Finite Group Databasebeta

← #29713977

Subgroups — #29713977

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×#16840044C211yesmaximal, characteristic, complemented
2C2×C4×#11758202C211yesmaximal, characteristic, complemented
3#21450741C211yesmaximal, characteristic
4#21447012C212yesmaximal, complemented
5#21446827C212yesmaximal, complemented
6#21446781C211yesmaximal, characteristic, complemented
7#21446614C211yesmaximal, characteristic
8#21446131C212yesmaximal, complemented
9#21446018C212yesmaximal, complemented
10#21445824C211yesmaximal, characteristic, complemented
11C2×C61×#556C211yesmaximal, characteristic
12C22×#11758202C2211yescharacteristic
13C2×#14519756C2212yes—
14C2×C4×Dic122C2211yescharacteristic
15#16840044—24no—
16C4×#11758202—24no—
17C2×#14519710C2211yescharacteristic
18C2×#14519708C2212yes—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519701C2211yescharacteristic
21C2×#14519701C2212yes—
22C2×#14519700C2212yes—
23C2×#14519700C2211yescharacteristic
24C2×C22⋊#92418C2212yes—
25C2×C22⋊#92418C2212yes—
26C2×C4×#92418C2211yescharacteristic
27#16839685—22no—
28#16839665—21no—
29#16839658—21no—
30#16839632C2211yescharacteristic
31#16839622C2212yes—
32#16839524C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35C61×#556—24no—
36C61×#556—24no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×#126×C61—22no—
39C2×#124×C61—21no—
40C2×C22⋊Q8×C61C2212yes—
41C2×C4×Q8×C61—21no—
42C2×C4×Q8×C61C2211yescharacteristic
43C2×C4×D4×C61C2211yescharacteristic
44C2×#117×C61C2212yes—
45C22×Dic122C2311yescharacteristic
46C2×#11758202—22no—
47C2×#11758202—22no—
48C2×#11758202—22no—
49C22×#92418C2311yescharacteristic
50C22×#92418C2312yes—
51C22×#92418C2311yescharacteristic
52#14519756—24no—
53#14519756—42no—
54C4×Dic122—22no—
55C4×Dic122—22no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2312yes—
60C2×#11758196—21no—
61C2×#11758196—22no—
62C2×C22⋊Dic61—22no—
63C2×C22⋊Dic61—22no—
64C2×C4×Dic61—21no—
65#14519710—22no—
66#14519708—24no—
67C4⋊#92418—22no—
68#14519701—22no—
69#14519701—42no—
70#14519700—24no—
71#14519700—22no—
72C22⋊#92418—24no—
73C22⋊#92418—42no—
74C4×#92418—22no—
75#14519684—22no—
76C22×Q8×C61—21no—
77C22×Q8×C61C2311yescharacteristic
78C22×D4×C61D411yescharacteristic
79C23×C4×C61C2312yes—
80C4⋊Q8×C61—22no—
81C4⋊Q8×C61D414yes—
82#126×C61—216no—
83#124×C61—28no—
84C22⋊Q8×C61—24no—
85C22⋊Q8×C61—24no—
86C4×Q8×C61—28no—
87C4×Q8×C61—22no—
88C4×Q8×C61D414yes—
89C4×D4×C61—22no—
90C4×D4×C61—22no—
91#117×C61—24no—
92#117×C61—24no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—22no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61D411yescharacteristic
97C2×C4⋊C4×C61C2312yes—
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61—22no—
101C2×C22⋊C4×C61D412yes—
102C2×C42×C61—22no—
103C2×C42×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C24×C61C2×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×C61C2×D412yes—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—22no—
134C2×Dic122—21no—
135C2×Dic122—21no—
136C2×Dic122—21no—
137C2×Dic122—21no—
138C2×Dic122—21no—
139C2×Dic122#4112yes—
140C4⋊C4×C61—28no—
141C4⋊C4×C61—24no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61C2×D412yes—
147C4⋊C4×C61C2×D414yes—
148C22⋊C4×C61—28no—
149C22⋊C4×C61—28no—
150C22⋊C4×C61—24no—
151C42×C61—28no—
152C42×C61—24no—
153C42×C61C2×D412yes—
154C42×C61C2×D412yes—
155C22⋊Dic61—44no—
