Finite Group Databasebeta

← #26376604

Subgroups — #26376604

Order 12032.

3038 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094003C211yesmaximal, characteristic
2C2×#15689600C211yesmaximal, characteristic, complemented
3#20246415C214yesmaximal, complemented
4#20246393C214yesmaximal, complemented
5#20246258C214yesmaximal, complemented
6#20246233C214yesmaximal
7#20246070C214yesmaximal, complemented
8#20246004C211yesmaximal, characteristic, complemented
9#20245972C211yesmaximal, characteristic, complemented
10C4×#13094003C214yesmaximal, complemented
11#20245771C211yesmaximal, characteristic
12#20245744C211yesmaximal, characteristic, complemented
13C22×#117×C47C211yesmaximal, characteristic
14C23×Dic94C2211yescharacteristic
15C2×#13094003C4116yescomplemented
16C2×#13094003C2212yes—
17C2×#13094003C2212yes—
18C2×#13094003C2212yes—
19C2×#13094003C2212yes—
20#15689600—28no—
21C2×C4×Dic94C2214yescomplemented
22C2×#13093989C2214yescomplemented
23C2×#13093983C2211yescharacteristic
24C2×#13093983C2211yescharacteristic
25C2×#13093980C2214yes—
26C22×C4⋊Dic47C2211yescharacteristic
27C22×Dic47⋊C4C2212yes—
28C22×C22⋊Dic47C2212yes—
29#15689029C2214yescomplemented
30#15689028C2214yescomplemented
31#15689021C2214yescomplemented
32#15689019C2214yescomplemented
33#15689018C2218yescomplemented
34#15689017C2218yes—
35#15689011C2218yes—
36#15689007C2218yescomplemented
37#15689004C2218yescomplemented
38#15688996C2218yes—
39#15688986C2212yes—
40#15688984C2214yes—
41#15688983C2212yes—
42#15688983C2212yes—
43#15688981C2214yes—
44#15688979C2212yes—
45C4⋊C4×Dic47C2214yescomplemented
46C4×Dic47⋊C4C2218yescomplemented
47C4×C4⋊Dic47C2214yescomplemented
48C22⋊C4×Dic47C2214yes—
49C4×C22⋊Dic47C2212yes—
50C4×C22⋊Dic47C2212yes—
51C4×C22⋊Dic47C2218yes—
52C24×C4×C47C2211yescharacteristic
53C2×#117×C47—28no—
54C2×#117×C47C2212yes—
55C2×#117×C47C2212yes—
56C2×#117×C47C2211yescharacteristic
57C2×#117×C47C2212yes—
58C2×#117×C47C2211yescharacteristic
59C22×C4⋊C4×C47C2212yes—
60C22×C22⋊C4×C47C2212yes—
61C22×C42×C47C2212yes—
62C22×Dic94—24no—
63C22×Dic94—24no—
64C22×Dic94C2×C418yescomplemented
65C22×Dic94C2312yes—
66C22×Dic94C2312yes—
67C23×Dic47C2312yes—
68C23×Dic47C2312yes—
69#13094003C2×C4116yes—
70#13094003C2×C4116yes—
71#13094003C2×C4116yescomplemented
72#13094003C2×C4116yescomplemented
73C4×Dic94—216no—
74#13093989—216no—
75#13093983—24no—
76#13093983—24no—
77#13093980—216no—
78C2×C4⋊Dic47C2×C418yescomplemented
79C2×C4⋊Dic47C2312yes—
80C2×C4⋊Dic47C2312yes—
81C2×C4⋊Dic47C2312yes—
82C2×C4⋊Dic47C2312yes—
83C2×Dic47⋊C4C2×C4116yescomplemented
84C2×Dic47⋊C4C2318yes—
85C2×Dic47⋊C4C2318yes—
86C2×Dic47⋊C4C2314yes—
87C2×Dic47⋊C4C2314yes—
88C2×C22⋊Dic47C2×C4116yes—
89C2×C22⋊Dic47C2314yes—
90C2×C22⋊Dic47C2312yes—
91C2×C22⋊Dic47C2312yes—
92C2×C22⋊Dic47C2314yes—
93C2×C22⋊Dic47C2314yes—
94C2×C22⋊Dic47C2314yes—
95C2×C4×Dic47C2312yes—
96C2×C4×Dic47C2312yes—
97C2×C4×Dic47C2318yes—
98C2×C4×Dic47C2318yes—
