Finite Group Databasebeta

← #29696770

Subgroups — #29696770

Order 15616.

1172 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840458C211yesmaximal, characteristic
2#21448698C212yesmaximal, complemented
3#21446654C212yesmaximal, complemented
4#21446653C214yesmaximal, complemented
5#21445806C212yesmaximal, complemented
6C4×#14519692C211yesmaximal, characteristic, complemented
7#21445606C212yesmaximal, complemented
8C61×#1799C211yesmaximal, characteristic, complemented
9#16840458—22no—
10#16840458—24no—
11#16840458—22no—
12C2×#14519772C412yescomplemented
13C2×#14519769C411yescharacteristic, complemented
14C2×#14519759C412yescomplemented
15C2×#14519751C411yescharacteristic, complemented
16C2×#14519751C2211yescharacteristic
17C2×#14519749C2212yes—
18C2×C4×D244C411yescharacteristic, complemented
19C2×#14519737C2212yes—
20C2×#14519736C411yescharacteristic, complemented
21C22×C4⋊Dic61C2211yescharacteristic
22C2×#14519692C2212yes—
23C2×#14519691C2214yes—
24C2×#14519691C2214yes—
25#16839658—22no—
26#16839647—24no—
27#16839620C2214yes—
28C4⋊C4×Dic61—22no—
29C22⋊C4×Dic61C2214yes—
30C4×C22⋊Dic61C2212yes—
31#16839529C2211yescharacteristic
32#16839529C2211yescharacteristic
33C2×C4×D4×C61C2211yescharacteristic
34C2×#99×C61C2212yes—
35C61×#380C2212yes—
36C61×#342C2212yes—
37C22×D244C2×C411yescharacteristic
38C22×C22⋊D61C2×C412yes—
39C22×C22⋊D61C2312yes—
40C23×Dic61C2312yes—
41C22×C4×D61C2×C412yes—
42#14519772—24no—
43#14519772—24no—
44#14519769—22no—
45#14519769—22no—
46#14519759—24no—
47#14519759—24no—
48#14519751—22no—
49#14519751—22no—
50#14519751—22no—
51#14519751—22no—
52#14519749—24no—
53#14519749—24no—
54C4×D244—22no—
55C4×D244—21no—
56C4×D244—21no—
57#14519737—24no—
58#14519737—24no—
59#14519736—22no—
60#14519736—21no—
61#14519736—21no—
62C2×C4⋊Dic61C2×C411yescharacteristic
63C2×C4⋊Dic61C2×C411yescharacteristic
64C2×C4⋊Dic61C2×C411yescharacteristic
65C2×C4⋊Dic61C2×C412yes—
66C2×C4⋊Dic61D412yes—
67C2×C4⋊Dic61D414yes—
68C2×C4⋊Dic61D412yes—
69C2×#11758196—24no—
70C2×#11758196C2×C412yes—
71C2×#11758196C2×C412yes—
72C2×C22⋊Dic61C2×C412yes—
73C2×C22⋊Dic61C2312yes—
74C2×C22⋊Dic61C2312yes—
75C2×#11758198C2×C412yes—
76C2×#11758198C2×C412yes—
77C2×#11758198C2×C412yes—
78C2×C4×Dic61—22no—
79C2×C4×Dic61—24no—
80C2×C4×Dic61C2×C412yes—
81#14519692—24no—
82#14519691—28no—
83#14519691—44no—
84#14519684—22no—
85#14519684—24no—
86#14519684—24no—
87C22×D4×C61C2311yescharacteristic
88C23×C4×C61C2×C411yescharacteristic
89C23×C4×C61C2311yescharacteristic
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92C2×C4⋊C4×C61—22no—
93C2×C4⋊C4×C61C2×C411yescharacteristic
94C2×C22⋊C4×C61C2×C411yescharacteristic
95C2×C22⋊C4×C61C2314yes—
96C2×C22⋊C4×C61C2311yescharacteristic
97C2×C42×C61C2×C411yescharacteristic
98#99×C61—42no—
99#95×C61—22no—
100C24×C61C22×C411yescharacteristic
101C24×C61C2411yescharacteristic
102C23×D61C22×C412yes—
103C2×D4×C61—22no—
104C2×D4×C61—22no—
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—24no—
