Finite Group Databasebeta

← #29681385

Subgroups — #29681385

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×#14519807C211yesmaximal, characteristic
2C22×C4×Dic122C211yesmaximal, characteristic, complemented
3C2×#16839763C211yesmaximal, characteristic
4C2×#16839770C211yesmaximal, characteristic, complemented
5#21446177C214yesmaximal, complemented
6#21446163C214yesmaximal, complemented
7#21445540C211yesmaximal, characteristic, complemented
8#21445513C211yesmaximal, characteristic, complemented
9C61×#2297C211yesmaximal, characteristic
10C23×Dic122C2211yescharacteristic
11C2×#14519807—24no—
12C2×#14519807—24no—
13C2×#14519807C418yescomplemented
14C22×#11758202C2212yes—
15C22×#11758200C2212yes—
16C22×#11758194C2211yescharacteristic
17C22×#11758194C2211yescharacteristic
18C2×C4×Dic122—24no—
19C2×C4×Dic122—22no—
20C2×C4×Dic122—24no—
21C2×C4×Dic122C2214yes—
22C22×C4⋊Dic61C2211yescharacteristic
23C22×#11758196—21no—
24C22×C4×Dic61—21no—
25#16839770—24no—
26#16839763—24no—
27C2×#14519701C2212yes—
28C2×#14519700C2214yes—
29C2×#14519698C2212yes—
30#16839524C2214yes—
31#16839517C2214yes—
32C22×#41×C61C2211yescharacteristic
33C2×#117×C61C2211yescharacteristic
34C22×C42×C61C2211yescharacteristic
35C61×#391C2212yes—
36C61×#390C2212yes—
37C61×#386—21no—
38C61×#386—21no—
39C4×C22⋊C4×C61—22no—
40C22×Dic122—21no—
41C22×Dic122—22no—
42C22×Dic122—22no—
43C22×Dic122—22no—
44C22×Dic122C2×C414yes—
45C22×Dic122D414yes—
46C22×Dic122D414yes—
47C22×Dic122C2312yes—
48C23×Dic61—21no—
49C23×Dic61—21no—
50#14519807—232no—
51C2×#11758202—24no—
52C2×#11758202—24no—
53C2×#11758202C2×C418yescomplemented
54C2×#11758200—24no—
55C2×#11758200—24no—
56C2×#11758200C2×C418yescomplemented
57C2×#11758194—22no—
58C2×#11758194—22no—
59C2×#11758194—22no—
60C2×#11758194—22no—
61C2×#11758194—22no—
62C2×#11758194C2×C414yes—
63C22×#92418C2312yes—
64C22×#92418C2312yes—
65C4×Dic122—216no—
66C4×Dic122—28no—
67C4×Dic122—28no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61D414yes—
71C2×C4⋊Dic61C2312yes—
72C2×#11758196—28no—
73C2×#11758196—24no—
74C2×C4×Dic61—28no—
75C2×C4×Dic61—24no—
76#14519701—24no—
77#14519700—28no—
78#14519698—24no—
79C2×#41×C61—22no—
80C2×#41×C61—22no—
81C2×#41×C61—22no—
82C2×#41×C61—22no—
83C2×#41×C61—22no—
84C2×#41×C61C2×C414yes—
85C22×Q8×C61C2311yescharacteristic
86C22×D4×C61—21no—
87C22×D4×C61C2311yescharacteristic
88C23×C4×C61—21no—
89C23×C4×C61D411yescharacteristic
90C23×C4×C61D411yescharacteristic
91C23×C4×C61C2311yescharacteristic
92#117×C61—22no—
93#117×C61—22no—
94C2×C4⋊C4×C61C2312yes—
95C2×C22⋊C4×C61—21no—
96C2×C22⋊C4×C61—22no—
97C2×C22⋊C4×C61—22no—
98C2×C42×C61—22no—
99C2×C42×C61—22no—
100C2×C42×C61—21no—
101C2×C42×C61D412yes—
102C2×C42×C61C2312yes—
103#95×C61—22no—
104#95×C61—22no—
105C24×C61—21no—
106C24×C61C2×D411yescharacteristic
107#41×C61—28no—
108#41×C61—28no—
109#41×C61—44no—
