Finite Group Databasebeta

← #29699885

Subgroups — #29699885

Order 15616.

838 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839860C211yesmaximal, characteristic
2#21448784C211yesmaximal, characteristic, complemented
3#21448770C211yesmaximal, characteristic
4#21446697C211yesmaximal, characteristic
5#21446670C211yesmaximal, characteristic
6#21446463C211yesmaximal, characteristic, complemented
7#21446456C211yesmaximal, characteristic, complemented
8#21446444C211yesmaximal, characteristic
9#21446437C211yesmaximal, characteristic
10#21445985C211yesmaximal, characteristic, complemented
11#21445830C211yesmaximal, characteristic
12#21445715C211yesmaximal, characteristic, complemented
13#21445711C211yesmaximal, characteristic, complemented
14#21445696C211yesmaximal, characteristic, complemented
15C61×#1803C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×#14519735—21no—
19#16839860C418yescomplemented
20C2×#14519710C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519688C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32#16839669C2211yescharacteristic
33#16839669C2211yescharacteristic
34#16839663C2211yescharacteristic
35#16839663C2211yescharacteristic
36#16839652—21no—
37#16839652—21no—
38#16839648—21no—
39#16839648—21no—
40#16839630—21no—
41#16839628C2211yescharacteristic
42#16839619—21no—
43C4⋊C4×Dic61C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4⋊C4×Dic61C2211yescharacteristic
46C4⋊C4×Dic61C2211yescharacteristic
47C4×C4⋊Dic61C2211yescharacteristic
48#16839608—21no—
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4×Q8×C61C2211yescharacteristic
54C2×C4⋊C8×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C61×#384C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C2×#11758201—22no—
63C2×#11758201—22no—
64C2×#11758201C2×C414yes—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C4×D244—22no—
68C4×D244C2×C414yes—
69#14519735—44no—
70C2×C4⋊Dic61D411yescharacteristic
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519710—22no—
87#14519710—41no—
88#14519710C2×C414yes—
89#14519708—22no—
90#14519708—41no—
91#14519701C2×C414yes—
92#14519699—41no—
93#14519699—41no—
94#14519697C2×C414yes—
95#14519688C2×C414yes—
96#14519687—22no—
97C8×Dic61—22no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684D411yescharacteristic
109#14519684D411yescharacteristic
110C22×Q8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C4×Q8×C61—22no—
114C4×Q8×C61C2×C414yes—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C42×C61—21no—
122C2×C42×C61C2311yescharacteristic
123C4⋊C8×C61—22no—
124#103×C61—41no—
125#103×C61—41no—
126C23×D61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61C22×C412yes—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244C22×C412yes—
153C2×C4×D61—24no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—42no—
169C4⋊C4×C61—81no—
170C4⋊C4×C61C22×C412yes—
171C42×C61—24no—
172C42×C61C22×C412yes—
173C42×C61#4112yes—
174#11758201—28no—
175#11758198—24no—
176#11758198—42no—
177#11758198—42no—
178#11758198—81no—
179#11758198—81no—
180C4⋊Dic61—21no—
181C4⋊Dic61—22no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—41no—
187C4⋊Dic61C22×C412yes—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—42no—
193C4×Dic61—42no—
194C4×Dic61—42no—
195C4×Dic61—42no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418C22×C412yes—
199C2×#92418C22×C412yes—
200C2×#92418#4112yes—
201C2×#92418#4112yes—
202C23×C61C22×D411yescharacteristic
203C22×D61—22no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210Q8×C61—22no—
211Q8×C61—22no—
