Finite Group Databasebeta

← #29707740

Subgroups — #29707740

Order 15616.

1606 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840331C211yesmaximal, characteristic
2C2×#16839761C211yesmaximal, characteristic, complemented
3C2×#16839766C211yesmaximal, characteristic, complemented
4C2×#16839756C211yesmaximal, characteristic, complemented
5C22×#14519697C211yesmaximal, characteristic, complemented
6#21447060C214yesmaximal, complemented
7#21447053C214yesmaximal, complemented
8#21445516C211yesmaximal, characteristic, complemented
9C61×#2317C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×D244C2211yescharacteristic
12C22×#11758194C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14#16840331—24no—
15#16840331—24no—
16C2×#14519774C2212yes—
17C2×#14519765C2212yes—
18C2×#14519752—21no—
19C2×#14519752—21no—
20C2×#14519747—22no—
21#16839766—24no—
22#16839761—24no—
23#16839756—24no—
24C2×#14519700C2212yescomplemented
25C2×#14519700C2212yescomplemented
26C2×#14519697—24no—
27C2×#14519697C2214yes—
28C2×#14519697C2214yes—
29C2×#14519697C2212yescomplemented
30C2×#14519697C2212yescomplemented
31#16839517C2214yescomplemented
32C2×#117×C61C2211yescharacteristic
33C2×#117×C61C2211yescharacteristic
34C22×C4⋊C4×C61C2211yescharacteristic
35C61×#389C2214yescomplemented
36C61×#385—22no—
37C61×#384—21no—
38C61×#382—21no—
39C24×D61—21no—
40C2×#11758222—22no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222D414yes—
46C22×Dic122C2311yescharacteristic
47C22×D244—22no—
48C22×D244—22no—
49C22×D244—22no—
50C22×D244—22no—
51C22×D244—21no—
52C22×D244—22no—
53C22×D244—22no—
54C22×D244—22no—
55C22×D244D414yes—
56C22×D244C2312yes—
57C22×D244C2311yescharacteristic
58C22×C22⋊D61—21no—
59C22×C4×D61—21no—
60C2×#11758194—22no—
61C2×#11758194—22no—
62C2×#11758194—22no—
63C22×#92418—22no—
64C22×#92418—22no—
65C22×#92418—22no—
66C22×#92418C2312yes—
67C22×#92418C2312yes—
68#14519774—24no—
69#14519774—24no—
70#14519765—24no—
71#14519765—24no—
72#14519752—28no—
73#14519752—28no—
74#14519747—216no—
75C2×#11758198—22no—
76C2×#11758198—22no—
77#14519700—24no—
78#14519700—24no—
79#14519697—28no—
80#14519697—24no—
81#14519697—28no—
82#14519697—24no—
83C23×C4×C61—21no—
84C23×C4×C61C2311yescharacteristic
85#117×C61—22no—
86#117×C61—22no—
87#117×C61—22no—
88#117×C61D414yescomplemented
89C2×C4⋊C4×C61—22no—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—24no—
93C2×C4⋊C4×C61D411yescharacteristic
94C2×C4⋊C4×C61D411yescharacteristic
95C2×C4⋊C4×C61C2312yes—
96C2×C4⋊C4×C61C2312yes—
97C2×C22⋊C4×C61—21no—
98C2×C22⋊C4×C61D412yes—
99C2×C42×C61—21no—
100C2×C42×C61C2312yes—
101#95×C61—24no—
102C24×C61C2×D411yescharacteristic
103C23×D61—21no—
104C23×D61—22no—
105C23×D61—24no—
106C23×D61—42no—
107C23×D61—42no—
108C23×D61—82no—
109C23×D61—82no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—42no—
113C22×C4×C61—82no—
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×D412yes—
119C22×C4×C61C2×D412yes—
120C22×C4×C61#4112yes—
121C22×C4×C61#4112yes—
122C22×C4×C61#4112yes—
123C22×C4×C61C2411yescharacteristic
124C2×C22⋊D61—24no—
125C2×C22⋊D61—42no—
