Finite Group Databasebeta

← #29714762

Subgroups — #29714762

Order 15616.

1150 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840482C211yesmaximal, characteristic, complemented
2C2×#16839968C211yesmaximal, characteristic, complemented
3C2×C4×#11758199C211yesmaximal, characteristic
4#21450742C211yesmaximal, characteristic, complemented
5#21447059C212yesmaximal, complemented
6#21446898C212yesmaximal
7#21446754C211yesmaximal, characteristic, complemented
8#21446620C211yesmaximal, characteristic, complemented
9#21446175C212yesmaximal, complemented
10#21446074C212yesmaximal, complemented
11C61×#1802C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13C22×#11758199C2211yescharacteristic
14#16840482—24no—
15#16840482—24no—
16C2×#14519777C2211yescharacteristic
17C2×#14519770—22no—
18C2×#14519764C2212yes—
19C2×#14519757—21no—
20C2×#14519756—21no—
21C2×C4×Dic122C2211yescharacteristic
22C2×#14519742C2212yes—
23C2×C4×D244C2211yescharacteristic
24C2×#14519736—21no—
25#16839968—24no—
26C4×#11758199—24no—
27C2×#14519710C2211yescharacteristic
28C2×#14519702C2212yes—
29C2×#14519700C2212yes—
30C2×#14519700C2212yes—
31C2×#14519698C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×C4×#92418C2211yescharacteristic
34#16839684—22no—
35#16839663C2212yes—
36#16839661C2212yes—
37#16839619—21no—
38#16839516C2212yes—
39#16839516C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×#102×C61C2212yes—
43C61×#384C2211yescharacteristic
44C61×#340C2212yes—
45C22×Dic122D411yescharacteristic
46C22×Dic122C2312yes—
47C22×D244C2311yescharacteristic
48C22×C4×D61—21no—
49C2×#11758190—24no—
50C2×#11758190—24no—
51C2×#11758190—42no—
52C2×#11758199—22no—
53C2×#11758199—22no—
54C2×#11758199—22no—
55C22×#92418C2311yescharacteristic
56C22×#92418C2311yescharacteristic
57#14519777—22no—
58#14519777D414yes—
59#14519770—216no—
60#14519764—24no—
61#14519764—24no—
62#14519757—44no—
63#14519756—28no—
64C4×Dic122—22no—
65C4×Dic122—22no—
66#14519742—24no—
67#14519742—24no—
68C4×D244—22no—
69C4×D244—22no—
70#14519736—28no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—22no—
74C2×C4⋊Dic61D411yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—22no—
77C2×#11758196D412yes—
78C2×C22⋊Dic61—22no—
79C2×C22⋊Dic61—22no—
80C2×#11758198—21no—
81C2×#11758198—22no—
82C2×#11758198—22no—
83C2×C4×Dic61—22no—
84C2×C4×Dic61C2312yes—
85#14519710—22no—
86#14519702—24no—
87#14519700—24no—
88#14519700—42no—
89#14519698—22no—
90#14519697—22no—
91C4×#92418—22no—
92#14519684—22no—
93C22×D4×C61C2311yescharacteristic
94C23×C4×C61—21no—
95C22×C8×C61C2312yes—
96C4×D4×C61—22no—
97C4×D4×C61D414yes—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61C2312yes—
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61—22no—
105#102×C61—42no—
106C24×C61—21no—
107C23×D61—21no—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61C2×D412yes—
114C22×C4×C61—21no—
115C22×C4×C61—24no—
116C22×C4×C61—22no—
117C22×C4×C61—42no—
118C22×C4×C61—42no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—22no—
123C22×Dic61—21no—
124C22×Dic61C2×D412yes—
125C22×Dic61C2×D412yes—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244#4112yes—
132C2×C4×D61—42no—
133C2×C4×D61—42no—
134C2×C4×D61—42no—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122—22no—
140C2×Dic122—22no—
141C2×Dic122—22no—
142C2×Dic122—22no—
143C2×Dic122C2×D414yes—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—42no—
151C4⋊C4×C61—81no—
152C4⋊C4×C61C2×D412yes—
153C22⋊C4×C61—24no—
154C42×C61C2×D412yes—
155C42×C61C2×D412yes—
156C22⋊Dic61—28no—
157C22⋊Dic61—28no—
158#11758199—28no—
159#11758198—44no—
160#11758198—42no—
161#11758198—82no—
162C4⋊Dic61—24no—
163C4⋊Dic61—24no—
164C4⋊Dic61—22no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—44no—
168#11758196—28no—
169#11758196—24no—
170C4×Dic61—28no—
171C4×Dic61—22no—
