Finite Group Databasebeta

← #29686886

Subgroups — #29686886

Order 15616.

946 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840032C211yesmaximal, characteristic
2#21454632C212yesmaximal, complemented
3C2×#16839844C211yesmaximal, characteristic, complemented
4#131×Dic61C211yesmaximal, characteristic, complemented
5#21446727C211yesmaximal, characteristic, complemented
6#21446602C211yesmaximal, characteristic, complemented
7#21446479C212yesmaximal, complemented
8#21446325C212yesmaximal, complemented
9#108×Dic61C211yesmaximal, characteristic, complemented
10#21445610C212yesmaximal, complemented
11C61×#1816C211yesmaximal, characteristic, complemented
12C2×#14519802C2212yes—
13C22×D488C2211yescharacteristic
14C2×#14519784C2211yescharacteristic
15#41×Dic61C2211yescharacteristic
16C2×#14519739C2212yes—
17#16840032C418yescomplemented
18#16839871—22no—
19#16839865—22no—
20#16839847—22no—
21#16839844—21no—
22#16839844—21no—
23#16839844—21no—
24#16839844—21no—
25#16839842—22no—
26C2×#14519713C2211yescharacteristic
27#16839802C2212yes—
28#16839768C2212yes—
29C2×#14519697C2212yes—
30C2×#14519693C2212yes—
31C2×#14519693C2212yes—
32#16839720—21no—
33#16839720C2211yescharacteristic
34#16839717—21no—
35#16839717C2211yescharacteristic
36M16×Dic61—21no—
37M16×Dic61C2212yes—
38M16×Dic61C2211yescharacteristic
39C2×C8×Dic61—21no—
40C2×C8×Dic61C2211yescharacteristic
41#16839531C2212yes—
42#16839526C2212yes—
43C2×#131×C61C2211yescharacteristic
44C2×#108×C61C2211yescharacteristic
45C2×#107×C61C2211yescharacteristic
46C61×#417C2212yes—
47C61×#343C2212yes—
48C22×D244C2312yes—
49C22×C4×D61—22no—
50#14519802—24no—
51#14519802C2×C418yescomplemented
52C2×#11758201—22no—
53C2×D488—21no—
54C2×D488—21no—
55C2×D488—22no—
56C2×D488C2×C414yescomplemented
57C2×#11758199—22no—
58#14519784—22no—
59#14519784—22no—
60C2×#11758194—21no—
61C2×#11758194C2312yes—
62C2×#11758194C2311yescharacteristic
63C22×#92418—21no—
64C22×#92418C2311yescharacteristic
65Q8×Dic61D411yescharacteristic
66Q8×Dic61D411yescharacteristic
67D4×Dic61—21no—
68D4×Dic61—21no—
69D4×Dic61D411yescharacteristic
70D4×Dic61D411yescharacteristic
71#14519739—22no—
72#14519739—22no—
73#14519739—22no—
74#14519739—22no—
75#14519736—21no—
76#14519736C2311yescharacteristic
77C2×#11758198—22no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61C2311yescharacteristic
80#14519713C2×C414yescomplemented
81#14519710—22no—
82#14519708—22no—
83#14519699—22no—
84#14519699—22no—
85#14519697—22no—
86#14519697—22no—
87#14519693—24no—
88#14519693C2×C418yescomplemented
89#14519687—21no—
90#14519687—21no—
91#14519687—22no—
92#14519687—21no—
93C8×Dic61—21no—
94C8×Dic61—21no—
95C8×Dic61—22no—
96C8×Dic61—21no—
97C8×Dic61D411yescharacteristic
98C8×Dic61D411yescharacteristic
99C8×Dic61D411yescharacteristic
100C8×Dic61D411yescharacteristic
101C2×#41×C61C2311yescharacteristic
102#131×C61—22no—
103#131×C61C2×C414yescomplemented
104C2×M16×C61—21no—
105C2×M16×C61C2312yes—
106C2×M16×C61C2311yescharacteristic
107C22×C8×C61—21no—
108C22×C8×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2312yes—
110#108×C61—22no—
111#107×C61—21no—
