Finite Group Databasebeta

← #29648367

Subgroups — #29648367

Order 15616.

1018 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2#431×D61C211yesmaximal, characteristic
3#423×D61C211yesmaximal, characteristic
4#21444506C214yesmaximal, complemented
5#21444505C211yesmaximal, characteristic, complemented
6#21444503C211yesmaximal, characteristic, complemented
7#21444500C211yesmaximal, characteristic, complemented
8#21444473C214yesmaximal, complemented
9C61×#1548C211yesmaximal, characteristic, complemented
10C2×M16×D61—21no—
11#16840317—21no—
12#16840317—21no—
13#16840317—21no—
14#16840317—21no—
15#16840317C412yes—
16#16840317C412yes—
17#16840317C2212yes—
18#16840317C2212yes—
19C2×#14519777C2211yescharacteristic
20C2×#14519768—21no—
21C2×#14519757—21no—
22C2×#14519753C412yes—
23C2×Q8×Dic61C412yes—
24C2×#14519742—21no—
25C2×#14519735—21no—
26C2×C42×D61C2211yescharacteristic
27#16839815C2211yescharacteristic
28#16839814C2212yes—
29#16839813—21no—
30#16839812—21no—
31#16839786C2211yescharacteristic
32#16839785—21no—
33#16839784—21no—
34#16839784C2212yes—
35C4⋊C8×D61—21no—
36C4⋊C8×D61C2212yes—
37C4⋊C8×D61C2212yes—
38C22⋊C8×D61—21no—
39C4×C8×D61—21no—
40#16839511C2214yescomplemented
41#16839509C2214yescomplemented
42#16839507C2214yescomplemented
43C2×C4⋊Q8×C61C2211yescharacteristic
44C61×#431C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C61×#333C2214yescomplemented
47C22×Dic122—21no—
48C22×C4×D61C2×C411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D411yescharacteristic
51C2×#11758194—21no—
52C2×#11758189—22no—
53M16×D61—44no—
54C2×C8×D61—22no—
55C2×C8×D61—22no—
56#14519777—21no—
57#14519777—21no—
58#14519777C2×C412yes—
59#14519777C2312yes—
60#14519768—22no—
61#14519768—22no—
62#14519768—22no—
63#14519768—22no—
64#14519757—22no—
65#14519757—22no—
66#14519757—22no—
67#14519757—22no—
68#14519753—22no—
69#14519753—22no—
70#14519753C2×C414yes—
71#14519753C2×C414yes—
72Q8×Dic61—22no—
73Q8×Dic61—22no—
74Q8×Dic61C2×C414yes—
75Q8×Dic61C2×C414yes—
76#14519742—22no—
77#14519742—22no—
78#14519742—22no—
79#14519742—22no—
80#14519735—22no—
81#14519735—22no—
82#14519735—22no—
83#14519735—22no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—22no—
86C2×C4⋊Dic61C2×C412yes—
87C2×#11758196—22no—
88C2×#11758198—22no—
89C2×C4×Dic61—22no—
90C2×C4×Dic61C2×C411yescharacteristic
91C2×C4×Dic61D411yescharacteristic
92C2×C4×Dic61D411yescharacteristic
93C42×D61C2×C411yescharacteristic
94C42×D61C2×C411yescharacteristic
95C42×D61D411yescharacteristic
96C42×D61D411yescharacteristic
97C42×D61D411yescharacteristic
98C42×D61D411yescharacteristic
99C42×D61C2311yescharacteristic
100C42×D61C2311yescharacteristic
101C4⋊#92418—21no—
102C4⋊#92418C2312yes—
103C4⋊#92418C2312yes—
104#14519706—22no—
105#14519706—22no—
106#14519706—22no—
107C22⋊#92418—21no—
108#14519689—22no—
109C4×#92418—21no—
110C8×Dic61—22no—
111C22×Q8×C61C2×C412yes—
112C2×M16×C61—21no—
113C4⋊Q8×C61—21no—
114C4⋊Q8×C61—21no—
115C4⋊Q8×C61C2×C412yes—
116C4⋊Q8×C61C2312yes—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61—21no—
119C2×C42×C61C2311yescharacteristic
120C4⋊C8×C61—21no—
121C4⋊C8×C61C2312yes—
122C4⋊C8×C61C2312yes—
123C22⋊C8×C61—21no—
