Finite Group Databasebeta

← #29707828

Subgroups — #29707828

Order 15616.

1454 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519805C211yesmaximal, characteristic
2#21454336C211yesmaximal, characteristic, complemented
3C2×#16839766C211yesmaximal, characteristic, complemented
4C2×#16839744C211yesmaximal, characteristic, complemented
5C22×#14519693C211yesmaximal, characteristic, complemented
6#21447082C214yesmaximal, complemented
7#21447075C214yesmaximal, complemented
8#21445546C211yesmaximal, characteristic, complemented
9C61×#2905C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×D244C2211yescharacteristic
12C2×#14519805—22no—
13C2×#14519805—22no—
14C2×#14519805—24no—
15C2×#14519805—24no—
16C22×#11758190C2212yes—
17C22×D488C2212yes—
18C22×#11758194C2211yescharacteristic
19C22×#11758194C2211yescharacteristic
20#16839770C2212yescomplemented
21#16839766—22no—
22#16839766C2214yes—
23#16839766C2212yescomplemented
24#16839761—22no—
25#16839756—22no—
26C2×#14519697—22no—
27#16839744—24no—
28C2×#14519695C2212yescomplemented
29C2×#14519693—24no—
30C2×#14519693C2214yes—
31C2×#14519693C2212yescomplemented
32#16839527C2214yescomplemented
33C22×M16×C61C2211yescharacteristic
34C22×M16×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C61×#424—22no—
37C61×#424C2214yescomplemented
38C61×#423—22no—
39C24×D61—21no—
40C2×#11758222—22no—
41C2×#11758222—21no—
42C2×#11758222—21no—
43C2×#11758222—21no—
44C2×#11758222—21no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222D414yes—
48C22×Dic122C2311yescharacteristic
49C22×D244—21no—
50C22×D244—21no—
51C22×D244—22no—
52C22×D244—21no—
53C22×D244—21no—
54C22×D244—22no—
55C22×D244—21no—
56C22×D244—22no—
57C22×D244D414yes—
58C22×D244C2312yes—
59C22×D244C2311yescharacteristic
60C22×C22⋊D61—21no—
61C22×C4×D61—21no—
62#14519805—216no—
63#14519805—216no—
64C2×#11758190—24no—
65C2×#11758190—24no—
66C2×#11758190—24no—
67C2×D488—24no—
68C2×D488—24no—
69C2×D488—24no—
70C2×#11758194—21no—
71C2×#11758194—21no—
72C2×#11758194—22no—
73C2×#11758194—22no—
74C2×#11758194—22no—
75C2×#11758194—22no—
76C2×#11758194C2312yes—
77C2×#11758194C2312yes—
78C22×#92418—21no—
79C22×#92418—21no—
80#14519700—24no—
81#14519697—28no—
82#14519697—24no—
83#14519695—24no—
84#14519693—28no—
85#14519693—24no—
86C23×C4×C61C2311yescharacteristic
87C2×M16×C61—22no—
88C2×M16×C61—22no—
89C2×M16×C61—21no—
90C2×M16×C61—22no—
91C2×M16×C61—22no—
92C2×M16×C61D412yes—
93C2×M16×C61D412yes—
94C2×M16×C61C2312yes—
95C22×C8×C61—21no—
96C22×C8×C61C2312yes—
97#117×C61—21no—
98#117×C61—21no—
99#117×C61D412yes—
100#117×C61C2312yes—
101C2×C4⋊C4×C61D412yes—
102C2×C22⋊C4×C61—21no—
103C2×C42×C61—21no—
104C4⋊C8×C61—28no—
105C24×C61#4111yescharacteristic
106C23×D61—24no—
107C23×D61—21no—
108C23×D61—22no—
109C23×D61—41no—
110C23×D61—41no—
111C23×D61—42no—
112C23×D61—84no—
113C22×C4×C61—22no—
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61#4111yescharacteristic
121C22×C4×C61#4112yes—
122C22×C4×C61#4112yes—
123C22×C4×C61C2411yescharacteristic
124C2×C22⋊D61—24no—
125C2×C22⋊D61—41no—
126C2×C22⋊D61—41no—
127C2×C22⋊D61—42no—
128C22×Dic61—21no—
129#11758222—24no—
130#11758222—24no—
131#11758222—24no—
132#11758222—24no—
133#11758222—42no—
134#11758222—42no—
135#11758222—42no—
136#11758222—42no—
137C2×D244—24no—
