Finite Group Databasebeta

← #29721595

Subgroups — #29721595

Order 15616.

1086 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839974C212yesmaximal, complemented
2C2×#16839954C211yesmaximal, characteristic
3C2×C4×#11758201C211yesmaximal, characteristic, complemented
4C4⋊C4×D244C211yesmaximal, characteristic, complemented
5#21447193C212yesmaximal, complemented
6#21447055C212yesmaximal, complemented
7#21446626C211yesmaximal, characteristic, complemented
8#21446538C211yesmaximal, characteristic, complemented
9#21446170C212yesmaximal, complemented
10#21445824C211yesmaximal, characteristic, complemented
11C61×#2548C211yesmaximal, characteristic, complemented
12C22×#11758201C2211yescharacteristic
13C22×#11758199C2212yes—
14C2×C4×D244C2212yes—
15C2×C4×D244C2211yescharacteristic
16C2×C4⋊C4×D61—21no—
17#16839974—28no—
18#16839954—24no—
19C4×#11758201—24no—
20C2×#14519708C2211yescharacteristic
21C2×C4⋊#92418C2212yes—
22C2×C4⋊#92418C2211yescharacteristic
23C2×#14519701C2212yes—
24C2×#14519701C2211yescharacteristic
25C2×#14519698C2212yes—
26C2×#14519698C2212yes—
27C2×#14519698C2212yes—
28C2×#14519697C2211yescharacteristic
29C2×C4×#92418C2211yescharacteristic
30#16839660C2212yes—
31#16839636—22no—
32#16839632C2211yescharacteristic
33#16839619—21no—
34#16839524C2212yes—
35#16839517C2212yes—
36C2×#127×C61C2211yescharacteristic
37C2×#124×C61C2212yes—
38C2×C4×Q8×C61C2211yescharacteristic
39C61×#394—22no—
40C61×#390—21no—
41C61×#390—21no—
42C61×#390C2212yes—
43C4×C4⋊C4×C61C2211yescharacteristic
44C22×D244C2311yescharacteristic
45C22×C4×D61—21no—
46C22×C4×D61—22no—
47C2×#11758201—22no—
48C2×#11758201—22no—
49C2×#11758201D414yescomplemented
50C2×#11758199—24no—
51C2×#11758199—24no—
52C2×#11758199—24no—
53C22×#92418C2311yescharacteristic
54C22×#92418C2312yes—
55C22×#92418C2311yescharacteristic
56C4×D244—22no—
57C4×D244—24no—
58C4×D244—42no—
59C4×D244D414yes—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61C2312yes—
63C2×C4⋊Dic61C2311yescharacteristic
64C2×#11758196—22no—
65C4⋊C4×D61—44no—
66C2×#11758198—21no—
67C2×#11758198—22no—
68#14519708—22no—
69C4⋊#92418—22no—
70C4⋊#92418—24no—
71#14519701—24no—
72#14519701—22no—
73#14519698—24no—
74#14519698—24no—
75#14519698—42no—
76#14519697—22no—
77C4×#92418—22no—
78#14519684—22no—
79C22×Q8×C61C2311yescharacteristic
80C22×D4×C61—21no—
81C22×D4×C61—21no—
82C22×D4×C61C2312yes—
83#127×C61—22no—
84#127×C61—22no—
85#124×C61—24no—
86#124×C61—42no—
87C4×Q8×C61—22no—
88C4×Q8×C61—22no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61D411yescharacteristic
91C2×C4⋊C4×C61D411yescharacteristic
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C22⋊C4×C61—22no—
94C2×C22⋊C4×C61—22no—
95C2×C22⋊C4×C61—22no—
96C2×C22⋊C4×C61—22no—
97C2×C42×C61—21no—
98C2×C42×C61C2312yes—
99C2×C42×C61C2311yescharacteristic
100#95×C61D412yes—
101C24×C61—22no—
102C23×D61—21no—
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C2×Q8×C61—21no—
106C2×Q8×C61—21no—
107C2×Q8×C61—21no—
108C2×Q8×C61C2×D412yes—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—42no—
116C2×D4×C61—42no—
117C2×D4×C61—42no—
118C2×D4×C61—42no—
119C2×D4×C61—81no—
120C2×D4×C61—81no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C22×Dic61—21no—
129C22×Dic61—22no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244C2×D412yes—
134C2×D244C2×D412yes—
135C2×D244#4112yes—
136C2×C4×D61—24no—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×C4×D61—44no—
140C2×C4×D61—44no—
141C2×C4×D61—82no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—42no—
147C22⋊C4×C61—44no—
148C22⋊C4×C61—44no—
149C22⋊C4×C61—82no—
150C22⋊C4×C61—82no—
151C42×C61—22no—
152C42×C61—22no—
153C42×C61—42no—
154C42×C61C2×D412yes—
155C42×C61#4112yes—
156#11758201—28no—
157#11758199—48no—
158#11758198—24no—
159#11758198—44no—
160C4⋊Dic61—21no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61—42no—
164C4⋊Dic61—81no—
165C4⋊Dic61C2×D412yes—