212Q8×C61—41no—
213Q8×C61—41no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61C4×D412yes—
231C2×C4×C61C4×D412yes—
232C2×C4×C61C23×C411yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61H(5,𝔽2)11yescharacteristic
237C2×C4×C612−1+411yescharacteristic
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—22no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61AGL(1,ℤ/8)12yes—
259D244—22no—
260D244—22no—
261D244—41no—
262D244—41no—
263C4×D61—48no—
264C8×C61—41no—
265C8×C61—41no—
266#92418—22no—
267#92418#11714yescomplemented
268#22820—611nomaximal, Sylow
269C22×C61C2×C4×D411yescharacteristic
270C22×C61#54711yescharacteristic
271C22×C61#55011yescharacteristic
272C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C61#57511yescharacteristic
274C22×C61#57711yescharacteristic
275C22×C61#57811yescharacteristic
276C2×D61—42no—
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C4×C61—22no—
284C4×C61—22no—
285C4×C61—42no—
286C4×C61—42no—
287C4×C61—81no—
288C4×C61—81no—
289C4×C61C2×#11711yescharacteristic
290C4×C61#52011yescharacteristic
291C4×C61#55511yescharacteristic
292C4×C61#55611yescharacteristic
293Dic61—22no—
294Dic61—41no—
295Dic61—41no—
296Dic61—41no—
297Dic61—42no—
298Dic61—81no—
299Dic61—81no—
300C2×#440—611no—
301#2356—611no—
302#2354—611no—
303#1973—611no—
304#1968—611no—
305#1928—611no—
306#1927—611no—
307#1926—611no—
308#1925—611no—
309#1847—611no—
310#1804—611no—
311#1803D6111yescharacteristic, complemented
312#1776—611no—
313C4×#107—611no—
314#1771—611no—
315C2×C61#235411yescharacteristic
316C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
317C2×C61#294711yescharacteristic
318C2×C61#334311yescharacteristic
319C2×C61#334811yescharacteristic
320C2×C61#339711yescharacteristic
321C2×C61#340211yescharacteristic
322D61—82no—
323C22×SD16—611no—
324C2×C4×Q8C2×D6111yescharacteristic
325C2×C4×D4—611no—
326C2×#117—1221no—
327#440—618no—
328C2×#107—611no—
329C2×#107—611no—
330C2×#107—611no—
331C2×#106—611no—
332C2×#106—611no—
333C2×C4⋊C8C2×D6111yescharacteristic
334C2×#103—611no—
335C2×#103C2×D6111yescharacteristic
336C2×#103C2×D6111yescharacteristic
337C2×#102—611no—
338C2×#102—611no—
339C2×#102—611no—
340C2×#97—611no—
341C2×#97—611no—
342C2×C4×C8—611no—
343#391—611no—
344#391—611no—
345#390—611no—
346#390—611no—
347#388—1221no—
348#388—1221no—
349#387—1221no—
350#387—1221no—
351#384C2×D6111yescharacteristic
352#383—1221no—
353#382—611no—
354#380—1221no—
355C4×C4⋊C4—611no—
356C4×C4⋊C4—611no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—611no—
359C4×C4⋊C4—611no—
360#376—1221no—
361#340—611no—
362#340—611no—
363#340C2×D6111yescharacteristic
364#340C2×D6111yescharacteristic
365#336—611no—
366#336—611no—
367C61#2282011yescharacteristic, complemented, Sylow
368C22×Q8C22×D6111yescharacteristic
369C22×D4—611no—
370C23×C4—1221no—
371C2×SD16—614no—
372C2×SD16—1222no—
373C2×SD16—1222no—
374C22×C8—611no—
375C22×C8—611no—
376C22×C8C22×D6111yescharacteristic
377C22×C8C22×D6111yescharacteristic
378C4×Q8—22no—
379C4×Q8C4×D6114yes—
380C4×D4—614no—
381C4×D4—1222no—
382#117—2444no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C4⋊C4C22×D6111yescharacteristic
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C42—21no—
400C2×C42—611no—
401C2×C42—611no—
402C2×C42—1221no—
403C2×C42—1221no—
404C2×C42—1221no—
405C2×C42—1221no—
406C2×C42C22×D6111yescharacteristic
407#107—614no—
408#107—1222no—
409#107—2441no—
410#106—1222no—
411#106—2441no—
412C4⋊C8—22no—
413#103—41no—
414#103—41no—
415#103—614no—
416#102—614no—
417#102—2441no—
418#102—2441no—
419#97—614no—
