Finite Group Databasebeta

← #29681981

Subgroups — #29681981

Order 15616.

1654 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
1C22×#14519808C211yesmaximal, characteristic
2C2×#16840528C211yesmaximal, characteristic
3#21454336C211yesmaximal, characteristic
4C22×#14519696C211yesmaximal, characteristic, complemented
5#21446337C218yesmaximal, complemented
6#21445558C211yesmaximal, characteristic, complemented
7#21445546C211yesmaximal, characteristic, complemented
8C2×C61×#419C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C2×#14519808—22no—
11C2×#14519808—24no—
12C2×#14519808C418yescomplemented
13C2×#14519808C2214yes—
14C22×#11758202—21no—
15C22×#11758201—21no—
16#16840528—24no—
17#16840528—24no—
18C2×#14519785—22no—
19C2×#14519783C2214yes—
20C22×#11758194C2211yescharacteristic
21C22×#11758194C2211yescharacteristic
22C22×#11758194C2211yescharacteristic
23C2×M16×D61—22no—
24#16839770—21no—
25#16839766—21no—
26#16839761C2214yes—
27#16839756—22no—
28C2×#14519696—24no—
29C2×#14519696C2214yescomplemented
30#16839531C2214yescomplemented
31#16839527C2214yescomplemented
32C23×Q8×C61C2211yescharacteristic
33C22×M16×C61C2211yescharacteristic
34C2×#117×C61C2211yescharacteristic
35C61×#419—22no—
36C61×#419—22no—
37C2×#103×C61C2214yescomplemented
38C2×#11758222—21no—
39C2×#11758222—22no—
40C2×#11758222—22no—
41C2×#11758222—21no—
42C2×#11758222C2×C414yes—
43C2×#11758222D414yes—
44C2×#11758222D412yes—
45C2×#11758222C2312yes—
46C22×Dic122D411yescharacteristic
47C22×D244D411yescharacteristic
48C22×C22⋊D61—21no—
49C22×C4×D61—21no—
50#14519808—216no—
51#14519808—28no—
52#14519808C2×C4116yes—
53C2×#11758202—28no—
54C2×#11758202—24no—
55C2×#11758201—28no—
56C2×#11758201—24no—
57#14519785—216no—
58#14519783—28no—
59#14519783—28no—
60C2×#11758194—21no—
61C2×#11758194—22no—
62C2×#11758194—22no—
63C2×#11758194—21no—
64C2×#11758194—21no—
65C2×#11758194—22no—
66C2×#11758194—22no—
67C2×#11758194C2×C414yes—
68C2×#11758194D414yes—
69C2×#11758194C2312yes—
70C2×#11758194C2312yes—
71C2×#11758194C2312yes—
72C2×#11758189—24no—
73M16×D61—216no—
74C22×#92418—21no—
75C22×#92418—21no—
76C22×#92418—21no—
77C2×C8×D61—24no—
78#14519700—24no—
79#14519697—24no—
80#14519696—28no—
81C22×Q8×C61—22no—
82C22×Q8×C61—21no—
83C22×Q8×C61—21no—
84C22×Q8×C61—21no—
85C22×Q8×C61—22no—
86C22×Q8×C61—21no—
87C22×Q8×C61—21no—
88C22×Q8×C61—22no—
89C22×Q8×C61C2×C414yes—
90C22×Q8×C61C2312yes—
91C23×C4×C61—21no—
92C23×C4×C61C2311yescharacteristic
93C2×M16×C61—22no—
94C2×M16×C61—22no—
95C2×M16×C61D412yes—
96C2×M16×C61D412yes—
97C22×C8×C61C2312yes—
98#117×C61—21no—
99#117×C61—21no—
100#117×C61—21no—
101#117×C61—21no—
102C2×C4⋊C4×C61C2312yes—
103C2×C22⋊C4×C61—21no—
104C2×C42×C61—21no—
105#103×C61—28no—
106C24×C61#4111yescharacteristic
107C23×D61—21no—
108C2×Q8×C61—24no—
109C2×Q8×C61—22no—
110C2×Q8×C61—24no—
111C2×Q8×C61—24no—
112C2×Q8×C61—24no—
113C2×Q8×C61—24no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—22no—
117C2×Q8×C61—22no—
118C2×Q8×C61—42no—
119C2×Q8×C61—42no—
120C2×Q8×C61—41no—
121C2×Q8×C61—41no—
122C2×Q8×C61—42no—
123C2×Q8×C61—42no—
124C2×Q8×C61—42no—
125C2×Q8×C61C22×C414yes—
