Finite Group Databasebeta

← #26217372

Subgroups — #26217372

Order 12032.

1714 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689596C211yesmaximal, characteristic
2C22×#13093938C211yesmaximal, characteristic
3#20246002C211yesmaximal, characteristic, complemented
4#20246000C211yesmaximal, characteristic, complemented
5#20245963C211yesmaximal, characteristic
6#20243670C218yesmaximal, complemented
7C2×#15688889C211yesmaximal, characteristic, complemented
8C2×C47×#342C211yesmaximal, characteristic, complemented
9C23×C22⋊D47C2211yescharacteristic
10#15689596—22no—
11#15689596—22no—
12#15689596—22no—
13#15689596C2214yes—
14C2×C4×C22⋊D47—22no—
15C2×(C22⋊D47)⋊C4—22no—
16C2×#13093980—21no—
17C22×C22⋊Dic47C2211yescharacteristic
18C22×C22⋊Dic47C2211yescharacteristic
19C22×C22⋊Dic47C2211yescharacteristic
20C2×C22⋊C4×D47—21no—
21C2×#13093938C418yescomplemented
22C2×#13093938C2214yes—
23#15689002—22no—
24#15689000C2214yes—
25#15688998—22no—
26#15688998C2214yescomplemented
27#15688996—22no—
28#15688995C2214yescomplemented
29#15688984—21no—
30#15688983—22no—
31#15688981—22no—
32#15688978—21no—
33#15688889C2214yescomplemented
34C22×C22⋊C4×C47C2211yescharacteristic
35C22×C22⋊C4×C47C2211yescharacteristic
36C22×C22⋊C4×C47C2211yescharacteristic
37C47×#342C2214yescomplemented
38C24×D47D411yescharacteristic
39C22×C22⋊D47—21no—
40C22×C22⋊D47—21no—
41C22×C22⋊D47—21no—
42C22×C22⋊D47—21no—
43C22×C22⋊D47—22no—
44C22×C22⋊D47—22no—
45C22×C22⋊D47C2×C414yes—
46C22×C22⋊D47D412yes—
47C22×C22⋊D47Q814yes—
48C22×C22⋊D47C2312yes—
49C23×Dic47—21no—
50C23×Dic47D411yescharacteristic
51C23×Dic47D412yes—
52C23×Dic47C2312yes—
53C22×C4×D47—22no—
54C4×C22⋊D47—28no—
55C4×C22⋊D47—28no—
56(C22⋊D47)⋊C4—28no—
57(C22⋊D47)⋊C4—28no—
58#13093980—24no—
59#13093980—24no—
60C2×C4⋊Dic47—24no—
61C2×Dic47⋊C4—22no—
62C2×Dic47⋊C4—24no—
63C2×C22⋊Dic47—21no—
64C2×C22⋊Dic47—21no—
65C2×C22⋊Dic47—21no—
66C2×C22⋊Dic47—21no—
67C2×C22⋊Dic47—21no—
68C2×C22⋊Dic47—22no—
69C2×C22⋊Dic47—21no—
70C2×C22⋊Dic47—21no—
71C2×C22⋊Dic47—21no—
72C2×C22⋊Dic47—21no—
73C2×C22⋊Dic47—21no—
74C2×C22⋊Dic47—21no—
75C2×C22⋊Dic47C2×C414yescomplemented
76C2×C22⋊Dic47D412yes—
77C2×C22⋊Dic47C2312yes—
78C2×C22⋊Dic47C2312yes—
79C2×(C2×D47)⋊C4—22no—
80C2×(C2×D47)⋊C4—24no—
81C22⋊C4×D47—24no—
82C22⋊C4×D47—24no—
83C2×C4×Dic47—22no—
84#13093938C2×C4116yescomplemented
85C47⋊#95—24no—
86C47⋊#95—24no—
87C47⋊#95—24no—
88C25×C47C2311yescharacteristic
89C23×C4×C47—21no—
90C23×C4×C47—21no—
91C23×C4×C47D412yes—
92C2×C22⋊C4×C47—21no—
93C2×C22⋊C4×C47—21no—
94C2×C22⋊C4×C47—21no—
95C2×C22⋊C4×C47—21no—
96C2×C22⋊C4×C47—21no—
97C2×C22⋊C4×C47—22no—
98C2×C22⋊C4×C47—22no—
99C2×C22⋊C4×C47—22no—
100C2×C22⋊C4×C47—21no—
101C2×C22⋊C4×C47C2×C414yescomplemented
102C2×C22⋊C4×C47D412yes—
103C2×C22⋊C4×C47C2312yes—
104C2×C22⋊C4×C47C2312yes—
105C2×C22⋊C4×C47C2312yes—
106C24×C47—82no—
107C24×C47C22×C412yes—
108C24×C47C2×D411yescharacteristic
