47C2×D412yes—
138C22×C4×C47#4114yes—
139C22×C4×C47#4112yes—
140C22×C4×C47#4112yes—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—21no—
148C22×Dic47—22no—
149C22×Dic47—21no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—24no—
154C22×Dic47—22no—
155C22×Dic47—24no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47C22×C414yes—
159C22×Dic47C2×D414yes—
160C22×Dic47C2×D412yes—
161C22×Dic47C2×D412yes—
162C22×Dic47#4114yes—
163C2×D188—28no—
164C2×D188—28no—
165C2×D188—28no—
166C2×D188—44no—
167C2×D188—44no—
168C2×D188—44no—
169C2×D188—44no—
170C4⋊C4×C47—44no—
171C42×C47—44no—
172(C2×D47)⋊C4—232no—
173(C2×D47)⋊C4—416no—
174(C2×D47)⋊C4—48no—
175(C2×D47)⋊C4—48no—
176C4⋊Dic47—216no—
177C4⋊Dic47—28no—
178C4×Dic47—216no—
179C4×Dic47—28no—
180C4×Dic47—28no—
181C23×C47C23×C411yescharacteristic
182C23×C47C22×D411yescharacteristic
183C23×C47C22×D411yescharacteristic
184C23×C47C22×D411yescharacteristic
185C23×C47C22×D411yescharacteristic
186C23×C47C2×#4112yes—
187C23×C47C2×#4111yescharacteristic
188C23×C47C2×#4111yescharacteristic
189C23×C47C2×#4112yes—
190C23×C47H(5,𝔽2)12yes—
191C23×C472−1+412yes—
192C22×D47—24no—
193C22×D47—24no—
194C22×D47—24no—
195C22×D47—28no—
196C22×D47—28no—
197C22×D47—42no—
198C22×D47—42no—
199C22×D47—42no—
200C22×D47—42no—
201C22×D47—42no—
202C22×D47—44no—
203C22×D47—42no—
204C22×D47—42no—
205C22×D47—42no—
206C22×D47—44no—
207C22×D47—44no—
208C22×D47—44no—
209C22×D47—44no—
210C22×D47—44no—
211C22×D47—44no—
212C22×D47—44no—
213C22×D47—44no—
214C22×D47—44no—
215Q8×C47—232no—
216C2×C4×C47—24no—
217C2×C4×C47—24no—
218C2×C4×C47—24no—
219C2×C4×C47—24no—
220C2×C4×C47—24no—
221C2×C4×C47—24no—
222C2×C4×C47—24no—
223C2×C4×C47—24no—
224C2×C4×C47—24no—
225C2×C4×C47—42no—
226C2×C4×C47—42no—
227C2×C4×C47—42no—
228C2×C4×C47—42no—
229C2×C4×C47—42no—
230C2×C4×C47—42no—
231C2×C4×C47—42no—
232C2×C4×C47—44no—
233C2×C4×C47—44no—
234C2×C4×C47C2×C22⋊C418yes—
235C2×C4×C47C2×C22⋊C414yes—
236C2×C4×C47C4×D414yes—
237C2×C4×C47C4×D414yes—
238C2×C4×C47#12014yes—
239C2×C4×C47#12114yes—
240C2×C4×C47#12114yes—
241C2×C4×C47#12314yes—
242C2×Dic47—22no—
243C2×Dic47—22no—
244C2×Dic47—24no—
245C2×Dic47—24no—
246C2×Dic47—24no—
247C2×Dic47—24no—
248C2×Dic47—24no—
249C2×Dic47—24no—
250C2×Dic47—22no—
251C2×Dic47—22no—
252C2×Dic47—22no—
253C2×Dic47—28no—
254C2×Dic47—42no—
255C2×Dic47—42no—
256C2×Dic47—42no—
257C2×Dic47—42no—
258C2×Dic47—42no—
259C2×Dic47—42no—
260C2×Dic47—42no—
261C2×Dic47—42no—
262C2×Dic47—44no—
263C2×Dic47#12414yescomplemented
264D188—416no—
265#14026—471nomaximal, Sylow
266C22×C47C22×C22⋊C411yescharacteristic
