Finite Group Databasebeta

← #29686263

Subgroups — #29686263

Order 15616.

1260 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839957C211yesmaximal, characteristic
2C2×#16839872C212yesmaximal, complemented
3C4×#14519774C211yesmaximal, characteristic, complemented
4#21446746C211yesmaximal, characteristic, complemented
5#21446582C211yesmaximal, characteristic, complemented
6#21446446C212yesmaximal, complemented
7#21446167C212yesmaximal, complemented
8#21445819C211yesmaximal, characteristic, complemented
9#21445672C211yesmaximal, characteristic, complemented
10#21445647C212yesmaximal, complemented
11C42×D4×C61C211yesmaximal, characteristic, complemented
12C22×#11758199C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×C4×D244—21no—
15C2×C4×D244—21no—
16C2×C4×D244C2212yes—
17#16839957C418yescomplemented
18#16839872—28no—
19C2×#14519708C2212yes—
20C2×C4⋊#92418C2211yescharacteristic
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519698C2212yes—
23C2×#14519697C2212yes—
24C2×#14519697C2212yes—
25C2×#14519697C2212yes—
26C2×#14519688C2212yes—
27#16839664—21no—
28#16839664—21no—
29#16839664C2212yes—
30#16839632C2211yescharacteristic
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34C2×C4×D4×C61C2211yescharacteristic
35C2×C4×D4×C61C2211yescharacteristic
36C2×C4×D4×C61C2211yescharacteristic
37C22×C42×C61—21no—
38C4×C4⋊C4×C61C2211yescharacteristic
39C4×C4⋊C4×C61C2211yescharacteristic
40C4×C4⋊C4×C61C2211yescharacteristic
41C4×C22⋊C4×C61—21no—
42C4×C22⋊C4×C61—21no—
43C4×C22⋊C4×C61—21no—
44C43×C61C2211yescharacteristic
45C22×D244—21no—
46C22×D244—21no—
47C22×D244C2312yes—
48C22×C4×D61—22no—
49C2×#11758199—24no—
50C2×#11758199—24no—
51C2×#11758199C2×C418yescomplemented
52C22×#92418C2312yes—
53C22×#92418C2312yes—
54#14519774—22no—
55#14519774C2×C414yescomplemented
56C4×D244—24no—
57C4×D244—24no—
58C4×D244—44no—
59C4×D244—44no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61C2312yes—
63C2×#11758198—22no—
64C2×#11758198—22no—
65C2×#11758198—22no—
66#14519708—24no—
67C4⋊#92418—22no—
68C4⋊#92418C2×C414yescomplemented
69#14519698—24no—
70#14519697—24no—
71#14519697—24no—
72#14519697C2×C418yescomplemented
73#14519688—24no—
74C22×D4×C61C2311yescharacteristic
75C23×C4×C61—21no—
76C23×C4×C61—21no—
77C23×C4×C61—21no—
78C4×D4×C61—22no—
79C4×D4×C61—22no—
80C4×D4×C61—22no—
81C4×D4×C61—22no—
82C4×D4×C61C2×C414yescomplemented
83C4×D4×C61D414yescomplemented
84C2×C4⋊C4×C61C2311yescharacteristic
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C22⋊C4×C61—21no—
88C2×C22⋊C4×C61—21no—
89C2×C22⋊C4×C61—21no—
90C2×C42×C61—21no—
91C2×C42×C61—21no—
92C2×C42×C61—21no—
93C2×C42×C61—44no—
94C2×C42×C61D411yescharacteristic
95C2×C42×C61D411yescharacteristic
96C2×C42×C61D411yescharacteristic
97C2×C42×C61D411yescharacteristic
98C2×C42×C61C2311yescharacteristic
99C2×C42×C61C2311yescharacteristic
100C2×C42×C61C2311yescharacteristic
101#95×C61—21no—
102#95×C61—21no—
103#95×C61—21no—
104C24×C61—21no—
105C23×D61—22no—
106C2×D4×C61—21no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61C22×C412yes—
110C2×D4×C61C2×D412yes—
111C2×D4×C61#4112yes—
112C22×C4×C61—24no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—24no—
117C22×C4×C61—24no—
