Finite Group Databasebeta

← #29779834

Subgroups — #29779834

Order 15616.

978 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454215C211yesmaximal, characteristic, complemented
2#21448383C211yesmaximal, characteristic
3#21446956C211yesmaximal, characteristic, complemented
4#21446955C211yesmaximal, characteristic, complemented
5#21446952C211yesmaximal, characteristic, complemented
6#21446623C212yesmaximal
7#21445945C212yesmaximal, complemented
8#21445818C212yesmaximal, complemented
9#21445817C212yesmaximal, complemented
10#21445667C211yesmaximal, characteristic, complemented
11C61×#2023C211yesmaximal, characteristic, complemented
12C2×#14519777C2211yescharacteristic
13C2×#14519775—21no—
14C2×#14519775—21no—
15C2×#14519773C2211yescharacteristic
16C2×D4×Dic61C2211yescharacteristic
17C2×#14519739C2211yescharacteristic
18C2×#14519737—21no—
19C2×#14519737—21no—
20C2×#14519706C2212yes—
21C2×#14519699C2212yes—
22C2×#14519698C2212yes—
23C2×#14519697C2212yes—
24C2×C8×Dic61C2212yes—
25#16839664C2211yescharacteristic
26#16839662—21no—
27#16839662—21no—
28#16839631—21no—
29#16839631—21no—
30#16839631C2211yescharacteristic
31#16839631C2212yes—
32#16839631C2212yes—
33C42×Dic61C2211yescharacteristic
34#16839525C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839517C2211yescharacteristic
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×#102×C61C2212yes—
40C61×#390C2211yescharacteristic
41C61×#340C2212yes—
42C61×#336C2211yescharacteristic
43C22×Dic122—21no—
44C22×D244C2311yescharacteristic
45C22×C22⋊D61—21no—
46C22×C22⋊D61—21no—
47C23×Dic61—21no—
48C22×C4×D61—21no—
49C22×#92418C2312yes—
50#14519777—42no—
51#14519775—44no—
52#14519775—44no—
53#14519773—22no—
54#14519773Q814yescomplemented
55D4×Dic61—24no—
56#14519739—24no—
57#14519737—44no—
58#14519737—44no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61C2311yescharacteristic
61C2×C4⋊Dic61C2312yes—
62C2×#11758196—21no—
63C2×#11758196—21no—
64C2×#11758196—22no—
65C2×#11758196—22no—
66C2×#11758196—22no—
67C2×#11758196—22no—
68C2×C22⋊Dic61—21no—
69C2×#11758198—21no—
70C2×#11758198—21no—
71C2×#11758198—21no—
72C2×C4×Dic61D411yescharacteristic
73C2×C4×Dic61D411yescharacteristic
74C2×C4×Dic61D411yescharacteristic
75C2×C4×Dic61D411yescharacteristic
76C2×C4×Dic61C2311yescharacteristic
77C2×C4×Dic61C2311yescharacteristic
78#14519706—24no—
79#14519699—42no—
80#14519698—42no—
81#14519697—42no—
82C8×Dic61—24no—
83C22×Q8×C61—21no—
84C22×D4×C61C2311yescharacteristic
85C22×C8×C61C2312yes—
86C4⋊Q8×C61—42no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61C2312yes—
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C22⋊C4×C61—21no—
91C2×C22⋊C4×C61—21no—
92C2×C42×C61C2311yescharacteristic
93#102×C61—42no—
94C24×C61—21no—
95C23×D61—21no—
96C2×Q8×C61—42no—
97C2×Q8×C61—42no—
98C2×Q8×C61—42no—
99C2×D4×C61—22no—
100C2×D4×C61—21no—
101C2×D4×C61—22no—
102C2×D4×C61C2×Q812yes—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2411yescharacteristic
