Finite Group Databasebeta

← #26379098

Subgroups — #26379098

Order 12032.

2238 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094023C211yesmaximal, characteristic
2#20246566C214yesmaximal, complemented
3#20246475C212yesmaximal, complemented
4#20246474C212yesmaximal, complemented
5#20246259C211yesmaximal, characteristic
6#20246258C212yesmaximal
7#20246257C211yesmaximal, characteristic
8#20246255C211yesmaximal, characteristic
9#20246252C212yesmaximal
10#20246135C212yesmaximal, complemented
11#20246133C214yesmaximal, complemented
12#20246132C212yesmaximal, complemented
13#20245973C212yesmaximal
14#20245972C212yesmaximal
15#20245783C211yesmaximal, characteristic
16#20245782C211yesmaximal, characteristic
17C47×#2344C211yesmaximal, characteristic
18C23×Dic94C2212yes—
19C2×#13094023C4116yescomplemented
20C2×#13094023C2212yes—
21C2×#13094023C2211yescharacteristic
22C2×#13094023C2212yes—
23C2×#13094023C2211yescharacteristic
24C2×#13094023C2211yescharacteristic
25C2×#13094023C2211yescharacteristic
26C2×#13093983C2212yes—
27C2×#13093980C2212yes—
28C2×#13093980C2212yes—
29C22×C4⋊Dic47C2211yescharacteristic
30C22×Dic47⋊C4C2212yes—
31C22×C4×Dic47C2211yescharacteristic
32#15689029C2214yes—
33#15689029C2214yes—
34#15689028C2214yes—
35#15689028C2214yes—
36#15689027C2214yes—
37#15689027C2214yes—
38#15689027C2214yes—
39#15689027C2214yes—
40#15689022C2212yes—
41#15689021C2212yes—
42#15689020C2214yes—
43#15689019C2212yes—
44#15689018C2214yes—
45#15689010C2214yes—
46#15689009C2214yes—
47#15689008C2214yes—
48#15689007C2214yes—
49#15689006C2214yes—
50#15689005C2214yes—
51#15689004C2214yes—
52#15688990C2212yes—
53#15688990C2212yes—
54#15688988C2212yes—
55#15688987C2212yes—
56#15688986C2212yes—
57#15688984C2211yescharacteristic
58#15688984C2211yescharacteristic
59#15688984C2212yes—
60#15688984C2212yes—
61#15688984C2212yes—
62#15688983C2211yescharacteristic
63#15688981C2211yescharacteristic
64#15688981C2211yescharacteristic
65C4⋊C4×Dic47C2212yes—
66C4⋊C4×Dic47C2212yes—
67C4⋊C4×Dic47C2212yes—
68C4×Dic47⋊C4C2212yes—
69C4×Dic47⋊C4C2214yes—
70C4×C4⋊Dic47C2212yes—
71#15688974C2214yes—
72C22⋊C4×Dic47C2212yes—
73C4×C22⋊Dic47C2211yescharacteristic
74#15688969C2212yes—
75#15688968C2212yes—
76C2×#117×C47C2212yes—
77C22×C4⋊C4×C47C2211yescharacteristic
78C47×#389C2212yes—
79C47×#387C2214yes—
80C47×#381C2212yes—
81C47×#380C2211yescharacteristic
82C4×C4⋊C4×C47C2212yes—
83C4×C22⋊C4×C47C2211yescharacteristic
84C22×Dic94—44no—
85C22×Dic94—44no—
