Finite Group Databasebeta

← #26232898

Subgroups — #26232898

Order 12032.

1110 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689832C211yesmaximal, characteristic
2C2×#15689252C211yesmaximal, characteristic, complemented
3#20246300C211yesmaximal, characteristic, complemented
4#20244356C211yesmaximal, characteristic, complemented
5#20244355C211yesmaximal, characteristic, complemented
6#20244347C211yesmaximal, characteristic, complemented
7#20244344C211yesmaximal, characteristic, complemented
8#20244343C211yesmaximal, characteristic, complemented
9#20244334C211yesmaximal, characteristic, complemented
10#20243386C211yesmaximal, characteristic
11#20243386C211yesmaximal, characteristic
12#20243215C211yesmaximal, characteristic
13#20243213C211yesmaximal, characteristic
14#102×Dic47C211yesmaximal, characteristic
15C47×#1802C211yesmaximal, characteristic
16#15689832—24no—
17#15689832C418yescomplemented
18C2×#13094023C2211yescharacteristic
19C2×#13094016C2211yescharacteristic
20C2×#13094015C2211yescharacteristic
21C2×#13094003—21no—
22C2×#13094001—21no—
23C2×#13094001—21no—
24C2×C4×Dic94C2211yescharacteristic
25C2×#13093989—21no—
26C2×D4×Dic47C2211yescharacteristic
27C2×D4×Dic47C2211yescharacteristic
28C22×Dic47⋊C4—21no—
29#15689252—24no—
30C2×Dic47⋊C8C2211yescharacteristic
31C2×Dic47⋊C8C2211yescharacteristic
32C2×#13093945C2211yescharacteristic
33C2×#13093945C2211yescharacteristic
34C2×#13093935C2211yescharacteristic
35C2×#13093934C2211yescharacteristic
36C2×C8×Dic47C2211yescharacteristic
37#15689019—21no—
38#15689018—21no—
39#15689018—21no—
40#15688992C2211yescharacteristic
41#15688992C2211yescharacteristic
42C4⋊C4×Dic47C2211yescharacteristic
43C4⋊C4×Dic47C2211yescharacteristic
44C4×C4⋊Dic47C2211yescharacteristic
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53#15688885C2211yescharacteristic
54#15688885C2211yescharacteristic
55C2×C4×D4×C47C2211yescharacteristic
56C2×C4⋊C8×C47C2211yescharacteristic
57C2×#102×C47C2211yescharacteristic
58C2×#102×C47C2211yescharacteristic
59C47×#384C2211yescharacteristic
60C47×#340C2211yescharacteristic
61C47×#340C2211yescharacteristic
62C22×Dic94—21no—
63C22×Dic94D411yescharacteristic
64C23×Dic47—21no—
65C23×Dic47—21no—
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68#13094023—22no—
69#13094023C2×C414yes—
70#13094016—22no—
71#13094016C2×C414yes—
72#13094015—22no—
73#13094015C2×C414yes—
74#13094003—28no—
75#13094001—28no—
76#13094001—28no—
77C4×Dic94C2×C414yes—
78C4×Dic94D414yes—
79#13093989—44no—
80D4×Dic47—22no—
81D4×Dic47—22no—
82D4×Dic47C2×C414yes—
83D4×Dic47C2×C414yes—
84C2×C4⋊Dic47—21no—
85C2×C4⋊Dic47D411yescharacteristic
86C2×C4⋊Dic47C2311yescharacteristic
87C2×C4⋊Dic47C2311yescharacteristic
88C2×C4⋊Dic47C2311yescharacteristic
89C2×C4⋊Dic47C2311yescharacteristic
90C2×Dic47⋊C4—21no—
91C2×Dic47⋊C4—21no—
92C2×Dic47⋊C4—21no—
93C2×Dic47⋊C4—21no—
94C2×Dic47⋊C4—28no—
95C2×Dic47⋊C4—21no—
