Finite Group Databasebeta

← #29648625

Subgroups — #29648625

Order 15616.

1276 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840310C211yesmaximal, characteristic
2#21454447C211yesmaximal, characteristic, complemented
3#423×D61C211yesmaximal, characteristic, complemented
4#21444538C214yesmaximal, complemented
5#21444505C211yesmaximal, characteristic, complemented
6#21444503C211yesmaximal, characteristic, complemented
7#21444493C214yesmaximal, complemented
8#21444487C211yesmaximal, characteristic, complemented
9C61×#1545C211yesmaximal, characteristic, complemented
10C22×#11758224—21no—
11C2×M16×D61—21no—
12#16840310—21no—
13#16840310—21no—
14#16840310—21no—
15#16840310—21no—
16#16840310C412yes—
17#16840310C412yes—
18#16840310C2212yes—
19#16840310C2212yes—
20C2×#14519777C2211yescharacteristic
21C2×#14519764—21no—
22C2×#14519748—21no—
23C2×#14519748C412yes—
24C2×D4×Dic61—21no—
25C2×D4×Dic61C412yes—
26C2×C42×D61C2211yescharacteristic
27#16839818C2212yescomplemented
28#16839816—21no—
29#16839815C2211yescharacteristic
30#16839813—21no—
31#16839786C2211yescharacteristic
32#16839785—21no—
33#16839784—21no—
34#16839784C2212yescomplemented
35#16839784C2212yescomplemented
36C4⋊C8×D61—21no—
37C4⋊C8×D61C2212yescomplemented
38#16839724—21no—
39C4×#11758189—21no—
40#16839509C2214yescomplemented
41#16839507C2214yescomplemented
42#16839506C2214yescomplemented
43C2×#127×C61C2211yescharacteristic
44C61×#431C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C61×#331C2214yescomplemented
47C2×#11758224—22no—
48C2×#11758224—22no—
49C2×#11758224—22no—
50C2×#11758224—22no—
51C2×#11758224—48no—
52C22×Dic122—21no—
53C22×C22⋊D61—22no—
54C23×Dic61—22no—
55C22×C4×D61C2×C411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C2×#11758194—21no—
59C2×#11758189—22no—
60C2×#11758189—22no—
61M16×D61—44no—
62C2×C8×D61—22no—
63#14519777—21no—
64#14519777—21no—
65#14519777C2×C412yes—
66#14519777C2312yes—
67#14519764—22no—
68#14519764—22no—
69#14519764—22no—
70#14519764—22no—
71#14519748—22no—
72#14519748—22no—
73#14519748—22no—
74#14519748—22no—
75#14519748—22no—
76#14519748—22no—
77#14519748C2×C414yes—
78#14519748C2×C414yes—
79D4×Dic61—22no—
80D4×Dic61—22no—
81D4×Dic61—22no—
82D4×Dic61—22no—
83D4×Dic61—22no—
84D4×Dic61—22no—
85D4×Dic61C2×C414yes—
86D4×Dic61C2×C414yes—
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61C2×C412yes—
89C2×C22⋊Dic61—22no—
90C2×C22⋊Dic61—22no—
91C2×C4×Dic61C2×C411yescharacteristic
92C2×C4×Dic61D411yescharacteristic
93C2×C4×Dic61D411yescharacteristic
94C42×D61C2×C411yescharacteristic
95C42×D61C2×C411yescharacteristic
96C42×D61D411yescharacteristic
97C42×D61D411yescharacteristic
98C42×D61D411yescharacteristic
99C42×D61D411yescharacteristic
100C42×D61C2311yescharacteristic
101C42×D61C2311yescharacteristic
102C4⋊#92418—21no—
103C4⋊#92418C2312yes—
104C4⋊#92418C2312yes—
105#14519706—22no—
106#14519706—22no—
107#14519706—22no—
108C22⋊#92418—21no—
109#14519689—22no—
110#14519687—22no—
111C4×#92418—21no—
112C22×D4×C61—21no—
113C22×D4×C61—21no—
114C22×D4×C61C2×C412yes—
115C2×M16×C61—21no—
116#127×C61—21no—
117#127×C61—21no—
118#127×C61C2×C412yes—
119#127×C61C2312yes—
120C2×C42×C61C2311yescharacteristic
121C4⋊C8×C61—21no—
122C4⋊C8×C61C2312yes—
123C4⋊C8×C61C2312yes—
124C22⋊C8×C61—21no—
125C4×C8×C61—21no—
126C24×C61—22no—
127C23×D61C22⋊C411yescharacteristic
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—44no—