156C22⋊Dic61—82no—
157#11758202—28no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61—22no—
161C4⋊Dic61—22no—
162C4⋊Dic61—42no—
163C4⋊Dic61—81no—
164#11758196—44no—
165#11758196—42no—
166C4×Dic61—42no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418—21no—
170C2×#92418—22no—
171C2×#92418—22no—
172C2×#92418—21no—
173C2×#92418—22no—
174C2×#92418—21no—
175C2×#92418#4112yes—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61C22×D411yescharacteristic
184Q8×C61—22no—
185Q8×C61—22no—
186Q8×C61—41no—
187Q8×C61—41no—
188Q8×C61—48no—
189D4×C61—44no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—24no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61#12012yes—
209C2×C4×C61#12012yes—
210C2×C4×C61#13512yes—
211C2×C4×C61#13712yes—
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C2×#4111yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61H(5,𝔽2)11yescharacteristic
217C2×C4×C612−1+411yescharacteristic
218C2×Dic61—22no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227Dic122—22no—
228Dic122—22no—
229Dic122—41no—
230Dic122—41no—
231#92418—22no—
232#92418—41no—
233#92418—42no—
234#24123—611nomaximal, Sylow
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61C2×#12011yescharacteristic
243C22×C61#54611yescharacteristic
244C22×C61#54711yescharacteristic
245C22×C61C2×#13511yescharacteristic
246C22×C61C2×#13711yescharacteristic
247C22×C61#57711yescharacteristic
248C22×C61#57811yescharacteristic
249C4×C61—22no—
250C4×C61—42no—
251C4×C61—42no—
252C4×C61—44no—
253C4×C61—42no—
254C4×C61#45214yes—
255C4×C61#52511yescharacteristic
256C4×C61#53711yescharacteristic
257C4×C61#54811yescharacteristic
258C4×C61#55011yescharacteristic
259Dic61—42no—
260Dic61—81no—
261Dic61—82no—
262C2×#556D6111yescharacteristic
263C2×#475—611no—
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—82no—
274C2×C61#240711yescharacteristic
275C2×C61C2×#45211yescharacteristic
276C2×C61#301311yescharacteristic
277C2×C61#330411yescharacteristic
278C2×C61#332811yescharacteristic
279C2×C61#334211yescharacteristic
280C2×C61#335611yescharacteristic
281C22×Q16—611no—
282#556—24no—
283#556—24no—
284C2×C4⋊Q8C2×D6111yescharacteristic
285C2×#126—22no—
286C2×#124—21no—
287C2×C22⋊Q8—612no—
288C2×C22⋊Q8C2×D6112yes—
289C2×C4×Q8—21no—
290C2×C4×Q8—611no—
291C2×C4×Q8C2×D6111yescharacteristic
292C2×C4×D4C2×D6111yescharacteristic
293C2×#117C2×D6112yes—
294#475—1224no—
295C4×Q16—1224no—
296C2×#107—611no—
297C2×#106—612no—
298C2×C4⋊C8—611no—
299C2×#103—611no—
300C2×#103—612no—
301C2×#103—611no—
302C2×#103—612no—
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×D4C22⋊D6111yescharacteristic
319C23×C4C22×D6112yes—
320C2×Q16—1222no—
321C2×Q16—1222no—
322C2×Q16—1222no—
323C22×C8—611no—
324C22×C8—611no—
325C22×C8—612no—
326C4⋊Q8—22no—
327C4⋊Q8C22⋊D6114yes—
328#126—216no—
329#124—28no—
330C22⋊Q8—24no—
331C22⋊Q8—24no—
332C22⋊Q8—1224no—
333C22⋊Q8—2442no—
334C4×Q8—28no—
335C4×Q8—22no—
336C4×Q8—1222no—
337C4×Q8—1222no—
338C4×Q8C22⋊D6114yes—
339C4×D4—22no—
340C4×D4—22no—
341#117—24no—
342#117—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—612no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4—1222no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4C22⋊D6111yescharacteristic
353C2×C4⋊C4C22×D6111yescharacteristic
354C2×C4⋊C4C22×D6112yes—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—1222no—
358C2×C22⋊C4—1222no—
