Finite Group Databasebeta

← #30596657

Subgroups — #30596657

Order 15616.

1026 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462263C211yesmaximal, characteristic
2C2×#16840704C211yesmaximal, characteristic
3#21460682C214yesmaximal, complemented
4C2×#16839725C211yesmaximal, characteristic
5C2×#14519856—21no—
6C22×C4⋊#6923—21no—
7C22×C22⋊#6923—21no—
8#16840705C412yescomplemented
9#16840704—21no—
10#16840704—21no—
11#16840704—21no—
12#16840704C412yescomplemented
13#16840704C2212yes—
14#16840704C2212yes—
15#16840703—21no—
16C4×C22⋊#6923—21no—
17C4×C22⋊#6923C412yescomplemented
18C2×(C2×#6923)⋊C4C412yescomplemented
19C23×C4×D61C2211yescharacteristic
20C22×#11758194C411yescharacteristic, complemented
21#16839725—21no—
22#16839725—21no—
23#16839725—21no—
24#16839725—21no—
25C2×#14519689C412yescomplemented
26C2×#14519689C2212yes—
27C22×M16×C61C411yescharacteristic, complemented
28C23×#6923—22no—
29#14519856—42no—
30#14519856—42no—
31C2×C4⋊#6923—22no—
32C2×C4⋊#6923—22no—
33C2×C4⋊#6923—22no—
34C2×C4⋊#6923—42no—
35C2×C4⋊#6923—42no—
36C2×C22⋊#6923—22no—
37C2×C22⋊#6923—22no—
38C2×C22⋊#6923—22no—
39C2×C22⋊#6923—22no—
40C2×C22⋊#6923—22no—
41C2×C22⋊#6923—44no—
42C2×C4×#6923—22no—
43(C2×#6923)⋊C4—22no—
44(C2×#6923)⋊C4—22no—
45(C2×#6923)⋊C4C2×C414yes—
46(C2×#6923)⋊C4C2×C414yescomplemented
47C24×D61D411yescharacteristic
48C23×Dic61Q811yescharacteristic
49C22×C4×D61—42no—
50C22×C4×D61—42no—
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61C2×C411yescharacteristic
54C22×C4×D61C2×C411yescharacteristic
55C22×C4×D61C2×C411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C22×C4×D61Q811yescharacteristic
62C22×C4×D61C2311yescharacteristic
63C2×#11758194—21no—
64C2×#11758194—21no—
65C2×#11758194—42no—
66C22×#92418C2×C412yescomplemented
67#14519689—22no—
68#14519689—22no—
69#14519689—22no—
70#14519689—22no—
71C23×C4×C61C2×C411yescharacteristic
72C2×M16×C61—21no—
73C2×M16×C61—21no—
74C2×M16×C61—42no—
75C22×C8×C61C2×C412yescomplemented
76C24×C61C4⋊C411yescharacteristic
77C23×D61—22no—
78C23×D61—22no—
79C23×D61—42no—
80C23×D61—42no—
81C23×D61C22⋊C411yescharacteristic
82C23×D61C22⋊C411yescharacteristic
83C23×D61C22⋊C411yescharacteristic
84C23×D61C22⋊C411yescharacteristic
85C23×D61C4⋊C411yescharacteristic
86C23×D61C2×D411yescharacteristic
87C22×#6923—24no—
88C22×#6923—24no—
89C22×#6923—44no—
90C22×#6923—44no—
91C22×#6923—44no—
92C22×C4×C61—22no—
93C22×C4×C61—22no—
94C22×C4×C61C4211yescharacteristic
95C22×C4×C61C4211yescharacteristic
96C22×C4×C61C22⋊C411yescharacteristic
97C22×C4×C61C22⋊C411yescharacteristic
98C22×C4×C61C4⋊C411yescharacteristic
99C22×C4×C61C22×C411yescharacteristic
100C22×Dic61—42no—
101C22×Dic61C4⋊C411yescharacteristic
102C22×Dic61C4⋊C411yescharacteristic
103C22×Dic61C4⋊C411yescharacteristic
104C22×Dic61C4⋊C411yescharacteristic
105C22×Dic61C4⋊C411yescharacteristic
106C22×Dic61C2×Q811yescharacteristic
107C2×C4×D61—22no—
108C2×C4×D61—22no—
109C2×C4×D61—22no—
110C2×C4×D61—22no—
111C2×C4×D61—42no—
112C2×C4×D61—42no—
113C2×C4×D61—41no—
114C2×C4×D61—41no—
115C2×C4×D61—42no—
116C2×C4×D61—41no—
117C2×C4×D61—41no—
118C2×C4×D61—41no—
119C2×C4×D61—41no—
120C2×C4×D61—42no—
121C2×C4×D61—41no—
