Finite Group Databasebeta

← #29719006

Subgroups — #29719006

Order 15616.

1182 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839912C211yesmaximal, characteristic
2C22×#14519699C211yesmaximal, characteristic
3#21450305C211yesmaximal, characteristic, complemented
4#21448639C211yesmaximal, characteristic
5#21447236C212yesmaximal, complemented
6#21447133C212yesmaximal, complemented
7#21446150C212yesmaximal, complemented
8#21445997C211yesmaximal, characteristic, complemented
9#21445903C211yesmaximal, characteristic, complemented
10#21445828C212yesmaximal, complemented
11C61×#1892C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C22×#11758201C2211yescharacteristic
14C22×#11758201C2211yescharacteristic
15C2×#14519739C2212yes—
16C22×C4⋊Dic61C2211yescharacteristic
17C2×C22⋊C4×D61—21no—
18#16839912—24no—
19C2×#14519706C2212yescomplemented
20C2×#14519699—24no—
21C2×#14519699C2212yes—
22C2×#14519699C2212yes—
23C2×#14519697C2211yescharacteristic
24C2×#14519697C2211yescharacteristic
25C2×C22⋊#92418C2211yescharacteristic
26#16839701—22no—
27#16839669C2212yescomplemented
28#16839663C2212yes—
29#16839640—22no—
30#16839627C2212yescomplemented
31#16839623—21no—
32#16839623—21no—
33#16839617—21no—
34C4×C22⋊Dic61C2211yescharacteristic
35#16839524C2212yescomplemented
36#16839524C2212yescomplemented
37#16839516C2212yescomplemented
38C23×C8×C61C2211yescharacteristic
39C2×C22⋊Q8×C61C2211yescharacteristic
40C2×#103×C61C2212yescomplemented
41C61×#380C2211yescharacteristic
42C61×#340C2212yescomplemented
43C24×D61—21no—
44C22×D244—24no—
45C22×D244—21no—
46C22×D244—22no—
47C22×D244—42no—
48C22×D244D414yes—
49C22×D244C2311yescharacteristic
50C22×D244C2311yescharacteristic
51C23×Dic61—21no—
52C22×C4×D61—22no—
53C2×#11758201—22no—
54C2×#11758201—22no—
55C2×#11758201—24no—
56C2×#11758201—42no—
57C22×#92418C2311yescharacteristic
58C22×#92418C2311yescharacteristic
59#14519739—24no—
60#14519739—42no—
61C2×C4⋊Dic61—22no—
62C2×C4⋊Dic61—22no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61C2312yes—
65C2×C22⋊Dic61D411yescharacteristic
66C2×C22⋊Dic61D411yescharacteristic
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C22⋊C4×D61—44no—
71C2×C4×Dic61C2312yes—
72#14519706—42no—
73#14519699—24no—
74#14519699—24no—
75#14519697—22no—
76#14519697—22no—
77C22⋊#92418—22no—
78#14519684—22no—
79#14519684—22no—
80#14519684—22no—
81#14519684—22no—
82#14519684D412yes—
83C22×Q8×C61C2311yescharacteristic
84C23×C4×C61C2311yescharacteristic
85C22×C8×C61—22no—
86C22×C8×C61—42no—
87C22×C8×C61C2312yes—
88C22⋊Q8×C61—22no—
89C22⋊Q8×C61—22no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61C2312yes—
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C22⋊C4×C61—21no—
94C2×C22⋊C4×C61—21no—
95#103×C61—42no—
96#95×C61—21no—
97C24×C61C2×D411yescharacteristic
98C23×D61—21no—
99C23×D61—21no—
100C23×D61—24no—
101C23×D61—44no—
102C23×D61—42no—
103C23×D61—42no—
104C23×D61—82no—
105C2×Q8×C61—21no—
106C2×Q8×C61—21no—
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D412yes—
116C22×C4×C61#4112yes—
117C22×C4×C61C2411yescharacteristic
118C22×Dic61—22no—
119C22×Dic61—42no—
120C22×Dic61—42no—
121C22×Dic61—42no—
122C22×Dic61C2×D412yes—
123C22×Dic61C2×D412yes—
124C2×D244—21no—
125C2×D244—22no—
126C2×D244—22no—
127C2×D244—22no—
128C2×D244—28no—
129C2×D244—24no—
130C2×D244—21no—
131C2×D244—44no—
132C2×D244—48no—
133C2×D244—44no—
134C2×D244C2×D412yes—
135C2×D244C2×D412yes—
136C2×D244#4114yes—
137C2×C4×D61—44no—
138C2×C4×D61—44no—
139C2×C4×D61—82no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C2×C8×C61—44no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—42no—
149C22⋊C4×C61—42no—
150C22⋊C4×C61—42no—
151C22⋊Dic61—22no—
152C22⋊Dic61—22no—
153#11758201—44no—
154#11758201—44no—
155#11758198—44no—
156#11758198—44no—
157#11758198—82no—
158C4⋊Dic61—22no—
159C4⋊Dic61—22no—
160C4⋊Dic61—42no—
161C4×Dic61—22no—
