Finite Group Databasebeta

← #29707177

Subgroups — #29707177

Order 15616.

2146 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519808C211yesmaximal, characteristic, complemented
2C22×#14519805C211yesmaximal, characteristic
3C22×#14519766C211yesmaximal, characteristic, complemented
4C2×#16839766C211yesmaximal, characteristic, complemented
5#21448307C211yesmaximal, characteristic, complemented
6#21446914C214yesmaximal, complemented
7#21446897C214yesmaximal, complemented
8#21445512C211yesmaximal, characteristic, complemented
9C2×C61×#419C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×D244C2211yescharacteristic
12C2×#14519808—24no—
13C2×#14519808—24no—
14C2×#14519808—24no—
15C2×#14519805—24no—
16C2×#14519805—24no—
17C2×#14519805—24no—
18C22×#11758202C2212yescomplemented
19C22×#11758190C2212yes—
20C22×#11758201C2212yescomplemented
21C22×D488C2212yes—
22C22×#11758194C2211yescharacteristic
23C2×#14519766—22no—
24C2×#14519766—24no—
25C2×#14519766—24no—
26C2×#14519766C2214yes—
27C22×#11758198—22no—
28C22×C4×Dic61C2211yescharacteristic
29#16839766—24no—
30C2×#14519697C2214yes—
31#16839667C2212yescomplemented
32#16839663C2212yescomplemented
33#16839639—22no—
34C4×#11758198—22no—
35#16839516C2214yescomplemented
36C23×Q8×C61C2211yescharacteristic
37C22×M16×C61C2211yescharacteristic
38C2×#117×C61C2211yescharacteristic
39C61×#419—24no—
40C2×#103×C61C2214yescomplemented
41C24×D61—21no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222D414yes—
48C22×Dic122C2311yescharacteristic
49C22×D244—22no—
50C22×D244—22no—
51C22×D244—22no—
52C22×D244—22no—
53C22×D244—22no—
54C22×D244—21no—
55C22×D244D414yes—
56C22×D244C2312yes—
57C22×D244C2311yescharacteristic
58C22×C22⋊D61—21no—
59C23×Dic61D412yes—
60C22×C4×D61—21no—
61#14519808—232no—
62#14519805—232no—
63C2×#11758202—24no—
64C2×#11758202—24no—
65C2×#11758202—24no—
66C2×#11758190—24no—
67C2×#11758190—24no—
68C2×#11758190—24no—
69C2×#11758201—24no—
70C2×#11758201—24no—
71C2×#11758201—24no—
72C2×D488—24no—
73C2×D488—24no—
74C2×D488—24no—
75C2×#11758194—22no—
76C2×#11758194—22no—
77C2×#11758194D414yes—
78C22×#92418C2312yes—
79#14519766—216no—
80#14519766—28no—
81#14519766—28no—
82C2×#11758198—216no—
83C2×#11758198—28no—
84C2×#11758198—24no—
85C2×C4×Dic61—22no—
86C2×C4×Dic61—22no—
87C2×C4×Dic61D412yes—
88C2×C4×Dic61C2312yes—
89#14519697—28no—
90#14519684—24no—
91C22×Q8×C61—22no—
92C22×Q8×C61—21no—
93C22×Q8×C61—22no—
94C22×Q8×C61—22no—
95C22×Q8×C61—22no—
96C22×Q8×C61—22no—
97C22×Q8×C61D414yes—
98C22×Q8×C61C2312yes—
99C23×C4×C61—21no—
100C23×C4×C61C2311yescharacteristic
101C2×M16×C61—22no—
102C2×M16×C61—22no—
103C2×M16×C61D414yes—
104C22×C8×C61C2312yes—
105#117×C61—22no—
106#117×C61—22no—
107C2×C4⋊C4×C61C2312yes—
108C2×C22⋊C4×C61—21no—
109C2×C42×C61—21no—
110#103×C61—28no—
111C24×C61C2×D411yescharacteristic
112C23×D61—24no—
113C23×D61—21no—
114C23×D61—22no—
115C23×D61—48no—
116C23×D61—42no—
117C23×D61—42no—
118C2×Q8×C61—24no—
119C2×Q8×C61—22no—
120C2×Q8×C61—28no—
121C2×Q8×C61—28no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—44no—
127C2×Q8×C61—42no—
128C2×Q8×C61—44no—
129C2×Q8×C61—42no—
130C2×Q8×C61C2×D414yes—
131C22×C4×C61—24no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—42no—
135C22×C4×C61—42no—
