Finite Group Databasebeta

← #29686264

Subgroups — #29686264

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×#16839874C212yesmaximal, complemented
2#127×Dic61C211yesmaximal, characteristic, complemented
3C4×D4×Dic61C211yesmaximal, characteristic, complemented
4#21446747C211yesmaximal, characteristic, complemented
5#21446583C211yesmaximal, characteristic, complemented
6#21446448C212yesmaximal, complemented
7#21446169C212yesmaximal, complemented
8#21445819C211yesmaximal, characteristic, complemented
9#21445674C211yesmaximal, characteristic, complemented
10#21445646C212yesmaximal, complemented
11C2×C61×#459C211yesmaximal, characteristic
12C2×D4×Dic61—21no—
13C2×D4×Dic61—21no—
14C2×D4×Dic61C2212yes—
15C2×D4×Dic61C2211yescharacteristic
16C2×D4×Dic61C2211yescharacteristic
17C22×C4×Dic61—21no—
18#16839874—28no—
19C2×#14519709C2212yes—
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519698C2212yes—
22C2×#14519698C2212yes—
23C2×#14519698C2212yes—
24C2×#14519687C2212yes—
25#16839665—21no—
26#16839665—21no—
27#16839665C2212yes—
28#16839632C2211yescharacteristic
29C4⋊C4×Dic61C2211yescharacteristic
30C4⋊C4×Dic61C2211yescharacteristic
31C4×C4⋊Dic61C2211yescharacteristic
32C22⋊C4×Dic61—21no—
33C22⋊C4×Dic61—21no—
34C4×C22⋊Dic61—21no—
35C42×Dic61C2211yescharacteristic
36#16839525C2212yes—
37#16839524C2212yes—
38#16839516C2212yes—
39C22×D8×C61C2212yes—
40C2×#127×C61C2211yescharacteristic
41C2×C4×D4×C61C2211yescharacteristic
42C61×#459C418yescomplemented
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#102×C61C2212yes—
45C23×Dic61—21no—
46C23×Dic61—21no—
47C23×Dic61—22no—
48C22×#92418C2312yes—
49D4×Dic61—22no—
50D4×Dic61—22no—
51D4×Dic61—24no—
52D4×Dic61—22no—
53D4×Dic61—24no—
54D4×Dic61—44no—
55D4×Dic61—44no—
56D4×Dic61D414yescomplemented
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61C2312yes—
60C2×C4⋊Dic61C2311yescharacteristic
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C22⋊Dic61—21no—
63C2×C22⋊Dic61—21no—
64C2×C22⋊Dic61—22no—
65C2×C22⋊Dic61—22no—
66C2×C22⋊Dic61—22no—
67C2×C4×Dic61—21no—
68C2×C4×Dic61—21no—
69C2×C4×Dic61—21no—
70C2×C4×Dic61—44no—
71C2×C4×Dic61D411yescharacteristic
72C2×C4×Dic61D411yescharacteristic
73C2×C4×Dic61D411yescharacteristic
74C2×C4×Dic61D411yescharacteristic
75C2×C4×Dic61C2311yescharacteristic
76C2×C4×Dic61C2311yescharacteristic
77#14519709—24no—
78C4⋊#92418—22no—
79#14519698—24no—
80#14519698—24no—
81#14519698—24no—
82#14519687—24no—
83#14519684—21no—
84#14519684—21no—
85#14519684—21no—
86C22×D4×C61—21no—
87C22×D4×C61—21no—
88C22×D4×C61C2312yes—
89C22×D4×C61C2311yescharacteristic
90C23×C4×C61—21no—
91C2×D8×C61—24no—
92C2×D8×C61—24no—
93C2×D8×C61C2×C418yescomplemented
94C22×C8×C61C2312yes—
95#127×C61—22no—
96#127×C61C2×C414yescomplemented
97C4×D4×C61—22no—
98C4×D4×C61C2×C414yescomplemented
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C22⋊C4×C61—21no—
101C2×C42×C61C2311yescharacteristic
102C4⋊C8×C61C2×C414yescomplemented
103#102×C61C2×C418yescomplemented
104C24×C61—21no—
105C24×C61—22no—
106C2×D4×C61—24no—
107C2×D4×C61—24no—
