Finite Group Databasebeta

← #29714365

Subgroups — #29714365

Order 15616.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840312C211yesmaximal, characteristic
2C2×#16840048C211yesmaximal, characteristic, complemented
3C2×#16839963C211yesmaximal, characteristic, complemented
4C4×#14519756C211yesmaximal, characteristic, complemented
5#21447009C212yesmaximal, complemented
6#21446824C212yesmaximal, complemented
7#21446770C211yesmaximal, characteristic, complemented
8#21446745C211yesmaximal, characteristic, complemented
9#21446152C212yesmaximal, complemented
10#21446083C212yesmaximal, complemented
11C61×#1797C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13C22×#11758190C2212yes—
14#16840312—24no—
15#16840312—24no—
16C2×#14519777C2211yescharacteristic
17C2×#14519774C2211yescharacteristic
18C2×#14519765—21no—
19C2×#14519756C2211yescharacteristic
20C2×#14519751—21no—
21C2×#14519751C2212yes—
22C2×C4×Dic122C2211yescharacteristic
23C2×C4×D244—21no—
24C2×C4×D244C2212yes—
25#16840048—24no—
26#16839963—24no—
27C2×#14519709C2212yes—
28C2×#14519702C2212yes—
29C2×#14519699C2212yes—
30C2×#14519699C2212yes—
31C2×#14519699C2212yes—
32#16839668C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839646—21no—
35#16839635—21no—
36#16839622C2211yescharacteristic
37C4×#11758196—21no—
38C4×C4⋊Dic61C2211yescharacteristic
39C4×C22⋊Dic61—21no—
40#16839525C2212yes—
41#16839525C2212yes—
42C22×C42×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×C4⋊C8×C61C2211yescharacteristic
45C2×C22⋊C8×C61C2212yes—
46C61×#336C2212yes—
47C2×#11758222—28no—
48C2×#11758222—44no—
49C2×#11758222—44no—
50C22×Dic122—21no—
51C22×Dic122C2311yescharacteristic
52C22×D244—21no—
53C22×D244—21no—
54C22×D244C2312yes—
55C22×C22⋊D61—22no—
56C22×C4×D61—22no—
57C2×#11758190—24no—
58C2×#11758190—24no—
59C2×#11758190—42no—
60#14519777—22no—
61#14519777D414yes—
62#14519774—22no—
63#14519774D414yes—
64#14519765—28no—
65#14519756—22no—
66#14519756—22no—
67#14519751—28no—
68#14519751—24no—
69#14519751—24no—
70C4×Dic122—22no—
71C4×Dic122—22no—
72C4×D244—28no—
73C4×D244—24no—
74C4×D244—24no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2312yes—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×#11758198—22no—
86C2×#11758198—22no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89#14519709—24no—
90#14519702—24no—
91#14519699—24no—
92#14519699—24no—
93#14519699—42no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99C23×C4×C61D411yescharacteristic
100C23×C4×C61D411yescharacteristic
101C23×C4×C61C2311yescharacteristic
102C22×C8×C61C2312yes—
103C22×C8×C61C2312yes—
104C2×C42×C61—22no—
105C2×C42×C61—22no—
106C2×C42×C61D411yescharacteristic
107C2×C42×C61D411yescharacteristic
108C2×C42×C61C2311yescharacteristic
109C2×C42×C61C2311yescharacteristic
110C4⋊C8×C61—22no—
111C4⋊C8×C61—22no—
112C22⋊C8×C61—42no—
113C24×C61C2×D411yescharacteristic
114C23×D61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—41no—
125C22×C4×C61—41no—
126C22×C4×C61SD1614yescomplemented
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2411yescharacteristic
133C2×C22⋊D61—28no—
134C2×C22⋊D61—44no—
135C2×C22⋊D61—44no—
