Finite Group Databasebeta

← #30620136

Subgroups — #30620136

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×#16840742C212yesmaximal, complemented
2#21462327C212yesmaximal, complemented
3#21461958C212yesmaximal, complemented
4#21461842C214yesmaximal, complemented
5#21461679C211yesmaximal, characteristic, complemented
6#21461649C212yesmaximal, complemented
7C4×#14519835C211yesmaximal, characteristic, complemented
8C2×#16840412C211yesmaximal, characteristic
9C2×#14519856C2214yes—
10#16840742—28no—
11C2×#14519835C2211yescharacteristic
12C2×#14519835C2214yes—
13C2×#14519835C2212yes—
14C2×#14519835C2214yes—
15C2×#14519835C2211yescharacteristic
16#16840713—22no—
17#16840711—24no—
18#16840705C2214yes—
19#16840704C2212yes—
20C4⋊C4×#6923—22no—
21C4×C22⋊#6923C2212yes—
22#16840640C2212yes—
23#16840640C2212yes—
24C22×#11758224C2212yes—
25#16840412—24no—
26#16840412—22no—
27#16840412—22no—
28C2×#14519776C411yescharacteristic, complemented
29C2×#14519759C2212yes—
30C2×#14519758C412yescomplemented
31C2×#14519754C412yescomplemented
32C2×#14519748C2211yescharacteristic
33C2×D4×Dic61C411yescharacteristic, complemented
34C2×C4×C22⋊D61C2211yescharacteristic
35C22×C4⋊Dic61C411yescharacteristic, complemented
36C2×C4⋊C4×D61C2211yescharacteristic
37C2×#121×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#14519835—24no—
48#14519835—22no—
49#14519835—28no—
50#14519835—44no—
51#14519835D414yescomplemented
52(C2×#6923)⋊C4—22no—
53(C2×#6923)⋊C4—24no—
54(C2×#6923)⋊C4—24no—
55(C2×#6923)⋊C4—22no—
56C2×#11758224—24no—
57C2×#11758224—24no—
58C2×#11758224—24no—
59C2×#11758224—24no—
60C2×#11758224—44no—
61C22×Dic122C2×C412yes—
62C22×C22⋊D61C2312yes—
63C22×C22⋊D61C2311yescharacteristic
64C23×Dic61C2×C412yes—
65C23×Dic61C2×C412yes—
66C22×C4×D61C2312yes—
67C22×C4×D61C2311yescharacteristic
68#14519776—22no—
69#14519776—22no—
70#14519759—24no—
71#14519759—24no—
72#14519758—24no—
73#14519758—24no—
74#14519754—24no—
75#14519754—24no—
76#14519748—22no—
77#14519748—22no—
78D4×Dic61—22no—
79D4×Dic61—21no—
80D4×Dic61—21no—
81C4×C22⋊D61—22no—
82C4×C22⋊D61—22no—
83C2×C4⋊Dic61—22no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61C2×C411yescharacteristic
87C2×C4⋊Dic61C2×C412yes—
88C2×C4⋊Dic61C2×C411yescharacteristic
89C2×#11758196C2×C411yescharacteristic
90C2×#11758196C2×C412yes—
91C2×#11758196C2×C412yes—
92C4⋊C4×D61D414yes—
93C4⋊C4×D61D412yes—
94C4⋊C4×D61D412yes—
95C2×C22⋊Dic61C2×C411yescharacteristic
96C2×C22⋊Dic61C2×C412yes—
97C2×C22⋊Dic61C2×C412yes—
98C2×#11758198C2311yescharacteristic
99C2×C4×Dic61C2×C411yescharacteristic
100C22×D4×C61C2×C412yes—
101C22×D4×C61C2×C411yescharacteristic
102C23×C4×C61C2×C411yescharacteristic
103#121×C61—22no—
104#121×C61—22no—
105C2×C4⋊C4×C61C2×C411yescharacteristic
106C2×C22⋊C4×C61C2×C412yes—
107C24×C61C22×C412yes—
108C24×C61C22×C411yescharacteristic
109C23×D61C2411yescharacteristic
110C22×#6923—22no—
111C22×#6923—22no—
112C22×#6923—24no—
113C2×D4×C61—24no—
114C2×D4×C61—24no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—42no—
118C2×D4×C61—42no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—42no—
122C22×C4×C61C22×C412yes—
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C22×C411yescharacteristic
125C2×C22⋊D61—22no—
126C2×C22⋊D61—22no—
127C2×C22⋊D61—22no—
128C2×C22⋊D61—24no—
129C2×C22⋊D61—22no—
130C2×C22⋊D61—24no—
131C2×C22⋊D61C2×D412yes—
132C2×C22⋊D61#4112yes—
133C22×Dic61—24no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—24no—
139C22×Dic61—42no—
140C22×Dic61—42no—
141C22×Dic61C22×C412yes—
142C22×Dic61C22×C412yes—
143C22×Dic61C22×C411yescharacteristic
144C22×Dic61C22×C411yescharacteristic
145C22×Dic61C22×C411yescharacteristic
146#11758224—48no—
147#11758224—48no—
148#11758224—48no—
149#11758224—48no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—24no—
