Finite Group Databasebeta

← #29695670

Subgroups — #29695670

Order 15616.

1582 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
1C2×#16840433C211yesmaximal, characteristic
2C2×#16839840C211yesmaximal, characteristic, complemented
3C2×#16839761C212yesmaximal, complemented
4#21446746C211yesmaximal, characteristic, complemented
5#21446624C211yesmaximal, characteristic, complemented
6#21446255C212yesmaximal, complemented
7#21446136C212yesmaximal, complemented
8#21445821C211yesmaximal, characteristic, complemented
9#21445757C212yesmaximal, complemented
10C4×#14519700C211yesmaximal, characteristic, complemented
11D4×C4⋊C4×C61C211yesmaximal, characteristic, complemented
12C22×#11758222C2212yes—
13C22×#11758194C2212yes—
14#16840433—24no—
15#16840433C418yescomplemented
16C2×#14519774C2211yescharacteristic
17C2×#14519773—21no—
18C2×#14519751C2212yes—
19C2×#14519750—22no—
20C2×C4×Dic122C2211yescharacteristic
21C2×#14519739—21no—
22#16839840—24no—
23C2×C4⋊#92418C2211yescharacteristic
24#16839761—28no—
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2212yes—
27C2×#14519700C2211yescharacteristic
28C2×#14519697C2212yes—
29C2×#14519697C2212yes—
30C2×#14519697C2212yes—
31C2×#14519697C2212yes—
32C2×C22⋊#92418C2212yes—
33C2×C4×#92418C2211yescharacteristic
34#16839524C2211yescharacteristic
35#16839524C2211yescharacteristic
36#16839524C2212yes—
37C2×C4×D4×C61—21no—
38C2×C4×D4×C61C2211yescharacteristic
39C22×C4⋊C4×C61C2212yes—
40C61×#389—21no—
41C61×#385—22no—
42C61×#384C2211yescharacteristic
43C61×#380C2212yes—
44C4×C4⋊C4×C61C2211yescharacteristic
45C2×#11758222—24no—
46C2×#11758222—24no—
47C2×#11758222—24no—
48C2×#11758222—42no—
49C2×#11758222—42no—
50C2×#11758222C2×C418yescomplemented
51C22×Dic122C2311yescharacteristic
52C22×D244—21no—
53C22×D244—21no—
54C22×D244C2312yes—
55C22×C22⋊D61—22no—
56C22×C22⋊D61—22no—
57C22×C4×D61—22no—
58C2×#11758194—24no—
59C2×#11758194—24no—
60C2×#11758194—42no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2312yes—
63C22×#92418C2311yescharacteristic
64#14519774—22no—
65#14519774C2×C414yescomplemented
66#14519773—28no—
67#14519751—24no—
68#14519751C2×C418yescomplemented
69#14519750—216no—
70C4×Dic122—22no—
71C4×Dic122C2×C414yescomplemented
72#14519739—28no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758198—22no—
76C2×C4×Dic61—21no—
77C4⋊#92418—22no—
78#14519700—24no—
79#14519700—22no—
80#14519700D414yescomplemented
81#14519697—24no—
82#14519697—24no—
83#14519697—42no—
84#14519697—42no—
85C22⋊#92418—24no—
86C4×#92418—22no—
87C22×D4×C61D411yescharacteristic
88C23×C4×C61—22no—
89C23×C4×C61C2312yes—
90C4×D4×C61—44no—
91C4×D4×C61C2×C414yes—
92C4×D4×C61D414yes—
93C2×C4⋊C4×C61—24no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—42no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C4⋊C4×C61C2312yes—
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C22⋊C4×C61—22no—
103C2×C22⋊C4×C61D412yes—
104C2×C42×C61—21no—
105C2×C42×C61C2311yescharacteristic
106#95×C61—21no—
107#95×C61—22no—
108C24×C61C2×D412yes—
109C23×D61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61C22⋊C414yes—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61—44no—
121C22×C4×C61—44no—
122C22×C4×C61—82no—
