Finite Group Databasebeta

← #30615170

Subgroups — #30615170

Order 15616.

1866 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462263C212yesmaximal, complemented
2#21461714C212yesmaximal, complemented
3#21461681C214yesmaximal, complemented
4#21461645C212yesmaximal
5#21461566C214yesmaximal, complemented
6C22×#14519737C211yesmaximal, characteristic
7C2×#14519856—22no—
8C2×#14519835C2218yes—
9#16840705—22no—
10#16840704—22no—
11#16840704C2214yescomplemented
12#16840703C2214yes—
13C4×C22⋊#6923—22no—
14C4×C22⋊#6923C2214yescomplemented
15#16840640C2218yescomplemented
16C23×C22⋊D61C2211yescharacteristic
17C24×Dic61C411yescharacteristic, complemented
18C23×C4×D61C2211yescharacteristic
19C2×#14519737—22no—
20C2×#14519737—22no—
21C2×#14519737—22no—
22C2×#14519737—22no—
23C2×#14519737C412yescomplemented
24C2×#14519737C412yescomplemented
25C2×#14519737C2212yes—
26C2×#14519737C2212yes—
27C22×#11758196C411yescharacteristic, complemented
28C22×#11758198C2211yescharacteristic
29C22×C4×Dic61C411yescharacteristic, complemented
30C22×C22⋊C4×C61C411yescharacteristic, complemented
31#14519856—48no—
32C2×C4⋊#6923—24no—
33C2×C22⋊#6923—24no—
34C2×C22⋊#6923—24no—
35C2×C22⋊#6923C2318yes—
36C2×C4×#6923—24no—
37#14519835—216no—
38(C2×#6923)⋊C4—24no—
39(C2×#6923)⋊C4—24no—
40C24×D61C2311yescharacteristic
41C22×C22⋊D61—22no—
42C22×C22⋊D61—22no—
43C22×C22⋊D61—28no—
44C22×C22⋊D61C2×C412yes—
45C22×C22⋊D61C2312yes—
46C23×Dic61—21no—
47C23×Dic61—21no—
48C23×Dic61—21no—
49C23×Dic61—21no—
50C23×Dic61—44no—
51C23×Dic61C2×C411yescharacteristic
52C23×Dic61C2×C411yescharacteristic
53C23×Dic61C2×C411yescharacteristic
54C23×Dic61C2×C412yes—
55C23×Dic61C2×C412yes—
56C22×C4×D61—44no—
57C22×C4×D61C2×C411yescharacteristic
58C22×C4×D61C2×C411yescharacteristic
59C22×C4×D61D412yes—
60C22×C4×D61D412yes—
61C22×C4×D61D412yes—
62C22×C4×D61D412yes—
63C22×C4×D61C2311yescharacteristic
64C22×C4×D61C2311yescharacteristic
65#14519737—24no—
66#14519737—24no—
67#14519737—24no—
68#14519737—24no—
69#14519737—24no—
70#14519737—24no—
71#14519737—24no—
72#14519737—24no—
73C2×#11758196—22no—
74C2×#11758196—22no—
75C2×#11758196C2×C412yes—
76C2×#11758196C2×C412yes—
77C2×#11758198—22no—
78C2×#11758198—22no—
79C2×#11758198C2×C412yes—
80C2×#11758198C2312yes—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61C2×C412yes—
86C2×C4×Dic61C2×C412yes—
87C25×C61C2×C411yescharacteristic
88C23×C4×C61C2×C411yescharacteristic
89C23×C4×C61C2×C411yescharacteristic
90C2×C22⋊C4×C61—22no—
91C2×C22⋊C4×C61—22no—
92C2×C22⋊C4×C61C2×C412yes—
93C2×C22⋊C4×C61C2×C412yes—
94C24×C61—21no—
95C24×C61—21no—
96C24×C61—24no—
97C24×C61—42no—
98C24×C61—42no—
99C24×C61C22×C411yescharacteristic
100C24×C61C22×C412yes—
101C23×D61—24no—
102C23×D61—42no—
103C23×D61—42no—
104C23×D61C22×C411yescharacteristic
105C23×D61C2×D412yes—
106C23×D61C2×D412yes—
107C23×D61C2411yescharacteristic
108C22×#6923—28no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—42no—
112C22×C4×C61—42no—
113C22×C4×C61C22⋊C412yes—
114C22×C4×C61C22⋊C412yes—
115C22×C4×C61C22×C412yes—
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C22×C411yescharacteristic
118C2×C22⋊D61—216no—
119C2×C22⋊D61—216no—
120C2×C22⋊D61—24no—
121C2×C22⋊D61—24no—