108C2×C22⋊D61—24no—
109C2×C22⋊D61—24no—
110C2×C22⋊D61—42no—
111C2×C22⋊D61—42no—
112C2×C22⋊D61—42no—
113C2×C22⋊D61—42no—
114C22×Dic61—21no—
115C22×Dic61—22no—
116C22×Dic61—42no—
117C22×Dic61—42no—
118C22×Dic61—42no—
119C22×Dic61C2×D411yescharacteristic
120C22×Dic61C2×D411yescharacteristic
121C22×Dic61C2×D411yescharacteristic
122C22×Dic61C2×D411yescharacteristic
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—21no—
126C2×D244C2×Q812yes—
127C2×C4×D61—42no—
128C2×C4×D61—42no—
129C2×C4×D61—42no—
130C2×Dic122—42no—
131C2×Dic122—42no—
132C2×Dic122—42no—
133C2×C8×C61—22no—
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—42no—
140C22⋊C4×C61—42no—
141C22⋊C4×C61—42no—
142C42×C61—22no—
143C22⋊Dic61—42no—
144#11758198—42no—
145#11758198—42no—
146#11758198—42no—
147C4⋊Dic61—22no—
148C4⋊Dic61—22no—
149C4⋊Dic61—22no—
150C4⋊Dic61—42no—
151#11758196—44no—
152#11758196—42no—
153#11758196—44no—
154#11758196—44no—
155#11758196—42no—
156#11758196—44no—
157C4×Dic61—22no—
158C4×Dic61—22no—
159C4×Dic61D814yes—
160C4×Dic61SD1614yes—
161C4×Dic61C2×Q812yes—
162C4×Dic61#4112yes—
163C4×Dic61#4114yes—
164C2×#92418—22no—
165C2×#92418—22no—
166C2×#92418—22no—
167C23×C61—22no—
168C23×C61—41no—
169C23×C61—41no—
170C23×C61—41no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61—41no—
174C23×C61C22×D411yescharacteristic
175C22×D61—22no—
176C22×D61—41no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—41no—
182Q8×C61—84no—
183D4×C61—24no—
184D4×C61—42no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—41no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—42no—
195C2×C4×C61C2×D812yes—
196C2×C4×C61C2×SD1612yes—
197C2×C4×C61C22×Q811yescharacteristic
198C2×C4×C61C2×#4112yes—
199C2×C4×C61C2×#4111yescharacteristic
200C2×C4×C61H(5,𝔽2)12yes—
201C22⋊D61—48no—
202C22⋊D61—48no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—41no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—42no—
211C2×Dic61—84no—
212C2×Dic61C22⋊Q812yes—
213C2×Dic61C22⋊Q812yes—
214C2×Dic61#12412yes—
215C2×Dic61#12412yes—
216C2×Dic61C2×D812yes—
217C2×Dic61C2×D812yes—
218C2×Dic61C2×SD1612yes—
219C2×Dic61C2×SD1612yes—
220C2×Dic61AGL(1,ℤ/8)12yes—
221C2×Dic61#13712yes—
222D244—24no—
223D244—42no—
224C4×D61—84no—
225Dic122—84no—
226C8×C61—42no—
227#92418—42no—
228#29139—611nomaximal, Sylow
229C22×C61—42no—
230C22×C61—42no—
231C22×C61—42no—
232C22×C61—81no—
233C22×C61—81no—
234C22×C61—81no—
235C22×C61—81no—
236C22×C61C2×C22⋊Q811yescharacteristic
237C22×C61C2×#12411yescharacteristic
238C22×C61#53511yescharacteristic
239C22×C61C22×D811yescharacteristic
240C22×C61C22×SD1611yescharacteristic
241C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
242C22×C61C2×#13711yescharacteristic
243C2×D61—42no—
244C2×D61—42no—
245C2×D61—42no—
246C2×D61—81no—
247C2×D61—81no—
248C2×D61—81no—
249C2×D61—81no—
250C4×C61—22no—
251C4×C61—22no—
252C4×C61—81no—
253C4×C61—82no—
254C4×C61#55412yes—
255C4×C61#55512yes—