118C22×C4×C61—42no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61—42no—
123C22×C4×C61—42no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C22×Dic61—22no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—22no—
135C2×D244—22no—
136C2×D244—24no—
137C2×D244—24no—
138C2×D244—22no—
139C2×D244—42no—
140C2×D244—42no—
141C2×D244—42no—
142C2×D244—42no—
143C2×D244C22×C414yes—
144C2×C4×D61—44no—
145C2×C4×D61—44no—
146C2×C4×D61—44no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61C22×C412yes—
151C4⋊C4×C61C2×D412yes—
152C4⋊C4×C61#4112yes—
153C22⋊C4×C61—24no—
154C22⋊C4×C61—24no—
155C22⋊C4×C61—24no—
156C42×C61—22no—
157C42×C61—22no—
158C42×C61—22no—
159C42×C61—21no—
160C42×C61—21no—
161C42×C61—41no—
162C42×C61—42no—
163C42×C61—42no—
164C42×C61—42no—
165C42×C61C22×C412yes—
166C42×C61C2×D412yes—
167C42×C61#4112yes—
168C42×C61#4112yes—
169#11758199—216no—
170#11758198—44no—
171#11758198—44no—
172#11758198—44no—
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C4⋊Dic61—42no—
176C4⋊Dic61—42no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418C22×C414yes—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61C22×D411yescharacteristic
191C22×D61—24no—
192C22×D61—42no—
193C22×D61—42no—
194C22×D61—42no—
195C22×D61—42no—
196C22×D61—42no—
197C22×D61—42no—
198D4×C61—22no—
199D4×C61—22no—
200D4×C61—22no—
201D4×C61C4×D414yescomplemented
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—21no—
213C2×C4×C61—22no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—48no—
217C2×C4×C61—48no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—48no—
225C2×C4×C61C4×D412yes—
226C2×C4×C61C4×D412yes—
227C2×C4×C61#12112yes—
228C2×C4×C61#12112yes—
229C2×C4×C61AGL(1,ℤ/8)12yes—
230C2×C4×C61AGL(1,ℤ/8)12yes—
231C2×C4×C61AGL(1,ℤ/8)12yes—
232C2×C4×C61C23×C411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241D244—24no—
242D244—24no—
243D244—48no—
244D244—42no—
245D244—42no—
246D244—48no—
247C4×D61—88no—
248#92418—24no—
249#92418—42no—
250#21503—611nomaximal, Sylow
251C22×C61—24no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61C2×C4×D411yescharacteristic
259C22×C61C2×#12111yescharacteristic
260C22×C61#52511yescharacteristic
261C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C61#57511yescharacteristic
265C2×D61—44no—
266C2×D61—44no—
267C2×D61—44no—
268C2×D61—82no—
269C2×D61—82no—
270C2×D61—82no—
271C2×D61—82no—
272C4×C61—22no—
273C4×C61—22no—
274C4×C61—22no—
275C4×C61—22no—
276C4×C61—42no—
277C4×C61—41no—
278C4×C61—42no—
279C4×C61—42no—
280C4×C61#44214yescomplemented
281C4×C61C2×C4×D411yescharacteristic
282C4×C61C4×#4111yescharacteristic
283C4×C61#52911yescharacteristic
284C4×C61#53211yescharacteristic
285Dic61—82no—
286C2×#459—611no—
287C2×#442—612no—
288C4×#127—611no—
289C42×D4D6111yescharacteristic, complemented
290#1982—611no—
291#1948—611no—
292#1926—612no—
293#1879—612no—
294#1802—611no—
295#1767—611no—
296#1762—612no—
297C2×C61—44no—
298C2×C61C4×#12111yescharacteristic
299C2×C61C2×#44211yescharacteristic
300C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
301C2×C61#303611yescharacteristic
302C2×C61#303811yescharacteristic
303C2×C61#310911yescharacteristic
304C2×C61#311511yescharacteristic
305D61—84no—