126C22×C4×C61—24no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—41no—
130C22×C4×C61—41no—
131C22×C4×C61—42no—
132C22×C4×C61C22×C412yes—
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D412yes—
139C22×C4×C61#4111yescharacteristic
140C22×C4×C61#4112yes—
141C22×C4×C61C2411yescharacteristic
142C2×C22⋊D61—24no—
143C2×C22⋊D61—22no—
144C2×C22⋊D61—24no—
145C2×C22⋊D61—22no—
146C22×Dic61—21no—
147#11758222—24no—
148#11758222—24no—
149#11758222—24no—
150#11758222—24no—
151#11758222—24no—
152#11758222—22no—
153#11758222—22no—
154#11758222C22⋊C418yes—
155#11758222C22×C414yes—
156#11758222C2×D414yes—
157C2×D244—22no—
158C2×D244—21no—
159C2×D244C22⋊C414yes—
160C2×D244C2×D412yes—
161C2×C4×D61—24no—
162C2×C4×D61—24no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×Dic122—22no—
166C2×Dic122—21no—
167C2×Dic122C22⋊C414yes—
168C2×Dic122C2×D412yes—
169M16×C61—24no—
170M16×C61—24no—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C2×C8×C61C2×D414yes—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C22⋊C4×C61—41no—
177C22⋊C4×C61—41no—
178C42×C61—41no—
179C42×C61—41no—
180#11758202—216no—
181#11758201—216no—
182#11758194—24no—
183#11758194—24no—
184#11758194—22no—
185#11758194—22no—
186#11758194—22no—
187#11758194—22no—
188#11758194—42no—
189#11758194C22×C414yes—
190#11758194C2×D414yes—
191C2×#92418—24no—
192C2×#92418—24no—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418—22no—
196C2×#92418—42no—
197#11758189—216no—
198C8×D61—216no—
199C23×C61—42no—
200C23×C61C4×D412yes—
201C23×C61#12111yescharacteristic
202C23×C61#12111yescharacteristic
203C23×C61#12312yes—
204C23×C61C2×#4111yescharacteristic
205C22×D61—22no—
206C22×D61—24no—
207C22×D61—82no—
208Q8×C61—24no—
209Q8×C61—24no—
210Q8×C61—42no—
211Q8×C61—42no—
212Q8×C61—44no—
213Q8×C61—44no—
214C2×C4×C61—24no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—44no—
218C2×C4×C61—44no—
219C2×C4×C61—42no—
220C2×C4×C61—82no—
221C2×C4×C61C2×C22⋊C412yes—
222C2×C4×C61C2×C22⋊C412yes—
223C2×C4×C61C4×D412yes—
224C2×C4×C61C4×D412yes—
225C2×C4×C61C4×D412yes—
226C2×C4×C61#12012yes—
227C2×C4×C61#12012yes—
228C2×C4×C61#12112yes—
229C2×C4×C61#12112yes—
230C2×C4×C61#12111yescharacteristic
231C2×C4×C61#12111yescharacteristic
232C2×C4×C61#12312yes—
233C2×C4×C61C23×C411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C22⋊D61—24no—
240C22⋊D61—24no—
241C22⋊D61—28no—
242C2×Dic61—22no—
243C2×Dic61—24no—
244D244—22no—
245D244—24no—
246D244C2×C22⋊C414yes—
247C4×D61—24no—
248C4×D61—24no—
249C4×D61—28no—
250Dic122—22no—
251Dic122—24no—
252Dic122C2×C22⋊C414yes—
253C8×C61—24no—
254#92418—24no—
255#92418—24no—
256#92418—42no—
257#21006—611nomaximal, Sylow
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—22no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61#38612yes—
264C22×C61#38612yes—
265C22×C61C2×C4×D411yescharacteristic
266C22×C61C2×#12111yescharacteristic
267C22×C61C2×#12311yescharacteristic
268C2×D61—24no—
269C2×D61—44no—
270C2×D61—84no—
271C4×C61—44no—
272C4×C61—82no—
273C4×C61#38612yes—
274C4×C61#38612yes—
275C4×C61C22×C22⋊C411yescharacteristic
276C4×C61C2×C4×D411yescharacteristic
277C4×C61C2×#12011yescharacteristic
