Finite Group Databasebeta

← #29570909

Subgroups — #29570909

Order 15616.

804 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#121×Dic61C211yesmaximal, characteristic
2#21445583C211yesmaximal, characteristic, complemented
3C2×#16839496C211yesmaximal, characteristic
4#95×Dic61C211yesmaximal, characteristic, complemented
5#21443699C211yesmaximal, characteristic, complemented
6#21443698C211yesmaximal, characteristic, complemented
7C2×C61×#327C211yesmaximal, characteristic
8C2×D4×Dic61—21no—
9C2×D4×Dic61C411yescharacteristic, complemented
10C22×C4×Dic61—21no—
11C22×C4×Dic61C2211yescharacteristic
12C2×C22⋊#92418C2211yescharacteristic
13C2×#14519687—21no—
14#16839665C411yescharacteristic, complemented
15#16839661C411yescharacteristic, complemented
16#16839655—21no—
17#16839642—21no—
18#16839622C2211yescharacteristic
19C4⋊C4×Dic61C411yescharacteristic, complemented
20C22⋊C4×Dic61—21no—
21C2×#14519684—21no—
22C2×#14519684C2211yescharacteristic
23#16839516—21no—
24#16839496C414yescomplemented
25C2×#121×C61C2211yescharacteristic
26C2×C22⋊C8×C61C2211yescharacteristic
27C2×#95×C61C2211yescharacteristic
28C61×#327C414yescomplemented
29C23×Dic61—21no—
30C23×Dic61—21no—
31C23×Dic61D411yescharacteristic
32C23×Dic61D411yescharacteristic
33C22×#92418—21no—
34D4×Dic61—42no—
35D4×Dic61—42no—
36D4×Dic61—42no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61C2×C411yescharacteristic
39C2×C4⋊Dic61C2×C411yescharacteristic
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—21no—
43C2×C22⋊Dic61—21no—
44C2×C4×Dic61—24no—
45C2×C4×Dic61—21no—
46C2×C4×Dic61—42no—
47C2×C4×Dic61—42no—
48C2×C4×Dic61—42no—
49C2×C4×Dic61C2×C411yescharacteristic
50C2×C4×Dic61C2×C411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53C22⋊#92418—21no—
54C22⋊#92418C2×C412yescomplemented
55#14519687—42no—
56#14519684—24no—
57#14519684—21no—
58#14519684—21no—
59#14519684—21no—
60#14519684—21no—
61#14519684C2×C412yescomplemented
62C22×D4×C61—21no—
63C22×D4×C61C2×C411yescharacteristic
64C23×C4×C61—21no—
65C23×C4×C61C2311yescharacteristic
66C22×C8×C61—21no—
67#121×C61—21no—
68#121×C61—21no—
69#121×C61C2×C412yes—
70#121×C61C2×C412yes—
71C2×C4⋊C4×C61C2×C411yescharacteristic
72C2×C22⋊C4×C61—21no—
73C22⋊C8×C61—21no—
74C22⋊C8×C61C2×C412yescomplemented
75#95×C61—21no—
76#95×C61C2×C412yescomplemented
77C24×C61—21no—
78C24×C61C2×D411yescharacteristic
79C2×D4×C61—22no—
80C2×D4×C61—22no—
81C2×D4×C61—21no—
82C2×D4×C61—41no—
83C2×D4×C61—41no—
84C2×D4×C61—41no—
85C2×D4×C61—41no—
86C2×D4×C61—41no—
87C2×D4×C61—41no—
88C2×D4×C61C4212yescomplemented
89C22×C4×C61—22no—
90C22×C4×C61—22no—
91C22×C4×C61—21no—
92C22×C4×C61—22no—
93C22×C4×C61—41no—
94C22×C4×C61—41no—
95C22×C4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61—41no—
98C22×C4×C61—41no—
99C22×C4×C61C22×C411yescharacteristic
100C22×C4×C61C22×C411yescharacteristic
101C22×C4×C61C22×C411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61#4111yescharacteristic
104C22×C4×C61#4111yescharacteristic
105C22×Dic61—21no—
106C22×Dic61—21no—
107C22×Dic61—22no—
108C22×Dic61—22no—
