Finite Group Databasebeta

← #29715659

Subgroups — #29715659

Order 15616.

1546 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459583C211yesmaximal, characteristic
2C2×#16839951C212yesmaximal, complemented
3#21454150C212yesmaximal, complemented
4#21447070C212yesmaximal, complemented
5#21446950C212yesmaximal, complemented
6#21446655C212yesmaximal, complemented
7#21445807C211yesmaximal, characteristic, complemented
8#21445766C212yesmaximal, complemented
9C4×#123×C61C211yesmaximal, characteristic, complemented
10C2×#14519822C2212yes—
11#16840416—22no—
12#16840407—24no—
13D4×D244—22no—
14#16840343—21no—
15#16840324—21no—
16#16840324C2212yes—
17#16840312—21no—
18C2×#14519760C2214yescomplemented
19C2×#14519751—21no—
20C2×#14519751C2212yes—
21#16839951—28no—
22C2×#14519692C2212yes—
23C2×#14519691C2214yescomplemented
24C2×#14519691C2214yescomplemented
25#16839664—22no—
26#16839660—22no—
27#16839653—24no—
28#16839641C2214yescomplemented
29#16839622C2212yescomplemented
30#16839529C2212yescomplemented
31C2×#123×C61C2211yescharacteristic
32C2×C4×D4×C61C2211yescharacteristic
33C22×C42×C61C2211yescharacteristic, complemented
34C61×#388—21no—
35C61×#387—21no—
36C61×#386C2211yescharacteristic
37C61×#385C2211yescharacteristic, complemented
38C61×#381C2211yescharacteristic, complemented
39C4×C4⋊C4×C61—21no—
40C4×C22⋊C4×C61—21no—
41C4×C22⋊C4×C61C2211yescharacteristic, complemented
42#14519822—24no—
43#14519822—24no—
44#14519822—44no—
45C2×#11758222—22no—
46C2×#11758224—24no—
47C22×D244—21no—
48C22×D244D412yes—
49C22×C22⋊D61—22no—
50C22×C22⋊D61C2314yes—
51C2×D4×D61—24no—
52#14519777—21no—
53#14519774—21no—
54#14519765—22no—
55#14519761—24no—
56#14519761—24no—
57#14519760—28no—
58#14519760—44no—
59#14519757—24no—
60#14519752—24no—
61#14519751—22no—
62#14519751—22no—
63#14519751—22no—
64#14519751—22no—
65#14519751—22no—
66#14519751—22no—
67#14519751—22no—
68#14519751—22no—
69#14519747—24no—
70#14519747—24no—
71C4×D244—22no—
72C4×D244—22no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—22no—
75C2×C4⋊Dic61D412yes—
76C2×#11758196—24no—
77C2×C22⋊Dic61C2314yes—
78C2×#11758198—22no—
79C2×#11758198—24no—
80C2×#11758198—24no—
81C2×#11758198D414yes—
82#14519692—24no—
83#14519691—28no—
84#14519691—44no—
85#14519684—22no—
86#14519684—24no—
87C22×D4×C61C2311yescharacteristic
88C23×C4×C61D411yescharacteristic
89C23×C4×C61C2311yescharacteristic
90C23×C4×C61C2311yescharacteristic
91C23×C4×C61C2311yescharacteristic
92#123×C61—22no—
93#123×C61D414yescomplemented
94C4×D4×C61—22no—
95C4×D4×C61—22no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C22⋊C4×C61—21no—
100C2×C22⋊C4×C61—21no—
101C2×C22⋊C4×C61D411yescharacteristic
102C2×C22⋊C4×C61C2311yescharacteristic
103C2×C22⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61C2311yescharacteristic
105C2×C42×C61—21no—
106C2×C42×C61—21no—
107C2×C42×C61—21no—
108C2×C42×C61—22no—
109C2×C42×C61—22no—
110C2×C42×C61D411yescharacteristic
111#95×C61—21no—
112#95×C61—21no—
113#95×C61—21no—
114#95×C61—21no—
115#95×C61—21no—
116C24×C61C2×D411yescharacteristic
117C24×C61C2411yescharacteristic
118C23×D61—22no—
119C23×D61C2×D414yes—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61C2×D412yes—
123C2×D4×C61#4112yes—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—21no—
130C22×C4×C61—22no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—42no—
135C22×C4×C61—42no—
136C22×C4×C61—42no—
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D412yescomplemented
145C2×C22⋊D61—24no—
146C2×C22⋊D61—24no—
147C2×C22⋊D61—24no—
148C2×C22⋊D61—24no—
149C2×C22⋊D61—24no—
150C2×C22⋊D61—24no—
151C2×C22⋊D61—24no—
152C2×C22⋊D61—44no—
153C2×C22⋊D61—44no—
154C22×Dic61—22no—
155C22×Dic61—24no—
156C22×Dic61C2×D414yes—
157#11758224—48no—
158#11758224—48no—
159D4×D61—48no—
160D4×D61—48no—
161#11758222—44no—
162#11758222—44no—
163C2×D244—22no—
164C2×D244—22no—
165C2×D244—22no—
166C2×D244—22no—
167C2×D244—22no—
168C2×D244—44no—
169C2×D244—82no—
170C2×C4×D61—24no—
171C2×Dic122—22no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—42no—