86C22×Dic94C2×C4116yes—
87C22×Dic94C2312yes—
88C22×Dic94C2312yes—
89C22×Dic94C2312yes—
90C22×Dic94C2312yes—
91C23×Dic47C2311yescharacteristic
92C23×Dic47C2311yescharacteristic
93C23×Dic47C2312yes—
94#13094023C2×C4116yes—
95#13094023C2×C418yes—
96#13094023C2×C4116yes—
97#13094023C2×C418yes—
98#13094023C2×C418yes—
99#13094023C2×C418yes—
100#13093983—44no—
101#13093980—44no—
102#13093980—44no—
103C2×C4⋊Dic47C2×C418yes—
104C2×C4⋊Dic47C2312yes—
105C2×C4⋊Dic47C2314yes—
106C2×C4⋊Dic47C2311yescharacteristic
107C2×C4⋊Dic47C2311yescharacteristic
108C2×C4⋊Dic47C2312yes—
109C2×Dic47⋊C4C2×C4116yes—
110C2×Dic47⋊C4C2312yes—
111C2×Dic47⋊C4C2312yes—
112C2×Dic47⋊C4C2314yes—
113C2×Dic47⋊C4C2314yes—
114C2×Dic47⋊C4C2312yes—
115C2×Dic47⋊C4C2312yes—
116C2×Dic47⋊C4C2312yes—
117C2×Dic47⋊C4C2312yes—
118C2×C22⋊Dic47C2311yescharacteristic
119C2×C22⋊Dic47C2311yescharacteristic
120C2×C22⋊Dic47C2312yes—
121C2×C4×Dic47C2×C418yes—
122C2×C4×Dic47C2314yes—
123C2×C4×Dic47C2312yes—
124C2×C4×Dic47C2314yes—
125C2×C4×Dic47C2314yes—
126C2×C4×Dic47C2314yes—
127C2×C4×Dic47C2311yescharacteristic
128C2×C4×Dic47C2311yescharacteristic
129C2×C4×Dic47C2311yescharacteristic
130C2×C4×Dic47C2311yescharacteristic
131C47⋊#95C2312yes—
132C47⋊#95C2314yes—
133C47⋊#95C2312yes—
134C47⋊#95C2312yes—
135C47⋊#95C2312yes—
136C47⋊#95C2312yes—
137C47⋊#95C2312yes—
138C47⋊#95C2314yes—
139C47⋊#95C2314yes—
140C47⋊#95C2314yes—
141C47⋊#95C2312yes—
142C47⋊#95C2312yes—
143C47⋊#95C2314yes—
144C47⋊#95C2314yes—
145C47⋊#95C2314yes—
146C47⋊#95C2314yes—
147C47⋊#95C2314yes—
148C23×C4×C47C2311yescharacteristic
149C23×C4×C47C2312yes—
150#117×C47—44no—
151C2×C4⋊C4×C47C2×C418yes—
152C2×C4⋊C4×C47C2312yes—
153C2×C4⋊C4×C47C2314yes—
154C2×C4⋊C4×C47C2311yescharacteristic
155C2×C4⋊C4×C47C2312yes—
156C2×C4⋊C4×C47C2311yescharacteristic
157C2×C22⋊C4×C47C2312yes—
158C2×C22⋊C4×C47C2311yescharacteristic
159C2×C22⋊C4×C47C2311yescharacteristic
160C2×C42×C47C2312yes—
161C2×C42×C47C2314yes—
162#95×C47C2312yes—
163#95×C47C2314yes—
164#95×C47C2312yes—
165#95×C47C2314yes—
166C24×C47C2411yescharacteristic
167C22×C4×C47—24no—
168C22×C4×C47—24no—
169C22×C4×C47—24no—
170C22×C4×C47C22×C414yes—
171C22×C4×C47C22×C418yes—
172C22×C4×C47C2412yes—
173C22×C4×C47C2414yes—
174C22×C4×C47C2412yes—
175C22×C4×C47C2412yes—
176C22×C4×C47C2412yes—
177C22×C4×C47C2411yescharacteristic
178C22×C4×C47C2411yescharacteristic
179C22×Dic47—24no—
180C22×Dic47—24no—
181C22×Dic47—22no—
182C22×Dic47—22no—
183C22×Dic47—24no—
184C22×Dic47C22×C414yes—