126C2×C22⋊D61—42no—
127C22×Dic61—21no—
128#11758222—28no—
129#11758222—28no—
130#11758222—44no—
131#11758222—44no—
132C2×D244—28no—
133C2×D244—28no—
134C2×D244—22no—
135C2×D244—21no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—24no—
143C2×D244—22no—
144C2×D244—21no—
145C2×D244—28no—
146C2×D244—28no—
147C2×D244—44no—
148C2×D244—42no—
149C2×D244—44no—
150C2×D244—42no—
151C2×D244—84no—
152C2×D244—84no—
153C2×D244C2×D414yes—
154C2×D244C2×D412yes—
155C2×C4×D61—24no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122C2×D412yes—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—42no—
170C4⋊C4×C61—84no—
171C4⋊C4×C61C2×D414yes—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—24no—
174C42×C61—24no—
175C42×C61—22no—
176C42×C61—22no—
177#11758198—28no—
178#11758198—28no—
179#11758194—44no—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418—44no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61#12012yes—
190C23×C61#12112yes—
191C23×C61#12412yes—
192C23×C61C22×D411yescharacteristic
193C22×D61—24no—
194C22×D61—22no—
195C22×D61—41no—
196C22×D61—42no—
197C22×D61—41no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—44no—
201C22×D61—42no—
202C22×D61—44no—
203C22×D61—48no—
204C22×D61—84no—
205C22×D61—84no—
206C22×D61—88no—
207C22×D61—88no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—24no—
211C2×C4×C61—24no—
212C2×C4×C61—24no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—44no—
217C2×C4×C61—84no—
218C2×C4×C61#12012yes—
219C2×C4×C61#12114yes—
220C2×C4×C61#12114yes—
221C2×C4×C61#12112yes—
222C2×C4×C61#12314yes—
223C2×C4×C61#12314yes—
224C2×C4×C61#12412yes—
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C22⋊D61—48no—
232C2×Dic61—22no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235D244—24no—
236D244—22no—
237D244—24no—
238D244—22no—
239D244—42no—
240D244—41no—
241D244—42no—
242D244—41no—
243D244—48no—
244D244—88no—
245D244—88no—
246C4×D61—48no—
247Dic122—22no—
248Dic122—22no—
249Dic122—41no—
250Dic122—41no—
251#92418—42no—
252#92418—42no—
253#23509—611nomaximal, Sylow
254C22×C61—24no—
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—24no—
258C22×C61#39414yes—
259C22×C61C2×#12011yescharacteristic
260C22×C61C2×#12111yescharacteristic
261C22×C61C2×#12411yescharacteristic
262C22×C61#57612yes—
263C22×C61#57712yes—
264C2×D61—42no—
265C2×D61—44no—
266C2×D61—42no—
267C2×D61—44no—
268C2×D61—42no—
269C2×D61—44no—
270C2×D61—816no—
271C2×D61—82no—
272C2×D61—84no—
273C2×D61—82no—
274C2×D61—84no—
275C4×C61—42no—
276C4×C61—42no—
277C4×C61—44no—
278C4×C61—82no—
279C4×C61#39414yes—
280C4×C61C2×#12111yescharacteristic
281C4×C61C2×#12111yescharacteristic
282C4×C61C2×#12311yescharacteristic
283C4×C61C2×#12311yescharacteristic
284Dic61—42no—
285C2×#535—611no—
286C2×#417—611no—
287C2×#418—611no—
288C2×#416—611no—
289C22×#102—611no—
290#2317D6111yescharacteristic, complemented
291#2042—614no—
292#2041—614no—
293#1729—611no—
294C2×C61—24no—
295C2×C61C2×#39411yescharacteristic
296C2×C61#300811yescharacteristic
297C2×C61#300911yescharacteristic
298C2×C61#304512yes—
299C2×C61#313612yes—