172C4×Dic61—22no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418#4112yes—
179#11758190—48no—
180C23×C61—22no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61C22×D411yescharacteristic
188C22×D61—22no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195D4×C61—22no—
196D4×C61—22no—
197D4×C61—41no—
198D4×C61—41no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—22no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—48no—
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×C61AGL(1,ℤ/8)12yes—
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—22no—
220C2×Dic61—24no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230D244—22no—
231D244—22no—
232D244—41no—
233D244—41no—
234C4×D61—84no—
235Dic122—24no—
236Dic122—24no—
237Dic122—42no—
238Dic122—42no—
239Dic122—44no—
240C8×C61—42no—
241#92418—22no—
242#92418—41no—
243#24185—611nomaximal, Sylow
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—81no—
248C22×C61—81no—
249C22×C61—81no—
250C22×C61—81no—
251C22×C61C2×#12011yescharacteristic
252C22×C61#54811yescharacteristic
253C22×C61#55011yescharacteristic
254C22×C61C2×#13511yescharacteristic
255C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C61#57711yescharacteristic
257C22×C61#57811yescharacteristic
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—42no—
261C2×D61—81no—
262C2×D61—81no—
263C2×D61—81no—
264C2×D61—81no—
265C4×C61—22no—
266C4×C61—42no—
267C4×C61—82no—
268C4×C61#44914yes—
269C4×C61#52511yescharacteristic
270C4×C61#53711yescharacteristic
271C4×C61#54611yescharacteristic
272C4×C61#54711yescharacteristic
273Dic61—44no—
274Dic61—42no—
275Dic61—42no—
276Dic61—81no—
277C2×#556—611no—
278C2×#461—611no—
279C2×C4×D8—611no—
280#2538—611no—
281#2042—612no—
282#2012—612no—
283#1984—611no—
284#1956—611no—
285#1880—612no—
286#1863—612no—
287#1802D6111yescharacteristic, complemented
288C2×C61—82no—
289C2×C61#240711yescharacteristic
290C2×C61C2×#44911yescharacteristic
291C2×C61#301311yescharacteristic
292C2×C61#336011yescharacteristic
293C2×C61#336711yescharacteristic
294C2×C61#339311yescharacteristic
295C2×C61#339711yescharacteristic
296D61—82no—
297C22×SD16—612no—
298C22×D8—611no—
299#556—1224no—
300#556—1224no—
301C2×C4⋊Q8—611no—
302C2×#126—1222no—
303C2×#124—612no—
304C2×C22⋊Q8—1221no—
305C2×C22⋊Q8—1221no—
306C2×C4×Q8—611no—
307C2×C4×Q8—612no—
308C2×C4×D4—611no—
309C2×C4×D4C2×D6111yescharacteristic
310C2×#117—1221no—
311#461—1224no—
312C4×D8—1224no—
313C2×#107—611no—
314C2×C4⋊C8C2×D6111yescharacteristic
315C2×#103—612no—
316C2×#103—612no—
317C2×#103—612no—
318C2×#102—611no—
319C2×#102—611no—
320C2×#102C2×D6112yes—
321C2×C4×C8—611no—
322#396—1222no—
323#390—612no—
324#389—612no—
325#384C2×D6111yescharacteristic
326#380—1221no—
327#340C2×D6112yes—
328#336—612no—
329#336—612no—
330C61#2418511yescharacteristic, complemented, Sylow
331C22×Q8—611no—
332C22×Q8—612no—
333C22×D4—611no—
334C22×D4C22×D6111yescharacteristic
335C23×C4—21no—
336C23×C4—1221no—
337C2×SD16—1224no—
338C2×SD16—1224no—
339C2×SD16—2442no—
340C2×D8—1222no—
341C2×D8—1222no—
342C2×D8—1222no—
343C22×C8—611no—
344C22×C8—611no—
345C22×C8C22×D6112yes—
346C4⋊Q8—614no—
347C4⋊Q8—1222no—
348#126—12216no—
349#124—1224no—
350#124—1224no—
351C22⋊Q8—1228no—
352C22⋊Q8—2444no—
353C4×Q8—1224no—
354C4×Q8—1224no—
355C4×Q8—1222no—
356C4×Q8—1222no—
357C4×D4—22no—
358C4×D4—1222no—
359C4×D4—1222no—
360C4×D4D24414yes—
361#117—1228no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—612no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4C22×D6112yes—
372C2×C4⋊C4C22×D6111yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4—1222no—
378C2×C22⋊C4—1221no—
379C2×C42—612no—
380C2×C42—1222no—
381C2×C42C22×D6111yescharacteristic
382#107—1222no—
383C4⋊C8—22no—
384#103—1224no—
385#103—1224no—
386#103—2442no—
387#102—42no—
388#102—1222no—