185C22×Dic47C22×C414yes—
186C22×Dic47C22×C418yes—
187C22×Dic47C2412yes—
188C22×Dic47C2414yes—
189C22×Dic47C2412yes—
190C22×Dic47C2411yescharacteristic
191C22×Dic47C2411yescharacteristic
192C22×Dic47C2411yescharacteristic
193C22×Dic47C2412yes—
194C22×Dic47C2412yes—
195C22×Dic47C2411yescharacteristic
196C2×Dic94—432no—
197C2×Dic94—42no—
198C2×Dic94—42no—
199C2×Dic94—42no—
200C2×Dic94—42no—
201C2×Dic94—42no—
202C2×Dic94—42no—
203C2×Dic94—42no—
204C2×Dic94—42no—
205C2×Dic94C22×C418yes—
206C2×Dic94C22×C418yes—
207C2×Dic94C22×C418yes—
208C2×Dic94C22×C418yes—
209C4⋊C4×C47—24no—
210C4⋊C4×C47—44no—
211C4⋊C4×C47C22×C418yes—
212C4⋊C4×C47C22×C414yes—
213C4⋊C4×C47C22×C414yes—
214C22⋊C4×C47—22no—
215C22⋊C4×C47—24no—
216C22⋊C4×C47—22no—
217C42×C47—24no—
218C42×C47—44no—
219C22⋊Dic47—22no—
220C22⋊Dic47—24no—
221C22⋊Dic47—22no—
222C4⋊Dic47—24no—
223C4⋊Dic47—44no—
224C4⋊Dic47C22×C418yes—
225C4⋊Dic47C22×C414yes—
226C4⋊Dic47C22×C414yes—
227Dic47⋊C4—24no—
228Dic47⋊C4—24no—
229Dic47⋊C4—44no—
230Dic47⋊C4—44no—
231Dic47⋊C4C22×C418yes—
232Dic47⋊C4C22×C418yes—
233Dic47⋊C4C22×C418yes—
234Dic47⋊C4C22×C418yes—
235C4×Dic47—24no—
236C4×Dic47—28no—
237C4×Dic47—44no—
238C4×Dic47—44no—
239C4×Dic47—44no—
240C4×Dic47C22×C414yes—
241C4×Dic47C22×C414yes—
242C4×Dic47C22×C414yes—
243C4×Dic47C22×C414yes—
244C23×C47—22no—
245C23×C47—22no—
246C23×C47C23×C412yes—
247C23×C47H(5,𝔽2)12yes—
248C23×C47H(5,𝔽2)12yes—
249C23×C47C2511yescharacteristic
250C2×C4×C47—216no—
251C2×C4×C47—22no—
252C2×C4×C47—22no—
253C2×C4×C47—22no—
254C2×C4×C47—22no—
255C2×C4×C47—22no—
256C2×C4×C47—28no—
257C2×C4×C47—22no—
258C2×C4×C47—24no—
259C2×C4×C47—22no—
260C2×C4×C47—24no—
261C2×C4×C47—22no—
262C2×C4×C47—22no—
263C2×C4×C47—22no—
264C2×C4×C47—22no—
265C2×C4×C47—24no—
266C2×C4×C47C23×C412yes—
267C2×C4×C47C23×C414yes—
268C2×C4×C47C23×C414yes—
269C2×C4×C47C23×C412yes—
270C2×C4×C472−1+414yes—
271C2×Dic47—28no—
272C2×Dic47—216no—
273C2×Dic47—22no—
274C2×Dic47—22no—
275C2×Dic47—22no—
276C2×Dic47—22no—
277C2×Dic47—21no—
278C2×Dic47—21no—
279C2×Dic47—21no—
280C2×Dic47—21no—
281C2×Dic47—22no—
282C2×Dic47—28no—
283C2×Dic47—22no—
284C2×Dic47—24no—
285C2×Dic47—22no—
286C2×Dic47—24no—
287C2×Dic47—22no—
288C2×Dic47—22no—
289C2×Dic47—22no—
290C2×Dic47—22no—
291C2×Dic47—24no—
292C2×Dic47C23×C412yes—
293C2×Dic47C23×C412yes—
294C2×Dic47C23×C412yes—
295C2×Dic47C23×C414yes—
296C2×Dic47C23×C414yes—
297C2×Dic47C23×C412yes—
298C2×Dic472−1+414yes—
299Dic94—48no—
300Dic94—48no—
301Dic94—48no—
302Dic94—48no—
