Finite Group Databasebeta

← #29721544

Subgroups — #29721544

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×#16839957C211yesmaximal, characteristic, complemented
2C2×#16839955C212yesmaximal, complemented
3#21450765C211yesmaximal, characteristic, complemented
4D4×C4⋊Dic61C211yesmaximal, characteristic, complemented
5#21447202C212yesmaximal, complemented
6#21447052C212yesmaximal, complemented
7#21446625C211yesmaximal, characteristic, complemented
8#21446537C211yesmaximal, characteristic, complemented
9#21446165C212yesmaximal, complemented
10#21445819C211yesmaximal, characteristic, complemented
11C2×C4×D8×C61C211yesmaximal, characteristic, complemented
12C22×#11758199C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×#14519773C2212yes—
15C2×#14519751—21no—
16C2×#14519751—21no—
17C2×D4×Dic61C2212yes—
18C22×C4⋊Dic61—21no—
19#16839957—24no—
20#16839955—28no—
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519706C2212yes—
23C2×#14519699C2212yes—
24C2×#14519699C2212yes—
25C2×#14519698C2212yes—
26C2×#14519697C2212yes—
27#16839671—22no—
28#16839663C2212yes—
29#16839661C2212yes—
30#16839637—22no—
31#16839632C2211yescharacteristic
32#16839622—21no—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839524C2212yes—
35#16839516C2212yes—
36C22×D8×C61C2211yescharacteristic
37C2×C4×D4×C61C2211yescharacteristic
38C2×C4×D4×C61C2211yescharacteristic
39C4×D8×C61—24no—
40C2×#107×C61C2211yescharacteristic
41C2×#102×C61C2211yescharacteristic
42C2×#102×C61C2211yescharacteristic
43C2×C4×C8×C61C2211yescharacteristic
44C22×D244—21no—
45C22×D244—21no—
46C22×D244C2312yes—
47C22×C22⋊D61—22no—
48C22×C22⋊D61—22no—
49C23×Dic61—22no—
50C2×#11758199—24no—
51C2×#11758199—24no—
52C2×#11758199—24no—
53C22×#92418C2312yes—
54#14519774—22no—
55#14519774—22no—
56#14519773—24no—
57#14519773—42no—
58#14519751—44no—
59#14519751—44no—
60D4×Dic61—24no—
61D4×Dic61—42no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—22no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—44no—
66C2×C4⋊Dic61D411yescharacteristic
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61C2312yes—
69C2×C22⋊Dic61—22no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×C4×Dic61C2312yes—
73C4⋊#92418—22no—
74#14519706—24no—
75#14519699—24no—
76#14519699—42no—
77#14519698—24no—
78#14519697—24no—
79#14519684—22no—
80#14519684D412yes—
81C22×D4×C61C2311yescharacteristic
82C22×D4×C61C2311yescharacteristic
83C23×C4×C61—21no—
84C23×C4×C61—21no—
85C2×D8×C61—22no—
86C2×D8×C61—22no—
87C2×D8×C61D414yescomplemented
88C22×C8×C61C2311yescharacteristic
89C22×C8×C61C2311yescharacteristic
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93C4×D4×C61D414yescomplemented
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C22⋊C4×C61—21no—
97C2×C22⋊C4×C61—21no—
98C2×C42×C61C2311yescharacteristic
99#107×C61—22no—
100#102×C61—22no—
101#102×C61—22no—
102C4×C8×C61—22no—
103C24×C61—21no—
104C24×C61—21no—
105C23×D61—22no—
106C2×D4×C61—21no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61C2×D412yes—
115C2×D4×C61C2×D412yes—
116C2×D4×C61C2×D412yes—
117C2×D4×C61#4112yes—
118C22×C4×C61—21no—
119C22×C4×C61—24no—
120C22×C4×C61—21no—
121C22×C4×C61—42no—
122C22×C4×C61—42no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61—42no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—44no—
130C2×C22⋊D61—44no—
131C2×C22⋊D61—44no—
132C2×C22⋊D61—44no—
133C2×C22⋊D61—82no—
134C2×C22⋊D61—82no—
135C22×Dic61—22no—
136C22×Dic61—44no—
137C22×Dic61—44no—
138C22×Dic61—82no—
139C22×Dic61C2×D412yes—
140C22×Dic61C2×D412yes—
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—22no—
144C2×D244—22no—
145C2×D244—22no—
146C2×D244—22no—
147C2×D244—42no—
148C2×D244—42no—
149C2×D244—42no—
150C2×D244—42no—
151C2×D244—81no—
152C2×D244—81no—
153D8×C61—28no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61C2×D412yes—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61C2×D412yes—
164C22⋊C4×C61—24no—
165C22⋊C4×C61—42no—
166C42×C61C2×D412yes—
167C42×C61#4112yes—
168C22⋊Dic61—44no—
169#11758199—48no—
170#11758198—44no—
171#11758198—44no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C4⋊Dic61—42no—