134C2×D4×C61—44no—
135C2×D4×C61—42no—
136C2×D4×C61—42no—
137C2×D4×C61C22×C414yes—
138C22×C4×C61C22×C411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C2×C22⋊D61—24no—
142C2×C22⋊D61—24no—
143C2×C22⋊D61—48no—
144C22×Dic61—22no—
145C22×Dic61—24no—
146C22×Dic61—24no—
147C22×Dic61—48no—
148C22×Dic61C22⋊C411yescharacteristic
149C22×Dic61C22⋊C411yescharacteristic
150C22×Dic61C22⋊C411yescharacteristic
151#11758224—48no—
152#11758224—48no—
153#11758224—48no—
154#11758224—48no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—24no—
158C2×C4×D61—42no—
159C2×C4×D61C22⋊C411yescharacteristic
160C2×C4×D61C22⋊C411yescharacteristic
161C2×C4×D61C22⋊C411yescharacteristic
162C2×C4×D61C22⋊C411yescharacteristic
163C2×C4×D61C22⋊C411yescharacteristic
164C2×C4×D61C22⋊C411yescharacteristic
165C2×C4×D61C22×C411yescharacteristic
166C2×C4×D61C22×C411yescharacteristic
167C2×C4×D61C2×D411yescharacteristic
168C2×C4×D61C2×D411yescharacteristic
169C2×C4×D61C2×D411yescharacteristic
170C2×C4×D61C2×D411yescharacteristic
171C2×Dic122—22no—
172C2×Dic122—22no—
173C2×Dic122—44no—
174M16×C61—42no—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C42×C61C22×C411yescharacteristic
178C42×C61C2×D411yescharacteristic
179C42×C61C2×D411yescharacteristic
180C42×C61C2411yescharacteristic
181C22⋊Dic61—24no—
182C22⋊Dic61—24no—
183C22⋊Dic61—24no—
184C22⋊Dic61—24no—
185C4⋊Dic61—22no—
186C4⋊Dic61—22no—
187C4⋊Dic61—22no—
188C4⋊Dic61C22×C414yes—
189C4×Dic61C22⋊C411yescharacteristic
190C4×Dic61C22⋊C411yescharacteristic
191C4×Dic61C22⋊C411yescharacteristic
192C4×Dic61C22⋊C411yescharacteristic
193C4×Dic61C22⋊C411yescharacteristic
194C4×Dic61C22⋊C411yescharacteristic
195C4×Dic61C22×C411yescharacteristic
196C4×Dic61C22×C411yescharacteristic
197C4×Dic61C2×D411yescharacteristic
198C4×Dic61C2×D411yescharacteristic
199C4×Dic61C2×D411yescharacteristic
200C4×Dic61C2×D411yescharacteristic
201#11758194—42no—
202C2×#92418—22no—
203C2×#92418—22no—
204#11758189—44no—
205#11758189—44no—
206C8×D61—44no—
207C23×C61—24no—
208C23×C61—44no—
209C23×C61—44no—
210C23×C61—44no—
211C23×C61C2×C22⋊C411yescharacteristic
212C22×D61—22no—
213C22×D61—22no—
214C22×D61—41no—
215C22×D61C2×C22⋊C411yescharacteristic
216C22×D61C2×C22⋊C411yescharacteristic
217D4×C61—44no—
218D4×C61—44no—
219D4×C61—44no—
220D4×C61—42no—
221D4×C61—42no—
222D4×C61—44no—
223D4×C61—44no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61C2×C22⋊C411yescharacteristic
227C2×C4×C61C2×C22⋊C411yescharacteristic
228C2×C4×C61C2×C22⋊C411yescharacteristic
229C2×C4×C61AGL(1,ℤ/8)12yes—
230C2×C4×C61#13712yes—
231C2×C4×C61C23×C411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C22⋊D61—48no—
235C22⋊D61—48no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—24no—
239C2×Dic61—41no—
240C2×Dic61—48no—
241C2×Dic61—44no—
242C2×Dic61—48no—
243C2×Dic61C2×C22⋊C411yescharacteristic
244C2×Dic61C2×C22⋊C411yescharacteristic
245C2×Dic61C2×C22⋊C411yescharacteristic
246C2×Dic61C2×C22⋊C411yescharacteristic
247C2×Dic61C2×C22⋊C411yescharacteristic
248C2×Dic61C2×C22⋊C411yescharacteristic
249C4×D61—22no—
250C4×D61—21no—
251C4×D61—21no—
252C4×D61—22no—
253C4×D61—21no—
254C4×D61—21no—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—41no—
261C4×D61—41no—
262C4×D61AGL(1,ℤ/8)14yes—
263Dic122—44no—
264Dic122—44no—
265C8×C61—42no—
266C8×C61—42no—
267#92418—42no—
268#92418—42no—
269#17745—611nomaximal, Sylow
