Finite Group Databasebeta

← #29696209

Subgroups — #29696209

Order 15616.

1126 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840478C211yesmaximal, characteristic
2M16×D244C211yesmaximal, characteristic, complemented
3#21454554C211yesmaximal, characteristic, complemented
4#21446809C211yesmaximal, characteristic, complemented
5#21446281C212yesmaximal, complemented
6#21446281C212yesmaximal, complemented
7#21445901C212yesmaximal, complemented
8#21445808C211yesmaximal, characteristic, complemented
9C4×#14519693C211yesmaximal, characteristic, complemented
10#21445556C212yesmaximal, complemented
11C61×#1997C211yesmaximal, characteristic, complemented
12C2×M16×D61C2212yes—
13C2×M16×D61C2212yes—
14#16840478—21no—
15#16840478—21no—
16#16840478—22no—
17#16840478—22no—
18#16840478—21no—
19#16840478—21no—
20C2×#14519766C414yes—
21C2×#14519752C412yes—
22C2×#14519752C2212yes—
23C2×C4×D244C2211yescharacteristic
24C2×C4×D244C2211yescharacteristic
25C2×#14519735C412yes—
26C2×#14519735C2212yes—
27#16839841—21no—
28#16839839—21no—
29#16839835—22no—
30#16839832—22no—
31C8×D244—21no—
32#16839827—21no—
33#16839786C2211yescharacteristic
34C2×#14519693C2211yescharacteristic
35C2×#14519693C2211yescharacteristic
36C2×#14519693C2212yes—
37#16839725C2212yes—
38#16839725C2212yes—
39C4×#11758194C2211yescharacteristic
40#16839531C2212yes—
41#16839531C2212yes—
42#16839530C2211yescharacteristic
43#16839530C2212yes—
44#16839530C2211yescharacteristic
45C2×C4⋊Q8×C61C2211yescharacteristic
46C61×#423C2211yescharacteristic
47C4×M16×C61C2211yescharacteristic
48C61×#343C2212yes—
49C61×#343C2212yes—
50C22×D244C2×C414yes—
51C22×D244C2×C411yescharacteristic
52C22×D244C2311yescharacteristic
53C22×C4×D61C2×C412yes—
54C22×C4×D61C2312yes—
55C2×#11758194C2311yescharacteristic
56C2×#11758194C2312yes—
57C2×#11758194C2311yescharacteristic
58C2×#11758189—22no—
59C2×#11758189—22no—
60M16×D61—24no—
61M16×D61—24no—
62M16×D61—42no—
63M16×D61—42no—
64C2×C8×D61—22no—
65C2×C8×D61—22no—
66#14519766—24no—
67#14519766—24no—
68#14519766—24no—
69#14519766—24no—
70#14519752—22no—
71#14519752—22no—
72#14519752—24no—
73#14519752—22no—
74#14519752—22no—
75#14519752—22no—
76#14519752—22no—
77C4×D244—21no—
78C4×D244—21no—
79C4×D244—21no—
80C4×D244—21no—
81C4×D244D411yescharacteristic
82C4×D244D411yescharacteristic
83C4×D244D412yes—
84C4×D244D412yes—
85C4×D244D411yescharacteristic
86C4×D244D411yescharacteristic
87#14519735—22no—
88#14519735—22no—
89#14519735—22no—
90#14519735—22no—
91#14519735—24no—
92#14519735—22no—
93#14519735—22no—
94C2×C4⋊Dic61C2×C411yescharacteristic
95C2×C4⋊Dic61C2311yescharacteristic
96C2×#11758198C2×C414yes—
97C2×#11758198C2×C414yes—
98C2×#11758198C2×C412yes—
99C2×#11758198C2×C412yes—
100C2×C4×Dic61C2×C414yes—
101C4⋊#92418—21no—
102C4⋊#92418—21no—
103#14519693—22no—
104#14519693—22no—
105#14519693—42no—
106#14519689—22no—
107#14519689—22no—
108#14519689—22no—
109#14519689—22no—
110#14519688—21no—
111C4×#92418—21no—
112C22×Q8×C61C2×C412yes—
113C2×M16×C61C2311yescharacteristic
114C2×M16×C61C2312yes—
115C2×M16×C61C2311yescharacteristic
116C4⋊Q8×C61—21no—
117C4⋊Q8×C61—21no—
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C2×C4⋊C4×C61C2×C412yes—
121C2×C4⋊C4×C61C2312yes—
122C2×C42×C61C2311yescharacteristic