166#11758196—44no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418—21no—
170C2×#92418—22no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418C2×D412yes—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—42no—
183C23×C61C22×D411yescharacteristic
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—41no—
191Q8×C61—22no—
192Q8×C61—22no—
193Q8×C61—41no—
194Q8×C61—41no—
195D4×C61—42no—
196D4×C61—42no—
197D4×C61—42no—
198D4×C61—42no—
199D4×C61—84no—
200D4×C61—84no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61#12112yes—
216C2×C4×C61#12112yes—
217C2×C4×C61#13512yes—
218C2×C4×C61AGL(1,ℤ/8)12yes—
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61H(5,𝔽2)11yescharacteristic
224C2×C4×C612−1+411yescharacteristic
225C2×Dic61—22no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231D244—22no—
232D244—22no—
233D244—22no—
234D244#12114yes—
235C4×D61—48no—
236C4×D61—88no—
237#92418—22no—
238#92418—42no—
239#92418—41no—
240#24731—611nomaximal, Sylow
241C22×C61—82no—
242C22×C61—82no—
243C22×C61—82no—
244C22×C61—82no—
245C22×C61—82no—
246C22×C61—82no—
247C22×C61—82no—
248C22×C61C2×#12111yescharacteristic
249C22×C61D4211yescharacteristic
250C22×C61#54811yescharacteristic
251C22×C61C2×#13511yescharacteristic
252C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
253C22×C61#57611yescharacteristic
254C22×C61#57711yescharacteristic
255C2×D61—24no—
256C2×D61—42no—
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—42no—
261C2×D61—42no—
262C4×C61—22no—
263C4×C61—42no—
264C4×C61—42no—
265C4×C61—42no—
266C4×C61—81no—
267C4×C61#46314yescomplemented
268C4×C61C2×#12111yescharacteristic
269C4×C61#53711yescharacteristic
270C4×C61#54711yescharacteristic
271C4×C61#55311yescharacteristic
272Dic61—42no—
273Dic61—82no—
274C2×#463—612no—
275C2×#459—611no—
276C2×C4×SD16—611no—
277#2548D6111yescharacteristic, complemented
278D4×C4⋊C4—611no—
279#2065—612no—
280#2042—612no—
281#1957—611no—
282#1939—611no—
283#1879—612no—
284#1803—611no—
285C2×C61—162no—
286C2×C61#241611yescharacteristic
287C2×C61C2×#46311yescharacteristic
288C2×C61#304511yescharacteristic
289C2×C61#327411yescharacteristic
290C2×C61#328411yescharacteristic
291C2×C61#336111yescharacteristic
292C2×C61#337311yescharacteristic
293D61—44no—
294C22×SD16—611no—
295C22×D8—612no—
296C2×#127C2×D6111yescharacteristic
297C2×#124C2×D6112yes—
298C2×C4×Q8C2×D6111yescharacteristic
299C2×C4×D4—611no—
300C2×C4×D4—612no—
301C22×C4⋊C4—1221no—
302#463—1228no—
303#459—1224no—
304C4×SD16—1224no—
305C2×#106—611no—
306C2×C4⋊C8—611no—
307C2×C4⋊C8—612no—
308C2×#103—612no—
309C2×#103—611no—
310C2×#102—612no—
311C2×#102—612no—
312C2×#102—611no—
313C2×#102—612no—
314C2×C4×C8—611no—
315#394—22no—
316#390—21no—
317#390—21no—
318#390C2×D6112yes—
319#389—612no—
320#385—1222no—
321#384—611no—
322#380—1221no—
323C4×C4⋊C4C2×D6111yescharacteristic
324#340—612no—
325#336—612no—
326C61#2473111yescharacteristic, complemented, Sylow
327C22×Q8C22×D6111yescharacteristic
328C22×D4—21no—
329C22×D4—21no—
330C22×D4—611no—
331C22×D4C22×D6112yes—
332C23×C4—1221no—
333C23×C4—1222no—
334C2×SD16—614no—
335C2×SD16—1222no—
336C2×SD16—1222no—
337C2×D8—1224no—
338C2×D8—1224no—
339C2×D8—1224no—
340C22×C8—611no—
341C22×C8—612no—
342C22×C8—611no—
343#127—22no—
344#127—22no—
345#124—24no—
346#124—42no—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×D4—614no—
350C4×D4—1224no—
351C4×D4—1222no—
352C4×D4—2442no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—611no—
355C2×C4⋊C4—612no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—2444no—
360C2×C4⋊C4C22⋊D6111yescharacteristic
361C2×C4⋊C4C22⋊D6111yescharacteristic
362C2×C4⋊C4C22×D6111yescharacteristic
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—1221no—
368C2×C22⋊C4—1222no—
369C2×C42—21no—
370C2×C42C22×D6112yes—
371C2×C42C22×D6111yescharacteristic
372#106—1222no—
373C4⋊C8—1224no—
374C4⋊C8—1222no—
375#103—1222no—
376#103—1224no—
377#102—1222no—
378#102—1224no—
