Finite Group Databasebeta

← #29695483

Subgroups — #29695483

Order 15616.

1542 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840433C211yesmaximal, characteristic
2C2×#16839759C212yesmaximal, complemented
3#21448768C211yesmaximal, characteristic, complemented
4#21446745C211yesmaximal, characteristic, complemented
5#21446625C211yesmaximal, characteristic, complemented
6#21446247C212yesmaximal, complemented
7#21446139C212yesmaximal, complemented
8#21445825C211yesmaximal, characteristic, complemented
9#21445760C212yesmaximal, complemented
10C4×#14519702C211yesmaximal, characteristic, complemented
11C2×C8×D4×C61C211yesmaximal, characteristic, complemented
12C22×#11758222C2212yes—
13#16840433—24no—
14#16840433C418yescomplemented
15C2×#14519774C2211yescharacteristic
16C2×#14519773—21no—
17C2×#14519751C2212yes—
18C2×#14519750—22no—
19C2×C4×Dic122C2211yescharacteristic
20C2×#14519739—21no—
21C2×#14519736C2212yes—
22#16839759—28no—
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2212yes—
25C2×#14519702C2211yescharacteristic
26C2×#14519699C2212yes—
27C2×#14519699C2212yes—
28C2×#14519699C2212yes—
29C2×#14519699C2212yes—
30#16839665—21no—
31#16839655—22no—
32#16839632C2211yescharacteristic
33#16839622C2212yes—
34C4⋊C4×Dic61—21no—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839525C2211yescharacteristic
37#16839525C2211yescharacteristic
38#16839525C2212yes—
39C23×C8×C61C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C8×D4×C61—24no—
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×C22⋊C8×C61C2212yes—
44C2×C4×C8×C61C2211yescharacteristic
45C2×#11758222—24no—
46C2×#11758222—24no—
47C2×#11758222—24no—
48C2×#11758222—42no—
49C2×#11758222—42no—
50C2×#11758222C2×C418yes—
51C22×Dic122C2311yescharacteristic
52C22×D244—21no—
53C22×D244—21no—
54C22×D244C2312yes—
55C22×C22⋊D61—22no—
56C22×C22⋊D61—22no—
57C22×C4×D61—22no—
58#14519774—22no—
59#14519774C2×C414yes—
60#14519773—28no—
61#14519751—24no—
62#14519751C2×C418yes—
63#14519750—216no—
64C4×Dic122—22no—
65C4×Dic122C2×C414yes—
66#14519739—28no—
67#14519736—24no—
68#14519736—42no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×C22⋊Dic61—22no—
77C2×C22⋊Dic61—22no—
78C2×#11758198—22no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—22no—
81C2×C4×Dic61—21no—
82#14519702—24no—
83#14519702—22no—
84#14519702D414yescomplemented
85#14519699—24no—
86#14519699—24no—
87#14519699—42no—
88#14519699—42no—
89#14519684—21no—
90#14519684—22no—
91C22×D4×C61D411yescharacteristic
92C23×C4×C61C2312yes—
93C22×C8×C61—24no—
94C22×C8×C61—24no—
95C22×C8×C61—42no—
96C22×C8×C61C2312yes—
97C22×C8×C61C2311yescharacteristic
98C22×C8×C61C2311yescharacteristic
99C4×D4×C61C2×C414yes—
100C4×D4×C61D414yes—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C22⋊C4×C61D412yes—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61—22no—
105C22⋊C8×C61—24no—
106C4×C8×C61—22no—
107C24×C61C2×D412yes—
108C23×D61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61C22⋊C414yes—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61C22×C414yes—
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D414yes—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C2×C22⋊D61—28no—
