Finite Group Databasebeta

← #29681427

Subgroups — #29681427

Order 15616.

1766 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519807C211yesmaximal, characteristic
2C2×#16839758C211yesmaximal, characteristic
3C22×#14519700C211yesmaximal, characteristic, complemented
4#21448403C211yesmaximal, characteristic, complemented
5#21446160C214yesmaximal, complemented
6#21446153C214yesmaximal, complemented
7#21445538C211yesmaximal, characteristic, complemented
8#21445516C211yesmaximal, characteristic, complemented
9C2×C61×#520C211yesmaximal, characteristic
10C23×Dic122C2211yescharacteristic
11C2×#14519807—24no—
12C2×#14519807—24no—
13C2×#14519807C418yescomplemented
14C22×#11758202C2212yes—
15C22×#11758200C2212yes—
16C22×#11758194C2211yescharacteristic
17C23×#92418C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19#16839758—24no—
20C2×#14519701C2212yes—
21C2×#14519700—24no—
22C2×#14519700C2214yes—
23C2×#14519698C2212yes—
24#16839668C2214yes—
25#16839667—21no—
26#16839667—21no—
27#16839623—22no—
28#16839524C2214yes—
29#16839517C2214yes—
30C22×#41×C61C2211yescharacteristic
31C61×#520—24no—
32C61×#520—24no—
33C2×C4×Q8×C61C2212yes—
34C2×C4×D4×C61—22no—
35C2×C4×D4×C61C2212yes—
36C2×#117×C61—21no—
37C2×#117×C61C2211yescharacteristic
38C22×C4⋊C4×C61—21no—
39C22×C4⋊C4×C61C2211yescharacteristic
40C22×Dic122—22no—
41C22×Dic122—22no—
42C22×Dic122—22no—
43C22×Dic122—22no—
44C22×Dic122—21no—
45C22×Dic122—22no—
46C22×Dic122C2×C414yescomplemented
47C22×Dic122C2312yes—
48C23×Dic61—21no—
49#14519807—232no—
50C2×#11758202—24no—
51C2×#11758202—24no—
52C2×#11758202C2×C418yescomplemented
53C2×#11758200—24no—
54C2×#11758200—24no—
55C2×#11758200C2×C418yescomplemented
56C2×#11758194—22no—
57C2×#11758194—22no—
58C2×#11758194C2×C414yescomplemented
59C22×#92418—22no—
60C22×#92418—22no—
61C22×#92418—22no—
62C22×#92418C2312yes—
63C22×#92418C2312yes—
64#14519736—22no—
65#14519736—22no—
66C2×C4⋊Dic61C2312yes—
67C2×C22⋊Dic61—21no—
68C2×C4×Dic61—21no—
69#14519701—24no—
70#14519700—28no—
71#14519698—24no—
72#14519684—24no—
73#14519684—24no—
74C2×#41×C61—22no—
75C2×#41×C61—22no—
76C2×#41×C61—22no—
77C2×#41×C61C2×C414yes—
78C2×#41×C61D414yes—
79C2×#41×C61D414yes—
80C22×Q8×C61C2311yescharacteristic
81C22×D4×C61—21no—
82C22×D4×C61C2311yescharacteristic
83C23×C4×C61—21no—
84C23×C4×C61—21no—
85C23×C4×C61—22no—
86C23×C4×C61C2311yescharacteristic
87C4×Q8×C61—24no—
88C4×Q8×C61—24no—
89C4×D4×C61—216no—
90C4×D4×C61—24no—
91C4×D4×C61—24no—
92#117×C61—28no—
93#117×C61—22no—
94#117×C61—22no—
95C2×C4⋊C4×C61—28no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61—22no—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C4⋊C4×C61D414yes—
102C2×C4⋊C4×C61C2312yes—
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61D412yes—
105C2×C42×C61—22no—
106C2×C42×C61C2312yes—
107C24×C61—21no—
108C24×C61C2×D411yescharacteristic
109#41×C61—28no—
110#41×C61—28no—
111#41×C61—28no—
112#41×C61C22⋊C4116yes—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61C22×C412yes—
117C2×Q8×C61C2×D412yes—
118C2×Q8×C61C2×D412yes—
119C2×D4×C61—24no—
120C2×D4×C61—24no—
