Finite Group Databasebeta

← #29732212

Subgroups — #29732212

Order 15616.

1474 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840312C211yesmaximal, characteristic, complemented
2C2×#16839996C212yesmaximal, complemented
3#21447147C211yesmaximal, characteristic, complemented
4#21447053C211yesmaximal, characteristic, complemented
5#21446955C211yesmaximal, characteristic, complemented
6#21446889C211yesmaximal, characteristic, complemented
7#21446245C212yesmaximal, complemented
8#21446170C212yesmaximal, complemented
9#21446065C212yesmaximal, complemented
10C2×#16839517C211yesmaximal, characteristic, complemented
11C61×#2854C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13C22×#11758199C2212yes—
14C23×#92418C2212yes—
15#16840312—28no—
16C2×#14519777C2211yescharacteristic
17C2×#14519774C2211yescharacteristic
18C2×#14519765—21no—
19C2×#14519751—21no—
20C2×#14519751C2212yes—
21C2×C4×D244—21no—
22C2×C4×D244C2212yes—
23#16839996—28no—
24C2×#14519710C2212yes—
25C2×#14519698C2212yes—
26C2×#14519697C2212yes—
27C2×#14519697C2212yes—
28C2×#14519697C2212yes—
29#16839524C2212yes—
30#16839524C2212yes—
31#16839517C2212yes—
32#16839517C2211yescharacteristic
33#16839517C2211yescharacteristic
34C2×#123×C61—21no—
35C2×#121×C61C2211yescharacteristic
36C22×C42×C61C2211yescharacteristic
37C61×#390C2211yescharacteristic
38C61×#390C2211yescharacteristic
39C61×#389C2211yescharacteristic
40C61×#389C2211yescharacteristic
41C61×#388—21no—
42C61×#388—21no—
43C61×#386—21no—
44C61×#380C2211yescharacteristic
45C2×#11758222—28no—
46C2×#11758222—44no—
47C2×#11758222—44no—
48C22×Dic122—21no—
49C22×D244—21no—
50C22×D244—21no—
51C22×D244C2312yes—
52C22×C22⋊D61—22no—
53C22×C4×D61—22no—
54C2×#11758199—24no—
55C2×#11758199—24no—
56C2×#11758199—42no—
57C22×#92418—24no—
58C22×#92418—24no—
59C22×#92418—42no—
60C22×#92418C2312yes—
61C22×#92418C2312yes—
62#14519777—24no—
63#14519774—24no—
64#14519765—28no—
65#14519751—28no—
66#14519751—24no—
67#14519751D418yes—
68C4×D244—28no—
69C4×D244—24no—
70C4×D244—24no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2312yes—
73C2×#11758198—22no—
74C2×#11758198—22no—
75#14519710—24no—
76#14519698—24no—
77#14519697—24no—
78#14519697—24no—
79#14519697—42no—
80C22×D4×C61—21no—
81C22×D4×C61C2311yescharacteristic
82C23×C4×C61D411yescharacteristic
83C23×C4×C61D411yescharacteristic
84C23×C4×C61C2311yescharacteristic
85#123×C61—44no—
86#121×C61—24no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C22⋊C4×C61—21no—
93C2×C22⋊C4×C61—21no—
94C2×C22⋊C4×C61—21no—
95C2×C22⋊C4×C61—21no—
96C2×C22⋊C4×C61—21no—
97C2×C22⋊C4×C61—21no—
98C2×C42×C61—24no—
99C2×C42×C61D411yescharacteristic
100C2×C42×C61D411yescharacteristic
101C2×C42×C61C2311yescharacteristic
102C2×C42×C61C2311yescharacteristic
103#95×C61—21no—
104#95×C61—21no—
105#95×C61—21no—
106C24×C61—21no—
107C24×C61C2×D411yescharacteristic
108C23×D61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—42no—
114C2×D4×C61—42no—
115C2×D4×C61—42no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—41no—
127C22×C4×C61D814yescomplemented
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D414yes—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C2×C22⋊D61—28no—
136C2×C22⋊D61—44no—
137C2×C22⋊D61—44no—
138C22×Dic61—22no—
139#11758222—432no—
140C2×D244—22no—
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—22no—
144C2×D244—24no—
145C2×D244—24no—
146C2×D244—22no—
147C2×D244—42no—
148C2×D244—42no—
149C2×D244—42no—
150C2×D244—42no—
151C2×D244C2×D414yes—
152C2×C4×D61—28no—
153C2×C4×D61—44no—
154C2×C4×D61—44no—
155C2×Dic122—24no—
156C2×Dic122—42no—
157C2×Dic122—42no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—42no—
163C4⋊C4×C61—42no—
164C22⋊C4×C61—42no—
165C22⋊C4×C61—42no—
166C22⋊C4×C61—42no—
167C22⋊C4×C61—42no—
168C22⋊C4×C61—42no—
169C22⋊C4×C61—42no—
170C42×C61—22no—
171C42×C61—22no—
172C42×C61—22no—
173C42×C61—22no—
174#11758199—48no—