109C22×Dic61—22no—
110C22×Dic61—22no—
111C22×Dic61—22no—
112C22×Dic61—21no—
113C22×Dic61—21no—
114C22×Dic61—22no—
115C22×Dic61—21no—
116C22×Dic61—21no—
117C22×Dic61—41no—
118C22×Dic61—42no—
119C22×Dic61—42no—
120C22×Dic61—41no—
121C22×Dic61—41no—
122C22×Dic61—41no—
123C22×Dic61—41no—
124C22×Dic61—42no—
125C22×Dic61C22⋊C411yescharacteristic
126C22×Dic61C22⋊C411yescharacteristic
127C22×Dic61C22⋊C411yescharacteristic
128C22×Dic61C22⋊C411yescharacteristic
129C2×C8×C61—22no—
130C2×C8×C61—41no—
131C2×C8×C61—41no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61C4212yescomplemented
134C22⋊C4×C61—22no—
135C22⋊C4×C61—22no—
136C22⋊Dic61—42no—
137C22⋊Dic61—41no—
138C22⋊Dic61—41no—
139C22⋊Dic61—42no—
140C22⋊Dic61—42no—
141C4⋊Dic61—42no—
142C4⋊Dic61—41no—
143C4⋊Dic61—41no—
144C4×Dic61—22no—
145C4×Dic61—22no—
146C4×Dic61—41no—
147C4×Dic61—41no—
148C4×Dic61—48no—
149C4×Dic61—42no—
150C2×#92418—22no—
151C2×#92418—41no—
152C2×#92418—41no—
153C23×C61—21no—
154C23×C61—21no—
155C23×C61—21no—
156C23×C61—21no—
157C23×C61—22no—
158C23×C61—41no—
159C23×C61—41no—
160C23×C61—41no—
161C23×C61—41no—
162C23×C61—41no—
163C23×C61—41no—
164C23×C61C2×C22⋊C411yescharacteristic
165C23×C61C4×D411yescharacteristic
166C23×C61C4×D411yescharacteristic
167C23×C61AGL(1,ℤ/8)11yescharacteristic
168C23×C61AGL(1,ℤ/8)11yescharacteristic
169C23×C61AGL(1,ℤ/8)11yescharacteristic
170C23×C61#13711yescharacteristic
171D4×C61—42no—
172D4×C61—42no—
173D4×C61—42no—
174D4×C61—42no—
175D4×C61—42no—
176D4×C61—42no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—24no—
182C2×C4×C61—22no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—42no—
195C2×C4×C61C2×C4211yescharacteristic
196C2×C4×C61#11711yescharacteristic
197C2×C4×C61C4×D411yescharacteristic
198C2×C4×C61C4×D411yescharacteristic
199C2×Dic61—24no—
200C2×Dic61—24no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—24no—
208C2×Dic61—21no—
209C2×Dic61—21no—
210C2×Dic61—21no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—84no—
226C2×Dic61#9912yes—
227C2×Dic61#10412yes—
228C2×Dic61#10412yes—
229C8×C61—42no—
230#92418—42no—
231#8218—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—21no—
238C22×C61—21no—
239C22×C61—21no—
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—21no—
243C22×C61—21no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—81no—
248C22×C61—81no—
249C22×C61—81no—
250C22×C61—81no—
251C22×C61C4×C22⋊C411yescharacteristic
252C22×C61C2×#9911yescharacteristic
253C22×C61#41011yescharacteristic
254C22×C61#41711yescharacteristic
255C22×C61#41811yescharacteristic
256C22×C61C2×#10411yescharacteristic
257C22×C61C2×#10411yescharacteristic
258C22×C61#44011yescharacteristic
259C22×C61#44011yescharacteristic
260C22×C61#44211yescharacteristic
261C22×C61#44211yescharacteristic
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—21no—
266C4×C61—44no—
267C4×C61—42no—
268Dic61—22no—
269Dic61—22no—
270Dic61—22no—
271Dic61—44no—
272Dic61—41no—
273Dic61—81no—
274Dic61—81no—
275C4×#121—611no—
276#1749—611no—
277C2×#327—611no—
278C2×#327D6111yescharacteristic