175C22⋊C4×C61—24no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—42no—
181C22⋊C4×C61C2×D412yescomplemented
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—42no—
185C42×C61—42no—
186C42×C61—41no—
187C42×C61—41no—
188C22⋊Dic61—24no—
189C22⋊Dic61—24no—
190#11758198—24no—
191#11758198—24no—
192#11758198—24no—
193#11758198—24no—
194#11758198—48no—
195#11758198—48no—
196C4⋊Dic61—22no—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61—22no—
200C4⋊Dic61—44no—
201#11758196—48no—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61#12112yes—
209C23×C61C22×D411yescharacteristic
210C23×C61C22×D412yes—
211C23×C61C22×D411yescharacteristic
212C23×C61C2×#4111yescharacteristic
213C23×C61H(5,𝔽2)12yes—
214C22×D61—24no—
215C22×D61—24no—
216C22×D61—24no—
217C22×D61—44no—
218C22×D61—44no—
219C22×D61—44no—
220C22×D61—82no—
221C22×D61—82no—
222D4×C61—24no—
223D4×C61—24no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—24no—
227C2×C4×C61—22no—
228C2×C4×C61—24no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—41no—
236C2×C4×C61—42no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—42no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—81no—
252C2×C4×C61#12012yes—
253C2×C4×C61#12011yescharacteristic
254C2×C4×C61#12011yescharacteristic
255C2×C4×C61#12112yes—
256C22⋊D61—48no—
257C22⋊D61—44no—
258C22⋊D61—44no—
259C22⋊D61—44no—
260C22⋊D61—44no—
261C2×Dic61—24no—
262C2×Dic61—24no—
263C2×Dic61—24no—
264C2×Dic61—44no—
265C2×Dic61—44no—
266C2×Dic61—44no—
267C2×Dic61—44no—
268D244—44no—
269D244—44no—
270D244—82no—
271D244—82no—
272D244—82no—
273C4×D61—48no—
274Dic122—44no—
275#24261—611nomaximal, Sylow
276C22×C61—22no—
277C22×C61—21no—
278C22×C61—22no—
279C22×C61—21no—
280C22×C61—22no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—81no—
288C22×C61C2×#12011yescharacteristic
289C22×C61C2×#12111yescharacteristic
290C22×C61#53511yescharacteristic
291C22×C61D4212yes—
292C22×C61#54612yes—
293C2×D61—44no—
294C2×D61—44no—
295C2×D61—44no—
296C2×D61—84no—
297C2×D61—84no—
298C2×D61—164no—
299C4×C61—22no—
300C4×C61—22no—
301C4×C61—21no—
302C4×C61—42no—
303C4×C61—42no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—42no—
307C4×C61—81no—
308C4×C61#45812yescomplemented
309Dic61—44no—
310Dic61—84no—
311#3496—611no—
312C2×#458—612no—
313#2854—612no—
314C4×#123D6111yescharacteristic, complemented
315#2044—612no—
316#2022—612no—
317#1964—612no—
318#1799—611no—
319#1790—612no—
320C2×C61—22no—
321C2×C61—21no—
322C2×C61—21no—
323C2×C61—22no—
324C2×C61—42no—
325C2×C61#240811yescharacteristic
326C2×C61C2×#45811yescharacteristic
327C2×C61#302511yescharacteristic
328D61—84no—
329C2×H(5,𝔽2)—612no—
330#547—1222no—
331#546—1224no—
332D42—1222no—
333#537—1221no—
334#534—612no—
335#534—1221no—
336#532—1221no—
337C2×#123—614no—
338C2×#123C2×D6111yescharacteristic
339C2×#121—612no—
340C2×#121—1221no—
341C2×C4×D4C2×D6111yescharacteristic
342C22×C42C2×D6111yescharacteristic, complemented
343#458—1228no—
344C2×#99—612no—
345C2×#99—614no—
346C2×#99—614no—
347#390—1222no—
348#389—1222no—
349#388—21no—
350#388—1224no—
351#387—21no—
352#386—614no—
353#386C2×D6111yescharacteristic
354#385C2×D6111yescharacteristic, complemented
355#381C2×D6111yescharacteristic, complemented
356#380—612no—
357C4×C4⋊C4—21no—
358C4×C22⋊C4—21no—
359C4×C22⋊C4C2×D6111yescharacteristic, complemented
360#342—612no—
361C61#2426111yescharacteristic, complemented, Sylow
362H(5,𝔽2)—1224no—
363H(5,𝔽2)—1224no—
364H(5,𝔽2)—2444no—
365C2×#41—1224no—
366C2×#41—1222no—
367C22×D4—614no—
368C22×D4—612no—
369C22×D4—1221no—
370C22×D4—1224no—
371C22×D4—1222no—
372C22×D4C22×D6111yescharacteristic
373C23×C4C22⋊D6111yescharacteristic
374C23×C4C22×D6111yescharacteristic
375C23×C4C22×D6111yescharacteristic
376C23×C4C22×D6111yescharacteristic
377C4⋊Q8—1221no—
378#127—1221no—
379#124—1222no—
380#123—22no—
381#123—1224no—
382#123—1224no—
383#123—1228no—
384#123—2444no—
385#123D24414yescomplemented
386C22⋊Q8—1224no—
387#121—1224no—
388#121—1222no—
389#121—1222no—
390#121—1222no—
391#121—1222no—
392#121—1222no—
393#121—1222no—
394#121—1222no—
395#121—1222no—
396#120—1224no—