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—44no—
273C22×C61—44no—
274C22×C61—44no—
275C22×C61—88no—
276C22×C61#41712yes—
277C22×C61C22×C22⋊C411yescharacteristic
278C2×D61—21no—
279C2×D61—21no—
280C2×D61—21no—
281C2×D61—21no—
282C2×D61—21no—
283C2×D61—44no—
284C2×D61#41812yes—
285C4×C61—21no—
286C4×C61—21no—
287C4×C61#41312yes—
288C4×C61#41712yes—
289C4×C61C2×AGL(1,ℤ/8)12yes—
290C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
291C4×C61C2×#13711yescharacteristic
292Dic61—21no—
293Dic61—21no—
294Dic61—21no—
295Dic61—21no—
296Dic61—21no—
297Dic61—44no—
298Dic61#41812yes—
299C2×#532—611no—
300#2920—611no—
301C2×#423—611no—
302#1557—614no—
303#1549—611no—
304#1549—611no—
305#1547—614no—
306#1545—611no—
307#1545D6111yescharacteristic, complemented
308C2×C61—22no—
309C2×C61—84no—
310C2×C61C2×#41311yescharacteristic
311C2×C61C2×#41811yescharacteristic
312C2×C61C2×#41711yescharacteristic
313D61—22no—
314D61—41no—
315C22×#41—1221no—
316C22×M16—1221no—
317#532—612no—
318#532—612no—
319#532—612no—
320#532—612no—
321#532—1221no—
322#532—1221no—
323#532—1221no—
324#532—1221no—
325C2×C4⋊Q8—611no—
326C2×#127C2×D6111yescharacteristic
327C2×#124—1221no—
328C2×#121—612no—
329C2×#121—1221no—
330C2×C4×D4—612no—
331C2×C4×D4—1221no—
332C22×C42—611no—
333#432—612no—
334#432—1221no—
335#431—611no—
336#431—1221no—
337#431C2×D6111yescharacteristic
338#423—611no—
339#423—612no—
340#423—612no—
341#423—1221no—
342#423—1221no—
343#423C2×D6111yescharacteristic
344C2×C4⋊C8—612no—
345C2×C4⋊C8—1221no—
346#407—1221no—
347C4×M16—1221no—
348#332—614no—
349#332—614no—
350#331—614no—
351#331C2×D6114yescomplemented
352C61#1774511yescharacteristic, complemented, Sylow
353C2×#41—1222no—
354C2×#41—1222no—
355C2×#41—1222no—
356C2×#41—1222no—
357C2×#41—2448no—
358C22×Q8—1221no—
359C22×D4—21no—
360C22×D4—21no—
361C22×D4—1222no—
362C22×D4C4×D6112yes—
363C23×C4—611no—
364C23×C4—611no—
365C23×C4—611no—
366C23×C4—1222no—
367C2×M16—21no—
368C2×M16—1222no—
369C2×M16—1222no—
370C2×M16—1221no—
371C2×M16—2444no—
372C22×C8—1222no—
373C4⋊Q8—612no—
374C4⋊Q8—612no—
375C4⋊Q8—1221no—
376C4⋊Q8—1221no—
377#127—21no—
378#127—21no—
379#127C4×D6112yes—
380#127C22×D6112yes—
381#124—1222no—
382#124—1222no—
383#124—1222no—
384#124—1222no—
385#121—614no—
386#121—614no—
387#121—1222no—
388#121—1222no—
389#121—1222no—
390#121—1222no—
391#121—1222no—
392#121—1222no—
393C4×D4—614no—
394C4×D4—614no—
395C4×D4—1222no—
396C4×D4—1222no—
397C4×D4—1222no—
398C4×D4—1222no—
399C4×D4—1222no—
400C4×D4—1222no—
401C2×C4⋊C4—612no—
402C2×C4⋊C4—1221no—
403C2×C22⋊C4—1222no—
404C2×C22⋊C4—1222no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42C22×D6111yescharacteristic
417C4⋊C8—21no—
418C4⋊C8—612no—
419C4⋊C8—612no—
420C4⋊C8—1222no—
421C4⋊C8—1222no—
422C4⋊C8—1222no—
423C4⋊C8—1221no—
424C4⋊C8C22×D6112yes—
425C4⋊C8C22×D6112yes—
426C22⋊C8—21no—
427C22⋊C8—1222no—
428C22⋊C8—1221no—
429#97—1222no—
430C4×C8—21no—
431C4×C8—1221no—
432C24—22no—
433C24—611no—
434#41—2448no—
435#41—2448no—
436#41—2448no—
437#41—2448no—
438C2×Q8—1222no—
439C2×Q8—1222no—
440C2×Q8—2444no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—44no—
447C2×D4—44no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—1224no—
451C2×D4—1224no—
452C2×D4—2448no—
453C2×D4C2×C4×D6114yes—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—1224no—
470C22×C4—1224no—
471C22×C4—1224no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—2442no—
