Finite Group Databasebeta

← #29721538

Subgroups — #29721538

Order 15616.

1222 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×#16839957C212yesmaximal, complemented
2C2×#16839957C211yesmaximal, characteristic, complemented
3C2×C4×#11758199C211yesmaximal, characteristic, complemented
4#21450761C211yesmaximal, characteristic, complemented
5#21447204C212yesmaximal, complemented
6#21447053C212yesmaximal, complemented
7#21446624C211yesmaximal, characteristic, complemented
8#21446538C211yesmaximal, characteristic, complemented
9#21446167C212yesmaximal, complemented
10#21445816C211yesmaximal, characteristic, complemented
11D4×C4⋊C4×C61C211yesmaximal, characteristic, complemented
12C22×#11758199C2212yes—
13C22×#11758199C2211yescharacteristic
14C2×#14519774C2211yescharacteristic
15C2×#14519774C2212yes—
16C2×#14519752—21no—
17C2×#14519752—21no—
18C2×C4×D244C2211yescharacteristic
19#16839957—28no—
20#16839957—24no—
21C4×#11758199—24no—
22C2×#14519710C2211yescharacteristic
23C2×C4⋊#92418C2212yes—
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519697C2212yes—
27C2×#14519697C2212yes—
28C2×#14519697C2212yes—
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2212yes—
31C2×C4×#92418C2211yescharacteristic
32#16839672—22no—
33#16839664C2212yes—
34#16839524C2212yes—
35#16839517C2212yes—
36C2×C4×D4×C61C2212yes—
37C2×C4×D4×C61C2211yescharacteristic
38C22×C4⋊C4×C61—21no—
39C61×#389C2212yes—
40C61×#385—22no—
41C61×#384C2211yescharacteristic
42C61×#380—21no—
43C4×C4⋊C4×C61C2211yescharacteristic
44C22×D244—21no—
45C22×D244—22no—
46C22×D244—22no—
47C22×D244—21no—
48C22×D244C2312yes—
49C22×D244C2311yescharacteristic
50C22×C4×D61—21no—
51C2×#11758199—22no—
52C2×#11758199—22no—
53C2×#11758199—24no—
54C2×#11758199—24no—
55C2×#11758199—24no—
56C2×#11758199D414yescomplemented
57C22×#92418C2311yescharacteristic
58C22×#92418C2312yes—
59C22×#92418C2311yescharacteristic
60#14519774—24no—
61#14519774—22no—
62#14519774—22no—
63#14519774—42no—
64#14519752—44no—
65#14519752—44no—
66C4×D244—22no—
67C4×D244—22no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758198—21no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72#14519710—22no—
73C4⋊#92418—22no—
74C4⋊#92418—24no—
75#14519698—22no—
76#14519697—24no—
77#14519697—24no—
78#14519697—22no—
79#14519697—24no—
80#14519697—42no—
81C4×#92418—22no—
82C22×D4×C61C2311yescharacteristic
83C23×C4×C61—21no—
84C23×C4×C61—22no—
85C4×D4×C61—24no—
86C4×D4×C61—22no—
87C4×D4×C61—42no—
88C4×D4×C61D414yescomplemented
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—44no—
93C2×C4⋊C4×C61D411yescharacteristic
94C2×C4⋊C4×C61D411yescharacteristic
95C2×C4⋊C4×C61C2312yes—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C22⋊C4×C61—21no—
98C2×C22⋊C4×C61—22no—
99C2×C42×C61C2312yes—
100C2×C42×C61C2311yescharacteristic
101#95×C61—22no—
102#95×C61D412yes—
103C24×C61—21no—
104C23×D61—21no—
105C23×D61—22no—
106C2×D4×C61—21no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61C2×D412yes—
110C2×D4×C61C2×D412yes—
111C2×D4×C61#4112yes—
112C22×C4×C61—21no—
113C22×C4×C61—24no—
114C22×C4×C61—22no—
115C22×C4×C61—44no—
116C22×C4×C61—42no—
117C22×C4×C61—42no—
118C22×C4×C61—44no—