175#11758198—28no—
176#11758198—28no—
177C4⋊Dic61—24no—
178C4⋊Dic61—22no—
179C4⋊Dic61C2×D414yes—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—48no—
186C2×#92418#4114yescomplemented
187C23×C61—21no—
188C23×C61—21no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61#12012yes—
199C23×C61C2×D812yes—
200C23×C61AGL(1,ℤ/8)12yes—
201C23×C61C22×D411yescharacteristic
202C22×D61—24no—
203C22×D61—42no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—42no—
209D4×C61—42no—
210D4×C61—42no—
211D4×C61—84no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—28no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—44no—
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—41no—
235C2×C4×C61—41no—
236C2×C4×C61#12012yes—
237C2×C4×C61C2×D812yes—
238C2×C4×C61C2×D812yes—
239C2×C4×C61#13512yes—
240C2×C4×C61C22×D412yes—
241C2×C4×C61C2×#4112yes—
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C2×C4×C612−1+411yescharacteristic
244C22⋊D61—416no—
245C2×Dic61—24no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248D244—24no—
249D244—24no—
250D244—48no—
251D244—48no—
252D244—42no—
253D244—42no—
254C4×D61—416no—
255Dic122—48no—
256#92418—24no—
257#92418—42no—
258#25496—611nomaximal, Sylow
259C22×C61—22no—
260C22×C61—22no—
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61#44714yes—
273C22×C61C2×#12011yescharacteristic
274C22×C61#53211yescharacteristic
275C22×C61#53711yescharacteristic
276C22×C61C22×D811yescharacteristic
277C22×C61C2×#13511yescharacteristic
278C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C61#57511yescharacteristic
280C2×D61—44no—
281C2×D61—44no—
282C2×D61—44no—
283C2×D61—82no—
284C2×D61—82no—
285C2×D61—82no—
286C2×D61—82no—
287C4×C61—22no—
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—22no—
291C4×C61—41no—
292C4×C61—81no—
293C4×C61—81no—
294C4×C61—81no—
295C4×C61#54612yes—
296C4×C61#54712yes—
297Dic61—44no—
298C2×#532—611no—
299C2×#466—612no—
300#2854D6111yescharacteristic, complemented
301#2057—611no—
302#2041—611no—
303#2023—611no—
304#2011—611no—
305#1893—612no—
306#1879—612no—
307#1862—612no—
308C2×#336—611no—
309C2×C61—24no—
310C2×C61—161no—
311C2×C61#243011yescharacteristic
312C2×C61C2×#44711yescharacteristic
313C2×C61#300211yescharacteristic
314C2×C61#316911yescharacteristic
315C2×C61#318011yescharacteristic
316C2×C61#324511yescharacteristic
317C2×C61#324711yescharacteristic
318D61—84no—
319C22×#41—1221no—
320C22×D8—612no—
321C23×C8—612no—
322#532—1228no—
323C2×C4⋊Q8—611no—
324C2×#127—611no—
325C2×#124—1221no—
326C2×#123—21no—
327C2×#121—612no—
328C2×#121—1221no—
329C2×#121C2×D6111yescharacteristic
330C2×C4×D4—612no—
331C2×C4×D4—1221no—
332C22×C42C2×D6111yescharacteristic
333#466—1228no—
334C2×#107—612no—
335C2×#102—612no—
336C2×#102—612no—
337C2×#102—612no—
338C2×#102—612no—
339#390C2×D6111yescharacteristic
340#390C2×D6111yescharacteristic
341#389C2×D6111yescharacteristic
342#389C2×D6111yescharacteristic
343#388—21no—
344#388—21no—
345#386—21no—
346#380C2×D6111yescharacteristic
347#340—612no—
348#340—612no—
349#336—611no—
350#336—612no—
351#336—611no—
352C61#2549611yescharacteristic, complemented, Sylow
353C2×#41—1228no—
354C2×#41—2444no—
355C2×#41—2444no—
356C22×Q8—1221no—
357C22×D4—21no—
358C22×D4—612no—
359C22×D4—1222no—
360C22×D4—1221no—
361C22×D4—1221no—
362C22×D4C22×D6111yescharacteristic
363C23×C4—1222no—
364C23×C4C22⋊D6111yescharacteristic
365C23×C4C22⋊D6111yescharacteristic
366C23×C4C22×D6111yescharacteristic
367C2×D8—1224no—
368C2×D8—1224no—
369C2×D8—2442no—
370C22×C8—612no—
371C22×C8—612no—
372C22×C8—1224no—
373C22×C8—1224no—
374C22×C8—2442no—
375C4⋊Q8—1224no—
376#127—1224no—
377#124—1228no—
378#123—44no—
379#121—24no—
380#121—618no—
381#121—1228no—
382#121—1224no—
383C4×D4—1224no—
384C4×D4—1224no—
385C4×D4—1228no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—612no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—1222no—
400C2×C22⋊C4—1222no—
