Finite Group Databasebeta

← #30609196

Subgroups — #30609196

Order 15616.

2554 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
1C23×C4⋊#6923C211yesmaximal, characteristic, complemented
2C2×#16840710C212yesmaximal
3C2×#16840705C211yesmaximal, characteristic, complemented
4#21461004C214yesmaximal, complemented
5#21461002C214yesmaximal, complemented
6#21460992C212yesmaximal, complemented
7C2×#123×D61C211yesmaximal, characteristic
8C24×#6923C2212yes—
9C22×C4⋊#6923—28no—
10C22×C4⋊#6923—24no—
11C22×C4⋊#6923C2214yes—
12C22×C22⋊#6923C2212yes—
13C22×C22⋊#6923C2214yes—
14#16840710—28no—
15#16840705—24no—
16C2×(C2×#6923)⋊C4C2214yes—
17C2×(C2×#6923)⋊C4C2214yes—
18C2×(C2×#6923)⋊C4C2218yes—
19C22×D4×D61C2211yescharacteristic
20C23×C4×D61C2211yescharacteristic
21C2×#14519762C412yescomplemented
22C2×#14519761C411yescharacteristic, complemented
23C2×#14519760C411yescharacteristic, complemented
24C2×#14519759C412yescomplemented
25C2×#14519758C411yescharacteristic, complemented
26#123×D61—24no—
27#123×D61—24no—
28C2×C4⋊C4×D61C2212yes—
29C2×C22⋊C4×D61C2211yescharacteristic
30C2×C22⋊C4×D61C2212yes—
31C2×#123×C61C411yescharacteristic, complemented
32C23×#6923—24no—
33C23×#6923—24no—
34C23×#6923—48no—
35C23×#6923D418yes—
36C23×#6923C2314yes—
37C23×#6923C2314yes—
38C2×C4⋊#6923—232no—
39C2×C4⋊#6923—216no—
40C2×C4⋊#6923—28no—
41C2×C22⋊#6923—24no—
42C2×C22⋊#6923—24no—
43C2×C22⋊#6923—24no—
44C2×C22⋊#6923—28no—
45C2×C22⋊#6923—48no—
46(C2×#6923)⋊C4—28no—
47(C2×#6923)⋊C4—28no—
48(C2×#6923)⋊C4—48no—
49C24×D61C2311yescharacteristic
50C24×D61C2311yescharacteristic
51C22×D244C2×C411yescharacteristic
52C22×C22⋊D61C2×C411yescharacteristic
53C22×C22⋊D61C2×C411yescharacteristic
54C2×D4×D61—24no—
55C2×D4×D61—22no—
56C2×D4×D61—22no—
57C2×D4×D61—24no—
58C23×Dic61C2×C411yescharacteristic
59C22×C4×D61—24no—
60C22×C4×D61—24no—
61C22×C4×D61C2×C412yes—
62C22×C4×D61D414yes—
63C22×C4×D61D414yes—
64C22×C4×D61C2312yes—
65C22×C4×D61C2311yescharacteristic
66C22×C4×D61C2312yes—
67#14519762—28no—
68#14519761—24no—
69#14519760—24no—
70#14519759—28no—
71#14519758—24no—
72C2×C4⋊Dic61C2×C412yescomplemented
73C2×#11758196C2×C412yescomplemented
74C2×#11758196C2×C412yescomplemented
75C4⋊C4×D61—24no—
76C4⋊C4×D61—24no—
77C2×C22⋊Dic61C2×C411yescharacteristic
78C2×C22⋊Dic61C2×C412yescomplemented
79C2×#11758198C2×C412yes—
80C2×#11758198C2×C412yescomplemented
81C2×#11758198C2×C412yescomplemented
82C22⋊C4×D61—22no—
83C22⋊C4×D61—22no—
84C22⋊C4×D61—24no—
85C22⋊C4×D61—24no—
86C22×D4×C61C2×C411yescharacteristic
87C23×C4×C61C2×C411yescharacteristic
88#123×C61—24no—
89C2×C4⋊C4×C61C2×C412yescomplemented
90C2×C22⋊C4×C61C2×C411yescharacteristic
91C2×C22⋊C4×C61C2×C412yescomplemented
