Finite Group Databasebeta

← #29662257

Subgroups — #29662257

Order 15616.

802 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456248C211yesmaximal, characteristic, complemented
2#21455821C212yesmaximal, complemented
3C8×D4×D61C211yesmaximal, characteristic, complemented
4#21445181C211yesmaximal, characteristic, complemented
5#21445175C211yesmaximal, characteristic, complemented
6#21445174C211yesmaximal, characteristic, complemented
7#21445163C212yesmaximal, complemented
8#21445162C212yesmaximal, complemented
9#21444566C211yesmaximal, characteristic, complemented
10#21444467C212yesmaximal, complemented
11C61×#1669C211yesmaximal, characteristic, complemented
12C2×#14519800—21no—
13C2×#14519800—21no—
14C22×C8×D61—21no—
15#16840313C2212yes—
16C4×D4×D61C2211yescharacteristic
17#16840147—21no—
18#16840147—21no—
19#16840139C2211yescharacteristic, complemented
20#16840136C2212yescomplemented
21#16840077—22no—
22#16840039C2212yes—
23#16839958C2212yes—
24#16839957C2212yes—
25#16839897—22no—
26#16839844—22no—
27#16839842—22no—
28C8×D244C2211yescharacteristic
29D4×#92418C2211yescharacteristic
30#16839827C2211yescharacteristic
31C8×C22⋊D61—21no—
32#16839825—21no—
33#16839814C2212yes—
34C4⋊C8×D61C2211yescharacteristic
35C22⋊C8×D61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839515C2211yescharacteristic, complemented
38#16839512C2212yescomplemented
39#16839508C2212yescomplemented
40#16839505C2212yescomplemented
41#16839505C2212yescomplemented
42C61×#494C2211yescharacteristic, complemented
43C61×#474C2212yes—
44C8×D4×C61C2211yescharacteristic
45C61×#335C2211yescharacteristic, complemented
46C61×#332C2212yescomplemented
47C2×#11758226—22no—
48C2×D4×D61D411yescharacteristic
49C22×C4×D61—21no—
50#14519800—44no—
51#14519800—44no—
52C2×#11758201—22no—
53C2×#11758201—22no—
54C2×D488—22no—
55C2×D488—21no—
56C2×D488—21no—
57C2×#11758199—22no—
58C22×#92418—21no—
59C2×C8×D61—21no—
60C2×C8×D61—21no—
61C2×C8×D61—21no—
62C2×C8×D61—44no—
63C2×C8×D61D411yescharacteristic
64C2×C8×D61D411yescharacteristic
65#14519774C2312yes—
66#14519766—22no—
67#14519752—22no—
68#14519752—22no—
69C4×D244C2311yescharacteristic
70D4×Dic61D411yescharacteristic
71#14519739—22no—
72#14519739—22no—
73#14519739D411yescharacteristic
74C4×C22⋊D61—21no—
75#14519737—21no—
76C4⋊C4×D61D411yescharacteristic
77C22⋊C4×D61—21no—
78C42×D61C2311yescharacteristic
79#14519710—22no—
80C4⋊#92418C2312yes—
81C4⋊#92418C2311yescharacteristic
82#14519706—21no—
83#14519699—22no—
84#14519697—22no—
85#14519697—22no—
86C22⋊#92418—21no—
87#14519689—22no—
88#14519689—21no—
89#14519689—21no—
90C4×#92418C2311yescharacteristic
91C8×Dic61—22no—
92C8×Dic61D411yescharacteristic
93C8×Dic61D411yescharacteristic
94C2×Q16×C61—21no—
95C2×Q16×C61—21no—
96C22×C8×C61—21no—
97C4⋊Q8×C61C2312yes—
98C4×D4×C61C2311yescharacteristic
99#107×C61—22no—
100C4⋊C8×C61C2312yes—
101C4⋊C8×C61C2311yescharacteristic
102#102×C61—22no—
103C22⋊C8×C61—21no—
104C4×C8×C61C2311yescharacteristic
105C23×D61—21no—