112#107×C61—21no—
113#103×C61—22no—
114#102×C61—22no—
115C23×D61—22no—
116#41×C61—21no—
117#41×C61—21no—
118#41×C61C22×C412yes—
119#41×C61#4112yes—
120C2×Q8×C61C2×D411yescharacteristic
121C2×D4×C61—21no—
122C2×D4×C61C2×D411yescharacteristic
123C22×C4×C61—21no—
124C22×C4×C61—22no—
125C22×C4×C61C2411yescharacteristic
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61#4111yescharacteristic
129C22×Dic61#4111yescharacteristic
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—22no—
135C2×D244C22×C414yes—
136C2×C4×D61—42no—
137C2×C4×D61—42no—
138C2×C4×D61—84no—
139M16×C61—21no—
140M16×C61—24no—
141M16×C61—22no—
142M16×C61—22no—
143M16×C61C22×C412yes—
144C2×C8×C61—21no—
145C2×C8×C61—24no—
146C2×C8×C61—22no—
147C2×C8×C61C22×C412yes—
148C2×C8×C61C2×D411yescharacteristic
149C2×C8×C61C2×D411yescharacteristic
150C2×C8×C61#4112yes—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C22⋊Dic61—21no—
154C22⋊Dic61—21no—
155C22⋊Dic61—21no—
156#11758201—28no—
157#11758199—28no—
158#11758198—42no—
159#11758198—42no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162C4⋊Dic61C2×D411yescharacteristic
163C4⋊Dic61C2×D411yescharacteristic
164C4×Dic61—21no—
165C4×Dic61—21no—
166C4×Dic61—21no—
167C4×Dic61C2×D411yescharacteristic
168C4×Dic61C2×D411yescharacteristic
169C4×Dic61C2×D411yescharacteristic
170C4×Dic61C2×D411yescharacteristic
171#11758194—24no—
172#11758194—21no—
173#11758194—22no—
174#11758194C22×C414yes—
175#11758194#4112yes—
176C2×#92418—24no—
177C2×#92418—21no—
178C2×#92418—22no—
179C2×#92418—21no—
180C2×#92418—21no—
181C2×#92418C2×D411yescharacteristic
182C2×#92418C2×D411yescharacteristic
183D488—24no—
184D488—24no—
185C23×C61—21no—
186C23×C61C2×#4111yescharacteristic
187C22×D61—24no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—82no—
191C22×D61—82no—
192Q8×C61—21no—
193Q8×C61C4×D412yes—
194D4×C61—22no—
195D4×C61—22no—
196D4×C61—21no—
197D4×C61C4×D412yes—
198C2×C4×C61—21no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—42no—
202C2×C4×C61—82no—
203C2×C4×C61C23×C411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×Dic61—22no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—41no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61#12111yescharacteristic
216C2×Dic61#12111yescharacteristic
217C2×Dic61#12111yescharacteristic
218C2×Dic61#12111yescharacteristic
219D244—24no—
220D244—24no—
221D244—42no—
222D244—42no—
223C4×D61—84no—
224C4×D61—84no—
225C8×C61—22no—
226C8×C61—42no—
227C8×C61C4×D412yes—
228C8×C61C4×D412yes—
229#92418—24no—
230#92418—21no—
231#92418—22no—
232#92418—21no—
233C4×#472—611nomaximal, Sylow
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—22no—
237C22×C61—41no—
238C22×C61—41no—
239C22×C61C4×#4111yescharacteristic
240C22×C61C2×#12111yescharacteristic
241C22×C61#52911yescharacteristic
242C2×D61—44no—
243C2×D61—84no—
244C2×D61—82no—
245C2×D61—82no—
246C2×D61—84no—
247C4×C61—22no—
248C4×C61—82no—
249C4×C61C2×C4×D411yescharacteristic
250C4×C61C2×C4×D411yescharacteristic
251C4×C61#52511yescharacteristic
252C4×C61#53211yescharacteristic
253Dic61—21no—
254Dic61—21no—
255Dic61—21no—
256Dic61—41no—