122C2×C4×D61—41no—
123C2×C4×D61—42no—
124C2×C4×D61—42no—
125C2×C4×D61C4211yescharacteristic
126C2×C4×D61C4211yescharacteristic
127C2×C4×D61C22⋊C411yescharacteristic
128C2×C4×D61C22⋊C411yescharacteristic
129C2×C4×D61C22⋊C411yescharacteristic
130C2×C4×D61C22⋊C411yescharacteristic
131C2×C4×D61C22⋊C411yescharacteristic
132C2×C4×D61C22⋊C411yescharacteristic
133C2×C4×D61C4⋊C411yescharacteristic
134C2×C4×D61C4⋊C411yescharacteristic
135C2×C4×D61C4⋊C411yescharacteristic
136C2×C4×D61C4⋊C411yescharacteristic
137C2×C4×D61C22×C411yescharacteristic
138C2×C4×D61C22×C411yescharacteristic
139C2×C4×D61C2×D411yescharacteristic
140C2×C4×D61C2×D411yescharacteristic
141C22⋊#6923—44no—
142C22⋊#6923—44no—
143C22⋊#6923—44no—
144C22⋊#6923—42no—
145C22⋊#6923—44no—
146C22⋊#6923—42no—
147C4⋊#6923—44no—
148C4⋊#6923—42no—
149C4⋊#6923—42no—
150C4⋊#6923—42no—
151C4⋊#6923—42no—
152C4×#6923—44no—
153M16×C61—42no—
154M16×C61—42no—
155C2×C8×C61—22no—
156C2×C8×C61—42no—
157#11758194—42no—
158#11758194—42no—
159C2×#92418—22no—
160C2×#92418—42no—
161C23×C61—22no—
162C23×C61—22no—
163C23×C61—41no—
164C23×C61#9511yescharacteristic
165C23×C61#9511yescharacteristic
166C23×C61C2×C4⋊C411yescharacteristic
167C22×D61—22no—
168C22×D61—22no—
169C22×D61—21no—
170C22×D61—21no—
171C22×D61—22no—
172C22×D61—22no—
173C22×D61—22no—
174C22×D61—22no—
175C22×D61—22no—
176C22×D61—22no—
177C22×D61—21no—
178C22×D61—21no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—48no—
187C22×D61—48no—
188C22×D61#9511yescharacteristic
189C22×D61#9511yescharacteristic
190C22×D61C2×C22⋊C411yescharacteristic
191C22×D61C2×C22⋊C411yescharacteristic
192C22×D61AGL(1,ℤ/8)12yes—
193C22×D61#13712yes—
194C2×#6923—44no—
195C2×#6923—44no—
196C2×#6923—44no—
197C2×#6923—44no—
198C2×#6923—44no—
199C2×#6923—44no—
200C2×#6923—88no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61#9511yescharacteristic
209C2×C4×C61#9511yescharacteristic
210C2×C4×C61#9912yes—
211C2×C4×C61#9912yes—
212C2×C4×C61C2×C4211yescharacteristic
213C2×C4×C61C2×C22⋊C411yescharacteristic
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61#9511yescharacteristic
221C2×Dic61#9511yescharacteristic
222C2×Dic61C2×C4⋊C411yescharacteristic
223C2×Dic61C2×C4⋊C411yescharacteristic
224C4×D61—22no—
225C4×D61—22no—
226C4×D61—22no—
227C4×D61—22no—
228C4×D61—21no—
229C4×D61—21no—
230C4×D61—21no—
231C4×D61—21no—
232C4×D61—21no—
233C4×D61—21no—
234C4×D61—21no—
235C4×D61—21no—
236C4×D61—41no—
237C4×D61—41no—
238C4×D61—41no—
239C4×D61—41no—
240C4×D61—41no—
241C4×D61—41no—
242C4×D61—41no—
243C4×D61—41no—
244C8×C61—42no—
245#92418—42no—
246#7983—611nomaximal, Sylow
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—44no—
255C22×C61#34212yes—
256C22×C61C2×#9511yescharacteristic
257C22×C61#42812yes—
258C2×D61—24no—
259C2×D61—24no—
260C2×D61—21no—
261C2×D61—21no—
262C2×D61—21no—
263C2×D61—21no—
264C2×D61—21no—
265C2×D61—21no—
266C2×D61—21no—
267C2×D61—21no—
268C2×D61—21no—
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—44no—
272C2×D61—44no—
273C2×D61—44no—
274C2×D61—42no—
275C2×D61—42no—
276C2×D61H(3,ℤ/4)12yes—
277C2×D61#41712yes—
278C2×D61#41811yescharacteristic
279C2×D61#41911yescharacteristic