96C2×Dic47⋊C4—21no—
97C2×Dic47⋊C4—21no—
98C2×Dic47⋊C4—21no—
99C2×Dic47⋊C4D411yescharacteristic
100C2×Dic47⋊C4D411yescharacteristic
101C2×C22⋊Dic47—21no—
102C2×C22⋊Dic47—21no—
103C2×C4×Dic47—21no—
104C2×C4×Dic47—21no—
105C2×C4×Dic47—21no—
106C2×C4×Dic47C2311yescharacteristic
107C2×C4×Dic47C2311yescharacteristic
108Dic47⋊C8—22no—
109Dic47⋊C8—22no—
110#13093945—22no—
111#13093945Q814yescomplemented
112#13093935—22no—
113#13093934—22no—
114C8×Dic47—22no—
115C47⋊#95—21no—
116C47⋊#95—21no—
117C47⋊#95—21no—
118C22×D4×C47C2311yescharacteristic
119C23×C4×C47—21no—
120C22×C8×C47C2311yescharacteristic
121C22×C8×C47C2311yescharacteristic
122C4×D4×C47—22no—
123C4×D4×C47C2×C414yes—
124C2×C4⋊C4×C47—21no—
125C2×C4⋊C4×C47—21no—
126C2×C4⋊C4×C47C2311yescharacteristic
127C2×C4⋊C4×C47C2311yescharacteristic
128C2×C4⋊C4×C47C2311yescharacteristic
129C2×C22⋊C4×C47—21no—
130C2×C42×C47C2311yescharacteristic
131C4⋊C8×C47—22no—
132#102×C47—22no—
133#102×C47—22no—
134C24×C47—21no—
135C2×D4×C47—21no—
136C2×D4×C47—21no—
137C2×D4×C47—21no—
138C2×D4×C47C22×C412yes—
139C2×D4×C47C2×Q812yes—
140C2×D4×C47#4112yes—
141C22×C4×C47—21no—
142C22×C4×C47—24no—
143C22×C4×C47—21no—
144C22×C4×C47—21no—
145C22×C4×C47—81no—
146C22×C4×C47—81no—
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2411yescharacteristic
150C22×Dic47—21no—
151C22×Dic47—21no—
152C22×Dic47—21no—
153C22×Dic47—24no—
154C22×Dic47—24no—
155C22×Dic47—21no—
156C22×Dic47—42no—
157C22×Dic47—42no—
158C22×Dic47—81no—
159C22×Dic47—81no—
160C22×Dic47C2×D411yescharacteristic
161C22×Dic47C2×D411yescharacteristic
162C22×Dic47C2×D411yescharacteristic
163C22×Dic47C2×D411yescharacteristic
164C2×Dic94—22no—
165C2×Dic94—22no—
166C2×Dic94—24no—
167C2×Dic94—81no—
168C2×Dic94—81no—
169C2×Dic94C22⋊C414yes—
170C2×C8×C47—21no—
171C2×C8×C47—21no—
172C2×C8×C47—21no—
173C2×C8×C47—21no—
174C2×C8×C47—21no—
175C2×C8×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47—21no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47—42no—
182C4⋊C4×C47—81no—
183C4⋊C4×C47C22×C412yes—
184C22⋊C4×C47—24no—
185C42×C47C22×C412yes—
186C42×C47C2×D412yes—
187C22⋊Dic47—24no—
188C22⋊Dic47—24no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47—24no—
193C4⋊Dic47—21no—
194C4⋊Dic47—21no—
195C4⋊Dic47C22⋊C414yes—
196C4⋊Dic47C22×C412yes—
197C4⋊Dic47C22×C412yes—
198C4⋊Dic47C2×Q812yes—
199Dic47⋊C4—24no—
200Dic47⋊C4—24no—
201Dic47⋊C4—24no—
202Dic47⋊C4—24no—
203Dic47⋊C4—42no—
204Dic47⋊C4—42no—
205Dic47⋊C4—81no—
206Dic47⋊C4—81no—
207Dic47⋊C4C22⋊C414yes—
208Dic47⋊C4C22⋊C414yes—
209C4×Dic47—42no—
210C4×Dic47—42no—
211C4×Dic47—42no—
212C4×Dic47C22×C412yes—
213C4×Dic47C22×C412yes—
214C4×Dic47C2×D412yes—
215C4×Dic47C2×D412yes—
216C2×C47⋊C8—21no—
217C2×C47⋊C8—21no—
218C2×C47⋊C8—21no—
219C2×C47⋊C8—21no—
220C2×C47⋊C8—21no—
221C2×C47⋊C8C2×Q812yes—
222C23×C47—22no—
223C23×C47—22no—
224C23×C47—22no—
225C23×C47—41no—
226C23×C47—41no—
227C23×C47—41no—
228C23×C47—41no—
