Finite Group Databasebeta

← #29721556

Subgroups — #29721556

Order 15616.

1050 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839979C212yesmaximal
2C2×C4×#11758199C211yesmaximal, characteristic
3#21450815C211yesmaximal, characteristic, complemented
4D4×C4⋊Dic61C211yesmaximal, characteristic, complemented
5#21447210C212yesmaximal, complemented
6#21447057C212yesmaximal, complemented
7#21446622C211yesmaximal, characteristic, complemented
8#21446535C211yesmaximal, characteristic, complemented
9#21446155C212yesmaximal, complemented
10#21445820C211yesmaximal, characteristic, complemented
11C2×C61×#461C211yesmaximal, characteristic, complemented
12C22×#11758199C2211yescharacteristic
13C2×#14519766C2212yes—
14C2×C4×D244C2211yescharacteristic
15C2×D4×Dic61C2212yes—
16C22×C4⋊Dic61—21no—
17#16839979—28no—
18C4×#11758199—24no—
19C2×#14519710C2211yescharacteristic
20C2×#14519706C2212yes—
21C2×#14519701C2212yes—
22C2×#14519701C2212yes—
23C2×#14519699C2212yes—
24C2×#14519698C2211yescharacteristic
25C2×#14519697C2211yescharacteristic
26C2×C4×#92418C2211yescharacteristic
27#16839688—22no—
28#16839669C2212yes—
29#16839661C2212yes—
30#16839637—22no—
31#16839632C2211yescharacteristic
32#16839622—21no—
33#16839619—21no—
34#16839619—21no—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839525C2212yes—
37#16839516C2212yes—
38C22×SD16×C61C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4×D4×C61C2211yescharacteristic
41C61×#461—24no—
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C22×D244C2311yescharacteristic
45C23×Dic61—22no—
46C22×C4×D61—21no—
47C2×#11758199—22no—
48C2×#11758199—22no—
49C2×#11758199D414yes—
50C22×#92418C2311yescharacteristic
51C22×#92418C2311yescharacteristic
52#14519766—24no—
53#14519766—42no—
54C4×D244—22no—
55C4×D244—22no—
56D4×Dic61—24no—
57D4×Dic61—42no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—44no—
62C2×C4⋊Dic61D411yescharacteristic
63C2×C4⋊Dic61D411yescharacteristic
64C2×C4⋊Dic61C2312yes—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C22⋊Dic61—22no—
67C2×#11758198—21no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×C4×Dic61C2312yes—
71#14519710—22no—
72#14519706—24no—
73#14519701—24no—
74#14519701—42no—
75#14519699—24no—
76#14519698—22no—
77#14519697—22no—
78C4×#92418—22no—
79#14519684—22no—
80#14519684—22no—
81#14519684—22no—
82#14519684D412yes—
83C22×Q8×C61C2312yes—
84C22×D4×C61C2311yescharacteristic
85C23×C4×C61—21no—
86C2×SD16×C61—24no—
87C2×SD16×C61—24no—
88C2×SD16×C61—24no—
89C22×C8×C61C2312yes—
90C4⋊Q8×C61—22no—
91C4⋊Q8×C61—22no—
92C4×D4×C61—22no—
93C4×D4×C61D414yes—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C22⋊C4×C61—21no—
98C2×C42×C61C2311yescharacteristic
99C4⋊C8×C61—22no—
100#103×C61—24no—
101C24×C61—21no—
102C23×D61—21no—
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61C2×D412yes—
113C2×D4×C61C2×D412yes—
114C2×D4×C61#4112yes—
115C22×C4×C61—21no—
116C22×C4×C61—24no—
117C22×C4×C61—22no—
118C22×C4×C61—42no—
119C22×C4×C61—42no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C22×Dic61—21no—
124C22×Dic61—22no—
125C22×Dic61—44no—
126C22×Dic61—44no—
127C22×Dic61—82no—
128C22×Dic61C2×D412yes—
129C22×Dic61C2×D412yes—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244C2×D412yes—
136C2×C4×D61—42no—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139SD16×C61—48no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—42no—
145C4⋊C4×C61—42no—
146C4⋊C4×C61C2×D412yes—
147C22⋊C4×C61—24no—
148C42×C61C2×D412yes—
149C42×C61#4112yes—
150C22⋊Dic61—44no—
151#11758199—28no—
152#11758198—44no—
153#11758198—44no—
154#11758198—42no—
155C4⋊Dic61—22no—
156C4⋊Dic61—22no—
157C4⋊Dic61—21no—
158C4⋊Dic61—22no—
159C4⋊Dic61—21no—
160C4⋊Dic61—22no—
161C4⋊Dic61—44no—
162C4⋊Dic61—42no—
163C4⋊Dic61—81no—
164C4×Dic61—22no—
165C4×Dic61—22no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418C2×D412yes—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61C22×D411yescharacteristic
