Finite Group Databasebeta

← #29714087

Subgroups — #29714087

Order 15616.

1140 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840485C211yesmaximal, characteristic, complemented
2C2×#16840052C211yesmaximal, characteristic, complemented
3#21450716C211yesmaximal, characteristic
4#21447013C212yesmaximal, complemented
5#21446819C212yesmaximal, complemented
6#21446792C211yesmaximal, characteristic, complemented
7#21446617C211yesmaximal, characteristic, complemented
8#21446138C212yesmaximal, complemented
9#21446015C212yesmaximal, complemented
10#21445834C211yesmaximal, characteristic, complemented
11C2×C61×#440C211yesmaximal, characteristic, complemented
12#16840485—24no—
13#16840485—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519772—22no—
16C2×#14519765—21no—
17C2×#14519753C2212yes—
18C2×#14519748C2212yes—
19C2×C4×Dic122—21no—
20C2×C4×D244C2211yescharacteristic
21C2×#14519735C2212yes—
22#16840052—24no—
23C2×#14519708C2212yes—
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519701C2212yes—
26C2×#14519698C2212yes—
27C2×#14519689C2212yes—
28C2×#14519689C2212yes—
29#16839687—21no—
30#16839671—21no—
31#16839641—22no—
32#16839628C2211yescharacteristic
33#16839619C2212yes—
34#16839525C2212yes—
35#16839525C2212yes—
36#16839524C2212yes—
37C22×SD16×C61C2211yescharacteristic
38C2×C4×Q8×C61C2211yescharacteristic
39C2×C4×D4×C61C2211yescharacteristic
40C61×#440—24no—
41C2×#107×C61C2211yescharacteristic
42C2×#103×C61C2211yescharacteristic
43C2×#102×C61C2211yescharacteristic
44C2×#97×C61C2211yescharacteristic
45C22×Dic122—21no—
46C22×D244D411yescharacteristic
47C22×C22⋊D61—22no—
48C22×C4×D61C2312yes—
49C22×#92418C2312yes—
50#14519777—22no—
51#14519777D414yescomplemented
52#14519772—216no—
53#14519765—28no—
54#14519753—24no—
55#14519753—24no—
56#14519748—24no—
57#14519748—42no—
58C4×Dic122—28no—
59C4×D244—22no—
60C4×D244—22no—
61#14519735—24no—
62#14519735—24no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61D411yescharacteristic
65C2×C4⋊Dic61C2312yes—
66C2×C4⋊Dic61C2312yes—
67C2×#11758196—22no—
68C2×C22⋊Dic61—22no—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×#11758198D412yes—
73C2×C4×Dic61—22no—
74#14519708—24no—
75C4⋊#92418—22no—
76#14519701—42no—
77#14519698—24no—
78#14519689—24no—
79#14519689—42no—
80#14519684—21no—
81#14519684—21no—
82#14519684—22no—
83C22×Q8×C61C2311yescharacteristic
84C22×D4×C61C2311yescharacteristic
85C23×C4×C61—21no—
86C2×SD16×C61—22no—
87C2×SD16×C61—22no—
88C2×SD16×C61—22no—
89C22×C8×C61C2311yescharacteristic
90C22×C8×C61C2311yescharacteristic
91C4×Q8×C61—22no—
92C4×Q8×C61D414yescomplemented
93C4×D4×C61—22no—
94C4×D4×C61—22no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C22⋊C4×C61—21no—
99C2×C42×C61—21no—
100C2×C42×C61C2311yescharacteristic
101#107×C61—22no—
102#103×C61—22no—
103#102×C61—22no—
104#97×C61—22no—
105C24×C61—21no—
106C23×D61C2×D412yes—
107C2×Q8×C61—21no—
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61C2×D412yes—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61#4112yes—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—42no—
123C22×C4×C61—82no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—44no—
128C2×C22⋊D61—44no—
129C2×C22⋊D61—82no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61C2×D412yes—
133C2×D244—22no—
134C2×D244—42no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×Dic122—24no—
142C2×Dic122—44no—
143SD16×C61—28no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61#4112yes—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61C2×D412yes—
155C22⋊C4×C61—42no—
156C42×C61—24no—
157C42×C61C2×D412yes—
158C42×C61C2×D412yes—
159C22⋊Dic61—44no—
160#11758198—28no—
161#11758198—28no—
162#11758198—24no—
163#11758198—82no—
164C4⋊Dic61—24no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61C2×D414yes—