119C22×C4×C61—82no—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2411yescharacteristic
125C22×Dic61—21no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—21no—
135C2×D244—22no—
136C2×D244—22no—
137C2×D244—44no—
138C2×D244—44no—
139C2×D244—44no—
140C2×D244—42no—
141C2×D244—42no—
142C2×D244—44no—
143C2×D244—42no—
144C2×D244—42no—
145C2×D244—82no—
146C2×D244—81no—
147C2×D244—81no—
148C2×D244—82no—
149C2×D244C2×D412yes—
150C2×C4×D61—42no—
151C2×C4×D61—42no—
152C2×C4×D61—42no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—44no—
158C4⋊C4×C61—42no—
159C4⋊C4×C61—81no—
160C4⋊C4×C61D814yescomplemented
161C4⋊C4×C61C2×D412yes—
162C22⋊C4×C61—24no—
163C22⋊C4×C61—44no—
164C42×C61—22no—
165C42×C61—22no—
166C42×C61C2×D412yes—
167C42×C61#4112yes—
168#11758199—28no—
169#11758199—48no—
170#11758198—44no—
171#11758198—42no—
172#11758198—44no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—21no—
182C2×#92418—21no—
183C2×#92418C2×D412yes—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61C22×D411yescharacteristic
192C22×D61—22no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—42no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61—42no—
205C22×D61—42no—
206D4×C61—22no—
207D4×C61—22no—
208D4×C61—22no—
209D4×C61#12114yescomplemented
210C2×C4×C61—22no—
211C2×C4×C61—21no—
212C2×C4×C61—22no—
213C2×C4×C61—21no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—48no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—88no—
225C2×C4×C61#12112yes—
226C2×C4×C61#12112yes—
227C2×C4×C61C2×D814yes—
228C2×C4×C61C2×D812yes—
229C2×C4×C61AGL(1,ℤ/8)12yes—
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61H(5,𝔽2)11yescharacteristic
235C2×C4×C612−1+411yescharacteristic
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239D244—22no—
240D244—22no—
241D244—42no—
242D244—41no—
243D244—42no—
244D244—42no—
245D244—41no—
246D244—42no—
247D244—88no—
248D244—84no—
249D244—88no—
250D244—84no—
251C4×D61—84no—
252#92418—22no—
253#92418—42no—
254#92418—41no—
255#24726—611nomaximal, Sylow
256C22×C61—24no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61C2×#12111yescharacteristic
264C22×C61D4211yescharacteristic
265C22×C61#54811yescharacteristic
266C22×C61C22×D811yescharacteristic
267C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C61#57611yescharacteristic
269C22×C61#57711yescharacteristic
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C2×D61—81no—
277C2×D61—82no—
278C2×D61—82no—
279C2×D61—82no—
280C2×D61—82no—
281C2×D61—82no—
282C2×D61—82no—
283C2×D61—82no—
284C4×C61—24no—
285C4×C61—22no—
286C4×C61—42no—
287C4×C61—42no—
288C4×C61—82no—
289C4×C61#45914yescomplemented
290C4×C61C2×#12111yescharacteristic
291C4×C61#53711yescharacteristic
292C4×C61#54711yescharacteristic
293C4×C61#55311yescharacteristic
294Dic61—81no—
295C2×#459—612no—
296C2×#459—611no—
297C2×C4×D8—611no—
298#2541—611no—
299D4×C4⋊C4D6111yescharacteristic, complemented
300#2067—612no—
301#2041—612no—
302#1957—611no—
303#1939—611no—
304#1879—612no—
305#1802—611no—
306C2×C61—44no—
307C2×C61#241611yescharacteristic
308C2×C61C2×#45911yescharacteristic
309C2×C61#304511yescharacteristic
310C2×C61#327211yescharacteristic
311C2×C61D4×D811yescharacteristic