122C2×C22⋊D61—24no—
123C2×C22⋊D61—24no—
124C2×C22⋊D61—24no—
125C2×C22⋊D61#4118yes—
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—24no—
129C22×Dic61—24no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—44no—
137C22×Dic61—44no—
138C22×Dic61—44no—
139C22×Dic61—44no—
140C22×Dic61—44no—
141C22×Dic61—44no—
142C22×Dic61—42no—
143C22×Dic61C22⋊C411yescharacteristic
144C22×Dic61C22⋊C411yescharacteristic
145C22×Dic61C22⋊C411yescharacteristic
146C22×Dic61C22⋊C411yescharacteristic
147C22×Dic61C22×C411yescharacteristic
148C22×Dic61C22×C411yescharacteristic
149C22×Dic61C22×C412yes—
150C22×Dic61C22×C412yes—
151C2×C4×D61—28no—
152C2×C4×D61—42no—
153C2×C4×D61—42no—
154C2×C4×D61—44no—
155C2×C4×D61—44no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C2×C4×D61—44no—
159C2×C4×D61C22⋊C412yes—
160C2×C4×D61C22⋊C412yes—
161C2×C4×D61C22⋊C412yes—
162C2×C4×D61C22⋊C412yes—
163C2×C4×D61C2×D412yes—
164C2×C4×D61C2×D412yes—
165C2×C4×D61C2×D412yes—
166C2×C4×D61C2×D412yes—
167C22⋊#6923—216no—
168C22⋊#6923—48no—
169C22⋊#6923—48no—
170C4⋊#6923—48no—
171C4×#6923—48no—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—24no—
174#11758198—24no—
175#11758198—24no—
176#11758196—24no—
177#11758196—24no—
178C4×Dic61—22no—
179C4×Dic61—22no—
180C4×Dic61—22no—
181C4×Dic61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—24no—
186C23×C61—24no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—42no—
190C23×C61—416no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61C2×C22⋊C411yescharacteristic
197C23×C61C2×C22⋊C411yescharacteristic
198C23×C61#11714yescomplemented
199C23×C61C23×C411yescharacteristic
200C22×D61—22no—
201C22×D61—24no—
202C22×D61—24no—
203C22×D61—24no—
204C22×D61—24no—
205C22×D61—41no—
206C22×D61—42no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—42no—
210C22×D61—41no—
211C22×D61—416no—
212C22×D61C2×C22⋊C412yes—
213C22×D61C22×D412yes—
214C22×D61C2×#4114yes—
215C2×#6923—48no—
216C2×#6923—48no—
217C2×#6923—48no—
218C2×C4×C61—24no—
219C2×C4×C61—22no—
220C2×C4×C61—42no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61C2×C22⋊C412yes—
227C2×C4×C61C2×C22⋊C412yes—
228C22⋊D61—216no—
229C22⋊D61—216no—
230C2×Dic61—24no—
231C2×Dic61—24no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—44no—
241C2×Dic61—44no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—44no—
245C2×Dic61—44no—
246C2×Dic61—42no—
247C2×Dic61#9914yescomplemented
248C2×Dic61C2×C22⋊C411yescharacteristic
249C2×Dic61C2×C22⋊C411yescharacteristic
250C2×Dic61C2×C22⋊C411yescharacteristic
251C2×Dic61C2×C22⋊C411yescharacteristic
252C2×Dic61#11714yescomplemented
253C4×D61—24no—
254C4×D61—24no—
255C4×D61—24no—
256C4×D61—24no—
257C4×D61—42no—
258C4×D61—42no—
259C4×D61—42no—
260C4×D61—42no—
261#20722—611nomaximal, Sylow
262C22×C61—28no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—24no—
270C22×C61—44no—
271C22×C61—44no—
272C22×C61—48no—
273C22×C61—48no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61C2×#9912yes—
277C22×C61#41012yes—
278C22×C61C22×C22⋊C411yescharacteristic
279C22×C61C2×#11712yes—
280C2×D61—22no—
281C2×D61—22no—
282C2×D61—22no—
283C2×D61—22no—
284C2×D61—22no—
285C2×D61—44no—
286C2×D61—48no—
287C2×D61—48no—
288C2×D61—44no—