170#11758196—28no—
171C4×Dic61—28no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—22no—
175C23×C61—22no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61C22×D411yescharacteristic
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—22no—
188Q8×C61—22no—
189Q8×C61—22no—
190Q8×C61—41no—
191Q8×C61—41no—
192D4×C61—22no—
193D4×C61—22no—
194D4×C61—41no—
195D4×C61—41no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—42no—
203C2×C4×C61—41no—
204C2×C4×C61—42no—
205C2×C4×C61—84no—
206C2×C4×C61#12012yes—
207C2×C4×C61#12012yes—
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
214C22⋊D61—88no—
215C2×Dic61—22no—
216C2×Dic61—24no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—44no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223D244—44no—
224C4×D61—42no—
225C4×D61—42no—
226C4×D61—42no—
227C4×D61—42no—
228Dic122—48no—
229C8×C61—22no—
230C8×C61—41no—
231#92418—42no—
232#24131—611nomaximal, Sylow
233C22×C61—42no—
234C22×C61—44no—
235C22×C61—82no—
236C22×C61—81no—
237C22×C61—81no—
238C22×C61C2×#12011yescharacteristic
239C22×C61#54611yescharacteristic
240C22×C61#54711yescharacteristic
241C22×C61#57512yes—
242C22×C61#57712yes—
243C2×D61—42no—
244C2×D61—42no—
245C2×D61—42no—
246C2×D61—42no—
247C2×D61—42no—
248C2×D61—42no—
249C2×D61—42no—
250C4×C61—22no—
251C4×C61—22no—
252C4×C61—42no—
253C4×C61—42no—
254C4×C61—81no—
255C4×C61#52511yescharacteristic
256C4×C61#53711yescharacteristic
257C4×C61#54811yescharacteristic
258C4×C61#55011yescharacteristic
259Dic61—42no—
260Dic61—44no—
261Dic61—82no—
262C2×#556—611no—
263C2×#476—611no—
264C2×#440D6111yescharacteristic, complemented
265#2535—611no—
266#2035—612no—
267#2000—612no—
268#1993—611no—
269#1955—611no—
270#1875—612no—
271#1854—612no—
272#1805—611no—
273C2×C61—82no—
274C2×C61#240711yescharacteristic
275C2×C61#300511yescharacteristic
276C2×C61#301011yescharacteristic
277C2×C61#331112yes—
278C2×C61#335012yes—
279D61—82no—
280C22×SD16C2×D6111yescharacteristic
281#556—1224no—
282#556—1224no—
283C2×C4⋊Q8—611no—
284C2×#126—1222no—
285C2×#124—1221no—
286C2×C22⋊Q8—612no—
287C2×#121—612no—
288C2×C4×Q8—1221no—
289C2×C4×Q8C2×D6111yescharacteristic
290C2×C4×D4—611no—
291C2×C4×D4C2×D6111yescharacteristic
292C2×#117—612no—
293#476—1224no—
294#440—24no—
295C2×#107C2×D6111yescharacteristic
296C2×#106—612no—
297C2×C4⋊C8—611no—
298C2×#103—612no—
299C2×#103C2×D6111yescharacteristic
300C2×#102—612no—
301C2×#102C2×D6111yescharacteristic
302C2×C22⋊C8—612no—
303C2×C22⋊C8—612no—
304C2×#97C2×D6111yescharacteristic
305#397—1221no—
306#392—1221no—
307#386—1222no—
308#382—611no—
309#380—612no—
310#340—612no—
311#340—612no—
312#340—612no—
313C61#2413111yescharacteristic, complemented, Sylow
314C22×Q8—1221no—
315C22×Q8C22×D6111yescharacteristic
316C22×D4—611no—
317C22×D4—1222no—
318C22×D4C22×D6111yescharacteristic
319C23×C4—21no—
320C23×C4—612no—
321C2×SD16—22no—
322C2×SD16—22no—
323C2×SD16—22no—
324C22×C8—612no—
325C22×C8C22×D6111yescharacteristic
326C22×C8C22×D6111yescharacteristic
327C4⋊Q8—614no—
328C4⋊Q8—1222no—
329#126—12216no—
330#124—1228no—
331C22⋊Q8—1224no—
332C22⋊Q8—1224no—
333#121—1224no—
334#121—2442no—
335C4×Q8—22no—
336C4×Q8—1228no—
337C4×Q8C22⋊D6114yescomplemented
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—1222no—
341C4×D4—1222no—
342#117—1224no—
343#117—1224no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—612no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—612no—
348C2×C4⋊C4—1222no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4C22×D6111yescharacteristic
351C2×C4⋊C4C22×D6111yescharacteristic
352C2×C22⋊C4—21no—
353C2×C22⋊C4—612no—
354C2×C22⋊C4—1222no—
355C2×C22⋊C4—1222no—
356C2×C22⋊C4—1222no—
357C2×C22⋊C4—1222no—
358C2×C42—21no—
359C2×C42—1222no—
360C2×C42C22×D6111yescharacteristic