176C4⋊Dic61—44no—
177C4⋊Dic61—81no—
178C4⋊Dic61D814yescomplemented
179C4×Dic61—22no—
180C4×Dic61—22no—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61C22×D411yescharacteristic
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—22no—
210D4×C61—22no—
211D4×C61—41no—
212D4×C61—41no—
213D4×C61#12114yescomplemented
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—22no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—48no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—84no—
224C2×C4×C61#12112yes—
225C2×C4×C61#12112yes—
226C2×C4×C61C2×D812yes—
227C2×C4×C61AGL(1,ℤ/8)12yes—
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61H(5,𝔽2)11yescharacteristic
233C2×C4×C612−1+411yescharacteristic
234C22⋊D61—88no—
235C22⋊D61—88no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—88no—
245C2×Dic61C2×D814yes—
246D244—42no—
247D244—42no—
248D244—42no—
249D244—42no—
250D244—84no—
251D244—84no—
252C8×C61—22no—
253C8×C61—41no—
254#92418—42no—
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×C61—42no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61C2×#12111yescharacteristic
271C22×C61D4211yescharacteristic
272C22×C61#54811yescharacteristic
273C22×C61C22×D811yescharacteristic
274C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C61#57611yescharacteristic
276C22×C61#57711yescharacteristic
277C2×D61—82no—
278C2×D61—82no—
279C2×D61—82no—
280C2×D61—82no—
281C2×D61—82no—
282C2×D61—82no—
283C2×D61—82no—
284C4×C61—22no—
285C4×C61—42no—
286C4×C61—81no—
287C4×C61#45914yescomplemented
288C4×C61C2×#12111yescharacteristic
289C4×C61#53711yescharacteristic
290C4×C61#54711yescharacteristic
291C4×C61#55311yescharacteristic
292Dic61—24no—
293Dic61—42no—
294Dic61—82no—
295C2×#459—612no—
296C2×#459—611no—
297C2×C4×D8D6111yescharacteristic, complemented
298#2541—611no—
299D4×C4⋊C4—611no—
300#2067—612no—
301#2041—612no—
302#1957—611no—
303#1939—611no—
304#1879—612no—
305#1802—611no—
306C2×C61—44no—
307C2×C61—82no—
308C2×C61#241611yescharacteristic
309C2×C61C2×#45911yescharacteristic
310C2×C61#304511yescharacteristic
311C2×C61#327211yescharacteristic
312C2×C61D4×D811yescharacteristic
313C2×C61#335811yescharacteristic
314C2×C61#336611yescharacteristic
315D61—162no—
316C22×D8—612no—
317C22×D8C2×D6111yescharacteristic
318C2×#127—612no—
319C2×#127—611no—
320C2×#121—1221no—
321C2×#121—1221no—
322C2×C4×D4—612no—
323C2×C4×D4C2×D6111yescharacteristic
324C2×C4×D4C2×D6111yescharacteristic
325C22×C4⋊C4—1221no—
326#459—1228no—
327#459—1224no—
328C4×D8—24no—
329C2×#107C2×D6111yescharacteristic
330C2×C4⋊C8—612no—
331C2×C4⋊C8—611no—
332C2×#102—612no—
333C2×#102—612no—
334C2×#102—612no—
335C2×#102—612no—
336C2×#102C2×D6111yescharacteristic
337C2×#102C2×D6111yescharacteristic
338C2×C4×C8C2×D6111yescharacteristic
339#392—1222no—
340#390—612no—
341#389—612no—
342#385—1222no—
343#384—611no—
344#380—1221no—
345C4×C4⋊C4—611no—
346#340—612no—
347#336—612no—
348C61#2472611yescharacteristic, complemented, Sylow
349C22×D4—612no—
350C22×D4—1222no—
351C22×D4—1222no—
352C22×D4—1221no—
353C22×D4—1221no—
354C22×D4C22×D6111yescharacteristic
355C22×D4C22×D6111yescharacteristic
356C23×C4—21no—
357C23×C4—21no—
358C23×C4—1222no—
359C2×D8—22no—
360C2×D8—22no—
361C2×D8—1224no—
362C2×D8—1224no—
363C2×D8—1224no—
364C2×D8D24414yescomplemented
365C22×C8—612no—
366C22×C8C22×D6111yescharacteristic
367C22×C8C22×D6111yescharacteristic
368#127—1224no—
369#127—1222no—
370#127—1222no—
371#127—2442no—
372#121—2444no—
373#121—2444no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—1224no—
378C4×D4—2442no—
379C4×D4C22⋊D6114yescomplemented
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—612no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—2444no—
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—1222no—
392C2×C22⋊C4—1222no—
393C2×C22⋊C4—1222no—
394C2×C42—612no—
395C2×C42C22×D6111yescharacteristic
396#107—22no—
397C4⋊C8—1224no—
398C4⋊C8—1222no—
399#102—22no—
400#102—22no—
401#102—1224no—
402#102—1224no—
403#102—1224no—
404#102—2442no—
405C4×C8—22no—
406#95—612no—
407#95—1222no—
408C24—21no—
409C24—21no—
