Finite Group Databasebeta

← #29662871

Subgroups — #29662871

Order 15616.

766 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456272C211yesmaximal, characteristic
2C2×#16839962C211yesmaximal, characteristic, complemented
3Q8×#11758194C211yesmaximal, characteristic, complemented
4#21445301C212yesmaximal, complemented
5#21445293C212yesmaximal, complemented
6#21445177C212yesmaximal, complemented
7#21445169C212yesmaximal, complemented
8#21444572C211yesmaximal, characteristic, complemented
9#21444496C211yesmaximal, characteristic, complemented
10#21444486C211yesmaximal, characteristic, complemented
11C61×#3103C211yesmaximal, characteristic, complemented
12C2×#14519802—21no—
13C2×#14519802—21no—
14C22×#11758201—21no—
15C2×#14519774C2211yescharacteristic
16#16840152—22no—
17#16840138C2212yes—
18#16840136C2212yes—
19#16839962—21no—
20#16839962—21no—
21#16839962C2212yes—
22#16839962C2212yes—
23#16839893—22no—
24#16839893C2212yescomplemented
25Q8×#92418C2212yescomplemented
26#16839786—21no—
27#16839786C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519697—21no—
30C4×#11758194—21no—
31C4×#11758194C2211yescharacteristic
32#16839514C2212yescomplemented
33#16839512C2212yescomplemented
34#16839508C2212yescomplemented
35#16839506C2212yescomplemented
36#16839505C2212yes—
37#16839505C2212yes—
38C61×#532C2211yescharacteristic
39C2×C4×Q8×C61C2211yescharacteristic
40C61×#481—21no—
41C61×#475C2211yescharacteristic, complemented
42C61×#475C2211yescharacteristic, complemented
43C61×#460C2211yescharacteristic, complemented
44C61×#460C2211yescharacteristic, complemented
45C61×#448—21no—
46C4×SD16×C61—21no—
47C4×SD16×C61—21no—
48C61×#431C2211yescharacteristic
49C22×D244—21no—
50C22×D244—21no—
51C22×D244—21no—
52#14519802—44no—
53#14519802—44no—
54C2×#11758201—22no—
55C2×#11758201—22no—
56C2×#11758201—22no—
57C2×#11758201—22no—
58C2×#11758201—44no—
59C2×#11758199—22no—
60C2×#11758199—22no—
61C2×#11758194—21no—
62C2×#11758194D411yescharacteristic
63C2×#11758194D411yescharacteristic
64C22×#92418—21no—
65#14519774—21no—
66#14519774—21no—
67#14519774—21no—
68#14519774C2312yes—
69C4⋊#92418—21no—
70C4⋊#92418—21no—
71C4⋊#92418—22no—
72C4⋊#92418—22no—
73C4⋊#92418C2312yes—
74C4⋊#92418C2312yes—
75#14519697—22no—
76#14519697—22no—
77#14519688—22no—
78#14519688D412yes—
79C4×#92418—22no—
80C4×#92418C2312yes—
81C2×#41×C61—21no—
82C22×Q8×C61D411yescharacteristic
83C2×SD16×C61—21no—
84C2×SD16×C61—21no—
85C4⋊Q8×C61C2311yescharacteristic
86#127×C61C2311yescharacteristic
87#124×C61—21no—
88#121×C61—21no—
89#121×C61—21no—
90C4×Q8×C61—21no—
91C4×Q8×C61—21no—
92C4×Q8×C61D411yescharacteristic
93C4×Q8×C61D411yescharacteristic
94C4×Q8×C61C2311yescharacteristic
95C4×Q8×C61C2311yescharacteristic
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C42×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#106×C61—21no—
103#106×C61—21no—
104C4⋊C8×C61C2311yescharacteristic
105C4⋊C8×C61C2311yescharacteristic
106#103×C61—21no—
107#103×C61—21no—
108#102×C61—21no—
109#102×C61—21no—
110C22⋊C8×C61—21no—
111C4×C8×C61—21no—
