Finite Group Databasebeta

← #29689356

Subgroups — #29689356

Order 15616.

1398 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×#16840308C211yesmaximal, characteristic
2C2×#16839816C211yesmaximal, characteristic, complemented
3#21448221C211yesmaximal, characteristic, complemented
4#21446744C212yesmaximal, complemented
5#21446740C212yesmaximal, complemented
6#21446634C212yesmaximal, complemented
7#21446632C212yesmaximal, complemented
8#21445760C211yesmaximal, characteristic, complemented
9#21445759C211yesmaximal, characteristic, complemented
10#21445757C211yesmaximal, characteristic, complemented
11C61×#1788C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13#16840308—24no—
14#16840308C418yescomplemented
15C2×#14519775C2211yescharacteristic
16C2×#14519773C2211yescharacteristic
17C2×#14519766C2211yescharacteristic
18C2×#14519764C2211yescharacteristic
19C2×#14519749—21no—
20C2×#14519749—21no—
21C2×C4×C22⋊D61—21no—
22C2×C4×C22⋊D61—21no—
23C2×#14519733—21no—
24C22×C4×Dic61C2211yescharacteristic
25#16839816—24no—
26C2×#14519706C2212yes—
27C2×#14519702C2212yes—
28C2×#14519701C2212yes—
29C2×#14519700C2212yes—
30C2×#14519699C2212yes—
31C2×#14519698C2212yes—
32C2×#14519697C2212yes—
33C2×C22⋊#92418C2211yescharacteristic
34C2×#14519687C2212yes—
35#16839631C2212yes—
36#16839626—21no—
37#16839622C2211yescharacteristic
38#16839620—21no—
39C4⋊C4×Dic61C2212yes—
40#16839613—21no—
41C22×#41×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C2×#102×C61C2212yes—
44C2×C22⋊C8×C61C2211yescharacteristic
45C61×#380C2211yescharacteristic
46C2×#11758222—22no—
47C2×#11758222—42no—
48C2×#11758222—42no—
49C2×#11758222C2×C414yes—
50C22×Dic122C2311yescharacteristic
51C22×D244C2311yescharacteristic
52C22×C22⋊D61—21no—
53C22×C22⋊D61—21no—
54C22×C22⋊D61—21no—
55C23×Dic61D411yescharacteristic
56C23×Dic61D411yescharacteristic
57C22×C4×D61—21no—
58C22×#92418C2312yes—
59#14519775—22no—
60#14519775C2×C414yes—
61#14519773—22no—
62#14519773C2×C414yes—
63#14519766—22no—
64#14519766C2×C414yes—
65#14519764—22no—
66#14519764C2×C414yes—
67#14519749—28no—
68#14519749—28no—
69C4×C22⋊D61—28no—
70C4×C22⋊D61—28no—
71#14519733—44no—
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—22no—
76C2×#11758196—22no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61C2×C414yes—
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D414yes—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519706—24no—
90#14519702—42no—
91#14519701—42no—
92#14519700—42no—
93#14519699—42no—
94#14519698—42no—
95#14519697—42no—
96C22⋊#92418—22no—
97#14519687—24no—
98#14519684—21no—
99#14519684—22no—
100C2×#41×C61—22no—
101C2×#41×C61—42no—
102C2×#41×C61—42no—
103C2×#41×C61C2×C414yes—
104C22×Q8×C61C2311yescharacteristic
105C22×D4×C61—21no—
106C22×D4×C61C2311yescharacteristic
107C23×C4×C61—21no—
108C23×C4×C61C2311yescharacteristic
109C22×C8×C61C2312yes—
110C2×C4⋊C4×C61C2312yes—
111C2×C22⋊C4×C61—21no—
112#103×C61—42no—
113#102×C61—42no—
114C22⋊C8×C61—22no—
115#95×C61—21no—
116C24×C61—21no—
117C24×C61C2×D411yescharacteristic
118C23×D61—21no—
119#41×C61—44no—
120#41×C61—44no—
121#41×C61—48no—
122C2×Q8×C61—21no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61C22×C412yes—
