Finite Group Databasebeta

← #29779836

Subgroups — #29779836

Order 15616.

978 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454216C211yesmaximal, characteristic, complemented
2C4×#14519773C211yesmaximal, characteristic
3#21446954C212yesmaximal, complemented
4#21446953C212yesmaximal, complemented
5#21446623C212yesmaximal
6#21445945C212yesmaximal, complemented
7#21445820C211yesmaximal, characteristic, complemented
8#21445819C211yesmaximal, characteristic, complemented
9#21445816C211yesmaximal, characteristic, complemented
10C4⋊C8×Dic61C211yesmaximal, characteristic, complemented
11C61×#1802C211yesmaximal, characteristic, complemented
12C2×#14519776—21no—
13C2×#14519776—21no—
14C2×#14519776C2212yes—
15C2×#14519773C2211yescharacteristic
16C2×C4×D244C2211yescharacteristic
17C2×C4×C22⋊D61—21no—
18C2×C4×C22⋊D61—21no—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519699C2212yes—
21C2×#14519698C2212yes—
22C2×#14519697C2212yes—
23C2×C8×Dic61C2212yes—
24#16839665C2211yescharacteristic
25#16839663C2211yescharacteristic
26#16839662—21no—
27#16839662—21no—
28#16839632C2211yescharacteristic
29#16839631—21no—
30#16839631—21no—
31#16839631C2211yescharacteristic
32#16839631C2212yes—
33C42×Dic61C2211yescharacteristic
34#16839525C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839516C2212yes—
38C2×C4×D4×C61C2211yescharacteristic
39C2×C4⋊C8×C61C2211yescharacteristic
40C2×#102×C61C2212yes—
41C61×#384C2211yescharacteristic
42C61×#340C2212yes—
43C22×Dic122—22no—
44C22×D244C2311yescharacteristic
45C22×C22⋊D61—21no—
46C22×C22⋊D61—21no—
47C22×C4×D61—21no—
48C22×#92418C2312yes—
49#14519776—42no—
50#14519776—42no—
51#14519776—44no—
52#14519776—44no—
53#14519773—22no—
54#14519773Q814yescomplemented
55C4×D244—22no—
56C4×D244—22no—
57C4×C22⋊D61—44no—
58C4×C22⋊D61—44no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C4⋊Dic61C2312yes—
63C2×#11758196—21no—
64C2×#11758196—21no—
65C2×#11758196—22no—
66C2×#11758196—22no—
67C2×#11758196—22no—
68C2×C22⋊Dic61—21no—
69C2×C22⋊Dic61—21no—
70C2×#11758198—21no—
71C2×#11758198—21no—
72C2×#11758198—21no—
73C2×C4×Dic61D411yescharacteristic
74C2×C4×Dic61D411yescharacteristic
75C2×C4×Dic61D411yescharacteristic
76C2×C4×Dic61D411yescharacteristic
77C2×C4×Dic61C2311yescharacteristic
78C2×C4×Dic61C2311yescharacteristic
79C4⋊#92418—22no—
80#14519699—42no—
81#14519698—42no—
82#14519697—42no—
83C8×Dic61—24no—
84C22×D4×C61C2311yescharacteristic
85C23×C4×C61—21no—
86C22×C8×C61C2312yes—
87C4×D4×C61—22no—
88C4×D4×C61—22no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C4⋊C4×C61C2312yes—
93C2×C22⋊C4×C61—21no—
94C2×C42×C61C2311yescharacteristic
95C4⋊C8×C61—22no—
96#102×C61—42no—
97C24×C61—21no—
98C23×D61—21no—
99C2×D4×C61—21no—
100C2×D4×C61—21no—
101C2×D4×C61—21no—
102C2×D4×C61—21no—
103C2×D4×C61—21no—
104C2×D4×C61C2×Q812yes—
105C22×C4×C61—22no—
106C22×C4×C61—21no—
107C22×C4×C61—42no—
108C22×C4×C61—42no—
109C22×C4×C61—42no—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2411yescharacteristic
113C2×C22⋊D61—24no—
114C2×C22⋊D61—24no—
115C2×C22⋊D61—42no—