180C22×D61—22no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187Q8×C61—42no—
188Q8×C61—42no—
189Q8×C61—42no—
190Q8×C61—42no—
191D4×C61—22no—
192D4×C61—22no—
193D4×C61—22no—
194D4×C61#12114yes—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—22no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—48no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61#12112yes—
205C2×C4×C61#12112yes—
206C2×C4×C61#13512yes—
207C2×C4×C61AGL(1,ℤ/8)12yes—
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C61H(5,𝔽2)11yescharacteristic
213C2×C4×C612−1+411yescharacteristic
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—22no—
220C2×Dic61—41no—
221C2×Dic61—42no—
222C2×Dic61—41no—
223C2×Dic61—42no—
224C2×Dic61—41no—
225C2×Dic61—42no—
226C2×Dic61—88no—
227D244—22no—
228D244—22no—
229D244—41no—
230D244—41no—
231C4×D61—84no—
232C8×C61—42no—
233#92418—22no—
234#92418—41no—
235#24727—611nomaximal, Sylow
236C22×C61—24no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61—42no—
243C22×C61C2×#12111yescharacteristic
244C22×C61D4211yescharacteristic
245C22×C61#54811yescharacteristic
246C22×C61C2×#13511yescharacteristic
247C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
248C22×C61#57711yescharacteristic
249C22×C61#57811yescharacteristic
250C2×D61—42no—
251C2×D61—42no—
252C2×D61—42no—
253C2×D61—81no—
254C2×D61—81no—
255C2×D61—81no—
256C2×D61—81no—
257C4×C61—22no—
258C4×C61—42no—
259C4×C61—82no—
260C4×C61#46314yescomplemented
261C4×C61C2×#12111yescharacteristic
262C4×C61#53711yescharacteristic
263C4×C61#54711yescharacteristic
264C4×C61#55311yescharacteristic
265Dic61—42no—
266Dic61—42no—
267Dic61—82no—
268Dic61—81no—
269C2×#463—612no—
270C2×#461D6111yescharacteristic, complemented
271C2×C4×D8—611no—
272#2547—611no—
273D4×C4⋊C4—611no—
274#2068—612no—
275#2042—612no—
276#1956—611no—
277#1939—611no—
278#1877—612no—
279#1802—611no—
280C2×C61—44no—
281C2×C61#241611yescharacteristic
282C2×C61C2×#46311yescharacteristic
283C2×C61#304611yescharacteristic
284C2×C61#327211yescharacteristic
285C2×C61#328311yescharacteristic
286C2×C61#336011yescharacteristic
287C2×C61#336711yescharacteristic
288D61—82no—
289C22×SD16C2×D6112yes—
290C22×D8—611no—
291C2×C4⋊Q8C2×D6111yescharacteristic
292C2×#124—612no—
293C2×C4×D4—612no—
294C2×C4×D4—611no—
295C2×C4×D4C2×D6111yescharacteristic
296C22×C4⋊C4—1221no—
297#463—1228no—
298#461—24no—
299C4×D8—1224no—
300C2×#107—611no—
301C2×C4⋊C8—612no—
302C2×C4⋊C8C2×D6111yescharacteristic
303C2×#103—612no—
304C2×#103—612no—
305C2×#103C2×D6112yes—
306C2×#102—611no—
307C2×#102—612no—
308C2×#102—611no—
309C2×C4×C8—611no—
310#397—1222no—
311#391—612no—
312#389—612no—
313#385—1222no—
314#384—611no—
315#380—1221no—
316#380—1221no—
317#380—1221no—
318C4×C4⋊C4—611no—
319#340—612no—
320#336—612no—
321C61#2472711yescharacteristic, complemented, Sylow
322C22×Q8C22×D6112yes—
323C22×D4—611no—
324C22×D4C22×D6111yescharacteristic
325C23×C4—21no—
326C23×C4—1222no—
327C23×C4—1221no—
328C2×SD16—24no—
329C2×SD16—24no—
330C2×SD16—24no—
331C2×D8—614no—
332C2×D8—1222no—
333C2×D8—1222no—
334C22×C8—611no—
335C22×C8—611no—
336C22×C8C22×D6112yes—
337C4⋊Q8—22no—
338C4⋊Q8—22no—
339#124—1224no—
340#124—2442no—
341C4×D4—22no—
342C4×D4—1224no—
343C4×D4—1222no—
344C4×D4—1222no—
345C4×D4—2442no—
346C4×D4C22⋊D6114yes—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—611no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—612no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—2444no—
357C2×C4⋊C4C22×D6111yescharacteristic
358C2×C22⋊C4—21no—
359C2×C22⋊C4—1222no—
360C2×C22⋊C4—1222no—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4—1221no—
363C2×C42—612no—
364C2×C42C22×D6111yescharacteristic
365#107—1222no—
366C4⋊C8—22no—
367C4⋊C8—1224no—
368#103—24no—
369#103—1224no—
370#103—2442no—
371#102—1224no—
372#102—1222no—
373#102—1222no—
374C4×C8—1222no—