278C4×C61C2×#12111yescharacteristic
279Dic61—24no—
280C22×#137—611no—
281C2×#568—611no—
282#2896—611no—
283C2×#419D6111yescharacteristic, complemented
284C22×#101—611no—
285#1908—618no—
286#1742—611no—
287#1738—611no—
288C2×C61—22no—
289C2×C61—21no—
290C2×C61—21no—
291C2×C61—22no—
292C2×C61C2×#38611yescharacteristic
293C2×C61#306912yes—
294D61—44no—
295C22×#41—611no—
296C23×Q8C2×D6111yescharacteristic
297C2×#137—618no—
298C2×#137—614no—
299C2×#137—1224no—
300C2×#137—1222no—
301C22×Q16—1221no—
302C22×SD16—1221no—
303#568—1224no—
304#568—1224no—
305C2×#131—614no—
306C2×#131—1222no—
307C22×M16—611no—
308C22×M16—611no—
309C22×M16—611no—
310C22×M16—1222no—
311C22×M16C2×D6111yescharacteristic
312C2×#117C2×D6111yescharacteristic
313#419—22no—
314#419—22no—
315#419—1221no—
316#418—1221no—
317#417—614no—
318#416—1222no—
319C2×#103C2×D6114yescomplemented
320C2×#101—614no—
321C2×#101—1224no—
322#343—614no—
323#341—614no—
324C61#2100611yescharacteristic, complemented, Sylow
325C2×#41—614no—
326C2×#41—614no—
327C2×#41—612no—
328C2×#41—612no—
329C2×#41—1222no—
330C2×#41—1222no—
331C2×#41—1221no—
332C2×#41—1221no—
333C22×Q8—22no—
334C22×Q8—21no—
335C22×Q8—22no—
336C22×Q8—21no—
337C22×Q8—21no—
338C22×Q8—22no—
339C22×Q8—21no—
340C22×Q8—21no—
341C22×Q8—611no—
342C22×Q8C4×D6114yes—
343C22×Q8C22×D6112yes—
344C22×D4—611no—
345C22×D4—1221no—
346C23×C4—21no—
347C23×C4—1221no—
348C23×C4C22×D6111yescharacteristic
349#137—6116no—
350#137—1228no—
351#137—12216no—
352C2×Q16—1224no—
353C2×Q16—1228no—
354C2×SD16—1224no—
355C2×SD16—1228no—
356#131—1228no—
357#131—1228no—
358#131—12216no—
359C2×M16—22no—
360C2×M16—22no—
361C2×M16—614no—
362C2×M16—614no—
363C2×M16—612no—
364C2×M16—612no—
365C2×M16—612no—
366C2×M16—12216no—
367C2×M16—1222no—
368C2×M16—1221no—
369C2×M16—1222no—
370C2×M16—1221no—
371C2×M16—1221no—
372C2×M16—1224no—
373C2×M16—1222no—
374C2×M16—1222no—
375C2×M16C22⋊D6112yes—
376C2×M16C22⋊D6112yes—
377C22×C8—1221no—
378C22×C8—1221no—
379C22×C8—1224no—
380C22×C8—1221no—
381C22×C8C22×D6112yes—
382#117—21no—
383#117—21no—
384#117—21no—
385#117—21no—
386C2×C4⋊C4C22×D6112yes—
387C2×C22⋊C4—21no—
388C2×C42—21no—
389#103—28no—
390#103—1224no—
391#102—1224no—
392#101—1228no—
393C24—1221no—
394C24#1175822411yescharacteristic
395#41—614no—
396#41—614no—
397#41—618no—
398#41—1222no—
399#41—1222no—
400#41—1224no—
401#41—1224no—
402#41—1224no—
403#41—1224no—
404#41—1224no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—41no—
418C2×Q8—41no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—612no—
423C2×Q8—614no—
424C2×Q8—1221no—
425C2×Q8—1222no—
426C2×Q8C2×C4×D6114yes—
427C2×D4—612no—
428C2×D4—614no—
429C2×D4—1222no—
430C2×D4—1224no—
431C2×D4—1222no—
432C2×D4—1224no—
433C2×D4—1221no—
434C2×D4—1222no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—1221no—
442C22×C4—1222no—
443C22×C4—1224no—
444C22×C4—1224no—
445C22×C4—1222no—
446C22×C4C2×C4×D6112yes—
447C22×C4#1175822212yes—
448C22×C4D4×D6112yes—
449C22×C4D4×D6111yescharacteristic
450C22×C4D4×D6111yescharacteristic
451C22×C4D4×D6112yes—
452C22×C4#1175822411yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C2×C22⋊D6111yescharacteristic