306C22×D8—612no—
307C2×#127—611no—
308C2×C4×D4—612no—
309C2×C4×D4—1221no—
310C2×C4×D4—1221no—
311C2×C4×D4C2×D6111yescharacteristic
312C2×C4×D4C2×D6111yescharacteristic
313C2×C4×D4C2×D6111yescharacteristic
314C22×C42—21no—
315#459—618no—
316#442—1228no—
317C2×#106—612no—
318C2×C4⋊C8—611no—
319C2×C4⋊C8—611no—
320C2×#102—612no—
321C2×#102—612no—
322C2×#102—612no—
323C2×#102—612no—
324C2×#97—612no—
325#390—612no—
326#390—1221no—
327#390—1221no—
328#384—611no—
329C4×C4⋊C4C2×D6111yescharacteristic
330C4×C4⋊C4C2×D6111yescharacteristic
331C4×C4⋊C4C2×D6111yescharacteristic
332C4×C22⋊C4—21no—
333C4×C22⋊C4—21no—
334C4×C22⋊C4—21no—
335C43C2×D6111yescharacteristic
336#340—612no—
337#340—612no—
338#336—612no—
339C61#2150311yescharacteristic, complemented, Sylow
340C22×D4—612no—
341C22×D4—1221no—
342C22×D4—1221no—
343C22×D4C22×D6111yescharacteristic
344C23×C4—21no—
345C23×C4—21no—
346C23×C4—21no—
347C23×C4—1222no—
348C2×D8—618no—
349C2×D8—1224no—
350C2×D8—1224no—
351C22×C8—612no—
352C22×C8—612no—
353#127—614no—
354#127—1222no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—1224no—
360C4×D4—1224no—
361C4×D4—2444no—
362C4×D4—2444no—
363C4×D4C4×D6114yescomplemented
364C4×D4D24414yescomplemented
365C2×C4⋊C4—612no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4C22×D6111yescharacteristic
369C2×C4⋊C4C22×D6111yescharacteristic
370C2×C4⋊C4C22×D6111yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1222no—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—44no—
381C2×C42C22⋊D6111yescharacteristic
382C2×C42C22⋊D6111yescharacteristic
383C2×C42C22⋊D6111yescharacteristic
384C2×C42C22⋊D6111yescharacteristic
385C2×C42C22×D6111yescharacteristic
386C2×C42C22×D6111yescharacteristic
387C2×C42C22×D6111yescharacteristic
388#106—1224no—
389C4⋊C8—614no—
390C4⋊C8—1222no—
391#102—618no—
392#102—1224no—
393#102—1224no—
394#102—1224no—
395#97—1224no—
396#95—21no—
397#95—21no—
398#95—21no—
399C24—21no—
400C24—1222no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—614no—
405C2×D4—1222no—
406C2×D4—1222no—
407C2×D4—1222no—
408C2×D4—1222no—
409C2×D4—1222no—
410C2×D4—1224no—
411C2×D4—1224no—
412C2×D4—2442no—
413C2×D4—2442no—
414C2×D4—2442no—
415C2×D4—2442no—
416C2×D4C2×C4×D6112yes—
417C2×D4C2×D24412yes—
418C2×D4#1175822212yes—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—1222no—
432C22×C4—2444no—
433C22×C4—2444no—
434C22×C4—2444no—
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C2×C22⋊D6111yescharacteristic
437C22×C4C2×C22⋊D6111yescharacteristic
438C22×C4C2×C22⋊D6111yescharacteristic
439C22×C4C2×C22⋊D6111yescharacteristic
440C22×C4C2×C22⋊D6111yescharacteristic
441C22×C4C23×D6111yescharacteristic
442D8—12216no—
443C2×C8—614no—
444C2×C8—1222no—
445C2×C8—1222no—
446C2×C8—1222no—
447C2×C8—1222no—
448C2×C8—1222no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—1222no—
453C4⋊C4—1222no—
454C4⋊C4—2442no—
455C4⋊C4—2442no—
456C4⋊C4C2×C4×D6112yes—
457C4⋊C4C2×D24412yes—
458C4⋊C4#1175822212yes—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—2444no—
463C22⋊C4—2444no—
464C22⋊C4—2444no—
465C42—21no—
466C42—21no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—41no—
471C42—42no—
472C42—42no—
473C42—42no—
474C42C2×C4×D6112yes—
475C42C2×D24412yes—