138C2×D244—24no—
139C2×D244—24no—
140C2×D244—24no—
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—22no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—22no—
148C2×D244—21no—
149C2×D244—24no—
150C2×D244—22no—
151C2×D244—21no—
152C2×D244—42no—
153C2×D244—42no—
154C2×D244—41no—
155C2×D244—41no—
156C2×D244—42no—
157C2×D244—42no—
158C2×D244—42no—
159C2×D244C2×D414yes—
160C2×D244C2×D412yes—
161C2×C4×D61—24no—
162C2×C4×D61—41no—
163C2×C4×D61—41no—
164C2×C4×D61—42no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122C2×D412yes—
171M16×C61—24no—
172M16×C61—24no—
173M16×C61—22no—
174M16×C61—22no—
175M16×C61—42no—
176C2×C8×C61—22no—
177C2×C8×C61—22no—
178C2×C8×C61—22no—
179C2×C8×C61—42no—
180C2×C8×C61C2×D414yes—
181C4⋊C4×C61—22no—
182C4⋊C4×C61C2×D414yes—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C42×C61—22no—
186C42×C61—22no—
187#11758194—22no—
188#11758194—22no—
189#11758194—22no—
190#11758194—22no—
191#11758194—42no—
192#11758194—42no—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418—42no—
196C2×#92418—42no—
197D488—216no—
198#11758190—216no—
199C23×C61—42no—
200C23×C61#12111yescharacteristic
201C23×C61#12111yescharacteristic
202C23×C61#12312yes—
203C23×C61#12312yes—
204C23×C61C2×#4111yescharacteristic
205C22×D61—22no—
206C22×D61—24no—
207C22×D61—41no—
208C22×D61—42no—
209C22×D61—41no—
210C22×D61—42no—
211C22×D61—44no—
212C22×D61—44no—
213C22×D61—84no—
214C22×D61—84no—
215C22×D61—88no—
216C22×D61—82no—
217C22×D61—84no—
218C22×D61—88no—
219C2×C4×C61—24no—
220C2×C4×C61—82no—
221C2×C4×C61#12012yes—
222C2×C4×C61#12012yes—
223C2×C4×C61#12112yes—
224C2×C4×C61#12112yes—
225C2×C4×C61#12112yes—
226C2×C4×C61#12112yes—
227C2×C4×C61#12111yescharacteristic
228C2×C4×C61#12111yescharacteristic
229C2×C4×C61#12312yes—
230C2×C4×C61#12312yes—
231C2×C4×C61#12412yes—
232C2×C4×C61#12412yes—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C22⋊D61—44no—
240C22⋊D61—44no—
241C2×Dic61—22no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244D244—24no—
245D244—22no—
246D244—24no—
247D244—22no—
248D244—42no—
249D244—41no—
250D244—42no—
251D244—41no—
252D244—44no—
253D244—44no—
254C4×D61—44no—
255C4×D61—44no—
256Dic122—22no—
257Dic122—22no—
258Dic122—41no—
259Dic122—41no—
260C8×C61—24no—
261C8×C61—42no—
262#92418—42no—
263#92418—42no—
264#23521—611nomaximal, Sylow
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61#39412yes—
271C22×C61#39412yes—
272C22×C61C2×#12111yescharacteristic
273C22×C61C2×#12311yescharacteristic
274C22×C61C2×#12311yescharacteristic
275C2×D61—42no—
276C2×D61—44no—
277C2×D61—82no—
278C2×D61—88no—
279C2×D61—84no—
280C2×D61—88no—
281C2×D61—82no—
282C2×D61—84no—
283C2×D61—82no—
284C2×D61—84no—
285C4×C61—44no—
286C4×C61#39412yes—
287C4×C61#39412yes—
288C4×C61C2×#12011yescharacteristic
289C4×C61C2×#12111yescharacteristic
290C4×C61C2×#12111yescharacteristic
291C4×C61C2×#12411yescharacteristic
292Dic61—42no—
293C22×AGL(1,ℤ/8)—611no—
294#2905D6111yescharacteristic, complemented
295#2896—611no—
296C2×#418—611no—
297C2×#413—611no—
298C22×#100—611no—
299#2046—614no—
300#2045—614no—
301#1738—611no—
302C2×C61—21no—
303C2×C61—21no—
304C2×C61—22no—
305C2×C61—22no—
306C2×C61C2×#39411yescharacteristic
307C2×C61#306812yes—
308D61—82no—
309D61—84no—
310C22×#41—611no—
311C23×D4—611no—