280#6923—84no—
281#6923—84no—
282C4×C61—21no—
283C4×C61—21no—
284C4×C61#34212yes—
285C4×C61C2×#9911yescharacteristic
286C4×C61C2×#9911yescharacteristic
287Dic61—21no—
288Dic61—21no—
289Dic61—21no—
290Dic61—21no—
291C2×#407—611no—
292#2280—611no—
293C2×#380—611no—
294#1289—614no—
295C2×C61—22no—
296C2×C61—22no—
297C2×C61—41no—
298C2×C61C2×H(3,ℤ/4)11yescharacteristic
299C2×C61#173611yescharacteristic
300C2×C61C2×#34211yescharacteristic
301D61—21no—
302D61—22no—
303D61—22no—
304D61—21no—
305D61—41no—
306C24×C4—611no—
307C22×M16—611no—
308C22×M16#692311yescharacteristic, complemented
309C2×#117—1221no—
310C22×C4⋊C4—1221no—
311C22×C22⋊C4—1221no—
312#407—1221no—
313#407—1221no—
314#407—1221no—
315#407—1221no—
316C2×C22⋊C8—612no—
317C2×C22⋊C8—612no—
318#381—612no—
319#380—612no—
320#380—612no—
321#380—612no—
322#380—1221no—
323#380—1221no—
324#380—1221no—
325#379—1221no—
326C4×C22⋊C4—612no—
327C4×C22⋊C4—1221no—
328C2×#95—612no—
329C61#798311yescharacteristic, complemented, Sylow
330C25—611no—
331C23×C4—611no—
332C23×C4—611no—
333C23×C4—611no—
334C23×C4—611no—
335C23×C4—611no—
336C23×C4—611no—
337C23×C4—611no—
338C23×C4—611no—
339C23×C4—611no—
340C23×C4—611no—
341C23×C4—611no—
342C23×C4—611no—
343C23×C4—611no—
344C23×C4—1222no—
345C23×C4—2442no—
346C23×C4—2442no—
347C23×C4C2×#692311yescharacteristic
348C2×M16—21no—
349C2×M16—21no—
350C2×M16—42no—
351C2×M16—1221no—
352C2×M16—1221no—
353C2×M16—2442no—
354C22×C8—612no—
355C22×C8C2×#692312yescomplemented
356#117—2442no—
357#117—2442no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4—1222no—
361C2×C4⋊C4—2442no—
362C2×C4⋊C4—2442no—
363C2×C22⋊C4—1222no—
364C2×C22⋊C4—1222no—
365C2×C22⋊C4—1222no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—2444no—
369C2×C42—1222no—
370C22⋊C8—1222no—
371C22⋊C8—1222no—
372C22⋊C8—1222no—
373C22⋊C8—1222no—
374#95—614no—
375#95—614no—
376#95—1222no—
377#95—1222no—
378C24—611no—
379C24—611no—
380C24—611no—
381C24—611no—
382C24—611no—
383C24—611no—
384C24—1222no—
385C24—1222no—
386C24—2442no—
387C24—2442no—
388C24C4⋊#692311yescharacteristic
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—611no—
392C22×C4—611no—
393C22×C4—611no—
394C22×C4—611no—
395C22×C4—611no—
396C22×C4—611no—
397C22×C4—611no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—611no—
401C22×C4—611no—
402C22×C4—611no—
403C22×C4—611no—
404C22×C4—611no—
405C22×C4—611no—
406C22×C4—611no—
407C22×C4—611no—
408C22×C4—611no—
409C22×C4—611no—
410C22×C4—611no—
411C22×C4—611no—
412C22×C4—611no—
413C22×C4—1222no—
414C22×C4—1222no—
415C22×C4—1222no—
416C22×C4—1222no—
417C22×C4—1224no—
418C22×C4—1224no—
419C22×C4—2441no—
420C22×C4—2441no—
421C22×C4—2441no—
422C22×C4—2441no—
423C22×C4—2441no—
424C22×C4—2441no—
425C22×C4—2441no—
426C22×C4—2441no—
427C22×C4—2442no—
428C22×C4—2442no—
429C22×C4—2442no—
430C22×C4—2442no—
431C22×C4—2442no—
432C22×C4—2442no—
433C22×C4—2442no—
434C22×C4—2444no—
435C22×C4—2444no—
436C22×C4—2444no—
437C22×C4C4×#692311yescharacteristic
438C22×C4C4×#692311yescharacteristic
439C22×C4C4⋊#692311yescharacteristic
440C22×C4C22⋊#692311yescharacteristic
441C22×C4C22⋊#692311yescharacteristic
442C22×C4C22×#692311yescharacteristic
