Finite Group Databasebeta

← #29713985

Subgroups — #29713985

Order 15616.

1090 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840486C211yesmaximal, characteristic
2C2×C4×#11758202C211yesmaximal, characteristic, complemented
3#21450737C211yesmaximal, characteristic
4#21447017C212yesmaximal, complemented
5#21446828C212yesmaximal, complemented
6#21446781C211yesmaximal, characteristic, complemented
7#21446612C211yesmaximal, characteristic
8#21446128C212yesmaximal, complemented
9#21446017C212yesmaximal, complemented
10#21445825C211yesmaximal, characteristic, complemented
11C2×C61×#475C211yesmaximal, characteristic, complemented
12C22×#11758202C2211yescharacteristic
13#16840486—24no—
14#16840486—24no—
15C2×#14519772—22no—
16C2×#14519766—21no—
17C2×#14519757C2212yes—
18C2×#14519753C2212yes—
19C2×Q8×Dic61—21no—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×#14519735C2212yes—
23C4×#11758202—24no—
24C2×#14519710C2211yescharacteristic
25C2×#14519702C2212yes—
26C2×#14519701C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519689C2212yes—
29C2×#14519689C2212yes—
30C2×C4×#92418C2211yescharacteristic
31#16839684—22no—
32#16839664—21no—
33#16839660—21no—
34#16839632C2211yescharacteristic
35#16839619C2212yes—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4×Q8×C61C2211yescharacteristic
41C61×#475—24no—
42C2×#106×C61C2212yes—
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#103×C61C2212yes—
45C22×Dic122C2311yescharacteristic
46C22×D244D411yescharacteristic
47C22×C4×D61C2312yes—
48C2×#11758202—22no—
49C2×#11758202—22no—
50C2×#11758202—22no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2311yescharacteristic
53#14519772—216no—
54#14519766—28no—
55#14519757—24no—
56#14519757—24no—
57#14519753—24no—
58#14519753—42no—
59Q8×Dic61—28no—
60C4×Dic122—22no—
61C4×Dic122D414yes—
62C4×D244—22no—
63C4×D244—22no—
64#14519735—24no—
65#14519735—24no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61D411yescharacteristic
70C2×C4⋊Dic61C2312yes—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—22no—
74C2×#11758198—22no—
75C2×#11758198—22no—
76C2×#11758198—22no—
77C2×#11758198—22no—
78C2×#11758198D412yes—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—22no—
81#14519710—22no—
82#14519702—42no—
83#14519701—22no—
84#14519700—22no—
85#14519689—24no—
86#14519689—42no—
87C4×#92418—22no—
88#14519684—22no—
89C22×Q8×C61—21no—
90C22×Q8×C61C2311yescharacteristic
91C22×C8×C61C2312yes—
92C4⋊Q8×C61—22no—
93C4⋊Q8×C61D414yes—
94C4×Q8×C61—22no—
95C4×Q8×C61—22no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C4⋊C4×C61C2312yes—
100C2×C42×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#106×C61—24no—
103C4⋊C8×C61—22no—
104#103×C61—24no—
105C23×D61C2×D412yes—
106C2×Q8×C61—24no—
107C2×Q8×C61—21no—
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—42no—
113C2×Q8×C61—42no—
114C2×Q8×C61#4112yes—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2411yescharacteristic
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61—22no—
124C22×Dic61C2×D412yes—
125C2×D244—22no—
126C2×D244—22no—
127C2×D244—22no—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×Dic122—21no—
135C2×Dic122—21no—
136C2×Dic122—21no—
137C2×Dic122—21no—
138C2×Dic122—21no—
139C2×Dic122C2×D412yes—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C4⋊C4×C61—24no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—42no—
148C4⋊C4×C61C2×D414yes—
149C42×C61—42no—
150C42×C61C2×D412yes—
151C42×C61C2×D412yes—
152#11758202—28no—
153#11758198—28no—
154#11758198—28no—
155#11758198—24no—
156#11758198—44no—
157#11758198—82no—
158C4⋊Dic61—28no—
159C4⋊Dic61—22no—
160C4⋊Dic61—21no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61—42no—
164C4⋊Dic61—81no—
165C4⋊Dic61C2×D412yes—
166#11758196—24no—
167#11758196—44no—
168C4×Dic61—28no—