257Dic61—41no—
258Dic61#47212yescomplemented
259C2×#472—611no—
260C4×AGL(1,ℤ/8)—612no—
261C2×C4×D8—611no—
262C4×#131—611no—
263#1978—611no—
264#1952—611no—
265#1931—612no—
266#1907—612no—
267#1816D6111yescharacteristic, complemented
268C4×#108—611no—
269C4×#100—612no—
270C2×C61—21no—
271C2×C61—21no—
272C2×C61—42no—
273C2×C61C4×#12111yescharacteristic
274C2×C61C2×#47211yescharacteristic
275C2×C61#306711yescharacteristic
276D61—84no—
277C2×AGL(1,ℤ/8)—612no—
278C22×D8—611no—
279C2×#131—611no—
280C2×#131C2×D6111yescharacteristic
281C4×#41—611no—
282C2×C4×D4—612no—
283#472—618no—
284#442—1222no—
285#440—1222no—
286C4×SD16—1222no—
287C4×D8—1221no—
288C4×D8—1221no—
289C4×D8—1221no—
290C4×D8—1221no—
291C4×D8—1222no—
292C2×#108—611no—
293C2×#108C2×D6111yescharacteristic
294#428—612no—
295C2×#107C2×D6111yescharacteristic
296#418—612no—
297#417C2×D6112yes—
298C2×#102—612no—
299C2×#100—612no—
300C2×#100—612no—
301#405—611no—
302#405—611no—
303#405—1221no—
304#405—1221no—
305C4×M16—611no—
306C4×M16—612no—
307C4×M16—1221no—
308C2×C4×C8—611no—
309C2×C4×C8—1221no—
310#343—612no—
311#343C2×D6112yes—
312#341—612no—
313C61C4×#47211yescharacteristic, complemented, Sylow
314C2×#41C22×D6111yescharacteristic
315C22×D4—612no—
316C23×C4—1222no—
317AGL(1,ℤ/8)—618no—
318AGL(1,ℤ/8)—1224no—
319C2×SD16—1222no—
320C2×D8—614no—
321C2×D8—1222no—
322C2×D8—1221no—
323C2×D8—1221no—
324C2×D8—1222no—
325#131—22no—
326#131—1222no—
327#131—1222no—
328#131C4×D6114yescomplemented
329C2×M16—21no—
330C2×M16—611no—
331C2×M16—612no—
332C2×M16—1221no—
333C2×M16C22×D6112yes—
334C2×M16C22×D6111yescharacteristic
335C22×C8—21no—
336C22×C8—611no—
337C22×C8—1221no—
338C22×C8C22×D6111yescharacteristic
339C4×Q8—611no—
340C4×Q8—611no—
341C4×D4—611no—
342C4×D4—611no—
343C4×D4—1222no—
344C4×D4—1222no—
345C4×D4—1222no—
346C4×D4—1222no—
347C4×D4—1221no—
348C4×D4—1221no—
349#117—611no—
350#117—1221no—
351C2×C4⋊C4C22×D6112yes—
352C2×C22⋊C4—1222no—
353C2×C42—611no—
354C2×C42—1221no—
355#108—22no—
356#108—614no—
357#107—21no—
358#107—21no—
359#107—1222no—
360#106—1222no—
361#103—22no—
362#102—22no—
363#102—1222no—
364#102—1222no—
365#102—1222no—
366#102—1222no—
367#100—618no—
368#100—1224no—
369#97—1221no—
370#97—1222no—
371#97—1221no—
372#97—1221no—
373C4×C8—611no—
374C4×C8—611no—
375C4×C8—611no—
376C4×C8—611no—
377C4×C8—1221no—
378C4×C8—1222no—
379C4×C8—1221no—
380C4×C8—1221no—
381C24—1222no—
382#41—21no—
383#41—21no—
384#41C2×C4×D6112yes—
385#41#1175822612yes—
386C2×Q8D4×D6111yescharacteristic
387C2×D4—21no—
388C2×D4—614no—
389C2×D4—1222no—
390C2×D4—1222no—
391C2×D4—1222no—
392C2×D4—1222no—
393C2×D4—1222no—
394C2×D4D4×D6111yescharacteristic
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—611no—
398C22×C4—611no—
399C22×C4—1221no—
400C22×C4—1221no—
401C22×C4—4884no—
402C22×C4—2442no—
403C22×C4—2442no—
404C22×C4C23×D6111yescharacteristic
405SD16—1228no—
406D8—1228no—
407D8—1224no—
408D8—1224no—
409M16—21no—
410M16—24no—
411M16—22no—
412M16—22no—
413M16—614no—
414M16—612no—
415M16—1224no—
416M16—1221no—
417M16—1222no—
418M16C2×C4×D6112yes—
419C2×C8—21no—
420C2×C8—24no—
421C2×C8—22no—