126C2×D4×C61—24no—
127C2×D4×C61—21no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—44no—
131C2×D4×C61C22×C412yes—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—21no—
135C22×C4×C61—24no—
136C22×C4×C61—22no—
137C22×C4×C61—44no—
138C22×C4×C61C22×C412yes—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2411yescharacteristic
142C2×C22⋊D61—24no—
143C2×C22⋊D61—24no—
144C2×C22⋊D61—24no—
145C2×C22⋊D61—44no—
146C2×C22⋊D61—44no—
147C2×C22⋊D61—42no—
148C2×C22⋊D61—42no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—21no—
152C22×Dic61—22no—
153C22×Dic61—42no—
154C22×Dic61C22⋊C414yes—
155C22×Dic61C22⋊C414yes—
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160#11758222—44no—
161#11758222—44no—
162#11758222—48no—
163C2×D244—22no—
164C2×D244—22no—
165C2×D244—21no—
166C2×D244C22×C412yes—
167C2×C4×D61—24no—
168C2×C4×D61—44no—
169C2×Dic122—22no—
170C2×Dic122—22no—
171C2×Dic122—21no—
172C2×Dic122C22×C412yes—
173C2×C8×C61—22no—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C22⋊C4×C61—42no—
179C22⋊Dic61—24no—
180C22⋊Dic61—24no—
181C22⋊Dic61—42no—
182#11758198—24no—
183#11758198—24no—
184C4⋊Dic61—22no—
185C4⋊Dic61—22no—
186#11758196—24no—
187#11758196—24no—
188#11758196—44no—
189#11758196—44no—
190C4×Dic61—44no—
191C4×Dic61C22⋊C414yes—
192C4×Dic61C22⋊C414yes—
193C4×Dic61C22×C412yes—
194C4×Dic61C22×C412yes—
195C4×Dic61C2×D412yes—
196C4×Dic61C2×D412yes—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C23×C61—22no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61—22no—
204C23×C61—22no—
205C23×C61—42no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—42no—
209C23×C61C2×C22⋊C412yes—
210C23×C61C22×D411yescharacteristic
211C22×D61—22no—
212C22×D61—42no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—42no—
216Q8×C61—22no—
217Q8×C61—22no—
218Q8×C61—42no—
219D4×C61—22no—
220D4×C61—22no—
221D4×C61—48no—
222D4×C61—42no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—24no—
227C2×C4×C61—22no—
228C2×C4×C61—48no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61C2×C22⋊C412yes—
234C2×C4×C61C23×C411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61H(5,𝔽2)12yes—
237C2×C4×C61H(5,𝔽2)12yes—
238C22⋊D61—48no—
239C22⋊D61—48no—
240C22⋊D61—48no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—28no—
246C2×Dic61—22no—
247C2×Dic61—44no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61C2×C22⋊C412yes—
253C2×Dic61C2×C22⋊C412yes—
254C2×Dic61C2×C22⋊C412yes—
255C2×Dic61C2×C22⋊C412yes—
256C2×Dic61AGL(1,ℤ/8)12yes—
257C2×Dic61#13712yes—
258D244—22no—
259D244—22no—
260D244—42no—
261C4×D61—48no—
262Dic122—22no—
263Dic122—22no—
264Dic122—42no—
265C8×C61—42no—
266#92418—42no—
267#21809—611nomaximal, Sylow
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61—82no—
277C22×C61—82no—
278C22×C61C22×C22⋊C411yescharacteristic
279C22×C61#53512yes—
280C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C61C2×#13711yescharacteristic
282C22×C61#57512yes—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—42no—
286C2×D61—82no—
287C2×D61—82no—
288C4×C61—22no—
289C4×C61—42no—
290C4×C61—82no—
291C4×C61#51811yescharacteristic