312C2×C61#335811yescharacteristic
313C2×C61#336611yescharacteristic
314D61—82no—
315D61—162no—
316C22×D8—611no—
317C22×D8—612no—
318C2×#127—611no—
319C2×#127—612no—
320C2×#121—1221no—
321C2×#121—1221no—
322C2×C4×D4—611no—
323C2×C4×D4C2×D6112yes—
324C2×C4×D4C2×D6111yescharacteristic
325C22×C4⋊C4—21no—
326#459—1228no—
327#459—1224no—
328C4×D8—1224no—
329C2×#107—611no—
330C2×C4⋊C8—611no—
331C2×C4⋊C8—612no—
332C2×#102—611no—
333C2×#102—612no—
334C2×#102—612no—
335C2×#102—612no—
336C2×#102—611no—
337C2×#102—612no—
338C2×C4×C8—611no—
339#392—1222no—
340#390—612no—
341#389C2×D6112yes—
342#385—22no—
343#384C2×D6111yescharacteristic
344#380—21no—
345C4×C4⋊C4C2×D6111yescharacteristic
346#340—612no—
347#336—612no—
348C61#2472611yescharacteristic, complemented, Sylow
349C22×D4—611no—
350C22×D4—612no—
351C22×D4—1221no—
352C22×D4—1222no—
353C22×D4—1222no—
354C22×D4—1221no—
355C22×D4C22×D6111yescharacteristic
356C23×C4—21no—
357C23×C4—22no—
358C23×C4—1221no—
359C2×D8—614no—
360C2×D8—1222no—
361C2×D8—1222no—
362C2×D8—1224no—
363C2×D8—1224no—
364C2×D8—1224no—
365C22×C8—611no—
366C22×C8—612no—
367C22×C8—611no—
368#127—1224no—
369#127—1222no—
370#127—1222no—
371#127—2442no—
372#121—2444no—
373#121—2444no—
374C4×D4—24no—
375C4×D4—22no—
376C4×D4—42no—
377C4×D4—1222no—
378C4×D4—1222no—
379C4×D4D24414yescomplemented
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—44no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4C22⋊D6111yescharacteristic
386C2×C4⋊C4C22⋊D6111yescharacteristic
387C2×C4⋊C4C22×D6112yes—
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1222no—
393C2×C22⋊C4—1222no—
394C2×C42C22×D6112yes—
395C2×C42C22×D6111yescharacteristic
396#107—1222no—
397C4⋊C8—1224no—
398C4⋊C8—1222no—
399#102—1224no—
400#102—1224no—
401#102—1222no—
402#102—1222no—
403#102—1224no—
404#102—2442no—
405C4×C8—1222no—
406#95—22no—
407#95C22⋊D6112yes—
408C24—21no—
409C24—1221no—
410C24—1222no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—612no—
415C2×D4—1222no—
416C2×D4—1221no—
417C2×D4—1222no—
418C2×D4—1221no—
419C2×D4—1222no—
420C2×D4—1221no—
421C2×D4—1222no—
422C2×D4—1221no—
423C2×D4—1222no—
424C2×D4—1221no—
425C2×D4—1222no—
426C2×D4—4881no—
427C2×D4—4882no—
428C2×D4—4881no—
429C2×D4—4882no—
430C2×D4—2444no—
431C2×D4—2444no—
432C2×D4—2444no—
433C2×D4—2442no—
434C2×D4—2442no—
435C2×D4—2444no—
436C2×D4—2442no—
437C2×D4—2442no—
438C2×D4C2×D24412yes—
439C2×D4D4×D6112yes—
440C2×D4#1175822612yes—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—82no—
449C22×C4—1221no—
450C22×C4—2442no—
451C22×C4—2442no—
452C22×C4—2442no—
453C22×C4C2×C22⋊D6112yes—
454C22×C4C2×C22⋊D6111yescharacteristic
455C22×C4C2×C22⋊D6112yes—
456C22×C4C2×C22⋊D6111yescharacteristic
457C22×C4C23×D6111yescharacteristic
458D8—1228no—
459D8—2448no—
460C2×C8—612no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1222no—
464C2×C8—1221no—
465C2×C8—1222no—
466C2×C8—1222no—
467C2×C8—1221no—
468C2×C8—1221no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—81no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4#1175819914yescomplemented
479C4⋊C4C2×D24412yes—
480C22⋊C4—24no—
481C22⋊C4—44no—
482C22⋊C4—2444no—
483C22⋊C4—2442no—
484C22⋊C4—2444no—
485C42—22no—
486C42—22no—