162C4×Dic61—22no—
163C2×#92418—21no—
164C2×#92418—21no—
165C2×#92418—22no—
166C2×#92418—22no—
167C23×C61—22no—
168C23×C61—22no—
169C23×C61#12112yes—
170C23×C61#13512yes—
171C23×C61AGL(1,ℤ/8)12yes—
172C23×C61C22×D411yescharacteristic
173C22×D61—22no—
174C22×D61—24no—
175C22×D61—22no—
176C22×D61—48no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—42no—
180C22×D61—42no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—42no—
184C22×D61—48no—
185C22×D61—84no—
186C22×D61—88no—
187C22×D61—88no—
188Q8×C61—41no—
189Q8×C61—41no—
190Q8×C61—42no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—24no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61#12112yes—
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C2×#4112yes—
205C2×C4×C61C2×#4111yescharacteristic
206C2×C4×C61H(5,𝔽2)12yes—
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—22no—
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—84no—
217D244—22no—
218D244—22no—
219D244—22no—
220D244—24no—
221D244—48no—
222D244—42no—
223D244#12314yes—
224C4×D61—88no—
225C8×C61—42no—
226#92418—41no—
227#92418—41no—
228#24527—611nomaximal, Sylow
229C22×C61—22no—
230C22×C61—24no—
231C22×C61—22no—
232C22×C61—24no—
233C22×C61#46314yes—
234C22×C61C2×#12111yescharacteristic
235C22×C61#54612yes—
236C22×C61C2×#13511yescharacteristic
237C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
238C22×C61#57712yes—
239C2×D61—24no—
240C2×D61—42no—
241C2×D61—44no—
242C2×D61—42no—
243C2×D61—44no—
244C2×D61—42no—
245C2×D61—42no—
246C2×D61—42no—
247C2×D61—816no—
248C2×D61—82no—
249C2×D61—82no—
250C4×C61—22no—
251C4×C61—81no—
252C4×C61—81no—
253C4×C61—82no—
254C4×C61C2×#12311yescharacteristic
255C4×C61#53411yescharacteristic
256C4×C61#55112yes—
257Dic61—42no—
258Dic61—42no—
259Dic61—82no—
260C2×#450—611no—
261C22×#102—611no—
262#2503—611no—
263D4×C22⋊C4—611no—
264#2073—612no—
265#2055—612no—
266#1892D6111yescharacteristic, complemented
267#1877—612no—
268#1850—611no—
269#1823—611no—
270#1804—612no—
271C2×C61—24no—
272C2×C61#241211yescharacteristic
273C2×C61C2×#46311yescharacteristic
274C2×C61#304211yescharacteristic
275C2×C61#329612yes—
276C2×C61#331012yes—
277D61—44no—
278D61—82no—
279C23×D4—611no—
280C22×SD16—611no—
281C22×SD16—611no—
282C23×C8C2×D6111yescharacteristic
283C2×C22⋊Q8C2×D6111yescharacteristic
284C2×C4×D4—612no—
285C22×C4⋊C4—611no—
286C22×C22⋊C4—1221no—
287#450—1224no—
288C2×C4⋊C8—612no—
289C2×#103C2×D6112yescomplemented
290C2×#102—612no—
291C2×#102—612no—
292C2×#102—611no—
293C2×#102—611no—
294C2×#102—1224no—
295C2×C22⋊C8—611no—
296#400—1222no—
297#391—612no—
298#390—612no—
299#386—1222no—
300#382—612no—
301#381—1221no—
302#381—1221no—
303#380C2×D6111yescharacteristic
304#379—1221no—
305C4×C22⋊C4—611no—
306#340—612no—
307#340—612no—
308#340C2×D6112yescomplemented
309#336—612no—
310C61#2452711yescharacteristic, complemented, Sylow
311C25—1221no—
312C22×Q8C22×D6111yescharacteristic
313C22×D4—614no—
314C22×D4—611no—
315C22×D4—611no—
316C22×D4—1224no—
317C22×D4—1222no—
318C22×D4—1221no—
319C22×D4—2442no—
320C23×C4—1222no—
321C23×C4—1221no—
322C23×C4C22×D6111yescharacteristic
323C2×SD16—1222no—
324C2×SD16—1222no—
325C2×SD16—1224no—
326C2×SD16—2442no—
327C22×C8—22no—
328C22×C8—42no—
329C22×C8—611no—
330C22×C8—611no—
331C22×C8C22×D6112yes—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C4×D4—1224no—
335C4×D4—2442no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—1222no—
339C2×C4⋊C4—1222no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4C22×D6112yes—
342C2×C4⋊C4C22×D6111yescharacteristic
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—611no—
346C2×C22⋊C4—611no—
347C2×C22⋊C4—1222no—
348C2×C22⋊C4—1222no—
349C2×C22⋊C4—1222no—
350C2×C22⋊C4—2444no—
351C2×C42—612no—
352C4⋊C8—2442no—
353#103—42no—
354#102—1222no—
355#102—1222no—
356#102—1224no—
357#102—1224no—
358C22⋊C8—1222no—
359#95—21no—
360#95—612no—