110C22×C4×C61—42no—
111C22×C4×C61—42no—
112C22×C4×C61C22×C411yescharacteristic
113C22×C4×C61C22×C411yescharacteristic
114C22×C4×C61C22×C411yescharacteristic
115C22×C4×C61C22×C411yescharacteristic
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×D412yes—
119C2×C22⋊D61—24no—
120C2×C22⋊D61—24no—
121C2×C22⋊D61—22no—
122C2×C22⋊D61—22no—
123C2×C22⋊D61—24no—
124C2×C22⋊D61—82no—
125C22×Dic61—22no—
126C22×Dic61—22no—
127C22×Dic61—82no—
128C22×Dic61C22×C412yes—
129C22×Dic61C22×C412yes—
130C22×Dic61C22×C412yes—
131C22×Dic61C2×D414yes—
132C22×Dic61C2×D414yes—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244—42no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—82no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—44no—
141C22⋊C4×C61—22no—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—24no—
144C22⋊C4×C61—24no—
145C42×C61—21no—
146C42×C61—21no—
147C22⋊Dic61—22no—
148C22⋊Dic61—22no—
149C22⋊Dic61—22no—
150C22⋊Dic61—22no—
151C22⋊Dic61—22no—
152C22⋊Dic61#4114yes—
153#11758198—24no—
154#11758198—24no—
155#11758198—22no—
156#11758198—22no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61C22⋊C412yes—
160C4⋊Dic61C22⋊C411yescharacteristic
161C4⋊Dic61C22⋊C411yescharacteristic
162C4⋊Dic61C22⋊C412yes—
163C4⋊Dic61C22⋊C412yes—
164C4⋊Dic61C22⋊C414yes—
165C4⋊Dic61C22⋊C414yes—
166#11758196—24no—
167#11758196—24no—
168#11758196—84no—
169C4×Dic61—22no—
170C4×Dic61—22no—
171C4×Dic61—44no—
172C4×Dic61—84no—
173C23×C61—21no—
174C23×C61—21no—
175C23×C61—21no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61C23×C411yescharacteristic
181C23×C61C22×D412yes—
182C23×C61C2×#4112yes—
183C23×C61H(5,𝔽2)11yescharacteristic
184C23×C612−1+411yescharacteristic
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—42no—
189C22×D61—42no—
190D4×C61—42no—
191D4×C61—42no—
192C2×C4×C61—24no—
193C2×C4×C61—22no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—41no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—44no—
206C2×C4×C61—44no—
207C2×C4×C61—44no—
208C2×C4×C61C2×C22⋊C411yescharacteristic
209C2×C4×C61C2×C22⋊C412yes—
210C2×C4×C61C2×C22⋊C411yescharacteristic
211C22⋊D61—44no—
212C22⋊D61—44no—
213C22⋊D61—84no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—24no—
218C2×Dic61—44no—
219C2×Dic61—44no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61C2×C22⋊C412yes—
228C2×Dic61C2×C22⋊C412yes—
229C2×Dic61C2×C22⋊C414yes—
230D244—42no—
231D244—42no—
232C4×D61—82no—
233C4×D61—82no—
234#22535—611nomaximal, Sylow
235C22×C61—22no—
236C22×C61—21no—
237C22×C61—22no—
238C22×C61—21no—
239C22×C61—42no—
240C22×C61—41no—
241C22×C61—41no—
242C22×C61—41no—
243C22×C61—41no—
244C22×C61—41no—
245C22×C61—81no—
246C22×C61—81no—
247C22×C61C22×C22⋊C411yescharacteristic
248C22×C61C4×#4111yescharacteristic
249C22×C61#52011yescharacteristic
250C22×C61#54812yes—
251C22×C61#55012yes—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—82no—
256C2×D61—82no—
257C4×C61—21no—
258C4×C61—22no—
259C4×C61—41no—