303#34365—471nomaximal, Sylow
304C22×C47—22no—
305C22×C47—22no—
306C22×C47—24no—
307C22×C47—24no—
308C22×C47#51814yes—
309C22×C47C24×C411yescharacteristic
310C22×C47C2×H(5,𝔽2)11yescharacteristic
311C22×C47C2×H(5,𝔽2)11yescharacteristic
312C22×C47C2×2−1+412yes—
313C22×C47C2×2−1+412yes—
314C4×C47—24no—
315C4×C47—24no—
316C4×C47—22no—
317C4×C47—44no—
318C4×C47—42no—
319C4×C47—42no—
320C4×C47#51914yes—
321Dic47—22no—
322Dic47—22no—
323Dic47—24no—
324Dic47—24no—
325Dic47—22no—
326Dic47—42no—
327Dic47—44no—
328Dic47—42no—
329Dic47#51914yes—
330C22×C4⋊Q8—471no—
331#2381—474no—
332#2369—472no—
333#2369—472no—
334#2344—472no—
335#2344—471no—
336#2344—471no—
337#2344—472no—
338#2344—471no—
339#2344D4711yescharacteristic
340#2331—472no—
341#2331—474no—
342#2331—472no—
343#2310—472no—
344#2310—472no—
345#2283—471no—
346#2283—471no—
347C2×C47—24no—
348C2×C47C2×#51811yescharacteristic
349C2×C47C2×#51911yescharacteristic
350C2×C47C2×#51911yescharacteristic
351C2×C47#354412yes—
352C2×C47#354412yes—
353C23×Q8—472no—
354C2×C4⋊Q8—4716no—
355C2×C4⋊Q8—472no—
356C2×C4⋊Q8—471no—
357C2×C4⋊Q8—471no—
358C2×C4⋊Q8—471no—
359C2×C4⋊Q8—471no—
360C2×C4⋊Q8—472no—
361C2×#117—472no—
362C2×#117—472no—
363C2×#117—472no—
364C2×#117C2×D4712yes—
365C22×C4⋊C4—472no—
366C22×C4⋊C4—471no—
367C22×C4⋊C4C2×D4711yescharacteristic
368C22×C42—471no—
369#391—474no—
370#391—474no—
371#391—474no—
372#391—474no—
373#391—474no—
374#391—474no—
375#391—474no—
376#391—474no—
377#389—474no—
378#389—472no—
379#389—472no—
380#389—472no—
381#389—474no—
382#389C2×D4712yes—
383#387—474no—
384#387—474no—
385#387—474no—
386#387—474no—
387#387—474no—
388#387—474no—
389#387—474no—
390#387C2×D4714yes—
391#383—472no—
392#383—472no—
393#382—472no—
394#382—472no—
395#381—472no—
396#381—471no—
397#381—472no—
398#381—471no—
399#381—472no—
400#381—472no—
401#381C2×D4712yes—
402#380—471no—
403#380—471no—
404#380—471no—
405#380C2×D4711yescharacteristic
406C4×C4⋊C4—472no—
407C4×C4⋊C4—472no—
408C4×C4⋊C4—472no—
409C4×C4⋊C4—472no—
410C4×C4⋊C4—474no—
411C4×C4⋊C4—472no—
412C4×C4⋊C4—474no—
413C4×C4⋊C4C2×D4712yes—
414C4×C22⋊C4—471no—
415C4×C22⋊C4—472no—
416C4×C22⋊C4C2×D4711yescharacteristic
417#376—472no—
418#376—472no—
419C47#3436511yescharacteristic, complemented, Sylow
420C22×Q8—4716no—
421C22×Q8—472no—
422C22×Q8—472no—
423C22×Q8—472no—
424C22×Q8—472no—
425C22×Q8—1884no—
426C22×Q8—1884no—
427C23×C4—472no—
428C23×C4—471no—
429C23×C4—471no—
430C23×C4C22×D4712yes—
431C23×C4C22×D4711yescharacteristic