136C22×Dic61—22no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141#11758222—432no—
142C2×D244—22no—
143C2×D244—22no—
144C2×D244—22no—
145C2×D244—22no—
146C2×D244—24no—
147C2×D244—24no—
148C2×D244—22no—
149C2×D244—42no—
150C2×D244—42no—
151C2×D244—42no—
152C2×D244—42no—
153C2×D244C2×D414yes—
154C2×C4×D61—28no—
155C2×C4×D61—44no—
156C2×C4×D61—44no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—24no—
162C2×Dic122—21no—
163C2×Dic122—42no—
164C2×Dic122—42no—
165C2×Dic122#4112yescomplemented
166C2×C8×C61—22no—
167C2×C8×C61—22no—
168C2×C8×C61—22no—
169C2×C8×C61—22no—
170C2×C8×C61—22no—
171C2×C8×C61—22no—
172C42×C61—22no—
173C42×C61—22no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61C2×D412yes—
177C42×C61C2×D412yes—
178C22⋊Dic61—24no—
179C22⋊Dic61—42no—
180#11758198—28no—
181#11758198—28no—
182C4⋊Dic61—24no—
183C4⋊Dic61—22no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61C2×D414yes—
190C4⋊Dic61#4112yescomplemented
191#11758196—24no—
192#11758196—42no—
193#11758196—42no—
194C4×Dic61—42no—
195C4×Dic61—42no—
196#11758190—48no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61C2×SD1612yes—
204C23×C61C22×D411yescharacteristic
205C22×D61—24no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—42no—
211C22×D61—42no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—28no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—44no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61#12012yes—
229C2×C4×C61#12012yes—
230C2×C4×C61C2×SD1612yes—
231C2×C4×C61C2×SD1612yes—
232C2×C4×C61AGL(1,ℤ/8)12yes—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C2×C4×C612−1+411yescharacteristic
239C22⋊D61—416no—
240C2×Dic61—24no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—42no—
246C2×Dic61—41no—
247C2×Dic61—42no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255D244—24no—
256D244—24no—
257D244—48no—
258D244—42no—
259D244—42no—
260D244—48no—
261C4×D61—416no—
262Dic122—22no—
263Dic122—22no—
264Dic122—41no—
265Dic122—41no—
266Dic122—48no—
267C8×C61—42no—
268C8×C61—42no—
269#24154—611nomaximal, Sylow
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—22no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61C2×#12011yescharacteristic
278C22×C61#52511yescharacteristic
279C22×C61#53711yescharacteristic
280C22×C61C22×SD1611yescharacteristic
281C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C61#57511yescharacteristic
283C22×C61#57511yescharacteristic
284C2×D61—44no—
285C2×D61—44no—
286C2×D61—44no—
287C2×D61—82no—
288C2×D61—82no—
289C2×D61—82no—
290C2×D61—82no—
291C4×C61—22no—
292C4×C61—22no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61#45014yes—
296C4×C61#54611yescharacteristic
297C4×C61#54711yescharacteristic
298C4×C61#54811yescharacteristic
299C4×C61#55011yescharacteristic
300Dic61—42no—
301Dic61—42no—
302Dic61—44no—
303Dic61—81no—
304Dic61—81no—
305C2×#532—611no—
306C2×#475—611no—
307C2×#460—611no—
308C4×C22⋊Q8—611no—
309#2034—612no—
310#2000—612no—
311#1988—611no—
312#1982—611no—
313#1877—612no—
314#1864—612no—
315#1797D6111yescharacteristic, complemented
316C2×C61—42no—
317C2×C61#240711yescharacteristic
318C2×C61C2×#45011yescharacteristic
319C2×C61#300511yescharacteristic
