Finite Group Databasebeta

← #30620133

Subgroups — #30620133

Order 15616.

1474 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840741C212yesmaximal, complemented
2#21462327C212yesmaximal, complemented
3#21461959C212yesmaximal, complemented
4#21461843C214yesmaximal, complemented
5#21461679C211yesmaximal, characteristic, complemented
6#21461648C212yesmaximal, complemented
7C4×#14519835C211yesmaximal, characteristic, complemented
8C2×#16840413C211yesmaximal, characteristic
9C2×#14519856C2214yes—
10#16840741—28no—
11C2×#14519836C2214yes—
12C2×#14519836C2214yes—
13C2×#14519835C2211yescharacteristic
14C2×#14519835C2212yes—
15C2×#14519835C2211yescharacteristic
16#16840713—22no—
17#16840711—24no—
18#16840705C2214yes—
19#16840704C2212yes—
20C4⋊C4×#6923—22no—
21C4×C22⋊#6923C2212yes—
22#16840640C2212yes—
23#16840640C2212yes—
24C22×#11758222C2212yes—
25#16840413—24no—
26#16840413—22no—
27#16840413—22no—
28C2×#14519776C411yescharacteristic, complemented
29C2×#14519762C2212yes—
30C2×#14519760C412yescomplemented
31C2×#14519755C412yescomplemented
32C2×#14519752C2211yescharacteristic
33C2×#14519751C411yescharacteristic, complemented
34C2×#14519739C411yescharacteristic, complemented
35C2×C4×C22⋊D61C2211yescharacteristic
36C2×C4⋊C4×D61C2211yescharacteristic
37C22×C4⋊C4×C61C411yescharacteristic, complemented
38#14519856—28no—
39#14519856—44no—
40C2×C4⋊#6923—22no—
41C2×C4⋊#6923—24no—
42C2×C22⋊#6923C2314yes—
43C2×C22⋊#6923C2312yes—
44C2×C22⋊#6923C2312yes—
45C2×C4×#6923—24no—
46C2×C4×#6923—22no—
47#14519836—28no—
48#14519836—44no—
49#14519835—24no—
50#14519835—22no—
51#14519835D414yescomplemented
52(C2×#6923)⋊C4—22no—
53(C2×#6923)⋊C4—24no—
54(C2×#6923)⋊C4—24no—
55(C2×#6923)⋊C4—22no—
56C2×#11758222—24no—
57C2×#11758222—24no—
58C2×#11758222—24no—
59C2×#11758222—24no—
60C2×#11758222—44no—
61C22×Dic122C2×C412yes—
62C22×D244C2×C411yescharacteristic
63C22×D244C2312yes—
64C22×C22⋊D61C2×C412yes—
65C22×C22⋊D61C2311yescharacteristic
66C22×C4×D61C2×C412yes—
67C22×C4×D61C2312yes—
68C22×C4×D61C2311yescharacteristic
69#14519776—22no—
70#14519776—22no—
71#14519762—24no—
72#14519762—24no—
73#14519760—24no—
74#14519760—24no—
75#14519755—24no—
76#14519755—24no—
77#14519752—22no—
78#14519752—22no—
79#14519751—22no—
80#14519751—22no—
81#14519739—22no—
82#14519739—21no—
83#14519739—21no—
84C4×C22⋊D61—22no—
85C4×C22⋊D61—22no—
86C2×C4⋊Dic61C2×C411yescharacteristic
87C2×#11758196C2×C411yescharacteristic
88C2×#11758196C2×C412yes—
89C2×#11758196C2×C412yes—
90C4⋊C4×D61D414yes—
91C4⋊C4×D61D412yes—
92C4⋊C4×D61D412yes—
93C2×C22⋊Dic61C2×C411yescharacteristic
94C2×#11758198C2×C412yes—
95C2×#11758198C2×C412yes—
96C2×#11758198C2×C412yes—
97C2×#11758198C2311yescharacteristic
98C2×C4×Dic61C2×C411yescharacteristic
99C23×C4×C61C2×C412yes—
100C23×C4×C61C2×C411yescharacteristic
101C2×C4⋊C4×C61—22no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61C2×C411yescharacteristic
105C2×C4⋊C4×C61C2×C412yes—
106C2×C4⋊C4×C61C2×C411yescharacteristic
107C24×C61C22×C411yescharacteristic
108C23×D61C22×C412yes—
109C23×D61C2411yescharacteristic
110C22×#6923—22no—
111C22×#6923—22no—
112C22×#6923—24no—
113C22×C4×C61—24no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—42no—
119C22×C4×C61—42no—
120C22×C4×C61C22×C412yes—
121C22×C4×C61C22×C412yes—