99C47⋊#95C2314yes—
100C47⋊#95C2318yes—
101C47⋊#95C2314yes—
102C47⋊#95C2318yes—
103C47⋊#95C2318yes—
104C47⋊#95C2318yes—
105C25×C47C2311yescharacteristic
106C23×C4×C47—24no—
107C23×C4×C47—24no—
108C23×C4×C47C2×C418yes—
109C23×C4×C47C2314yes—
110C23×C4×C47C2311yescharacteristic
111C23×C4×C47C2311yescharacteristic
112C23×C4×C47C2311yescharacteristic
113C23×C4×C47C2311yescharacteristic
114C23×C4×C47C2311yescharacteristic
115C23×C4×C47C2311yescharacteristic
116#117×C47—24no—
117#117×C47—28no—
118#117×C47—28no—
119#117×C47—28no—
120#117×C47D418yes—
121C2×C4⋊C4×C47—28no—
122C2×C4⋊C4×C47C2314yes—
123C2×C4⋊C4×C47C2312yes—
124C2×C4⋊C4×C47C2312yes—
125C2×C22⋊C4×C47—28no—
126C2×C22⋊C4×C47C2312yes—
127C2×C22⋊C4×C47C2314yes—
128C2×C22⋊C4×C47C2312yes—
129C2×C42×C47—28no—
130C2×C42×C47C2314yes—
131C2×C42×C47C2312yes—
132C2×C42×C47C2312yes—
133C24×C47—22no—
134C24×C47—22no—
135C24×C47—24no—
136C24×C47—42no—
137C24×C47C22×C414yes—
138C24×C47C2411yescharacteristic
139C24×C47C2411yescharacteristic
140C24×C47C2411yescharacteristic
141C22×C4×C47—22no—
142C22×C4×C47—22no—
143C22×C4×C47—232no—
144C22×C4×C47—22no—
145C22×C4×C47—22no—
146C22×C4×C47—22no—
147C22×C4×C47—22no—
148C22×C4×C47—22no—
149C22×C4×C47—22no—
150C22×C4×C47—22no—
151C22×C4×C47—22no—
152C22×C4×C47—28no—
153C22×C4×C47—28no—
154C22×C4×C47—44no—
155C22×C4×C47C22×C414yes—
156C22×C4×C47C22×C414yes—
157C22×C4×C47C22×C414yes—
158C22×C4×C47C22×C414yes—
159C22×C4×C47C22×C414yes—
160C22×C4×C47C22×C414yes—
161C22×C4×C47C2×D414yes—
162C22×C4×C47#4114yes—
163C22×C4×C47C2414yes—
164C22×C4×C47C2411yescharacteristic
165C22×C4×C47C2411yescharacteristic
166C22×C4×C47C2414yes—
167C22×C4×C47C2411yescharacteristic
168C22×C4×C47C2411yescharacteristic
169C22×Dic47—24no—
170C22×Dic47—24no—
171C22×Dic47—44no—
172C22×Dic47C22×C418yes—
173C22×Dic47C22×C418yes—
174C22×Dic47C2414yes—
175C22×Dic47C2414yes—
176C22×Dic47C2414yes—
177C22×Dic47C2414yes—
178C2×Dic94—232no—
179C2×Dic94—24no—
180C2×Dic94—24no—
181C2×Dic94—24no—
182C2×Dic94—24no—
183C2×Dic94C22×C418yes—
184C2×Dic94C22×C418yes—
185C4⋊C4×C47—28no—
186C4⋊C4×C47—24no—
187C4⋊C4×C47C2×D418yes—
188C22⋊C4×C47—24no—
189C22⋊C4×C47—28no—
190C22⋊C4×C47C2×D418yes—
191C42×C47—28no—
192C42×C47—24no—
193C42×C47C2×D418yes—
194C22⋊Dic47—24no—
195C22⋊Dic47—24no—
196C22⋊Dic47—28no—
197C22⋊Dic47—44no—
198C22⋊Dic47C22×C4116yes—
199C22⋊Dic47C22×C4116yes—
200C4⋊Dic47—24no—
201C4⋊Dic47—24no—
202C4⋊Dic47C22×C418yes—
203C4⋊Dic47C22×C418yes—
204Dic47⋊C4—216no—
205Dic47⋊C4—48no—
206Dic47⋊C4C22×C4116yes—
207Dic47⋊C4C22×C4116yes—
208C4×Dic47—216no—
209C4×Dic47—24no—
210C4×Dic47—24no—
211C4×Dic47—48no—
212C23×C47—21no—
213C23×C47—21no—
214C23×C47—22no—
215C23×C47—22no—
216C23×C47—22no—
217C23×C47—28no—
218C23×C47—28no—
219C23×C47—21no—
220C23×C47—21no—
221C23×C47—22no—
222C23×C47—22no—
223C23×C47—44no—
224C23×C47—48no—
225C23×C47—48no—
226C23×C47C23×C412yes—
227C23×C47C23×C412yes—
228C23×C47C23×C412yes—
229C23×C47C22×D412yes—