92C24×C61C22×C411yescharacteristic
93C24×C61C22×C411yescharacteristic
94C23×D61—24no—
95C23×D61—24no—
96C23×D61—22no—
97C23×D61—21no—
98C23×D61—21no—
99C23×D61—22no—
100C23×D61—24no—
101C23×D61—24no—
102C23×D61C22×C411yescharacteristic
103C23×D61C22×C411yescharacteristic
104C23×D61C2×D412yes—
105C23×D61C2×D412yes—
106C23×D61C2×D414yes—
107C23×D61#4114yes—
108C23×D61C2411yescharacteristic
109C22×#6923—216no—
110C22×#6923—24no—
111C22×#6923—24no—
112C22×#6923—28no—
113C22×#6923—28no—
114C22×#6923—24no—
115C22×#6923—24no—
116C22×#6923—24no—
117C22×#6923—28no—
118C22×#6923—28no—
119C22×#6923—416no—
120C22×#6923—416no—
121C22×#6923—416no—
122C22×#6923C2×D418yes—
123C2×D4×C61—22no—
124C2×D4×C61—24no—
125C22×C4×C61—24no—
126C22×C4×C61C22⋊C414yescomplemented
127C22×C4×C61C22×C411yescharacteristic
128C22×C4×C61C22×C412yes—
129C22×C4×C61C22×C412yes—
130C2×C22⋊D61—22no—
131C2×C22⋊D61—24no—
132C2×C22⋊D61—22no—
133C2×C22⋊D61—24no—
134C22×Dic61—24no—
135C22×Dic61C22⋊C414yescomplemented
136C22×Dic61C22×C411yescharacteristic
137C22×Dic61C22×C412yes—
138C22×Dic61C22×C412yes—
139D4×D61—44no—
140D4×D61—48no—
141D4×D61—44no—
142C2×D244—22no—
143C2×D244—24no—
144C2×C4×D61—28no—
145C2×C4×D61—28no—
146C2×C4×D61—216no—
147C2×C4×D61—28no—
148C2×C4×D61—24no—
149C2×C4×D61—24no—
150C2×C4×D61—22no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—24no—
157C2×C4×D61—24no—
158C2×C4×D61C22⋊C418yescomplemented
159C2×C4×D61C2×D414yes—
160C2×C4×D61C2×D414yes—
161C22⋊#6923—416no—
162C22⋊#6923—48no—
163C4⋊#6923—232no—
164C4⋊C4×C61—24no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—24no—
167C22⋊Dic61—22no—
168C22⋊Dic61—24no—
169#11758198—24no—
170#11758198—24no—
171#11758198—24no—
172C4⋊Dic61—24no—
173#11758196—24no—
174#11758196—24no—
175C23×C61—24no—
176C23×C61—21no—
177C23×C61—22no—
178C23×C61—24no—
179C23×C61C2×C22⋊C412yes—
180C23×C61C4×D414yes—
181C23×C61C23×C411yescharacteristic
182C22×D61—24no—
183C22×D61—28no—
184C22×D61—24no—
185C22×D61—24no—
186C22×D61—28no—
187C22×D61—28no—
188C22×D61—216no—
189C22×D61—21no—
190C22×D61—22no—
191C22×D61—24no—
192C22×D61—41no—
193C22×D61—42no—
194C22×D61—41no—
195C22×D61—44no—
196C22×D61—42no—
197C22×D61—42no—
198C22×D61—44no—
199C22×D61—44no—
200C22×D61—48no—
201C22×D61C2×C22⋊C412yes—
202C22×D61C4×D414yes—
203C22×D61#12014yes—
204C22×D61#12118yes—
205C22×D61#12314yes—
206C22×D61C22×D412yes—
207C22×D61C22×D411yescharacteristic
208C22×D61C22×D411yescharacteristic
209C22×D61C2×#4112yes—
210C22×D61H(5,𝔽2)14yes—
211C22×D612−1+414yes—
212C2×#6923—28no—
213C2×#6923—28no—
214C2×#6923—28no—
215C2×#6923—432no—
216C2×#6923—48no—
217C2×#6923—48no—
218C2×#6923—44no—
219C2×#6923—44no—
220C2×#6923—44no—
221C2×#6923—48no—
222C2×#6923—48no—