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D412yes—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D412yes—
142C22×C4×C61#4112yes—
143C22×C4×C61C2411yescharacteristic
144C2×C22⋊D61—24no—
145C2×C22⋊D61—42no—
146C2×C22⋊D61—42no—
147C22×Dic61—24no—
148C22×Dic61—21no—
149C22×Dic61—44no—
150C22×Dic61C2×D414yes—
151#11758222—28no—
152#11758222—28no—
153#11758222—44no—
154#11758222—44no—
155C2×D244—28no—
156C2×D244—28no—
157C2×D244—22no—
158C2×D244—22no—
159C2×D244—22no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—22no—
165C2×D244—24no—
166C2×D244—22no—
167C2×D244—21no—
168C2×D244—44no—
169C2×D244—44no—
170C2×D244—42no—
171C2×D244—42no—
172C2×D244C2×D414yes—
173C2×D244C2×D412yes—
174C2×C4×D61—24no—
175C2×C4×D61—42no—
176C2×C4×D61—42no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122C2×D412yes—
183M16×C61—28no—
184C2×C8×C61—22no—
185C2×C8×C61—22no—
186C2×C8×C61C2×D414yes—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—22no—
189C22⋊C4×C61—42no—
190C42×C61—42no—
191#11758202—216no—
192#11758201—216no—
193#11758198—232no—
194#11758198—48no—
195C4×Dic61—24no—
196C4×Dic61—22no—
197C4×Dic61—22no—
198#11758194—28no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418C2×D414yes—
202D488—216no—
203#11758190—216no—
204C23×C61—24no—
205C23×C61#12012yes—
206C23×C61#12112yes—
207C23×C61#12712yes—
208C23×C61C22×D411yescharacteristic
209C22×D61—24no—
210C22×D61—22no—
211C22×D61—416no—
212C22×D61—41no—
213C22×D61—42no—
214C22×D61—41no—
215C22×D61—42no—
216C22×D61—44no—
217C22×D61—48no—
218C22×D61—48no—
219C22×D61—88no—
220C22×D61—88no—
221Q8×C61—24no—
222Q8×C61—24no—
223Q8×C61—42no—
224Q8×C61—42no—
225Q8×C61—48no—
226C2×C4×C61—24no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—48no—
230C2×C4×C61—42no—
231C2×C4×C61—44no—
232C2×C4×C61#12014yes—
233C2×C4×C61#12014yes—
234C2×C4×C61#12012yes—
235C2×C4×C61#12114yes—
236C2×C4×C61#12114yes—
237C2×C4×C61#12112yes—
238C2×C4×C61#12712yes—
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C22⋊D61—48no—
246C2×Dic61—24no—
247C2×Dic61—22no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—48no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253D244—24no—
254D244—22no—
255D244—22no—
256D244—24no—
257D244—42no—
258D244—42no—
259D244—41no—
260D244—41no—
261D244—48no—
262C4×D61—48no—
263Dic122—22no—
264Dic122—22no—
265Dic122—41no—
266Dic122—41no—
267C8×C61—24no—
268#92418—24no—
269#23431—611nomaximal, Sylow
270C22×C61—24no—
271C22×C61—24no—
272C22×C61—24no—
273C22×C61#39214yes—
274C22×C61C2×#12011yescharacteristic
275C22×C61C2×#12111yescharacteristic
276C22×C61C2×#12711yescharacteristic
277C22×C61#57514yes—
278C2×D61—42no—
279C2×D61—44no—
280C2×D61—48no—
281C2×D61—44no—
282C2×D61—816no—
283C2×D61—82no—
284C2×D61—84no—
285C2×D61—82no—
286C2×D61—84no—
287C4×C61—44no—
288C4×C61—82no—
289C4×C61#39214yes—
290C4×C61C2×#12011yescharacteristic
291C4×C61C2×#12011yescharacteristic
292C4×C61C2×#12111yescharacteristic
293C4×C61C2×#12111yescharacteristic
294Dic61—42no—
295Dic61—44no—
296C22×#137—611no—
297C22×AGL(1,ℤ/8)—611no—
298C22×#124—611no—
299C2×#419D6111yescharacteristic, complemented
300C2×#418—611no—
301#2297—611no—
302#2015—614no—
303#2012—614no—
304#1728—611no—
305C2×C61—24no—
306C2×C61C2×#39211yescharacteristic