230C23×C47C2×#4112yes—
231C23×C472−1+414yes—
232C23×C47C2511yescharacteristic
233C2×C4×C47—28no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—24no—
239C2×C4×C47—24no—
240C2×C4×C47—28no—
241C2×C4×C47—28no—
242C2×C4×C47—28no—
243C2×C4×C47—28no—
244C2×C4×C47—24no—
245C2×C4×C47—24no—
246C2×C4×C47—24no—
247C2×C4×C47—416no—
248C2×C4×C47C4×D4116yes—
249C2×C4×C47C23×C412yes—
250C2×C4×C47C23×C412yes—
251C2×C4×C47C23×C412yes—
252C2×C4×C47C23×C412yes—
253C2×C4×C47C22×D418yes—
254C2×C4×C47C22×D412yes—
255C2×C4×C47C22×D412yes—
256C2×C4×C47C2×#4112yes—
257C2×C4×C47C2×#4112yes—
258C2×Dic47—216no—
259C2×Dic47—24no—
260C2×Dic47—24no—
261C2×Dic47—28no—
262C2×Dic47—24no—
263C2×Dic47—24no—
264C2×Dic47—24no—
265C2×Dic47—24no—
266C2×Dic47—24no—
267C2×Dic47—24no—
268C2×Dic47—48no—
269C2×Dic47C23×C418yes—
270C2×Dic47C23×C418yes—
271Dic94—216no—
272Dic94—216no—
273#34203—471nomaximal, Sylow
274C22×C47—22no—
275C22×C47—22no—
276C22×C47—22no—
277C22×C47—22no—
278C22×C47—24no—
279C22×C47—24no—
280C22×C47—24no—
281C22×C47—24no—
282C22×C47—22no—
283C22×C47—22no—
284C22×C47—22no—
285C22×C47—416no—
286C22×C47—42no—
287C22×C47—42no—
288C22×C47C2×C4×D414yes—
289C22×C47#51914yescomplemented
290C22×C47C24×C411yescharacteristic
291C22×C47C23×D411yescharacteristic
292C22×C47C22×#4111yescharacteristic
293C22×C47C2×2−1+412yes—
294C22×C47#58512yes—
295C4×C47—22no—
296C4×C47—22no—
297C4×C47—28no—
298C4×C47—44no—
299C4×C47C2×C4×D414yes—
300C4×C47C2×C4×D414yes—
301Dic47—28no—
302Dic47—28no—
303Dic47—44no—
304Dic47—44no—
305C22×C22⋊Q8—471no—
306C2×#519—471no—
307C22×#117D4711yescharacteristic
308#2362—474no—
309#2360—474no—
310#2344—474no—
311#2342—474no—
312#2323—474no—
313#2316—471no—
314#2310—471no—
315C4×C22⋊Q8—474no—
316#2280—471no—
317#2274—471no—
318C2×C47—24no—
319C2×C47—24no—
320C2×C47—42no—
321C2×C47C22×C4×D411yescharacteristic
322C2×C47C2×#51911yescharacteristic
323C2×C47#342811yescharacteristic
324C2×C47#348912yes—
325C2×C47#349512yes—
326C23×Q8—471no—
327C24×C4C2×D4711yescharacteristic
328C2×C22⋊Q8—4716no—
329C2×C22⋊Q8—472no—
330C2×C22⋊Q8—472no—
331C2×C22⋊Q8—472no—
332C2×C22⋊Q8—472no—
333#519—948no—
334C2×C4×Q8—474no—
335C2×C4×Q8—474no—
336C2×#117—28no—
337C2×#117—471no—
338C2×#117—471no—
339C2×#117—474no—
340C2×#117C2×D4711yescharacteristic
341C2×#117C2×D4712yes—
342C2×#117C2×D4712yes—
343C2×#117C2×D4712yes—
344C2×#117C2×D4711yescharacteristic
345C22×C4⋊C4—471no—
346C22×C4⋊C4—472no—
347C22×C4⋊C4C2×D4712yes—
348C22×C22⋊C4—472no—
349C22×C22⋊C4C2×D4712yes—
350C22×C42C2×D4712yes—
351#391—474no—
352#391—474no—
353#389—474no—
354#389—478no—
355#389—474no—
356#388—478no—
357#388—478no—
358#387—478no—
359#387—478no—
360#385—478no—
361#381—472no—
362#381—474no—
363#380—472no—
364#380—472no—
365#380—474no—
366#379—472no—
367C4×C4⋊C4—474no—
368C4×C4⋊C4—474no—
369C4×C4⋊C4—478no—
370C4×C22⋊C4—478no—
371C4×C22⋊C4—472no—
372C4×C22⋊C4—472no—
373C4×C22⋊C4—474no—
374C47#3420311yescharacteristic, complemented, Sylow
