Finite Group Databasebeta

← #29575139

Subgroups — #29575139

Order 15616.

834 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839901C211yesmaximal, characteristic
2#21448522C211yesmaximal, characteristic
3#21445754C211yesmaximal, characteristic, complemented
4C2×#16839495C211yesmaximal, characteristic
5#21443698C211yesmaximal, characteristic, complemented
6#21443662C211yesmaximal, characteristic, complemented
7C61×#1446C211yesmaximal, characteristic, complemented
8C22×#11758201—21no—
9C2×#14519751C2211yescharacteristic
10C2×C4×C22⋊D61—21no—
11#16839901—22no—
12#16839901C414yescomplemented
13C2×#14519701—21no—
14C2×#14519701—21no—
15C2×#14519701—21no—
16C2×C22⋊#92418C2211yescharacteristic
17C2×C22⋊#92418C2211yescharacteristic
18#16839654—21no—
19#16839647—21no—
20#16839641—21no—
21#16839640—21no—
22#16839636—21no—
23#16839622C2211yescharacteristic
24#16839495—22no—
25C23×Q8×C61C2211yescharacteristic
26C2×#95×C61—21no—
27C2×#95×C61—21no—
28C2×#95×C61C2211yescharacteristic
29C2×#95×C61C2211yescharacteristic
30C22×D244D411yescharacteristic
31C22×C22⋊D61—21no—
32C22×C4×D61—21no—
33C2×#11758201—24no—
34C2×#11758201—42no—
35C2×#11758201—42no—
36C22×#92418—21no—
37C22×#92418—21no—
38#14519751—21no—
39#14519751—21no—
40#14519751C2×C412yes—
41#14519751D412yes—
42C4×C22⋊D61—42no—
43C4×C22⋊D61—42no—
44C2×C4⋊Dic61D411yescharacteristic
45C2×C4⋊Dic61D411yescharacteristic
46C2×C4⋊Dic61D411yescharacteristic
47C2×#11758196—21no—
48C2×C22⋊Dic61—21no—
49C2×#11758198—21no—
50C2×#11758198—21no—
51C2×#11758198—21no—
52C2×#11758198—21no—
53C2×C4×Dic61—21no—
54#14519701—24no—
55#14519701—42no—
56#14519701—42no—
57C22⋊#92418—21no—
58C22⋊#92418—21no—
59C22⋊#92418—21no—
60C22⋊#92418C2×C412yescomplemented
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64C22×Q8×C61—21no—
65C22×Q8×C61—21no—
66C22×Q8×C61—21no—
67C22×Q8×C61—21no—
68C22×Q8×C61—21no—
69C22×Q8×C61—42no—
70C22×Q8×C61—42no—
71C22×Q8×C61C2×C412yescomplemented
72C23×C4×C61—21no—
73C23×C4×C61—21no—
74C23×C4×C61D411yescharacteristic
75C23×C4×C61D411yescharacteristic
76C23×C4×C61C2311yescharacteristic
77#95×C61—21no—
78#95×C61—21no—
79#95×C61—21no—
80#95×C61—21no—
81#95×C61—42no—
82#95×C61—42no—
83C24×C61C2×D411yescharacteristic
84C23×D61—21no—
85C2×Q8×C61—22no—
86C2×Q8×C61—22no—
87C2×Q8×C61—22no—
88C2×Q8×C61—22no—
89C2×Q8×C61—22no—
90C2×Q8×C61—22no—
91C2×Q8×C61—44no—
92C2×Q8×C61—41no—
93C2×Q8×C61—41no—
94C2×Q8×C61—41no—
95C2×Q8×C61—41no—
96C2×Q8×C61—44no—
97C2×Q8×C61—44no—
98C2×Q8×C61—44no—
99C2×Q8×C61—41no—
100C2×Q8×C61—41no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—22no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—41no—
111C22×C4×C61—41no—
112C22×C4×C61—41no—
113C22×C4×C61—41no—
114C22×C4×C61—41no—
115C22×C4×C61—41no—
116C22×C4×C61—41no—
117C22×C4×C61—41no—
118C22×C4×C61—41no—
119C22×C4×C61—41no—
120C22×C4×C61—41no—
121C22×C4×C61—41no—
122C22×C4×C61—41no—
123C22×C4×C61—41no—
124C22×C4×C61—41no—
125C22×C4×C61—41no—
126C22×C4×C61SD1612yescomplemented
127C22×C4×C61C22×C411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61#4111yescharacteristic
132C22×C4×C61#4111yescharacteristic
133C2×C22⋊D61—22no—
134C2×C22⋊D61—22no—
135C2×C22⋊D61—41no—
136C2×C22⋊D61—41no—
137C2×C22⋊D61—41no—
138C2×C22⋊D61—41no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C2×D244—21no—
143C2×D244—41no—
144C2×D244—41no—
145C2×D244C22⋊C412yes—
146C2×C4×D61—42no—
147C2×C4×D61—81no—
148C2×C4×D61—81no—
149C22⋊Dic61—41no—
150C22⋊Dic61—41no—
151#11758201—48no—
152#11758198—22no—
153#11758198—22no—
154#11758198—42no—
155#11758198—81no—
156#11758198—81no—
157C4⋊Dic61—21no—
158C4⋊Dic61—41no—
159C4⋊Dic61—41no—
160C4⋊Dic61C22⋊C412yes—
161#11758196—42no—
162C4×Dic61—42no—
163C2×#92418—22no—
164C2×#92418—41no—
165C2×#92418—41no—
166C2×#92418—41no—
167C2×#92418—41no—