123C4⋊C8×C61—21no—
124C4⋊C8×C61—21no—
125#97×C61—21no—
126C4×C8×C61—21no—
127C23×D61C22×C412yes—
128C23×D61C22×C412yes—
129C2×Q8×C61—22no—
130C2×Q8×C61—22no—
131C2×Q8×C61—82no—
132C22×C4×C61C22×C411yescharacteristic
133C22×C4×C61C22×C414yes—
134C22×C4×C61C22×C411yescharacteristic
135C22×C4×C61C2411yescharacteristic
136C22×Dic61C22×C412yes—
137C22×Dic61C22×C412yes—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—24no—
145C2×D244—24no—
146C2×D244—42no—
147C2×D244—84no—
148C2×D244C22⋊C411yescharacteristic
149C2×D244C22⋊C411yescharacteristic
150C2×D244C22⋊C412yes—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—44no—
158C2×C4×D61C2×D412yes—
159C2×C4×D61C2×D412yes—
160C2×C4×D61C2×D412yes—
161C2×C4×D61C2×D412yes—
162M16×C61—21no—
163M16×C61—21no—
164M16×C61—22no—
165M16×C61—22no—
166M16×C61—21no—
167M16×C61#4112yes—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C42×C61C2×D411yescharacteristic
176C42×C61C2×D411yescharacteristic
177C42×C61C2×D411yescharacteristic
178C42×C61C2×D411yescharacteristic
179#11758198—24no—
180#11758198—24no—
181#11758198—22no—
182#11758198—22no—
183#11758198—24no—
184#11758198—24no—
185#11758198C22⋊C412yes—
186#11758198C22⋊C412yes—
187#11758198C22⋊C412yes—
188#11758198C22⋊C412yes—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61C22⋊C412yes—
192C4⋊Dic61C22⋊C411yescharacteristic
193C4⋊Dic61C22⋊C411yescharacteristic
194C4×Dic61—24no—
195C4×Dic61—24no—
196#11758194—21no—
197#11758194—21no—
198#11758194—22no—
199#11758194—22no—
200#11758194—21no—
201#11758194#4112yes—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—22no—
205#11758189—44no—
206#11758189—44no—
207C8×D61—44no—
208C8×D61—44no—
209C23×C61C23×C411yescharacteristic
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—22no—
213C22×D61—22no—
214C22×D61—22no—
215C22×D61—42no—
216C22×D61—42no—
217C22×D61—42no—
218C22×D61—42no—
219C22×D61C2×C22⋊C412yes—
220C22×D61C2×C22⋊C412yes—
221Q8×C61—82no—
222Q8×C61—82no—
223C2×C4×C61—22no—
224C2×C4×C61—24no—
225C2×C4×C61—41no—
226C2×C4×C61—44no—
227C2×C4×C61C2×C22⋊C411yescharacteristic
228C2×C4×C61C2×C22⋊C411yescharacteristic
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61H(5,𝔽2)11yescharacteristic
236C2×C4×C612−1+411yescharacteristic
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—42no—
241C2×Dic61C2×C22⋊C412yes—
242C2×Dic61C2×C22⋊C412yes—
243D244—22no—
244D244—22no—
245D244—42no—
246D244—41no—
247D244—41no—
248D244—41no—
249D244—41no—
250D244—84no—
251D244—84no—
252C4×D61—22no—
253C4×D61—22no—
254C4×D61—22no—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—44no—
261C4×D61—44no—
262C8×C61—22no—
263C8×C61—42no—
264C8×C61—41no—
265#92418—22no—
266#92418—42no—
267#92418—41no—
268#22483—611nomaximal, Sylow
269C22×C61—21no—
270C22×C61—21no—
271C22×C61C22×C22⋊C411yescharacteristic
272C22×C61C4×#4111yescharacteristic
273C22×C61#52011yescharacteristic
274C2×D61—22no—
275C2×D61—22no—
276C2×D61—22no—
277C2×D61—22no—
278C2×D61—22no—
279C2×D61—22no—
280C2×D61—42no—
281C2×D61—42no—
282C2×D61—42no—
283C2×D61—82no—
284C2×D61—82no—
285C2×D61—82no—
286C2×D61—82no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—44no—
291C4×C61#41312yes—
292C4×C61#54611yescharacteristic
293C4×C61#54611yescharacteristic