223D4×C61—44no—
224C2×C4×C61—24no—
225C2×C4×C61—28no—
226C2×C4×C61—21no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61C2×C22⋊C414yes—
231C22⋊D61—44no—
232C22⋊D61—44no—
233C2×Dic61—24no—
234C2×Dic61—28no—
235C2×Dic61—21no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—24no—
239C2×Dic61C2×C22⋊C414yes—
240D244—44no—
241C4×D61—24no—
242C4×D61—28no—
243C4×D61—24no—
244C4×D61—216no—
245C4×D61—41no—
246C4×D61—42no—
247C4×D61—41no—
248C4×D61—42no—
249C4×D61—42no—
250C4×D61—44no—
251#14256—611nomaximal, Sylow
252C22×C61—24no—
253C22×C61—28no—
254C22×C61—42no—
255C22×C61—44no—
256C22×C61—41no—
257C22×C61#38614yes—
258C22×C61C22×C22⋊C411yescharacteristic
259C22×C61C2×C4×D412yes—
260C22×C61#52014yes—
261C2×D61—24no—
262C2×D61—28no—
263C2×D61—24no—
264C2×D61—24no—
265C2×D61—216no—
266C2×D61—28no—
267C2×D61—42no—
268C2×D61—44no—
269C2×D61—41no—
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—84no—
273C2×D61—82no—
274C2×D61#38614yes—
275C2×D61C2×#12011yescharacteristic
276C2×D61C2×#12112yes—
277C2×D61C2×#12311yescharacteristic
278C2×D61#53414yes—
279C2×D61#53512yes—
280C2×D61#53614yes—
281C2×D61#53612yes—
282C2×D61#53714yes—
283C2×D61#53918yes—
284#6923—48no—
285#6923—84no—
286C4×C61—24no—
287C4×C61—42no—
288C4×C61—41no—
289Dic61—24no—
290Dic61—42no—
291Dic61—41no—
292C22×#123—611no—
293C23×C4⋊C4—611no—
294C2×#386—612no—
295C2×#381—611no—
296#1441—614no—
297#1439—614no—
298C22⋊#95—612no—
299C2×C61—24no—
300C2×C61—81no—
301C2×C61C2×#38611yescharacteristic
302C2×C61#231112yes—
303C2×C61#232214yes—
304D61—24no—
305D61—81no—
306D61#261311yescharacteristic
307D61#261411yescharacteristic
308D61#261814yes—
309D61#262312yes—
310C23×D4—611no—
311C24×C4—611no—
312C24×C4—612no—
313C2×#123—611no—
314C2×#123—611no—
315C2×#123—612no—
316C2×#123—611no—
317C2×#123—612no—
318C2×#123—1224no—
319C2×#123—1224no—
320C2×#123#692311yescharacteristic, complemented
321C22×C4⋊C4—612no—
322C22×C4⋊C4—614no—
323C22×C4⋊C4—1224no—
324C22×C4⋊C4—1228no—
325C22×C22⋊C4—611no—
326C22×C22⋊C4—612no—
327C22×C22⋊C4—612no—
328C22×C22⋊C4—614no—
329#386—1228no—
330#381—1224no—
331C2×#95—614no—
332C2×#95—614no—
333C2×#95—618no—
334C61#1425611yescharacteristic, complemented, Sylow
335C25—611no—
336C25—611no—
337C22×D4—611no—
338C22×D4—611no—
339C22×D4—611no—
340C22×D4—1222no—
341C22×D4—1222no—
342C22×D4—1224no—
343C22×D4—1224no—
344C22×D4C2×#692311yescharacteristic
345C23×C4—611no—
346C23×C4—614no—
347C23×C4—614no—
348C23×C4—612no—
349C23×C4—612no—
350C23×C4—614no—
351C23×C4—618no—
352C23×C4—612no—
353C23×C4—611no—
354C23×C4—614no—
355C23×C4—1224no—
356C23×C4—1224no—
357C23×C4—1224no—
358C23×C4—1224no—
359C23×C4—2448no—
360C23×C4C2×#692311yescharacteristic