397#120—1224no—
398C4×D4—22no—
399C4×D4—22no—
400C4×D4—1222no—
401C4×D4—1222no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—612no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1224no—
407C2×C4⋊C4—1222no—
408C2×C4⋊C4C22⋊D6111yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—614no—
412C2×C22⋊C4—614no—
413C2×C22⋊C4—1222no—
414C2×C22⋊C4—1224no—
415C2×C22⋊C4—1224no—
416C2×C22⋊C4C22⋊D6111yescharacteristic
417C2×C22⋊C4C22×D6111yescharacteristic
418C2×C22⋊C4C22×D6111yescharacteristic
419C2×C22⋊C4C22×D6111yescharacteristic
420C2×C42—22no—
421C2×C42—22no—
422C2×C42—21no—
423C2×C42—21no—
424C2×C42—21no—
425C2×C42C22⋊D6111yescharacteristic
426#99—1224no—
427#99—1228no—
428#99—2444no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—1224no—
435#95—1222no—
436C24—614no—
437C24—1222no—
438C24C2×C22⋊D6111yescharacteristic
439C24C23×D6111yescharacteristic
440#41—2444no—
441#41—2448no—
442#41—2448no—
443#41—2444no—
444C2×Q8—1222no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—1222no—
448C2×D4—1222no—
449C2×D4—1222no—
450C2×D4—1224no—
451C2×D4—1224no—
452C2×D4—1224no—
453C2×D4—1224no—
454C2×D4—1224no—
455C2×D4—1224no—
456C2×D4—1224no—
457C2×D4—1222no—
458C2×D4—1222no—
459C2×D4—4882no—
460C2×D4—2444no—
461C2×D4—2448no—
462C2×D4—2448no—
463C2×D4—2444no—
464C2×D4—2444no—
465C2×D4C2×D24412yes—
466C2×D4#1175822212yes—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—614no—
481C22×C4—1224no—
482C22×C4—1222no—
483C22×C4—1224no—
484C22×C4C2×D24412yescomplemented
485C22×C4D4×D6112yes—
486C22×C4D4×D6112yes—
487C22×C4C2×C22⋊D6111yescharacteristic
488C22×C4C2×C22⋊D6111yescharacteristic
489C22×C4C2×C22⋊D6111yescharacteristic
490C22×C4C2×C22⋊D6111yescharacteristic
491C22×C4C2×C22⋊D6111yescharacteristic
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—42no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—2448no—
500C4⋊C4—2444no—
501C22⋊C4—22no—
502C22⋊C4—21no—
503C22⋊C4—24no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—42no—
507C22⋊C4—1224no—
508C22⋊C4—1224no—
509C22⋊C4—1224no—
510C22⋊C4—1224no—
511C22⋊C4—1224no—
512C22⋊C4—1224no—
513C22⋊C4—2448no—
514C22⋊C4—2448no—
515C22⋊C4C2×D24412yescomplemented
516C42—21no—
517C42—21no—
518C42—42no—
519C42—41no—
520C42—42no—
521C42—41no—
522C23—21no—
523C23—21no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—1224no—
529C23—1224no—
530C23—1224no—
531C23—4882no—
532C23—4882no—
533C23—2444no—
534C23—2444no—
535C23—2444no—
536C23#1451975112yes—
537C23C2×D4×D6112yes—
538C23C22×C22⋊D6111yescharacteristic
539C23C22×D24411yescharacteristic
540C23C2×#1175822211yescharacteristic
541C23#1451982312yes—
542Q8—2444no—
543D4—24no—
544D4—24no—
545D4—4882no—
546D4—4882no—
547D4—4882no—
548D4—2444no—
549D4—2448no—
550D4—2444no—
551D4—2444no—
552D4—2444no—
553D4—2444no—
554D4—2444no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—81no—
583C2×C4—1224no—
584C2×C4—1224no—
585C2×C4—1224no—
586C2×C4—2448no—
587C2×C4—2444no—
588C2×C4—2444no—
589C2×C4—2444no—
590C2×C4—2444no—
591C2×C4#1451974511yescharacteristic
592C2×C4#1451974512yes—
593C2×C4#1451974511yescharacteristic
594C2×C4#1451975112yes—
595C22—21no—
596C22—22no—
597C22—21no—
598C22—22no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—42no—
603C22—42no—
604C22—41no—
605C22—41no—
606C22—81no—
607C22—9764no—
608C22—4884no—
609C22—4884no—
610C22—2444no—
611C22—2444no—
612C22—2444no—
613C22C2×#1451974511yescharacteristic
614C22C2×#1451975111yescharacteristic
615C22#1684032611yescharacteristic
616C22D4×D24412yes—
617C22#1684040812yes—
618C4—21no—
619C4—22no—
620C4—22no—
621C4—42no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—42no—
626C4—81no—
627C4—4884no—
628C4—2444no—
629C4#1683995112yescomplemented
630C2—21no—
631C2—21no—
632C2—22no—
633C2—22no—
634C2—42no—
635C2—4884no—
636C2#2144917511yescharacteristic
637C2C2×#1683995111yescharacteristic
638C2#2145521011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC4×#123×C61C2
socleC22×C61C2×#120
prime radical#21459583C2
hypercenterC4×#123D61
hypocenterC61#24261
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×#123