123C22×C4×C61C22×C414yes—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D414yes—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—28no—
130C2×C22⋊D61—28no—
131C2×C22⋊D61—44no—
132C2×C22⋊D61—44no—
133C2×C22⋊D61—82no—
134C2×C22⋊D61—82no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137#11758222—216no—
138#11758222—48no—
139#11758222—48no—
140#11758222—48no—
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—22no—
144C2×D244—22no—
145C2×D244—24no—
146C2×D244—22no—
147C2×D244—24no—
148C2×D244—42no—
149C2×D244—42no—
150C2×D244—81no—
151C2×D244—81no—
152C2×D244C22×C414yes—
153C2×C4×D61—28no—
154C2×C4×D61—44no—
155C2×C4×D61—44no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122C22×C412yes—
160C2×Dic122C2×D412yes—
161C2×Dic122#4112yes—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—44no—
168C4⋊C4×C61—42no—
169C4⋊C4×C61—81no—
170C4⋊C4×C61C22⋊C414yes—
171C4⋊C4×C61C2×D412yes—
172C22⋊C4×C61—44no—
173C22⋊C4×C61C22⋊C418yes—
174C42×C61—42no—
175C42×C61C22×C412yes—
176C42×C61C2×D412yes—
177#11758198—28no—
178C4⋊Dic61—21no—
179C4⋊Dic61C22×C412yes—
180#11758196—24no—
181C4×Dic61—24no—
182#11758194—48no—
183C2×#92418—21no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—22no—
191C2×#92418C2×D412yes—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61C2×C22⋊C414yes—
199C23×C61C22×D411yescharacteristic
200C22×D61—24no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—42no—
207D4×C61—28no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—24no—
213C2×C4×C61—24no—
214C2×C4×C61—24no—
215C2×C4×C61—24no—
216C2×C4×C61—22no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—42no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—41no—
225C2×C4×C61—88no—
226C2×C4×C61C2×C22⋊C412yes—
227C2×C4×C61C2×C22⋊C414yes—
228C2×C4×C61C2×C22⋊C412yes—
229C2×C4×C61AGL(1,ℤ/8)12yes—
230C2×C4×C61#13712yes—
231C2×C4×C61C23×C411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61H(5,𝔽2)11yescharacteristic
236C2×C4×C61H(5,𝔽2)11yescharacteristic
237C22⋊D61—416no—
238C22⋊D61—88no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245D244—24no—
246D244—24no—
247D244—42no—
248D244—42no—
249D244—48no—
250D244—84no—
251C4×D61—416no—
252Dic122—22no—
253Dic122—22no—
254Dic122—22no—
255Dic122C4×D414yescomplemented
256#92418—22no—
257#92418—42no—
258#92418—41no—
259#22432—611nomaximal, Sylow
260C22×C61—24no—
261C22×C61—24no—
262C22×C61—24no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61C22×C22⋊C411yescharacteristic
268C22×C61D4211yescharacteristic
269C22×C61#54611yescharacteristic
270C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C61C2×#13711yescharacteristic
272C22×C61#57611yescharacteristic
273C22×C61#57711yescharacteristic
274C2×D61—44no—
275C2×D61—44no—
276C2×D61—44no—
277C2×D61—82no—
278C2×D61—82no—
279C2×D61—82no—
280C2×D61—82no—
281C4×C61—24no—
282C4×C61—22no—
283C4×C61—42no—
284C4×C61—81no—
285C4×C61—82no—
286C4×C61#41714yescomplemented
287C4×C61C2×C4×D411yescharacteristic
288C4×C61#51811yescharacteristic
289C4×C61D4211yescharacteristic
290C4×C61#54611yescharacteristic
291Dic61—24no—
292Dic61—42no—
293C2×#550—611no—
294C2×#436—611no—
295C2×#417—612no—
296D4×C4⋊C4D6111yescharacteristic, complemented
297#1982—611no—
298#1957—611no—
299#1895—612no—
300#1874—612no—
301#1803—611no—
302#1788—612no—