420#97—1222no—
421C4×C8—1222no—
422#95—611no—
423#95—611no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434C24—1221no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8C2×C4×D6112yes—
441C2×D4—612no—
442C2×D4—1221no—
443C2×D4—1221no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—1221no—
456C22×C4—1224no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—2442no—
460C22×C4—2442no—
461C22×C4D4×D6111yescharacteristic
462C22×C4D4×D6111yescharacteristic
463C22×C4C23×D6111yescharacteristic
464SD16—1228no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—612no—
472C2×C8—612no—
473C2×C8—612no—
474C2×C8—612no—
475C2×C8—1221no—
476C2×C8—1221no—
477C4⋊C4—21no—
478C4⋊C4—24no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—42no—
484C4⋊C4—81no—
485C4⋊C4—612no—
486C4⋊C4—1222no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—2441no—
493C4⋊C4C2×C4×D6112yes—
494C22⋊C4—1224no—
495C22⋊C4—4881no—
496C22⋊C4—4881no—
497C22⋊C4—2442no—
498C22⋊C4—2442no—
499C42—24no—
500C42—1221no—
501C42—1221no—
502C42—1221no—
503C42—1221no—
504C42—2442no—
505C42—2442no—
506C42—2442no—
507C42—2442no—
508C42C2×C4×D6112yes—
509C42#1175822412yes—
510C23—1222no—
511C23—2441no—
512C23—2441no—
513C23—2441no—
514C23—2441no—
515C23—2441no—
516C23—2441no—
517C23C2×D4×D6111yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522D4—1222no—
523D4—1222no—
524D4—2441no—
525D4—2441no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—612no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1222no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—2448no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4#1451973712yes—
565C2×C4#1451973712yes—
566C2×C4C22×C4×D6111yescharacteristic
567C2×C4#41×D6111yescharacteristic
568C2×C4#41×D6111yescharacteristic
569C2×C4C2×#1175822411yescharacteristic
570C2×C4#1451982211yescharacteristic
571C2×C4#1451982511yescharacteristic
572C8—41no—
573C8—41no—
574C8—614no—
575C8—1222no—
576C22—4881no—
577C22—4881no—
578C22—4881no—
579C22—4881no—
580C22—2442no—
581C22—2442no—
582C22—2442no—
583C22C2×#1451973711yescharacteristic
584C22#1684041411yescharacteristic
585C22#1684043011yescharacteristic
586C22AGL(1,ℤ/8)×D6111yescharacteristic
587C22#1684056111yescharacteristic
588C22#1684056811yescharacteristic
589C22#1684057511yescharacteristic
590C4—22no—
591C4—22no—
592C4—42no—
593C4—42no—
594C4—81no—
595C4—81no—
596C4—1222no—
597C4—4881no—
598C4—4881no—
599C4—2442no—
600C4—2441no—
601C4—2441no—
602C4—2441no—
603C4#117×D6111yescharacteristic
604C4#1684024311yescharacteristic
605C4#1684047611yescharacteristic
606C4#1684048311yescharacteristic
607C2—4882no—
608C2#2144876311yescharacteristic
609C2#2145463211yescharacteristic
610C2#2145465611yescharacteristic
611C2#2145825611yescharacteristic
612C2#2145834111yescharacteristic
613C2#2145883411yescharacteristic
614C2#2145889311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1803D61
hypocenterC61#22820
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. #21448784C2semidirect
6. #21446463C2semidirect
7. #21446456C2semidirect
10. #21445985C2semidirect
12. #21445715C2semidirect
13. #21445711C2semidirect
14. #21445696C2semidirect
15. C61×#1803C2semidirect
19. #16839860C4semidirect
267. #92418#117semidirect
311. #1803D61semidirect
367. C61#22820semidirect

Maximal subgroups

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

subgroupindex
C2×#168398602
#214487842
#214487702
#214466972
#214466702
#214464632
#214464562
#214464442
#214464372
#214459852
#214458302
#214457152
#214457112
#214456962
C61×#18032
#2282061