Complements

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

NH ≅ G/Ntype
2. C2×#16839951C2semidirect
3. #21454150C2semidirect
4. #21447070C2semidirect
5. #21446950C2semidirect
6. #21446655C2semidirect
7. #21445807C2semidirect
8. #21445766C2semidirect
9. C4×#123×C61C2semidirect
18. C2×#14519760C22semidirect
23. C2×#14519691C22semidirect
24. C2×#14519691C22semidirect
28. #16839641C22semidirect
29. #16839622C22semidirect
30. #16839529C22semidirect
33. C22×C42×C61C22semidirect
37. C61×#385C22semidirect
38. C61×#381C22semidirect
41. C4×C22⋊C4×C61C22semidirect
93. #123×C61D4semidirect
144. C22×C4×C61C2×D4semidirect
181. C22⋊C4×C61C2×D4semidirect
308. C4×C61#458semidirect
314. C4×#123D61semidirect
342. C22×C42C2×D61semidirect
354. #385C2×D61semidirect
355. #381C2×D61semidirect
359. C4×C22⋊C4C2×D61semidirect
361. C61#24261semidirect
385. #123D244semidirect
484. C22×C4C2×D244semidirect
515. C22⋊C4C2×D244semidirect
629. C4#16839951semidirect

Maximal subgroups

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

subgroupindex
#214595832
C2×#168399512
#214541502
#214470702
#214469502
#214466552
#214458072
#214457662
C4×#123×C612
#2426161