294C4×C61#54711yescharacteristic
295C4×C61#54711yescharacteristic
296Dic61—22no—
297Dic61—22no—
298Dic61—42no—
299C2×#555—611no—
300D4×M16—611no—
301D4×M16—611no—
302#1997—611no—
303#1997D6111yescharacteristic, complemented
304#1900—612no—
305#1900—612no—
306#1822—612no—
307#1800—611no—
308C4×#100—611no—
309#1742—612no—
310C2×C61—21no—
311C2×C61—21no—
312C2×C61#233911yescharacteristic
313C2×C61C2×#41311yescharacteristic
314C2×C61#288811yescharacteristic
315D61—42no—
316D61—42no—
317D61—82no—
318C22×M16—612no—
319C22×M16—612no—
320#555—1221no—
321#555—1221no—
322#555—1222no—
323#555—1222no—
324#555—1221no—
325#555—1221no—
326C2×C4⋊Q8C2×D6111yescharacteristic
327C2×#124—614no—
328C2×#121—612no—
329C2×#121—612no—
330C2×C4×D4—611no—
331C2×C4×D4—611no—
332C2×#117—612no—
333C2×#117—612no—
334#436—1221no—
335#436—1221no—
336#435—1222no—
337#435—1222no—
338C8×D4—1221no—
339C8×D4—1221no—
340#423—611no—
341#423C2×D6111yescharacteristic
342C2×#100—611no—
343C2×#100—611no—
344C2×#100—612no—
345#407—612no—
346#407—612no—
347C4×M16—611no—
348C4×M16C2×D6111yescharacteristic
349#343—612no—
350#343—611no—
351#343—611no—
352#343—612no—
353#343—612no—
354#343C2×D6112yes—
355#343C2×D6112yes—
356C61#2248311yescharacteristic, complemented, Sylow
357C22×Q8C4×D6112yes—
358C22×D4—611no—
359C22×D4—614no—
360C22×D4—611no—
361C23×C4—612no—
362C23×C4—612no—
363C2×M16—611no—
364C2×M16—612no—
365C2×M16—611no—
366C2×M16—1222no—
367C2×M16—1222no—
368C2×M16—1224no—
369C2×M16—1224no—
370C2×M16—2442no—
371C2×M16—2442no—
372C2×M16C22×D6112yes—
373C2×M16C22×D6111yescharacteristic
374C2×M16C22×D6111yescharacteristic
375C22×C8—1222no—
376C22×C8—1222no—
377C4⋊Q8—21no—
378C4⋊Q8—21no—
379C4⋊Q8—21no—
380C4⋊Q8—21no—
381#124—1224no—
382#124—1224no—
383#124—1224no—
384#124—1224no—
385#121—1222no—
386#121—1222no—
387#121—1222no—
388#121—1222no—
389#121—1222no—
390#121—1222no—
391#121—1224no—
392C4×D4—611no—
393C4×D4—611no—
394C4×D4—612no—
395C4×D4—612no—
396C4×D4—611no—
397C4×D4—611no—
398C4×D4—1221no—
399C4×D4—1221no—
400C4×D4—1221no—
401C4×D4—1221no—
402#117—1224no—
403#117—1222no—
404#117—1222no—
405#117—1222no—
406#117—1222no—
407#117—1222no—
408#117—1222no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4C4×D6112yes—
412C2×C4⋊C4C22×D6112yes—
413C2×C22⋊C4—612no—
414C2×C22⋊C4—614no—
415C2×C22⋊C4—614no—
416C2×C22⋊C4—612no—
417C2×C42—614no—
418C2×C42C22×D6111yescharacteristic
419C4⋊C8—21no—
420C4⋊C8—21no—
421C4⋊C8—1221no—
422C4⋊C8—1221no—
423#100—1222no—
424#100—1222no—
425#100—2442no—
426C22⋊C8—1222no—
427C22⋊C8—1222no—
428C22⋊C8—1222no—
429C22⋊C8—1222no—
430#97—21no—
431#97—1221no—
432C4×C8—21no—
433C4×C8—1221no—
434C24—612no—
435C24—612no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—82no—
439C2×D4—611no—
440C2×D4—611no—
441C2×D4—612no—
442C2×D4—1224no—
443C2×D4—1224no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—1221no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—4884no—
451C2×D4—2442no—
452C22×C4—612no—
453C22×C4—612no—
454C22×C4—612no—
455C22×C4—612no—
456C22×C4—612no—
457C22×C4—612no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—2444no—
465C22×C4C2×C4×D6111yescharacteristic
466C22×C4C2×C4×D6114yes—
467C22×C4C2×C4×D6111yescharacteristic
468C22×C4C23×D6111yescharacteristic