123C2×C22⋊D61—28no—
124C2×C22⋊D61—44no—
125C2×C22⋊D61—44no—
126C2×C22⋊D61—82no—
127C2×C22⋊D61—82no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—22no—
133#11758222—216no—
134#11758222—48no—
135#11758222—48no—
136#11758222—48no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—22no—
141C2×D244—24no—
142C2×D244—22no—
143C2×D244—24no—
144C2×D244—42no—
145C2×D244—42no—
146C2×D244—81no—
147C2×D244—81no—
148C2×D244C22×C414yes—
149C2×C4×D61—28no—
150C2×C4×D61—44no—
151C2×C4×D61—44no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122C22×C412yes—
156C2×Dic122C2×D412yes—
157C2×Dic122#4112yes—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—22no—
162C2×C8×C61—22no—
163C2×C8×C61—21no—
164C2×C8×C61—22no—
165C2×C8×C61—21no—
166C2×C8×C61—48no—
167C2×C8×C61C2×D412yes—
168C4⋊C4×C61C22⋊C414yes—
169C22⋊C4×C61C22⋊C418yes—
170C42×C61C22×C412yes—
171C42×C61C2×D412yes—
172C22⋊Dic61—44no—
173C22⋊Dic61—82no—
174#11758198—28no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—22no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—22no—
181C4⋊Dic61—42no—
182C4⋊Dic61—81no—
183C4⋊Dic61C22×C412yes—
184C4⋊Dic61C2×D412yes—
185#11758196—24no—
186C4×Dic61—24no—
187C4×Dic61—44no—
188C4×Dic61—42no—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61—22no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61C2×C22⋊C414yes—
196C23×C61C22×D411yescharacteristic
197C22×D61—24no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61—42no—
204D4×C61—28no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—24no—
210C2×C4×C61—24no—
211C2×C4×C61—24no—
212C2×C4×C61—24no—
213C2×C4×C61C2×C22⋊C412yes—
214C2×C4×C61C2×C22⋊C414yes—
215C2×C4×C61C2×C22⋊C412yes—
216C2×C4×C61#13512yes—
217C2×C4×C61#13512yes—
218C2×C4×C61C23×C411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61H(5,𝔽2)11yescharacteristic
223C2×C4×C61H(5,𝔽2)11yescharacteristic
224C22⋊D61—416no—
225C22⋊D61—88no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—42no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—41no—
241D244—24no—
242D244—24no—
243D244—42no—
244D244—42no—
245D244—48no—
246D244—84no—
247C4×D61—416no—
248Dic122—22no—
249Dic122—22no—
250Dic122—22no—
251Dic122C4×D414yes—
252C8×C61—22no—
253C8×C61—42no—
254C8×C61—41no—
255#22418—611nomaximal, Sylow
256C22×C61—24no—
257C22×C61—24no—
258C22×C61—24no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61C22×C22⋊C411yescharacteristic
264C22×C61D4211yescharacteristic
265C22×C61#54611yescharacteristic
266C22×C61C2×#13511yescharacteristic
267C22×C61C2×#13511yescharacteristic
268C22×C61#57611yescharacteristic
269C22×C61#57711yescharacteristic
270C2×D61—44no—
271C2×D61—44no—
272C2×D61—44no—
273C2×D61—82no—
274C2×D61—82no—
275C2×D61—82no—
276C2×D61—82no—
277C4×C61—24no—
278C4×C61—22no—
279C4×C61#41614yescomplemented
280C4×C61C2×C4×D411yescharacteristic
281C4×C61#51811yescharacteristic
282C4×C61D4211yescharacteristic
283C4×C61#54611yescharacteristic
284Dic61—24no—
285Dic61—42no—
286Dic61—42no—
287Dic61—82no—
288Dic61—81no—
289C2×#550—611no—
290C2×C8×D4D6111yescharacteristic, complemented
291C2×#416—612no—
292#2354—611no—
293#1982—611no—
294#1957—611no—
295#1893—612no—
296#1875—612no—
297#1803—611no—