256Dic61—22no—
257Dic61—81no—
258Dic61—82no—
259Dic61#47414yes—
260Dic61#47614yes—
261Dic61#48614yes—
262#2868—611no—
263C4×#127—611no—
264#2023—611no—
265#2023—611no—
266#2023—611no—
267#2023D6111yescharacteristic, complemented
268#1957—612no—
269#1834—612no—
270#1802—612no—
271#1802—612no—
272C4×C4⋊C8—611no—
273C2×C61—82no—
274C2×C61#256311yescharacteristic
275C2×C61C2×#47411yescharacteristic
276C2×C61C2×#47611yescharacteristic
277C2×C61C2×#48611yescharacteristic
278C2×C61#314111yescharacteristic
279C2×C61#320511yescharacteristic
280C2×C61#320611yescharacteristic
281D61—82no—
282C2×C4⋊Q8—611no—
283C2×C4⋊Q8—1221no—
284C2×C4⋊Q8—1221no—
285C2×C4⋊Q8C2×D6111yescharacteristic
286C2×#127—611no—
287C2×C4×D4—611no—
288C2×C4×D4—611no—
289C2×C4×D4—1221no—
290C2×C4×D4—1221no—
291C2×C4⋊C8—612no—
292C2×#102—612no—
293C2×#102—612no—
294C2×#102—612no—
295C2×#102C2×D6112yes—
296C2×C4×C8—612no—
297#390—611no—
298#390—1221no—
299#390—1221no—
300#390C2×D6111yescharacteristic
301#384—611no—
302#384—612no—
303#384—612no—
304#384—1221no—
305#384—1221no—
306C43—611no—
307#340—612no—
308#340—612no—
309#340—612no—
310#340C2×D6112yes—
311#336—611no—
312#336C2×D6111yescharacteristic
313C61#2913911yescharacteristic, complemented, Sylow
314C22×Q8—21no—
315C22×Q8—1221no—
316C22×D4—611no—
317C22×D4—1221no—
318C22×D4—1221no—
319C22×D4C22×D6111yescharacteristic
320C23×C4—1221no—
321C23×C4—1221no—
322C22×C8—612no—
323C22×C8C22×D6112yes—
324C4⋊Q8—42no—
325C4⋊Q8—2442no—
326C4⋊Q8—2444no—
327C4⋊Q8—2444no—
328#127—614no—
329#127—1222no—
330C4×D4—1224no—
331C4×D4—1224no—
332C4×D4—2444no—
333C4×D4—2444no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—612no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4—1222no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4C22×D6111yescharacteristic
345C2×C4⋊C4C22×D6112yes—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—1221no—
349C2×C22⋊C4—1221no—
350C2×C22⋊C4—1221no—
351C2×C22⋊C4—1221no—
352C2×C42—611no—
353C2×C42—611no—
354C2×C42—611no—
355C2×C42—611no—
356C2×C42—611no—
357C2×C42—611no—
358C2×C42C22×D6111yescharacteristic
359C4⋊C8—1224no—
360#102—42no—
361#102—2442no—
362#102—2442no—
363#102—2442no—
364C4×C8—1224no—
365C24—21no—
366C24—1221no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—2442no—
371C2×Q8—2442no—
372C2×Q8—2442no—
373C2×D4—22no—
374C2×D4—21no—
375C2×D4—22no—
376C2×D4—612no—
377C2×D4—1224no—
378C2×D4—1224no—
379C2×D4—1222no—
380C2×D4—1221no—
381C2×D4—1222no—
382C2×D4—2442no—
383C2×D4—2442no—
384C2×D4—2442no—
385C2×D4—2442no—
386C2×D4Q8×D6112yes—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—611no—
390C22×C4—611no—
391C22×C4—611no—
392C22×C4—611no—
393C22×C4—1221no—
394C22×C4—1222no—
395C22×C4—2442no—
396C22×C4—2442no—
397C22×C4—2442no—
398C22×C4—2442no—
399C22×C4—2442no—
400C22×C4—2442no—
401C22×C4D4×D6111yescharacteristic
402C22×C4D4×D6111yescharacteristic
403C22×C4C23×D6111yescharacteristic
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—1222no—