108C2×D4×C61—22no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—42no—
117C2×D4×C61—42no—
118C2×D4×C61—42no—
119C2×D4×C61—42no—
120C2×D4×C61C22×C414yes—
121C2×D4×C61C22×C412yes—
122C2×D4×C61C2×D412yes—
123C2×D4×C61#4112yes—
124C22×C4×C61—21no—
125C22×C4×C61—24no—
126C22×C4×C61—42no—
127C22×C4×C61—42no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C22×Dic61—24no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—24no—
135C22×Dic61—22no—
136C22×Dic61—42no—
137C22×Dic61—42no—
138C22×Dic61—42no—
139C22×Dic61—44no—
140C22×Dic61—44no—
141C22×Dic61—42no—
142C22×Dic61—44no—
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147D8×C61—216no—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C2×C8×C61C22×C414yes—
151C4⋊C4×C61—21no—
152C4⋊C4×C61C22×C412yes—
153C22⋊C4×C61—24no—
154C42×C61C22×C412yes—
155C42×C61#4112yes—
156C22⋊Dic61—24no—
157C22⋊Dic61—24no—
158C22⋊Dic61—44no—
159C22⋊Dic61—44no—
160C22⋊Dic61—44no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61—22no—
164C4⋊Dic61—22no—
165C4⋊Dic61—42no—
166C4⋊Dic61—42no—
167C4⋊Dic61C2×D412yes—
168C4⋊Dic61#4112yes—
169C4×Dic61—22no—
170C4×Dic61—22no—
171C4×Dic61—21no—
172C4×Dic61—21no—
173C4×Dic61—22no—
174C4×Dic61—41no—
175C4×Dic61—42no—
176C4×Dic61—42no—
177C4×Dic61—42no—
178C4×Dic61C2×D412yes—
179C4×Dic61#4112yes—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—24no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61C22×D411yescharacteristic
198D4×C61—24no—
199D4×C61—22no—
200D4×C61—24no—
201D4×C61—22no—
202D4×C61—22no—
203D4×C61—48no—
204D4×C61—42no—
205D4×C61—42no—
206D4×C61—48no—
207D4×C61C4×D414yescomplemented
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—22no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—48no—
214C2×C4×C61C4×D412yes—
215C2×C4×C61C4×D412yes—
216C2×C4×C61AGL(1,ℤ/8)12yes—
217C2×C4×C61AGL(1,ℤ/8)12yes—
218C2×C4×C61C23×C411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—22no—
231C2×Dic61—21no—
232C2×Dic61—22no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—48no—
236C2×Dic61—48no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—88no—
245C2×Dic61#12112yes—
246C2×Dic61#12112yes—
247C2×Dic61AGL(1,ℤ/8)12yes—
248C8×C61—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×C61—44no—
259C22×C61—44no—
260C22×C61—44no—
261C22×C61—82no—
262C22×C61—82no—
263C22×C61—82no—
264C22×C61—82no—
265C22×C61C2×C4×D411yescharacteristic
266C22×C61C2×#12111yescharacteristic
267C22×C61#52511yescharacteristic
268C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C61#57511yescharacteristic
272C4×C61—22no—
273C4×C61—42no—
274C4×C61#44214yescomplemented
275C4×C61C2×C4×D411yescharacteristic
276C4×C61C4×#4111yescharacteristic
277C4×C61#52911yescharacteristic
278C4×C61#53211yescharacteristic
279Dic61—22no—
280Dic61—22no—
281Dic61—22no—
282Dic61—42no—
283Dic61—41no—
284Dic61—42no—
285Dic61—82no—
286C2×#459D6111yescharacteristic
287C2×#442—612no—
288C4×#127—611no—
289C42×D4—611no—
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×C61—84no—
299C2×C61C4×#12111yescharacteristic
300C2×C61C2×#44211yescharacteristic