359C2×C22⋊C4C22⋊D6112yes—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42—1221no—
363C2×C42C22×D6111yescharacteristic
364#107—1222no—
365#106—1224no—
366C4⋊C8—1222no—
367#103—1222no—
368#103—1222no—
369#103—1224no—
370#103—2442no—
371C22⋊C8—1224no—
372C22⋊C8—2442no—
373C4×C8—1222no—
374#95—1222no—
375C24C2×C22⋊D6112yes—
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×Q8C2×C22⋊D6112yes—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—1221no—
404C22×C4—1221no—
405C22×C4—1222no—
406C22×C4C2×C22⋊D6112yes—
407C22×C4C2×C22⋊D6111yescharacteristic
408C22×C4C2×C22⋊D6111yescharacteristic
409C22×C4C23×D6111yescharacteristic
410Q16—1228no—
411C2×C8—612no—
412C2×C8—1221no—
413C2×C8—1221no—
414C2×C8—1221no—
415C2×C8—1222no—
416C2×C8—1222no—
417C2×C8—1221no—
418C2×C8—1222no—
419C2×C8—1221no—
420C4⋊C4—28no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—1221no—
427C4⋊C4—1221no—
428C4⋊C4—1222no—
429C4⋊C4—1222no—
430C4⋊C4—4881no—
431C4⋊C4—2444no—
432C4⋊C4—2442no—
433C4⋊C4—2442no—
434C4⋊C4C2×C22⋊D6114yes—
435C4⋊C4C2×C22⋊D6112yes—
436C22⋊C4—28no—
437C22⋊C4—28no—
438C22⋊C4—24no—
439C22⋊C4—4882no—
440C22⋊C4—2444no—
441C42—28no—
442C42—24no—
443C42—2442no—
444C42C2×C22⋊D6112yes—
445C42C2×C22⋊D6112yes—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23C22×C22⋊D6111yescharacteristic
454Q8—22no—
455Q8—22no—
456Q8—41no—
457Q8—41no—
458Q8—48no—
459Q8—1222no—
460Q8—1222no—
461Q8—2441no—
462Q8—2441no—
463D4—44no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—1222no—
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#1451974612yes—
492C2×C4#1451974612yes—
493C2×C4#1451979712yes—
494C2×C4#1451980712yes—
495C2×C4C22×C22⋊D6111yescharacteristic
496C2×C4C22×C22⋊D6111yescharacteristic
497C2×C4#41×D6111yescharacteristic
498C2×C4#41×D6111yescharacteristic
499C2×C4#1451982311yescharacteristic
500C2×C4#1451982411yescharacteristic
501C8—1222no—
502C8—2441no—
503C8—2442no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22C2×#1451974611yescharacteristic
512C22#1684040011yescharacteristic
513C22#1684041111yescharacteristic
514C22C2×#1451979711yescharacteristic
515C22C2×#1451980711yescharacteristic
516C22#1684057011yescharacteristic
517C22#1684057711yescharacteristic
518C4—22no—
519C4—42no—
520C4—44no—
521C4—42no—
522C4—42no—
523C4—4881no—
524C4—4882no—
525C4—2442no—
526C4#1683992414yes—
527C4#1684026911yescharacteristic
528C4#1684034011yescharacteristic
529C4#1684041911yescharacteristic
530C4#1684043111yescharacteristic
531C2—82no—
532C2#2144915811yescharacteristic
533C2C2×#1683992411yescharacteristic
534C2#2145513411yescharacteristic
535C2#2145775211yescharacteristic
536C2#2145809611yescharacteristic
537C2#2145824511yescharacteristic
538C2#2145842811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#556C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenterC2×#556D61
hypocenterC61#24123
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. C2×#16840044C2semidirect
2. C2×C4×#11758202C2semidirect
4. #21447012C2semidirect
5. #21446827C2semidirect
6. #21446781C2semidirect
8. #21446131C2semidirect
9. #21446018C2semidirect
10. #21445824C2semidirect
314. C61#24123semidirect

Maximal subgroups

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

subgroupindex
C2×#168400442
C2×C4×#117582022
#214507412
#214470122
#214468272
#214467812
#214466142
#214461312
#214460182
#214458242
C2×C61×#5562
#2412361