312C2×AGL(1,ℤ/8)—1224no—
313C2×AGL(1,ℤ/8)—1224no—
314C2×AGL(1,ℤ/8)—1222no—
315C2×AGL(1,ℤ/8)—1222no—
316C22×SD16—612no—
317C22×D8—612no—
318C22×M16—611no—
319C22×M16—611no—
320C22×M16C2×D6111yescharacteristic
321C22×M16C2×D6111yescharacteristic
322C2×#117C2×D6111yescharacteristic
323#424—22no—
324#424C2×D6114yescomplemented
325#423—22no—
326#419—612no—
327#418—614no—
328#418—612no—
329#418—1222no—
330#417—1222no—
331#416—1222no—
332C2×#102—1222no—
333#413—1224no—
334C2×#101—612no—
335C2×#100—614no—
336C2×#100—612no—
337C2×#100—1224no—
338#341—614no—
339C61#2352111yescharacteristic, complemented, Sylow
340C25—1221no—
341C2×#41—614no—
342C2×#41—1222no—
343C2×#41—1221no—
344C2×#41—1221no—
345C2×#41—1222no—
346C2×#41—1222no—
347C2×#41—1221no—
348C2×#41—1221no—
349C22×Q8—611no—
350C22×D4—614no—
351C22×D4—611no—
352C22×D4—612no—
353C22×D4—1221no—
354C22×D4—1221no—
355C22×D4—1221no—
356C22×D4—1221no—
357C22×D4—1222no—
358C22×D4—1222no—
359C22×D4—1222no—
360C22×D4—1221no—
361C22×D4—1221no—
362C23×C4—1221no—
363C23×C4C22×D6111yescharacteristic
364AGL(1,ℤ/8)—12216no—
365AGL(1,ℤ/8)—12216no—
366C2×SD16—1224no—
367C2×SD16—1224no—
368C2×SD16—1224no—
369C2×D8—1224no—
370C2×D8—1224no—
371C2×D8—1224no—
372C2×M16—22no—
373C2×M16—22no—
374C2×M16—21no—
375C2×M16—22no—
376C2×M16—22no—
377C2×M16—612no—
378C2×M16—612no—
379C2×M16—1221no—
380C2×M16—1221no—
381C2×M16—1222no—
382C2×M16—1222no—
383C2×M16—1222no—
384C2×M16—1222no—
385C2×M16C22⋊D6112yes—
386C2×M16C22⋊D6112yes—
387C2×M16C22×D6112yes—
388C22×C8—21no—
389C22×C8—1221no—
390C22×C8—1221no—
391C22×C8C22×D6112yes—
392#117—21no—
393#117—21no—
394#117D24412yes—
395#117C22×D6112yes—
396C2×C4⋊C4D24412yes—
397C2×C22⋊C4—21no—
398C2×C42—21no—
399C4⋊C8—28no—
400#103—1224no—
401#102—1228no—
402#102—1224no—
403#101—1224no—
404#100—1228no—
405#100—1224no—
406C24—1221no—
407C24—1222no—
408C24—1224no—
409C24—4884no—
410C24—2441no—
411C24—2441no—
412C24—2442no—
413C24#1175822211yescharacteristic
414#41—1224no—
415#41—1224no—
416#41—1224no—
417#41—1224no—
418#41—2442no—
419#41—2442no—
420#41—2442no—
421#41—2442no—
422C2×Q8—612no—
423C2×Q8—1221no—
424C2×Q8—1221no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×D4—612no—
429C2×D4—614no—
430C2×D4—1224no—
431C2×D4—1221no—
432C2×D4—1222no—
433C2×D4—1221no—
434C2×D4—1222no—
435C2×D4—1222no—
436C2×D4—1221no—
437C2×D4—1222no—
438C2×D4—1221no—
439C2×D4—1222no—
440C2×D4—1221no—
441C2×D4—1224no—
442C2×D4—1224no—
443C2×D4—1224no—
444C2×D4—1224no—
445C2×D4—1224no—
446C2×D4—2441no—
447C2×D4—2441no—
448C2×D4—2442no—
449C2×D4—2441no—
450C2×D4—2441no—
451C2×D4—2442no—
452C2×D4—2442no—
453C2×D4—2442no—
454C2×D4—2442no—
455C2×D4—2442no—
456C22×C4—22no—
457C22×C4—1221no—
458C22×C4—1224no—
459C22×C4—2441no—
460C22×C4—2441no—
461C22×C4—2442no—
462C22×C4C2×D24411yescharacteristic
463C22×C4C2×D24411yescharacteristic
464C22×C4#1175822211yescharacteristic
465C22×C4D4×D6112yes—
466C22×C4D4×D6112yes—
467C22×C4#1175822612yes—
468C22×C4#1175822612yes—
469C22×C4C2×C22⋊D6111yescharacteristic
470C22×C4C2×C22⋊D6111yescharacteristic
471C22×C4C23×D6111yescharacteristic
472SD16—12216no—
473D8—12216no—
474M16—24no—
475M16—24no—
476M16—22no—
477M16—22no—
478M16—42no—
479M16—1222no—
480M16—1222no—
481M16—1222no—
482M16—1222no—
483M16—2442no—