301C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
302C2×C61#303611yescharacteristic
303C2×C61#303811yescharacteristic
304C2×C61#310911yescharacteristic
305C2×C61#311511yescharacteristic
306C22×D8C2×D6112yes—
307C2×#127C2×D6111yescharacteristic
308C2×C4×D4—611no—
309C2×C4×D4—612no—
310C2×C4×D4—611no—
311C2×C4×D4—1221no—
312C2×C4×D4—1221no—
313C2×C4×D4C2×D6111yescharacteristic
314C22×C42—1221no—
315#459Dic6118yescomplemented
316#442—1228no—
317C2×#106—612no—
318C2×C4⋊C8—611no—
319C2×C4⋊C8C2×D6111yescharacteristic
320C2×#102—612no—
321C2×#102—612no—
322C2×#102—612no—
323C2×#102C2×D6112yes—
324C2×#97—612no—
325#390—612no—
326#390—1221no—
327#390—1221no—
328#384—611no—
329C4×C4⋊C4—611no—
330C4×C4⋊C4—611no—
331C4×C4⋊C4—611no—
332C4×C22⋊C4—1221no—
333C4×C22⋊C4—1221no—
334C4×C22⋊C4—1221no—
335C43—611no—
336#340—612no—
337#340—612no—
338#336—612no—
339C61#2150311yescharacteristic, complemented, Sylow
340C22×D4—21no—
341C22×D4—21no—
342C22×D4C22×D6112yes—
343C22×D4C22×D6111yescharacteristic
344C23×C4—21no—
345C23×C4—1221no—
346C23×C4—1221no—
347C23×C4—1222no—
348C2×D8—24no—
349C2×D8—24no—
350C2×D8C2×Dic6118yescomplemented
351C22×C8—612no—
352C22×C8C22×D6112yes—
353#127—22no—
354#127C2×Dic6114yescomplemented
355C4×D4—22no—
356C4×D4—614no—
357C4×D4—1222no—
358C4×D4—1224no—
359C4×D4—1224no—
360C4×D4—1222no—
361C4×D4—1222no—
362C4×D4—2444no—
363C4×D4—2444no—
364C4×D4C2×Dic6114yescomplemented
365C2×C4⋊C4—611no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4C22×D6111yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—1221no—
373C2×C22⋊C4—1221no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1222no—
377C2×C42—611no—
378C2×C42—611no—
379C2×C42—611no—
380C2×C42—611no—
381C2×C42—611no—
382C2×C42—611no—
383C2×C42—1221no—
384C2×C42—1221no—
385C2×C42—1221no—
386C2×C42—2444no—
387C2×C42C22×D6111yescharacteristic
388#106—1224no—
389C4⋊C8—1222no—
390C4⋊C8C2×Dic6114yescomplemented
391#102—1224no—
392#102—1224no—
393#102—1224no—
394#102C2×Dic6118yescomplemented
395#97—1224no—
396#95—1221no—
397#95—1221no—
398#95—1221no—
399C24—21no—
400C24—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4D4×D6112yes—
416C2×D4#1175822412yes—
417C2×D4C22×Dic6114yes—
418C2×D4C22×Dic6112yes—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—1224no—
428C22×C4—1224no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—1222no—
432C22×C4—2444no—
433C22×C4—2442no—
434C22×C4—2444no—
435C22×C4—2444no—
436C22×C4—2442no—
437C22×C4—2442no—
438C22×C4—2442no—
439C22×C4D4×D6111yescharacteristic
440C22×C4D4×D6111yescharacteristic
441C22×C4C23×D6111yescharacteristic
442D8—216no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—1222no—
446C2×C8—1222no—
447C2×C8—1222no—
448C2×C8C22×Dic6114yes—
449C4⋊C4—21no—
450C4⋊C4—612no—
451C4⋊C4—612no—
452C4⋊C4—1221no—
453C4⋊C4—1221no—
454C4⋊C4—1222no—
455C4⋊C4—1222no—
456C4⋊C4—2442no—
457C4⋊C4—2442no—
458C4⋊C4C22×Dic6112yes—
459C22⋊C4—24no—
460C22⋊C4—1224no—
461C22⋊C4—1224no—
462C22⋊C4—2444no—
463C22⋊C4—2444no—
464C22⋊C4—2444no—
465C42—612no—
466C42—612no—
467C42—1222no—
468C42—1221no—
469C42—1221no—
470C42—1222no—