116C2×C22⋊D61—42no—
117C2×C22⋊D61—42no—
118C2×C22⋊D61—42no—
119C22×Dic61—21no—
120C22×Dic61—22no—
121C22×Dic61C2×D411yescharacteristic
122C22×Dic61C2×D411yescharacteristic
123C22×Dic61C2×D411yescharacteristic
124C22×Dic61C2×D411yescharacteristic
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244C2×Q812yes—
131C2×C4×D61—42no—
132C2×C4×D61—42no—
133C2×C4×D61—42no—
134C2×Dic122—44no—
135C2×Dic122—44no—
136C2×Dic122—44no—
137C2×C8×C61—22no—
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—42no—
145C4⋊C4×C61—42no—
146C22⋊C4×C61—42no—
147C42×C61#4112yes—
148C42×C61#4112yes—
149C22⋊Dic61—42no—
150C22⋊Dic61—42no—
151#11758198—42no—
152#11758198—42no—
153#11758198—42no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—22no—
157C4⋊Dic61—22no—
158C4⋊Dic61—42no—
159C4⋊Dic61—42no—
160#11758196—44no—
161#11758196—42no—
162#11758196—44no—
163#11758196—42no—
164#11758196—44no—
165C4×Dic61—21no—
166C4×Dic61—21no—
167C4×Dic61—22no—
168C4×Dic61—22no—
169C4×Dic61D814yes—
170C4×Dic61SD1614yes—
171C4×Dic61C2×Q812yes—
172C4×Dic61#4112yes—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—22no—
176C23×C61—22no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61C22×D411yescharacteristic
184C22×D61—22no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—41no—
191D4×C61—22no—
192D4×C61—22no—
193D4×C61—41no—
194D4×C61—41no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—41no—
198C2×C4×C61—42no—
199C2×C4×C61—41no—
200C2×C4×C61—42no—
201C2×C4×C61—41no—
202C2×C4×C61—42no—
203C2×C4×C61—84no—
204C2×C4×C61#12412yes—
205C2×C4×C61#12412yes—
206C2×C4×C61C2×D812yes—
207C2×C4×C61C2×SD1612yes—
208C2×C4×C61C22×Q811yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C61H(5,𝔽2)11yescharacteristic
213C2×C4×C61H(5,𝔽2)11yescharacteristic
214C22⋊D61—48no—
215C22⋊D61—48no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—41no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—42no—
226C2×Dic61C22⋊Q812yes—
227C2×Dic61C22⋊Q812yes—
228C2×Dic61C2×D812yes—
229C2×Dic61C2×D812yes—
230C2×Dic61C2×SD1612yes—
231C2×Dic61C2×SD1612yes—
232C2×Dic61AGL(1,ℤ/8)12yes—
233C2×Dic61#13712yes—
234D244—22no—
235D244—22no—
236D244—41no—
237D244—41no—
238C4×D61—84no—
239Dic122—88no—
240C8×C61—42no—
241#92418—42no—
242#29139—611nomaximal, Sylow
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—81no—
247C22×C61—81no—
248C22×C61—81no—
249C22×C61—81no—
250C22×C61C2×C22⋊Q811yescharacteristic
251C22×C61C2×#12411yescharacteristic
252C22×C61#53511yescharacteristic
253C22×C61C22×D811yescharacteristic
254C22×C61C22×SD1611yescharacteristic
255C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C61C2×#13711yescharacteristic
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—81no—
261C2×D61—81no—
262C2×D61—81no—
263C2×D61—81no—
264C4×C61—22no—
265C4×C61—81no—
266C4×C61—82no—
267C4×C61#48614yes—
268C4×C61#55411yescharacteristic
269C4×C61#55411yescharacteristic
270C4×C61#55511yescharacteristic
271C4×C61#55511yescharacteristic
272Dic61—22no—
273Dic61—22no—
274Dic61—81no—
275Dic61—82no—
276Dic61#47414yes—
277Dic61#47614yes—
278#2868—611no—