303C4×#103—611no—
304C2×C61—44no—
305C2×C61D4×C22⋊C411yescharacteristic
306C2×C61C2×#41711yescharacteristic
307C2×C61#289711yescharacteristic
308C2×C61#327411yescharacteristic
309C2×C61#327811yescharacteristic
310C2×C61#333211yescharacteristic
311C2×C61#333311yescharacteristic
312D61—84no—
313C22×#41—612no—
314C22×M16—612no—
315#550—618no—
316#550—1224no—
317C2×#127—611no—
318C2×#127—1221no—
319C2×#121—612no—
320C2×#121—1222no—
321C2×C4×Q8—611no—
322C2×C4×D4—21no—
323C2×C4×D4—1221no—
324C2×C4×D4C2×D6111yescharacteristic
325C22×C4⋊C4C2×D6112yes—
326#436—1224no—
327C2×C4⋊C8—611no—
328#417—1228no—
329C2×#103—611no—
330C2×#103—612no—
331C2×#103—611no—
332C2×#102—612no—
333C2×#102—612no—
334C2×#102—612no—
335C2×#102—612no—
336C2×C22⋊C8—612no—
337C2×C4×C8—611no—
338#389—21no—
339#385—22no—
340#384C2×D6111yescharacteristic
341#380C2×D6112yes—
342C4×C4⋊C4C2×D6111yescharacteristic
343#340—612no—
344#340—611no—
345#340—611no—
346C61#2243211yescharacteristic, complemented, Sylow
347C2×#41—618no—
348C2×#41—1224no—
349C2×#41—1224no—
350C2×#41—1224no—
351C2×#41—2442no—
352C2×#41—2442no—
353C22×Q8—611no—
354C22×D4—612no—
355C22×D4—1222no—
356C22×D4—1222no—
357C22×D4—1221no—
358C22×D4—1221no—
359C22×D4C22⋊D6111yescharacteristic
360C23×C4—22no—
361C23×C4—1222no—
362C23×C4C22×D6112yes—
363C2×M16—1224no—
364C2×M16—1224no—
365C2×M16—2442no—
366C22×C8—611no—
367C22×C8—612no—
368C22×C8—611no—
369#127—614no—
370#127—1228no—
371#127—1222no—
372#121—618no—
373#121—1224no—
374#121—12216no—
375C4×Q8—614no—
376C4×Q8—1222no—
377C4×D4—44no—
378C4×D4—1228no—
379C4×D4C4×D6114yes—
380C4×D4D24414yes—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—42no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4C22⋊D6111yescharacteristic
389C2×C4⋊C4C22×D6112yes—
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C22⋊C4—22no—
393C2×C22⋊C4—1222no—
394C2×C22⋊C4C22⋊D6112yes—
395C2×C42—21no—
396C2×C42—1221no—
397C2×C42C22×D6111yescharacteristic
398C4⋊C8—1222no—
399#103—614no—
400#103—1222no—
401#103—1224no—
402#102—1224no—
403#102—1224no—
404#102—2442no—
405#102—2442no—
406C22⋊C8—1224no—
407C4×C8—1222no—
408#95—21no—
409#95—22no—
410C24—1222no—
411C24C2×C22⋊D6112yes—
412#41—12216no—
413#41—2448no—
414#41—2448no—
415#41—2448no—
416C2×Q8—612no—
417C2×Q8—612no—
418C2×Q8—612no—
419C2×Q8—1221no—
420C2×Q8—1221no—
421C2×Q8—1221no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—614no—
425C2×D4—1228no—
426C2×D4—1228no—
427C2×D4—1222no—
428C2×D4—1222no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1222no—
432C2×D4—1224no—
433C2×D4—1224no—
434C2×D4—4881no—
435C2×D4—4881no—
436C2×D4—4882no—
437C2×D4—4882no—
438C2×D4—2444no—
439C2×D4—2444no—
440C2×D4—2442no—
441C2×D4—2442no—
442C2×D4#1175819814yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—82no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1228no—
456C22×C4—2444no—
457C22×C4—2444no—
458C22×C4C2×C4×D6114yes—
459C22×C4C2×D24414yes—
460C22×C4C2×C22⋊D6111yescharacteristic
461C22×C4C2×C22⋊D6112yes—
462C22×C4C2×C22⋊D6111yescharacteristic
463C22×C4C23×D6111yescharacteristic
464M16—2448no—
465C2×C8—612no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1222no—
469C2×C8—1221no—