476C42#1175822212yes—
477C42#1175822212yes—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—1224no—
486C23—2442no—
487C23—2442no—
488C23—2442no—
489C23—2442no—
490C23—2442no—
491C23—2442no—
492C23C22×C22⋊D6111yescharacteristic
493D4—22no—
494D4—22no—
495D4—22no—
496D4—1224no—
497D4—1224no—
498D4—2448no—
499D4—2442no—
500D4—2442no—
501D4—2448no—
502D4C4×D24414yescomplemented
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—48no—
518C2×C4—48no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—48no—
526C2×C4—4888no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4C4×C22⋊D6112yes—
531C2×C4C4×C22⋊D6112yes—
532C2×C4#1451975112yes—
533C2×C4#1451975112yes—
534C2×C4#1451980212yes—
535C2×C4#1451980212yes—
536C2×C4#1451980312yes—
537C2×C4C22×C4×D6111yescharacteristic
538C2×C4C22×D24411yescharacteristic
539C2×C4C2×#1175822211yescharacteristic
540C2×C4C2×#1175822211yescharacteristic
541C2×C4C2×#1175822211yescharacteristic
542C2×C4C2×#1175822211yescharacteristic
543C8—1224no—
544C8—2442no—
545C22—24no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—4882no—
553C22—4882no—
554C22—4882no—
555C22—4882no—
556C22—2444no—
557C22—2444no—
558C22—2444no—
559C22C2×C4×C22⋊D6111yescharacteristic
560C22C2×#1451975111yescharacteristic
561C22#1684027311yescharacteristic
562C22C2×#1451980211yescharacteristic
563C22C2×#1451980211yescharacteristic
564C22C2×#1451980311yescharacteristic
565C22#1684056011yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—42no—
571C4—42no—
572C4—41no—
573C4—42no—
574C4—4882no—
575C4#1683987214yescomplemented
576C4C2×C4×D24411yescharacteristic
577C4C4×#1175822211yescharacteristic
578C4#1684029611yescharacteristic
579C4#1684031211yescharacteristic
580C2—44no—
581C2—4884no—
582C2C4×#1451975111yescharacteristic
583C2C2×#1683987211yescharacteristic
584C2C4×#1451980211yescharacteristic
585C2#2145528311yescharacteristic
586C2#2145529911yescharacteristic
587C2#2145591011yescharacteristic
588C2#2145597011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC42×D4×C61C2
socleC23×C61C22×D4
prime radicalC2×#16839957C2
cosocleC22×C4×C61C24
hypercenterC42×D4D61
hypocenterC61#21503
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C43 ◁ C42×D4

Complements

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

NH ≅ G/Ntype
2. C2×#16839872C2semidirect
3. C4×#14519774C2semidirect
4. #21446746C2semidirect
5. #21446582C2semidirect
6. #21446446C2semidirect
7. #21446167C2semidirect
8. #21445819C2semidirect
9. #21445672C2semidirect
10. #21445647C2semidirect
11. C42×D4×C61C2semidirect
17. #16839957C4semidirect
51. C2×#11758199C2×C4semidirect
55. #14519774C2×C4semidirect
68. C4⋊#92418C2×C4semidirect
72. #14519697C2×C4semidirect
82. C4×D4×C61C2×C4semidirect
83. C4×D4×C61D4semidirect
201. D4×C61C4×D4semidirect
280. C4×C61#442semidirect
289. C42×D4D61semidirect
339. C61#21503semidirect
363. C4×D4C4×D61semidirect
364. C4×D4D244semidirect
502. D4C4×D244semidirect
575. C4#16839872semidirect

Maximal subgroups

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

subgroupindex
C2×#168399572
C2×#168398722
C4×#145197742
#214467462
#214465822
#214464462
#214461672
#214458192
#214456722
#214456472
C42×D4×C612
#2150361