229C23×C47C22×D411yescharacteristic
230D4×C47—22no—
231D4×C47—22no—
232D4×C47—22no—
233D4×C47C4×Q814yescomplemented
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—22no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—84no—
248C2×C4×C47C2×C22⋊C412yes—
249C2×C4×C47C2×C22⋊C412yes—
250C2×C4×C47C23×C411yescharacteristic
251C2×C4×C47C22×D411yescharacteristic
252C2×C4×C47C22×Q811yescharacteristic
253C2×C4×C47C2×#4111yescharacteristic
254C2×C4×C472−1+411yescharacteristic
255C2×C4×C472−1+411yescharacteristic
256C2×Dic47—21no—
257C2×Dic47—21no—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—21no—
261C2×Dic47—21no—
262C2×Dic47—22no—
263C2×Dic47—22no—
264C2×Dic47—22no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—48no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47—84no—
276C2×Dic47C2×C22⋊C412yes—
277C2×Dic47C2×C22⋊C412yes—
278C2×Dic47C2×C22⋊C412yes—
279C2×Dic47C2×C22⋊C412yes—
280C2×Dic47AGL(1,ℤ/8)12yes—
281C2×Dic47#13712yes—
282Dic94—28no—
283Dic94—84no—
284C8×C47—41no—
285C8×C47—41no—
286C47⋊C8—22no—
287C47⋊C8—41no—
288#22522—471nomaximal, Sylow
289C22×C47—24no—
290C22×C47—42no—
291C22×C47—42no—
292C22×C47—42no—
293C22×C47—42no—
294C22×C47—42no—
295C22×C47—42no—
296C22×C47C22×C22⋊C411yescharacteristic
297C22×C47#54811yescharacteristic
298C22×C47D4×Q811yescharacteristic
299C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C47C2×#13711yescharacteristic
301C22×C47#57611yescharacteristic
302C22×C47#57711yescharacteristic
303C4×C47—22no—
304C4×C47—22no—
305C4×C47—42no—
306C4×C47—81no—
307C4×C47—81no—
308C4×C47C2×C4×Q811yescharacteristic
309C4×C47#51911yescharacteristic
310C4×C47#54811yescharacteristic
311C4×C47D4×Q811yescharacteristic
312Dic47—22no—
313Dic47—22no—
314Dic47—22no—
315Dic47—42no—
316Dic47—42no—
317Dic47—42no—
318Dic47—81no—
319Dic47#41714yes—
320C2×#554—471no—
321C2×#446—471no—
322Q8×C4⋊C4—471no—
323#1992—471no—
324#1992—471no—
325#1990—471no—
326#1989—471no—
327#1989—471no—
328#1987—471no—
329#1802—471no—
330#1802—471no—
331#1802D4711yescharacteristic
332#1762—471no—
333#1762—471no—
334C4×#102—471no—
335C2×C47—44no—
336C2×C47Q8×C22⋊C411yescharacteristic
337C2×C47C2×#41711yescharacteristic
338C2×C47#289711yescharacteristic
339C2×C47#336611yescharacteristic
340C2×C47#337211yescharacteristic
341C2×C47#338411yescharacteristic
342C2×C47#338511yescharacteristic
343#554—478no—
344#554—944no—
345C2×C4⋊Q8—471no—
346C2×#125—471no—
347C2×#125—471no—
348C2×C22⋊Q8—941no—
349C2×C22⋊Q8—941no—
350C2×C22⋊Q8—941no—
351C2×C4×Q8—471no—
352C2×C4×Q8—941no—
353C2×C4×D4—471no—
354C2×C4×D4—471no—
355C2×C4×D4C2×D4711yescharacteristic
356C22×C4⋊C4—941no—
357#446—944no—
358C2×C4⋊C8—471no—
359C2×C4⋊C8—471no—
360C2×C4⋊C8C2×D4711yescharacteristic
361C2×#102—471no—
362C2×#102—471no—
363C2×#102C2×D4711yescharacteristic
364C2×#102C2×D4711yescharacteristic
365C2×#97—471no—
366C2×#97—471no—
367C2×C4×C8—471no—
368#389—941no—
369#389—941no—
370#389—941no—