124C4×C8×C61—21no—
125C23×D61C22⋊C411yescharacteristic
126C2×Q8×C61—22no—
127C2×Q8×C61—44no—
128C2×Q8×C61C22×C414yes—
129C22×C4×C61—22no—
130C22×C4×C61C22×C411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×Dic61—22no—
134C22×Dic61C22⋊C411yescharacteristic
135C22×Dic61C22⋊C411yescharacteristic
136C22×Dic61C22⋊C411yescharacteristic
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×C4×D61C22⋊C411yescharacteristic
140C2×C4×D61C22⋊C411yescharacteristic
141C2×C4×D61C22⋊C411yescharacteristic
142C2×C4×D61C22⋊C411yescharacteristic
143C2×C4×D61C22⋊C411yescharacteristic
144C2×C4×D61C22⋊C411yescharacteristic
145C2×C4×D61SD1614yes—
146C2×C4×D61Q1614yes—
147C2×C4×D61C22×C411yescharacteristic
148C2×C4×D61C22×C411yescharacteristic
149C2×C4×D61C2×D411yescharacteristic
150C2×C4×D61C2×D411yescharacteristic
151C2×C4×D61C2×D411yescharacteristic
152C2×C4×D61C2×D411yescharacteristic
153C2×Dic122—22no—
154C2×Dic122—22no—
155C2×Dic122—44no—
156M16×C61—42no—
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C42×C61C22×C411yescharacteristic
164C42×C61C2×D411yescharacteristic
165C42×C61C2×D411yescharacteristic
166C42×C61C2411yescharacteristic
167#11758198—24no—
168#11758198—24no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—24no—
173C4⋊Dic61—24no—
174C4⋊Dic61C22×C414yes—
175#11758196—24no—
176#11758196—24no—
177C4×Dic61—24no—
178C4×Dic61—24no—
179C4×Dic61C22⋊C411yescharacteristic
180C4×Dic61C22⋊C411yescharacteristic
181C4×Dic61C22⋊C411yescharacteristic
182C4×Dic61C22⋊C411yescharacteristic
183C4×Dic61C22⋊C411yescharacteristic
184C4×Dic61C22⋊C411yescharacteristic
185C4×Dic61C22×C411yescharacteristic
186C4×Dic61C22×C411yescharacteristic
187C4×Dic61C2×D411yescharacteristic
188C4×Dic61C2×D411yescharacteristic
189C4×Dic61C2×D411yescharacteristic
190C4×Dic61C2×D411yescharacteristic
191#11758194—42no—
192C2×#92418—22no—
193C2×#92418—22no—
194#11758189—44no—
195C8×D61—44no—
196C8×D61—44no—
197C23×C61C2×C22⋊C411yescharacteristic
198C22×D61—41no—
199C22×D61—41no—
200C22×D61#10314yes—
201C22×D61C2×C22⋊C411yescharacteristic
202C22×D61C2×C22⋊C411yescharacteristic
203Q8×C61—44no—
204Q8×C61—44no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—24no—
208C2×C4×C61—44no—
209C2×C4×C61C2×C22⋊C411yescharacteristic
210C2×C4×C61C2×C22⋊C411yescharacteristic
211C2×C4×C61C2×C22⋊C411yescharacteristic
212C2×C4×C61C2×SD1612yes—
213C2×C4×C61C2×Q1612yes—
214C2×C4×C61C23×C411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×Dic61—24no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—44no—
221C2×Dic61#10314yes—
222C2×Dic61C2×C22⋊C411yescharacteristic
223C2×Dic61C2×C22⋊C411yescharacteristic
224C2×Dic61C2×C22⋊C411yescharacteristic
225C2×Dic61C2×C22⋊C411yescharacteristic
226C2×Dic61C2×C22⋊C411yescharacteristic
227C2×Dic61C2×C22⋊C411yescharacteristic
228C4×D61—22no—
229C4×D61—22no—
230C4×D61—22no—
231C4×D61—22no—
232C4×D61—41no—
233C4×D61—41no—
234C4×D61—41no—
235C4×D61—41no—
236C4×D61#10314yes—
237C4×D61#10314yes—
238C4×D61C2×SD1612yes—
239C4×D61C2×SD1612yes—
240C4×D61C2×Q1612yes—
241C4×D61C2×Q1612yes—
242Dic122—44no—
243Dic122—44no—
244C8×C61—42no—
245C8×C61—42no—
246#92418—42no—
247#92418—42no—
248#17708—611nomaximal, Sylow