298#1788—612no—
299C4×#103—611no—
300C2×C61—44no—
301C2×C61D4×C22⋊C411yescharacteristic
302C2×C61C2×#41611yescharacteristic
303C2×C61#289711yescharacteristic
304C2×C61#328411yescharacteristic
305C2×C61#328711yescharacteristic
306C2×C61#330411yescharacteristic
307C2×C61#330511yescharacteristic
308D61—84no—
309C22×#41—612no—
310C23×C8C2×D6112yes—
311#550—618no—
312#550—1224no—
313C2×#127—611no—
314C2×#127—1221no—
315C2×#121—612no—
316C2×#121—1222no—
317C2×C4×Q8—611no—
318C2×C4×D4—1221no—
319C2×C4×D4C2×D6111yescharacteristic
320C2×#117—612no—
321C8×D4—24no—
322C2×C4⋊C8C2×D6111yescharacteristic
323#416—1228no—
324C2×#103—611no—
325C2×#103—612no—
326C2×#103—611no—
327C2×#102—612no—
328C2×#102—612no—
329C2×#102—612no—
330C2×#102—612no—
331C2×C22⋊C8C2×D6112yes—
332C2×C4×C8C2×D6111yescharacteristic
333#390—1221no—
334#388—1222no—
335#384—611no—
336#380—612no—
337C4×C4⋊C4—611no—
338C4×C4⋊C4—1221no—
339#340—612no—
340#340—611no—
341#340—611no—
342C61#2241811yescharacteristic, complemented, Sylow
343C2×#41—618no—
344C2×#41—1224no—
345C2×#41—1224no—
346C2×#41—1224no—
347C2×#41—2442no—
348C2×#41—2442no—
349C22×Q8—611no—
350C22×D4—612no—
351C22×D4—1222no—
352C22×D4—1222no—
353C22×D4—1221no—
354C22×D4—1221no—
355C22×D4D24411yescharacteristic
356C23×C4—1222no—
357C23×C4C22×D6112yes—
358C22×C8—24no—
359C22×C8—24no—
360C22×C8—42no—
361C22×C8C22×D6112yes—
362C22×C8C22×D6111yescharacteristic
363C22×C8C22×D6111yescharacteristic
364#127—614no—
365#127—1228no—
366#127—1222no—
367#121—618no—
368#121—1224no—
369#121—12216no—
370C4×Q8—614no—
371C4×Q8—1222no—
372C4×D4—1228no—
373C4×D4C4×D6114yes—
374C4×D4C22⋊D6114yes—
375#117—1224no—
376#117—2442no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—612no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4D24411yescharacteristic
385C2×C22⋊C4—1222no—
386C2×C22⋊C4—1222no—
387C2×C22⋊C4—1222no—
388C2×C22⋊C4D24412yes—
389C2×C42—1221no—
390C2×C42—1222no—
391C2×C42—1221no—
392C2×C42C22×D6111yescharacteristic
393C4⋊C8—22no—
394#103—614no—
395#103—1222no—
396#103—1224no—
397#102—1224no—
398#102—1224no—
399#102—2442no—
400#102—2442no—
401C22⋊C8—24no—
402C4×C8—22no—
403#95—1221no—
404#95—1222no—
405C24—1222no—
406C24C2×D24412yes—
407#41—12216no—
408#41—2448no—
409#41—2448no—
410#41—2448no—
411C2×Q8—612no—
412C2×Q8—612no—
413C2×Q8—612no—
414C2×Q8—1221no—
415C2×Q8—1221no—
416C2×Q8—1221no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—614no—
420C2×D4—1228no—
421C2×D4—1228no—
422C2×D4—1222no—
423C2×D4—1222no—
424C2×D4—1222no—
425C2×D4—1222no—
426C2×D4—1222no—
427C2×D4—1224no—
428C2×D4—1224no—
429C2×D4—4881no—
430C2×D4—4881no—
431C2×D4—4882no—
432C2×D4—4882no—
433C2×D4—2444no—
434C2×D4—2444no—
435C2×D4—2442no—
436C2×D4—2442no—
437C2×D4#1175819814yes—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1228no—
446C22×C4—1221no—
447C22×C4—1222no—
448C22×C4—2444no—
449C22×C4—2444no—
450C22×C4C2×C4×D6114yes—
451C22×C4C2×D24412yes—
452C22×C4C2×D24411yescharacteristic
453C22×C4C2×D24411yescharacteristic
454C22×C4C2×C22⋊D6114yes—
455C22×C4C23×D6111yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—22no—
459C2×C8—21no—
460C2×C8—22no—
461C2×C8—21no—
462C2×C8—22no—
463C2×C8—21no—