469M16—21no—
470M16—21no—
471M16—22no—
472M16—22no—
473M16—21no—
474M16—612no—
475M16—1221no—
476M16—1221no—
477M16—1222no—
478M16—1222no—
479M16—1221no—
480M16—2444no—
481M16—2444no—
482M16#1175822612yes—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—22no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1222no—
489C2×C8—2444no—
490C2×C8—2444no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—612no—
496C4⋊C4—611no—
497C4⋊C4—611no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C22⋊C4—612no—
501C22⋊C4—612no—
502C22⋊C4—612no—
503C22⋊C4—612no—
504C22⋊C4—1224no—
505C22⋊C4—1224no—
506C22⋊C4—1224no—
507C22⋊C4—1224no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C42—1224no—
511C42—1224no—
512C42D4×D6111yescharacteristic
513C42D4×D6111yescharacteristic
514C42D4×D6111yescharacteristic
515C42D4×D6111yescharacteristic
516C23—612no—
517C23—612no—
518C23—1222no—
519C23—1222no—
520C23—1222no—
521C23—1222no—
522C23—1222no—
523C23—2442no—
524C23—2442no—
525C23—2442no—
526C23—2442no—
527C23C22×C4×D6111yescharacteristic
528Q8—82no—
529Q8—82no—
530D4—1222no—
531D4—1222no—
532D4—4884no—
533D4—4884no—
534D4—2442no—
535D4—2441no—
536D4—2441no—
537D4—2441no—
538D4—2441no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—2444no—
557C2×C4—2444no—
558C2×C4—2442no—
559C2×C4C22⋊C4×D6111yescharacteristic
560C2×C4C22⋊C4×D6111yescharacteristic
561C2×C4C22⋊C4×D6111yescharacteristic
562C2×C4C22⋊C4×D6111yescharacteristic
563C2×C4C2×D4×D6111yescharacteristic
564C2×C4C2×D4×D6111yescharacteristic
565C2×C4#41×D6111yescharacteristic
566C2×C4C2×#1175822611yescharacteristic
567C2×C4#1451982311yescharacteristic
568C2×C4#1451982511yescharacteristic
569C8—22no—
570C8—42no—
571C8—41no—
572C8—1222no—
573C8—2442no—
574C8—2441no—
575C22—21no—
576C22—21no—
577C22—1222no—
578C22—1222no—
579C22—1222no—
580C22—1222no—
581C22—1222no—
582C22—1222no—
583C22—4882no—
584C22—4882no—
585C22—4882no—
586C22—4882no—
587C22—2442no—
588C22—2442no—
589C22—2442no—
590C22C2×C22⋊C4×D6111yescharacteristic
591C22C4×#1175822611yescharacteristic
592C22#1684024411yescharacteristic
593C4—21no—
594C4—21no—
595C4—21no—
596C4—44no—
597C4—1222no—
598C4—1222no—
599C4—2442no—
600C4#1683974512yes—
601C4#1684040311yescharacteristic
602C4#1684040511yescharacteristic
603C4#1684041411yescharacteristic
604C4#1684041511yescharacteristic
605C2—21no—
606C2—21no—
607C2—4882no—
608C2—2442no—
609C2—2442no—
610C2#2144864811yescharacteristic
611C2C2×#1683974511yescharacteristic
612C2#2145429511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1997C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840478C2
cosocleC22×C4×C61C24
hypercenter#1997D61
hypocenterC61#22483
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
2. M16×D244C2semidirect
3. #21454554C2semidirect
4. #21446809C2semidirect
5. #21446281C2semidirect
6. #21446281C2semidirect
7. #21445901C2semidirect
8. #21445808C2semidirect
9. C4×#14519693C2semidirect
10. #21445556C2semidirect
11. C61×#1997C2semidirect
303. #1997D61semidirect
356. C61#22483semidirect

Maximal subgroups

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

subgroupindex
C2×#168404782
M16×D2442
#214545542
#214468092
#214462812
#214459012
#214458082
C4×#145196932
#214455562
C61×#19972
#2248361