Finite Group Databasebeta

← #29681382

Subgroups — #29681382

Order 15616.

1802 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519808C211yesmaximal, characteristic
2C2×#16839761C211yesmaximal, characteristic
3C2×#16839771C211yesmaximal, characteristic, complemented
4#21448309C211yesmaximal, characteristic
5#21446174C214yesmaximal, complemented
6#21446160C214yesmaximal, complemented
7#21445538C211yesmaximal, characteristic, complemented
8#21445513C211yesmaximal, characteristic, complemented
9C22×C4×Q8×C61C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C2×#14519808—24no—
12C2×#14519808—24no—
13C2×#14519808C418yescomplemented
14C22×#11758202C2212yes—
15C22×#11758201C2212yes—
16C22×#11758194C2211yescharacteristic
17C22×#11758194C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19#16839771—24no—
20#16839761—24no—
21C2×#14519701C2214yes—
22C2×#14519700C2212yes—
23C2×#14519697C2212yes—
24#16839668C2212yes—
25#16839664C2212yes—
26#16839641—21no—
27#16839641—21no—
28C4×#11758198—22no—
29#16839524C2214yes—
30#16839517C2214yes—
31C23×Q8×C61C2211yescharacteristic
32C2×C4×Q8×C61—24no—
33C2×C4×Q8×C61—24no—
34C2×C4×Q8×C61—22no—
35C2×C4×Q8×C61C2214yes—
36C22×C4⋊C4×C61—21no—
37C22×C4⋊C4×C61C2211yescharacteristic
38C22×C42×C61—21no—
39C22×C42×C61C2211yescharacteristic
40C2×#11758222—22no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222C2×C414yes—
46C22×Dic122C2311yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C22⋊D61—21no—
49C22×C4×D61—21no—
50#14519808—232no—
51C2×#11758202—24no—
52C2×#11758202—24no—
53C2×#11758202C2×C418yescomplemented
54C2×#11758201—24no—
55C2×#11758201—24no—
56C2×#11758201C2×C418yescomplemented
57C2×#11758194—22no—
58C2×#11758194—22no—
59C2×#11758194—22no—
60C2×#11758194—22no—
61C2×#11758194—22no—
62C2×#11758194C2×C414yes—
63C22×#92418C2312yes—
64C22×#92418C2312yes—
65#14519736—22no—
66#14519736—22no—
67C2×C4⋊Dic61C2312yes—
68C2×C22⋊Dic61—21no—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C2×C4×Dic61—21no—
72#14519701—28no—
73#14519700—24no—
74#14519697—24no—
75#14519684—22no—
76#14519684—22no—
77C22×Q8×C61—22no—
78C22×Q8×C61—21no—
79C22×Q8×C61—22no—
80C22×Q8×C61—22no—
81C22×Q8×C61C2×C414yes—
82C22×Q8×C61D414yes—
83C22×Q8×C61D414yes—
84C22×Q8×C61C2312yes—
85C23×C4×C61—21no—
86C23×C4×C61—21no—
87C23×C4×C61D411yescharacteristic
88C23×C4×C61D411yescharacteristic
89C23×C4×C61C2311yescharacteristic
90C4×Q8×C61—216no—
91C4×Q8×C61—28no—
92C4×Q8×C61—28no—
93C2×C4⋊C4×C61—28no—
94C2×C4⋊C4×C61—24no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61D414yes—
98C2×C4⋊C4×C61C2312yes—
99C2×C42×C61—28no—
100C2×C42×C61—24no—
101C2×C42×C61—22no—
102C2×C42×C61—22no—
103C2×C42×C61D412yes—
104C2×C42×C61C2312yes—
105C24×C61C2×D411yescharacteristic
106C23×D61—21no—
107C2×Q8×C61—24no—
108C2×Q8×C61—28no—
109C2×Q8×C61—24no—
110C2×Q8×C61—24no—
111C2×Q8×C61—28no—
112C2×Q8×C61—28no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61C22⋊C4116yescomplemented
117C2×Q8×C61C22×C414yes—
118C2×Q8×C61C2×D414yes—
119C2×Q8×C61C2×D414yes—
120C22×C4×C61—24no—
121C22×C4×C61—22no—
122C22×C4×C61—24no—