249C22×C61—21no—
250C22×C61—21no—
251C22×C61C2×#10312yes—
252C22×C61C22×C22⋊C411yescharacteristic
253C2×D61—21no—
254C2×D61—21no—
255C2×D61—21no—
256C2×D61—21no—
257C2×D61—44no—
258C2×D61C2×#10312yes—
259C2×D61C2×#10312yes—
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—44no—
263C4×C61#41312yes—
264C4×C61C2×#10312yes—
265C4×C61C22×SD1611yescharacteristic
266C4×C61C22×Q1611yescharacteristic
267C4×C61#57512yes—
268Dic61—21no—
269Dic61—21no—
270Dic61—21no—
271Dic61—21no—
272Dic61—44no—
273Dic61C2×#10312yes—
274Dic61C2×#10312yes—
275C2×#533—611no—
276C2×#431—611no—
277C2×#423—611no—
278#1550—614no—
279#1549—611no—
280#1549—611no—
281#1548—611no—
282#1548D6111yescharacteristic, complemented
283#1542—614no—
284C2×C61—22no—
285C2×C61C2×#41311yescharacteristic
286C2×C61C22×#10311yescharacteristic
287C2×C61#289611yescharacteristic
288D61—22no—
289D61—22no—
290C22×M16—1221no—
291#533—612no—
292#533—612no—
293#533—612no—
294#533—612no—
295#533—1221no—
296#533—1221no—
297#533—1221no—
298#533—1221no—
299C2×C4⋊Q8—611no—
300C2×C4⋊Q8C2×D6111yescharacteristic
301C2×#125—1221no—
302C2×C22⋊Q8—612no—
303C2×C22⋊Q8—1221no—
304C2×C4×Q8—612no—
305C2×C4×Q8—1221no—
306C2×#117—1221no—
307C22×C42—611no—
308#431—611no—
309#431—612no—
310#431—1221no—
311#431—1221no—
312#431C2×D6111yescharacteristic
313#423—611no—
314#423—612no—
315#423—1221no—
316#423—1221no—
317#423C2×D6111yescharacteristic
318C2×C4⋊C8—612no—
319C2×C4⋊C8—612no—
320C2×C4⋊C8—1221no—
321C2×C22⋊C8—1221no—
322C2×C4×C8—1221no—
323#333—614no—
324#333C2×D6114yescomplemented
325#332—614no—
326#332—614no—
327C61#1770811yescharacteristic, complemented, Sylow
328C22×Q8—1221no—
329C22×Q8C4×D6112yes—
330C23×C4—611no—
331C23×C4—611no—
332C23×C4—611no—
333C2×M16—21no—
334C2×M16—1222no—
335C2×M16—1221no—
336C2×M16—2444no—
337C22×C8—1222no—
338C22×C8—1222no—
339C4⋊Q8—21no—
340C4⋊Q8—21no—
341C4⋊Q8—612no—
342C4⋊Q8—612no—
343C4⋊Q8—1221no—
344C4⋊Q8—1221no—
345C4⋊Q8C4×D6112yes—
346C4⋊Q8C22×D6112yes—
347#125—1222no—
348#125—1222no—
349#125—1222no—
350#125—1222no—
351C22⋊Q8—614no—
352C22⋊Q8—614no—
353C22⋊Q8—1222no—
354C22⋊Q8—1222no—
355C22⋊Q8—1222no—
356C22⋊Q8—1222no—
357C22⋊Q8—1222no—
358C22⋊Q8—1222no—
359C4×Q8—614no—
360C4×Q8—614no—
361C4×Q8—1222no—
362C4×Q8—1222no—
363C4×Q8—1222no—
364C4×Q8—1222no—
365C4×Q8—1222no—
366C4×Q8—1222no—
367#117—1222no—
368#117—1222no—
369#117—1222no—
370#117—1222no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—612no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1221no—
377C2×C22⋊C4—1222no—
378C2×C42—611no—
379C2×C42—611no—
380C2×C42—611no—
381C2×C42—611no—
382C2×C42—611no—
383C2×C42—611no—
384C2×C42—611no—
385C2×C42—611no—
386C2×C42—611no—
387C2×C42—611no—
388C2×C42—611no—
389C2×C42—1222no—
390C2×C42C22×D6111yescharacteristic
391C4⋊C8—21no—
392C4⋊C8—612no—
393C4⋊C8—612no—
394C4⋊C8—1222no—
395C4⋊C8—1222no—
396C4⋊C8—1222no—
397C4⋊C8—1221no—
398C4⋊C8C22×D6112yes—
399C4⋊C8C22×D6112yes—
400C22⋊C8—21no—
401C22⋊C8—1222no—
402C22⋊C8—1221no—
403C4×C8—21no—
404C4×C8—1222no—
405C4×C8—1221no—
406C24—611no—
407C2×Q8—22no—
408C2×Q8—44no—
409C2×Q8—1222no—
410C2×Q8—1222no—
411C2×Q8—2444no—
412C2×Q8C2×C4×D6114yes—
