Finite Group Databasebeta

← #29714327

Subgroups — #29714327

Order 15616.

1198 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic, complemented
2C2×#16840052C211yesmaximal, characteristic, complemented
3C4×#14519748C211yesmaximal, characteristic
4#21447013C212yesmaximal, complemented
5#21446819C212yesmaximal
6#21446766C211yesmaximal, characteristic, complemented
7#21446743C211yesmaximal, characteristic, complemented
8#21446176C212yesmaximal, complemented
9#21446085C212yesmaximal, complemented
10#21445796C211yesmaximal, characteristic, complemented
11C2×C61×#461C211yesmaximal, characteristic, complemented
12#16840317—24no—
13#16840317—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519768—21no—
16C2×#14519757—21no—
17C2×#14519753C2212yes—
18C2×#14519748C2211yescharacteristic
19C2×Q8×Dic61C2212yes—
20C2×#14519742—21no—
21C2×C4×C22⋊D61—21no—
22C2×#14519735—21no—
23C2×C42×D61C2211yescharacteristic
24#16840052—24no—
25C2×#14519708C2212yes—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519701C2212yes—
28C2×#14519701C2212yes—
29C2×#14519698C2212yes—
30C2×#14519689C2212yes—
31#16839665C2211yescharacteristic
32#16839658C2211yescharacteristic
33#16839650—21no—
34#16839638—21no—
35#16839619C2211yescharacteristic
36C4×C4⋊Dic61C2211yescharacteristic
37C4×C22⋊Dic61—21no—
38#16839525C2212yes—
39#16839525C2212yes—
40#16839516C2212yes—
41C22×SD16×C61C2212yes—
42C2×C4⋊Q8×C61C2211yescharacteristic
43C2×C4×D4×C61C2211yescharacteristic
44C61×#461—24no—
45C2×C4⋊C8×C61C2211yescharacteristic
46C2×#103×C61C2212yes—
47C22×Dic122—21no—
48C22×C22⋊D61—21no—
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D411yescharacteristic
51C22×C4×D61C2311yescharacteristic
52C22×#92418C2312yes—
53#14519777—22no—
54#14519777D414yescomplemented
55#14519768—28no—
56#14519757—28no—
57#14519753—24no—
58#14519753—24no—
59#14519748—22no—
60#14519748—22no—
61Q8×Dic61—24no—
62Q8×Dic61—24no—
63#14519742—28no—
64C4×C22⋊D61—44no—
65#14519735—28no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—22no—
72C2×#11758196—21no—
73C2×C22⋊Dic61—21no—
74C2×C22⋊Dic61—21no—
75C2×C22⋊Dic61—21no—
76C2×#11758198—22no—
77C2×#11758198—21no—
78C2×C4×Dic61—22no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61C2311yescharacteristic
83C42×D61—22no—
84C42×D61—22no—
85#14519708—24no—
86C4⋊#92418—22no—
87#14519701—24no—
88#14519701—42no—
89#14519698—24no—
90#14519689—42no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94C22×Q8×C61C2312yes—
95C22×D4×C61C2311yescharacteristic
96C23×C4×C61—21no—
97C2×SD16×C61—24no—
98C2×SD16×C61—24no—
99C2×SD16×C61—42no—
100C22×C8×C61C2312yes—
101C4⋊Q8×C61—22no—
102C4⋊Q8×C61D414yescomplemented
103C4×D4×C61—22no—
104C4×D4×C61—22no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C22⋊C4×C61—21no—
109C2×C42×C61C2311yescharacteristic
110C4⋊C8×C61—22no—
111#103×C61—24no—
112C24×C61—21no—
113C23×D61C2×D411yescharacteristic
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—22no—
117C2×Q8×C61—22no—
118C2×Q8×C61—22no—
119C2×Q8×C61C2×D414yes—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61#4112yes—
126C22×C4×C61—22no—
127C22×C4×C61—21no—
128C22×C4×C61—42no—
129C22×C4×C61—42no—
130C22×C4×C61—42no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C2×C22⋊D61—24no—
135C2×C22⋊D61—42no—
136C2×C22⋊D61—42no—
137C22×Dic61—21no—
138C22×Dic61—22no—
139C22×Dic61—21no—
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—21no—
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61—21no—
151C2×C4×D61—21no—
152C2×C4×D61—41no—
153C2×C4×D61—41no—
154C2×C4×D61SD1614yescomplemented
155C2×Dic122—24no—
156C2×Dic122—42no—
157C2×Dic122—42no—
158SD16×C61—48no—
159C2×C8×C61—22no—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—24no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C22⋊C4×C61—42no—
167C42×C61C2×D412yes—
168C42×C61C2×D412yes—
169C22⋊Dic61—24no—
170C22⋊Dic61—42no—
171C22⋊Dic61—42no—
172#11758198—28no—
173#11758198—42no—
174C4⋊Dic61—28no—
175C4⋊Dic61—24no—
176C4⋊Dic61—22no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61C2×D414yes—
181C4⋊Dic61#4112yes—
182#11758196—28no—
183#11758196—42no—
184C4×Dic61—28no—