375C25C22×D4711yescharacteristic
376C22×Q8—478no—
377C22×Q8—472no—
378C22×Q8—472no—
379C22×Q8—944no—
380C22×Q8—944no—
381C23×C4—24no—
382C23×C4—24no—
383C23×C4—472no—
384C23×C4—472no—
385C23×C4C4×D4718yes—
386C23×C4C22×D4714yes—
387C23×C4C22×D4711yescharacteristic
388C23×C4C22×D4711yescharacteristic
389C23×C4C22×D4711yescharacteristic
390C23×C4C22×D4711yescharacteristic
391C23×C4C22×D4711yescharacteristic
392C23×C4C22×D4711yescharacteristic
393C22⋊Q8—4716no—
394C22⋊Q8—4716no—
395C22⋊Q8—4716no—
396C22⋊Q8—4716no—
397C4×Q8—9416no—
398C4×Q8—9416no—
399#117—24no—
400#117—28no—
401#117—28no—
402#117—28no—
403#117—944no—
404#117—944no—
405#117—9416no—
406#117C22⋊D4718yes—
407C2×C4⋊C4—28no—
408C2×C4⋊C4—4716no—
409C2×C4⋊C4—478no—
410C2×C4⋊C4—474no—
411C2×C4⋊C4—474no—
412C2×C4⋊C4—472no—
413C2×C4⋊C4—472no—
414C2×C4⋊C4—478no—
415C2×C4⋊C4—472no—
416C2×C4⋊C4—472no—
417C2×C4⋊C4—478no—
418C2×C4⋊C4C22×D4714yes—
419C2×C4⋊C4C22×D4712yes—
420C2×C4⋊C4C22×D4712yes—
421C2×C22⋊C4—28no—
422C2×C22⋊C4—4716no—
423C2×C22⋊C4—474no—
424C2×C22⋊C4—474no—
425C2×C22⋊C4—472no—
426C2×C22⋊C4—472no—
427C2×C22⋊C4—474no—
428C2×C22⋊C4—474no—
429C2×C22⋊C4C22×D4712yes—
430C2×C22⋊C4C22×D4714yes—
431C2×C22⋊C4C22×D4712yes—
432C2×C42—28no—
433C2×C42—472no—
434C2×C42—478no—
435C2×C42—472no—
436C2×C42—478no—
437C2×C42C22×D4714yes—
438C2×C42C22×D4712yes—
439C2×C42C22×D4712yes—
440#95—474no—
441#95—478no—
442#95—474no—
443#95—478no—
444#95—478no—
445#95—478no—
446C24—22no—
447C24—22no—
448C24—24no—
449C24—42no—
450C24C2×C4×D4714yes—
451C24C23×D4711yescharacteristic
452C24C23×D4711yescharacteristic
453C24C23×D4711yescharacteristic
454C2×Q8—478no—
455C2×Q8—478no—
456C2×Q8—9432no—
457C2×Q8—944no—
458C2×Q8—944no—
459C2×Q8—944no—
460C2×Q8—944no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—232no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—28no—
473C22×C4—28no—
474C22×C4—44no—
475C22×C4—478no—
476C22×C4—474no—
477C22×C4—474no—
478C22×C4—478no—
479C22×C4—474no—
480C22×C4—474no—
481C22×C4—944no—
482C22×C4—944no—
483C22×C4—1884no—
484C22×C4C2×C4×D4714yes—
485C22×C4C2×C4×D4714yes—
486C22×C4C2×C4×D4714yes—
487C22×C4C2×C4×D4714yes—
488C22×C4C2×C4×D4714yes—
489C22×C4C2×C4×D4714yes—
490C22×C4#1062535014yes—
491C22×C4C2×C22⋊D4714yes—
492C22×C4C23×D4714yes—
493C22×C4C23×D4711yescharacteristic
494C22×C4C23×D4711yescharacteristic
495C22×C4C23×D4714yes—
496C22×C4C23×D4711yescharacteristic
497C22×C4C23×D4711yescharacteristic
498C4⋊C4—28no—
499C4⋊C4—24no—
500C4⋊C4—4716no—
501C4⋊C4—4716no—
502C4⋊C4—478no—
503C4⋊C4—478no—
504C4⋊C4—9416no—
505C4⋊C4—944no—
506C4⋊C4—944no—
507C4⋊C4—1888no—
508C4⋊C4C2×C22⋊D4718yes—
509C22⋊C4—24no—
510C22⋊C4—28no—
511C22⋊C4—4716no—
512C22⋊C4—4716no—
513C22⋊C4—944no—
514C22⋊C4—944no—
515C22⋊C4—948no—
516C22⋊C4—1884no—
517C22⋊C4C2×C22⋊D4718yes—
518C42—28no—
519C42—24no—
520C42—9416no—
521C42—944no—
522C42—944no—
523C42—1888no—
524C42C2×C22⋊D4718yes—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—28no—