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C22×C411yescharacteristic
124C2×C22⋊D61—22no—
125C2×C22⋊D61—22no—
126C2×C22⋊D61—24no—
127C2×C22⋊D61—24no—
128C2×C22⋊D61—42no—
129C2×C22⋊D61C2×D412yes—
130C2×C22⋊D61#4112yes—
131C22×Dic61C22×C412yes—
132C22×Dic61C22×C411yescharacteristic
133C22×Dic61C22×C411yescharacteristic
134C22×Dic61C22×C411yescharacteristic
135#11758222—48no—
136#11758222—48no—
137#11758222—48no—
138#11758222—48no—
139C2×D244—22no—
140C2×D244—24no—
141C2×D244—22no—
142C2×D244—24no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—42no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—24no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—24no—
152C2×C4×D61—42no—
153C2×C4×D61—42no—
154C2×C4×D61C2×D414yes—
155C2×C4×D61C2×D414yes—
156C2×C4×D61C2×D412yes—
157C2×C4×D61C2×D412yes—
158C2×Dic122—24no—
159C2×Dic122—24no—
160C2×Dic122—42no—
161C22⋊#6923—22no—
162C22⋊#6923—24no—
163C22⋊#6923—22no—
164C22⋊#6923—22no—
165C22⋊#6923—24no—
166C22⋊#6923C2×D414yes—
167C4⋊#6923—48no—
168C4⋊#6923—44no—
169C4×#6923—48no—
170C4×#6923—44no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61C22⋊C412yes—
176C4⋊C4×C61C22⋊C411yescharacteristic
177C4⋊C4×C61C22⋊C411yescharacteristic
178C22⋊Dic61—22no—
179#11758198—22no—
180#11758198—22no—
181#11758198—22no—
182#11758198—24no—
183#11758198—24no—
184C4⋊Dic61C22⋊C412yes—
185C4⋊Dic61C22⋊C412yes—
186#11758196—22no—
187#11758196—24no—
188#11758196C22⋊C414yes—
189#11758196C22⋊C414yes—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61C4×D412yes—
197C23×C61C23×C411yescharacteristic
198C22×D61—22no—
199C22×D61—22no—
200C22×D61—22no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—42no—
204C22×D61—42no—
205C22×D61C22×D412yes—
206C22×D61C22×D411yescharacteristic
207C22×D61C2×#4111yescharacteristic
208C22×D61H(5,𝔽2)12yes—
209C2×#6923—24no—
210C2×#6923—42no—
211C2×#6923—42no—
212C2×#6923—42no—
213C2×#6923—44no—
214C2×#6923—42no—
215C2×#6923—44no—
216C2×#6923—44no—
217C2×#6923—42no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—21no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—44no—
226C2×C4×C61—44no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—41no—
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C2×C4×C61C2×C22⋊C412yes—
232C2×C4×C61C2×C22⋊C412yes—
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C22⋊D61—24no—
235C22⋊D61—24no—
236C22⋊D61—48no—
237C2×Dic61—21no—
238C2×Dic61—22no—
239C2×Dic61—21no—
240C2×Dic61—22no—
241C2×Dic61—21no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61C2×C22⋊C412yes—
245C2×Dic61C2×C22⋊C414yes—
246C2×Dic61C4×D412yes—
247D244—42no—
248D244—44no—
249D244—44no—
250D244—42no—
251C4×D61—24no—
252C4×D61—24no—
253C4×D61—24no—
254C4×D61—24no—
255C4×D61—24no—
256C4×D61—42no—
257C4×D61—44no—
258C4×D61—44no—
259Dic122—48no—
260#22406—611nomaximal, Sylow
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—22no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61C22×C22⋊C411yescharacteristic
269C22×C61C2×C4×D411yescharacteristic
270C22×C61#51811yescharacteristic
271C2×D61—22no—
272C2×D61—22no—
273C2×D61—41no—
274C2×D61—41no—
275C2×D61—42no—
276C2×D61—42no—
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—82no—
280C2×D61—82no—
281C2×D61D4212yes—
282C2×D61#54612yes—
283#6923—44no—
284#6923—84no—
