Finite Group Databasebeta

← #29721320

Subgroups — #29721320

Order 15616.

912 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
1C2×#16840039C211yesmaximal, characteristic, complemented
2C2×#16839818C211yesmaximal, characteristic, complemented
3#21454166C211yesmaximal, characteristic, complemented
4C4×#14519752C211yesmaximal, characteristic
5#21447251C212yesmaximal, complemented
6#21447011C212yesmaximal
7#21446582C211yesmaximal, characteristic, complemented
8#21446561C212yesmaximal, complemented
9#21446514C212yesmaximal, complemented
10#21445820C211yesmaximal, characteristic, complemented
11C61×#1988C211yesmaximal, characteristic, complemented
12C2×#14519753—21no—
13C2×#14519752C2211yescharacteristic
14C2×C4×D244—21no—
15C2×C4×D244C2211yescharacteristic
16C2×C42×D61C2211yescharacteristic
17#16840039—24no—
18#16839818—24no—
19C2×#14519710C2212yes—
20C2×#14519708C2212yes—
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519699C2212yes—
23C2×#14519697C2212yes—
24C2×#14519689C2212yes—
25C2×#14519687C2212yes—
26#16839669—21no—
27#16839663C2211yescharacteristic
28#16839661C2211yescharacteristic
29#16839658—21no—
30#16839658C2212yes—
31#16839654—21no—
32#16839640—21no—
33#16839632C2211yescharacteristic
34#16839619—21no—
35#16839619C2211yescharacteristic
36#16839619C2212yes—
37C4×#11758198—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C4×C4⋊C4×C61C2211yescharacteristic
43C61×#340C2212yes—
44C61×#340C2212yes—
45C22×D244—21no—
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C22×#92418C2312yes—
52#14519753—44no—
53#14519752—22no—
54#14519752—22no—
55C4×D244—22no—
56C4×D244—22no—
57C4×D244—44no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C4⋊Dic61C2312yes—
63C2×#11758196—22no—
64C2×#11758198—21no—
65C2×#11758198—21no—
66C2×#11758198—22no—
67C2×#11758198—21no—
68C2×#11758198—21no—
69C2×#11758198—21no—
70C2×C4×Dic61D411yescharacteristic
71C2×C4×Dic61D411yescharacteristic
72C2×C4×Dic61C2311yescharacteristic
73C42×D61—22no—
74C42×D61—22no—
75#14519710—24no—
76#14519708—24no—
77C4⋊#92418D414yescomplemented
78#14519699—24no—
79#14519697—24no—
80#14519689—24no—
81#14519687—24no—
82#14519684—21no—
83#14519684—21no—
84#14519684—22no—
85#14519684—22no—
86C22×Q8×C61—21no—
87C22×C8×C61C2312yes—
88C4⋊Q8×C61—22no—
89C4⋊Q8×C61—22no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61C2312yes—
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C42×C61—21no—
95C2×C42×C61C2311yescharacteristic
96C4⋊C8×C61—22no—
97#95×C61—21no—
98C23×D61—21no—
99C23×D61C2×D411yescharacteristic
100C2×Q8×C61—42no—
101C2×Q8×C61—42no—
102C2×Q8×C61—42no—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—22no—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2411yescharacteristic
109C22×Dic61—21no—
110C22×Dic61—22no—
111C22×Dic61C2×D411yescharacteristic
112C22×Dic61C2×D411yescharacteristic
113C22×Dic61C2×D411yescharacteristic
114C2×D244—21no—
115C2×D244—21no—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244—24no—
119C2×D244—21no—
120C2×D244—42no—
121C2×D244—42no—
122C2×D244#4112yes—
123C2×C4×D61—22no—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—21no—
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×C4×D61—41no—
135C2×C4×D61—42no—
136C2×C4×D61—42no—
137C2×C4×D61—42no—
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C2×C8×C61—22no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—42no—
147C4⋊C4×C61#4112yes—
148C42×C61—42no—
149C42×C61C2×D412yes—
150C42×C61#4112yes—
151#11758198—24no—
152#11758198—42no—
153#11758198—42no—
154#11758198—42no—
155#11758198—44no—
156#11758198—42no—
157C4⋊Dic61—21no—
158C4⋊Dic61—22no—
159C4⋊Dic61—22no—
160C4⋊Dic61—21no—
161C4⋊Dic61—42no—
162C4⋊Dic61—42no—
163C4⋊Dic61—42no—
164#11758196—44no—
165C4×Dic61—22no—
166C4×Dic61—22no—
167C4×Dic61—21no—
168C4×Dic61—21no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418C2×D414yes—
172C23×C61C22×D411yescharacteristic
173C22×D61—21no—
174C22×D61—21no—
175C22×D61—21no—
176C22×D61—21no—
177C22×D61—21no—
178C22×D61—21no—