168C2×#92418—41no—
169C23×C61—21no—
170C23×C61—21no—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61C4×D411yescharacteristic
174C23×C61#12011yescharacteristic
175C23×C61#12111yescharacteristic
176C23×C61C2×SD1611yescharacteristic
177C23×C61#13511yescharacteristic
178C23×C61AGL(1,ℤ/8)11yescharacteristic
179C23×C61#13711yescharacteristic
180C22×D61—22no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187Q8×C61—24no—
188Q8×C61—24no—
189Q8×C61—42no—
190Q8×C61—42no—
191Q8×C61—42no—
192Q8×C61—42no—
193Q8×C61—42no—
194Q8×C61—42no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—24no—
200C2×C4×C61—22no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—81no—
222C2×C4×C61—81no—
223C2×C4×C61—81no—
224C2×C4×C61—81no—
225C2×C4×C61—81no—
226C2×C4×C61—81no—
227C2×C4×C61C2×C22⋊C411yescharacteristic
228C2×C4×C61C4×D411yescharacteristic
229C2×C4×C61#12111yescharacteristic
230C2×C4×C61#12311yescharacteristic
231C22⋊D61—42no—
232C22⋊D61—42no—
233C22⋊D61—42no—
234C22⋊D61—42no—
235C2×Dic61—22no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244D244—42no—
245D244—81no—
246C4×D61—84no—
247#92418—42no—
248#92418—81no—
249#9039—611nomaximal, Sylow
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61#38611yescharacteristic
263C22×C61C4×SD1611yescharacteristic
264C22×C61#44011yescharacteristic
265C22×C61#45011yescharacteristic
266C22×C61#45211yescharacteristic
267C22×C61#45511yescharacteristic
268C22×C61#45611yescharacteristic
269C22×C61B(4,𝔽2)11yescharacteristic
270C22×C61B(4,𝔽2)11yescharacteristic
271C22×C61#46111yescharacteristic
272C22×C61#46311yescharacteristic
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—42no—
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—21no—
284C4×C61—42no—
285C4×C61—44no—
286C4×C61—82no—
287C4×C61—81no—
288Dic61—42no—
289Dic61—81no—
290Dic61—81no—
291C2×#448—611no—
292#2325—611no—
293#1787—611no—
294C2×#327—611no—
295#1446D6111yescharacteristic, complemented
296#1294—611no—
297#1280—611no—
298C2×C61—21no—
299C2×C61—21no—
300C2×C61—21no—
301C2×C61—21no—
302C2×C61#160111yescharacteristic
303C2×C61#160311yescharacteristic
304C2×C61#160411yescharacteristic
305C2×C61#160611yescharacteristic
306C2×C61#187711yescharacteristic
307C2×C61#188211yescharacteristic
308C2×C61#188911yescharacteristic
309D61—82no—
310C23×Q8C2×D6111yescharacteristic
311C22×SD16—1221no—
312C2×#121—611no—
313C2×C4×D4—1221no—
314#448—614no—
315#448—1222no—
316C2×#103—1221no—
317C2×#103—1221no—
318C2×#103—1221no—
319C2×C22⋊C8—611no—
320C2×C22⋊C8—611no—
321#388—1221no—
322#387—1221no—
323#386—1221no—
324#386—1221no—
325#385—1221no—
326#380—611no—
327C2×#95—21no—
328C2×#95—21no—
329C2×#95C2×D6111yescharacteristic
330C2×#95C2×D6111yescharacteristic
331#327—1222no—
332C61#903911yescharacteristic, complemented, Sylow
333C22×Q8—21no—
334C22×Q8—21no—
335C22×Q8—21no—
336C22×Q8—21no—
337C22×Q8—21no—
338C22×Q8—42no—
339C22×Q8—42no—
340C22×Q8C4×D6112yescomplemented
341C22×D4—611no—
342C22×D4—1221no—
343C23×C4—21no—
344C23×C4—21no—
345C23×C4—1221no—
346C23×C4C22⋊D6111yescharacteristic
347C23×C4C22⋊D6111yescharacteristic
348C23×C4C22×D6111yescharacteristic
349C2×SD16—1224no—
350C2×SD16—2442no—
351C2×SD16—2442no—
352C22×C8—1221no—
353C22×C8—1221no—
354#121—612no—
355#121—612no—
356#121—1221no—
357#121—1221no—
358C4×D4—2442no—
359C4×D4—2442no—
360C2×C4⋊C4—611no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—1221no—
364C2×C22⋊C4—1221no—
365C2×C22⋊C4—1221no—
366C2×C22⋊C4—1221no—
367C2×C22⋊C4—1221no—
368C2×C22⋊C4—1221no—
369C2×C42—1221no—
370#103—1224no—
371#103—2442no—
372#103—2442no—
373C22⋊C8—612no—
374C22⋊C8—1221no—
375C22⋊C8—1221no—
376C22⋊C8—1221no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—42no—
382#95—42no—
383#95—1221no—
384#95—1221no—
385#95—1221no—
386C24—1221no—