285#6923—82no—
286C4×C61—22no—
287C4×C61—22no—
288C4×C61—21no—
289C4×C61—42no—
290C4×C61—41no—
291C4×C61#41012yescomplemented
292Dic61—22no—
293Dic61—24no—
294Dic61—22no—
295Dic61—41no—
296C2×#547—611no—
297C2×#410—612no—
298#2344—612no—
299#1964—612no—
300#1899—614no—
301#1799—611no—
302#1782—612no—
303C4×#99—611no—
304C2×C61—21no—
305C2×C61—21no—
306C2×C61—42no—
307C2×C61D4×C22⋊C411yescharacteristic
308C2×C61C2×#41011yescharacteristic
309C2×C61#288411yescharacteristic
310D61—22no—
311D61—22no—
312D61—82no—
313C22×#41—612no—
314#547—1222no—
315#547—1224no—
316#547—1222no—
317C2×C4⋊Q8—611no—
318C2×#123—612no—
319C2×#123—612no—
320C2×C22⋊Q8—612no—
321C2×#121—611no—
322C2×#121—611no—
323C2×C4×D4—611no—
324C2×C4×D4—611no—
325C2×#117—614no—
326C22×C4⋊C4—611no—
327C22×C4⋊C4#692311yescharacteristic, complemented
328#410—1228no—
329C2×#99—611no—
330C2×#99—612no—
331C2×#99—611no—
332C2×#99—614no—
333C2×#99—614no—
334#389—1222no—
335#387—1224no—
336#381—614no—
337#380—612no—
338C4×C4⋊C4—1222no—
339C4×C22⋊C4—612no—
340#342—612no—
341#342—612no—
342C61#2240611yescharacteristic, complemented, Sylow
343C2×#41—1224no—
344C2×#41—1224no—
345C2×#41—1224no—
346C2×#41—1224no—
347C2×#41—2444no—
348C22×Q8—612no—
349C22×D4—611no—
350C22×D4—612no—
351C22×D4—612no—
352C22×D4—611no—
353C23×C4—612no—
354C23×C4—611no—
355C23×C4—612no—
356C23×C4C2×#692312yes—
357C23×C4C2×#692311yescharacteristic
358C4⋊Q8—1222no—
359C4⋊Q8—1222no—
360#123—1224no—
361#123—1224no—
362#123—1224no—
363#123—1224no—
364C22⋊Q8—1224no—
365C22⋊Q8—1224no—
366#121—1222no—
367#121—1222no—
368#121—1222no—
369#121—1222no—
370C4×D4—1222no—
371C4×D4—1222no—
372C4×D4—1222no—
373C4×D4—1221no—
374C4×D4—1221no—
375#117—1228no—
376#117—2444no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—612no—
382C2×C4⋊C4—612no—
383C2×C4⋊C4—614no—
384C2×C4⋊C4—612no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—612no—
387C2×C4⋊C4—1224no—
388C2×C4⋊C4—1222no—
389C2×C4⋊C4C2×#692311yescharacteristic
390C2×C4⋊C4C2×#692312yes—
391C2×C4⋊C4C2×#692311yescharacteristic
392C2×C22⋊C4—612no—
393C2×C22⋊C4—611no—
394C2×C22⋊C4—611no—
395C2×C22⋊C4—612no—
396C2×C22⋊C4—612no—
397C2×C22⋊C4—612no—
398C2×C22⋊C4—614no—
399C2×C22⋊C4—612no—
400C2×C42—611no—
401C2×C42—1224no—
402C2×C42—1222no—
403#99—614no—
404#99—1228no—
405#99—1222no—
406#99—1224no—
407#99—2444no—
408#95—1222no—
409#95—1224no—
410#95—1222no—
411#95—1224no—
412C24—611no—
413C24—612no—
414C24C22×#692311yescharacteristic
415#41—2448no—
416#41—2448no—
417#41—2448no—
418#41—2448no—
419C2×Q8—1224no—
420C2×Q8—1224no—
421C2×Q8—2442no—
422C2×D4—612no—
423C2×D4—612no—
424C2×D4—1222no—
425C2×D4—1224no—
426C2×D4—1224no—
427C2×D4—1222no—
428C2×D4—1224no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1224no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—2442no—
435C2×D4—2442no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—611no—
444C22×C4—612no—
445C22×C4—611no—
446C22×C4—614no—
447C22×C4—614no—
448C22×C4—612no—
449C22×C4—612no—
450C22×C4—611no—
451C22×C4—1224no—
452C22×C4—1222no—
453C22×C4—1224no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1224no—
460C22×C4—2442no—
461C22×C4—2442no—
462C22×C4C22×#692312yes—
463C22×C4C22×#692311yescharacteristic
464C22×C4C22×#692312yes—
465C22×C4C22×#692311yescharacteristic