169C4×Dic61—24no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418#4112yes—
176C23×C61C22×D411yescharacteristic
177C22×D61—22no—
178C22×D61—22no—
179C22×D61—22no—
180C22×D61—22no—
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—22no—
184Q8×C61—22no—
185Q8×C61—22no—
186Q8×C61—41no—
187Q8×C61—41no—
188Q8×C61—48no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—24no—
192C2×C4×C61—22no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61#12012yes—
201C2×C4×C61#12012yes—
202C2×C4×C61#13512yes—
203C2×C4×C61#13712yes—
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×C4×C61H(5,𝔽2)11yescharacteristic
209C2×C4×C612−1+411yescharacteristic
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222D244—44no—
223C4×D61—42no—
224C4×D61—42no—
225C4×D61—42no—
226C4×D61—42no—
227Dic122—22no—
228Dic122—22no—
229Dic122—41no—
230Dic122—41no—
231C8×C61—42no—
232#92418—22no—
233#92418—41no—
234#24123—611nomaximal, Sylow
235C22×C61C2×#12011yescharacteristic
236C22×C61#54611yescharacteristic
237C22×C61#54711yescharacteristic
238C22×C61C2×#13511yescharacteristic
239C22×C61C2×#13711yescharacteristic
240C22×C61#57711yescharacteristic
241C22×C61#57811yescharacteristic
242C2×D61—42no—
243C2×D61—42no—
244C2×D61—42no—
245C2×D61—42no—
246C2×D61—42no—
247C2×D61—42no—
248C2×D61—42no—
249C4×C61—22no—
250C4×C61—42no—
251C4×C61—44no—
252C4×C61—81no—
253C4×C61#45214yes—
254C4×C61#52511yescharacteristic
255C4×C61#53711yescharacteristic
256C4×C61#54811yescharacteristic
257C4×C61#55011yescharacteristic
258Dic61—42no—
259Dic61—42no—
260Dic61—42no—
261Dic61—82no—
262C2×#556—611no—
263C2×#475D6111yescharacteristic, complemented
264C2×C4×Q16—611no—
265#2538—611no—
266#2035—612no—
267#2001—612no—
268#1990—611no—
269#1954—611no—
270#1873—612no—
271#1854—612no—
272#1803—611no—
273C2×C61#240711yescharacteristic
274C2×C61C2×#45211yescharacteristic
275C2×C61#301311yescharacteristic
276C2×C61#330411yescharacteristic
277C2×C61#332811yescharacteristic
278C2×C61#334211yescharacteristic
279C2×C61#335611yescharacteristic
280D61—82no—
281C22×Q16—611no—
282#556—1224no—
283#556—1224no—
284C2×C4⋊Q8C2×D6111yescharacteristic
285C2×#126—1222no—
286C2×#124—1221no—
287C2×C22⋊Q8—612no—
288C2×C22⋊Q8—612no—
289C2×C4×Q8—611no—
290C2×C4×Q8—1221no—
291C2×C4×Q8C2×D6111yescharacteristic
292C2×C4×D4—611no—
293C2×#117—612no—
294#475—24no—
295C4×Q16—1224no—
296C2×#107—611no—
297C2×#106C2×D6112yes—
298C2×C4⋊C8C2×D6111yescharacteristic
299C2×#103—611no—
300C2×#103—611no—
301C2×#103—612no—
302C2×#103C2×D6112yes—
303C2×C22⋊C8—612no—
304C2×C22⋊C8—612no—
305C2×C4×C8—611no—
306#396—1222no—
307#390—1221no—
308#389—1221no—
309#384—611no—
310#380—612no—
311#340—612no—
312#340—612no—
313#340—612no—
314C61#2412311yescharacteristic, complemented, Sylow
315C22×Q8—21no—
316C22×Q8—611no—
317C22×Q8C22×D6111yescharacteristic
318C22×D4—611no—
319C23×C4—612no—
320C2×Q16—1222no—
321C2×Q16—1222no—
322C2×Q16—1222no—
323C22×C8—611no—
324C22×C8—611no—
325C22×C8C22×D6112yes—
326C4⋊Q8—22no—
327C4⋊Q8C22⋊D6114yes—
328#126—12216no—
329#124—1228no—
330C22⋊Q8—1224no—
331C22⋊Q8—1224no—
332C22⋊Q8—1224no—
333C22⋊Q8—2442no—
334C4×Q8—22no—
335C4×Q8—22no—
336C4×Q8—614no—
337C4×Q8—1228no—
338C4×Q8—1222no—
339C4×D4—1222no—
340C4×D4—1222no—
341#117—1224no—
342#117—1224no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—611no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—612no—
348C2×C4⋊C4—1222no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—1222no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4C22×D6111yescharacteristic
354C2×C4⋊C4C22×D6112yes—
355C2×C22⋊C4—612no—
356C2×C22⋊C4—1222no—
357C2×C22⋊C4—1222no—
358C2×C22⋊C4—1222no—
359C2×C22⋊C4—1222no—
360C2×C42—21no—
361C2×C42—1222no—
362C2×C42—1221no—
363C2×C42C22×D6111yescharacteristic
364#107—1222no—
365#106—24no—
366C4⋊C8—22no—