112C23×D61—41no—
113#41×C61—42no—
114#41×C61—42no—
115C2×Q8×C61—21no—
116C2×Q8×C61—22no—
117C2×Q8×C61—21no—
118C2×Q8×C61—41no—
119C2×Q8×C61C2×D411yescharacteristic
120C2×Q8×C61C2×D411yescharacteristic
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—41no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—41no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61#4111yescharacteristic
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—42no—
135C2×D244—42no—
136C2×D244—42no—
137C2×D244—42no—
138C2×D244—41no—
139C2×D244—41no—
140C2×D244—41no—
141C2×D244—82no—
142SD16×C61—42no—
143SD16×C61—42no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61C2×D411yescharacteristic
154C4⋊C4×C61C2×D411yescharacteristic
155C22⋊C4×C61—41no—
156C22⋊C4×C61—41no—
157C42×C61—21no—
158C42×C61—21no—
159C42×C61—21no—
160C42×C61—21no—
161C42×C61C2×D411yescharacteristic
162C42×C61C2×D411yescharacteristic
163C42×C61#4111yescharacteristic
164C42×C61C2411yescharacteristic
165#11758201—44no—
166#11758201—44no—
167#11758201—44no—
168#11758201—44no—
169#11758199—44no—
170#11758199—44no—
171#11758194—22no—
172#11758194—42no—
173#11758194SD1614yescomplemented
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—42no—
177C2×#92418C2×D412yes—
178C2×#92418C2×D412yes—
179C23×C61—41no—
180C23×C61#12111yescharacteristic
181C22×D61—42no—
182C22×D61—82no—
183C22×D61—82no—
184C22×D61—82no—
185Q8×C61—21no—
186Q8×C61—21no—
187Q8×C61—21no—
188Q8×C61—42no—
189Q8×C61—41no—
190Q8×C61—41no—
191Q8×C61AGL(1,ℤ/8)12yescomplemented
192D4×C61—41no—
193D4×C61—41no—
194D4×C61—41no—
195D4×C61—41no—
196D4×C61—81no—
197D4×C61—82no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—41no—
205C2×C4×C61—42no—
206C2×C4×C61—41no—
207C2×C4×C61—81no—
208C2×C4×C61—82no—
209C2×C4×C61#12111yescharacteristic
210C2×C4×C61#12111yescharacteristic
211C2×C4×C61#12111yescharacteristic
212C2×C4×C61C2×SD1612yes—
213C2×C4×C61AGL(1,ℤ/8)12yes—
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217D244—42no—
218D244—42no—
219D244—42no—
220D244—42no—
221D244—82no—
222D244—82no—
223D244—82no—
224D244—82no—
225C8×C61—41no—
226C8×C61—41no—
227#92418—22no—
228#92418—42no—
229#92418—42no—
230#92418C2×SD1614yescomplemented
231#19170—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—81no—
235C22×C61—81no—
236C22×C61—81no—
237C22×C61#46012yes—
238C22×C61C2×#12111yescharacteristic
239C2×D61—82no—
240C2×D61—82no—
241C2×D61—82no—
242C2×D61—164no—
243C4×C61—21no—
244C4×C61—22no—
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—41no—
248C4×C61—41no—
249C4×C61—41no—
250C4×C61—81no—
251C4×C61#46012yes—
252C4×C61C22×SD1611yescharacteristic
253C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
254C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
255C4×C61#57511yescharacteristic
256#3151—611no—
257#3103D6111yescharacteristic, complemented
258C2×#460—611no—
259Q8×M16—611no—
260#1688—612no—
261#1686—612no—
262#1668—612no—
263#1666—612no—
264#1565—611no—
265#1547—611no—
266#1545—611no—
267C2×C61—22no—
268C2×C61—161no—