106C2×Q8×C61—22no—
107C2×D4×C61C2×D411yescharacteristic
108C22×C4×C61—21no—
109C2×C22⋊D61—21no—
110C22×Dic61—21no—
111#11758226—44no—
112#11758226—44no—
113D4×D61—22no—
114D4×D61—22no—
115C2×D244—22no—
116C2×D244—22no—
117C2×D244—22no—
118C2×D244—42no—
119C2×D244C2×D411yescharacteristic
120C2×C4×D61—21no—
121C2×C4×D61—21no—
122C2×C4×D61—21no—
123C2×C4×D61—42no—
124C2×C4×D61—42no—
125C2×C4×D61—42no—
126C2×C4×D61C2×D411yescharacteristic
127C2×C4×D61C2×D411yescharacteristic
128C2×C4×D61#4111yescharacteristic
129Q16×C61—42no—
130Q16×C61—42no—
131C2×C8×C61—21no—
132C2×C8×C61—22no—
133C2×C8×C61—42no—
134C2×C8×C61C2×D411yescharacteristic
135C2×C8×C61C2×D411yescharacteristic
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61C2×D411yescharacteristic
139C22⋊C4×C61—21no—
140C42×C61C2411yescharacteristic
141C22⋊Dic61—21no—
142#11758201—44no—
143#11758201—44no—
144#11758199—44no—
145#11758198—21no—
146#11758198—21no—
147#11758198—42no—
148#11758198—42no—
149C4⋊Dic61C2×D411yescharacteristic
150#11758196—21no—
151C4×Dic61C2×D411yescharacteristic
152C4×Dic61C2×D411yescharacteristic
153C4×Dic61#4111yescharacteristic
154C2×#92418—22no—
155C2×#92418—21no—
156C2×#92418—42no—
157C2×#92418C2×D411yescharacteristic
158C2×#92418C2×D411yescharacteristic
159D488—42no—
160D488—44no—
161D488—42no—
162C8×D61—22no—
163C8×D61—42no—
164C8×D61—42no—
165C8×D61—42no—
166C8×D61D814yescomplemented
167C23×C61—21no—
168C22×D61—24no—
169C22×D61—21no—
170C22×D61—42no—
171C22×D61—42no—
172C22×D61—42no—
173C22×D61#12111yescharacteristic
174Q8×C61—42no—
175Q8×C61—42no—
176D4×C61—21no—
177D4×C61#13512yescomplemented
178C2×C4×C61—21no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61—42no—
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C22×D411yescharacteristic
184C2×C4×C61C2×#4111yescharacteristic
185C22⋊D61—24no—
186C2×Dic61—21no—
187C2×Dic61—42no—
188C2×Dic61#12111yescharacteristic
189C2×Dic61#12111yescharacteristic
190C2×Dic61#12111yescharacteristic
191D244—21no—
192D244—42no—
193D244—42no—
194D244—42no—
195D244—42no—
196D244—82no—
197D244—84no—
198D244#13512yescomplemented
199C4×D61—21no—
200C4×D61—21no—
201C4×D61—41no—
202C4×D61—42no—
203C4×D61—41no—
204C4×D61—41no—
205C4×D61—42no—
206C4×D61—84no—
207C4×D61C2×D812yes—
208C4×D61AGL(1,ℤ/8)12yes—
209C8×C61—21no—
210C8×C61—41no—
211C8×C61—42no—
212C8×C61C2×D812yes—
213#92418—21no—
214#92418—41no—
215#92418—42no—
216#92418C2×D812yes—
217#19101—611nomaximal, Sylow
218C22×C61—22no—
219C22×C61—41no—
220C22×C61—41no—
221C22×C61C2×#12111yescharacteristic
222C2×D61—21no—
223C2×D61—21no—
224C2×D61—22no—
225C2×D61—48no—
226C2×D61—41no—
227C2×D61—41no—
228C2×D61—82no—
229C2×D61—82no—
230C2×D61—82no—
231C2×D61#45912yes—
232C4×C61—21no—
233C4×C61—41no—
234C4×C61—82no—
235C4×C61C22×D811yescharacteristic
236C4×C61C2×#13511yescharacteristic
237C4×C61C2×#13511yescharacteristic
238C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
239Dic61—21no—
240Dic61—21no—
241Dic61—41no—
242Dic61#45912yes—