292C4×C61#51811yescharacteristic
293C4×C61#53512yes—
294Dic61—22no—
295Dic61—22no—
296Dic61—22no—
297Dic61—42no—
298Dic61—82no—
299Dic61#41714yes—
300C2×#532—611no—
301C2×#432—611no—
302#2285—611no—
303#1982—612no—
304#1981—612no—
305#1959—612no—
306#1959—612no—
307#1788—611no—
308#1788—611no—
309#1788—611no—
310#1788D6111yescharacteristic, complemented
311C2×C61—42no—
312C2×C61—82no—
313C2×C61#230911yescharacteristic
314C2×C61C2×#41711yescharacteristic
315C2×C61#289611yescharacteristic
316C2×C61#321412yes—
317C2×C61#321612yes—
318D61—82no—
319C22×#41—611no—
320C22×#41C2×D6111yescharacteristic
321#532—618no—
322#532—1224no—
323C2×C4⋊Q8—611no—
324C2×#127—611no—
325C2×#124—611no—
326C2×#124—611no—
327C2×#121—1221no—
328C2×#121—1221no—
329C2×C4×D4—1221no—
330C2×C4×D4—1221no—
331C2×#117—1221no—
332C22×C42—611no—
333#432—1224no—
334C2×C4⋊C8—612no—
335C2×#103—612no—
336C2×#103—612no—
337C2×#103—612no—
338C2×#103C2×D6112yes—
339C2×#102—612no—
340C2×#102—612no—
341C2×#102—612no—
342C2×#102C2×D6112yes—
343C2×C22⋊C8—611no—
344C2×C22⋊C8C2×D6111yescharacteristic
345C2×#97—612no—
346#384—612no—
347#382—1221no—
348#380—611no—
349#380—1221no—
350#380C2×D6111yescharacteristic
351C4×C4⋊C4—612no—
352C4×C4⋊C4—1221no—
353C61#2180911yescharacteristic, complemented, Sylow
354C2×#41—22no—
355C2×#41—42no—
356C2×#41—42no—
357C2×#41—614no—
358C2×#41—1222no—
359C2×#41—2442no—
360C2×#41—2442no—
361C2×#41C4×D6114yes—
362C22×Q8—611no—
363C22×Q8C22×D6111yescharacteristic
364C22×D4—21no—
365C22×D4—611no—
366C22×D4—1221no—
367C22×D4—1221no—
368C22×D4—1221no—
369C22×D4C22×D6111yescharacteristic
370C23×C4—21no—
371C23×C4—611no—
372C23×C4—611no—
373C23×C4—1221no—
374C23×C4C22×D6111yescharacteristic
375C22×C8—612no—
376C22×C8C22×D6112yes—
377C4⋊Q8—614no—
378C4⋊Q8—1222no—
379#127—614no—
380#127—1222no—
381#124—614no—
382#124—614no—
383#124—1222no—
384#124—1222no—
385#121—1228no—
386#121—1228no—
387C4×D4—1228no—
388C4×D4—1228no—
389#117—2444no—
390C2×C4⋊C4—612no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1222no—
394C2×C4⋊C4—1222no—
395C2×C4⋊C4C22×D6112yes—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C42—614no—
403C2×C42—611no—
404C2×C42—611no—
405C2×C42—614no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—1222no—
409C4⋊C8—1224no—
410#103—42no—
411#103—2442no—
412#103—2442no—
413#103—2442no—
414#102—42no—
415#102—2442no—
416#102—2442no—
417#102—2442no—
418C22⋊C8—22no—
419C22⋊C8—1222no—
420#97—1224no—
421#95—21no—
422#95—1222no—
423#95—1221no—
424C24—21no—
425C24—1221no—
426C24D4×D6111yescharacteristic
427#41—44no—
428#41—44no—
429#41—48no—
430#41—2444no—
431#41—2444no—
432#41—2448no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—612no—
437C2×Q8—1221no—
438C2×Q8—1222no—
439C2×Q8—1222no—
440C2×Q8C2×C4×D6112yes—
441C2×D4—24no—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—44no—
446C2×D4—612no—
447C2×D4—1224no—
448C2×D4—1224no—
449C2×D4—1224no—
450C2×D4—1221no—
451C2×D4—1222no—
452C2×D4—1222no—
453C2×D4—2442no—
454C2×D4—2442no—
455C2×D4—2444no—
456C2×D4—2444no—
457C2×D4C2×C4×D6112yes—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—44no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—614no—