484M16—2442no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—42no—
489C2×C8—1222no—
490C2×C8—1222no—
491C2×C8—2442no—
492C2×C8—2442no—
493C2×C8C2×C22⋊D6114yes—
494C4⋊C4—22no—
495C4⋊C4C2×D24414yes—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C42—22no—
499C42—22no—
500C23—42no—
501C23—1222no—
502C23—1224no—
503C23—4882no—
504C23—4884no—
505C23—4884no—
506C23—4884no—
507C23—4888no—
508C23—4888no—
509C23—2441no—
510C23—2442no—
511C23—2441no—
512C23—2442no—
513C23—2444no—
514C23—2444no—
515C23#1451975111yescharacteristic
516C23#1451975111yescharacteristic
517C23#1451976212yes—
518C23#1451976212yes—
519C23C2×#1175822211yescharacteristic
520Q8—1222no—
521Q8—1222no—
522Q8—2441no—
523Q8—2441no—
524D4—1222no—
525D4—1224no—
526D4—1224no—
527D4—1222no—
528D4—2444no—
529D4—2444no—
530D4—2442no—
531D4—2441no—
532D4—2442no—
533D4—2441no—
534D4—2444no—
535D4—2444no—
536C2×C4—24no—
537C2×C4—82no—
538C2×C4—1222no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2444no—
542C2×C4—2444no—
543C2×C4#1451974512yes—
544C2×C4#1451974512yes—
545C2×C4#1451975111yescharacteristic
546C2×C4#1451975111yescharacteristic
547C2×C4#1451975212yes—
548C2×C4#1451975212yes—
549C2×C4#1451975212yes—
550C2×C4#1451975212yes—
551C2×C4#1451976212yes—
552C2×C4#1451976212yes—
553C2×C4#1451976612yes—
554C2×C4#1451976612yes—
555C2×C4C2×D4×D6111yescharacteristic
556C2×C4C2×D4×D6111yescharacteristic
557C2×C4C22×C22⋊D6111yescharacteristic
558C2×C4C22×D24411yescharacteristic
559C2×C4C2×#1175822611yescharacteristic
560C2×C4C2×#1175822611yescharacteristic
561C8—24no—
562C8—42no—
563C8—2442no—
564C8—2442no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—42no—
569C22—42no—
570C22—4882no—
571C22—4884no—
572C22—4884no—
573C22—4888no—
574C22—4888no—
575C22—4882no—
576C22—4884no—
577C22—4882no—
578C22—2442no—
579C22—2444no—
580C22#1683967912yes—
581C22#1683967912yes—
582C22C2×#1451975111yescharacteristic
583C22C2×#1451976211yescharacteristic
584C22C2×#1451976211yescharacteristic
585C4—44no—
586C4—2442no—
587C4#1683967912yes—
588C4#1683967912yes—
589C4C2×#1451974511yescharacteristic
590C4C2×#1451975211yescharacteristic
591C4C2×#1451975211yescharacteristic
592C4C2×#1451976611yescharacteristic
593C2—21no—
594C2—21no—
595C2—22no—
596C2—22no—
597C2—4884no—
598C2—4882no—
599C2C2×#1683967911yescharacteristic
600C2#2145554712yes—
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2905C2
socleC22×C61C2×#121
prime radicalC22×#14519805C2
hypercenter#2905D61
hypocenterC61#23521
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2905

Complements

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

NH ≅ G/Ntype
2. #21454336C2semidirect
3. C2×#16839766C2semidirect
4. C2×#16839744C2semidirect
5. C22×#14519693C2semidirect
6. #21447082C2semidirect
7. #21447075C2semidirect
8. #21445546C2semidirect
9. C61×#2905C2semidirect
20. #16839770C22semidirect
23. #16839766C22semidirect
28. C2×#14519695C22semidirect
31. C2×#14519693C22semidirect
32. #16839527C22semidirect
37. C61×#424C22semidirect
294. #2905D61semidirect
324. #424C2×D61semidirect
339. C61#23521semidirect

Maximal subgroups

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

subgroupindex
C22×#145198052
#214543362
C2×#168397662
C2×#168397442
C22×#145196932
#214470822
#214470752
#214455462
C61×#29052
#2352161