110#41×C61—44no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61C22×C412yes—
117C2×D4×C61—24no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—42no—
124C2×D4×C61—42no—
125C2×D4×C61C22×C412yes—
126C22×C4×C61—22no—
127C22×C4×C61—24no—
128C22×C4×C61—22no—
129C22×C4×C61—21no—
130C22×C4×C61—24no—
131C22×C4×C61—22no—
132C22×C4×C61—42no—
133C22×C4×C61—42no—
134C22×C4×C61—42no—
135C22×C4×C61C22×C412yes—
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D412yes—
139C22×C4×C61C2×D412yes—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D412yes—
142C22×C4×C61#4112yes—
143C22×C4×C61#4112yes—
144C22×C4×C61C2411yescharacteristic
145C22×Dic61—24no—
146C22×Dic61—24no—
147C22×Dic61—24no—
148C22×Dic61—22no—
149C22×Dic61—24no—
150C22×Dic61—22no—
151C22×Dic61—44no—
152C2×Dic122—28no—
153C2×Dic122—24no—
154C2×Dic122—24no—
155C2×Dic122—28no—
156C2×Dic122—24no—
157C2×Dic122—28no—
158C2×Dic122—22no—
159C2×Dic122—22no—
160C2×Dic122—22no—
161C2×Dic122C22⋊C4116yescomplemented
162C2×Dic122C22×C414yes—
163C2×Dic122C2×D414yes—
164C2×Dic122C2×D414yes—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C22⋊C4×C61—44no—
168C22⋊C4×C61—42no—
169C22⋊C4×C61—44no—
170C42×C61—24no—
171C42×C61—22no—
172C42×C61—22no—
173C42×C61—42no—
174#11758202—216no—
175#11758200—216no—
176C4⋊Dic61—24no—
177C4⋊Dic61—22no—
178C4⋊Dic61C2×D414yes—
179#11758196—216no—
180C4×Dic61—216no—
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×C61—22no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61C4×D412yes—
197C23×C61#12012yes—
198C23×C61#12112yes—
199C23×C61C22×D411yescharacteristic
200Q8×C61—22no—
201Q8×C61—22no—
202Q8×C61—41no—
203Q8×C61—41no—
204D4×C61—22no—
205D4×C61—22no—
206D4×C61—48no—
207D4×C61—41no—
208D4×C61—41no—
209C2×C4×C61—28no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—44no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—48no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61C2×C22⋊C414yes—
224C2×C4×C61C4×D414yes—
225C2×C4×C61C4×D412yes—
226C2×C4×C61#12012yes—
227C2×C4×C61#12114yes—
228C2×C4×C61#12112yes—
229C2×C4×C61#12314yes—
230C2×C4×C61#13714yescomplemented
231C2×C4×C61C23×C411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×Dic61—24no—
238C2×Dic61—24no—
239C2×Dic61—24no—
240C2×Dic61—216no—
241C2×Dic61—24no—
242C2×Dic61—48no—
243C2×Dic61—44no—
244Dic122—24no—
245Dic122—24no—
246Dic122—24no—
247Dic122—216no—
248Dic122C2×C22⋊C418yes—
249#92418—24no—
250#92418—42no—
251#20926—611nomaximal, Sylow
252C22×C61—24no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—24no—
256C22×C61—44no—
257C22×C61—42no—
258C22×C61—81no—
259C22×C61—81no—
260C22×C61—82no—
261C22×C61#38614yes—
262C22×C61C2×C4×D411yescharacteristic
263C22×C61C2×#12011yescharacteristic
264C22×C61C2×#12111yescharacteristic
265C22×C61C2×#13712yes—
266C22×C61#57512yes—
267C4×C61—24no—
268C4×C61—42no—
269C4×C61—42no—
270C4×C61—82no—
271C4×C61#38614yes—