471C42—1222no—
472C42—2441no—
473C42—2442no—
474C42—2442no—
475C42—2442no—
476C42#1175822412yes—
477C42C22×Dic6112yes—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23C2×D4×D6111yescharacteristic
493D4—24no—
494D4—22no—
495D4—24no—
496D4—22no—
497D4—22no—
498D4—48no—
499D4—42no—
500D4—42no—
501D4—48no—
502D4D4×Dic6114yescomplemented
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—48no—
509C2×C4—612no—
510C2×C4—612no—
511C2×C4—612no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1222no—
519C2×C4—1221no—
520C2×C4—1222no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—4888no—
524C2×C4—2448no—
525C2×C4—2448no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4D4×Dic6112yes—
534C2×C4D4×Dic6112yes—
535C2×C4#1451980112yes—
536C2×C4#1451980112yes—
537C2×C4C23×Dic6111yescharacteristic
538C2×C4C2×D4×D6111yescharacteristic
539C2×C4#41×D6111yescharacteristic
540C2×C4#41×D6111yescharacteristic
541C2×C4C2×#1175822411yescharacteristic
542C2×C4C2×#1175822411yescharacteristic
543C8—24no—
544C8—2442no—
545C22—24no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22C2×D4×Dic6111yescharacteristic
560C22#121×D6111yescharacteristic
561C22#1684027211yescharacteristic
562C22AGL(1,ℤ/8)×D6111yescharacteristic
563C22C2×#1451980111yescharacteristic
564C22C2×#1451980111yescharacteristic
565C22#1684055811yescharacteristic
566C4—22no—
567C4—42no—
568C4—1222no—
569C4—1222no—
570C4—1222no—
571C4—4882no—
572C4—2442no—
573C4—2441no—
574C4—2442no—
575C4#1683987414yescomplemented
576C4C2×D4×Dic6111yescharacteristic
577C4#41×Dic6111yescharacteristic
578C4#1684029311yescharacteristic
579C4#1684030711yescharacteristic
580C2—44no—
581C2—84no—
582C2#121×Dic6111yescharacteristic
583C2C2×#1683987411yescharacteristic
584C2AGL(1,ℤ/8)×Dic6111yescharacteristic
585C2#2145527811yescharacteristic
586C2#2145529411yescharacteristic
587C2#2145590911yescharacteristic
588C2#2145596711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#459C2
socleC23×C61C22×D4
cosocleC22×C4×C61C24
hypercenterC2×#459D61
hypocenterC61#21503
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. C2×#16839874C2semidirect
2. #127×Dic61C2semidirect
3. C4×D4×Dic61C2semidirect
4. #21446747C2semidirect
5. #21446583C2semidirect
6. #21446448C2semidirect
7. #21446169C2semidirect
8. #21445819C2semidirect
9. #21445674C2semidirect
10. #21445646C2semidirect
42. C61×#459C4semidirect
56. D4×Dic61D4semidirect
93. C2×D8×C61C2×C4semidirect
96. #127×C61C2×C4semidirect
98. C4×D4×C61C2×C4semidirect
102. C4⋊C8×C61C2×C4semidirect
103. #102×C61C2×C4semidirect
207. D4×C61C4×D4semidirect
274. C4×C61#442semidirect
315. #459Dic61semidirect
339. C61#21503semidirect
350. C2×D8C2×Dic61semidirect
354. #127C2×Dic61semidirect
364. C4×D4C2×Dic61semidirect
390. C4⋊C8C2×Dic61semidirect
394. #102C2×Dic61semidirect
502. D4D4×Dic61semidirect
575. C4#16839874semidirect

Maximal subgroups

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

subgroupindex
C2×#168398742
#127×Dic612
C4×D4×Dic612
#214467472
#214465832
#214464482
#214461692
#214458192
#214456742
#214456462
C2×C61×#4592
#2150361