Finite Group Databasebeta

← #29713970

Subgroups — #29713970

Order 15616.

1134 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840469C211yesmaximal, characteristic
2C2×#16840058C211yesmaximal, characteristic, complemented
3C2×C4×Dic244C211yesmaximal, characteristic, complemented
4#21450741C211yesmaximal, characteristic
5#21447023C212yesmaximal, complemented
6#21446829C212yesmaximal, complemented
7#21446780C211yesmaximal, characteristic, complemented
8#21446124C212yesmaximal, complemented
9#21446009C212yesmaximal, complemented
10#21445825C211yesmaximal, characteristic, complemented
11C61×#1954C211yesmaximal, characteristic
12C22×Dic244C2211yescharacteristic
13#16840469—24no—
14#16840469—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519768—21no—
17C2×#14519756C2212yes—
18C2×#14519756C2212yes—
19C2×#14519754—22no—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×D4×Dic61—21no—
23C22×C4⋊Dic61C2212yes—
24#16840058—24no—
25C4×Dic244—24no—
26C2×#14519711C2212yes—
27C2×#14519711C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2212yes—
30C2×#14519702C2211yescharacteristic
31C2×#14519702C2212yes—
32#16839685—22no—
33#16839665—21no—
34#16839658—21no—
35#16839632C2211yescharacteristic
36#16839622C2212yes—
37#16839525C2212yes—
38#16839525C2212yes—
39#16839525C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×C22⋊C8×C61C2212yes—
43C2×C22⋊C8×C61C2212yes—
44C2×C4×C8×C61C2211yescharacteristic
45C22×Dic122—21no—
46C22×Dic122C2311yescharacteristic
47C22×Dic122C2311yescharacteristic
48C23×Dic61—22no—
49C2×Dic244—22no—
50C2×Dic244—22no—
51C2×Dic244—22no—
52#14519777—22no—
53#14519777D414yes—
54#14519768—28no—
55#14519756—24no—
56#14519756—24no—
57#14519756—24no—
58#14519756—42no—
59#14519754—216no—
60C4×Dic122—22no—
61C4×Dic122—22no—
62C4×Dic122—22no—
63C4×Dic122D414yes—
64D4×Dic61—28no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—24no—
68C2×C4⋊Dic61—24no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2312yes—
75C2×#11758196—21no—
76C2×#11758196—22no—
77C2×#11758196—21no—
78C2×#11758196—22no—
79C2×C22⋊Dic61—22no—
80C2×C22⋊Dic61—22no—
81C2×C22⋊Dic61—22no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84#14519711—24no—
85#14519711—22no—
86#14519702—22no—
87#14519702—24no—
88#14519702—22no—
89#14519702—42no—
90#14519684—22no—
91C22×D4×C61D411yescharacteristic
92C23×C4×C61C2312yes—
93C22×C8×C61C2311yescharacteristic
94C22×C8×C61C2312yes—
95C22×C8×C61C2311yescharacteristic
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C22⋊C4×C61D412yes—
100C2×C42×C61C2311yescharacteristic
101C4⋊C8×C61—22no—
102C22⋊C8×C61—24no—
103C22⋊C8×C61—42no—
104C4×C8×C61—22no—
105C24×C61C2×D412yes—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61Q1614yescomplemented
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2411yescharacteristic
119C22×Dic61—21no—
120C22×Dic61—22no—
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61—28no—
124C22×Dic61—21no—
125C22×Dic61—22no—
126C22×Dic61—82no—
127C22×Dic61—82no—
128C2×Dic122—21no—
129C2×Dic122—21no—
130C2×Dic122—21no—
131C2×Dic122—21no—
132C2×Dic122—21no—
133C2×Dic122—21no—
134C2×Dic122—21no—
135C2×Dic122—21no—
136C2×Dic122—21no—
137C2×Dic122—21no—
138C2×Dic122—24no—
139C2×Dic122—81no—
140C2×Dic122—81no—
141C2×Dic122C2×D412yes—
142C2×Dic122#4112yes—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—21no—
149C2×C8×C61—22no—
150C2×C8×C61—21no—
151C2×C8×C61#4112yes—
152C4⋊C4×C61—22no—
153C22⋊C4×C61—24no—
154C42×C61C2×D412yes—
155C42×C61C2×D412yes—
156C22⋊Dic61—28no—
157C22⋊Dic61—44no—
158C22⋊Dic61—82no—
159C4⋊Dic61—28no—
160C4⋊Dic61—24no—
161C4⋊Dic61—21no—
162C4⋊Dic61—22no—
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61—42no—
168C4⋊Dic61—81no—
169C4⋊Dic61C2×D412yes—
170C4⋊Dic61C2×D414yes—
171#11758196—28no—
172#11758196—24no—
173#11758196—44no—
174#11758196—42no—
175C4×Dic61—24no—
176C4×Dic61—42no—