272C4×C61C22×C22⋊C411yescharacteristic
273C4×C61C2×C4×D411yescharacteristic
274C4×C61C2×#12111yescharacteristic
275C4×C61C2×#12311yescharacteristic
276Dic61—28no—
277Dic61—44no—
278C22×#137—611no—
279C22×C4×Q8—611no—
280C2×#417—611no—
281C2×#419—611no—
282#2297D6111yescharacteristic
283#1880—614no—
284#1878—614no—
285#1736—611no—
286#1728—611no—
287C2×C61—24no—
288C2×C61—82no—
289C2×C61C2×#38611yescharacteristic
290C2×C61C4×#13712yes—
291C2×C61#300411yescharacteristic
292C2×C61#300511yescharacteristic
293C2×C61#303712yes—
294C22×#41C2×D6111yescharacteristic
295C23×Q8—611no—
296C2×#137—618no—
297C2×#137—1224no—
298C2×#137—1224no—
299C22×Q16—612no—
300C22×SD16—612no—
301C22×M16—611no—
302C22×M16—611no—
303C2×C4×Q8—614no—
304C2×C4×Q8—1224no—
305C2×C4×Q8—1224no—
306C2×C4×Q8—1222no—
307C2×#117C2×D6111yescharacteristic
308C22×C4⋊C4—611no—
309C22×C4⋊C4—1221no—
310C22×C42—1221no—
311C22×C42C2×D6111yescharacteristic
312#419—1224no—
313#417—1224no—
314C2×#103—612no—
315C2×#103—614no—
316C2×#102—612no—
317#391C2×D6112yes—
318#390C2×D6112yes—
319#386—21no—
320#386—21no—
321C4×C22⋊C4—22no—
322#340—614no—
323#336—614no—
324C61#2092611yescharacteristic, complemented, Sylow
325C2×#41—22no—
326C2×#41—22no—
327C2×#41—22no—
328C2×#41—22no—
329C2×#41—22no—
330C2×#41C4×D6114yes—
331C22×Q8—614no—
332C22×Q8—614no—
333C22×Q8—614no—
334C22×Q8—612no—
335C22×Q8—1222no—
336C22×Q8—1222no—
337C22×Q8—1222no—
338C22×Q8—1221no—
339C22×Q8C22×D6111yescharacteristic
340C22×D4—21no—
341C22×D4C22×D6111yescharacteristic
342C23×C4—21no—
343C23×C4—1221no—
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—1228no—
364C4×Q8—1228no—
365C4×Q8—12216no—
366#117—22no—
367#117—22no—
368C2×C4⋊C4—614no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—1228no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1224no—
374C2×C4⋊C4C22×D6112yes—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—21no—
381C2×C42—1228no—
382C2×C42—1224no—
383C2×C42C22⋊D6112yes—
384C2×C42C22×D6112yes—
385#103—1228no—
386#103—1224no—
387#102—1224no—
388#95—22no—
389#95—22no—
390C24—21no—
391C24C2×C22⋊D6111yescharacteristic
392#41—28no—
393#41—28no—
394#41—44no—
395#41—44no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—614no—
402C2×Q8—614no—
403C2×Q8—614no—
404C2×Q8—6116no—
405C2×Q8—1222no—
406C2×Q8—1222no—
407C2×Q8—1222no—
408C2×Q8—1228no—
409C2×Q8—1224no—
410C2×Q8—1224no—
411C2×Q8—1224no—
412C2×Q8—1228no—
413C2×Q8—1228no—
414C2×Q8C2×C4×D6112yes—
415C2×D4—24no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4C2×C4×D6112yes—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—1222no—
434C22×C4—1224no—
435C22×C4—1224no—
436C22×C4—1224no—
437C22×C4—1222no—
438C22×C4—1224no—
439C22×C4—2444no—
440C22×C4C2×C4×D6112yes—
441C22×C4#1175822212yes—
442C22×C4D4×D6112yes—
443C22×C4D4×D6112yes—
444C22×C4D4×D6112yes—
445C22×C4#1175822412yes—
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—22no—