361#123—24no—
362#123—1228no—
363#123—1224no—
364#123—1228no—
365#123—1224no—
366#123—1224no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—612no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—1224no—
371C2×C4⋊C4—1224no—
372C2×C4⋊C4—1228no—
373C2×C4⋊C4—12216no—
374C2×C4⋊C4—12232no—
375C2×C4⋊C4C2×#692312yescomplemented
376C2×C22⋊C4—611no—
377C2×C22⋊C4—612no—
378C2×C22⋊C4—612no—
379C2×C22⋊C4—612no—
380C2×C22⋊C4—612no—
381C2×C22⋊C4—1224no—
382C2×C22⋊C4—1224no—
383C2×C22⋊C4—1222no—
384C2×C22⋊C4—1222no—
385C2×C22⋊C4—1228no—
386C2×C22⋊C4—1224no—
387C2×C22⋊C4—1224no—
388C2×C22⋊C4—1224no—
389C2×C22⋊C4—2448no—
390C2×C22⋊C4C2×#692311yescharacteristic
391C2×C22⋊C4C2×#692312yescomplemented
392#95—1228no—
393#95—1228no—
394#95—2448no—
395C24—611no—
396C24—612no—
397C24—612no—
398C24—614no—
399C24—614no—
400C24—611no—
401C24—611no—
402C24—1224no—
403C24—1224no—
404C24—1221no—
405C24—1221no—
406C24—1222no—
407C24—1224no—
408C24—1222no—
409C24—1224no—
410C24C22×#692311yescharacteristic
411C24C22×#692311yescharacteristic
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—1222no—
415C2×D4—1224no—
416C2×D4—1222no—
417C2×D4—1224no—
418C2×D4—1222no—
419C2×D4—1224no—
420C2×D4—2444no—
421C2×D4—2448no—
422C2×D4—2444no—
423C22×C4—24no—
424C22×C4—614no—
425C22×C4—612no—
426C22×C4—614no—
427C22×C4—614no—
428C22×C4—618no—
429C22×C4—618no—
430C22×C4—612no—
431C22×C4—611no—
432C22×C4—1228no—
433C22×C4—1224no—
434C22×C4—12216no—
435C22×C4—1228no—
436C22×C4—1224no—
437C22×C4—1228no—
438C22×C4—1222no—
439C22×C4—1224no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—1222no—
445C22×C4—1224no—
446C22×C4—1224no—
447C22×C4—1228no—
448C22×C4—1228no—
449C22×C4—1224no—
450C22×C4—1224no—
451C22×C4—12216no—
452C22×C4—1224no—
453C22×C4—1224no—
454C22×C4—1228no—
455C22×C4—1224no—
456C22×C4—1228no—
457C22×C4—24416no—
458C22×C4—24416no—
459C22×C4—24416no—
460C22×C4C22⋊#692314yescomplemented
461C22×C4C22×#692312yes—
462C22×C4C22×#692311yescharacteristic
463C22×C4C22×#692312yes—
464C4⋊C4—24no—
465C4⋊C4—1224no—
466C4⋊C4—1224no—
467C4⋊C4—12232no—
468C4⋊C4—1224no—
469C22⋊C4—22no—
470C22⋊C4—24no—
471C22⋊C4—1224no—
472C22⋊C4—1224no—
473C22⋊C4—1224no—
474C22⋊C4—1222no—
475C22⋊C4—1224no—
476C22⋊C4—24416no—
477C22⋊C4—2448no—
478C23—24no—
479C23—21no—
480C23—22no—
481C23—24no—
482C23—611no—
483C23—611no—
484C23—612no—
485C23—612no—
486C23—614no—
487C23—614no—
488C23—614no—
489C23—618no—
490C23—614no—
491C23—612no—
492C23—614no—
493C23—1224no—
494C23—1228no—
495C23—1228no—
496C23—1224no—
497C23—1228no—
498C23—12216no—
499C23—1224no—
500C23—1221no—
501C23—1222no—
502C23—1224no—
503C23—2444no—
504C23—2441no—
505C23—2442no—
506C23—2441no—
507C23—2442no—
508C23—2444no—