109C24×C47C2×D411yescharacteristic
110C24×C47C2×D411yescharacteristic
111C24×C47C2×D411yescharacteristic
112C24×C47C2×D411yescharacteristic
113C24×C47C2×D411yescharacteristic
114C24×C47C2×Q812yes—
115C24×C47#4112yes—
116C24×C47#4112yes—
117C24×C47C2411yescharacteristic
118C23×D47—21no—
119C23×D47—21no—
120C23×D47—22no—
121C23×D47—82no—
122C23×D47C4⋊C414yescomplemented
123C23×D47C2×D412yes—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—24no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—42no—
130C22×C4×C47—82no—
131C22×C4×C47—82no—
132C22×C4×C47—82no—
133C22×C4×C47C2×D414yes—
134C2×C22⋊D47—24no—
135C2×C22⋊D47—24no—
136C2×C22⋊D47—24no—
137C2×C22⋊D47—24no—
138C2×C22⋊D47—22no—
139C2×C22⋊D47—22no—
140C2×C22⋊D47—24no—
141C2×C22⋊D47—24no—
142C2×C22⋊D47—24no—
143C2×C22⋊D47—24no—
144C2×C22⋊D47—24no—
145C2×C22⋊D47—22no—
146C2×C22⋊D47—22no—
147C2×C22⋊D47—22no—
148C2×C22⋊D47—22no—
149C2×C22⋊D47C4⋊C418yescomplemented
150C2×C22⋊D47C22×C414yes—
151C2×C22⋊D47C2×Q814yes—
152C22×Dic47—21no—
153C22×Dic47—21no—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—24no—
158C22×Dic47—22no—
159C22×Dic47—22no—
160C22×Dic47—22no—
161C22×Dic47—22no—
162C22×Dic47—82no—
163C22×Dic47—82no—
164C22×Dic47—82no—
165C22×Dic47C4⋊C414yescomplemented
166C22×Dic47C2×D412yes—
167C22×Dic47C2×D414yes—
168C2×C4×D47—28no—
169C2×C4×D47—84no—
170C22⋊C4×C47—24no—
171C22⋊C4×C47—22no—
172C22⋊C4×C47—22no—
173C22⋊C4×C47—22no—
174C22⋊C4×C47—22no—
175C22⋊C4×C47—22no—
176C22⋊C4×C47—22no—
177C22⋊C4×C47—22no—
178C22⋊C4×C47—42no—
179C22⋊C4×C47C22×C414yescomplemented
180C22⋊Dic47—24no—
181C22⋊Dic47—22no—
182C22⋊Dic47—22no—
183C22⋊Dic47—22no—
184C22⋊Dic47—22no—
185C22⋊Dic47—22no—
186C22⋊Dic47—41no—
187C22⋊Dic47—41no—
188C22⋊Dic47—41no—
189C22⋊Dic47—41no—
190C22⋊Dic47C22×C414yescomplemented
191(C2×D47)⋊C4—28no—
192(C2×D47)⋊C4—84no—
193C4⋊Dic47—84no—
194Dic47⋊C4—28no—
195Dic47⋊C4—84no—
196C4×Dic47—28no—
197C23×C47—42no—
198C23×C47—42no—
199C23×C47—42no—
200C23×C47—42no—
201C23×C47—42no—
202C23×C47—42no—
203C23×C47—42no—
204C23×C47—82no—
205C23×C47—82no—
206C23×C47—82no—
207C23×C47—82no—
208C23×C47C2×C4⋊C412yes—
209C23×C47C2×C4⋊C412yes—
210C23×C47C4×D412yes—
211C23×C47C4×D412yes—
212C23×C47C4×D412yes—
213C23×C47C4×D412yes—
214C23×C47#12011yescharacteristic
215C23×C47#12011yescharacteristic
216C23×C47#12011yescharacteristic
217C23×C47#12011yescharacteristic
218C23×C47C22⋊Q812yes—
219C23×C47C22⋊Q812yes—
220C23×C47C22⋊Q812yes—
221C23×C47C22⋊Q812yes—
222C23×C47#12312yes—
223C23×C47#12312yes—
224C23×C47C23×C411yescharacteristic
225C23×C47C22×D411yescharacteristic
226C23×C47C22×D411yescharacteristic
227C23×C47C22×D411yescharacteristic
228C23×C47C22×Q811yescharacteristic
229C23×C47C2×#4111yescharacteristic
230C23×C47C2×#4111yescharacteristic
231C22×D47—22no—
232C22×D47—24no—
233C22×D47—22no—
234C22×D47—24no—
235C22×D47—44no—
236C22×D47—44no—