179C22×D61—22no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61#13712yes—
187Q8×C61—84no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—22no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—42no—
194C2×C4×C61—41no—
195C2×C4×C61—42no—
196C2×C4×C61—41no—
197C2×C4×C61—42no—
198C2×C4×C61—41no—
199C2×C4×C61#12112yes—
200C2×C4×C61#12112yes—
201C2×C4×C61AGL(1,ℤ/8)12yes—
202C2×C4×C61AGL(1,ℤ/8)12yes—
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C2×#4111yescharacteristic
205C2×C4×C61C2×#4111yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C612−1+411yescharacteristic
208C2×C4×C612−1+411yescharacteristic
209C2×Dic61—21no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—21no—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—41no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—42no—
224D244—22no—
225D244—22no—
226D244—48no—
227D244—41no—
228D244—41no—
229C4×D61—44no—
230C4×D61—41no—
231C4×D61—41no—
232C4×D61—41no—
233C4×D61—41no—
234C4×D61—44no—
235C4×D61—84no—
236C8×C61—42no—
237#92418—24no—
238#24709—611nomaximal, Sylow
239C22×C61C2×#12111yescharacteristic
240C22×C61#52511yescharacteristic
241C22×C61#53611yescharacteristic
242C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
243C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
244C22×C61C2×#13711yescharacteristic
245C22×C61#57511yescharacteristic
246C2×D61—22no—
247C2×D61—22no—
248C2×D61—22no—
249C2×D61—41no—
250C2×D61—41no—
251C2×D61—41no—
252C2×D61—41no—
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—42no—
256C2×D61—81no—
257C2×D61—81no—
258C2×D61—81no—
259C2×D61—81no—
260C4×C61—22no—
261C4×C61—42no—
262C4×C61—81no—
263C4×C61—82no—
264C4×C61#46914yes—
265C4×C61#54711yescharacteristic
266C4×C61#54811yescharacteristic
267C4×C61#55211yescharacteristic
268C4×C61#55611yescharacteristic
269Dic61—41no—
270Dic61—41no—
271Dic61—41no—
272Dic61—82no—
273Dic61—81no—
274C2×#474—611no—
275C2×#432—611no—
276#2857—611no—
277C4×#121—611no—
278#2076—612no—
279#2034—612no—
280#1988D6111yescharacteristic, complemented
281#1948—611no—
282#1944—612no—
283#1936—612no—
284#1802—611no—
285C2×C61#241511yescharacteristic
286C2×C61C2×#46911yescharacteristic
287C2×C61#305611yescharacteristic
288C2×C61#311011yescharacteristic
289C2×C61#311511yescharacteristic
290C2×C61#323511yescharacteristic
291C2×C61#323911yescharacteristic
292D61—42no—
293D61—82no—
294C2×C4⋊Q8C2×D6111yescharacteristic
295C2×C22⋊Q8—1221no—
296C2×#121—611no—
297C2×C4×D4—611no—
298C2×C4×D4—1221no—
299C22×C42—611no—
300#474—1224no—
301#432—1224no—
302C2×#107—612no—
303C2×#106—612no—
304C2×C4⋊C8—611no—
305C2×C4⋊C8C2×D6111yescharacteristic
306C2×#102—612no—
307C2×#102—612no—
308C2×C22⋊C8—612no—
309C2×#97—612no—
310#391—1221no—
311#390—611no—
312#389—612no—
313#389—611no—
314#389—1221no—
315#388—1221no—
316#386—1221no—
317#384—611no—
318#380—611no—
319#380—612no—
320#380—1221no—
321C4×C4⋊C4C2×D6111yescharacteristic
322C4×C22⋊C4—1221no—
323#340—612no—
324#340—612no—
325#340C2×D6112yes—
326#340C2×D6112yes—
327C61#2470911yescharacteristic, complemented, Sylow
328C22×Q8—21no—
329C22×D4—611no—
330C22×D4—1221no—
331C23×C4—611no—
332C23×C4—611no—
333C23×C4—611no—
334C23×C4—1221no—
335C22×C8—612no—
336C22×C8C22×D6112yes—
337C4⋊Q8—22no—
338C4⋊Q8—22no—
339C22⋊Q8—2444no—
340#121—1222no—
341#121—1222no—
342C4×D4—1222no—
343C4×D4—1222no—
344C4×D4—2444no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—612no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4C22×D6112yes—
353C2×C4⋊C4C22×D6111yescharacteristic
354C2×C4⋊C4C22×D6111yescharacteristic
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1221no—
357C2×C22⋊C4—1221no—
358C2×C22⋊C4—1221no—
359C2×C22⋊C4—1222no—
360C2×C22⋊C4—1221no—
361C2×C42—21no—
362C2×C42—611no—
363C2×C42—611no—
364C2×C42—611no—
365C2×C42—1222no—
366C2×C42—1222no—
367C2×C42C22×D6111yescharacteristic
368#107—1224no—
369#106—1224no—
370C4⋊C8—22no—
371C4⋊C8—614no—
372#102—1224no—
373#102—1224no—
374C22⋊C8—1224no—
375#97—1224no—
376#95—21no—
377#95—1221no—
378#95—1221no—
379#95—1222no—
380#95—1222no—
381C24—611no—