185C4×Dic61—22no—
186C4×Dic61—22no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C4×Dic61—42no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—22no—
193C23×C61—22no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61C22×D411yescharacteristic
201C22×D61—21no—
202C22×D61—21no—
203C22×D61—21no—
204C22×D61—21no—
205C22×D61—21no—
206C22×D61—21no—
207C22×D61C2×SD1612yes—
208Q8×C61—24no—
209Q8×C61—24no—
210Q8×C61—42no—
211Q8×C61—42no—
212D4×C61—22no—
213D4×C61—22no—
214D4×C61—41no—
215D4×C61—41no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—24no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—84no—
225C2×C4×C61#12012yes—
226C2×C4×C61#12012yes—
227C2×C4×C61C2×SD1612yes—
228C2×C4×C61#13712yes—
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61H(5,𝔽2)11yescharacteristic
234C2×C4×C612−1+411yescharacteristic
235C22⋊D61—48no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—24no—
241C2×Dic61—22no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—41no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61C2×SD1612yes—
254C4×D61—28no—
255C4×D61—44no—
256C4×D61—41no—
257C4×D61—41no—
258C4×D61—41no—
259C4×D61—41no—
260Dic122—48no—
261C8×C61—42no—
262#92418—42no—
263#24151—611nomaximal, Sylow
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61C2×#12011yescharacteristic
272C22×C61#52511yescharacteristic
273C22×C61#53711yescharacteristic
274C22×C61C22×SD1611yescharacteristic
275C22×C61C2×#13711yescharacteristic
276C22×C61#57511yescharacteristic
277C22×C61#57511yescharacteristic
278C2×D61—22no—
279C2×D61—22no—
280C2×D61—22no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—41no—
285C4×C61—22no—
286C4×C61—44no—
287C4×C61—81no—
288C4×C61#44814yes—
289C4×C61#54611yescharacteristic
290C4×C61#54711yescharacteristic
291C4×C61#54811yescharacteristic
292C4×C61#55011yescharacteristic
293Dic61—22no—
294Dic61—42no—
295Dic61—41no—
296Dic61—44no—
297Dic61—41no—
298Dic61—81no—
299C2×#533—611no—
300C2×#476—611no—
301C2×#461D6111yescharacteristic, complemented
302C4×#121—611no—
303#2035—612no—
304#2000—612no—
305#1987—611no—
306#1981—611no—
307#1880—612no—
308#1865—612no—
309#1797—611no—
310C2×C61—82no—
311C2×C61#240711yescharacteristic
312C2×C61C2×#44811yescharacteristic
313C2×C61#300511yescharacteristic
314C2×C61#310111yescharacteristic
315C2×C61#311611yescharacteristic
316C2×C61#324611yescharacteristic
317C2×C61#325211yescharacteristic
318D61—42no—
319C22×SD16C2×D6112yes—
320#533—1224no—
321#533—1224no—
322C2×C4⋊Q8—611no—
323C2×C4⋊Q8C2×D6111yescharacteristic
324C2×#125—1221no—
325C2×C22⋊Q8—612no—
326C2×C22⋊Q8—1221no—
327C2×#121—611no—
328C2×C4×Q8—612no—
329C2×C4×Q8—1221no—
330C2×C4×D4—1221no—
331C2×C4×D4C2×D6111yescharacteristic
332C2×#117—1221no—
333C22×C42—611no—
334#476—1224no—
335#461—24no—
336C2×#106—612no—
337C2×C4⋊C8—611no—
338C2×C4⋊C8C2×D6111yescharacteristic
339C2×#103—612no—
340C2×#103—612no—
341C2×#103C2×D6112yes—
342C2×#102—612no—
343C2×C22⋊C8—612no—
344#390—611no—
345#389—611no—
346#388—1221no—
347#386—1221no—
348#380—611no—
349C4×C4⋊C4—611no—
350C4×C22⋊C4—1221no—
351#340—612no—
352#340—612no—
353#336—612no—
354C61#2415111yescharacteristic, complemented, Sylow
355C22×Q8—1221no—
356C22×Q8C22×D6112yes—
357C22×D4—1221no—
358C22×D4C22×D6111yescharacteristic
359C23×C4—21no—
360C23×C4—611no—
361C23×C4—611no—
362C23×C4—611no—
363C2×SD16—24no—
364C2×SD16—24no—
365C2×SD16—42no—
366C22×C8—612no—
367C22×C8C22×D6112yes—
368C4⋊Q8—22no—
369C4⋊Q8—614no—
370C4⋊Q8—1222no—
371C4⋊Q8C22⋊D6114yescomplemented
372#125—1228no—
373C22⋊Q8—1228no—
374C22⋊Q8—1224no—
375C22⋊Q8—1224no—
376#121—1222no—
377#121—1222no—
378C4×Q8—1228no—
379C4×Q8—1224no—
380C4×Q8—1224no—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—2444no—
384#117—1228no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—612no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—1222no—
391C2×C4⋊C4—1222no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1222no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C42—611no—
402C2×C42—611no—
403C2×C42—611no—
404C2×C42—1222no—
405C2×C42—1222no—
406C2×C42—1222no—
407C2×C42—1221no—