367#103—24no—
368#103—1222no—
369#103—1222no—
370#103—2442no—
371C22⋊C8—1224no—
372C22⋊C8—2442no—
373C4×C8—1222no—
374#95—1222no—
375C24—612no—
376C2×Q8—21no—
377C2×Q8—24no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—612no—
385C2×Q8—1221no—
386C2×Q8—1221no—
387C2×Q8—1221no—
388C2×Q8—1221no—
389C2×Q8—1221no—
390C2×Q8#1175822612yes—
391C2×D4—1222no—
392C2×D4—1222no—
393C2×D4—1222no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—612no—
398C22×C4—1221no—
399C22×C4—1221no—
400C22×C4—1222no—
401C22×C4—1222no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1222no—
407C22×C4D4×D6111yescharacteristic
408C22×C4D4×D6111yescharacteristic
409C22×C4C23×D6111yescharacteristic
410Q16—1228no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—612no—
415C2×C8—1221no—
416C2×C8—1221no—
417C2×C8—1221no—
418C2×C8—1221no—
419C2×C8—1221no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—42no—
425C4⋊C4—612no—
426C4⋊C4—1228no—
427C4⋊C4—1221no—
428C4⋊C4—1222no—
429C4⋊C4—1224no—
430C4⋊C4—1222no—
431C4⋊C4—1222no—
432C4⋊C4—4881no—
433C4⋊C4—2444no—
434C4⋊C4—2442no—
435C4⋊C4C2×C22⋊D6114yes—
436C22⋊C4—1228no—
437C22⋊C4—1228no—
438C22⋊C4—1224no—
439C22⋊C4—4882no—
440C22⋊C4—2444no—
441C42—42no—
442C42—1228no—
443C42—1224no—
444C42D4×D6112yes—
445C42C2×C22⋊D6112yes—
446C23—1222no—
447C23—1222no—
448C23—1222no—
449C23—1222no—
450C23—1222no—
451C23—1222no—
452C23—1222no—
453C23C2×D4×D6111yescharacteristic
454Q8—22no—
455Q8—22no—
456Q8—41no—
457Q8—41no—
458Q8—48no—
459Q8—1222no—
460Q8—1222no—
461Q8—2441no—
462Q8—2441no—
463D4—2444no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—1222no—
476C2×C4—1222no—
477C2×C4—1222no—
478C2×C4—1222no—
479C2×C4—1222no—
480C2×C4—2442no—
481C2×C4—2442no—
482C2×C4—2442no—
483C2×C4—2441no—
484C2×C4—2441no—
485C2×C4—2441no—
486C2×C4—2441no—
487C2×C4—2442no—
488C2×C4—2442no—
489C2×C4—2442no—
490C2×C4—2442no—
491C2×C4#1451974512yes—
492C2×C4#1451974512yes—
493C2×C4#1451979812yes—
494C2×C4#1451980612yes—
495C2×C4C2×D4×D6111yescharacteristic
496C2×C4C22×C22⋊D6111yescharacteristic
497C2×C4#41×D6111yescharacteristic
498C2×C4C2×#1175822611yescharacteristic
499C2×C4#1451982311yescharacteristic
500C2×C4#1451982511yescharacteristic
501C8—42no—
502C8—1222no—
503C8—2441no—
504C22—2442no—
505C22—2442no—
506C22—2442no—
507C22—2442no—
508C22—2442no—
509C22—2442no—
510C22—2442no—
511C22C2×#1451974511yescharacteristic
512C22#1684040311yescharacteristic
513C22#1684041411yescharacteristic
514C22C2×#1451979811yescharacteristic
515C22C2×#1451980611yescharacteristic
516C22#1684057111yescharacteristic
517C22#1684057811yescharacteristic
518C4—22no—
519C4—44no—
520C4—42no—
521C4—81no—
522C4—4882no—
523C4—2442no—
524C4—2442no—
525C4—2442no—
526C4#1683992114yes—
527C4#1684027511yescharacteristic
528C4#1684034211yescharacteristic
529C4#1684042311yescharacteristic
530C4#1684043111yescharacteristic
531C2—4882no—
532C2#2144916611yescharacteristic
533C2C2×#1683992111yescharacteristic
534C2#2145513311yescharacteristic
535C2#2145775511yescharacteristic
536C2#2145809911yescharacteristic
537C2#2145824811yescharacteristic
538C2#2145843111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#475C2
socleC23×C61C22×D4
prime radicalC22×D244D4
hypercenterC2×#475D61
hypocenterC61#24123
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11758202C2semidirect
4. #21447017C2semidirect
5. #21446828C2semidirect
6. #21446781C2semidirect
8. #21446128C2semidirect
9. #21446017C2semidirect
10. #21445825C2semidirect
11. C2×C61×#475C2semidirect
263. C2×#475D61semidirect
314. C61#24123semidirect

Maximal subgroups

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

subgroupindex
C2×#168404862
C2×C4×#117582022
#214507372
#214470172
#214468282
#214467812
#214466122
#214461282
#214460172
#214458252
C2×C61×#4752
#2412361