410C24—1222no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—1222no—
420C2×D4—1222no—
421C2×D4—1222no—
422C2×D4—1222no—
423C2×D4—1222no—
424C2×D4—1222no—
425C2×D4—4881no—
426C2×D4—4881no—
427C2×D4—4882no—
428C2×D4—4882no—
429C2×D4—2444no—
430C2×D4—2444no—
431C2×D4—2444no—
432C2×D4—2444no—
433C2×D4—2442no—
434C2×D4—2442no—
435C2×D4—2442no—
436C2×D4—2442no—
437C2×D4C2×D24412yes—
438C2×D4C2×D24412yes—
439C2×D4#1175822212yes—
440C2×D4C2×C22⋊D6112yes—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—612no—
450C22×C4—612no—
451C22×C4—1222no—
452C22×C4—4882no—
453C22×C4—2444no—
454C22×C4—2444no—
455C22×C4D4×D6111yescharacteristic
456C22×C4D4×D6111yescharacteristic
457C22×C4C23×D6111yescharacteristic
458D8—28no—
459D8—2448no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—1222no—
466C2×C8—1222no—
467C2×C8—1222no—
468C2×C8C2×D24412yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—614no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—4881no—
477C4⋊C4—2444no—
478C4⋊C4—2442no—
479C4⋊C4C2×C22⋊D6112yes—
480C22⋊C4—24no—
481C22⋊C4—42no—
482C22⋊C4—2444no—
483C22⋊C4—2444no—
484C22⋊C4—2444no—
485C42—1222no—
486C42—1222no—
487C42#1175822412yes—
488C42C2×C22⋊D6112yes—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—2442no—
504C23—2442no—
505C23—2442no—
506C23—2442no—
507C23—2442no—
508C23—2442no—
509C23—2442no—
510C23C2×D4×D6111yescharacteristic
511D4—22no—
512D4—22no—
513D4—22no—
514D4—22no—
515D4—22no—
516D4—41no—
517D4—41no—
518D4—4884no—
519D4—4888no—
520D4—4888no—
521D4—4884no—
522D4—2442no—
523D4—2442no—
524D4—2442no—
525D4—2442no—
526D4#1451975114yescomplemented
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—48no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—84no—
537C2×C4—614no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—4888no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4#1451974912yes—
548C2×C4#1451974912yes—
549C2×C4D8×D6112yes—
550C2×C4#1451980112yes—
551C2×C4C22×C22⋊D6111yescharacteristic
552C2×C4C22×D24411yescharacteristic
553C2×C4C2×#1175822411yescharacteristic
554C2×C4C2×#1175822211yescharacteristic
555C2×C4#1451982311yescharacteristic
556C2×C4#1451982411yescharacteristic
557C8—22no—
558C8—41no—
559C8—2442no—
560C22—24no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—4882no—
575C22—4882no—
576C22—4882no—
577C22—4882no—
578C22—4882no—
579C22—4882no—
580C22—4882no—
581C22C2×#1451974911yescharacteristic
582C22D4×D24411yescharacteristic
583C22#1684042111yescharacteristic
584C22C2×D8×D6111yescharacteristic
585C22C2×#1451980111yescharacteristic
586C22#1684056411yescharacteristic
587C22#1684056911yescharacteristic
588C4—22no—
589C4—42no—
590C4—81no—
591C4—1224no—
592C4—4882no—
593C4—2442no—
594C4#1683995514yescomplemented
595C4C2×#1451975111yescharacteristic
596C4#1684034011yescharacteristic
597C4#1684041611yescharacteristic
598C4#1684045811yescharacteristic
599C2—44no—
600C2—82no—
601C2—9762no—
602C2#2144925911yescharacteristic
603C2C2×#1683995511yescharacteristic
604C2#2145535611yescharacteristic
605C2#2145742011yescharacteristic
606C2D8×D24411yescharacteristic
607C2#2145845211yescharacteristic
608C2#2145852411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C4×D8×C61C2
socleC23×C61C22×D4
hypercenterC2×C4×D8D61
hypocenterC61#24726
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16839957C2semidirect
2. C2×#16839955C2semidirect
3. #21450765C2semidirect
4. D4×C4⋊Dic61C2semidirect
5. #21447202C2semidirect
6. #21447052C2semidirect
7. #21446625C2semidirect
8. #21446537C2semidirect
9. #21446165C2semidirect
10. #21445819C2semidirect
11. C2×C4×D8×C61C2semidirect
87. C2×D8×C61D4semidirect
93. C4×D4×C61D4semidirect
178. C4⋊Dic61D8semidirect
213. D4×C61#121semidirect
287. C4×C61#459semidirect
297. C2×C4×D8D61semidirect
348. C61#24726semidirect
364. C2×D8D244semidirect
379. C4×D4C22⋊D61semidirect
526. D4#14519751semidirect
594. C4#16839955semidirect

Maximal subgroups

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

subgroupindex
C2×#168399572
C2×#168399552
#214507652
D4×C4⋊Dic612
#214472022
#214470522
#214466252
#214465372
#214461652
#214458192
C2×C4×D8×C612
#2472661