387C24C2×C22⋊D6111yescharacteristic
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—44no—
395C2×Q8—41no—
396C2×Q8—41no—
397C2×Q8—41no—
398C2×Q8—41no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—41no—
403C2×Q8—41no—
404C2×D4—612no—
405C2×D4—1222no—
406C2×D4—1222no—
407C2×D4—1221no—
408C2×D4—2441no—
409C2×D4—2441no—
410C2×D4—2441no—
411C2×D4—2441no—
412C2×D4—2441no—
413C2×D4—2441no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—4881no—
443C22×C4—4881no—
444C22×C4—2442no—
445C22×C4#1175820112yescomplemented
446C22×C4C2×C4×D6111yescharacteristic
447C22×C4#1175822211yescharacteristic
448C22×C4D4×D6111yescharacteristic
449C22×C4D4×D6111yescharacteristic
450C22×C4D4×D6111yescharacteristic
451C22×C4#1175822411yescharacteristic
452SD16—2448no—
453C2×C8—1222no—
454C2×C8—2441no—
455C2×C8—2441no—
456C2×C8—2441no—
457C2×C8—2441no—
458C2×C8—2441no—
459C4⋊C4—612no—
460C4⋊C4—1221no—
461C4⋊C4—2441no—
462C4⋊C4—2441no—
463C4⋊C4—2442no—
464C22⋊C4—1222no—
465C22⋊C4—1222no—
466C22⋊C4—4881no—
467C22⋊C4—4881no—
468C22⋊C4—2442no—
469C22⋊C4—2441no—
470C22⋊C4—2441no—
471C42—2442no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—1222no—
477C23—2441no—
478C23—2441no—
479C23—2441no—
480C23—2441no—
481C23—2441no—
482C23—2441no—
483C23C4×C22⋊D6111yescharacteristic
484C23#1451974511yescharacteristic
485C23#1451974811yescharacteristic
486C23C2×#1175820111yescharacteristic
487C23#1451979711yescharacteristic
488C23#1451980211yescharacteristic
489C23#1451980811yescharacteristic
490Q8—24no—
491Q8—24no—
492Q8—42no—
493Q8—42no—
494Q8—42no—
495Q8—42no—
496Q8—42no—
497Q8—42no—
498D4—4881no—
499D4—2442no—
500D4—2442no—
501D4—2442no—
502D4—2442no—
503D4—2442no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—1222no—
537C2×C4—4884no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4C22⋊C4×D6111yescharacteristic
547C2×C4#1451973711yescharacteristic
548C2×C4#1451975011yescharacteristic
549C2×C4#1451975911yescharacteristic
550C8—4881no—
551C8—2442no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—4881no—
565C22—4881no—
566C22—4881no—
567C22—4881no—
568C22—2442no—
569C22—2442no—
570C22—2442no—
571C22#1683963811yescharacteristic
572C22C4×#1175820111yescharacteristic
573C22#1683986011yescharacteristic
574C22#1683991211yescharacteristic
575C22#1683992211yescharacteristic
576C22#1683993911yescharacteristic
577C22#1683994411yescharacteristic
578C22#1683994811yescharacteristic
579C22#1683994811yescharacteristic
580C22#1683996411yescharacteristic
581C22#1683997411yescharacteristic
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—44no—
587C4—42no—
588C4—82no—
589C4—81no—
590C4—4881no—
591C4—4881no—
592C4—2442no—
593C2—21no—
594C2—21no—
595C2—21no—
596C2—21no—
597C2—4882no—
598C2#2144477011yescharacteristic
599C2#2144478011yescharacteristic
600C2#2144478711yescharacteristic
601C2#2144479711yescharacteristic
602C2#2144615611yescharacteristic
603C2#2144619111yescharacteristic
604C2#2144622411yescharacteristic
subgroupG/H
centerC23#14519745
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1446C2
socleC23×C61#120
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1446D61
hypocenterC61#9039
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1446

Complements

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

NH ≅ G/Ntype
3. #21445754C2semidirect
5. #21443698C2semidirect
6. #21443662C2semidirect
7. C61×#1446C2semidirect
12. #16839901C4semidirect
60. C22⋊#92418C2×C4semidirect
71. C22×Q8×C61C2×C4semidirect
126. C22×C4×C61SD16semidirect
295. #1446D61semidirect
332. C61#9039semidirect
340. C22×Q8C4×D61semidirect
445. C22×C4#11758201semidirect

Maximal subgroups

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

subgroupindex
C2×#168399012
#214485222
#214457542
C2×#168394952
#214436982
#214436622
C61×#14462
#903961