123C22×C4×C61—24no—
124C22×C4×C61—22no—
125C22×C4×C61—24no—
126C22×C4×C61—22no—
127C22×C4×C61—24no—
128C22×C4×C61—22no—
129C22×C4×C61—42no—
130C22×C4×C61—44no—
131C22×C4×C61C22×C412yes—
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D412yes—
138C22×C4×C61#4112yes—
139C22×C4×C61#4112yes—
140C22×C4×C61C2411yescharacteristic
141C2×C22⋊D61—24no—
142C2×C22⋊D61—42no—
143C2×C22⋊D61—42no—
144C22×Dic61—21no—
145C22×Dic61—22no—
146#11758222—28no—
147#11758222—28no—
148#11758222—44no—
149#11758222—44no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244C22×C412yes—
156C2×C4×D61—24no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122C22×C412yes—
165C4⋊C4×C61—216no—
166C4⋊C4×C61—24no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61C2×D414yes—
169C42×C61—216no—
170C42×C61—24no—
171C42×C61—22no—
172C42×C61—22no—
173C22⋊Dic61—42no—
174#11758202—216no—
175#11758201—216no—
176#11758198—44no—
177#11758198—44no—
178C4⋊Dic61—22no—
179C4⋊Dic61—22no—
180C4×Dic61—42no—
181#11758194—28no—
182#11758194—44no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418C22×C414yes—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61C4×D412yes—
192C23×C61#12012yes—
193C23×C61#12112yes—
194C23×C61C22×D411yescharacteristic
195C22×D61—22no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—42no—
199C22×D61—42no—
200Q8×C61—24no—
201Q8×C61—24no—
202Q8×C61—24no—
203Q8×C61—216no—
204Q8×C61C2×C22⋊C418yes—
205C2×C4×C61—28no—
206C2×C4×C61—24no—
207C2×C4×C61—22no—
208C2×C4×C61—24no—
209C2×C4×C61—24no—
210C2×C4×C61—22no—
211C2×C4×C61—216no—
212C2×C4×C61—22no—
213C2×C4×C61—24no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—44no—
217C2×C4×C61—48no—
218C2×C4×C61—44no—
219C2×C4×C61C2×C22⋊C414yes—
220C2×C4×C61C4×D414yes—
221C2×C4×C61C4×D412yes—
222C2×C4×C61#12012yes—
223C2×C4×C61#12114yes—
224C2×C4×C61#12112yes—
225C2×C4×C61#12314yes—
226C2×C4×C61#13714yescomplemented
227C2×C4×C61C23×C411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C22⋊D61—48no—
234C2×Dic61—22no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240D244—22no—
241D244—22no—
242D244—41no—
243D244—41no—
244C4×D61—48no—
245Dic122—22no—
246Dic122—22no—
247Dic122—41no—
248Dic122—41no—
249#92418—24no—
250#92418—42no—
251#20926—611nomaximal, Sylow
252C22×C61—24no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—24no—
256C22×C61#38614yes—
257C22×C61C2×C4×D411yescharacteristic
258C22×C61C2×#12011yescharacteristic
259C22×C61C2×#12111yescharacteristic
260C22×C61C2×#13712yes—
261C22×C61#57512yes—
262C2×D61—44no—
263C2×D61—42no—
264C2×D61—81no—
265C2×D61—81no—
266C2×D61—82no—
267C4×C61—24no—
268C4×C61—28no—
269C4×C61—42no—
270C4×C61—44no—
271C4×C61#38614yes—
272C4×C61C22×C22⋊C411yescharacteristic
273C4×C61C2×C4×D411yescharacteristic
274C4×C61C2×#12111yescharacteristic
275C4×C61C2×#12311yescharacteristic
276Dic61—42no—
277Dic61—82no—
278C22×#137—611no—
279C22×C4×Q8D6111yescharacteristic, complemented
280C2×#417—611no—
281C2×#419—611no—
282#2297—611no—
283#1880—614no—
284#1878—614no—
285#1736—611no—
286#1728—611no—
287C2×C61—24no—
288C2×C61C2×#38611yescharacteristic