121C2×D4×C61—24no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61C22×C412yes—
126C2×D4×C61C2×D412yes—
127C2×D4×C61C2×D412yes—
128C22×C4×C61—21no—
129C22×C4×C61—24no—
130C22×C4×C61—24no—
131C22×C4×C61—24no—
132C22×C4×C61—22no—
133C22×C4×C61—28no—
134C22×C4×C61—24no—
135C22×C4×C61—84no—
136C22×C4×C61—82no—
137C22×C4×C61C22×C412yes—
138C22×C4×C61C2×D412yes—
139C22×C4×C61C2×D412yes—
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D412yes—
144C22×C4×C61#4112yes—
145C22×C4×C61#4112yes—
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—24no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—42no—
151C22×Dic61—42no—
152C2×Dic122—28no—
153C2×Dic122—28no—
154C2×Dic122—22no—
155C2×Dic122—22no—
156C2×Dic122—22no—
157C2×Dic122—22no—
158C2×Dic122—24no—
159C2×Dic122—22no—
160C2×Dic122—42no—
161C2×Dic122—44no—
162C2×Dic122—42no—
163C2×Dic122—44no—
164C2×Dic122C22×C414yes—
165C4⋊C4×C61—28no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—24no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61C2×D414yes—
171C22⋊C4×C61—28no—
172C22⋊C4×C61—24no—
173C42×C61—28no—
174C42×C61—22no—
175C42×C61—22no—
176C22⋊Dic61—42no—
177#11758202—216no—
178#11758200—216no—
179C4⋊Dic61—22no—
180C4⋊Dic61—22no—
181C4×Dic61—42no—
182#11758194—28no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—44no—
189C2×#92418C22×C414yes—
190C23×C61—22no—
191C23×C61—22no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61C4×D412yes—
198C23×C61#12012yes—
199C23×C61#12112yes—
200C23×C61C22×D411yescharacteristic
201Q8×C61—22no—
202Q8×C61—22no—
203Q8×C61—22no—
204Q8×C61C2×C22⋊C414yes—
205D4×C61—216no—
206D4×C61—22no—
207D4×C61—22no—
208D4×C61—22no—
209D4×C61C2×C22⋊C414yes—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—216no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—24no—
217C2×C4×C61—24no—
218C2×C4×C61—24no—
219C2×C4×C61—44no—
220C2×C4×C61—84no—
221C2×C4×C61—88no—
222C2×C4×C61C2×C22⋊C414yes—
223C2×C4×C61C4×D414yes—
224C2×C4×C61C4×D412yes—
225C2×C4×C61#12012yes—
226C2×C4×C61#12114yes—
227C2×C4×C61#12112yes—
228C2×C4×C61#12314yes—
229C2×C4×C61C23×C411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×Dic61—24no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—48no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242Dic122—24no—
243Dic122—24no—
244Dic122—42no—
245Dic122—42no—
246Dic122—48no—
247#92418—24no—
248#92418—42no—
249#20931—611nomaximal, Sylow
250C22×C61—24no—
251C22×C61—22no—
252C22×C61—22no—
253C22×C61—24no—
254C22×C61—24no—
255C22×C61—44no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—44no—
259C22×C61#38614yes—
260C22×C61C2×C4×D411yescharacteristic
261C22×C61C2×#12011yescharacteristic
262C22×C61C2×#12111yescharacteristic
263C22×C61#57712yes—
264C22×C61#57812yes—
265C4×C61—24no—
266C4×C61—42no—
267C4×C61—44no—
268C4×C61—42no—
269C4×C61#38614yes—
270C4×C61C22×C22⋊C411yescharacteristic
271C4×C61C2×C4×D411yescharacteristic
272C4×C61C2×#12111yescharacteristic
273C4×C61C2×#12311yescharacteristic
274Dic61—44no—
275Dic61—82no—
276C22×#137—611no—
277C2×#520D6111yescharacteristic
278C2×#416—611no—
279C22×#103—611no—