279C4×#95—611no—
280#1294—611no—
281#1294—611no—
282C2×C61—21no—
283C2×C61—21no—
284C2×C61—21no—
285C2×C61—21no—
286C2×C61—82no—
287C2×C61#150411yescharacteristic
288C2×C61#150511yescharacteristic
289C2×C61#150711yescharacteristic
290C2×C61#150911yescharacteristic
291C2×C61C4×#9911yescharacteristic
292C2×C61C4×#10411yescharacteristic
293C2×C61#176211yescharacteristic
294C2×#121C2×D6111yescharacteristic
295C2×C4×D4—611no—
296C2×C4×D4—1221no—
297C22×C42—611no—
298C22×C42—1221no—
299C2×C22⋊C8—611no—
300C2×C22⋊C8C2×D6111yescharacteristic
301C2×#97—1221no—
302#390—611no—
303#389—611no—
304#388—1221no—
305#386—1221no—
306#380—611no—
307C4×C4⋊C4—611no—
308C4×C22⋊C4—1221no—
309C2×#95—611no—
310C2×#95—1221no—
311C2×#95C2×D6111yescharacteristic
312#336—1221no—
313#327—614no—
314#327Dic6114yescomplemented
315C61#821811yescharacteristic, complemented, Sylow
316C22×D4—21no—
317C22×D4C4×D6111yescharacteristic
318C23×C4—21no—
319C23×C4—611no—
320C23×C4—611no—
321C23×C4—1221no—
322C23×C4—1221no—
323C23×C4C22×D6111yescharacteristic
324C22×C8—21no—
325C22×C8—1221no—
326#121—21no—
327#121—21no—
328#121C4×D6112yes—
329#121C2×Dic6112yes—
330C4×D4—2442no—
331C4×D4—2442no—
332C4×D4—2442no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4C4×D6111yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—1221no—
339C2×C22⋊C4—1221no—
340C2×C22⋊C4—1221no—
341C2×C22⋊C4—1221no—
342C2×C42—611no—
343C2×C42—611no—
344C2×C42—611no—
345C2×C42—611no—
346C2×C42—1221no—
347C2×C42—1224no—
348C2×C42—2442no—
349C2×C42—2442no—
350C2×C42—2442no—
351C22⋊C8—21no—
352C22⋊C8—612no—
353C22⋊C8—1221no—
354C22⋊C8C2×Dic6112yescomplemented
355#97—2442no—
356#95—21no—
357#95—612no—
358#95—1221no—
359#95—1221no—
360#95—1224no—
361#95—1221no—
362#95—1221no—
363#95C2×Dic6112yescomplemented
364C24—21no—
365C24D4×D6111yescharacteristic
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4C4×Dic6112yescomplemented
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—611no—
387C22×C4—611no—
388C22×C4—611no—
389C22×C4—611no—
390C22×C4—1222no—
391C22×C4—1222no—
392C22×C4—1222no—
393C22×C4—1221no—
394C22×C4—1221no—
395C22×C4—1222no—
396C22×C4—1222no—
397C22×C4—1222no—
398C22×C4—1221no—
399C22×C4—1221no—
400C22×C4—1221no—
401C22×C4—1221no—
402C22×C4—2442no—
403C22×C4—2441no—
404C22×C4—2442no—
405C22×C4—2442no—
406C22×C4—2441no—
407C22×C4—2441no—
408C22×C4—2441no—
409C22×C4—2441no—
410C22×C4C2×C4×D6111yescharacteristic
411C22×C4C2×C4×D6111yescharacteristic
412C22×C4D4×D6111yescharacteristic
413C22×C4#1175822411yescharacteristic
414C22×C4#1175822411yescharacteristic
415C22×C4C22×Dic6111yescharacteristic
416C2×C8—22no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—1222no—
420C2×C8—2441no—
421C2×C8—2441no—
422C4⋊C4—21no—
423C4⋊C4—2441no—
424C4⋊C4—2441no—
425C4⋊C4—2442no—
426C4⋊C4C4×Dic6112yescomplemented
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—2442no—
430C22⋊C4—2442no—
431C22⋊C4—2442no—
432C22⋊C4—2441no—
433C22⋊C4—2441no—
434C42—1222no—
435C42—1222no—
436C42—2441no—
437C42—2441no—
438C42—2448no—