408C2×C42C22×D6111yescharacteristic
409#106—1224no—
410C4⋊C8—22no—
411C4⋊C8—1222no—
412#103—24no—
413#103—1224no—
414#103—2442no—
415#102—1224no—
416C22⋊C8—2442no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420C24—21no—
421C24—611no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—1224no—
428C2×Q8—2442no—
429C2×Q8—2442no—
430C2×Q8C2×C22⋊D6114yes—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—1224no—
437C2×D4—2442no—
438C2×D4—2442no—
439C2×D4#1175822412yes—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—611no—
446C22×C4—614no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1221no—
452C22×C4—1222no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1222no—
460C22×C4—1221no—
461C22×C4—2441no—
462C22×C4—2441no—
463C22×C4D4×D6111yescharacteristic
464C22×C4D4×D6111yescharacteristic
465C22×C4C23×D6111yescharacteristic
466SD16—48no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—1222no—
471C2×C8—1222no—
472C2×C8—1222no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—614no—
478C4⋊C4—612no—
479C4⋊C4—1228no—
480C4⋊C4—1228no—
481C4⋊C4—1222no—
482C4⋊C4—1221no—
483C4⋊C4—1224no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—2442no—
487C22⋊C4—42no—
488C22⋊C4—1228no—
489C22⋊C4—1224no—
490C22⋊C4—2442no—
491C22⋊C4—2442no—
492C22⋊C4—2442no—
493C42—1228no—
494C42—1221no—
495C42—1221no—
496C42—1222no—
497C42—1222no—
498C42—2442no—
499C42D4×D6112yes—
500C42C2×C22⋊D6112yes—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—612no—
509C23—1221no—
510C23—1221no—
511C23—1221no—
512C23—1221no—
513C23—1221no—
514C23—1221no—
515C23C2×D4×D6111yescharacteristic
516Q8—24no—
517Q8—24no—
518Q8—42no—
519Q8—42no—
520Q8—2448no—
521D4—22no—
522D4—22no—
523D4—41no—
524D4—41no—
525D4—2448no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—84no—
535C2×C4—612no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1228no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1224no—
542C2×C4—1221no—
543C2×C4—1222no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—2444no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4#1451974512yes—
560C2×C4#1451974512yes—
561C2×C4SD16×D6112yes—
562C2×C4#1451980612yes—
563C2×C4C2×D4×D6111yescharacteristic
564C2×C4C22×C22⋊D6111yescharacteristic
565C2×C4#41×D6111yescharacteristic
566C2×C4C2×#1175822411yescharacteristic
567C2×C4#1451982211yescharacteristic
568C2×C4#1451982411yescharacteristic
569C8—42no—
570C8—2442no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—1222no—
579C22—1222no—
580C22—1222no—
581C22—2441no—
582C22—2441no—
583C22—2441no—
584C22—2441no—
585C22C2×#1451974511yescharacteristic
586C22#1684027211yescharacteristic
587C22#1684034111yescharacteristic
588C22C2×SD16×D6111yescharacteristic
589C22C2×#1451980611yescharacteristic
590C22#1684056111yescharacteristic
591C22#1684056111yescharacteristic
592C4—22no—
593C4—44no—
594C4—81no—
595C4—1222no—
596C4—4881no—
597C4—2441no—
598C4—2442no—
599C4—2441no—
600C4—2444no—
601C4#1683989914yes—
602C4#1684040611yescharacteristic
603C4#1684041111yescharacteristic
604C4#1684041811yescharacteristic
605C4#1684043011yescharacteristic
606C2—82no—
607C2—2442no—
608C2#2144916011yescharacteristic
609C2C2×#1683989911yescharacteristic
610C2#2145506011yescharacteristic
611C2#2145582611yescharacteristic
612C2#2145597811yescharacteristic
613C2#2145713211yescharacteristic
614C2#2145718811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#461C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
hypercenterC2×#461D61
hypocenterC61#24151
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. C2×#16840317C2semidirect
2. C2×#16840052C2semidirect
4. #21447013C2semidirect
6. #21446766C2semidirect
7. #21446743C2semidirect
8. #21446176C2semidirect
9. #21446085C2semidirect
10. #21445796C2semidirect
11. C2×C61×#461C2semidirect
54. #14519777D4semidirect
102. C4⋊Q8×C61D4semidirect
154. C2×C4×D61SD16semidirect
301. C2×#461D61semidirect
354. C61#24151semidirect
371. C4⋊Q8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168400522
C4×#145197482
#214470132
#214468192
#214467662
#214467432
#214461762
#214460852
#214457962
C2×C61×#4612
#2415161