476C22×C4—2448no—
477C22×C4C2×C4×D6111yescharacteristic
478C22×C4D4×D6111yescharacteristic
479C22×C4D4×D6111yescharacteristic
480M16—42no—
481M16—2444no—
482M16—2444no—
483M16—2442no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—1222no—
487C2×C8—1222no—
488C2×C8—2444no—
489C4⋊C4—614no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C22⋊C4—1224no—
494C22⋊C4—1224no—
495C22⋊C4—1224no—
496C22⋊C4—1224no—
497C42—611no—
498C42—611no—
499C42—611no—
500C42—611no—
501C42—611no—
502C42—611no—
503C42—611no—
504C42—611no—
505C42—611no—
506C42—611no—
507C42—611no—
508C42—611no—
509C42C2×C4×D6111yescharacteristic
510C42D4×D6111yescharacteristic
511C42D4×D6111yescharacteristic
512C42C23×D6111yescharacteristic
513C23—24no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—611no—
518C23—611no—
519C23—1222no—
520C23—1222no—
521C23—2441no—
522C23C22⋊C4×D6111yescharacteristic
523Q8—2444no—
524Q8—2444no—
525D4—44no—
526D4—44no—
527D4—44no—
528D4—44no—
529D4—44no—
530D4—42no—
531D4—42no—
532D4—2448no—
533D4—2448no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—611no—
537C2×C4—614no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—611no—
541C2×C4—611no—
542C2×C4—611no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1224no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2448no—
560C2×C4—2441no—
561C2×C4—2448no—
562C2×C4—2444no—
563C2×C4C22⋊C4×D6111yescharacteristic
564C2×C4C22⋊C4×D6111yescharacteristic
565C2×C4C22⋊C4×D6111yescharacteristic
566C2×C4#1451980112yes—
567C2×C4#1451980612yes—
568C2×C4C22×C4×D6111yescharacteristic
569C2×C4C2×D4×D6111yescharacteristic
570C2×C4C2×D4×D6111yescharacteristic
571C8—42no—
572C8—42no—
573C8—2442no—
574C8—2442no—
575C22—21no—
576C22—21no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—88no—
581C22—612no—
582C22—1221no—
583C22—1221no—
584C22—1221no—
585C22—1221no—
586C22—1221no—
587C22—2444no—
588C22#1683976012yes—
589C22C2×C22⋊C4×D6111yescharacteristic
590C4—21no—
591C4—21no—
592C4—612no—
593C4—1221no—
594C4—1221no—
595C4—1221no—
596C4—1221no—
597C4—1221no—
598C4—2444no—
599C4#1683974312yes—
600C4#1683976012yes—
601C4AGL(1,ℤ/8)×D6112yes—
602C4C2×#1451980111yescharacteristic
603C4C2×#1451980611yescharacteristic
604C2—22no—
605C2—84no—
606C2—1222no—
607C2—2441no—
608C2C2×#1683974311yescharacteristic
609C2#418×D6111yescharacteristic
610C2C2×#1683976011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1545C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840310C2
cosocleC42×C61C24
hypercenter#1545D61
hypocenterC61#17745
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1545

Complements

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

NH ≅ G/Ntype
2. #21454447C2semidirect
3. #423×D61C2semidirect
4. #21444538C2semidirect
5. #21444505C2semidirect
6. #21444503C2semidirect
7. #21444493C2semidirect
8. #21444487C2semidirect
9. C61×#1545C2semidirect
27. #16839818C22semidirect
34. #16839784C22semidirect
35. #16839784C22semidirect
37. C4⋊C8×D61C22semidirect
40. #16839509C22semidirect
41. #16839507C22semidirect
42. #16839506C22semidirect
46. C61×#331C22semidirect
307. #1545D61semidirect
351. #331C2×D61semidirect
352. C61#17745semidirect

Maximal subgroups

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

subgroupindex
C2×#168403102
#214544472
#423×D612
#214445382
#214445052
#214445032
#214444932
#214444872
C61×#15452
#1774561