439C42—2442no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—22no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23C22⋊C4×D6111yescharacteristic
452C23#1451973711yescharacteristic
453C23D4×Dic6111yescharacteristic
454C23#1451980111yescharacteristic
455C23#1451980111yescharacteristic
456C23#1451980411yescharacteristic
457C23#1451980611yescharacteristic
458D4—42no—
459D4—42no—
460D4—42no—
461D4—42no—
462D4—42no—
463D4—42no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—612no—
483C2×C4—612no—
484C2×C4—612no—
485C2×C4—1224no—
486C2×C4—1224no—
487C2×C4—1221no—
488C2×C4—1221no—
489C2×C4—1221no—
490C2×C4—1221no—
491C2×C4—1221no—
492C2×C4—1221no—
493C2×C4—1224no—
494C2×C4—1221no—
495C2×C4—1221no—
496C2×C4—1221no—
497C2×C4—4884no—
498C2×C4—2442no—
499C2×C4—2442no—
500C2×C4—2441no—
501C2×C4—2441no—
502C2×C4—2441no—
503C2×C4—2441no—
504C2×C4—2441no—
505C2×C4—2441no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4—2442no—
512C2×C4C2×C4×Dic6111yescharacteristic
513C2×C4#1451973311yescharacteristic
514C2×C4#1451973711yescharacteristic
515C2×C4D4×Dic6111yescharacteristic
516C8—42no—
517C8—2442no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22C22⋊C4×Dic6111yescharacteristic
538C22#99×D6111yescharacteristic
539C22#1683973411yescharacteristic
540C22#1683976011yescharacteristic
541C22#1683976511yescharacteristic
542C22#104×D6111yescharacteristic
543C22#104×D6111yescharacteristic
544C22#1683985911yescharacteristic
545C22#1683986311yescharacteristic
546C22#1683987111yescharacteristic
547C22#1683987411yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—44no—
553C4—42no—
554C4—1222no—
555C4—1222no—
556C4—1222no—
557C4—4881no—
558C4—4881no—
559C4—2444no—
560C4—2441no—
561C2—21no—
562C2—21no—
563C2—21no—
564C2—21no—
565C2—82no—
566C2#2144432011yescharacteristic
567C2#2144432511yescharacteristic
568C2#2144433311yescharacteristic
569C2#2144434111yescharacteristic
570C2#99×Dic6111yescharacteristic
571C2#104×Dic6111yescharacteristic
572C2#2144564611yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC2×C4×C61C2×C42
FrattiniC23×C4C22×D61
FittingC2×C61×#327C2
socleC23×C61C2×C22⋊C4
prime radicalC2×#121×C61C22
cosocleC23×C4×C61C23
hypercenterC2×#327D61
hypocenterC61#8218
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. #21445583C2semidirect
4. #95×Dic61C2semidirect
5. #21443699C2semidirect
6. #21443698C2semidirect
9. C2×D4×Dic61C4semidirect
14. #16839665C4semidirect
15. #16839661C4semidirect
19. C4⋊C4×Dic61C4semidirect
24. #16839496C4semidirect
28. C61×#327C4semidirect
54. C22⋊#92418C2×C4semidirect
61. #14519684C2×C4semidirect
74. C22⋊C8×C61C2×C4semidirect
76. #95×C61C2×C4semidirect
88. C2×D4×C61C42semidirect
133. C4⋊C4×C61C42semidirect
314. #327Dic61semidirect
315. C61#8218semidirect
354. C22⋊C8C2×Dic61semidirect
363. #95C2×Dic61semidirect
375. C2×D4C4×Dic61semidirect
426. C4⋊C4C4×Dic61semidirect

Maximal subgroups

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

subgroupindex
#121×Dic612
#214455832
C2×#168394962
#95×Dic612
#214436992
#214436982
C2×C61×#3272
#821861