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—614no—
471C4⋊C4—614no—
472C4⋊C4—612no—
473C4⋊C4—612no—
474C4⋊C4—1222no—
475C4⋊C4—1224no—
476C4⋊C4—2448no—
477C4⋊C4—2444no—
478C4⋊C4C22⋊#692312yes—
479C4⋊C4C22⋊#692311yescharacteristic
480C4⋊C4C22⋊#692311yescharacteristic
481C22⋊C4—614no—
482C22⋊C4—1224no—
483C22⋊C4—1224no—
484C22⋊C4—1222no—
485C22⋊C4—1222no—
486C22⋊C4—1222no—
487C22⋊C4—1222no—
488C22⋊C4—1222no—
489C22⋊C4—1224no—
490C22⋊C4—1222no—
491C22⋊C4—1222no—
492C22⋊C4—1224no—
493C42—1221no—
494C42—1221no—
495C42—2448no—
496C42—2444no—
497C23—21no—
498C23—21no—
499C23—41no—
500C23—41no—
501C23—612no—
502C23—611no—
503C23—611no—
504C23—612no—
505C23—1222no—
506C23—1222no—
507C23—1222no—
508C23—2441no—
509C23—2441no—
510C23—2442no—
511C23—2442no—
512C23D4×#692312yes—
513C23C23×#692311yescharacteristic
514Q8—2448no—
515D4—1224no—
516D4—1224no—
517D4—2442no—
518D4—2444no—
519D4—2444no—
520D4—2448no—
521D4—2442no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—612no—
535C2×C4—614no—
536C2×C4—612no—
537C2×C4—1224no—
538C2×C4—1224no—
539C2×C4—1224no—
540C2×C4—1224no—
541C2×C4—1221no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1221no—
545C2×C4—1224no—
546C2×C4—1224no—
547C2×C4—1221no—
548C2×C4—2442no—
549C2×C4—2444no—
550C2×C4—2444no—
551C2×C4—2441no—
552C2×C4—2444no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2444no—
556C2×C4—2442no—
557C2×C4—2444no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2441no—
561C2×C4C2×C22⋊#692311yescharacteristic
562C2×C4C2×C22⋊#692312yes—
563C2×C4C2×C22⋊#692312yes—
564C2×C4C2×C22⋊#692311yescharacteristic
565C22—21no—
566C22—21no—
567C22—22no—
568C22—42no—
569C22—42no—
570C22—41no—
571C22—41no—
572C22—612no—
573C22—612no—
574C22—1222no—
575C22—1222no—
576C22—4882no—
577C22—4882no—
578C22—2441no—
579C22—2441no—
580C22—2442no—
581C22—2442no—
582C22—2442no—
583C22—2442no—
584C22C22×C22⋊#692311yescharacteristic
585C22C2×D4×#692311yescharacteristic
586C22#1684082911yescharacteristic
587C4—22no—
588C4—22no—
589C4—21no—
590C4—42no—
591C4—41no—
592C4—1224no—
593C4—1222no—
594C4—1222no—
595C4—4884no—
596C4—4882no—
597C4—2444no—
598C4—2441no—
599C4#1684074112yescomplemented
600C2—21no—
601C2—21no—
602C2—42no—
603C2—1222no—
604C2—1222no—
605C2—4882no—
606C2D4×C22⋊#692311yescharacteristic
607C2C2×#1684074111yescharacteristic
608C2#2146242011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC22×C4⋊C4×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840413C2
cosocleC23×D61C24
hypercenterC22×C4⋊C4#6923
hypocenterC61#22406
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16840741C2semidirect
2. #21462327C2semidirect
3. #21461959C2semidirect
4. #21461843C2semidirect
5. #21461679C2semidirect
6. #21461648C2semidirect
7. C4×#14519835C2semidirect
28. C2×#14519776C4semidirect
30. C2×#14519760C4semidirect
31. C2×#14519755C4semidirect
33. C2×#14519751C4semidirect
34. C2×#14519739C4semidirect
37. C22×C4⋊C4×C61C4semidirect
51. #14519835D4semidirect
291. C4×C61#410semidirect
327. C22×C4⋊C4#6923semidirect
342. C61#22406semidirect
599. C4#16840741semidirect

Maximal subgroups

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

subgroupindex
C2×#168407412
#214623272
#214619592
#214618432
#214616792
#214616482
C4×#145198352
C2×#168404132
#2240661