300D61—82no—
301D61—84no—
302C22×#41—611no—
303C23×D4—611no—
304C22×M16—611no—
305C23×C8—611no—
306#535—1224no—
307#535—1224no—
308C2×#127—612no—
309C2×#124—612no—
310C2×#121—1221no—
311C2×#121—1221no—
312C2×#120—1222no—
313C2×#117C2×D6111yescharacteristic
314C2×#117C2×D6111yescharacteristic
315C22×C4⋊C4C2×D6111yescharacteristic
316#418—1224no—
317#417—1224no—
318#416—1224no—
319C2×#103—612no—
320C2×#103—612no—
321C2×#102—614no—
322C2×#102—614no—
323C2×#102—612no—
324C2×#102—612no—
325C2×#102—1224no—
326#389C2×D6114yescomplemented
327#385—22no—
328#384—21no—
329#382—21no—
330#336—614no—
331C61#2350911yescharacteristic, complemented, Sylow
332C25—1221no—
333C2×#41—614no—
334C2×#41—1222no—
335C2×#41—1222no—
336C2×#41—1222no—
337C2×#41—1222no—
338C2×#41—1222no—
339C22×Q8—611no—
340C22×D4—614no—
341C22×D4—612no—
342C22×D4—611no—
343C22×D4—1221no—
344C22×D4—1222no—
345C22×D4—1222no—
346C22×D4—1222no—
347C22×D4—1222no—
348C22×D4—1222no—
349C22×D4—1222no—
350C22×D4—1222no—
351C22×D4—1221no—
352C23×C4—21no—
353C23×C4—1221no—
354C23×C4C22×D6111yescharacteristic
355C2×M16—1222no—
356C2×M16—1222no—
357C2×M16—1222no—
358C22×C8—612no—
359C22×C8—612no—
360C22×C8—1222no—
361C22×C8—1222no—
362C22×C8—1222no—
363#127—1224no—
364#127—1224no—
365#124—1224no—
366#124—1224no—
367#121—1228no—
368#121—1228no—
369#120—12216no—
370#117—22no—
371#117—22no—
372#117—22no—
373#117D24414yescomplemented
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4C22⋊D6111yescharacteristic
379C2×C4⋊C4C22⋊D6111yescharacteristic
380C2×C4⋊C4C22×D6112yes—
381C2×C4⋊C4C22×D6112yes—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—1222no—
384C2×C22⋊C4—1222no—
385C2×C22⋊C4C22⋊D6112yes—
386C2×C42—21no—
387C2×C42C22×D6112yes—
388#103—1224no—
389#103—1224no—
390#102—1228no—
391#102—1224no—
392#102—1224no—
393#102—1228no—
394#95—24no—
395C24—1224no—
396C24—1222no—
397C24—1221no—
398C24—4882no—
399C24—4882no—
400C24—2442no—
401C24—2442no—
402C24C2×C22⋊D6111yescharacteristic
403#41—1228no—
404#41—1228no—
405#41—2444no—
406#41—2444no—
407C2×Q8—612no—
408C2×Q8—1221no—
409C2×Q8—1221no—
410C2×Q8—1221no—
411C2×Q8—1221no—
412C2×Q8—1221no—
413C2×D4—614no—
414C2×D4—612no—
415C2×D4—1228no—
416C2×D4—1228no—
417C2×D4—1224no—
418C2×D4—1222no—
419C2×D4—1221no—
420C2×D4—1222no—
421C2×D4—1221no—
422C2×D4—1222no—
423C2×D4—1221no—
424C2×D4—1222no—
425C2×D4—1222no—
426C2×D4—1221no—
427C2×D4—1221no—
428C2×D4—1224no—
429C2×D4—1228no—
430C2×D4—1228no—
431C2×D4—4884no—
432C2×D4—4884no—
433C2×D4—2442no—
434C2×D4—2442no—
435C2×D4—2442no—
436C2×D4—2444no—
437C2×D4—2444no—
438C2×D4—2442no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—42no—
442C22×C4—82no—
443C22×C4—1221no—
444C22×C4—1224no—
445C22×C4—2442no—
446C22×C4—2442no—
447C22×C4C2×D24412yes—
448C22×C4#1175822212yes—
449C22×C4D4×D6112yes—
450C22×C4D4×D6112yes—
451C22×C4#1175822612yes—
452C22×C4#1175822612yes—
453C22×C4C2×C22⋊D6112yes—
454C22×C4C2×C22⋊D6112yes—
455C22×C4C2×C22⋊D6111yescharacteristic
456C22×C4C23×D6111yescharacteristic
457M16—2444no—
458C2×C8—1222no—
459C2×C8—1222no—
460C2×C8—1222no—
461C2×C8—1222no—