408C2×C8—1222no—
409C2×C8—1222no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—42no—
414C4⋊C4—1222no—
415C4⋊C4—1222no—
416C4⋊C4—1222no—
417C4⋊C4—2442no—
418C4⋊C4—2442no—
419C4⋊C4—2442no—
420C4⋊C4—2444no—
421C4⋊C4—2444no—
422C4⋊C4—2444no—
423C4⋊C4—2444no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—2442no—
427C22⋊C4—2442no—
428C22⋊C4—2442no—
429C22⋊C4—2442no—
430C42—22no—
431C42—612no—
432C42—612no—
433C42—614no—
434C42—614no—
435C42—614no—
436C42—1222no—
437C42—1222no—
438C23—22no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—1222no—
446C23—2441no—
447C23—2441no—
448C23—2441no—
449C23—2441no—
450C23—2441no—
451C23—2441no—
452C23C2×D4×D6111yescharacteristic
453Q8—84no—
454Q8—4884no—
455D4—24no—
456D4—42no—
457D4—1224no—
458D4—2448no—
459D4—2448no—
460D4—2442no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—612no—
472C2×C4—612no—
473C2×C4—612no—
474C2×C4—612no—
475C2×C4—612no—
476C2×C4—612no—
477C2×C4—612no—
478C2×C4—612no—
479C2×C4—612no—
480C2×C4—612no—
481C2×C4—1221no—
482C2×C4—1221no—
483C2×C4—4884no—
484C2×C4—4884no—
485C2×C4—2441no—
486C2×C4—2442no—
487C2×C4—2441no—
488C2×C4—2441no—
489C2×C4—2442no—
490C2×C4—2442no—
491C2×C4D8×D6112yes—
492C2×C4SD16×D6112yes—
493C2×C4C2×Q8×D6111yescharacteristic
494C2×C4#41×D6112yes—
495C2×C4#41×D6111yescharacteristic
496C2×C4#1451982212yes—
497C8—42no—
498C8—2442no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—4881no—
507C22—4881no—
508C22—4881no—
509C22—4881no—
510C22—2442no—
511C22—2442no—
512C22—2442no—
513C22C22⋊Q8×D6111yescharacteristic
514C22#124×D6111yescharacteristic
515C22#1684032711yescharacteristic
516C22C2×D8×D6111yescharacteristic
517C22C2×SD16×D6111yescharacteristic
518C22AGL(1,ℤ/8)×D6111yescharacteristic
519C22#137×D6111yescharacteristic
520C4—22no—
521C4—22no—
522C4—82no—
523C4—81no—
524C4—614no—
525C4—614no—
526C4—614no—
527C4—1222no—
528C4—4882no—
529C4—4881no—
530C4#1684046612yes—
531C4#1684047412yes—
532C2—82no—
533C2—4882no—
534C2#2145098711yescharacteristic
535C2#474×D6111yescharacteristic
536C2#476×D6111yescharacteristic
537C2#486×D6111yescharacteristic
538C2#2145621111yescharacteristic
539C2#2145673511yescharacteristic
540C2#2145674211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2023C2
socleC23×C61C22×D4
prime radicalC2×#14519773C22
hypercenter#2023D61
hypocenterC61#29139
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
1. #21454215C2semidirect
3. #21446956C2semidirect
4. #21446955C2semidirect
5. #21446952C2semidirect
7. #21445945C2semidirect
8. #21445818C2semidirect
9. #21445817C2semidirect
10. #21445667C2semidirect
11. C61×#2023C2semidirect
54. #14519773Q8semidirect
267. #2023D61semidirect
313. C61#29139semidirect

Maximal subgroups

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

subgroupindex
#214542152
#214483832
#214469562
#214469552
#214469522
#214466232
#214459452
#214458182
#214458172
#214456672
C61×#20232
#2913961