269C2×C61C2×#46011yescharacteristic
270C2×C61#305511yescharacteristic
271C2×C61#306811yescharacteristic
272D61—162no—
273C2×AGL(1,ℤ/8)—1221no—
274C2×AGL(1,ℤ/8)—1221no—
275C22×SD16—1221no—
276#532C2×D6111yescharacteristic
277C2×#127—611no—
278C2×C4×Q8C2×D6111yescharacteristic
279#496—1222no—
280#493—612no—
281#492—612no—
282#481—21no—
283#475C2×D6111yescharacteristic, complemented
284#475C2×D6111yescharacteristic, complemented
285#460—612no—
286#460—612no—
287#460—1221no—
288#460—1221no—
289#460C2×D6111yescharacteristic, complemented
290#460C2×D6111yescharacteristic, complemented
291#448—21no—
292#446—612no—
293#446—1222no—
294C8×Q8—612no—
295C4×SD16—21no—
296C4×SD16—21no—
297#431C2×D6111yescharacteristic
298#423—611no—
299#423—1221no—
300C2×C4⋊C8—611no—
301C2×#102—1221no—
302C4×M16—611no—
303C4×M16—1221no—
304#335—612no—
305#334—612no—
306#332—612no—
307#331—612no—
308#331—612no—
309#331—612no—
310C61#1917011yescharacteristic, complemented, Sylow
311C2×#41—21no—
312C22×Q8C22⋊D6111yescharacteristic
313C22×D4—1221no—
314C22×D4—1221no—
315C22×D4—1221no—
316AGL(1,ℤ/8)—2444no—
317AGL(1,ℤ/8)—2444no—
318C2×SD16—21no—
319C2×SD16—21no—
320C2×SD16—1222no—
321C2×SD16—1222no—
322C2×SD16—1222no—
323C2×SD16—1222no—
324C2×SD16—2444no—
325C2×D8—1222no—
326C2×D8—1222no—
327C2×M16—611no—
328C2×M16—611no—
329C2×M16—1221no—
330C22×C8—1221no—
331C4⋊Q8C22×D6111yescharacteristic
332#127—612no—
333#127—1221no—
334#127—1221no—
335#127—1221no—
336#127C22×D6111yescharacteristic
337#124—21no—
338#121—21no—
339#121—21no—
340C4×Q8—21no—
341C4×Q8—21no—
342C4×Q8C22⋊D6111yescharacteristic
343C4×Q8C22⋊D6111yescharacteristic
344C4×Q8C22×D6111yescharacteristic
345C4×Q8C22×D6111yescharacteristic
346C4×D4—21no—
347C4×D4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4C22⋊D6111yescharacteristic
350C2×C42—21no—
351C2×C42C22×D6111yescharacteristic
352#106—21no—
353#106—21no—
354C4⋊C8—612no—
355C4⋊C8—612no—
356C4⋊C8—1222no—
357C4⋊C8—1222no—
358C4⋊C8—1221no—
359C4⋊C8—1221no—
360C4⋊C8C22×D6111yescharacteristic
361C4⋊C8C22×D6111yescharacteristic
362#103—21no—
363#103—21no—
364#102—21no—
365#102—21no—
366#102—1222no—
367#102—1222no—
368C22⋊C8—21no—
369#97—612no—
370#97—1222no—
371C4×C8—21no—
372C4×C8—612no—
373C4×C8—1222no—
374C24—2441no—
375#41—42no—
376#41—42no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—21no—
380C2×Q8—41no—
381C2×Q8C2×C22⋊D6111yescharacteristic
382C2×Q8C2×C22⋊D6111yescharacteristic
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—1222no—
388C2×D4—1222no—
389C2×D4—1222no—
390C2×D4—4882no—
391C2×D4—2442no—
392C2×D4—2442no—
393C2×D4—2441no—
394C2×D4—2441no—
395C2×D4—2441no—
396C2×D4—2442no—
397C2×D4—2442no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—41no—
401C22×C4D4×D6111yescharacteristic
402C22×C4#1175822411yescharacteristic
403C22×C4C2×C22⋊D6111yescharacteristic
404SD16—42no—
405SD16—42no—
406SD16—2444no—
407SD16—2444no—
408SD16—2444no—
409SD16—2444no—
410D8—2444no—
411D8—2444no—
412M16—614no—
413M16—1222no—
414M16—2442no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—612no—