243#3147—611no—
244#3100—612no—
245C2×C8×D4—611no—
246#1669—611no—
247#1669D6111yescharacteristic, complemented
248#1667—611no—
249#1667—611no—
250#1665—612no—
251#1665—612no—
252#1564—611no—
253#1540—612no—
254C2×C61—42no—
255C2×C61C2×#45911yescharacteristic
256C2×C61#305011yescharacteristic
257C2×C61#306711yescharacteristic
258D61—22no—
259D61—42no—
260D61—162no—
261C2×#135—1221no—
262C2×#135—1221no—
263C23×C8—1221no—
264#532—612no—
265C2×C4×D4—611no—
266#495—1221no—
267#495—1221no—
268#494C2×D6111yescharacteristic, complemented
269#493—611no—
270#492—612no—
271#480—1222no—
272#474—612no—
273#474C2×D6112yes—
274#459—612no—
275#459—612no—
276#447—1222no—
277C4×D8—1222no—
278C4×D8—1222no—
279C8×D4—611no—
280C8×D4—611no—
281C8×D4—611no—
282C8×D4—1221no—
283C8×D4—1221no—
284C8×D4C2×D6111yescharacteristic
285#431—612no—
286C2×C4⋊C8—611no—
287C2×C22⋊C8—1221no—
288C2×C4×C8—611no—
289#335—611no—
290#335C2×D6111yescharacteristic, complemented
291#334—612no—
292#332—612no—
293#332C2×D6112yescomplemented
294#331—612no—
295#331—612no—
296C61#1910111yescharacteristic, complemented, Sylow
297C2×#41—1222no—
298C22×D4—611no—
299C23×C4—1221no—
300#135—2444no—
301#135—2444no—
302C2×Q16—21no—
303C2×Q16—21no—
304C2×SD16—1222no—
305C2×SD16—1222no—
306C2×D8—1222no—
307C2×D8—1222no—
308C2×D8—1221no—
309C2×D8—1221no—
310C22×C8—21no—
311C22×C8—611no—
312C22×C8—611no—
313C22×C8—1221no—
314C22×C8—1221no—
315C22×C8—1221no—
316C22×C8—1221no—
317C22×C8—2444no—
318C4⋊Q8C22×D6112yes—
319#127—612no—
320#124—1222no—
321#121—1222no—
322#121—1222no—
323C4×D4—611no—
324C4×D4—611no—
325C4×D4—611no—
326C4×D4—1221no—
327C4×D4—1221no—
328C4×D4—1222no—
329C4×D4—1222no—
330C4×D4C22×D6111yescharacteristic
331C2×C4⋊C4—611no—
332C2×C22⋊C4—1221no—
333C2×C42—611no—
334#107—22no—
335#107—1222no—
336C4⋊C8—611no—
337C4⋊C8—612no—
338C4⋊C8—1221no—
339C4⋊C8C22×D6111yescharacteristic
340C4⋊C8C22×D6112yes—
341#102—22no—
342#102—1222no—
343#102—1222no—
344#102—1222no—
345C22⋊C8—21no—
346C22⋊C8—1221no—
347C22⋊C8—1222no—
348C22⋊C8—1221no—
349C22⋊C8—1221no—
350C4×C8—611no—
351C4×C8—611no—
352C4×C8—611no—
353C4×C8—1222no—
354C4×C8C22×D6111yescharacteristic
355C24—1221no—
356#41—2444no—
357#41—2444no—
358C2×Q8—22no—
359C2×D4—611no—
360C2×D4—1221no—
361C2×D4—1222no—
362C2×D4—1222no—
363C2×D4—1222no—
364C2×D4—1222no—
365C2×D4—1222no—
366C2×D4—2442no—
367C2×D4D4×D6111yescharacteristic
368C22×C4—21no—
369C22×C4—611no—
370C22×C4—611no—
371C22×C4—611no—
372C22×C4—1221no—
373C22×C4—1221no—
374C22×C4—1221no—
375C22×C4—1221no—
376C22×C4—2442no—
377C22×C4—2442no—
378C22×C4—2442no—
379Q16—42no—
380Q16—42no—
381SD16—2444no—
382SD16—2444no—
383D8—2442no—
384D8—2444no—
385D8—2444no—
386D8—2442no—
387C2×C8—22no—
388C2×C8—21no—
389C2×C8—42no—
390C2×C8—611no—
391C2×C8—614no—
392C2×C8—611no—
393C2×C8—1222no—
394C2×C8—1222no—
395C2×C8—1221no—
396C2×C8—2442no—
397C2×C8—2442no—
398C2×C8—2442no—
399C2×C8—2442no—
400C2×C8D4×D6111yescharacteristic
401C2×C8D4×D6111yescharacteristic