470C2×C8—1222no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1222no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—81no—
482C4⋊C4—612no—
483C4⋊C4—1221no—
484C4⋊C4—1224no—
485C4⋊C4#1175819814yes—
486C4⋊C4D4×D6112yes—
487C22⋊C4—44no—
488C22⋊C4—1228no—
489C22⋊C4#1175819818yes—
490C42—42no—
491C42—1224no—
492C42C2×C4×D6112yes—
493C42C2×D24412yes—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—1224no—
501C23—2442no—
502C23—2442no—
503C23—2442no—
504C23—2442no—
505C23—2442no—
506C23—2442no—
507C23C2×#1175819814yes—
508C23C22×C22⋊D6111yescharacteristic
509Q8—614no—
510Q8—1222no—
511Q8—1222no—
512Q8—1222no—
513D4—28no—
514D4—1224no—
515D4—1224no—
516D4—4884no—
517D4—4888no—
518D4—24416no—
519D4—2448no—
520D4—2442no—
521D4—2442no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—88no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—24416no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4C2×#1175819812yes—
548C2×C4C2×#1175819814yes—
549C2×C4C2×#1175819812yes—
550C2×C4#1451980312yes—
551C2×C4#1451980812yes—
552C2×C4C22×C4×D6111yescharacteristic
553C2×C4C2×D4×D6111yescharacteristic
554C2×C4C22×D24411yescharacteristic
555C2×C4#41×D6111yescharacteristic
556C2×C4#1451982211yescharacteristic
557C2×C4#1451982311yescharacteristic
558C8—1222no—
559C8—2442no—
560C8—2441no—
561C22—24no—
562C22—24no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—4882no—
569C22—4882no—
570C22—4882no—
571C22—4882no—
572C22—2444no—
573C22—2444no—
574C22—2444no—
575C22C22×#1175819811yescharacteristic
576C22D4×C22⋊D6111yescharacteristic
577C22#1684040011yescharacteristic
578C22C2×#1451980311yescharacteristic
579C22C2×#1451980811yescharacteristic
580C22#1684056511yescharacteristic
581C22#1684057011yescharacteristic
582C4—22no—
583C4—24no—
584C4—42no—
585C4—82no—
586C4—81no—
587C4—1224no—
588C4—2442no—
589C4#1683976114yescomplemented
590C4C4×D4×D6111yescharacteristic
591C4#1684023411yescharacteristic
592C4D4×D24411yescharacteristic
593C4#1684040711yescharacteristic
594C2—44no—
595C2—4884no—
596C2D4×#1175819811yescharacteristic
597C2C2×#1683976111yescharacteristic
598C2#2145434211yescharacteristic
599C2#2145743511yescharacteristic
600C2#2145748111yescharacteristic
601C2#2145814011yescharacteristic
602C2#2145815111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingD4×C4⋊C4×C61C2
socleC23×C61C22×D4
prime radicalC2×#16840433C2
cosocleC22×C4×C61C24
hypercenterD4×C4⋊C4D61
hypocenterC61#22432
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16839840C2semidirect
3. C2×#16839761C2semidirect
4. #21446746C2semidirect
5. #21446624C2semidirect
6. #21446255C2semidirect
7. #21446136C2semidirect
8. #21445821C2semidirect
9. #21445757C2semidirect
10. C4×#14519700C2semidirect
11. D4×C4⋊C4×C61C2semidirect
15. #16840433C4semidirect
50. C2×#11758222C2×C4semidirect
65. #14519774C2×C4semidirect
68. #14519751C2×C4semidirect
71. C4×Dic122C2×C4semidirect
80. #14519700D4semidirect
255. Dic122C4×D4semidirect
286. C4×C61#417semidirect
296. D4×C4⋊C4D61semidirect
346. C61#22432semidirect
589. C4#16839761semidirect

Maximal subgroups

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

subgroupindex
C2×#168404332
C2×#168398402
C2×#168397612
#214467462
#214466242
#214462552
#214461362
#214458212
#214457572
C4×#145197002
D4×C4⋊C4×C612
#2243261