389#102—1222no—
390C4×C8—1222no—
391#95—1222no—
392C24—21no—
393C24—1221no—
394C2×Q8—614no—
395C2×Q8—1222no—
396C2×Q8—1222no—
397C2×Q8—1222no—
398C2×Q8—1222no—
399C2×Q8—1222no—
400C2×Q8—1222no—
401C2×Q8—1222no—
402C2×Q8—1222no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—612no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4—1221no—
413C2×D4—1221no—
414C2×D4C2×D24412yes—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—612no—
421C22×C4—612no—
422C22×C4—1222no—
423C22×C4—1221no—
424C22×C4—2442no—
425C22×C4—2442no—
426C22×C4—2442no—
427C22×C4D4×D6111yescharacteristic
428C22×C4D4×D6111yescharacteristic
429C22×C4C23×D6111yescharacteristic
430SD16—2448no—
431D8—1228no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—612no—
436C2×C8—1221no—
437C2×C8—1221no—
438C2×C8—1221no—
439C2×C8—1221no—
440C2×C8—1221no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—42no—
445C4⋊C4—81no—
446C4⋊C4—1228no—
447C4⋊C4—1224no—
448C4⋊C4—1222no—
449C4⋊C4—1224no—
450C4⋊C4—1224no—
451C4⋊C4—1221no—
452C4⋊C4—1221no—
453C4⋊C4—2444no—
454C4⋊C4C2×D24412yes—
455C22⋊C4—24no—
456C22⋊C4—1228no—
457C22⋊C4—1228no—
458C22⋊C4—4882no—
459C22⋊C4—2444no—
460C22⋊C4—2442no—
461C42—1228no—
462C42—1222no—
463C42—1222no—
464C42C2×D24412yes—
465C42D4×D6112yes—
466C23—22no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—1222no—
474C23—2441no—
475C23—2441no—
476C23—2441no—
477C23—2441no—
478C23—2441no—
479C23—2441no—
480C23C2×D4×D6111yescharacteristic
481Q8—1224no—
482Q8—1224no—
483Q8—2442no—
484Q8—2442no—
485Q8—2444no—
486D4—22no—
487D4—22no—
488D4—41no—
489D4—41no—
490D4—1222no—
491D4—1222no—
492D4—2441no—
493D4—2441no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—48no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—1222no—
504C2×C4—1222no—
505C2×C4—1224no—
506C2×C4—1222no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—4884no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4#1451974712yes—
517C2×C4#1451974712yes—
518C2×C4#1451979812yes—
519C2×C4#1451980112yes—
520C2×C4C2×D4×D6111yescharacteristic
521C2×C4C22×D24411yescharacteristic
522C2×C4#41×D6111yescharacteristic
523C2×C4#41×D6111yescharacteristic
524C2×C4#1451982211yescharacteristic
525C2×C4#1451982511yescharacteristic
526C8—42no—
527C8—1222no—
528C8—2441no—
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—
543C22C2×#1451974711yescharacteristic
544C22#1684042211yescharacteristic
545C22#1684043011yescharacteristic
546C22C2×#1451979811yescharacteristic
547C22C2×#1451980111yescharacteristic
548C22#1684057211yescharacteristic
549C22#1684057811yescharacteristic
550C4—22no—
551C4—42no—
552C4—82no—
553C4—4881no—
554C4—2442no—
555C4—2442no—
556C4—2444no—
557C4#1683990814yes—
558C4#1684027411yescharacteristic
559C4#1684034311yescharacteristic
560C4#1684040911yescharacteristic
561C4#1684041411yescharacteristic
562C2—82no—
563C2—4882no—
564C2#2144916511yescharacteristic
565C2C2×#1683990811yescharacteristic
566C2#2145513611yescharacteristic
567C2#2145846711yescharacteristic
568C2#2145853611yescharacteristic
569C2#2145880611yescharacteristic
570C2#2145884311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1802C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenter#1802D61
hypocenterC61#24185
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. C2×#16840482C2semidirect
2. C2×#16839968C2semidirect
4. #21450742C2semidirect
5. #21447059C2semidirect
7. #21446754C2semidirect
8. #21446620C2semidirect
9. #21446175C2semidirect
10. #21446074C2semidirect
11. C61×#1802C2semidirect
287. #1802D61semidirect
330. C61#24185semidirect

Maximal subgroups

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

subgroupindex
C2×#168404822
C2×#168399682
C2×C4×#117581992
#214507422
#214470592
#214468982
#214467542
#214466202
#214461752
#214460742
C61×#18022
#2418561