401C2×C42—24no—
402C2×C42C22⋊D6111yescharacteristic
403C2×C42C22⋊D6111yescharacteristic
404C2×C42C22×D6111yescharacteristic
405C2×C42C22×D6111yescharacteristic
406#107—1224no—
407#102—1224no—
408#102—1224no—
409#102—1224no—
410#102—2442no—
411#95—21no—
412#95—21no—
413#95—21no—
414C24—21no—
415C24—1222no—
416C24C2×C22⋊D6111yescharacteristic
417#41—24432no—
418C2×Q8—1224no—
419C2×Q8—2442no—
420C2×Q8—2442no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—614no—
429C2×D4—1224no—
430C2×D4—1224no—
431C2×D4—1228no—
432C2×D4—1222no—
433C2×D4—1222no—
434C2×D4—1222no—
435C2×D4—1222no—
436C2×D4—1222no—
437C2×D4—2442no—
438C2×D4—2442no—
439C2×D4—2444no—
440C2×D4—2444no—
441C2×D4—2442no—
442C2×D4—2442no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—41no—
454C22×C4—1228no—
455C22×C4—1222no—
456C22×C4—2444no—
457C22×C4—2444no—
458C22×C4#1175819914yescomplemented
459C22×C4D4×D6114yes—
460C22×C4C2×C22⋊D6111yescharacteristic
461C22×C4C2×C22⋊D6111yescharacteristic
462C22×C4C2×C22⋊D6111yescharacteristic
463C22×C4C2×C22⋊D6111yescharacteristic
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C23×D6111yescharacteristic
466D8—2448no—
467C2×C8—614no—
468C2×C8—1222no—
469C2×C8—1222no—
470C2×C8—1222no—
471C2×C8—1222no—
472C2×C8—1222no—
473C2×C8—2448no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—614no—
481C4⋊C4—1224no—
482C4⋊C4—1222no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—1228no—
490C22⋊C4—1228no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—1224no—
507C23—2442no—
508C23—2442no—
509C23—2442no—
510C23—2442no—
511C23—2442no—
512C23—2442no—
513C23#1451974512yes—
514C23C2×#1175819912yes—
515C23#1451980212yes—
516C23C22×C22⋊D6111yescharacteristic
517Q8—2448no—
518D4—42no—
519D4—42no—
520D4—84no—
521D4—1224no—
522D4—1224no—
523D4—2448no—
524D4—2448no—
525D4—24416no—
526D4—2442no—
527D4—2442no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—1224no—
553C2×C4—24416no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4#1451974512yes—
557C2×C4C2×#1175819912yes—
558C2×C4C2×#1175819912yes—
559C2×C4#1451979712yes—
560C2×C4C2×D4×D6112yes—
561C2×C4#41×D6112yes—
562C2×C4#1451982311yescharacteristic
563C2×C4#1451982411yescharacteristic
564C8—1224no—
565C8—2442no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—4882no—
580C22—4882no—
581C22—4882no—
582C22—4882no—
583C22—2444no—
584C22—2444no—
585C22—2444no—
586C22#1683989514yes—
587C22C2×#1451974511yescharacteristic
588C22#1684030811yescharacteristic
589C22#1684034011yescharacteristic
590C22C22×#1175819911yescharacteristic
591C22C2×#1451979711yescharacteristic
592C22C2×#1451980211yescharacteristic
593C22#1684056011yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—41no—
599C4—81no—
600C4—81no—
601C4—81no—
602C4—2444no—
603C4#1684040412yes—
604C4#1684041012yes—
605C2—24no—
606C2—161no—
607C2—4884no—
608C2#2144943111yescharacteristic
609C2C2×#1683989511yescharacteristic
610C2#2145503511yescharacteristic
611C2#2145639411yescharacteristic
612C2#2145651811yescharacteristic
613C2#2145712611yescharacteristic
614C2#2145714511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2854C2
socleC23×C61C22×D4
hypercenter#2854D61
hypocenterC61#25496
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
1. C2×#16840312C2semidirect
2. C2×#16839996C2semidirect
3. #21447147C2semidirect
4. #21447053C2semidirect
5. #21446955C2semidirect
6. #21446889C2semidirect
7. #21446245C2semidirect
8. #21446170C2semidirect
9. #21446065C2semidirect
10. C2×#16839517C2semidirect
11. C61×#2854C2semidirect
127. C22×C4×C61D8semidirect
186. C2×#92418#41semidirect
300. #2854D61semidirect
352. C61#25496semidirect
458. C22×C4#11758199semidirect

Maximal subgroups

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

subgroupindex
C2×#168403122
C2×#168399962
#214471472
#214470532
#214469552
#214468892
#214462452
#214461702
#214460652
C2×#168395172
C61×#28542
#2549661