464C2×C8—48no—
465C2×C8D4×D6112yes—
466C4⋊C4—612no—
467C4⋊C4—612no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1224no—
471C4⋊C4—1222no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1222no—
475C4⋊C4—4881no—
476C4⋊C4—2442no—
477C4⋊C4#1175819814yes—
478C22⋊C4—1228no—
479C22⋊C4—4882no—
480C22⋊C4—2444no—
481C22⋊C4#1175819818yes—
482C42—1224no—
483C42—2444no—
484C42—2442no—
485C42C2×C4×D6112yes—
486C42C2×C22⋊D6112yes—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—1224no—
494C23—2442no—
495C23—2442no—
496C23—2442no—
497C23—2442no—
498C23—2442no—
499C23—2442no—
500C23C2×#1175819814yes—
501C23C22×D24411yescharacteristic
502Q8—614no—
503Q8—1222no—
504Q8—1222no—
505Q8—1222no—
506D4—28no—
507D4—1224no—
508D4—1224no—
509D4—4884no—
510D4—4888no—
511D4—24416no—
512D4—2448no—
513D4—2442no—
514D4—2442no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—24416no—
529C2×C4—2441no—
530C2×C4—2442no—
531C2×C4—2441no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4C2×#1175819812yes—
540C2×C4C2×#1175819814yes—
541C2×C4C2×#1175819812yes—
542C2×C4#1451979912yes—
543C2×C4#1451979912yes—
544C2×C4C22×C4×D6111yescharacteristic
545C2×C4C2×D4×D6111yescharacteristic
546C2×C4C22×C22⋊D6111yescharacteristic
547C2×C4#41×D6111yescharacteristic
548C2×C4#1451982211yescharacteristic
549C2×C4#1451982311yescharacteristic
550C8—22no—
551C8—42no—
552C8—41no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—4882no—
561C22—4882no—
562C22—4882no—
563C22—4882no—
564C22—2444no—
565C22—2444no—
566C22—2444no—
567C22C22×#1175819811yescharacteristic
568C22D4×D24411yescharacteristic
569C22#1684040711yescharacteristic
570C22C2×#1451979911yescharacteristic
571C22C2×#1451979911yescharacteristic
572C22#1684056711yescharacteristic
573C22#1684057411yescharacteristic
574C4—22no—
575C4—24no—
576C4—1224no—
577C4—4881no—
578C4—4882no—
579C4—2442no—
580C4—2442no—
581C4#1683975914yescomplemented
582C4C4×D4×D6111yescharacteristic
583C4#1684023411yescharacteristic
584C4D4×C22⋊D6111yescharacteristic
585C4#1684040011yescharacteristic
586C2—44no—
587C2—4884no—
588C2D4×#1175819811yescharacteristic
589C2C2×#1683975911yescharacteristic
590C2#2145434611yescharacteristic
591C2#2145755111yescharacteristic
592C2#2145757911yescharacteristic
593C2#2145775911yescharacteristic
594C2#2145777011yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C8×D4×C61C2
socleC23×C61C22×D4
prime radicalC2×#16840433C2
cosocleC22×C4×C61C24
hypercenterC2×C8×D4D61
hypocenterC61#22418
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
2. C2×#16839759C2semidirect
3. #21448768C2semidirect
4. #21446745C2semidirect
5. #21446625C2semidirect
6. #21446247C2semidirect
7. #21446139C2semidirect
8. #21445825C2semidirect
9. #21445760C2semidirect
10. C4×#14519702C2semidirect
11. C2×C8×D4×C61C2semidirect
14. #16840433C4semidirect
84. #14519702D4semidirect
279. C4×C61#416semidirect
290. C2×C8×D4D61semidirect
342. C61#22418semidirect
581. C4#16839759semidirect

Maximal subgroups

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

subgroupindex
C2×#168404332
C2×#168397592
#214487682
#214467452
#214466252
#214462472
#214461392
#214458252
#214457602
C4×#145197022
C2×C8×D4×C612
#2241861