418C2×C8—612no—
419C2×C8—1222no—
420C2×C8—1222no—
421C2×C8—2442no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4C2×C22⋊D6111yescharacteristic
430C4⋊C4C2×C22⋊D6111yescharacteristic
431C22⋊C4—41no—
432C22⋊C4—41no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42D4×D6111yescharacteristic
438C42#1175822411yescharacteristic
439C42C2×C22⋊D6111yescharacteristic
440C42C23×D6111yescharacteristic
441C23—41no—
442C23—4882no—
443C23—4882no—
444C23—4882no—
445C23—2442no—
446C23#1451974911yescharacteristic
447Q8—21no—
448Q8—21no—
449Q8—21no—
450Q8—42no—
451Q8—41no—
452Q8—41no—
453Q8#1451980212yescomplemented
454D4—41no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—81no—
459D4—82no—
460D4—4882no—
461D4—4882no—
462D4—4882no—
463D4—4882no—
464D4—2442no—
465D4—2442no—
466D4—2442no—
467D4—2442no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4#1451974911yescharacteristic
480C2×C4#1451974911yescharacteristic
481C2×C4#1451974911yescharacteristic
482C2×C4SD16×D6112yes—
483C2×C4#1451980412yes—
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4C22×C22⋊D6111yescharacteristic
486C2×C4C2×#1175822411yescharacteristic
487C8—41no—
488C8—41no—
489C8—614no—
490C8—1222no—
491C8—2442no—
492C8—2442no—
493C22—21no—
494C22—21no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—9764no—
499C22—4882no—
500C22—4882no—
501C22—4882no—
502C22#1683996012yes—
503C22C2×#1451974911yescharacteristic
504C4—21no—
505C4—22no—
506C4—21no—
507C4—21no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—81no—
512C4#1683996012yes—
513C4C2×SD16×D6111yescharacteristic
514C4C2×#1451980211yescharacteristic
515C4C2×#1451980411yescharacteristic
516C4#1684056011yescharacteristic
517C2—22no—
518C2—161no—
519C2—9762no—
520C2C2×#1683996011yescharacteristic
521C2#2145544011yescharacteristic
522C2#2145554911yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3103C2
socleC22×C61C2×#121
prime radical#21456272C2
hypercenter#3103D61
hypocenterC61#19170
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
2. C2×#16839962C2semidirect
3. Q8×#11758194C2semidirect
4. #21445301C2semidirect
5. #21445293C2semidirect
6. #21445177C2semidirect
7. #21445169C2semidirect
8. #21444572C2semidirect
9. #21444496C2semidirect
10. #21444486C2semidirect
11. C61×#3103C2semidirect
24. #16839893C22semidirect
25. Q8×#92418C22semidirect
32. #16839514C22semidirect
33. #16839512C22semidirect
34. #16839508C22semidirect
35. #16839506C22semidirect
41. C61×#475C22semidirect
42. C61×#475C22semidirect
43. C61×#460C22semidirect
44. C61×#460C22semidirect
173. #11758194SD16semidirect
191. Q8×C61AGL(1,ℤ/8)semidirect
230. #92418C2×SD16semidirect
257. #3103D61semidirect
283. #475C2×D61semidirect
284. #475C2×D61semidirect
289. #460C2×D61semidirect
290. #460C2×D61semidirect
310. C61#19170semidirect
453. Q8#14519802semidirect

Maximal subgroups

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

subgroupindex
#214562722
C2×#168399622
Q8×#117581942
#214453012
#214452932
#214451772
#214451692
#214445722
#214444962
#214444862
C61×#31032
#1917061