237C22×D47—84no—
238C22×D47—82no—
239C22×D47—84no—
240C22×D47C2×C4⋊C414yes—
241C2×C4×C47—24no—
242C2×C4×C47—24no—
243C2×C4×C47—44no—
244C2×C4×C47—44no—
245C2×C4×C47—44no—
246C2×C4×C47—44no—
247C2×C4×C47—42no—
248C2×C4×C47—82no—
249C2×C4×C47—84no—
250C2×C4×C47—84no—
251C22⋊D47—24no—
252C22⋊D47—24no—
253C22⋊D47—24no—
254C22⋊D47—24no—
255C22⋊D47—28no—
256C22⋊D47—28no—
257C2×Dic47—22no—
258C2×Dic47—22no—
259C2×Dic47—24no—
260C2×Dic47—24no—
261C2×Dic47—24no—
262C2×Dic47—24no—
263C2×Dic47—44no—
264C2×Dic47—44no—
265C2×Dic47—44no—
266C2×Dic47—44no—
267C2×Dic47—44no—
268C2×Dic47—44no—
269C2×Dic47—42no—
270C2×Dic47—82no—
271C2×Dic47—84no—
272C2×Dic47C2×C4⋊C414yes—
273C4×D47—88no—
274#20889—471nomaximal, Sylow
275C22×C47—22no—
276C22×C47—22no—
277C22×C47—22no—
278C22×C47—22no—
279C22×C47—22no—
280C22×C47—22no—
281C22×C47—22no—
282C22×C47—42no—
283C22×C47—42no—
284C22×C47—42no—
285C22×C47—42no—
286C22×C47—42no—
287C22×C47—42no—
288C22×C47—42no—
289C22×C47—42no—
290C22×C47—42no—
291C22×C47—42no—
292C22×C47—42no—
293C22×C47—42no—
294C22×C47—41no—
295C22×C47—41no—
296C22×C47—41no—
297C22×C47—41no—
298C22×C47#38512yes—
299C22×C47#38512yes—
300C22×C47#38512yes—
301C22×C47#38512yes—
302C22×C47C22×C4⋊C411yescharacteristic
303C22×C47C2×C4×D411yescharacteristic
304C22×C47C2×C4×D411yescharacteristic
305C22×C47C2×#12011yescharacteristic
306C22×C47C2×C22⋊Q811yescharacteristic
307C22×C47C2×C22⋊Q811yescharacteristic
308C22×C47C2×#12311yescharacteristic
309C2×D47—24no—
310C2×D47—48no—
311C2×D47—44no—
312C2×D47—44no—
313C2×D47—82no—
314C2×D47—84no—
315C4×C47—44no—
316C4×C47—44no—
317C4×C47—82no—
318Dic47—24no—
319Dic47—44no—
320Dic47—44no—
321Dic47—82no—
322C2×#518—471no—
323C22×#99—471no—
324#2315—471no—
325#2315—471no—
326#2308—471no—
327#1867—478no—
328C2×#342—471no—
329C2×#342D4711yescharacteristic, complemented
330C2×C47—21no—
331C2×C47—21no—
332C2×C47—21no—
333C2×C47—21no—
334C2×C47—22no—
335C2×C47—22no—
336C2×C47—21no—
337C2×C47—21no—
338C2×C47—22no—
339C2×C47—22no—
340C2×C47C2×#38511yescharacteristic
341C2×C47#302612yes—
342D47—44no—
343C23×D4—471no—
344#518—474no—
345#518—942no—
346#518—942no—
347#518—942no—
348C2×C4×D4—942no—
349C2×C4×D4—942no—
350C2×#117—941no—
351C22×C22⋊C4—471no—
352C22×C22⋊C4—471no—
353C22×C22⋊C4—471no—
354C22×C22⋊C4—941no—
355C22×C22⋊C4C2×D4711yescharacteristic
356C22×C22⋊C4C2×D4711yescharacteristic
357C22×C22⋊C4C2×D4711yescharacteristic
358C2×#99—478no—
359C2×#99—474no—
360#386—474no—
361#386—942no—
362#385—474no—
363#385—474no—
364#385—942no—
365#385—942no—
366#381—941no—
367#380—942no—
368#380—942no—
369#379—941no—
370#342—474no—
371#342C2×D4714yescomplemented
372C47#2088911yescharacteristic, complemented, Sylow
373C25—471no—
374C25C22×D4711yescharacteristic
375C22×D4—474no—
376C22×D4—472no—
377C22×D4—474no—
378C22×D4—472no—
379C22×D4—941no—
380C22×D4—941no—
381C22×D4—942no—
382C22×D4—941no—