361#95—1222no—
362#95—1222no—
363#95—1222no—
364#95—1222no—
365C24—1221no—
366C24—1224no—
367C24—1221no—
368C24—4882no—
369C24—2444no—
370C24—2442no—
371C24—2442no—
372C24D4×D6111yescharacteristic
373C2×Q8—21no—
374C2×Q8—22no—
375C2×Q8—21no—
376C2×Q8—22no—
377C2×D4—612no—
378C2×D4—612no—
379C2×D4—614no—
380C2×D4—1228no—
381C2×D4—1222no—
382C2×D4—1221no—
383C2×D4—1221no—
384C2×D4—1222no—
385C2×D4—1222no—
386C2×D4—1224no—
387C2×D4—2444no—
388C2×D4—2448no—
389C2×D4—2444no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—612no—
396C22×C4—612no—
397C22×C4—1222no—
398C22×C4—4882no—
399C22×C4—2444no—
400C22×C4—2444no—
401C22×C4—2442no—
402C22×C4—2442no—
403C22×C4—2442no—
404C22×C4#1175822212yes—
405C22×C4D4×D6112yes—
406C22×C4D4×D6111yescharacteristic
407C22×C4C23×D6111yescharacteristic
408SD16—2444no—
409SD16—2444no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—44no—
414C2×C8—1221no—
415C2×C8—1221no—
416C2×C8—1222no—
417C2×C8—1222no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—42no—
423C4⋊C4—1222no—
424C4⋊C4—1222no—
425C4⋊C4—2442no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—1222no—
429C22⋊C4—1222no—
430C22⋊C4—4882no—
431C22⋊C4—2444no—
432C22⋊C4—2444no—
433C42—1222no—
434C42—1222no—
435C23—22no—
436C23—22no—
437C23—1222no—
438C23—1224no—
439C23—1222no—
440C23—4888no—
441C23—4888no—
442C23—4884no—
443C23—2448no—
444C23—2448no—
445C23—2441no—
446C23—2441no—
447C23—2442no—
448C23—2442no—
449C23—2441no—
450C23—2441no—
451C23—2442no—
452C23#1451975012yes—
453C23#1451980012yes—
454C23#1451980412yes—
455C23C2×D4×D6111yescharacteristic
456Q8—41no—
457Q8—41no—
458Q8—42no—
459D4—614no—
460D4—1222no—
461D4—1222no—
462D4—1222no—
463D4—1224no—
464D4—2448no—
465D4—2442no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—1222no—
478C2×C4—1222no—
479C2×C4—1222no—
480C2×C4—1222no—
481C2×C4—1222no—
482C2×C4—1222no—
483C2×C4—4888no—
484C2×C4—4884no—
485C2×C4—2442no—
486C2×C4—2442no—
487C2×C4—2442no—
488C2×C4#1451975012yes—
489C2×C4C2×D4×D6111yescharacteristic
490C2×C4#41×D6112yes—
491C2×C4C2×#1175822211yescharacteristic
492C2×C4#1451982212yes—
493C8—42no—
494C8—2441no—
495C8—2441no—
496C22—22no—
497C22—24no—
498C22—22no—
499C22—24no—
500C22—1224no—
501C22—48816no—
502C22—4882no—
503C22—4882no—
504C22—2442no—
505C22—2442no—
506C22—2442no—
507C22—2442no—
508C22—2444no—
509C22—2442no—
510C22—2444no—
511C22#1683998014yes—
512C22C2×#1451975011yescharacteristic
513C22#1684040112yes—
514C22C2×#1451980011yescharacteristic
515C22C2×#1451980411yescharacteristic
516C22#1684056912yes—
517C4—22no—
518C4—81no—
519C4—82no—
520C4—81no—
521C4—4882no—
522C4—2442no—
523C4—2442no—
524C4#123×D6111yescharacteristic
525C4#1684032211yescharacteristic
526C4#1684044112yes—
527C2—24no—
528C2—4882no—
529C2—2444no—
530C2#2144921811yescharacteristic
531C2C2×#1683998011yescharacteristic
532C2#2145533411yescharacteristic
533C2#2145766012yes—
534C2#2145784612yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1892C2
socleC23×C61C22×D4
prime radicalC23×D244C22
hypercenter#1892D61
hypocenterC61#24527
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
3. #21450305C2semidirect
5. #21447236C2semidirect
6. #21447133C2semidirect
7. #21446150C2semidirect
8. #21445997C2semidirect
9. #21445903C2semidirect
10. #21445828C2semidirect
11. C61×#1892C2semidirect
19. C2×#14519706C22semidirect
27. #16839669C22semidirect
30. #16839627C22semidirect
35. #16839524C22semidirect
36. #16839524C22semidirect
37. #16839516C22semidirect
40. C2×#103×C61C22semidirect
42. C61×#340C22semidirect
266. #1892D61semidirect
289. C2×#103C2×D61semidirect
308. #340C2×D61semidirect
310. C61#24527semidirect

Maximal subgroups

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

subgroupindex
C2×#168399122
C22×#145196992
#214503052
#214486392
#214472362
#214471332
#214461502
#214459972
#214459032
#214458282
C61×#18922
#2452761