267C22×C47C2×C4×D411yescharacteristic
268C22×C47C2×C4×D411yescharacteristic
269C22×C47C4×#4112yes—
270C22×C47#52012yes—
271C22×C47C2×#12011yescharacteristic
272C22×C47C2×#12111yescharacteristic
273C22×C47C2×#12111yescharacteristic
274C22×C47C2×#12311yescharacteristic
275C22×C47C2×#12411yescharacteristic
276C22×C47#52912yes—
277C22×C47#53211yescharacteristic
278C22×C47#53511yescharacteristic
279C22×C47#53611yescharacteristic
280C22×C47#54212yes—
281C22×C47#54612yes—
282C22×C47#54812yes—
283C22×C47#54812yes—
284C22×C47#54812yes—
285C22×C47#55012yes—
286C22×C47#55012yes—
287C22×C47#55612yes—
288C22×C47#55912yes—
289C2×D47—44no—
290C2×D47—48no—
291C2×D47—48no—
292C2×D47—44no—
293C2×D47—44no—
294C2×D47—84no—
295C2×D47—82no—
296C2×D47—82no—
297C2×D47—84no—
298C2×D47—82no—
299C2×D47—82no—
300C4×C47—28no—
301C4×C47—44no—
302C4×C47—82no—
303C4×C47#38618yes—
304Dic47—24no—
305Dic47—44no—
306Dic47—42no—
307Dic47—44no—
308C22×#124—471no—
309C22×C4×Q8—471no—
310C2×#391D4711yescharacteristic, complemented
311C2×#390—471no—
312C2×#390—471no—
313#1439—474no—
314#1437—474no—
315C4⋊#95—471no—
316C4×#95—471no—
317C2×C47C2×#38611yescharacteristic
318C2×C47C4×#12411yescharacteristic
319C2×C47#232611yescharacteristic
320C2×C47#234112yes—
321C2×C47#236712yes—
322C2×C47#242612yes—
323C2×C47#243111yescharacteristic
324C2×C47#243611yescharacteristic
325C2×C47#244812yes—
326C2×C47#254511yescharacteristic
327C2×C47#254811yescharacteristic
328D47—84no—
329C23×Q8C2×D4711yescharacteristic
330C23×D4—471no—
331C2×#124—478no—
332C2×#124—944no—
333C2×#124—944no—
334C2×C4×Q8—944no—
335C2×C4×Q8—944no—
336C2×C4×Q8—944no—
337C22×C4⋊C4—471no—
338C22×C4⋊C4—472no—
339C22×C4⋊C4C2×D4711yescharacteristic
340C22×C22⋊C4—474no—
341C22×C22⋊C4—472no—
342C22×C22⋊C4—474no—
343C22×C22⋊C4—472no—
344C22×C42—471no—
345C22×C42—471no—
346C22×C42—472no—
347C22×C42C2×D4711yescharacteristic
348#391—24no—
349#390—944no—
350#390—944no—
351C2×#95—472no—
352C2×#95—472no—
353C2×#95—474no—
354C2×#95C2×D4714yes—
355C47#1402611yescharacteristic, complemented, Sylow
356C25—472no—
357C22×Q8—22no—
358C22×Q8—22no—
359C22×Q8—22no—
360C22×Q8—22no—
361C22×Q8C4×D4714yes—
362C22×Q8D18814yes—
363C22×Q8C22⋊D4714yes—
364C22×D4—474no—
365C22×D4—942no—
366C22×D4—942no—
367C22×D4—942no—
368C22×D4—942no—
369C22×D4—942no—
370C22×D4—942no—
371C23×C4—471no—
372C23×C4—471no—
373C23×C4—472no—
374C23×C4—472no—
375C23×C4C22×D4712yes—
376C23×C4C22×D4712yes—
377C23×C4C22×D4712yes—
378C23×C4C22×D4711yescharacteristic
379#124—9432no—
380C4×Q8—9432no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—474no—
385C2×C4⋊C4—944no—