487C42C2×D24412yes—
488C42#1175822212yes—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—1222no—
497C23—2442no—
498C23—2442no—
499C23—2442no—
500C23—2442no—
501C23—2442no—
502C23—2442no—
503C23—2442no—
504C23—2441no—
505C23—2441no—
506C23—2441no—
507C23—2441no—
508C23—2441no—
509C23—2441no—
510C23C22×C22⋊D6111yescharacteristic
511D4—22no—
512D4—22no—
513D4—22no—
514D4—1222no—
515D4—1222no—
516D4—4884no—
517D4—4888no—
518D4—4888no—
519D4—4884no—
520D4—2442no—
521D4—2441no—
522D4—2442no—
523D4—2442no—
524D4—2441no—
525D4—2442no—
526D4#1451975214yescomplemented
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—48no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—88no—
542C2×C4—4884no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4#1451975112yes—
547C2×C4#1451975112yes—
548C2×C4C2×#1175819914yes—
549C2×C4C2×#1175819912yes—
550C2×C4#1451980212yes—
551C2×C4C2×D4×D6111yescharacteristic
552C2×C4C22×D24411yescharacteristic
553C2×C4C2×#1175822211yescharacteristic
554C2×C4C2×#1175822611yescharacteristic
555C2×C4#1451982211yescharacteristic
556C2×C4#1451982511yescharacteristic
557C8—1222no—
558C8—2442no—
559C8—2441no—
560C22—24no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—4882no—
568C22—4882no—
569C22—4882no—
570C22—4882no—
571C22—4882no—
572C22—4882no—
573C22—4882no—
574C22—4881no—
575C22—4881no—
576C22—4881no—
577C22—4881no—
578C22—2442no—
579C22—2442no—
580C22—2442no—
581C22C2×#1451975111yescharacteristic
582C22D4×C22⋊D6111yescharacteristic
583C22#1684042311yescharacteristic
584C22C22×#1175819911yescharacteristic
585C22C2×#1451980211yescharacteristic
586C22#1684056511yescharacteristic
587C22#1684057011yescharacteristic
588C4—24no—
589C4—22no—
590C4—42no—
591C4—42no—
592C4—82no—
593C4—4881no—
594C4#1683995714yescomplemented
595C4C2×#1451975211yescharacteristic
596C4#1684034311yescharacteristic
597C4#1684041311yescharacteristic
598C4#1684046011yescharacteristic
599C2—44no—
600C2—9762no—
601C2—4882no—
602C2#2144925311yescharacteristic
603C2C2×#1683995711yescharacteristic
604C2#2145535811yescharacteristic
605C2#2145741911yescharacteristic
606C2D4×#1175819911yescharacteristic
607C2#2145845411yescharacteristic
608C2#2145852611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingD4×C4⋊C4×C61C2
socleC23×C61C22×D4
hypercenterD4×C4⋊C4D61
hypocenterC61#24726
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16839957C2semidirect
2. C2×#16839957C2semidirect
3. C2×C4×#11758199C2semidirect
4. #21450761C2semidirect
5. #21447204C2semidirect
6. #21447053C2semidirect
7. #21446624C2semidirect
8. #21446538C2semidirect
9. #21446167C2semidirect
10. #21445816C2semidirect
11. D4×C4⋊C4×C61C2semidirect
56. C2×#11758199D4semidirect
88. C4×D4×C61D4semidirect
160. C4⋊C4×C61D8semidirect
209. D4×C61#121semidirect
289. C4×C61#459semidirect
299. D4×C4⋊C4D61semidirect
348. C61#24726semidirect
379. C4×D4D244semidirect
478. C4⋊C4#11758199semidirect
526. D4#14519752semidirect
594. C4#16839957semidirect

Maximal subgroups

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

subgroupindex
C2×#168399572
C2×C4×#117581992
#214507612
#214472042
#214470532
#214466242
#214465382
#214461672
#214458162
D4×C4⋊C4×C612
#2472661