Finite Group Databasebeta

← #29687209

Subgroups — #29687209

Order 15616.

964 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840094C211yesmaximal, characteristic
2C4×#14519757C211yesmaximal, characteristic
3C4×C4⋊C4×D61C211yesmaximal, characteristic
4#21446459C211yesmaximal, characteristic, complemented
5#21446458C211yesmaximal, characteristic, complemented
6#21446400C211yesmaximal, characteristic, complemented
7#21446399C211yesmaximal, characteristic, complemented
8#21446125C211yesmaximal, characteristic
9#21446005C211yesmaximal, characteristic
10#21445904C211yesmaximal, characteristic
11#21445717C211yesmaximal, characteristic, complemented
12#21445652C211yesmaximal, characteristic, complemented
13#21445650C211yesmaximal, characteristic, complemented
14#21445582C211yesmaximal, characteristic
15C61×#1776C211yesmaximal, characteristic, complemented
16C2×#14519757C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×C4⋊C4×D61C2211yescharacteristic
19C2×C4⋊C4×D61C2211yescharacteristic
20C2×C42×D61—21no—
21C2×C42×D61C2211yescharacteristic
22#16840094C418yescomplemented
23C2×#14519710C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519689C2211yescharacteristic
30C2×#14519689C2211yescharacteristic
31C2×#14519688C2211yescharacteristic
32#16839667C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839647—21no—
35#16839636—21no—
36#16839619C2211yescharacteristic
37C4⋊C4×Dic61C2211yescharacteristic
38C4⋊C4×Dic61C2211yescharacteristic
39C4×#11758196—21no—
40C4×C4⋊Dic61—21no—
41C4×C4⋊Dic61C2211yescharacteristic
42#16839613—21no—
43C4×#11758198—21no—
44#95×D61—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#106×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×C4×D61—21no—
62C22×C4×D61—21no—
63C22×C4×D61D411yescharacteristic
64C22×C4×D61D411yescharacteristic
65C22×C4×D61C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519757—22no—
69#14519757C2×C414yes—
70C4×Dic122—22no—
71C4×Dic122—22no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C4⋊C4×D61—22no—
81C4⋊C4×D61—22no—
82C4⋊C4×D61—22no—
83C4⋊C4×D61C2×C414yes—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61D411yescharacteristic
91C2×C4×Dic61D411yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93C42×D61—22no—
94C42×D61—22no—
95C42×D61—44no—
96#14519710—22no—
97#14519708C2×C414yescomplemented
98#14519702—22no—
99#14519702C2×C414yescomplemented
100#14519700—22no—
101#14519700C2×C414yescomplemented
102#14519689—22no—
103#14519689C2×C414yes—
104#14519688—22no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C42×C61—21no—
119C2×C42×C61—21no—
120C2×C42×C61C2311yescharacteristic
121#107×C61—22no—
122#106×C61C2×C414yescomplemented
123#97×C61—22no—
124#95×C61—21no—
125#95×C61—21no—
126C23×D61C2×D411yescharacteristic
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—21no—
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—22no—
148C2×C4×D61—24no—
149C2×C4×D61—24no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—41no—
153C2×C4×D61—42no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×C4×D61—42no—
157C2×C4×D61C22×C412yes—
158C2×C4×D61#4112yes—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122C22×C412yes—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61C22×C412yes—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61C22×C412yes—
177C4⋊C4×C61C22×C412yes—
178C4⋊C4×C61#4112yes—
179C42×C61—42no—
180C42×C61—42no—
181C42×C61#4112yes—
182C42×C61#4112yes—
183#11758198—24no—
184#11758198—42no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—24no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—42no—
191C4⋊Dic61C22×C412yes—
192C4⋊Dic61#4112yes—
193#11758196—24no—
194#11758196—24no—
195#11758196—42no—
196C4×Dic61—22no—
197C4×Dic61—22no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—42no—
201C4×Dic61—42no—
202C4×Dic61—42no—
203C4×Dic61—42no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418C22×C412yes—
210C23×C61C22×D411yescharacteristic
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61—21no—