361#107—22no—
362#106—1224no—
363C4⋊C8—1222no—
364#103—22no—
365#103—2442no—
366#102—22no—
367#102—1224no—
368C22⋊C8—1224no—
369C22⋊C8—2442no—
370#97—22no—
371#95—1222no—
372#95—1221no—
373#95—1221no—
374C24—21no—
375C24—612no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—1224no—
382C2×Q8—2444no—
383C2×Q8C2×C22⋊D6112yes—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—1222no—
390C2×D4—4882no—
391C2×D4—2442no—
392C2×D4—2444no—
393C2×D4—2444no—
394C2×D4#1175822212yes—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—42no—
399C22×C4—82no—
400C22×C4—612no—
401C22×C4—1222no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1222no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4D4×D6111yescharacteristic
410C22×C4D4×D6111yescharacteristic
411C22×C4C23×D6111yescharacteristic
412SD16—28no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—1222no—
419C2×C8—1222no—
420C2×C8—1222no—
421C2×C8#1175822212yes—
422C4⋊C4—21no—
423C4⋊C4—24no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—614no—
427C4⋊C4—1228no—
428C4⋊C4—1222no—
429C4⋊C4—1222no—
430C4⋊C4—1224no—
431C4⋊C4—1222no—
432C4⋊C4—1222no—
433C4⋊C4C2×C22⋊D6112yes—
434C22⋊C4—42no—
435C22⋊C4—1228no—
436C22⋊C4—1228no—
437C22⋊C4—1224no—
438C22⋊C4—4882no—
439C22⋊C4—2444no—
440C42—24no—
441C42—1228no—
442C42D4×D6112yes—
443C42C2×C22⋊D6112yes—
444C23—22no—
445C23—41no—
446C23—42no—
447C23—42no—
448C23—41no—
449C23—1222no—
450C23—1222no—
451C23—1222no—
452C23—1222no—
453C23—1222no—
454C23—1222no—
455C23—1222no—
456C23C2×D4×D6111yescharacteristic
457Q8—22no—
458Q8—22no—
459Q8—41no—
460Q8—41no—
461Q8—2448no—
462D4—22no—
463D4—22no—
464D4—41no—
465D4—41no—
466D4—4888no—
467D4—2444no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—84no—
478C2×C4—1222no—
479C2×C4—1224no—
480C2×C4—1222no—
481C2×C4—1222no—
482C2×C4—2442no—
483C2×C4—2442no—
484C2×C4—2442no—
485C2×C4—2444no—
486C2×C4—2442no—
487C2×C4—2442no—
488C2×C4—2442no—
489C2×C4—2442no—
490C2×C4#1451974512yes—
491C2×C4#1451974512yes—
492C2×C4C2×D4×D6111yescharacteristic
493C2×C4C22×C22⋊D6111yescharacteristic
494C2×C4C2×#1175822211yescharacteristic
495C2×C4C2×#1175822211yescharacteristic
496C2×C4#1451982311yescharacteristic
497C2×C4#1451982411yescharacteristic
498C8—22no—
499C8—41no—
500C8—2442no—
501C22—44no—
502C22—42no—
503C22—81no—
504C22—82no—
505C22—81no—
506C22—2442no—
507C22—2442no—
508C22—2442no—
509C22—2442no—
510C22—2442no—
511C22—2442no—
512C22—2442no—
513C22C2×#1451974511yescharacteristic
514C22#1684040811yescharacteristic
515C22#1684041711yescharacteristic
516C22#1684056112yes—
517C22#1684057212yes—
518C4—22no—
519C4—22no—
520C4—42no—
521C4—42no—
522C4—81no—
523C4—4882no—
524C4—2442no—
525C4—2444no—
526C4#1684027311yescharacteristic
527C4#1684034011yescharacteristic
528C4#1684042111yescharacteristic
529C4#1684043311yescharacteristic
530C2—82no—
531C2—4882no—
532C2#2144916411yescharacteristic
533C2#2145506011yescharacteristic
534C2#2145510411yescharacteristic
535C2#2145786012yes—
536C2#2145836912yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#440C2
socleC23×C61C22×D4
prime radical#21450716C2
hypercenterC2×#440D61
hypocenterC61#24131
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. C2×#16840485C2semidirect
2. C2×#16840052C2semidirect
4. #21447013C2semidirect
5. #21446819C2semidirect
6. #21446792C2semidirect
7. #21446617C2semidirect
8. #21446138C2semidirect
9. #21446015C2semidirect
10. #21445834C2semidirect
11. C2×C61×#440C2semidirect
51. #14519777D4semidirect
92. C4×Q8×C61D4semidirect
264. C2×#440D61semidirect
313. C61#24131semidirect
337. C4×Q8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168404852
C2×#168400522
#214507162
#214470132
#214468192
#214467922
#214466172
#214461382
#214460152
#214458342
C2×C61×#4402
#2413161