382C24—1221no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×D4—612no—
387C2×D4—1224no—
388C2×D4—1221no—
389C2×D4—1221no—
390C2×D4—1221no—
391C2×D4—1221no—
392C2×D4—1221no—
393C2×D4—2442no—
394C2×D4—2442no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—611no—
401C22×C4—1222no—
402C22×C4—1221no—
403C22×C4—1221no—
404C22×C4—1221no—
405C22×C4—1221no—
406C22×C4—1221no—
407C22×C4—1221no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1222no—
412C22×C4—1221no—
413C22×C4—1222no—
414C22×C4—2441no—
415C22×C4—2442no—
416C22×C4—2442no—
417C22×C4—2442no—
418C22×C4D4×D6111yescharacteristic
419C22×C4D4×D6111yescharacteristic
420C22×C4C23×D6111yescharacteristic
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—614no—
425C2×C8—1222no—
426C2×C8—1222no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—42no—
433C4⋊C4—1221no—
434C4⋊C4—1222no—
435C4⋊C4—1221no—
436C4⋊C4—1222no—
437C4⋊C4—2442no—
438C4⋊C4—2444no—
439C4⋊C4—2442no—
440C4⋊C4—2442no—
441C4⋊C4#1175822612yes—
442C22⋊C4—1224no—
443C22⋊C4—2442no—
444C22⋊C4—2442no—
445C22⋊C4—2442no—
446C22⋊C4—2444no—
447C22⋊C4—2442no—
448C42—42no—
449C42—1221no—
450C42—1221no—
451C42—1222no—
452C42—1222no—
453C42#1175822212yes—
454C42D4×D6112yes—
455C23—612no—
456C23—1221no—
457C23—1221no—
458C23—1221no—
459C23—1221no—
460C23—1221no—
461C23—1221no—
462C23—1222no—
463C23—2441no—
464C23—2441no—
465C23—2441no—
466C23—2441no—
467C23—2441no—
468C23—2441no—
469C23C2×D4×D6111yescharacteristic
470Q8—84no—
471D4—1222no—
472D4—1222no—
473D4—2448no—
474D4—2441no—
475D4—2441no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—1221no—
488C2×C4—1221no—
489C2×C4—1221no—
490C2×C4—1221no—
491C2×C4—1221no—
492C2×C4—1221no—
493C2×C4—1221no—
494C2×C4—1221no—
495C2×C4—1221no—
496C2×C4—4884no—
497C2×C4—2444no—
498C2×C4—2441no—
499C2×C4—2442no—
500C2×C4—2441no—
501C2×C4—2442no—
502C2×C4—2441no—
503C2×C4—2442no—
504C2×C4—2441no—
505C2×C4—2441no—
506C2×C4—2444no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4#1451975012yes—
510C2×C4#1451975012yes—
511C2×C4#1451980112yes—
512C2×C4#1451980412yes—
513C2×C4C2×D4×D6111yescharacteristic
514C2×C4#41×D6111yescharacteristic
515C2×C4C2×#1175822211yescharacteristic
516C2×C4C2×#1175822611yescharacteristic
517C2×C4#1451982411yescharacteristic
518C2×C4#1451982511yescharacteristic
519C8—42no—
520C8—1224no—
521C22—1222no—
522C22—1222no—
523C22—1222no—
524C22—4881no—
525C22—4881no—
526C22—4881no—
527C22—4881no—
528C22—2442no—
529C22—2442no—
530C22—2442no—
531C22—2441no—
532C22—2441no—
533C22—2441no—
534C22—2441no—
535C22C2×#1451975011yescharacteristic
536C22#1684027511yescharacteristic
537C22#1684033511yescharacteristic
538C22C2×#1451980111yescharacteristic
539C22C2×#1451980411yescharacteristic
540C22#137×D6111yescharacteristic
541C22#1684056111yescharacteristic
542C4—22no—
543C4—42no—
544C4—82no—
545C4—81no—
546C4—4881no—
547C4—4882no—
548C4—2441no—
549C4—2441no—
550C4—2441no—
551C4#1684001414yes—
552C4#1684041511yescharacteristic
553C4#1684042011yescharacteristic
554C4#1684045311yescharacteristic
555C4#1684048511yescharacteristic
556C2—4882no—
557C2—2442no—
558C2#2144925111yescharacteristic
559C2C2×#1684001411yescharacteristic
560C2#2145544711yescharacteristic
561C2#2145592011yescharacteristic
562C2#2145597311yescharacteristic
563C2#2145701611yescharacteristic
564C2#2145705311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1988C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
hypercenter#1988D61
hypocenterC61#24709
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
1. C2×#16840039C2semidirect
2. C2×#16839818C2semidirect
3. #21454166C2semidirect
5. #21447251C2semidirect
7. #21446582C2semidirect
8. #21446561C2semidirect
9. #21446514C2semidirect
10. #21445820C2semidirect
11. C61×#1988C2semidirect
77. C4⋊#92418D4semidirect
280. #1988D61semidirect
327. C61#24709semidirect

Maximal subgroups

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

subgroupindex
C2×#168400392
C2×#168398182
#214541662
C4×#145197522
#214472512
#214470112
#214465822
#214465612
#214465142
#214458202
C61×#19882
#2470961