280#2310—611no—
281#1878—614no—
282#1877—614no—
283#1736—611no—
284#1729—611no—
285C2×C61—24no—
286C2×C61—44no—
287C2×C61C2×#38611yescharacteristic
288C2×C61#294712yes—
289C2×C61#300911yescharacteristic
290C2×C61#301111yescharacteristic
291C2×C61#304612yes—
292C22×#41C2×D6111yescharacteristic
293C23×Q8—611no—
294C2×#137—618no—
295C2×#137—1224no—
296C2×#137—1224no—
297C22×Q16—612no—
298C22×SD16—612no—
299C22×M16—611no—
300C23×C8—611no—
301#520—24no—
302#520—24no—
303C2×C4×Q8C2×D6112yes—
304C2×C4×D4—22no—
305C2×C4×D4C2×D6112yes—
306C2×#117—21no—
307C2×#117—611no—
308C2×#117C2×D6111yescharacteristic
309C22×C4⋊C4—21no—
310C22×C4⋊C4C2×D6111yescharacteristic
311#416—1224no—
312C2×#103—612no—
313C2×#103—614no—
314C2×#103—1224no—
315C2×#102—612no—
316#391—614no—
317#391—1221no—
318#391—1221no—
319#381—1222no—
320#340—614no—
321#336—614no—
322C61#2093111yescharacteristic, complemented, Sylow
323C2×#41—22no—
324C2×#41—22no—
325C2×#41—22no—
326C2×#41C4×D6114yes—
327C2×#41D24414yes—
328C2×#41C22⋊D6114yes—
329C22×Q8—614no—
330C22×Q8—612no—
331C22×Q8—1222no—
332C22×Q8—1221no—
333C22×Q8—1222no—
334C22×Q8—1222no—
335C22×Q8—1222no—
336C22×Q8—1222no—
337C22×Q8C22×D6111yescharacteristic
338C22×D4—21no—
339C22×D4C22×D6111yescharacteristic
340C23×C4—21no—
341C23×C4—21no—
342C23×C4—22no—
343C23×C4—1221no—
344C23×C4C22×D6111yescharacteristic
345#137—12232no—
346C2×Q16—618no—
347C2×Q16—1224no—
348C2×Q16—1224no—
349C2×SD16—618no—
350C2×SD16—1224no—
351C2×SD16—1224no—
352C2×M16—614no—
353C2×M16—1222no—
354C2×M16—1222no—
355C22×C8—612no—
356C22×C8—612no—
357C22×C8—1222no—
358C22×C8—1222no—
359C22×C8—1222no—
360C4×Q8—24no—
361C4×Q8—24no—
362C4×D4—216no—
363C4×D4—24no—
364C4×D4—24no—
365#117—28no—
366#117—22no—
367#117—22no—
368#117—1222no—
369#117—1222no—
370C2×C4⋊C4—28no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—612no—
375C2×C4⋊C4C22⋊D6111yescharacteristic
376C2×C4⋊C4C22⋊D6111yescharacteristic
377C2×C4⋊C4C22⋊D6114yes—
378C2×C4⋊C4C22×D6112yes—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—1221no—
381C2×C22⋊C4C22⋊D6112yes—
382C2×C42—22no—
383C2×C42—1221no—
384C2×C42C22×D6112yes—
385#103—1224no—
386#103—1228no—
387#102—1224no—
388#95—1224no—
389#95—1224no—
390C24—21no—
391C24C2×C22⋊D6111yescharacteristic
392#41—28no—
393#41—28no—
394#41—28no—
395#41#11758198116yes—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—614no—
400C2×Q8—1222no—
401C2×Q8—1224no—
402C2×Q8—1222no—
403C2×Q8—1222no—
404C2×Q8—1222no—
405C2×Q8—1222no—
406C2×Q8—1228no—
407C2×Q8—1228no—
408C2×Q8—2442no—
409C2×Q8—2442no—
410C2×Q8—2444no—
411C2×Q8—2444no—
412C2×Q8C2×C4×D6112yes—
413C2×Q8C2×D24412yes—
414C2×Q8C2×C22⋊D6112yes—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4C2×C4×D6112yes—
422C2×D4C2×D24412yes—
423C2×D4C2×C22⋊D6112yes—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—28no—
430C22×C4—24no—
431C22×C4—84no—
432C22×C4—82no—
433C22×C4—1222no—
434C22×C4—1224no—
435C22×C4—1222no—
436C22×C4—2442no—
437C22×C4—2442no—
438C22×C4C2×C4×D6112yes—
439C22×C4C2×D24412yes—
440C22×C4#1175822212yes—
441C22×C4#1175822212yes—
442C22×C4C2×C22⋊D6112yes—