386C2×C4⋊C4—944no—
387C2×C4⋊C4—944no—
388C2×C4⋊C4—942no—
389C2×C4⋊C4—942no—
390C2×C22⋊C4—4716no—
391C2×C22⋊C4—948no—
392C2×C22⋊C4—948no—
393C2×C22⋊C4—948no—
394C2×C22⋊C4—948no—
395C2×C22⋊C4—944no—
396C2×C22⋊C4—944no—
397C2×C22⋊C4—948no—
398C2×C22⋊C4—1884no—
399C2×C22⋊C4—1884no—
400C2×C42—22no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—474no—
404C2×C42—474no—
405C2×C42—944no—
406C2×C42—944no—
407C2×C42—944no—
408C2×C42—942no—
409C2×C42—942no—
410C2×C42—942no—
411C2×C42—942no—
412#95—28no—
413#95—944no—
414#95—944no—
415#95—1884no—
416C24—474no—
417C24—942no—
418C24—942no—
419C24—942no—
420C24—942no—
421C24—944no—
422C24—942no—
423C24—942no—
424C24—944no—
425C24—944no—
426C24—944no—
427C24C23×D4711yescharacteristic
428C2×Q8—28no—
429C2×Q8—28no—
430C2×Q8—28no—
431C2×Q8—28no—
432C2×Q8—28no—
433C2×Q8—28no—
434C2×Q8(C2×D47)⋊C4116yescomplemented
435C2×D4—948no—
436C2×D4—948no—
437C2×D4—948no—
438C2×D4—1884no—
439C2×D4—1884no—
440C2×D4—1884no—
441C2×D4—1884no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—472no—
458C22×C4—474no—
459C22×C4—474no—
460C22×C4—472no—
461C22×C4—474no—
462C22×C4—941no—
463C22×C4—941no—
464C22×C4—942no—
465C22×C4—941no—
466C22×C4—942no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—942no—
470C22×C4—941no—
471C22×C4—942no—
472C22×C4—942no—
473C22×C4—942no—
474C22×C4—944no—
475C22×C4—942no—
476C22×C4—944no—
477C22×C4—942no—
478C22×C4—942no—
479C22×C4C2×C4×D4714yes—
480C22×C4C2×C4×D4712yes—
481C22×C4C2×D18814yes—
482C22×C4C2×D18812yes—
483C22×C4D4×D4712yes—
484C22×C4D4×D4712yes—
485C22×C4#1062535212yes—
486C22×C4#1062535212yes—
487C22×C4#1062535414yes—
488C22×C4C2×C22⋊D4714yes—
489C22×C4C2×C22⋊D4712yes—
490C4⋊C4—44no—
491C4⋊C4—9416no—
492C4⋊C4—948no—
493C22⋊C4—9432no—
494C22⋊C4—18816no—
495C22⋊C4—1888no—
496C22⋊C4—1888no—
497C42—44no—
498C42—9416no—
499C42—948no—
500C42—948no—
501C23—944no—
502C23—944no—
503C23—944no—
504C23—948no—
505C23—948no—
506C23—1884no—
507C23—1884no—
508C23—1884no—
509C23—1884no—
510C23—1884no—
511C23—1884no—
512C23—1884no—
513C23—1884no—
514C23—1884no—
515C23—1882no—
516C23—1882no—
517C23—1882no—
518C23—1882no—
519C23—1882no—
520C23—1884no—
521C23—1882no—
522C23—1882no—
523C23—1882no—
524C23C22×C4×D4711yescharacteristic
525C23C2×D4×D4711yescharacteristic
526C23C2×D4×D4711yescharacteristic
527C23C22×C22⋊D4711yescharacteristic
528C23C22×D18811yescharacteristic
529C23#41×D4712yes—
530C23C2×#1062535211yescharacteristic
531C23C2×#1062535211yescharacteristic
532C23C2×#1062535412yes—
533C23#1309407012yes—
534C23#1309407212yes—
535Q8—232no—