177Dic244—28no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61C2×Q1614yes—
185C23×C61C22×D411yescharacteristic
186D4×C61—28no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61#12012yes—
197C2×C4×C61#12012yes—
198C2×C4×C61C2×Q1612yes—
199C2×C4×C61#13712yes—
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C2×#4111yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C2×C4×C61H(5,𝔽2)11yescharacteristic
205C2×C4×C612−1+411yescharacteristic
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—24no—
209C2×Dic61—22no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—42no—
215C2×Dic61—41no—
216C2×Dic61—42no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—88no—
225Dic122—22no—
226Dic122—22no—
227Dic122—22no—
228Dic122—22no—
229Dic122—41no—
230Dic122—41no—
231Dic122—41no—
232Dic122—41no—
233Dic122—84no—
234C8×C61—22no—
235C8×C61—41no—
236C8×C61—42no—
237#24122—611nomaximal, Sylow
238C22×C61—24no—
239C22×C61—24no—
240C22×C61—24no—
241C22×C61—42no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61C2×#12011yescharacteristic
246C22×C61#54611yescharacteristic
247C22×C61#54711yescharacteristic
248C22×C61C22×Q1611yescharacteristic
249C22×C61C2×#13711yescharacteristic
250C22×C61#57611yescharacteristic
251C22×C61#57711yescharacteristic
252C4×C61—22no—
253C4×C61—42no—
254C4×C61#45114yes—
255C4×C61#52511yescharacteristic
256C4×C61#53711yescharacteristic
257C4×C61#54811yescharacteristic
258C4×C61#55011yescharacteristic
259Dic61—42no—
260Dic61—42no—
261Dic61—44no—
262Dic61—42no—
263Dic61—82no—
264Dic61—81no—
265C2×#554—611no—
266C2×#477—611no—
267C2×C4×Q16—611no—
268#2538—611no—
269#2036—612no—
270#2001—612no—
271#1990—611no—
272#1954D6111yescharacteristic
273#1872—612no—
274#1852—612no—
275#1803—611no—
276C2×C61—44no—
277C2×C61#240711yescharacteristic
278C2×C61C2×#45111yescharacteristic
279C2×C61#301211yescharacteristic
280C2×C61#329911yescharacteristic
281C2×C61#333311yescharacteristic
282C2×C61#333811yescharacteristic
283C2×C61#335611yescharacteristic
284C22×Q16—611no—
285#554—1224no—
286#554—1224no—
287C2×C4⋊Q8—611no—
288C2×#125—1221no—
289C2×C22⋊Q8—612no—
290C2×C22⋊Q8—612no—
291C2×C22⋊Q8—1222no—
292C2×C4×Q8—611no—
293C2×C4×Q8—611no—
294C2×C4×D4—1221no—
295C2×C4×D4C2×D6111yescharacteristic
296C22×C4⋊C4—612no—
297#477—1224no—
298C4×Q16—1224no—
299C2×#107—612no—
300C2×#107—611no—
301C2×C4⋊C8C2×D6111yescharacteristic
302C2×#103—612no—
303C2×#103—611no—
304C2×#103—611no—
305C2×#103—612no—
306C2×C22⋊C8C2×D6112yes—
307C2×C22⋊C8C2×D6112yes—
308C2×C4×C8C2×D6111yescharacteristic
309#396—1222no—
310#390—1221no—
311#389—1221no—
312#384—611no—
313#380—612no—
314#340—612no—
315#340—612no—
316#340—612no—
317C61#2412211yescharacteristic, complemented, Sylow
318C22×Q8—611no—
319C22×Q8—611no—
320C22×Q8—1221no—
321C22×D4D24411yescharacteristic
322C23×C4—1222no—
323C23×C4C22×D6112yes—
324C2×Q16—1222no—
325C2×Q16—1222no—
326C2×Q16—1222no—
327C22×C8C22×D6111yescharacteristic
328C22×C8C22×D6112yes—
329C22×C8C22×D6111yescharacteristic
330C4⋊Q8—614no—
331C4⋊Q8—1222no—
332#125—1228no—
333C22⋊Q8—12216no—
334C22⋊Q8—1224no—
335C22⋊Q8—1224no—
336C22⋊Q8—1224no—
337C22⋊Q8—2442no—
338C4×Q8—614no—
339C4×Q8—1222no—
340C4×Q8—1222no—
341C4×Q8—1222no—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—1228no—
345C2×C4⋊C4—611no—
346C2×C4⋊C4—612no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—612no—
349C2×C4⋊C4—1224no—
350C2×C4⋊C4—1224no—
351C2×C4⋊C4—1222no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4—1221no—
359C2×C4⋊C4D24411yescharacteristic
360C2×C22⋊C4—1222no—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4—1222no—
363C2×C22⋊C4D24412yes—
364C2×C42—1221no—
365C2×C42—1221no—
366C2×C42C22×D6111yescharacteristic
367#107—1222no—
368#107—1224no—
369C4⋊C8—22no—
370#103—1222no—
371#103—1224no—
372#103—1222no—
373#103—2442no—
374C22⋊C8—24no—
375C22⋊C8—42no—
376C4×C8—22no—