443C22×C4C2×C22⋊D6112yes—
444C22×C4C2×C22⋊D6112yes—
445C22×C4C2×C22⋊D6112yes—
446C22×C4C2×C22⋊D6111yescharacteristic
447C22×C4C23×D6111yescharacteristic
448Q16—12216no—
449SD16—12216no—
450M16—1228no—
451C2×C8—614no—
452C2×C8—1222no—
453C2×C8—1222no—
454C2×C8—1222no—
455C2×C8—1222no—
456C2×C8—1222no—
457C2×C8—2444no—
458C4⋊C4—28no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—1222no—
464C4⋊C4—1222no—
465C4⋊C4C2×C22⋊D6114yes—
466C22⋊C4—28no—
467C22⋊C4—24no—
468C22⋊C4—2442no—
469C42—28no—
470C42—22no—
471C42—22no—
472C42—2442no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—42no—
479C23—42no—
480C23C4×C22⋊D6112yes—
481C23#1451974612yes—
482C23#1451975112yes—
483C23C22×C22⋊D6111yescharacteristic
484Q8—22no—
485Q8—22no—
486Q8—22no—
487Q8—1224no—
488Q8—1224no—
489Q8—2442no—
490Q8—2442no—
491Q8—2448no—
492Q8C2×#1175819814yes—
493D4—216no—
494D4—22no—
495D4—22no—
496D4—22no—
497D4C2×#1175819814yes—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—216no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—44no—
508C2×C4—84no—
509C2×C4—88no—
510C2×C4—1224no—
511C2×C4—2442no—
512C2×C4—2442no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2448no—
517C2×C4C2×#1175819814yes—
518C2×C4C4×C22⋊D6114yes—
519C2×C4C4×C22⋊D6112yes—
520C2×C4#1451974612yes—
521C2×C4#1451975114yes—
522C2×C4#1451975112yes—
523C2×C4#1451976014yes—
524C2×C4C22×C4×D6111yescharacteristic
525C2×C4C22×C22⋊D6111yescharacteristic
526C2×C4C22×C22⋊D6111yescharacteristic
527C2×C4C22×D24411yescharacteristic
528C2×C4C2×#1175822211yescharacteristic
529C2×C4C2×#1175822211yescharacteristic
530C8—1224no—
531C8—2442no—
532C22—24no—
533C22—22no—
534C22—24no—
535C22—22no—
536C22—24no—
537C22—44no—
538C22—42no—
539C22—44no—
540C22—42no—
541C22#1683964114yes—
542C22C2×C4×C22⋊D6111yescharacteristic
543C22C2×#1451974611yescharacteristic
544C22C2×#1451975111yescharacteristic
545C22#1684057312yes—
546C22#1684057712yes—
547C4—24no—
548C4—42no—
549C4—42no—
550C4—44no—
551C4—4882no—
552C4—2444no—
553C4#1683964114yes—
554C4C22×#1175819811yescharacteristic
555C4C2×C4×C22⋊D6111yescharacteristic
556C4C2×#1451975111yescharacteristic
557C4C2×#1451976011yescharacteristic
558C2—24no—
559C2—44no—
560C2C2×#1683964111yescharacteristic
561C2#2145465812yes—
562C2#2145509611yescharacteristic
563C2#2145511411yescharacteristic
564C2#2145536912yes—
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#520C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenterC2×#520D61
hypocenterC61#20931
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
3. C22×#14519700C2semidirect
4. #21448403C2semidirect
5. #21446160C2semidirect
6. #21446153C2semidirect
7. #21445538C2semidirect
8. #21445516C2semidirect
13. C2×#14519807C4semidirect
46. C22×Dic122C2×C4semidirect
52. C2×#11758202C2×C4semidirect
55. C2×#11758200C2×C4semidirect
58. C2×#11758194C2×C4semidirect
322. C61#20931semidirect

Maximal subgroups

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

subgroupindex
C22×#145198072
C2×#168397582
C22×#145197002
#214484032
#214461602
#214461532
#214455382
#214455162
C2×C61×#5202
#2093161