320C2×C61#310311yescharacteristic
321C2×C61#311511yescharacteristic
322C2×C61#324611yescharacteristic
323C2×C61#325111yescharacteristic
324D61—84no—
325C22×#41—1221no—
326C22×SD16—612no—
327#532—1224no—
328#532—1224no—
329C2×C4⋊Q8—611no—
330C2×#127—611no—
331C2×#124—1221no—
332C2×C22⋊Q8—611no—
333C2×#121—612no—
334C2×#121—1221no—
335C2×C4×Q8—611no—
336C2×C4×D4—612no—
337C2×C4×D4—1221no—
338C22×C42C2×D6111yescharacteristic
339#475—1224no—
340#460—1224no—
341C2×#106—612no—
342C2×C4⋊C8C2×D6111yescharacteristic
343C2×C4⋊C8C2×D6111yescharacteristic
344C2×#103—612no—
345C2×#102—612no—
346C2×#102—612no—
347C2×#102—612no—
348C2×C22⋊C8C2×D6112yes—
349#391—611no—
350#389—611no—
351#387—1221no—
352#385—1221no—
353#380—611no—
354C4×C4⋊C4—611no—
355C4×C4⋊C4—1221no—
356C4×C22⋊C4—1221no—
357#340—612no—
358#340—612no—
359#336C2×D6112yes—
360C61#2415411yescharacteristic, complemented, Sylow
361C2×#41—1228no—
362C2×#41—2444no—
363C2×#41—2444no—
364C22×Q8—611no—
365C22×Q8—1221no—
366C22×D4—612no—
367C22×D4—1222no—
368C22×D4—1221no—
369C22×D4—1221no—
370C23×C4—1222no—
371C23×C4D24411yescharacteristic
372C23×C4D24411yescharacteristic
373C23×C4C22×D6111yescharacteristic
374C2×SD16—1224no—
375C2×SD16—1224no—
376C2×SD16—2442no—
377C22×C8C22×D6112yes—
378C22×C8C22×D6112yes—
379C4⋊Q8—614no—
380C4⋊Q8—1222no—
381#127—614no—
382#127—1222no—
383#124—1228no—
384C22⋊Q8—1222no—
385C22⋊Q8—1222no—
386#121—1224no—
387#121—1224no—
388#121—1228no—
389C4×Q8—1222no—
390C4×Q8—1222no—
391C4×D4—1224no—
392C4×D4—1224no—
393C4×D4—1228no—
394C2×C4⋊C4—612no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1222no—
404C2×C22⋊C4—1222no—
405C2×C22⋊C4—1221no—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42—1221no—
409C2×C42—1221no—
410C2×C42D24411yescharacteristic
411C2×C42D24411yescharacteristic
412C2×C42C22×D6111yescharacteristic
413C2×C42C22×D6111yescharacteristic
414#106—1224no—
415C4⋊C8—22no—
416C4⋊C8—22no—
417#103—1224no—
418#102—1224no—
419#102—1224no—
420#102—2442no—
421C22⋊C8—42no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427C24—1222no—
428C24C2×D24411yescharacteristic
429#41—24432no—
430C2×Q8—612no—
431C2×Q8—1221no—
432C2×Q8—1224no—
433C2×Q8—1221no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×Q8—1221no—
437C2×Q8—2442no—
438C2×Q8—2442no—
439C2×D4—614no—
440C2×D4—1228no—
441C2×D4—1222no—
442C2×D4—1222no—
443C2×D4—1222no—
444C2×D4—1222no—
445C2×D4—1222no—
446C2×D4—1224no—
447C2×D4—1224no—
448C2×D4—2442no—
449C2×D4—2442no—
450C2×D4—2444no—
451C2×D4—2444no—
452C2×D4—2442no—
453C2×D4—2442no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—1228no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1222no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—2444no—
472C22×C4—2444no—
473C22×C4#1175819014yescomplemented
474C22×C4C2×D24411yescharacteristic
475C22×C4C2×D24411yescharacteristic
476C22×C4C2×D24411yescharacteristic
477C22×C4C2×D24411yescharacteristic
478C22×C4C2×D24411yescharacteristic
479C22×C4C23×D6111yescharacteristic
480SD16—2448no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C4⋊C4—614no—
488C4⋊C4—612no—
489C4⋊C4—1224no—
490C4⋊C4—1222no—
491C4⋊C4—1221no—