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×C61C22⋊C412yes—
172C4⋊C4×C61C22⋊C412yes—
173C22⋊C4×C61—24no—
174C22⋊Dic61—22no—
175C22⋊Dic61—22no—
176C22⋊Dic61—22no—
177C22⋊Dic61—24no—
178#11758198—22no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—22no—
182C4⋊Dic61—22no—
183C4⋊Dic61C22⋊C412yes—
184C4⋊Dic61C22⋊C411yescharacteristic
185C4⋊Dic61C22⋊C411yescharacteristic
186#11758196—24no—
187#11758196—22no—
188#11758196C22⋊C414yes—
189#11758196C22⋊C414yes—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61C4×D412yes—
202C23×C61C23×C411yescharacteristic
203C22×D61—41no—
204C22×D61—41no—
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—
218D4×C61—48no—
219D4×C61—42no—
220D4×C61—42no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—21no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—41no—
228C2×C4×C61C2×C22⋊C411yescharacteristic
229C2×C4×C61C2×C22⋊C414yes—
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C22⋊D61—24no—
232C22⋊D61—24no—
233C22⋊D61—44no—
234C22⋊D61—44no—
235C2×Dic61—21no—
236C2×Dic61—22no—
237C2×Dic61—21no—
238C2×Dic61—22no—
239C2×Dic61—21no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—41no—
244C2×Dic61—44no—
245C2×Dic61—44no—
246C2×Dic61—44no—
247C2×Dic61—44no—
248C2×Dic61—41no—
249C2×Dic61C2×C22⋊C412yes—
250C2×Dic61C2×C22⋊C412yes—
251C2×Dic61C2×C22⋊C412yes—
252C2×Dic61C4×D412yes—
253C4×D61—24no—
254C4×D61—24no—
255C4×D61—24no—
256C4×D61—24no—
257C4×D61—24no—
258C4×D61—42no—
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×C61—42no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—82no—
272C22×C61—82no—
273C22×C61C22×C22⋊C411yescharacteristic
274C22×C61C2×C4×D411yescharacteristic
275C22×C61#51811yescharacteristic
276C2×D61—22no—
277C2×D61—22no—
278C2×D61—41no—
279C2×D61—41no—
280C2×D61—42no—
281C2×D61D4212yes—
282C2×D61#54612yes—
283#6923—44no—
284#6923—84no—
285#6923—82no—
286C4×C61—24no—
287C4×C61—21no—
288C4×C61—41no—
289C4×C61#41012yescomplemented
290Dic61—22no—
291Dic61—22no—
292Dic61—22no—
293Dic61—22no—
294Dic61—42no—
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×C61—82no—
308C2×C61D4×C22⋊C411yescharacteristic
309C2×C61C2×#41011yescharacteristic
310C2×C61#288411yescharacteristic
311D61—22no—
312D61—22no—
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#692311yescharacteristic, complemented
323C2×C4×D4—611no—
324C2×C4×D4—611no—
325C2×#117—614no—
326C22×C4⋊C4—611no—
327C22×C4⋊C4—611no—
328#410—1228no—
329C2×#99—611no—
330C2×#99—612no—
331C2×#99—614no—
332C2×#99—611no—
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×D4C2×#692312yes—
352C22×D4C2×#692311yescharacteristic
353C23×C4—612no—
354C23×C4—611no—
355C23×C4—612no—
356C23×C4—612no—
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—22no—
367#121—22no—
368#121—1222no—
369#121—1222no—
370C4×D4—1222no—
371C4×D4—1221no—
372C4×D4—1221no—
373C4×D4—1222no—
374C4×D4—1222no—
375#117—1228no—
376#117—2444no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—612no—
379C2×C4⋊C4—612no—
380C2×C4⋊C4—614no—
381C2×C4⋊C4—612no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—612no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—612no—
386C2×C4⋊C4—1222no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1224no—
390C2×C4⋊C4—1222no—
391C2×C4⋊C4C2×#692311yescharacteristic
392C2×C22⋊C4—612no—
393C2×C22⋊C4—612no—
394C2×C22⋊C4—611no—
395C2×C22⋊C4—612no—
396C2×C22⋊C4—611no—
397C2×C22⋊C4—614no—
398C2×C22⋊C4—612no—
399C2×C22⋊C4C2×#692312yes—
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—
413C24C22×#692312yes—
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—24no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—24no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—612no—