531C23—28no—
532C23—21no—
533C23—21no—
534C23—22no—
535C23—22no—
536C23—44no—
537C23—48no—
538C23—48no—
539C23C22×C4×D4712yes—
540C23C22×C4×D4712yes—
541C23C22×C4×D4712yes—
542C23C22×C22⋊D4712yes—
543C23C2×#1062535012yes—
544C23#1309407114yes—
545C23C24×D4711yescharacteristic
546Q8—9416no—
547Q8—9416no—
548C2×C4—28no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—416no—
563C2×C4—478no—
564C2×C4—478no—
565C2×C4—9416no—
566C2×C4—944no—
567C2×C4—944no—
568C2×C4—948no—
569C2×C4—944no—
570C2×C4—944no—
571C2×C4—944no—
572C2×C4—944no—
573C2×C4—944no—
574C2×C4—944no—
575C2×C4—1888no—
576C2×C4C4×C22⋊D47116yes—
577C2×C4C22×C4×D4712yes—
578C2×C4C22×C4×D4712yes—
579C2×C4C22×C4×D4712yes—
580C2×C4C22×C4×D4712yes—
581C2×C4C22×C22⋊D4718yes—
582C2×C4C22×C22⋊D4712yes—
583C2×C4C22×C22⋊D4712yes—
584C2×C4C2×#1062535012yes—
585C2×C4C2×#1062535012yes—
586C22—24no—
587C22—24no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—24no—
593C22—24no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—416no—
598C22—42no—
599C22—42no—
600C22C2×C4×C22⋊D4714yes—
601C22#1568960014yescomplemented
602C22C23×C4×D4711yescharacteristic
603C22C23×C22⋊D4711yescharacteristic
604C22C22×#1062535011yescharacteristic
605C22C2×#1309407112yes—
606C22#1568995912yes—
607C4—28no—
608C4—22no—
609C4—22no—
610C4—44no—
611C4—948no—
612C4—948no—
613C4—1884no—
614C4—1884no—
615C4C2×C4×C22⋊D4714yes—
616C4C2×C4×C22⋊D4714yes—
617C2—24no—
618C2—24no—
619C2—42no—
620C2C22×C4×C22⋊D4711yescharacteristic
621C2C2×#1568960011yescharacteristic
622C2#2025658911yescharacteristic
623C2#2025706812yes—
624C2#2025714012yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC22×#117×C47C2
socle, cosocleC23×C47C25
prime radicalC25×C47C23
hypercenterC22×#117D47
hypocenterC47#34203
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. C2×#15689600C2semidirect
3. #20246415C2semidirect
4. #20246393C2semidirect
5. #20246258C2semidirect
7. #20246070C2semidirect
8. #20246004C2semidirect
9. #20245972C2semidirect
10. C4×#13094003C2semidirect
12. #20245744C2semidirect
15. C2×#13094003C4semidirect
21. C2×C4×Dic94C22semidirect
22. C2×#13093989C22semidirect
29. #15689029C22semidirect
30. #15689028C22semidirect
31. #15689021C22semidirect
32. #15689019C22semidirect
33. #15689018C22semidirect
36. #15689007C22semidirect
37. #15689004C22semidirect
45. C4⋊C4×Dic47C22semidirect
46. C4×Dic47⋊C4C22semidirect
47. C4×C4⋊Dic47C22semidirect
64. C22×Dic94C2×C4semidirect
71. #13094003C2×C4semidirect
72. #13094003C2×C4semidirect
78. C2×C4⋊Dic47C2×C4semidirect
83. C2×Dic47⋊C4C2×C4semidirect
289. C22×C47#519semidirect
374. C47#34203semidirect
601. C22#15689600semidirect

Maximal subgroups

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

subgroupindex
C22×#130940032
C2×#156896002
#202464152
#202463932
#202462582
#202462332
#202460702
#202460042
#202459722
C4×#130940032
#202457712
#202457442
C22×#117×C472
#3420347