279C4×#127—611no—
280#2023—612no—
281#2023—612no—
282#1957—612no—
283#1834—612no—
284#1802—611no—
285#1802—611no—
286#1802—611no—
287#1802D6111yescharacteristic, complemented
288C4×C4⋊C8—611no—
289C2×C61—82no—
290C2×C61#256311yescharacteristic
291C2×C61C2×#47411yescharacteristic
292C2×C61C2×#47611yescharacteristic
293C2×C61C2×#48611yescharacteristic
294C2×C61#314111yescharacteristic
295C2×C61#320511yescharacteristic
296C2×C61#320611yescharacteristic
297D61—82no—
298C2×C4⋊Q8—612no—
299C2×C4⋊Q8—1221no—
300C2×C4⋊Q8—1221no—
301C2×#127—611no—
302C2×C4×D4—611no—
303C2×C4×D4—1221no—
304C2×C4×D4—1221no—
305C2×C4×D4C2×D6111yescharacteristic
306C2×C4⋊C8—611no—
307C2×C4⋊C8C2×D6111yescharacteristic
308C2×#102—612no—
309C2×#102—612no—
310C2×#102—612no—
311C2×#102C2×D6112yes—
312C2×C4×C8—612no—
313#390—611no—
314#390—611no—
315#390—1221no—
316#390—1221no—
317#384—611no—
318#384—612no—
319#384—611no—
320#384—1221no—
321#384—1221no—
322#384C2×D6111yescharacteristic
323C43—611no—
324#340—612no—
325#340—612no—
326#340—612no—
327#340C2×D6112yes—
328#336—612no—
329C61#2913911yescharacteristic, complemented, Sylow
330C22×Q8—1222no—
331C22×D4—611no—
332C22×D4—1221no—
333C22×D4—1221no—
334C22×D4C22×D6111yescharacteristic
335C23×C4—21no—
336C23×C4—1221no—
337C22×C8—612no—
338C22×C8C22×D6112yes—
339C4⋊Q8—2444no—
340C4⋊Q8—2442no—
341C4⋊Q8—2442no—
342C4⋊Q8—2444no—
343#127—614no—
344#127—1222no—
345C4×D4—22no—
346C4×D4—22no—
347C4×D4—1222no—
348C4×D4—1222no—
349C4×D4—2444no—
350C4×D4—2444no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—611no—
354C2×C4⋊C4—612no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4—1221no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4C22×D6111yescharacteristic
363C2×C4⋊C4C22×D6112yes—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—1221no—
366C2×C22⋊C4—1221no—
367C2×C22⋊C4—1221no—
368C2×C22⋊C4—1221no—
369C2×C22⋊C4—1221no—
370C2×C42—611no—
371C2×C42—611no—
372C2×C42—611no—
373C2×C42—611no—
374C2×C42—611no—
375C2×C42—611no—
376C2×C42C22×D6111yescharacteristic
377C4⋊C8—22no—
378C4⋊C8—1222no—
379#102—42no—
380#102—2442no—
381#102—2442no—
382#102—2442no—
383C4×C8—1224no—
384C24—21no—
385C24—1221no—
386C2×Q8—2444no—
387C2×Q8—2444no—
388C2×Q8—2444no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—612no—
395C2×D4—1224no—
396C2×D4—1224no—
397C2×D4—1221no—
398C2×D4—1221no—
399C2×D4—1221no—
400C2×D4—1221no—
401C2×D4—1221no—
402C2×D4—2442no—
403C2×D4—2442no—
404C2×D4—2442no—
405C2×D4—2442no—
406C2×D4Q8×D6112yes—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—611no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—1221no—
417C22×C4—1222no—
418C22×C4—2442no—
419C22×C4—2442no—
420C22×C4—2442no—
421C22×C4D4×D6111yescharacteristic
422C22×C4D4×D6111yescharacteristic
423C22×C4C23×D6111yescharacteristic
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—1222no—
428C2×C8—1222no—
429C2×C8—1222no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—1221no—
437C4⋊C4—1221no—
438C4⋊C4—1222no—
439C4⋊C4—1222no—
440C4⋊C4—2442no—
441C4⋊C4—2442no—
442C4⋊C4—2442no—