371#384—471no—
372#384—471no—
373#384C2×D4711yescharacteristic
374C4×C4⋊C4—471no—
375C4×C4⋊C4—471no—
376C4×C4⋊C4—471no—
377#340—471no—
378#340—471no—
379#340—471no—
380#340—471no—
381#340—471no—
382#340—471no—
383#340—471no—
384#340—471no—
385#340—471no—
386#340—471no—
387#340C2×D4711yescharacteristic
388#340C2×D4711yescharacteristic
389C47#2252211yescharacteristic, complemented, Sylow
390C22×Q8—471no—
391C22×Q8—941no—
392C22×D4C22×D4711yescharacteristic
393C23×C4—21no—
394C23×C4—941no—
395C23×C4—941no—
396C22×C8—471no—
397C22×C8—471no—
398C22×C8C22×D4711yescharacteristic
399C22×C8C22×D4711yescharacteristic
400C4⋊Q8—474no—
401C4⋊Q8—942no—
402#125—474no—
403#125—474no—
404#125—942no—
405#125—942no—
406C22⋊Q8—948no—
407C22⋊Q8—948no—
408C22⋊Q8—948no—
409C4×Q8—474no—
410C4×Q8—474no—
411C4×Q8—1884no—
412C4×D4—22no—
413C4×D4—474no—
414C4×D4—474no—
415C4×D4—942no—
416C4×D4—942no—
417C4×D4C4×D4714yes—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—471no—
421C2×C4⋊C4—471no—
422C2×C4⋊C4—471no—
423C2×C4⋊C4—471no—
424C2×C4⋊C4—471no—
425C2×C4⋊C4—471no—
426C2×C4⋊C4—471no—
427C2×C4⋊C4—948no—
428C2×C4⋊C4—941no—
429C2×C4⋊C4—941no—
430C2×C4⋊C4—941no—
431C2×C4⋊C4—941no—
432C2×C4⋊C4—941no—
433C2×C4⋊C4—941no—
434C2×C4⋊C4—941no—
435C2×C4⋊C4—941no—
436C2×C4⋊C4—941no—
437C2×C4⋊C4C22×D4711yescharacteristic
438C2×C4⋊C4C22×D4711yescharacteristic
439C2×C4⋊C4C22×D4711yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—941no—
442C2×C22⋊C4—941no—
443C2×C42—471no—
444C2×C42—471no—
445C2×C42—941no—
446C2×C42—941no—
447C2×C42—941no—
448C2×C42C22×D4711yescharacteristic
449C4⋊C8—22no—
450C4⋊C8—942no—
451C4⋊C8—942no—
452#102—22no—
453#102—22no—
454#102—474no—
455#102—942no—
456#97—942no—
457#97—942no—
458C4×C8—942no—
459#95—941no—
460#95—941no—
461#95—941no—
462C24—21no—
463C2×Q8—474no—
464C2×Q8—942no—
465C2×Q8—942no—
466C2×Q8—944no—
467C2×Q8—3761no—
468C2×Q8—3761no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C2×C4×D4712yes—
473C2×D4#1062535212yes—
474C2×D4Q8×D4712yes—
475C22×C4—21no—
476C22×C4—24no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4—471no—
482C22×C4—471no—
483C22×C4—471no—
484C22×C4—471no—
485C22×C4—944no—
486C22×C4—944no—
487C22×C4—941no—
488C22×C4—941no—
489C22×C4—941no—
490C22×C4—941no—
491C22×C4—3761no—
492C22×C4—3761no—
493C22×C4—1882no—
494C22×C4—1882no—
495C22×C4D4×D4711yescharacteristic
496C22×C4D4×D4711yescharacteristic
497C22×C4C23×D4711yescharacteristic
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—472no—
505C2×C8—941no—
506C2×C8—941no—
507C2×C8—941no—
508C2×C8—941no—
509C2×C8—941no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—42no—
516C4⋊C4—81no—
517C4⋊C4—474no—
518C4⋊C4—474no—
519C4⋊C4—474no—
520C4⋊C4—472no—
521C4⋊C4—472no—
522C4⋊C4—472no—
523C4⋊C4—944no—
524C4⋊C4—944no—
525C4⋊C4—944no—
526C4⋊C4—944no—
527C4⋊C4—944no—
528C4⋊C4—941no—
529C4⋊C4—941no—
530C4⋊C4—941no—
531C4⋊C4—941no—
532C4⋊C4—941no—