379#102—1224no—
380#102—2442no—
381C4×C8—1222no—
382#95—1222no—
383#95C22⋊D6112yes—
384C24—22no—
385C24—1221no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8D4×D6112yes—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—81no—
403C2×D4—81no—
404C2×D4—612no—
405C2×D4—612no—
406C2×D4—612no—
407C2×D4—1221no—
408C2×D4—1221no—
409C2×D4—1221no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—1221no—
413C22×C4—1224no—
414C22×C4—1222no—
415C22×C4—4882no—
416C22×C4—2444no—
417C22×C4—2444no—
418C22×C4—2442no—
419C22×C4—2442no—
420C22×C4C2×C22⋊D6112yes—
421C22×C4C2×C22⋊D6112yes—
422C22×C4C2×C22⋊D6111yescharacteristic
423C22×C4C2×C22⋊D6111yescharacteristic
424C22×C4C23×D6111yescharacteristic
425SD16—1228no—
426D8—2448no—
427C2×C8—612no—
428C2×C8—1221no—
429C2×C8—1221no—
430C2×C8—1221no—
431C2×C8—1222no—
432C2×C8—1222no—
433C2×C8—1222no—
434C2×C8—1221no—
435C2×C8—1221no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—42no—
441C4⋊C4—612no—
442C4⋊C4—1221no—
443C4⋊C4—1222no—
444C4⋊C4—1222no—
445C4⋊C4—4881no—
446C4⋊C4—2444no—
447C4⋊C4—2442no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—82no—
451C22⋊C4—82no—
452C22⋊C4—1224no—
453C22⋊C4—2444no—
454C42—22no—
455C42—22no—
456C42—42no—
457C42D4×D6112yes—
458C42#1175822412yes—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—1222no—
467C23—1222no—
468C23—1222no—
469C23—2441no—
470C23—2441no—
471C23—2441no—
472C23—2441no—
473C23C22×C22⋊D6111yescharacteristic
474Q8—22no—
475Q8—22no—
476Q8—41no—
477Q8—41no—
478D4—42no—
479D4—42no—
480D4—42no—
481D4—42no—
482D4—84no—
483D4—84no—
484D4—614no—
485D4—1222no—
486D4—1222no—
487D4—1222no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—1222no—
503C2×C4—4888no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4—2448no—
510C2×C4#1451974812yes—
511C2×C4#1451974812yes—
512C2×C4#1451979712yes—
513C2×C4#1451980212yes—
514C2×C4C2×D4×D6111yescharacteristic
515C2×C4C2×D4×D6111yescharacteristic
516C2×C4#41×D6111yescharacteristic
517C2×C4C2×#1175822411yescharacteristic
518C2×C4#1451982211yescharacteristic
519C2×C4#1451982411yescharacteristic
520C8—1222no—
521C8—2442no—
522C8—2441no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—1224no—
531C22—2442no—
532C22—2442no—
533C22—2442no—
534C22—2442no—
535C22—2442no—
536C22—2442no—
537C22C2×#1451974811yescharacteristic
538C22D4×C22⋊D6111yescharacteristic
539C22#1684041911yescharacteristic
540C22C2×#1451979711yescharacteristic
541C22C2×#1451980211yescharacteristic
542C22#1684056511yescharacteristic
543C22#1684057311yescharacteristic
544C4—22no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—81no—
549C4—4882no—
550C4—2442no—
551C4#1683997414yescomplemented
552C4#121×D6111yescharacteristic
553C4#1684034111yescharacteristic
554C4#1684041211yescharacteristic
555C4#1684045911yescharacteristic
556C2—162no—
557C2—2444no—
558C2#2144925511yescharacteristic
559C2C2×#1683997411yescharacteristic
560C2#2145535411yescharacteristic
561C2#2145743911yescharacteristic
562C2#2145754911yescharacteristic
563C2#2145847511yescharacteristic
564C2#2145860711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2548C2
socleC23×C61C22×D4
prime radicalC2×#16839954C2
hypercenter#2548D61
hypocenterC61#24731
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. C2×#16839974C2semidirect
3. C2×C4×#11758201C2semidirect
4. C4⋊C4×D244C2semidirect
5. #21447193C2semidirect
6. #21447055C2semidirect
7. #21446626C2semidirect
8. #21446538C2semidirect
9. #21446170C2semidirect
10. #21445824C2semidirect
11. C61×#2548C2semidirect
49. C2×#11758201D4semidirect
267. C4×C61#463semidirect
277. #2548D61semidirect
326. C61#24731semidirect
551. C4#16839974semidirect

Maximal subgroups

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

subgroupindex
C2×#168399742
C2×#168399542
C2×C4×#117582012
C4⋊C4×D2442
#214471932
#214470552
#214466262
#214465382
#214461702
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