260C4×C61—41no—
261C4×C61—41no—
262C4×C61—84no—
263C4×C61#41012yes—
264Dic61—22no—
265Dic61—22no—
266Dic61—24no—
267Dic61—44no—
268Dic61—42no—
269Dic61—82no—
270C2×#553—611no—
271#2345—612no—
272#1964—614no—
273#1964—612no—
274#1799—612no—
275#1799D6111yescharacteristic, complemented
276C4×#99—612no—
277C4×#99—611no—
278C2×C61—22no—
279C2×C61—22no—
280C2×C61—21no—
281C2×C61—21no—
282C2×C61—81no—
283C2×C61#234111yescharacteristic
284C2×C61C2×#41011yescharacteristic
285C2×C61#288311yescharacteristic
286D61—82no—
287#553—1222no—
288#553—1224no—
289#553—1222no—
290C2×#126—612no—
291C2×#125—611no—
292C2×#123—612no—
293C2×#121—612no—
294C2×#121—611no—
295C2×#121—611no—
296C2×C4×D4—612no—
297C2×C4×D4—611no—
298C2×C4×D4C2×D6111yescharacteristic
299C2×#117—611no—
300C22×C4⋊C4—611no—
301C2×#99—612no—
302C2×#99—614no—
303C2×#99—614no—
304C2×#99C2×D6112yes—
305#389—1222no—
306#387—1224no—
307#380—614no—
308#380C2×D6112yes—
309C4×C4⋊C4—1222no—
310C4×C22⋊C4—612no—
311C4×C22⋊C4—614no—
312#342—611no—
313#342—611no—
314#342C2×D6112yes—
315C61#2253511yescharacteristic, complemented, Sylow
316C22×D4—612no—
317C22×D4—612no—
318C22×D4—611no—
319C22×D4C22×D6111yescharacteristic
320C23×C4—612no—
321C23×C4—612no—
322C23×C4C4×D6111yescharacteristic
323C23×C4C22×D6111yescharacteristic
324#126—1224no—
325#126—1224no—
326#125—1222no—
327#125—1222no—
328#123—1224no—
329#123—1224no—
330#121—1222no—
331#121—1222no—
332#121—1224no—
333#121—1224no—
334#121—1222no—
335#121—1222no—
336C4×D4—22no—
337C4×D4—22no—
338C4×D4—1222no—
339C4×D4—1221no—
340C4×D4—1221no—
341C4×D4—1224no—
342C4×D4—1224no—
343#117—1222no—
344#117—1221no—
345#117—1221no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—612no—
350C2×C4⋊C4—612no—
351C2×C4⋊C4—614no—
352C2×C4⋊C4—612no—
353C2×C4⋊C4—611no—
354C2×C4⋊C4—612no—
355C2×C4⋊C4—612no—
356C2×C4⋊C4—1224no—
357C2×C4⋊C4C4×D6111yescharacteristic
358C2×C22⋊C4—612no—
359C2×C22⋊C4—612no—
360C2×C22⋊C4—612no—
361C2×C22⋊C4—612no—
362C2×C22⋊C4—612no—
363C2×C22⋊C4—612no—
364C2×C22⋊C4C4×D6111yescharacteristic
365C2×C22⋊C4C22×D6114yes—
366C2×C22⋊C4C22×D6111yescharacteristic
367C2×C42—612no—
368C2×C42—1224no—
369C2×C42—1222no—
370C2×C42C4×D6111yescharacteristic
371#99—42no—
372#99—1228no—
373#99—1224no—
374#99—2444no—
375#95—22no—
376#95—1224no—
377#95—1224no—
378#95—1222no—
379C24—612no—
380C24C2×C4×D6111yescharacteristic
381C24C23×D6111yescharacteristic
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—1224no—
387C2×D4—1224no—
388C2×D4—1224no—
389C2×D4—1222no—
390C2×D4—1222no—
391C2×D4—1221no—
392C2×D4—1221no—
393C2×D4—4882no—
394C2×D4—2442no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—612no—
401C22×C4—612no—
402C22×C4—614no—
403C22×C4—614no—
404C22×C4—612no—
405C22×C4—1222no—
406C22×C4—1222no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4—4882no—
410C22×C4—4882no—
411C22×C4C2×C4×D6111yescharacteristic
412C22×C4C2×C4×D6111yescharacteristic
413C22×C4C2×C4×D6111yescharacteristic