377#95—1222no—
378C24C2×D24412yes—
379C2×Q8—612no—
380C2×Q8—612no—
381C2×Q8—1221no—
382C2×Q8—1221no—
383C2×Q8—1224no—
384C2×Q8—1221no—
385C2×Q8—1221no—
386C2×Q8—1221no—
387C2×Q8—1221no—
388C2×Q8—1221no—
389C2×Q8—1221no—
390C2×Q8—1221no—
391C2×Q8—1221no—
392C2×Q8—4881no—
393C2×Q8—4881no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4Dic24414yescomplemented
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—1228no—
404C22×C4—1221no—
405C22×C4—1221no—
406C22×C4—1221no—
407C22×C4—1222no—
408C22×C4—1221no—
409C22×C4—1222no—
410C22×C4—4882no—
411C22×C4—4882no—
412C22×C4C2×D24412yes—
413C22×C4C2×D24411yescharacteristic
414C22×C4C2×D24411yescharacteristic
415C22×C4C23×D6111yescharacteristic
416Q16—1228no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—21no—
423C2×C8—22no—
424C2×C8—21no—
425C2×C8#1175822412yes—
426C4⋊C4—22no—
427C4⋊C4—612no—
428C4⋊C4—614no—
429C4⋊C4—1228no—
430C4⋊C4—1228no—
431C4⋊C4—1221no—
432C4⋊C4—1222no—
433C4⋊C4—1224no—
434C4⋊C4—1224no—
435C4⋊C4—1221no—
436C4⋊C4—1221no—
437C4⋊C4—1222no—
438C4⋊C4—1222no—
439C4⋊C4—4881no—
440C4⋊C4—2444no—
441C4⋊C4—2442no—
442C4⋊C4—2442no—
443C22⋊C4—24no—
444C22⋊C4—1228no—
445C22⋊C4—4882no—
446C22⋊C4—2444no—
447C42—1224no—
448C42—2442no—
449C42D4×D6112yes—
450C42D4×D6112yes—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23C2×Dic24414yes—
458C23C22×D24411yescharacteristic
459Q8—1222no—
460Q8—1222no—
461Q8—1222no—
462Q8—1222no—
463Q8—4884no—
464Q8—2441no—
465Q8—2441no—
466Q8—2441no—
467Q8—2441no—
468D4—28no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—1222no—
479C2×C4—1222no—
480C2×C4—1224no—
481C2×C4—1222no—
482C2×C4—4888no—
483C2×C4—2441no—
484C2×C4—2442no—
485C2×C4—2441no—
486C2×C4—2442no—
487C2×C4—2441no—
488C2×C4—2442no—
489C2×C4—2441no—
490C2×C4—2441no—
491C2×C4—2442no—
492C2×C4—2441no—
493C2×C4—2442no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4—2441no—
497C2×C4#1451974712yes—
498C2×C4#1451974712yes—
499C2×C4C2×Dic24412yes—
500C2×C4#1451981012yes—
501C2×C4C2×D4×D6111yescharacteristic
502C2×C4C2×D4×D6111yescharacteristic
503C2×C4#41×D6111yescharacteristic
504C2×C4C2×#1175822411yescharacteristic
505C2×C4#1451982211yescharacteristic
506C2×C4#1451982411yescharacteristic
507C8—22no—
508C8—41no—
509C8—42no—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22C2×#1451974711yescharacteristic
518C22#1684040711yescharacteristic
519C22#1684041611yescharacteristic
520C22C22×Dic24411yescharacteristic
521C22C2×#1451981011yescharacteristic
522C22#1684056711yescharacteristic
523C22#1684057411yescharacteristic
524C4—22no—
525C4—42no—
526C4—4881no—
527C4—4882no—
528C4—2442no—
529C4—2444no—
530C4—2442no—
531C4—2442no—
532C4#1683992014yes—
533C4#1684027211yescharacteristic
534C4#1684034111yescharacteristic
535C4#1684042011yescharacteristic
536C4#1684043211yescharacteristic
537C2—44no—
538C2#2144916211yescharacteristic
539C2C2×#1683992011yescharacteristic
540C2#2145512611yescharacteristic
541C2#2145769611yescharacteristic
542C2#2145815811yescharacteristic
543C2#2145821211yescharacteristic
544C2#2145843511yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1954C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenter#1954D61
hypocenterC61#24122
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1954

Complements

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

NH ≅ G/Ntype
2. C2×#16840058C2semidirect
3. C2×C4×Dic244C2semidirect
5. #21447023C2semidirect
6. #21446829C2semidirect
7. #21446780C2semidirect
8. #21446124C2semidirect
9. #21446009C2semidirect
10. #21445825C2semidirect
108. C2×D4×C61Q16semidirect
317. C61#24122semidirect
396. C2×D4Dic244semidirect

Maximal subgroups

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

subgroupindex
C2×#168404692
C2×#168400582
C2×C4×Dic2442
#214507412
#214470232
#214468292
#214467802
#214461242
#214460092
#214458252
C61×#19542
#2412261