413C22×C4—22no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—611no—
417C22×C4—611no—
418C22×C4—611no—
419C22×C4—614no—
420C22×C4—614no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—1222no—
432C22×C4—2442no—
433C22×C4—2442no—
434C22×C4C2×C4×D6111yescharacteristic
435C22×C4D4×D6111yescharacteristic
436C22×C4D4×D6111yescharacteristic
437M16—42no—
438M16—2444no—
439M16—2442no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—1222no—
443C2×C8—1222no—
444C2×C8—2444no—
445C2×C8—2444no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—614no—
451C4⋊C4—1224no—
452C4⋊C4—1224no—
453C4⋊C4—1224no—
454C4⋊C4—1224no—
455C4⋊C4—1222no—
456C4⋊C4—1222no—
457C4⋊C4—1222no—
458C22⋊C4—1224no—
459C22⋊C4—1224no—
460C42—611no—
461C42—611no—
462C42—611no—
463C42—611no—
464C42—611no—
465C42—611no—
466C42—611no—
467C42—611no—
468C42—611no—
469C42—611no—
470C42—611no—
471C42—611no—
472C42—1224no—
473C42—1224no—
474C42C2×C4×D6111yescharacteristic
475C42D4×D6111yescharacteristic
476C42D4×D6111yescharacteristic
477C42C23×D6111yescharacteristic
478C23—611no—
479C23—611no—
480C23—614no—
481C23—2441no—
482C23—2441no—
483C23C22⋊C4×D6111yescharacteristic
484Q8—44no—
485Q8—44no—
486Q8—2444no—
487Q8—2444no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—44no—
492C2×C4—611no—
493C2×C4—611no—
494C2×C4—614no—
495C2×C4—614no—
496C2×C4—614no—
497C2×C4—612no—
498C2×C4—612no—
499C2×C4—612no—
500C2×C4—612no—
501C2×C4—611no—
502C2×C4—611no—
503C2×C4—611no—
504C2×C4—611no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—1224no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2444no—
517C2×C4C22⋊C4×D6111yescharacteristic
518C2×C4C22⋊C4×D6111yescharacteristic
519C2×C4C22⋊C4×D6111yescharacteristic
520C2×C4SD16×D6112yes—
521C2×C4Q16×D6112yes—
522C2×C4C22×C4×D6111yescharacteristic
523C2×C4C2×D4×D6111yescharacteristic
524C2×C4C2×D4×D6111yescharacteristic
525C8—42no—
526C8—42no—
527C8—2442no—
528C8—2442no—
529C22—21no—
530C22—21no—
531C22—612no—
532C22—612no—
533C22—1221no—
534C22—1221no—
535C22—1221no—
536C22—1221no—
537C22—2444no—
538C22#103×D6112yes—
539C22C2×C22⋊C4×D6111yescharacteristic
540C4—21no—
541C4—21no—
542C4—44no—
543C4—612no—
544C4—612no—
545C4—1221no—
546C4—1221no—
547C4—1221no—
548C4—1221no—
549C4—2444no—
550C4#1683974312yes—
551C4#103×D6112yes—
552C4C2×SD16×D6111yescharacteristic
553C4C2×Q16×D6111yescharacteristic
554C4#1684056112yes—
555C2—22no—
556C2—1222no—
557C2—1222no—
558C2C2×#1683974311yescharacteristic
559C2C2×#103×D6111yescharacteristic
560C2#2145433711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1548C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC42×C61C24
hypercenter#1548D61
hypocenterC61#17708
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
4. #21444506C2semidirect
5. #21444505C2semidirect
6. #21444503C2semidirect
7. #21444500C2semidirect
8. #21444473C2semidirect
9. C61×#1548C2semidirect
40. #16839511C22semidirect
41. #16839509C22semidirect
42. #16839507C22semidirect
46. C61×#333C22semidirect
282. #1548D61semidirect
324. #333C2×D61semidirect
327. C61#17708semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
#431×D612
#423×D612
#214445062
#214445052
#214445032
#214445002
#214444732
C61×#15482
#1770861