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—611no—
405C4⋊C4—1221no—
406C4⋊C4D4×D6111yescharacteristic
407C22⋊C4—21no—
408C22⋊C4—1221no—
409C22⋊C4—1221no—
410C22⋊C4—1221no—
411C22⋊C4—2442no—
412C22⋊C4—2442no—
413C42—611no—
414C42—611no—
415C42—611no—
416C42C23×D6111yescharacteristic
417C23—21no—
418C23—611no—
419C23—1224no—
420C23—1221no—
421C23—2442no—
422C23—2442no—
423C23—2442no—
424Q8—42no—
425Q8—42no—
426D4—21no—
427D4—612no—
428D4—1224no—
429D4—1221no—
430D4—4882no—
431D4—4884no—
432D4—2442no—
433D4—2442no—
434D4—2442no—
435D4—2442no—
436D4#1451980012yescomplemented
437C2×C4—21no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—611no—
442C2×C4—612no—
443C2×C4—612no—
444C2×C4—611no—
445C2×C4—611no—
446C2×C4—1221no—
447C2×C4—1221no—
448C2×C4—1221no—
449C2×C4—4884no—
450C2×C4—2441no—
451C2×C4—2442no—
452C2×C4—2442no—
453C2×C4—2442no—
454C2×C4—2441no—
455C2×C4—2441no—
456C2×C4C2×D4×D6111yescharacteristic
457C2×C4C2×D4×D6111yescharacteristic
458C2×C4#41×D6111yescharacteristic
459C8—21no—
460C8—41no—
461C8—42no—
462C8—612no—
463C8—1221no—
464C8—2441no—
465C8—2442no—
466C8D8×D6112yes—
467C22—22no—
468C22—41no—
469C22—41no—
470C22—612no—
471C22—1221no—
472C22—1221no—
473C22—1222no—
474C22—4882no—
475C22—4882no—
476C22—4882no—
477C22—2448no—
478C22—2441no—
479C22—2441no—
480C22#121×D6111yescharacteristic
481C4—21no—
482C4—41no—
483C4—82no—
484C4—612no—
485C4—1221no—
486C4—1221no—
487C4—2441no—
488C4C2×D8×D6111yescharacteristic
489C4#135×D6111yescharacteristic
490C4C2×#1451980011yescharacteristic
491C4AGL(1,ℤ/8)×D6111yescharacteristic
492C2—42no—
493C2—1222no—
494C2—9762no—
495C2—2442no—
496C2#459×D6111yescharacteristic
497C2#2145540211yescharacteristic
498C2#2145554111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1669C2
socleC22×C61C2×#121
hypercenter#1669D61
hypocenterC61#19101
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1669

Complements

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

NH ≅ G/Ntype
1. #21456248C2semidirect
2. #21455821C2semidirect
3. C8×D4×D61C2semidirect
4. #21445181C2semidirect
5. #21445175C2semidirect
6. #21445174C2semidirect
7. #21445163C2semidirect
8. #21445162C2semidirect
9. #21444566C2semidirect
10. #21444467C2semidirect
11. C61×#1669C2semidirect
19. #16840139C22semidirect
20. #16840136C22semidirect
37. #16839515C22semidirect
38. #16839512C22semidirect
39. #16839508C22semidirect
40. #16839505C22semidirect
41. #16839505C22semidirect
42. C61×#494C22semidirect
45. C61×#335C22semidirect
46. C61×#332C22semidirect
166. C8×D61D8semidirect
177. D4×C61#135semidirect
198. D244#135semidirect
247. #1669D61semidirect
268. #494C2×D61semidirect
290. #335C2×D61semidirect
293. #332C2×D61semidirect
296. C61#19101semidirect
436. D4#14519800semidirect

Maximal subgroups

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

subgroupindex
#214562482
#214558212
C8×D4×D612
#214451812
#214451752
#214451742
#214451632
#214451622
#214445662
#214444672
C61×#16692
#1910161