289C2×D61#52514yes—
290#6923—88no—
291C4×C61—22no—
292C4×C61—22no—
293C4×C61—42no—
294Dic61—21no—
295Dic61—21no—
296Dic61—24no—
297Dic61—42no—
298Dic61C2×#9912yes—
299Dic61C2×#9912yes—
300C22×C4×D4—611no—
301#2280—612no—
302#1819—612no—
303#1799—614no—
304#1781—612no—
305C4×#99—614no—
306C2×C61—22no—
307C2×C61—22no—
308C2×C61—22no—
309C2×C61—44no—
310C2×C61#228011yescharacteristic
311C2×C61C22×#9911yescharacteristic
312C2×C61C2×#41011yescharacteristic
313D61—24no—
314D61—42no—
315C23×D4—611no—
316C24×C4—611no—
317C24×C4—611no—
318C2×C4×D4—612no—
319C2×C4×D4—612no—
320C2×C4×D4—612no—
321C2×C4×D4—612no—
322C2×C4×D4—1222no—
323C2×C4×D4—1222no—
324C2×C4×D4—1222no—
325C2×C4×D4—1222no—
326C2×#117—1222no—
327C22×C4⋊C4—611no—
328C22×C22⋊C4—611no—
329C22×C22⋊C4#692311yescharacteristic, complemented
330C22×C42—611no—
331C2×#99—618no—
332#381—1222no—
333#380—614no—
334#380—1222no—
335#379—614no—
336C4×C22⋊C4—614no—
337C4×C22⋊C4—1222no—
338#342—618no—
339C61#2072211yescharacteristic, complemented, Sylow
340C25—611no—
341C25C2×#692311yescharacteristic
342C22×D4—612no—
343C22×D4—612no—
344C22×D4—1228no—
345C22×D4—1222no—
346C22×D4—1222no—
347C23×C4—611no—
348C23×C4—612no—
349C23×C4—612no—
350C23×C4—612no—
351C23×C4—612no—
352C23×C4—611no—
353C23×C4—611no—
354C23×C4—611no—
355C23×C4—611no—
356C23×C4—611no—
357C23×C4—611no—
358C23×C4—612no—
359C23×C4—612no—
360C23×C4—1221no—
361C23×C4—1221no—
362C23×C4—1221no—
363C23×C4—1221no—
364C23×C4—2444no—
365C23×C4—2444no—
366C23×C4C2×#692311yescharacteristic
367C23×C4C2×#692311yescharacteristic
368C4×D4—1224no—
369C4×D4—1224no—
370C4×D4—1224no—
371C4×D4—1224no—
372C4×D4—1224no—
373C4×D4—1224no—
374C4×D4—1224no—
375C4×D4—1224no—
376#117—2448no—
377C2×C4⋊C4—612no—
378C2×C4⋊C4—612no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4—1222no—
381C2×C4⋊C4—1224no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—612no—
385C2×C22⋊C4—612no—
386C2×C22⋊C4—618no—
387C2×C22⋊C4—1222no—
388C2×C22⋊C4—1222no—
389C2×C22⋊C4—1224no—
390C2×C22⋊C4—1224no—
391C2×C22⋊C4C2×#692312yes—
392C2×C22⋊C4C2×#692312yes—
393C2×C42—612no—
394C2×C42—612no—
395C2×C42—1221no—
396C2×C42—1221no—
397C2×C42—1221no—
398C2×C42—1221no—
399C2×C42—1224no—
400#99—12216no—
401#95—1224no—
402#95—1224no—
403C24—21no—
404C24—21no—
405C24—24no—
406C24—42no—
407C24—42no—
408C24—611no—
409C24—612no—
410C24—612no—
411C24—611no—
412C24—1224no—
413C24—2442no—
414C24—2442no—
415C24C22×#692311yescharacteristic
416C24C22×#692312yes—
417C2×D4—618no—
418C2×D4—1224no—
419C2×D4—1224no—
420C2×D4—1224no—
421C2×D4—1224no—
422C2×D4—12216no—
423C2×D4—12216no—
424C2×D4—1224no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—612no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—612no—
437C22×C4—612no—
438C22×C4—612no—
439C22×C4—612no—
440C22×C4—612no—
441C22×C4—612no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—612no—
445C22×C4—1228no—
446C22×C4—1224no—
447C22×C4—1224no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1228no—
457C22×C4—2444no—
458C22×C4—2444no—
459C22×C4—2444no—
460C22×C4—2442no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4—2442no—
464C22×C4—2444no—
465C22×C4—2444no—
466C22×C4—2444no—
467C22×C4—2444no—