289C2×C61C4×#13712yes—
290C2×C61#300411yescharacteristic
291C2×C61#300511yescharacteristic
292C2×C61#303712yes—
293D61—82no—
294C22×#41—611no—
295C23×Q8C2×D6111yescharacteristic
296C2×#137—618no—
297C2×#137—1224no—
298C2×#137—1224no—
299C22×Q16—612no—
300C22×SD16—612no—
301C22×M16—611no—
302C22×M16—611no—
303C2×C4×Q8—24no—
304C2×C4×Q8—24no—
305C2×C4×Q8—22no—
306C2×C4×Q8C2×D6114yes—
307C2×#117—611no—
308C22×C4⋊C4—21no—
309C22×C4⋊C4C2×D6111yescharacteristic
310C22×C42—21no—
311C22×C42C2×D6111yescharacteristic
312#419—1224no—
313#417—1224no—
314C2×#103—614no—
315C2×#103—612no—
316C2×#102—612no—
317#391—612no—
318#390—612no—
319#386—1221no—
320#386—1221no—
321C4×C22⋊C4—1222no—
322#340—614no—
323#336—614no—
324C61#2092611yescharacteristic, complemented, Sylow
325C2×#41—614no—
326C2×#41—1222no—
327C2×#41—1222no—
328C2×#41—1222no—
329C2×#41—1222no—
330C2×#41—1222no—
331C22×Q8—22no—
332C22×Q8—22no—
333C22×Q8—21no—
334C22×Q8—22no—
335C22×Q8—611no—
336C22×Q8C4×D6114yes—
337C22×Q8D24414yes—
338C22×Q8C22⋊D6114yes—
339C22×Q8C22×D6112yes—
340C22×D4—611no—
341C22×D4—1221no—
342C23×C4—21no—
343C23×C4—21no—
344C23×C4—1221no—
345C23×C4C22⋊D6111yescharacteristic
346C23×C4C22⋊D6111yescharacteristic
347C23×C4C22×D6111yescharacteristic
348#137—12232no—
349C2×Q16—618no—
350C2×Q16—1224no—
351C2×Q16—1224no—
352C2×SD16—618no—
353C2×SD16—1224no—
354C2×SD16—1224no—
355C2×M16—614no—
356C2×M16—1222no—
357C2×M16—1222no—
358C2×M16—1222no—
359C2×M16—1222no—
360C2×M16—1222no—
361C22×C8—612no—
362C22×C8—612no—
363C4×Q8—216no—
364C4×Q8—28no—
365C4×Q8—28no—
366#117—1222no—
367#117—1222no—
368C2×C4⋊C4—28no—
369C2×C4⋊C4—24no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—612no—
373C2×C4⋊C4C22⋊D6114yes—
374C2×C4⋊C4C22×D6112yes—
375C2×C22⋊C4—1221no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4—1222no—
378C2×C42—28no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—24no—
382C2×C42—1221no—
383C2×C42C22⋊D6112yes—
384C2×C42C22×D6112yes—
385#103—1228no—
386#103—1224no—
387#102—1224no—
388#95—1222no—
389#95—1222no—
390C24—1221no—
391C24C2×C22⋊D6111yescharacteristic
392#41—1228no—
393#41—1228no—
394#41—2444no—
395#41—2444no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—28no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—28no—
404C2×Q8—28no—
405C2×Q8—612no—
406C2×Q8—1221no—
407C2×Q8—1221no—
408C2×Q8—1221no—
409C2×Q8—1221no—
410C2×Q8—1221no—
411C2×Q8#11758198116yescomplemented
412C2×Q8C2×C4×D6114yes—
413C2×Q8C2×D24414yes—
414C2×Q8C2×C22⋊D6114yes—
415C2×D4—612no—
416C2×D4—1224no—
417C2×D4—1221no—
418C2×D4—1221no—
419C2×D4—1221no—
420C2×D4—1221no—
421C2×D4—1221no—
422C2×D4—2442no—
423C2×D4—2442no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—1221no—
436C22×C4—1224no—
437C22×C4—1222no—
438C22×C4—2442no—
439C22×C4—2442no—
440C22×C4C2×C4×D6112yes—
441C22×C4C2×D24412yes—
442C22×C4#1175822212yes—
443C22×C4#1175822212yes—
444C22×C4C2×C22⋊D6112yes—
445C22×C4C2×C22⋊D6112yes—
446C22×C4C2×C22⋊D6112yes—
447C22×C4C2×C22⋊D6112yes—
448C22×C4C2×C22⋊D6111yescharacteristic
449C22×C4C23×D6111yescharacteristic
450Q16—12216no—