461C4⋊C4—22no—
462C4⋊C4—614no—
463C4⋊C4—12216no—
464C4⋊C4—1222no—
465C4⋊C4—1224no—
466C22⋊C4—44no—
467C22⋊C4—42no—
468C22⋊C4—44no—
469C42—22no—
470C42—22no—
471C42—24no—
472C42—42no—
473C42—12216no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—42no—
481C23C4×C22⋊D6112yes—
482C23#1451974512yes—
483C23#1451974812yes—
484C23C22×C22⋊D6111yescharacteristic
485Q8—22no—
486Q8—22no—
487Q8—41no—
488Q8—41no—
489Q8—618no—
490Q8—1224no—
491Q8—1224no—
492Q8—1224no—
493Q8—12216no—
494D4—22no—
495D4—22no—
496D4—48no—
497D4—41no—
498D4—41no—
499C2×C4—28no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—44no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—48no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—1224no—
514C2×C4—1224no—
515C2×C4—1224no—
516C2×C4—12216no—
517C2×C4—1224no—
518C2×C4—2448no—
519C2×C4—2444no—
520C2×C4C22⋊C4×D6114yes—
521C2×C4#1451973714yes—
522C2×C4C4×C22⋊D6112yes—
523C2×C4#1451974512yes—
524C2×C4#1451974812yes—
525C2×C4#1451975014yes—
526C2×C4#1451975914yes—
527C2×C4#1451980714yescomplemented
528C2×C4C22×C4×D6111yescharacteristic
529C2×C4C2×D4×D6111yescharacteristic
530C2×C4C2×D4×D6111yescharacteristic
531C2×C4C2×D4×D6111yescharacteristic
532C2×C4C2×#1175822411yescharacteristic
533C2×C4C2×#1175822211yescharacteristic
534C8—1224no—
535C8—2442no—
536C22—24no—
537C22—24no—
538C22—22no—
539C22—22no—
540C22—42no—
541C22—44no—
542C22—82no—
543C22—81no—
544C22—81no—
545C22#1683963814yes—
546C22C2×C4×C22⋊D6111yescharacteristic
547C22C2×#1451974511yescharacteristic
548C22C2×#1451974811yescharacteristic
549C22C2×#1451980712yes—
550C22#1684056012yes—
551C4—24no—
552C4—42no—
553C4—42no—
554C4—82no—
555C4—1228no—
556C4—2444no—
557C4#1683963814yes—
558C4C2×C22⋊C4×D6111yescharacteristic
559C4C2×#1451973711yescharacteristic
560C4C2×#1451975011yescharacteristic
561C4C2×#1451975911yescharacteristic
562C2—24no—
563C2—82no—
564C2C2×#1683963811yescharacteristic
565C2C4×#1451980712yes—
566C2#2145505211yescharacteristic
567C2#2145505911yescharacteristic
568C2#2145528712yes—
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2297C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenter#2297D61
hypocenterC61#20926
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2297

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic122C2semidirect
4. C2×#16839770C2semidirect
5. #21446177C2semidirect
6. #21446163C2semidirect
7. #21445540C2semidirect
8. #21445513C2semidirect
13. C2×#14519807C4semidirect
53. C2×#11758202C2×C4semidirect
56. C2×#11758200C2×C4semidirect
161. C2×Dic122C22⋊C4semidirect
230. C2×C4×C61#137semidirect
324. C61#20926semidirect
527. C2×C4#14519807semidirect

Maximal subgroups

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

subgroupindex
C22×#145198072
C22×C4×Dic1222
C2×#168397632
C2×#168397702
#214461772
#214461632
#214455402
#214455132
C61×#22972
#2092661