383C22×D4—941no—
384C22×D4—942no—
385C23×C4—21no—
386C23×C4—21no—
387C23×C4—471no—
388C23×C4—472no—
389C23×C4—472no—
390C23×C4—942no—
391C23×C4—941no—
392C23×C4D18812yes—
393C4×D4—948no—
394C4×D4—948no—
395C4×D4—948no—
396C4×D4—948no—
397#117—944no—
398#117—944no—
399C2×C4⋊C4—942no—
400C2×C4⋊C4—944no—
401C2×C4⋊C4—944no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—474no—
412C2×C22⋊C4—472no—
413C2×C22⋊C4—472no—
414C2×C22⋊C4—472no—
415C2×C22⋊C4—944no—
416C2×C22⋊C4—944no—
417C2×C22⋊C4—942no—
418C2×C22⋊C4—941no—
419C2×C22⋊C4—941no—
420C2×C22⋊C4—941no—
421C2×C22⋊C4—942no—
422C2×C22⋊C4—941no—
423C2×C22⋊C4—941no—
424C2×C22⋊C4—944no—
425C2×C22⋊C4—941no—
426C2×C22⋊C4—941no—
427C2×C22⋊C4—941no—
428C2×C22⋊C4—941no—
429C2×C22⋊C4—941no—
430C2×C22⋊C4—941no—
431C2×C22⋊C4C4×D4714yescomplemented
432C2×C22⋊C4D18812yes—
433C2×C22⋊C4C22×D4712yes—
434C2×C22⋊C4C22×D4712yes—
435C2×C22⋊C4C22×D4712yes—
436C2×C42—942no—
437#99—4716no—
438#95—944no—
439#95—944no—
440#95—944no—
441C24—82no—
442C24—474no—
443C24—472no—
444C24—941no—
445C24—942no—
446C24—941no—
447C24—3762no—
448C24C2×C4×D4712yes—
449C24C2×D18811yescharacteristic
450C24C2×D18811yescharacteristic
451C24#1062535012yes—
452C24D4×D4711yescharacteristic
453C24D4×D4711yescharacteristic
454C24D4×D4711yescharacteristic
455C24D4×D4711yescharacteristic
456C24#1062535212yes—
457C24Q8×D4712yes—
458C24C23×D4711yescharacteristic
459C2×D4—478no—
460C2×D4—474no—
461C2×D4—474no—
462C2×D4—944no—
463C2×D4—944no—
464C2×D4—944no—
465C2×D4—944no—
466C2×D4—942no—
467C2×D4—942no—
468C2×D4—944no—
469C2×D4—944no—
470C2×D4—944no—
471C2×D4—944no—
472C2×D4—944no—
473C2×D4—942no—
474C2×D4—942no—
475C2×D4—942no—
476C2×D4—942no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—42no—
483C22×C4—82no—
484C22×C4—82no—
485C22×C4—82no—
486C22×C4—474no—
487C22×C4—472no—
488C22×C4—474no—
489C22×C4—948no—
490C22×C4—941no—
491C22×C4—941no—
492C22×C4—942no—
493C22×C4—944no—
494C22×C4—942no—
495C22×C4—942no—
496C22×C4—942no—
497C22×C4—942no—
498C22×C4—942no—
499C22×C4—942no—
500C22×C4—3764no—
501C22×C4—3762no—
502C22×C4—3762no—
503C22×C4—3762no—
504C22×C4C2×D18814yes—
505C4⋊C4—948no—
506C4⋊C4—3764no—
507C4⋊C4—3764no—
508C22⋊C4—24no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—42no—
517C22⋊C4—474no—
518C22⋊C4—948no—
519C22⋊C4—944no—
520C22⋊C4—942no—
521C22⋊C4—942no—
522C22⋊C4—942no—
523C22⋊C4—942no—
524C22⋊C4—942no—
525C22⋊C4—3764no—
526C22⋊C4—1881no—
527C22⋊C4—1881no—
528C22⋊C4—1881no—
529C22⋊C4—1881no—
530C22⋊C4C2×C4×D4714yescomplemented
531C42—948no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23—82no—
543C23—474no—
544C23—944no—
545C23—942no—
546C23—944no—
547C23—942no—
548C23—3762no—
549C23—3764no—
550C23—3764no—
551C23—1884no—
552C23—1884no—
553C23C4⋊C4×D4712yes—
554C23C4⋊C4×D4712yes—
555C23(C22⋊D47)⋊C412yes—
556C23(C22⋊D47)⋊C412yes—
557C23C4×D18812yes—
558C23C4×D18812yes—