307C2×C61#300512yes—
308C2×C61#304012yes—
309C2×C61#315312yes—
310D61—82no—
311D61—84no—
312C22×#41—611no—
313C23×Q8C2×D6111yescharacteristic
314C23×D4—611no—
315C2×#137—1224no—
316C2×#137—1224no—
317C2×#137—1224no—
318C2×AGL(1,ℤ/8)—1224no—
319C2×AGL(1,ℤ/8)—1224no—
320C2×AGL(1,ℤ/8)—1224no—
321C22×Q16—612no—
322C22×SD16—612no—
323C22×SD16—612no—
324C22×D8—612no—
325C22×M16—611no—
326C22×M16C2×D6111yescharacteristic
327C2×#124—614no—
328C2×#124—1222no—
329C2×#124—1224no—
330C2×#124—1224no—
331C2×#117C2×D6111yescharacteristic
332C22×C22⋊C4—1222no—
333C22×C42—611no—
334#419—24no—
335#418—1224no—
336C2×#103C2×D6114yescomplemented
337C2×#102—614no—
338#391—612no—
339#390—612no—
340#386—1222no—
341C4×C22⋊C4—1222no—
342#336—614no—
343C61#2343111yescharacteristic, complemented, Sylow
344C25—1221no—
345C2×#41—614no—
346C2×#41—1222no—
347C2×#41—1222no—
348C2×#41—1222no—
349C2×#41—1222no—
350C2×#41—1222no—
351C22×Q8—21no—
352C22×Q8—22no—
353C22×Q8—22no—
354C22×Q8—22no—
355C22×Q8—22no—
356C22×Q8—22no—
357C22×Q8—611no—
358C22×Q8D24414yes—
359C22×Q8C22×D6112yes—
360C22×D4—614no—
361C22×D4—612no—
362C22×D4—611no—
363C22×D4—1221no—
364C22×D4—1222no—
365C22×D4—1222no—
366C22×D4—1222no—
367C22×D4—1222no—
368C22×D4—1221no—
369C22×D4—1222no—
370C23×C4—21no—
371C23×C4—612no—
372C23×C4—1221no—
373C23×C4C22×D6111yescharacteristic
374#137—12232no—
375AGL(1,ℤ/8)—12232no—
376C2×Q16—1224no—
377C2×Q16—1224no—
378C2×Q16—1224no—
379C2×SD16—1224no—
380C2×SD16—1224no—
381C2×SD16—1224no—
382C2×SD16—1224no—
383C2×SD16—1224no—
384C2×SD16—1224no—
385C2×D8—1224no—
386C2×D8—1224no—
387C2×D8—1224no—
388C2×M16—22no—
389C2×M16—22no—
390C2×M16—614no—
391C2×M16—1222no—
392C2×M16—1222no—
393C2×M16C22⋊D6114yes—
394C22×C8—612no—
395C22×C8C22×D6112yes—
396#124—1228no—
397#124—12216no—
398#124—1228no—
399#117—22no—
400#117—22no—
401C2×C4⋊C4C22×D6112yes—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—1228no—
404C2×C22⋊C4—12216no—
405C2×C22⋊C4—1224no—
406C2×C42—21no—
407C2×C42—612no—
408C2×C42—612no—
409C2×C42—1222no—
410C2×C42—1222no—
411#103—28no—
412#102—1228no—
413#95—1224no—
414C24—1221no—
415C24—1222no—
416C24—1224no—
417C24—2448no—
418C24—2442no—
419C24—2442no—
420C24D4×D6111yescharacteristic
421#41—1228no—
422#41—1228no—
423#41—2444no—
424#41—2444no—
425C2×Q8—22no—
426C2×Q8—24no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—28no—
432C2×Q8—28no—
433C2×Q8—44no—
434C2×Q8—42no—
435C2×Q8—44no—
436C2×Q8—42no—
437C2×Q8—612no—
438C2×Q8—1221no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8—1221no—
443C2×Q8C2×D24414yes—
444C2×D4—614no—
445C2×D4—612no—
446C2×D4—1224no—
447C2×D4—1222no—
448C2×D4—1221no—
449C2×D4—1222no—
450C2×D4—1221no—
451C2×D4—1221no—
452C2×D4—1222no—
453C2×D4—1221no—
454C2×D4—1222no—
455C2×D4—1222no—
456C2×D4—1221no—
457C2×D4—1228no—
458C2×D4—1224no—
459C2×D4—1228no—
460C2×D4—2442no—
461C2×D4—2442no—
462C2×D4—2444no—
463C2×D4—2442no—
464C2×D4—2444no—
465C2×D4—2442no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—614no—
472C22×C4—1221no—
473C22×C4—1224no—
474C22×C4—1224no—
475C22×C4—2444no—
476C22×C4—2442no—
477C22×C4—2442no—
478C22×C4C2×D24412yes—
479C22×C4#1175822212yes—
480C22×C4D4×D6112yes—
481C22×C4D4×D6112yes—
482C22×C4D4×D6112yes—
483C22×C4D4×D6111yescharacteristic
484C22×C4C2×C22⋊D6112yes—