492C4⋊C4—1224no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—2442no—
498C4⋊C4—2442no—
499C22⋊C4—1228no—
500C22⋊C4—1228no—
501C22⋊C4—1224no—
502C22⋊C4—2442no—
503C42—21no—
504C42—21no—
505C42—22no—
506C42—22no—
507C42—2442no—
508C42—2442no—
509C42D4×D6112yes—
510C42D4×D6112yes—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—1224no—
518C23—2442no—
519C23—2442no—
520C23—2442no—
521C23—2442no—
522C23—2442no—
523C23—2442no—
524C23C2×#1175819012yes—
525C23C22×D24411yescharacteristic
526Q8—1222no—
527Q8—1222no—
528Q8—2441no—
529Q8—2448no—
530Q8—2441no—
531D4—1224no—
532D4—1224no—
533D4—2448no—
534D4—24416no—
535D4—2442no—
536D4—2448no—
537D4—2442no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—28no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—1224no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—24416no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2442no—
566C2×C4—2441no—
567C2×C4—2442no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4#1451974712yes—
571C2×C4#1451974712yes—
572C2×C4C2×#1175819012yes—
573C2×C4C2×#1175819012yes—
574C2×C4#1451980512yes—
575C2×C4C2×D4×D6111yescharacteristic
576C2×C4C2×D4×D6111yescharacteristic
577C2×C4#41×D6111yescharacteristic
578C2×C4#41×D6111yescharacteristic
579C2×C4#1451982211yescharacteristic
580C2×C4#1451982511yescharacteristic
581C8—42no—
582C8—42no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—4882no—
591C22—4882no—
592C22—4882no—
593C22—4882no—
594C22—2444no—
595C22—2444no—
596C22—2444no—
597C22C2×#1451974711yescharacteristic
598C22#1684027411yescharacteristic
599C22#1684034311yescharacteristic
600C22C22×#1175819011yescharacteristic
601C22C2×#1451980511yescharacteristic
602C22#1684056211yescharacteristic
603C22#1684056211yescharacteristic
604C4—22no—
605C4—22no—
606C4—41no—
607C4—41no—
608C4—4881no—
609C4—4881no—
610C4—2442no—
611C4—2444no—
612C4—2442no—
613C4#1683991514yes—
614C4#1684040911yescharacteristic
615C4#1684041411yescharacteristic
616C4#1684042211yescharacteristic
617C4#1684043011yescharacteristic
618C2—42no—
619C2—4884no—
620C2#2144916511yescharacteristic
621C2C2×#1683991511yescharacteristic
622C2#2145506411yescharacteristic
623C2#2145585111yescharacteristic
624C2#2145597111yescharacteristic
625C2#2145713611yescharacteristic
626C2#2145718211yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1797C2
socleC23×C61C22×D4
prime radicalC2×#16840312C2
hypercenter#1797D61
hypocenterC61#24154
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. C2×#16840048C2semidirect
3. C2×#16839963C2semidirect
4. C4×#14519756C2semidirect
5. #21447009C2semidirect
6. #21446824C2semidirect
7. #21446770C2semidirect
8. #21446745C2semidirect
9. #21446152C2semidirect
10. #21446083C2semidirect
11. C61×#1797C2semidirect
126. C22×C4×C61SD16semidirect
165. C2×Dic122#41semidirect
190. C4⋊Dic61#41semidirect
315. #1797D61semidirect
360. C61#24154semidirect
473. C22×C4#11758190semidirect

Maximal subgroups

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

subgroupindex
C2×#168403122
C2×#168400482
C2×#168399632
C4×#145197562
#214470092
#214468242
#214467702
#214467452
#214461522
#214460832
C61×#17972
#2415461