451SD16—12216no—
452M16—1228no—
453M16—2444no—
454C2×C8—614no—
455C2×C8—1222no—
456C2×C8—1222no—
457C2×C8—1222no—
458C2×C8—1222no—
459C2×C8—1222no—
460C4⋊C4—216no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—1222no—
464C4⋊C4—1222no—
465C4⋊C4C2×C22⋊D6114yes—
466C22⋊C4—2444no—
467C22⋊C4—2442no—
468C22⋊C4—2444no—
469C42—216no—
470C42—22no—
471C42—22no—
472C42—24no—
473C42—2442no—
474C23—22no—
475C23—22no—
476C23—1222no—
477C23—2441no—
478C23—2441no—
479C23—2442no—
480C23—2442no—
481C23C4×C22⋊D6112yes—
482C23#1451974612yes—
483C23#1451975112yes—
484C23C22×C22⋊D6111yescharacteristic
485Q8—24no—
486Q8—24no—
487Q8—24no—
488Q8—216no—
489Q8—1222no—
490Q8—1222no—
491Q8—2441no—
492Q8—2441no—
493Q8C2×#1175819818yes—
494D4—1222no—
495D4—1222no—
496D4—2448no—
497D4—2441no—
498D4—2441no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—216no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—44no—
511C2×C4—48no—
512C2×C4—44no—
513C2×C4—1222no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2448no—
520C2×C4C2×#1175819814yes—
521C2×C4C4×C22⋊D6114yes—
522C2×C4C4×C22⋊D6112yes—
523C2×C4#1451974612yes—
524C2×C4#1451975114yes—
525C2×C4#1451975112yes—
526C2×C4#1451976014yes—
527C2×C4#1451980814yescomplemented
528C2×C4C22×C4×D6111yescharacteristic
529C2×C4C22×C22⋊D6111yescharacteristic
530C2×C4C22×C22⋊D6111yescharacteristic
531C2×C4C22×D24411yescharacteristic
532C2×C4C2×#1175822211yescharacteristic
533C2×C4C2×#1175822211yescharacteristic
534C8—1224no—
535C8—2442no—
536C22—24no—
537C22—24no—
538C22—22no—
539C22—22no—
540C22—4882no—
541C22—4881no—
542C22—4881no—
543C22—2442no—
544C22—2444no—
545C22#1683964114yes—
546C22C2×C4×C22⋊D6111yescharacteristic
547C22C2×#1451974611yescharacteristic
548C22C2×#1451975111yescharacteristic
549C22C2×#1451980812yes—
550C22#1684056012yes—
551C4—28no—
552C4—24no—
553C4—42no—
554C4—44no—
555C4—4882no—
556C4—2442no—
557C4#1683964114yes—
558C4C22×#1175819811yescharacteristic
559C4C2×C4×C22⋊D6111yescharacteristic
560C4C2×#1451975111yescharacteristic
561C4C2×#1451976011yescharacteristic
562C2—24no—
563C2—4882no—
564C2C2×#1683964111yescharacteristic
565C2C4×#1451980812yes—
566C2#2145505311yescharacteristic
567C2#2145506111yescharacteristic
568C2#2145529112yes—
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC22×C4×Q8×C61C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenterC22×C4×Q8D61
hypocenterC61#20926
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
3. C2×#16839771C2semidirect
5. #21446174C2semidirect
6. #21446160C2semidirect
7. #21445538C2semidirect
8. #21445513C2semidirect
9. C22×C4×Q8×C61C2semidirect
13. C2×#14519808C4semidirect
53. C2×#11758202C2×C4semidirect
56. C2×#11758201C2×C4semidirect
116. C2×Q8×C61C22⋊C4semidirect
226. C2×C4×C61#137semidirect
279. C22×C4×Q8D61semidirect
324. C61#20926semidirect
411. C2×Q8#11758198semidirect
527. C2×C4#14519808semidirect

Maximal subgroups

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

subgroupindex
C22×#145198082
C2×#168397612
C2×#168397712
#214483092
#214461742
#214461602
#214455382
#214455132
C22×C4×Q8×C612
#2092661