375#95—612no—
376#95—1222no—
377#95—1222no—
378#95—1222no—
379C24—21no—
380C24—1221no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—612no—
391C2×D4—1221no—
392C2×D4—1221no—
393C2×D4—1221no—
394C2×D4—1221no—
395C2×D4—1221no—
396C2×D4D4×D6112yes—
397C2×D4#1175822412yes—
398C2×D4C2×C22⋊D6112yes—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—612no—
405C22×C4—612no—
406C22×C4—1221no—
407C22×C4—1222no—
408C22×C4—4882no—
409C22×C4—2444no—
410C22×C4—2444no—
411C22×C4—2442no—
412C22×C4—2442no—
413C22×C4—2442no—
414C22×C4D4×D6111yescharacteristic
415C22×C4D4×D6111yescharacteristic
416C22×C4C23×D6111yescharacteristic
417SD16—48no—
418D8—1228no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—612no—
423C2×C8—1221no—
424C2×C8—1221no—
425C2×C8—1221no—
426C2×C8—1221no—
427C2×C8—1221no—
428C4⋊C4—21no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—1222no—
432C4⋊C4—1222no—
433C4⋊C4—1222no—
434C4⋊C4—1221no—
435C4⋊C4—1221no—
436C4⋊C4—1222no—
437C4⋊C4—4881no—
438C4⋊C4—2444no—
439C4⋊C4—2442no—
440C4⋊C4C2×C22⋊D6112yes—
441C22⋊C4—24no—
442C22⋊C4—2444no—
443C22⋊C4—2444no—
444C22⋊C4—2444no—
445C22⋊C4—2442no—
446C42—1222no—
447C42—1222no—
448C42#1175822412yes—
449C42C2×C22⋊D6112yes—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—1222no—
458C23—2441no—
459C23—2441no—
460C23—2441no—
461C23—2441no—
462C23—2441no—
463C23—2441no—
464C23C2×D4×D6111yescharacteristic
465Q8—42no—
466Q8—42no—
467Q8—42no—
468Q8—42no—
469D4—22no—
470D4—22no—
471D4—22no—
472D4—1222no—
473D4—1222no—
474D4—2441no—
475D4—2441no—
476D4#1451974814yes—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—48no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—4888no—
493C2×C4—4884no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4—2442no—
497C2×C4—2441no—
498C2×C4—2442no—
499C2×C4—2442no—
500C2×C4#1451974912yes—
501C2×C4#1451974912yes—
502C2×C4#1451979812yes—
503C2×C4#1451980412yes—
504C2×C4C2×D4×D6111yescharacteristic
505C2×C4C22×C22⋊D6111yescharacteristic
506C2×C4C2×#1175822411yescharacteristic
507C2×C4C2×#1175822411yescharacteristic
508C2×C4#1451982311yescharacteristic
509C2×C4#1451982411yescharacteristic
510C8—42no—
511C8—1222no—
512C8—2441no—
513C22—24no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—4881no—
521C22—4881no—
522C22—4881no—
523C22—4881no—
524C22—2442no—
525C22—2442no—
526C22—2442no—
527C22C2×#1451974911yescharacteristic
528C22#1684039611yescharacteristic
529C22#1684041811yescharacteristic
530C22C2×#1451979811yescharacteristic
531C22C2×#1451980411yescharacteristic
532C22#1684057111yescharacteristic
533C22#1684057811yescharacteristic
534C4—22no—
535C4—42no—
536C4—82no—
537C4—4882no—
538C4—4881no—
539C4—2442no—
540C4—2442no—
541C4#1683997914yescomplemented
542C4C2×#1451974811yescharacteristic
543C4#1684034011yescharacteristic
544C4#1684041211yescharacteristic
545C4#1684046411yescharacteristic
546C2—44no—
547C2—4882no—
548C2#2144926011yescharacteristic
549C2C2×#1683997911yescharacteristic
550C2#2145537311yescharacteristic
551C2#2145742211yescharacteristic
552C2#2145753611yescharacteristic
553C2#2145847211yescharacteristic
554C2#2145854111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#461C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenterC2×#461D61
hypocenterC61#24727
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
3. #21450815C2semidirect
4. D4×C4⋊Dic61C2semidirect
5. #21447210C2semidirect
6. #21447057C2semidirect
7. #21446622C2semidirect
8. #21446535C2semidirect
9. #21446155C2semidirect
10. #21445820C2semidirect
11. C2×C61×#461C2semidirect
260. C4×C61#463semidirect
270. C2×#461D61semidirect
321. C61#24727semidirect
541. C4#16839979semidirect

Maximal subgroups

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

subgroupindex
C2×#168399792
C2×C4×#117581992
#214508152
D4×C4⋊Dic612
#214472102
#214470572
#214466222
#214465352
#214461552
#214458202
C2×C61×#4612
#2472761