432C4⋊Q8—4716no—
433C4⋊Q8—478no—
434C4⋊Q8—478no—
435C4⋊Q8—478no—
436C4⋊Q8—478no—
437C4⋊Q8—4716no—
438#117—44no—
439#117—1884no—
440#117—1884no—
441#117—1884no—
442C2×C4⋊C4—4716no—
443C2×C4⋊C4—478no—
444C2×C4⋊C4—472no—
445C2×C4⋊C4—472no—
446C2×C4⋊C4—472no—
447C2×C4⋊C4—472no—
448C2×C4⋊C4—471no—
449C2×C4⋊C4—471no—
450C2×C4⋊C4—472no—
451C2×C4⋊C4—472no—
452C2×C4⋊C4—472no—
453C2×C4⋊C4—472no—
454C2×C4⋊C4—474no—
455C2×C4⋊C4—474no—
456C2×C4⋊C4—474no—
457C2×C4⋊C4C4×D4718yes—
458C2×C4⋊C4C22×D4712yes—
459C2×C4⋊C4C22×D4714yes—
460C2×C4⋊C4C22×D4711yescharacteristic
461C2×C4⋊C4C22×D4711yescharacteristic
462C2×C4⋊C4C22×D4712yes—
463C2×C22⋊C4—471no—
464C2×C22⋊C4—471no—
465C2×C22⋊C4—472no—
466C2×C22⋊C4C22×D4711yescharacteristic
467C2×C22⋊C4C22×D4712yes—
468C2×C22⋊C4C22×D4711yescharacteristic
469C2×C42—478no—
470C2×C42—471no—
471C2×C42—471no—
472C2×C42—471no—
473C2×C42—471no—
474C2×C42—474no—
475C2×C42—472no—
476C2×C42—474no—
477C2×C42—474no—
478C2×C42—474no—
479C2×C42C22×D4712yes—
480C2×C42C22×D4714yes—
481#95—472no—
482#95—472no—
483#95—472no—
484#95—472no—
485#95—474no—
486#95—472no—
487#95—474no—
488#95—472no—
489#95—472no—
490#95—474no—
491#95—472no—
492#95—474no—
493#95—474no—
494#95—474no—
495#95—474no—
496#95—474no—
497#95—474no—
498#95C22×D4712yes—
499#95C22×D4712yes—
500#95C22×D4714yes—
501#95C22×D4714yes—
502C24C23×D4711yescharacteristic
503C2×Q8—478no—
504C2×Q8—478no—
505C2×Q8—478no—
506C2×Q8—478no—
507C2×Q8—18832no—
508C2×Q8—1882no—
509C2×Q8—1882no—
510C2×Q8—1882no—
511C2×Q8—1882no—
512C2×Q8—1882no—
513C2×Q8—1882no—
514C2×Q8—1882no—
515C2×Q8—1882no—
516C22×C4—24no—
517C22×C4—24no—
518C22×C4—24no—
519C22×C4—471no—
520C22×C4—474no—
521C22×C4—474no—
522C22×C4—478no—
523C22×C4—471no—
524C22×C4—471no—
525C22×C4—471no—
526C22×C4—472no—
527C22×C4—472no—
528C22×C4—472no—
529C22×C4—474no—
530C22×C4—472no—
531C22×C4—944no—
532C22×C4—944no—
533C22×C4—942no—
534C22×C4—944no—
535C22×C4—942no—
536C22×C4C2×C4×D4714yes—
537C22×C4C2×C4×D4718yes—
538C22×C4C23×D4712yes—
539C22×C4C23×D4714yes—
540C22×C4C23×D4712yes—
541C22×C4C23×D4711yescharacteristic
542C22×C4C23×D4712yes—
543C22×C4C23×D4712yes—
544C22×C4C23×D4711yescharacteristic
545C4⋊C4—24no—
546C4⋊C4—44no—
547C4⋊C4—478no—
548C4⋊C4—478no—
549C4⋊C4—478no—
550C4⋊C4—478no—
551C4⋊C4—478no—
552C4⋊C4—474no—
553C4⋊C4—474no—
554C4⋊C4—944no—
555C4⋊C4—944no—
556C4⋊C4—944no—
557C4⋊C4—1884no—
558C4⋊C4—1884no—
559C4⋊C4—1884no—
560C4⋊C4C2×C4×D4718yes—
561C4⋊C4C2×C4×D4714yes—
562C4⋊C4C2×C4×D4714yes—