443M16—42no—
444M16—42no—
445M16—2442no—
446M16—2442no—
447C2×C8—22no—
448C2×C8—42no—
449C2×C8—1222no—
450C2×C8—2442no—
451C4⋊C4—2442no—
452C4⋊C4—2442no—
453C4⋊C4—2444no—
454C4⋊C4—2442no—
455C4⋊C4—2442no—
456C22⋊C4—2444no—
457C22⋊C4—2442no—
458C22⋊C4—2442no—
459C22⋊C4—2444no—
460C22⋊C4—2444no—
461C22⋊C4—2444no—
462C42—2444no—
463C23—22no—
464C23—22no—
465C23—41no—
466C23—611no—
467C23—611no—
468C23—611no—
469C23—612no—
470C23—612no—
471C23—611no—
472C23—1221no—
473C23—1222no—
474C23—1221no—
475C23—1221no—
476C23—1222no—
477C23—1222no—
478C23—1222no—
479C23—1221no—
480C23—1222no—
481C23—1222no—
482C23—1222no—
483C23—1222no—
484C23—2441no—
485C23—2441no—
486C23—2441no—
487C23—2441no—
488C23—2441no—
489C23—2441no—
490C23—2441no—
491C23—2448no—
492C23—2448no—
493C23(C2×#6923)⋊C411yescharacteristic
494C23(C2×#6923)⋊C411yescharacteristic
495C23C2×C4⋊#692311yescharacteristic
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—611no—
504C2×C4—611no—
505C2×C4—611no—
506C2×C4—611no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—1221no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—4888no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2444no—
535C2×C4—2444no—
536C2×C4—2444no—
537C2×C4—2444no—
538C2×C4—2444no—
539C2×C4—2444no—
540C2×C4(C2×#6923)⋊C411yescharacteristic
541C2×C4(C2×#6923)⋊C411yescharacteristic
542C2×C4#1451983512yes—
543C2×C4#1451983612yes—
544C2×C4C2×C4×#692311yescharacteristic
545C2×C4C2×C22⋊#692311yescharacteristic
546C8—42no—
547C8—2442no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—44no—
556C22—611no—
557C22—612no—
558C22—611no—
559C22—612no—
560C22—1224no—
561C22—1224no—
562C22—1221no—
563C22—1221no—
564C22—1221no—
565C22—1221no—
566C22—1221no—
567C22—1221no—
568C22—1221no—
569C22—1221no—
570C22—1221no—
571C22—2442no—
572C22—2442no—
573C22—2444no—
574C22—2444no—
575C22—2444no—
576C22—2442no—
577C22—2442no—
578C22#1684064012yes—
579C22C2×(C2×#6923)⋊C411yescharacteristic
580C22#1684077312yes—
581C4—21no—
582C4—21no—
583C4—1221no—
584C4—1221no—
585C4—1221no—
586C4—1221no—
587C4—4884no—
588C4—4884no—
589C4#1684064012yes—
590C4C2×#1451983511yescharacteristic
591C4C2×#1451983611yescharacteristic
592C2—22no—
593C2—22no—
594C2—41no—
595C2—1221no—
596C2—1222no—
597C2—1221no—
598C2—1222no—
599C2—2441no—
600C2#2146149111yescharacteristic
601C2#2146150711yescharacteristic
602C2C2×#1684064011yescharacteristic
subgroupG/H
centerC22C2×(C2×#6923)⋊C4
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC22×M16×C61C4
socleC22×C61C2×#95
prime radicalC24×D61D4
cosocleC22×C4×D61C23
hypercenterC22×M16#6923
hypocenterC61#7983
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
3. #21460682C2semidirect
8. #16840705C4semidirect
12. #16840704C4semidirect
17. C4×C22⋊#6923C4semidirect
18. C2×(C2×#6923)⋊C4C4semidirect
20. C22×#11758194C4semidirect
25. C2×#14519689C4semidirect
27. C22×M16×C61C4semidirect
46. (C2×#6923)⋊C4C2×C4semidirect
66. C22×#92418C2×C4semidirect
75. C22×C8×C61C2×C4semidirect
308. C22×M16#6923semidirect
329. C61#7983semidirect
355. C22×C8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214622632
C2×#168407042
#214606822
C2×#168397252
#798361