215C22×D61C4×D412yes—
216C22×D61AGL(1,ℤ/8)12yes—
217C22×D61#13712yes—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61#12312yes—
235C2×C4×C61#12312yes—
236C2×C4×C61C23×C411yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—21no—
249C2×Dic61—22no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61C4×D412yes—
262C2×Dic61#13712yes—
263C4×D61—28no—
264C4×D61—22no—
265C4×D61—22no—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—44no—
269C4×D61—48no—
270C4×D61—48no—
271Dic122—22no—
272Dic122—22no—
273Dic122—41no—
274Dic122—41no—
275C8×C61—22no—
276C8×C61—41no—
277#92418—22no—
278#92418—41no—
279#21579—611nomaximal, Sylow
280C22×C61C2×C4×D411yescharacteristic
281C22×C61C2×#12311yescharacteristic
282C22×C61#52511yescharacteristic
283C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C61C2×#13711yescharacteristic
285C22×C61C2×#13711yescharacteristic
286C22×C61#57511yescharacteristic
287C2×D61—22no—
288C2×D61—22no—
289C2×D61—22no—
290C2×D61—22no—
291C2×D61—22no—
292C2×D61—22no—
293C2×D61#44014yes—
294C4×C61—22no—
295C4×C61—22no—
296C4×C61—42no—
297C4×C61—42no—
298C4×C61—42no—
299C4×C61—81no—
300C4×C61C4×#4111yescharacteristic
301C4×C61C4×#4111yescharacteristic
302C4×C61#52911yescharacteristic
303C4×C61#52911yescharacteristic
304Dic61—22no—
305Dic61—22no—
306Dic61—41no—
307Dic61—42no—
308Dic61—42no—
309Dic61—42no—
310Dic61—81no—
311C2×#483—611no—
312C4×C22⋊Q8—611no—
313C2×C4×C4⋊C4—611no—
314#1928—611no—
315#1928—611no—
316#1919—611no—
317#1919—611no—
318#1872—611no—
319#1852—611no—
320#1823—611no—
321#1776—611no—
322#1776D6111yescharacteristic, complemented
323#1763—611no—
324#1763—611no—
325#1749—611no—
326C2×C61C4×#12311yescharacteristic
327C2×C61C2×#44011yescharacteristic
328C2×C61C4×#13711yescharacteristic
329C2×C61#309211yescharacteristic
330C2×C61#309511yescharacteristic
331C2×C61#311311yescharacteristic
332C2×C61#311411yescharacteristic
333D61—24no—
334C2×C22⋊Q8—611no—
335C2×C4×Q8—611no—
336C22×C4⋊C4—611no—
337C22×C4⋊C4—611no—
338C22×C42—611no—
339C22×C42—1221no—
340#483—618no—
341C2×#107—611no—
342C2×#107C2×D6111yescharacteristic
343C2×#106—611no—
344C2×#106C2×D6111yescharacteristic
345C2×#103—611no—
346C2×#103—611no—
347C2×#103—611no—
348C2×#103—611no—
349C2×C22⋊C8—611no—
350C2×C22⋊C8—611no—
351C2×#97—611no—
352C2×#97C2×D6111yescharacteristic
353#391—611no—
354#389—611no—
355#387—1221no—
356#385—1221no—
357#380—611no—
358C4×C4⋊C4—611no—
359C4×C4⋊C4—611no—
360C4×C4⋊C4—611no—
361C4×C4⋊C4—1221no—
362C4×C4⋊C4—1221no—
363C4×C4⋊C4—1221no—
364C4×C4⋊C4C2×D6111yescharacteristic
365C4×C4⋊C4C2×D6111yescharacteristic
366C4×C22⋊C4—1221no—
367C2×#95—1221no—
368#340—611no—
369#340—611no—
370#340—611no—
371#340—611no—
372#340—611no—
373#340—611no—
374#340C2×D6111yescharacteristic
375#340C2×D6111yescharacteristic
376#336—611no—
377#336—611no—
378C61#2157911yescharacteristic, complemented, Sylow
379C22×Q8—611no—
380C23×C4—611no—
381C23×C4—611no—
382C23×C4—611no—
383C23×C4—1221no—
384C23×C4—1221no—
385C22×C8—611no—
386C22×C8—611no—
387C22×C8C22×D6111yescharacteristic
388C22×C8C22×D6111yescharacteristic
389C22⋊Q8—614no—
390C22⋊Q8—1222no—
391C4×Q8—1222no—
392C4×Q8—1222no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—614no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—1222no—
398C2×C4⋊C4—1222no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1222no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C4⋊C4C22×D6111yescharacteristic
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C4⋊C4C22×D6111yescharacteristic
409C2×C22⋊C4—1221no—
410C2×C22⋊C4—1221no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42—1222no—
417C2×C42—1222no—
418C2×C42—1221no—
419C2×C42—1221no—
420C2×C42—1221no—
421C2×C42—1221no—
422C2×C42—2444no—
423C2×C42C22×D6111yescharacteristic
424#107—22no—
425#107—1222no—
426#106—614no—
427#106C4×D6114yescomplemented
428#103—614no—
429#103—614no—
430#103—1222no—
431#103—1222no—
432C22⋊C8—614no—
433C22⋊C8—1222no—
434#97—22no—
435#97—1222no—