455C22×C4C23×D6111yescharacteristic
456Q16—12216no—
457SD16—12216no—
458M16—24no—
459M16—24no—
460M16—614no—
461M16—614no—
462M16—12216no—
463M16—1222no—
464M16—1222no—
465M16—1224no—
466M16—1224no—
467M16—1222no—
468M16—1222no—
469M16—2442no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—12216no—
473C2×C8—1222no—
474C2×C8—1222no—
475C2×C8—1224no—
476C2×C8—1224no—
477C2×C8—1222no—
478C2×C8—2442no—
479C2×C8C2×C22⋊D6114yes—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C42—41no—
485C42—41no—
486C23—42no—
487C23—1224no—
488C23—1222no—
489C23—4882no—
490C23#1451973712yes—
491C23#1451974811yescharacteristic
492C23#1451974811yescharacteristic
493C23#1451975912yes—
494C23C2×#1175822411yescharacteristic
495Q8—24no—
496Q8—24no—
497Q8—42no—
498Q8—42no—
499Q8—44no—
500Q8—44no—
501Q8—614no—
502Q8—1222no—
503Q8—1224no—
504D4—614no—
505D4—1224no—
506D4—1224no—
507D4—1228no—
508D4—1222no—
509D4—1224no—
510C2×C4—24no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—82no—
517C2×C4—1222no—
518C2×C4—1224no—
519C2×C4—1224no—
520C2×C4—1224no—
521C2×C4—1228no—
522C2×C4C22⋊C4×D6112yes—
523C2×C4C22⋊C4×D6112yes—
524C2×C4#1451973712yes—
525C2×C4C4×C22⋊D6112yes—
526C2×C4C4×C22⋊D6112yes—
527C2×C4#1451974512yes—
528C2×C4#1451974512yes—
529C2×C4#1451974811yescharacteristic
530C2×C4#1451974811yescharacteristic
531C2×C4#1451975012yes—
532C2×C4#1451975012yes—
533C2×C4#1451975912yes—
534C2×C4C22×C4×D6111yescharacteristic
535C2×C4C2×D4×D6111yescharacteristic
536C2×C4C2×D4×D6111yescharacteristic
537C2×C4C2×D4×D6111yescharacteristic
538C2×C4C22×C22⋊D6111yescharacteristic
539C2×C4C2×#1175822211yescharacteristic
540C8—24no—
541C8—1224no—
542C8—1224no—
543C8—2442no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—42no—
548C22—42no—
549C22—1224no—
550C22—4884no—
551C22—2444no—
552C22#1683963812yes—
553C22#1683963812yes—
554C22C2×#1451973711yescharacteristic
555C22C2×#1451974811yescharacteristic
556C22C2×#1451975911yescharacteristic
557C4—44no—
558C4—82no—
559C4—1224no—
560C4#1683963812yes—
561C4#1683963812yes—
562C4C2×C22⋊C4×D6111yescharacteristic
563C4C2×C4×C22⋊D6111yescharacteristic
564C4C2×#1451974511yescharacteristic
565C4C2×#1451975011yescharacteristic
566C2—22no—
567C2—21no—
568C2—21no—
569C2—22no—
570C2—2444no—
571C2C2×#1683963811yescharacteristic
572C2#2145555212yes—
subgroupG/H
centerC22C2×#14519748
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#419C2
socleC22×C61C2×#121
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenterC2×#419D61
hypocenterC61#21006
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
4. C22×#14519696C2semidirect
5. #21446337C2semidirect
6. #21445558C2semidirect
7. #21445546C2semidirect
8. C2×C61×#419C2semidirect
12. C2×#14519808C4semidirect
29. C2×#14519696C22semidirect
30. #16839531C22semidirect
31. #16839527C22semidirect
37. C2×#103×C61C22semidirect
283. C2×#419D61semidirect
319. C2×#103C2×D61semidirect
324. C61#21006semidirect

Maximal subgroups

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

subgroupindex
C22×#145198082
C2×#168405282
#214543362
C22×#145196962
#214463372
#214455582
#214455462
C2×C61×#4192
#2100661