462C2×C8—1222no—
463C2×C8—1222no—
464C2×C8—2444no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C4⋊C4—84no—
472C4⋊C4C2×D24414yes—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—1228no—
476C22⋊C4—1228no—
477C42—22no—
478C42—22no—
479C42—24no—
480C23—22no—
481C23—22no—
482C23—1224no—
483C23—1222no—
484C23—4888no—
485C23—4888no—
486C23—4884no—
487C23—4884no—
488C23—2441no—
489C23—2442no—
490C23—2441no—
491C23—2442no—
492C23—2442no—
493C23—2444no—
494C23—2442no—
495C23—2444no—
496C23—2448no—
497C23#1451974512yes—
498C23#1451975112yes—
499C23#1451976612yes—
500C23C22×C22⋊D6111yescharacteristic
501Q8—1222no—
502Q8—1222no—
503Q8—2441no—
504Q8—2441no—
505D4—1224no—
506D4—1222no—
507D4—1224no—
508D4—1222no—
509D4—4888no—
510D4—4888no—
511D4—2448no—
512D4—2442no—
513D4—2441no—
514D4—2442no—
515D4—2441no—
516D4—2448no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—84no—
527C2×C4—1222no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2448no—
531C2×C4#1451974512yes—
532C2×C4#1451975112yes—
533C2×C4#1451975214yes—
534C2×C4#1451975214yes—
535C2×C4#1451976214yes—
536C2×C4#1451976214yes—
537C2×C4#1451976612yes—
538C2×C4C2×D4×D6111yescharacteristic
539C2×C4C2×D4×D6111yescharacteristic
540C2×C4C22×D24411yescharacteristic
541C2×C4C2×#1175822211yescharacteristic
542C2×C4C2×#1175822611yescharacteristic
543C2×C4C2×#1175822611yescharacteristic
544C8—2442no—
545C8—2442no—
546C22—24no—
547C22—22no—
548C22—22no—
549C22—24no—
550C22—48816no—
551C22—4884no—
552C22—4882no—
553C22—4884no—
554C22—4882no—
555C22—2442no—
556C22—2444no—
557C22—2442no—
558C22—2444no—
559C22—2442no—
560C22—2444no—
561C22#1683967914yes—
562C22C2×#1451974511yescharacteristic
563C22C2×#1451975111yescharacteristic
564C22C2×#1451976611yescharacteristic
565C22#1684056512yes—
566C22#1684057012yes—
567C4—42no—
568C4—42no—
569C4—44no—
570C4—82no—
571C4—2442no—
572C4#1683967914yes—
573C4C2×#1451975211yescharacteristic
574C4C2×#1451975211yescharacteristic
575C4C2×#1451976211yescharacteristic
576C4C2×#1451976211yescharacteristic
577C2—24no—
578C2—4884no—
579C2—4882no—
580C2C2×#1683967911yescharacteristic
581C2#2145508511yescharacteristic
582C2#2145509411yescharacteristic
583C2#2145535812yes—
584C2#2145617812yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2317C2
socleC23×C61C22×D4
prime radicalC2×#16840331C2
hypercenter#2317D61
hypocenterC61#23509
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. C2×#16839761C2semidirect
3. C2×#16839766C2semidirect
4. C2×#16839756C2semidirect
5. C22×#14519697C2semidirect
6. #21447060C2semidirect
7. #21447053C2semidirect
8. #21445516C2semidirect
9. C61×#2317C2semidirect
24. C2×#14519700C22semidirect
25. C2×#14519700C22semidirect
29. C2×#14519697C22semidirect
30. C2×#14519697C22semidirect
31. #16839517C22semidirect
35. C61×#389C22semidirect
88. #117×C61D4semidirect
290. #2317D61semidirect
326. #389C2×D61semidirect
331. C61#23509semidirect
373. #117D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168403312
C2×#168397612
C2×#168397662
C2×#168397562
C22×#145196972
#214470602
#214470532
#214455162
C61×#23172
#2350961