536D4—18816no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—474no—
556C2×C4—944no—
557C2×C4—944no—
558C2×C4—942no—
559C2×C4—942no—
560C2×C4—948no—
561C2×C4—944no—
562C2×C4—944no—
563C2×C4—944no—
564C2×C4—944no—
565C2×C4—942no—
566C2×C4—942no—
567C2×C4—942no—
568C2×C4—1882no—
569C2×C4—1882no—
570C2×C4—1882no—
571C2×C4—1882no—
572C2×C4—1882no—
573C2×C4—1882no—
574C2×C4—1884no—
575C2×C4—1882no—
576C2×C4—1882no—
577C2×C4C2×(C2×D47)⋊C418yes—
578C2×C4C2×(C2×D47)⋊C414yes—
579C2×C4(C22⋊D47)⋊C414yes—
580C2×C4(C22⋊D47)⋊C414yes—
581C2×C4#1309399414yes—
582C2×C4#1309399614yes—
583C2×C4#1309399614yes—
584C2×C4#1309400814yes—
585C22—3762no—
586C22—3762no—
587C22—3762no—
588C22—3764no—
589C22—3764no—
590C22—3762no—
591C22—1884no—
592C22—1884no—
593C22—1884no—
594C22—1888no—
595C22—1888no—
596C22C22×(C2×D47)⋊C411yescharacteristic
597C22C2×(C22⋊D47)⋊C411yescharacteristic
598C22C2×(C22⋊D47)⋊C411yescharacteristic
599C22C4×#1062535412yes—
600C22#1568960512yes—
601C22C2×#1309399411yescharacteristic
602C22C2×#1309399611yescharacteristic
603C22C2×#1309399611yescharacteristic
604C22C2×#1309400811yescharacteristic
605C22#124×D4711yescharacteristic
606C22#1568965912yes—
607C22#1568967411yescharacteristic
608C22#1568969111yescharacteristic
609C22#1568969811yescharacteristic
610C22#1568974612yes—
611C22#1568976412yes—
612C22#1568978312yes—
613C22#1568978412yes—
614C22#1568978512yes—
615C22#1568979212yes—
616C22#1568979512yes—
617C22#1568984712yes—
618C22#1568985512yes—
619C4—28no—
620C4—44no—
621C4—82no—
622C4—944no—
623C4—1884no—
624C4—1884no—
625C4—1882no—
626C4#1568900018yes—
627C2—3764no—
628C2C2×#1568900011yescharacteristic
629C2#2024593111yescharacteristic
630C2#2024610211yescharacteristic
631C2#2024622612yes—
632C2#2024646412yes—
633C2#2024695612yes—
634C2#2024700911yescharacteristic
635C2#2024704511yescharacteristic
636C2#2024718412yes—
637C2#2024836511yescharacteristic
638C2#2024838811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C47×#391C2
socle, cosocleC24×C47C24
prime radicalC23×D188C22
hypercenterC2×#391D47
hypocenterC47#14026
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. C22×Q8×Dic47C2semidirect
5. #20241634C2semidirect
6. #20241624C2semidirect
7. #20241617C2semidirect
8. C4×C47⋊#95C2semidirect
9. C2×C47×#391C2semidirect
13. C2×#13094013C4semidirect
114. C2×Q8×C47C22⋊C4semidirect
263. C2×Dic47#124semidirect
310. C2×#391D47semidirect
355. C47#14026semidirect
434. C2×Q8(C2×D47)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#130940132
C22×Q8×Dic472
C2×#156890252
C2×#156890242
#202416342
#202416242
#202416172
C4×C47⋊#952
C2×C47×#3912
#1402647