563C22⋊C4—22no—
564C22⋊C4—24no—
565C22⋊C4—22no—
566C22⋊C4—942no—
567C22⋊C4—944no—
568C22⋊C4—942no—
569C42—24no—
570C42—44no—
571C42—474no—
572C42—474no—
573C42—474no—
574C42—474no—
575C42—948no—
576C42—944no—
577C42—1884no—
578C42—1884no—
579C42—1884no—
580C23—22no—
581C23—22no—
582C23C22×C4×D4712yes—
583C23#1309406912yes—
584C23#1309407012yes—
585C23C24×D4711yescharacteristic
586Q8—1888no—
587Q8—1888no—
588Q8—1888no—
589Q8—1888no—
590C2×C4—216no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—472no—
607C2×C4—474no—
608C2×C4—472no—
609C2×C4—472no—
610C2×C4—472no—
611C2×C4—474no—
612C2×C4—474no—
613C2×C4—948no—
614C2×C4—9416no—
615C2×C4—948no—
616C2×C4—942no—
617C2×C4—942no—
618C2×C4—941no—
619C2×C4—941no—
620C2×C4—942no—
621C2×C4—942no—
622C2×C4—941no—
623C2×C4—941no—
624C2×C4—942no—
625C2×C4—942no—
626C2×C4—944no—
627C2×C4—942no—
628C2×C4—944no—
629C2×C4—942no—
630C2×C4—942no—
631C2×C4—942no—
632C2×C4—942no—
633C2×C4—944no—
634C2×C4C22×C4×D4712yes—
635C2×C4C22×C4×D4714yes—
636C2×C4C22×C4×D4714yes—
637C2×C4C22×C4×D4712yes—
638C2×C4#1309407114yes—
639C22—22no—
640C22—22no—
641C22—24no—
642C22—24no—
643C22#1568959514yes—
644C22C23×C4×D4711yescharacteristic
645C22C2×#1309406911yescharacteristic
646C22C2×#1309407011yescharacteristic
647C222−1+4×D4712yes—
648C22C2×#1309407112yes—
649C4—24no—
650C4—24no—
651C4—22no—
652C4—42no—
653C4—44no—
654C4—42no—
655C4—474no—
656C4—942no—
657C4—942no—
658C4—944no—
659C4—944no—
660C4—942no—
661C4—1882no—
662C4—1884no—
663C4—1882no—
664C4#1568960014yes—
665C2—24no—
666C2C2×#1568959511yescharacteristic
667C2#519×D4711yescharacteristic
668C2C2×#1568960011yescharacteristic
669C2#2025772412yes—
670C2#2025773012yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC47×#2344C2
socle, cosocleC23×C47C25
prime radicalC24×C47C24
hypercenter#2344D47
hypocenterC47#34365
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2344

Complements

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

NH ≅ G/Ntype
2. #20246566C2semidirect
3. #20246475C2semidirect
4. #20246474C2semidirect
10. #20246135C2semidirect
11. #20246133C2semidirect
12. #20246132C2semidirect
19. C2×#13094023C4semidirect
419. C47#34365semidirect

Maximal subgroups

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

subgroupindex
C22×#130940232
#202465662
#202464752
#202464742
#202462592
#202462582
#202462572
#202462552
#202462522
#202461352
#202461332
#202461322
#202459732
#202459722
#202457832
#202457822
C47×#23442
#3436547