443C4⋊C4—2444no—
444C4⋊C4—2444no—
445C4⋊C4—2444no—
446C4⋊C4—2442no—
447C22⋊C4—42no—
448C22⋊C4—2442no—
449C22⋊C4—2442no—
450C22⋊C4—2442no—
451C22⋊C4—2442no—
452C22⋊C4—2442no—
453C42—612no—
454C42—612no—
455C42—614no—
456C42—614no—
457C42—1222no—
458C42—1222no—
459C42—1221no—
460C42—1221no—
461C42#1175822212yes—
462C42#1175822412yes—
463C23—22no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—1222no—
471C23—2441no—
472C23—2441no—
473C23—2441no—
474C23—2441no—
475C23—2441no—
476C23—2441no—
477C23C2×D4×D6111yescharacteristic
478Q8—4888no—
479D4—22no—
480D4—22no—
481D4—41no—
482D4—41no—
483D4—1222no—
484D4—1222no—
485D4—2448no—
486D4—2448no—
487D4—2441no—
488D4—2441no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—84no—
498C2×C4—612no—
499C2×C4—612no—
500C2×C4—612no—
501C2×C4—612no—
502C2×C4—612no—
503C2×C4—612no—
504C2×C4—612no—
505C2×C4—612no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—1221no—
509C2×C4—1221no—
510C2×C4—4884no—
511C2×C4—2441no—
512C2×C4—2442no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4#1451976312yes—
518C2×C4#1451976312yes—
519C2×C4D8×D6112yes—
520C2×C4SD16×D6112yes—
521C2×C4C2×Q8×D6111yescharacteristic
522C2×C4#41×D6111yescharacteristic
523C2×C4C2×#1175822411yescharacteristic
524C2×C4C2×#1175822211yescharacteristic
525C2×C4#1451982211yescharacteristic
526C2×C4#1451982311yescharacteristic
527C8—42no—
528C8—2442no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—4881no—
537C22—4881no—
538C22—4881no—
539C22—4881no—
540C22—2442no—
541C22—2442no—
542C22—2442no—
543C22C22⋊Q8×D6111yescharacteristic
544C22C2×#1451976311yescharacteristic
545C22#1684032811yescharacteristic
546C22C2×D8×D6111yescharacteristic
547C22C2×SD16×D6111yescharacteristic
548C22AGL(1,ℤ/8)×D6111yescharacteristic
549C22#137×D6111yescharacteristic
550C4—22no—
551C4—82no—
552C4—81no—
553C4—614no—
554C4—614no—
555C4—1222no—
556C4—1222no—
557C4—4882no—
558C4—4881no—
559C4#1684010714yes—
560C4#1684046711yescharacteristic
561C4#1684047011yescharacteristic
562C4#1684047511yescharacteristic
563C4#1684047611yescharacteristic
564C2—82no—
565C2—4882no—
566C2#2145098911yescharacteristic
567C2#474×D6111yescharacteristic
568C2#476×D6111yescharacteristic
569C2C2×#1684010711yescharacteristic
570C2#2145621011yescharacteristic
571C2#2145673611yescharacteristic
572C2#2145674311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1802C2
socleC23×C61C22×D4
prime radicalC2×#14519773C22
hypercenter#1802D61
hypocenterC61#29139
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
1. #21454216C2semidirect
3. #21446954C2semidirect
4. #21446953C2semidirect
6. #21445945C2semidirect
7. #21445820C2semidirect
8. #21445819C2semidirect
9. #21445816C2semidirect
10. C4⋊C8×Dic61C2semidirect
11. C61×#1802C2semidirect
54. #14519773Q8semidirect
287. #1802D61semidirect
329. C61#29139semidirect

Maximal subgroups

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

subgroupindex
#214542162
C4×#145197732
#214469542
#214469532
#214466232
#214459452
#214458202
#214458192
#214458162
C4⋊C8×Dic612
C61×#18022
#2913961