414C22×C4C2×C4×D6111yescharacteristic
415C22×C4C2×C4×D6111yescharacteristic
416C22×C4D4×D6112yes—
417C22×C4D4×D6112yes—
418C4⋊C4—22no—
419C4⋊C4—44no—
420C4⋊C4—612no—
421C4⋊C4—611no—
422C4⋊C4—611no—
423C4⋊C4—612no—
424C4⋊C4—612no—
425C4⋊C4—614no—
426C4⋊C4—614no—
427C4⋊C4—1224no—
428C4⋊C4—1224no—
429C4⋊C4—1221no—
430C4⋊C4—1221no—
431C4⋊C4—4884no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—614no—
437C22⋊C4—1222no—
438C22⋊C4—1222no—
439C22⋊C4—1224no—
440C22⋊C4—1222no—
441C22⋊C4—1222no—
442C22⋊C4—1222no—
443C22⋊C4—1224no—
444C22⋊C4—1222no—
445C22⋊C4—1222no—
446C42—21no—
447C42—21no—
448C42—1222no—
449C42—1222no—
450C42—4884no—
451C42—2444no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—1222no—
460C23—1222no—
461C23—1222no—
462C23—2442no—
463C23—2442no—
464C23C22×C4×D6111yescharacteristic
465C23C2×D4×D6112yes—
466C23#41×D6112yes—
467C23#1451982211yescharacteristic
468C23#1451982511yescharacteristic
469D4—42no—
470D4—42no—
471D4—4884no—
472D4—2442no—
473D4—2444no—
474D4—2444no—
475D4—2442no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—612no—
493C2×C4—612no—
494C2×C4—614no—
495C2×C4—1222no—
496C2×C4—1222no—
497C2×C4—1222no—
498C2×C4—1224no—
499C2×C4—4882no—
500C2×C4—4882no—
501C2×C4—2444no—
502C2×C4—2444no—
503C2×C4—2442no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4C22⋊C4×D6111yescharacteristic
511C2×C4C22⋊C4×D6112yes—
512C2×C4C22⋊C4×D6111yescharacteristic
513C22—22no—
514C22—21no—
515C22—22no—
516C22—21no—
517C22—42no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—81no—
524C22—81no—
525C22—4882no—
526C22—4882no—
527C22—2442no—
528C22—2442no—
529C22—2442no—
530C22C2×C22⋊C4×D6111yescharacteristic
531C22#1684023011yescharacteristic
532C22#1684024311yescharacteristic
533C22#1684042212yes—
534C22#1684043012yes—
535C4—21no—
536C4—22no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—84no—
541C4—1222no—
542C4—1222no—
543C4—1224no—
544C4—4882no—
545C4—2444no—
546C4—2442no—
547C4#1683973412yes—
548C2—22no—
549C2—22no—
550C2—21no—
551C2—21no—
552C2—81no—
553C2—4882no—
554C2#2144865611yescharacteristic
555C2C2×#1683973411yescharacteristic
556C2#2145427811yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1799C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840458C2
cosocleC24×C61C24
hypercenter#1799D61
hypocenterC61#22535
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #21448698C2semidirect
3. #21446654C2semidirect
4. #21446653C2semidirect
5. #21445806C2semidirect
6. C4×#14519692C2semidirect
7. #21445606C2semidirect
8. C61×#1799C2semidirect
12. C2×#14519772C4semidirect
13. C2×#14519769C4semidirect
14. C2×#14519759C4semidirect
15. C2×#14519751C4semidirect
18. C2×C4×D244C4semidirect
20. C2×#14519736C4semidirect
275. #1799D61semidirect
315. C61#22535semidirect

Maximal subgroups

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

subgroupindex
C2×#168404582
#214486982
#214466542
#214466532
#214458062
C4×#145196922
#214456062
C61×#17992
#2253561