468C22×C4—2444no—
469C22×C4—2444no—
470C22×C4—2442no—
471C22×C4C22⋊#692312yes—
472C22×C4C22⋊#692312yes—
473C22×C4C22×#692312yes—
474C22×C4C22×#692311yescharacteristic
475C22×C4C22×#692311yescharacteristic
476C4⋊C4—1224no—
477C4⋊C4—1224no—
478C4⋊C4—2448no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—1224no—
482C22⋊C4—1224no—
483C22⋊C4—12216no—
484C22⋊C4—2448no—
485C22⋊C4—2448no—
486C42—1222no—
487C42—1222no—
488C42—1222no—
489C42—1222no—
490C42—2448no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—416no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—612no—
506C23—614no—
507C23—612no—
508C23—1224no—
509C23—1224no—
510C23—1224no—
511C23—1224no—
512C23—1222no—
513C23—2442no—
514C23—2441no—
515C23—2441no—
516C23—2442no—
517C23—2441no—
518C23—2441no—
519C23—24416no—
520C23C2×C22⋊#692311yescharacteristic
521C23C2×C22⋊#692311yescharacteristic
522C23#1451985614yescomplemented
523C23C23×#692311yescharacteristic
524D4—12216no—
525D4—12216no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—614no—
535C2×C4—611no—
536C2×C4—611no—
537C2×C4—611no—
538C2×C4—611no—
539C2×C4—614no—
540C2×C4—1224no—
541C2×C4—1224no—
542C2×C4—1224no—
543C2×C4—1224no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1224no—
549C2×C4—1224no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2444no—
559C2×C4—2444no—
560C2×C4—2444no—
561C2×C4—2444no—
562C2×C4—2448no—
563C2×C4—2448no—
564C2×C4—2448no—
565C2×C4—2442no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4C2×C22⋊#692312yes—
569C2×C4C2×C22⋊#692312yes—
570C22—28no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—24no—
578C22—44no—
579C22—48no—
580C22—48no—
581C22—44no—
582C22—42no—
583C22—42no—
584C22—614no—
585C22—1222no—
586C22—1222no—
587C22—1222no—
588C22—1222no—
589C22—1222no—
590C22—2444no—
591C22—2448no—
592C22—2448no—
593C22—2444no—
594C22D61⋊#9912yes—
595C22#1684074212yes—
596C22C22×C22⋊#692311yescharacteristic
597C22C2×#1451985612yes—
598C4—22no—
599C4—22no—
600C4—42no—
601C4—612no—
602C4—612no—
603C4—1221no—
604C4—1221no—
605C4—1224no—
606C4—4888no—
607C4—2442no—
608C2—22no—
609C2—22no—
610C2—22no—
611C2—44no—
612C2—1224no—
613C2—2442no—
614C2#2146226311yescharacteristic
615C2C2×D61⋊#9911yescharacteristic
616C2C2×#1684074211yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC22×C22⋊C4×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC23×C22⋊D61C22
cosocleC23×D61C24
hypercenterC22×C22⋊C4#6923
hypocenterC61#20722
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #21462263C2semidirect
2. #21461714C2semidirect
3. #21461681C2semidirect
5. #21461566C2semidirect
11. #16840704C22semidirect
14. C4×C22⋊#6923C22semidirect
15. #16840640C22semidirect
17. C24×Dic61C4semidirect
23. C2×#14519737C4semidirect
24. C2×#14519737C4semidirect
27. C22×#11758196C4semidirect
29. C22×C4×Dic61C4semidirect
30. C22×C22⋊C4×C61C4semidirect
198. C23×C61#117semidirect
247. C2×Dic61#99semidirect
252. C2×Dic61#117semidirect
329. C22×C22⋊C4#6923semidirect
339. C61#20722semidirect
522. C23#14519856semidirect

Maximal subgroups

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

subgroupindex
#214622632
#214617142
#214616812
#214616452
#214615662
C22×#145197372
#2072261