533C4⋊C4—3761no—
534C4⋊C4—3761no—
535C4⋊C4—1882no—
536C4⋊C4—1882no—
537C4⋊C4C2×C4×D4712yes—
538C22⋊C4—24no—
539C22⋊C4—944no—
540C22⋊C4—944no—
541C42—472no—
542C42—472no—
543C42—472no—
544C42—472no—
545C42—1882no—
546C42—1882no—
547C42—1882no—
548C42C2×C4×D4712yes—
549C42D4×D4712yes—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23C2×D4×D4711yescharacteristic
558Q8—948no—
559Q8—3764no—
560D4—22no—
561D4—22no—
562D4—22no—
563D4#1309398914yescomplemented
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—84no—
578C2×C4—472no—
579C2×C4—472no—
580C2×C4—472no—
581C2×C4—472no—
582C2×C4—472no—
583C2×C4—472no—
584C2×C4—941no—
585C2×C4—941no—
586C2×C4—941no—
587C2×C4—941no—
588C2×C4—941no—
589C2×C4—941no—
590C2×C4—942no—
591C2×C4—942no—
592C2×C4—942no—
593C2×C4—3764no—
594C2×C4—1888no—
595C2×C4—1881no—
596C2×C4—1881no—
597C2×C4—1881no—
598C2×C4—1881no—
599C2×C4—1881no—
600C2×C4—1881no—
601C2×C4—1881no—
602C2×C4—1881no—
603C2×C4—1881no—
604C2×C4C22⋊C4×D4712yes—
605C2×C4C22⋊C4×D4712yes—
606C2×C4C22×C4×D4711yescharacteristic
607C2×C4C2×D4×D4711yescharacteristic
608C2×C4C2×Q8×D4711yescharacteristic
609C2×C4C2×#1062535211yescharacteristic
610C2×C4#1309407111yescharacteristic
611C2×C4#1309407111yescharacteristic
612C8—41no—
613C8—41no—
614C8—942no—
615C8—1881no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22C2×C22⋊C4×D4711yescharacteristic
624C22#1568977911yescharacteristic
625C22#1568978611yescharacteristic
626C22AGL(1,ℤ/8)×D4711yescharacteristic
627C22#137×D4711yescharacteristic
628C22#1568992411yescharacteristic
629C22#1568992911yescharacteristic
630C4—22no—
631C4—22no—
632C4—42no—
633C4—81no—
634C4—81no—
635C4—474no—
636C4—942no—
637C4—942no—
638C4—942no—
639C4—3761no—
640C4—1882no—
641C4—1882no—
642C4—1882no—
643C4C2×#1309398911yescharacteristic
644C4#1568960011yescharacteristic
645C4#1568977911yescharacteristic
646C4#1568978611yescharacteristic
647C2—44no—
648C2#2024622011yescharacteristic
649C2#417×D4711yescharacteristic
650C2#2025190911yescharacteristic
651C2#2025609011yescharacteristic
652C2#2025616411yescharacteristic
653C2#2025628211yescharacteristic
654C2#2025629311yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1802C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#1802D47
hypocenterC47#22522
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#15689252C2semidirect
3. #20246300C2semidirect
4. #20244356C2semidirect
5. #20244355C2semidirect
6. #20244347C2semidirect
7. #20244344C2semidirect
8. #20244343C2semidirect
9. #20244334C2semidirect
17. #15689832C4semidirect
111. #13093945Q8semidirect
233. D4×C47C4×Q8semidirect
389. C47#22522semidirect
563. D4#13093989semidirect

Maximal subgroups

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

subgroupindex
C2×#156898322
C2×#156892522
#202463002
#202443562
#202443552
#202443472
#202443442
#202443432
#202443342
#202433862
#202432152
#202432132
#102×Dic472
C47×#18022
#2252247