509C23—2442no—
510C23—2444no—
511C23—2448no—
512C23C2×C22⋊#692312yes—
513C23D4×#692314yes—
514C23C23×#692311yescharacteristic
515D4—44no—
516D4—2444no—
517D4—2444no—
518D4—2444no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—614no—
526C2×C4—1224no—
527C2×C4—1228no—
528C2×C4—1224no—
529C2×C4—12216no—
530C2×C4—1228no—
531C2×C4—1224no—
532C2×C4—1228no—
533C2×C4—1228no—
534C2×C4—1228no—
535C2×C4—1222no—
536C2×C4—1221no—
537C2×C4—1222no—
538C2×C4—1224no—
539C2×C4—2442no—
540C2×C4—2444no—
541C2×C4—2442no—
542C2×C4—2441no—
543C2×C4—2442no—
544C2×C4—2441no—
545C2×C4—2444no—
546C2×C4—2444no—
547C2×C4—2448no—
548C2×C4—2444no—
549C2×C4—2448no—
550C2×C4—2448no—
551C2×C4—2448no—
552C2×C4—24432no—
553C2×C4C2×C22⋊#692314yes—
554C22—24no—
555C22—28no—
556C22—42no—
557C22—44no—
558C22—41no—
559C22—612no—
560C22—614no—
561C22—612no—
562C22—614no—
563C22—614no—
564C22—618no—
565C22—611no—
566C22—612no—
567C22—611no—
568C22—614no—
569C22—1224no—
570C22—12216no—
571C22—1224no—
572C22—1228no—
573C22—1224no—
574C22—1228no—
575C22—4884no—
576C22—4881no—
577C22—4881no—
578C22—4882no—
579C22—2442no—
580C22—2444no—
581C22—2441no—
582C22#1684071014yes—
583C22C22×C22⋊#692311yescharacteristic
584C22C2×D4×#692312yes—
585C22#1684083114yes—
586C4—24no—
587C4—42no—
588C4—41no—
589C4—1224no—
590C4—4884no—
591C4—2448no—
592C4—2442no—
593C4—2441no—
594C2—24no—
595C2—81no—
596C2—614no—
597C2—611no—
598C2—611no—
599C2—612no—
600C2—1224no—
601C2—4881no—
602C2C2×#1684071011yescharacteristic
603C2#2146229412yes—
604C2#2146230514yes—
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC2×#123×C61C4
prime radicalC22×D4×D61C22
cosocleC23×D61C24
hypercenterC2×#123#6923
hypocenterC61#14256
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C23×C4⋊#6923C2semidirect
3. C2×#16840705C2semidirect
4. #21461004C2semidirect
5. #21461002C2semidirect
6. #21460992C2semidirect
21. C2×#14519762C4semidirect
22. C2×#14519761C4semidirect
23. C2×#14519760C4semidirect
24. C2×#14519759C4semidirect
25. C2×#14519758C4semidirect
31. C2×#123×C61C4semidirect
72. C2×C4⋊Dic61C2×C4semidirect
73. C2×#11758196C2×C4semidirect
74. C2×#11758196C2×C4semidirect
78. C2×C22⋊Dic61C2×C4semidirect
80. C2×#11758198C2×C4semidirect
81. C2×#11758198C2×C4semidirect
89. C2×C4⋊C4×C61C2×C4semidirect
91. C2×C22⋊C4×C61C2×C4semidirect
126. C22×C4×C61C22⋊C4semidirect
135. C22×Dic61C22⋊C4semidirect
158. C2×C4×D61C22⋊C4semidirect
320. C2×#123#6923semidirect
334. C61#14256semidirect
375. C2×C4⋊C4C2×#6923semidirect
391. C2×C22⋊C4C2×#6923semidirect
460. C22×C4C22⋊#6923semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊#69232
C2×#168407102
C2×#168407052
#214610042
#214610022
#214609922
C2×#123×D612
#1425661