436#95—21no—
437#95—21no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441#95—1221no—
442#95—1221no—
443#95—1221no—
444#95—1221no—
445C24—611no—
446C2×Q8—612no—
447C2×Q8—1221no—
448C2×Q8—1221no—
449C2×Q8—1221no—
450C2×Q8—1221no—
451C2×Q8—1221no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—612no—
459C22×C4—612no—
460C22×C4—611no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1224no—
464C22×C4—1222no—
465C22×C4—1224no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1222no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1222no—
475C22×C4—1221no—
476C22×C4—2441no—
477C22×C4—2442no—
478C22×C4—2442no—
479C22×C4—2442no—
480C22×C4—2442no—
481C22×C4D4×D6111yescharacteristic
482C22×C4D4×D6111yescharacteristic
483C22×C4C23×D6111yescharacteristic
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—612no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8C2×C4×D6112yes—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—612no—
502C4⋊C4—612no—
503C4⋊C4—1224no—
504C4⋊C4—1224no—
505C4⋊C4—1224no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—2442no—
511C4⋊C4—2442no—
512C4⋊C4C2×C4×D6112yes—
513C4⋊C4C2×C4×D6112yes—
514C4⋊C4#1175822212yes—
515C22⋊C4—1224no—
516C22⋊C4—2442no—
517C42—42no—
518C42—42no—
519C42—1221no—
520C42—1221no—
521C42—1222no—
522C42—1222no—
523C42—2442no—
524C42—2442no—
525C42—2442no—
526C42—2442no—
527C42#1175822212yes—
528C42#1175822612yes—
529C23—612no—
530C23—612no—
531C23—612no—
532C23—1221no—
533C23—1221no—
534C23—1221no—
535C23—1221no—
536C23C2×D4×D6111yescharacteristic
537Q8—1222no—
538Q8—1222no—
539Q8—2441no—
540Q8—2441no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—612no—
558C2×C4—612no—
559C2×C4—1228no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—1221no—
571C2×C4—1222no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—2444no—
575C2×C4—2448no—
576C2×C4—2448no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4#1451976212yes—
587C2×C4#1451976212yes—
588C2×C4C22×C4×D6111yescharacteristic
589C2×C4#41×D6111yescharacteristic
590C2×C4#41×D6111yescharacteristic
591C2×C4C2×#1175822211yescharacteristic
592C2×C4C2×#1175822211yescharacteristic
593C2×C4C2×#1175822611yescharacteristic
594C8—22no—
595C8—41no—
596C8—1222no—
597C8—2441no—
598C22—614no—
599C22—1222no—
600C22—1222no—
601C22—1222no—
602C22—1222no—
603C22—1222no—
604C22—1222no—
605C22C4×D4×D6111yescharacteristic
606C22C2×#1451976211yescharacteristic
607C22#1684027111yescharacteristic
608C22AGL(1,ℤ/8)×D6111yescharacteristic
609C22#137×D6111yescharacteristic
610C22#137×D6111yescharacteristic
611C22#1684055811yescharacteristic
612C4—22no—
613C4—22no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—81no—
618C4—1222no—
619C4—1222no—
620C4—4881no—
621C4—2442no—
622C4—2442no—
623C4—2442no—
624C4—2441no—
625C4C4×#1175822211yescharacteristic
626C4C4×#1175822611yescharacteristic
627C4#1684029011yescharacteristic
628C4#1684029911yescharacteristic
629C2—1224no—
630C2C4×#1451976211yescharacteristic
631C2#440×D6111yescharacteristic
632C2#2145464111yescharacteristic
633C2#2145574511yescharacteristic
634C2#2145576911yescharacteristic
635C2#2145594711yescharacteristic
636C2#2145595511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1776C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1776D61
hypocenterC61#21579
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #21446459C2semidirect
5. #21446458C2semidirect
6. #21446400C2semidirect
7. #21446399C2semidirect
11. #21445717C2semidirect
12. #21445652C2semidirect
13. #21445650C2semidirect
15. C61×#1776C2semidirect
22. #16840094C4semidirect
97. #14519708C2×C4semidirect
99. #14519702C2×C4semidirect
101. #14519700C2×C4semidirect
122. #106×C61C2×C4semidirect
322. #1776D61semidirect
378. C61#21579semidirect
427. #106C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400942
C4×#145197572
C4×C4⋊C4×D612
#214464592
#214464582
#214464002
#214463992
#214461252
#214460052
#214459042
#214457172
#214456522
#214456502
#214455822
C61×#17762
#2157961