485C22×C4C23×D6111yescharacteristic
486Q16—12216no—
487SD16—12216no—
488SD16—12216no—
489D8—12216no—
490M16—28no—
491M16—1228no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—614no—
495C2×C8—1222no—
496C2×C8—1222no—
497C2×C8C2×C22⋊D6114yes—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C22⋊C4—42no—
501C22⋊C4—12232no—
502C22⋊C4—2448no—
503C42—42no—
504C42—1222no—
505C42—1222no—
506C42—1224no—
507C23—24no—
508C23—1224no—
509C23—1222no—
510C23—4888no—
511C23—4888no—
512C23—24416no—
513C23—2441no—
514C23—2442no—
515C23—2441no—
516C23—2442no—
517C23—2444no—
518C23—2448no—
519C23—2448no—
520C23#1451974712yes—
521C23#1451975012yes—
522C23#1451977312yes—
523C23C2×D4×D6111yescharacteristic
524Q8—24no—
525Q8—24no—
526Q8—42no—
527Q8—42no—
528Q8—48no—
529Q8—1222no—
530Q8—1222no—
531Q8—2441no—
532Q8—2441no—
533D4—1222no—
534D4—1224no—
535D4—1222no—
536D4—1224no—
537D4—2448no—
538D4—2442no—
539D4—2441no—
540D4—2442no—
541D4—2441no—
542D4—2448no—
543C2×C4—24no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—48no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—1224no—
550C2×C4—1222no—
551C2×C4—1224no—
552C2×C4—1224no—
553C2×C4—2448no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2448no—
557C2×C4#1451974514yes—
558C2×C4#1451974714yes—
559C2×C4#1451974712yes—
560C2×C4#1451975014yes—
561C2×C4#1451975012yes—
562C2×C4#1451975114yes—
563C2×C4#1451977312yes—
564C2×C4C2×D4×D6111yescharacteristic
565C2×C4C2×D4×D6111yescharacteristic
566C2×C4C2×D4×D6111yescharacteristic
567C2×C4C22×C22⋊D6111yescharacteristic
568C2×C4C22×D24411yescharacteristic
569C2×C4C2×#1175822211yescharacteristic
570C8—24no—
571C8—1224no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—48816no—
576C22—4884no—
577C22—4882no—
578C22—4884no—
579C22—4882no—
580C22—2442no—
581C22—2444no—
582C22—2444no—
583C22—2448no—
584C22#1683967114yes—
585C22C2×#1451974711yescharacteristic
586C22C2×#1451975011yescharacteristic
587C22C2×#1451977311yescharacteristic
588C22#1684056114yes—
589C4—44no—
590C4—82no—
591C4—2442no—
592C4—2444no—
593C4#1683967114yes—
594C4C2×#1451974511yescharacteristic
595C4C2×#1451974711yescharacteristic
596C4C2×#1451975011yescharacteristic
597C4C2×#1451975111yescharacteristic
598C2—24no—
599C2—4884no—
600C2—4882no—
601C2C2×#1683967111yescharacteristic
602C2#2145506512yes—
603C2#2145531612yes—
604C2#2145628612yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#419C2
socleC23×C61C22×D4
prime radicalC22×#14519805C2
hypercenterC2×#419D61
hypocenterC61#23431
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
1. C22×#14519808C2semidirect
3. C22×#14519766C2semidirect
4. C2×#16839766C2semidirect
5. #21448307C2semidirect
6. #21446914C2semidirect
7. #21446897C2semidirect
8. #21445512C2semidirect
9. C2×C61×#419C2semidirect
18. C22×#11758202C22semidirect
20. C22×#11758201C22semidirect
31. #16839667C22semidirect
32. #16839663C22semidirect
35. #16839516C22semidirect
40. C2×#103×C61C22semidirect
299. C2×#419D61semidirect
336. C2×#103C2×D61semidirect
343. C61#23431semidirect

Maximal subgroups

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

subgroupindex
C22×#145198082
C22×#145198052
C22×#145197662
C2×#168397662
#214483072
#214469142
#214468972
#214455122
C2×C61×#4192
#2343161