559C23#1309399411yescharacteristic
560C23#1309399411yescharacteristic
561C23#1309399411yescharacteristic
562C23#1309399411yescharacteristic
563C23#1309400212yes—
564C23#1309400212yes—
565C23#1309400412yes—
566C23#1309400412yes—
567C23#1309400612yes—
568C23#1309400612yes—
569C23C22×C4×D4711yescharacteristic
570C23C2×D4×D4711yescharacteristic
571C23C2×D4×D4711yescharacteristic
572C23C22×D18811yescharacteristic
573C23C2×Q8×D4711yescharacteristic
574C23C2×#1062535211yescharacteristic
575C23C2×#1062535011yescharacteristic
576D4—948no—
577D4—948no—
578D4—944no—
579D4—944no—
580D4—944no—
581D4—944no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—82no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—474no—
593C2×C4—944no—
594C2×C4—944no—
595C2×C4—942no—
596C2×C4—942no—
597C2×C4—944no—
598C2×C4—944no—
599C2×C4—3768no—
600C2×C4—3762no—
601C2×C4—3764no—
602C2×C4—1884no—
603C2×C4—1884no—
604C2×C4—1884no—
605C2×C4—1882no—
606C2×C4—1884no—
607C2×C4—1884no—
608C2×C4—1884no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—944no—
633C22—3764no—
634C22—3762no—
635C22—1884no—
636C22—1888no—
637C22—1884no—
638C22#1568899712yes—
639C22#1568899712yes—
640C22#1568899712yes—
641C22#1568899712yes—
642C22C2×C4⋊C4×D4711yescharacteristic
643C22C2×(C22⋊D47)⋊C411yescharacteristic
644C22C2×C4×D18811yescharacteristic
645C22C2×#1309399411yescharacteristic
646C22C2×#1309400211yescharacteristic
647C22C2×#1309400411yescharacteristic
648C22C2×#1309400611yescharacteristic
649C4—44no—
650C4—44no—
651C4—82no—
652C4—944no—
653C4—3762no—
654C4—1884no—
655C4—1884no—
656C2—21no—
657C2—21no—
658C2—21no—
659C2—22no—
660C2—22no—
661C2—21no—
662C2—22no—
663C2—22no—
664C2—21no—
665C2—21no—
666C2—1884no—
667C2C2×#1568899711yescharacteristic
668C2#2025278712yes—
subgroupG/H
centerC22C2×#13093994
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC2×C47×#342C2
socleC22×C47C2×#120
prime radicalC23×C22⋊D47C22
cosocleC24×C47C24
hypercenterC2×#342D47
hypocenterC47#20889
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C25 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
3. #20246002C2semidirect
4. #20246000C2semidirect
6. #20243670C2semidirect
7. C2×#15688889C2semidirect
8. C2×C47×#342C2semidirect
21. C2×#13093938C4semidirect
26. #15688998C22semidirect
28. #15688995C22semidirect
33. #15688889C22semidirect
37. C47×#342C22semidirect
75. C2×C22⋊Dic47C2×C4semidirect
84. #13093938C2×C4semidirect
101. C2×C22⋊C4×C47C2×C4semidirect
122. C23×D47C4⋊C4semidirect
149. C2×C22⋊D47C4⋊C4semidirect
165. C22×Dic47C4⋊C4semidirect
179. C22⋊C4×C47C22×C4semidirect
190. C22⋊Dic47C22×C4semidirect
329. C2×#342D47semidirect
371. #342C2×D47semidirect
372. C47#20889semidirect
431. C2×C22⋊C4C4×D47semidirect
530. C22⋊C4C2×C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156895962
C22×#130939382
#202460022
#202460002
#202459632
#202436702
C2×#156888892
C2×C47×#3422
#2088947