469C22×C4—614no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1221no—
473C22×C4—1224no—
474C22×C4—1222no—
475C22×C4—2444no—
476C22×C4—2442no—
477C22×C4C2×C4×D6112yes—
478C22×C4D4×D6112yes—
479C22×C4D4×D6111yescharacteristic
480C22×C4C23×D6111yescharacteristic
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—1222no—
485C2×C8—1222no—
486C2×C8—1222no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—1224no—
490C4⋊C4—1224no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C4⋊C4—2444no—
494C4⋊C4—2444no—
495C22⋊C4—42no—
496C22⋊C4—1224no—
497C22⋊C4—1224no—
498C22⋊C4—1224no—
499C22⋊C4—1224no—
500C22⋊C4—2442no—
501C42—612no—
502C42—612no—
503C42—612no—
504C42—612no—
505C42—614no—
506C42—614no—
507C42—2444no—
508C23—21no—
509C23—21no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—1222no—
518C23—2441no—
519C23—2441no—
520C23—2442no—
521C23—2442no—
522C23C22⋊C4×D6112yes—
523C23C2×D4×D6111yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—42no—
527Q8—1222no—
528Q8—1222no—
529Q8—2442no—
530D4—22no—
531D4—22no—
532D4—48no—
533D4—42no—
534D4—1222no—
535D4—1222no—
536D4—2448no—
537D4—2448no—
538D4—2448no—
539D4—2442no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—48no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—612no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—612no—
556C2×C4—1228no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—2448no—
563C2×C4—2444no—
564C2×C4—2442no—
565C2×C4—2442no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4C22⋊C4×D6112yes—
569C2×C4C22×C4×D6111yescharacteristic
570C2×C4C2×D4×D6111yescharacteristic
571C2×C4#1451982212yes—
572C2×C4#1451982212yes—
573C8—42no—
574C8—2442no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—82no—
584C22—82no—
585C22—4882no—
586C22—4882no—
587C22—2442no—
588C22—2442no—
589C22—2442no—
590C22C2×C22⋊C4×D6111yescharacteristic
591C22#1684032712yes—
592C22AGL(1,ℤ/8)×D6111yescharacteristic
593C22#137×D6111yescharacteristic
594C22#1684055912yes—
595C4—22no—
596C4—42no—
597C4—82no—
598C4—614no—
599C4—1222no—
600C4—1222no—
601C4—1222no—
602C4—4882no—
603C4—2442no—
604C4#1684023511yescharacteristic
605C4#1684023511yescharacteristic
606C4#1684032712yes—
607C2—42no—
608C2—82no—
609C2—4882no—
610C2#2144840011yescharacteristic
611C2#417×D6111yescharacteristic
612C2#2145433411yescharacteristic
613C2#2145681112yes—
614C2#2145682712yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1788C2
socleC23×C61C22×D4
prime radicalC2×#16840308C2
cosocleC22×C4×C61C24
hypercenter#1788D61
hypocenterC61#21809
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
2. C2×#16839816C2semidirect
3. #21448221C2semidirect
4. #21446744C2semidirect
5. #21446740C2semidirect
6. #21446634C2semidirect
7. #21446632C2semidirect
8. #21445760C2semidirect
9. #21445759C2semidirect
10. #21445757C2semidirect
11. C61×#1788C2semidirect
14. #16840308C4semidirect
310. #1788D61semidirect
353. C61#21809semidirect

Maximal subgroups

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

subgroupindex
C2×#168403082
C2×#168398162
#214482212
#214467442
#214467402
#214466342
#214466322
#214457602
#214457592
#214457572
C61×#17882
#2180961