422C2×C8—611no—
423C2×C8—611no—
424C2×C8—1224no—
425C2×C8—1221no—
426C2×C8—1222no—
427C2×C8—1221no—
428C2×C8—1221no—
429C2×C8C2×C4×D6112yes—
430C2×C8D4×D6111yescharacteristic
431C2×C8D4×D6111yescharacteristic
432C2×C8#1175822412yes—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—611no—
436C4⋊C4—611no—
437C4⋊C4—1221no—
438C4⋊C4—1221no—
439C22⋊C4—1221no—
440C22⋊C4—1221no—
441C22⋊C4—1221no—
442C22⋊C4—2442no—
443C22⋊C4—2442no—
444C42—611no—
445C42—611no—
446C42—611no—
447C42—611no—
448C42—1221no—
449C42—1221no—
450C42—1221no—
451C23—21no—
452C23—1224no—
453C23—4882no—
454C23—4882no—
455C23—2442no—
456C23—2442no—
457C23#41×D6111yescharacteristic
458Q8—21no—
459Q8#1451973912yes—
460D4—22no—
461D4—22no—
462D4—21no—
463D4—1224no—
464D4—1224no—
465D4—2442no—
466D4—2442no—
467D4#1451973912yes—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—42no—
472C2×C4—82no—
473C2×C4—611no—
474C2×C4—611no—
475C2×C4—611no—
476C2×C4—611no—
477C2×C4—1222no—
478C2×C4—1221no—
479C2×C4—1221no—
480C2×C4—4884no—
481C2×C4—4884no—
482C2×C4—2442no—
483C2×C4—2441no—
484C2×C4—2442no—
485C2×C4C22×C4×D6111yescharacteristic
486C2×C4C2×D4×D6111yescharacteristic
487C2×C4C2×D4×D6111yescharacteristic
488C2×C4#41×D6111yescharacteristic
489C2×C4C2×#1175822411yescharacteristic
490C2×C4C2×#1175822611yescharacteristic
491C8—22no—
492C8—42no—
493C8—1224no—
494C8—1221no—
495C8—1221no—
496C8—1222no—
497C8#1451973712yes—
498C8#1451973712yes—
499C22—21no—
500C22—21no—
501C22—22no—
502C22—41no—
503C22—41no—
504C22—4884no—
505C22—4884no—
506C22—4882no—
507C22—4882no—
508C22—2444no—
509C22#1684023011yescharacteristic
510C22#121×D6111yescharacteristic
511C22#1684029811yescharacteristic
512C4—22no—
513C4—82no—
514C4—612no—
515C4—1221no—
516C4—1221no—
517C4—1221no—
518C4—2441no—
519C4—2441no—
520C4C2×#1451973711yescharacteristic
521C4C2×#1451973911yescharacteristic
522C4#1684027511yescharacteristic
523C4#1684030711yescharacteristic
524C2—21no—
525C2—21no—
526C2—42no—
527C2—4884no—
528C2#2144832511yescharacteristic
529C2#472×D6111yescharacteristic
530C2#2145554111yescharacteristic
subgroupG/H
centerC22#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1816C2
socleC22×C61C2×#121
prime radicalC2×#16840032C2
cosocleC22×C4×C61C24
hypercenter#1816D61
hypocenterC61C4×#472
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1816

Complements

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

NH ≅ G/Ntype
2. #21454632C2semidirect
3. C2×#16839844C2semidirect
4. #131×Dic61C2semidirect
5. #21446727C2semidirect
6. #21446602C2semidirect
7. #21446479C2semidirect
8. #21446325C2semidirect
9. #108×Dic61C2semidirect
10. #21445610C2semidirect
11. C61×#1816C2semidirect
17. #16840032C4semidirect
51. #14519802C2×C4semidirect
56. C2×D488C2×C4semidirect
80. #14519713C2×C4semidirect
88. #14519693C2×C4semidirect
103. #131×C61C2×C4semidirect
258. Dic61#472semidirect
267. #1816D61semidirect
313. C61C4×#472semidirect
328. #131C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400322
#214546322
C2×#168398442
#131×Dic612
#214467272
#214466022
#214464792
#214463252
#108×Dic612
#214456102
C61×#18162
C4×#47261