429C2×D4—612no—
430C2×D4—1222no—
431C2×D4—1222no—
432C2×D4—1222no—
433C2×D4—1224no—
434C2×D4—1224no—
435C2×D4—1222no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—611no—
440C22×C4—612no—
441C22×C4—611no—
442C22×C4—614no—
443C22×C4—614no—
444C22×C4—612no—
445C22×C4—612no—
446C22×C4—612no—
447C22×C4—611no—
448C22×C4—1222no—
449C22×C4—1224no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1224no—
456C22×C4—1222no—
457C22×C4—1224no—
458C22×C4—1222no—
459C22×C4—1224no—
460C22×C4—2442no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4C22×#692311yescharacteristic
464C22×C4C22×#692312yes—
465C22×C4C22×#692311yescharacteristic
466C4⋊C4—612no—
467C4⋊C4—611no—
468C4⋊C4—611no—
469C4⋊C4—614no—
470C4⋊C4—614no—
471C4⋊C4—1222no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1224no—
477C4⋊C4—2448no—
478C4⋊C4—2444no—
479C4⋊C4C22⋊#692312yes—
480C4⋊C4C22⋊#692312yes—
481C22⋊C4—24no—
482C22⋊C4—614no—
483C22⋊C4—1222no—
484C22⋊C4—1222no—
485C22⋊C4—1222no—
486C22⋊C4—1222no—
487C22⋊C4—1224no—
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—22no—
500C23—22no—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—612no—
507C23—611no—
508C23—611no—
509C23—612no—
510C23—2441no—
511C23—2441no—
512C23D4×#692312yes—
513C23C23×#692311yescharacteristic
514Q8—2448no—
515D4—48no—
516D4—42no—
517D4—42no—
518D4—1224no—
519D4—1224no—
520D4—2444no—
521D4—2444no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—612no—
530C2×C4—612no—
531C2×C4—612no—
532C2×C4—612no—
533C2×C4—1224no—
534C2×C4—1224no—
535C2×C4—1224no—
536C2×C4—1224no—
537C2×C4—1221no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1221no—
541C2×C4—1224no—
542C2×C4—1224no—
543C2×C4—1221no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—2442no—
548C2×C4—2441no—
549C2×C4—2444no—
550C2×C4—2444no—
551C2×C4—2444no—
552C2×C4—2444no—
553C2×C4—2444no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2444no—
557C2×C4—2442no—
558C2×C4—2444no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2441no—
562C2×C4C2×C22⋊#692311yescharacteristic
563C2×C4C2×C22⋊#692314yes—
564C2×C4C2×C22⋊#692311yescharacteristic
565C22—21no—
566C22—21no—
567C22—22no—
568C22—42no—
569C22—42no—
570C22—41no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—82no—
576C22—82no—
577C22—612no—
578C22—612no—
579C22—1222no—
580C22—1222no—
581C22—2441no—
582C22—2441no—
583C22—2442no—
584C22C22×C22⋊#692311yescharacteristic
585C22C2×D4×#692311yescharacteristic
586C22#1684082911yescharacteristic
587C4—24no—
588C4—21no—
589C4—41no—
590C4—1222no—
591C4—1222no—
592C4—1222no—
593C4—1222no—
594C4—4884no—
595C4—4882no—
596C4—2444no—
597C4—2442no—
598C4—2441no—
599C4#1684074212yescomplemented
600C2—21no—
601C2—21no—
602C2—42no—
603C2—82no—
604C2—1222no—
605C2—1222no—
606C2D4×C22⋊#692311yescharacteristic
607C2C2×#1684074211yescharacteristic
608C2#2146242111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#121×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840412C2
cosocleC23×D61C24
hypercenterC2×#121#6923
hypocenterC61#22406
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16840742C2semidirect
2. #21462327C2semidirect
3. #21461958C2semidirect
4. #21461842C2semidirect
5. #21461679C2semidirect
6. #21461649C2semidirect
7. C4×#14519835C2semidirect
28. C2×#14519776C4semidirect
30. C2×#14519758C4semidirect
31. C2×#14519754C4semidirect
33. C2×D4×Dic61C4semidirect
35. C22×C4⋊Dic61C4semidirect
37. C2×#121×C61C4semidirect
51. #14519835D4semidirect
289. C4×C61#410semidirect
322. C2×#121#6923semidirect
342. C61#22406semidirect
599. C4#16840742semidirect

Maximal subgroups

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

subgroupindex
C2×#168407422
#214623272
#214619582
#214618422
#214616792
#214616492
C4×#145198352
C2×#168404122
#2240661