Finite Group Databasebeta

← #29686896

Subgroups — #29686896

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×#16840029C211yesmaximal, characteristic
2#21454623C212yesmaximal, complemented
3C2×D8×Dic61C211yesmaximal, characteristic, complemented
4#21454241C211yesmaximal, characteristic, complemented
5#21446604C211yesmaximal, characteristic, complemented
6#21446484C212yesmaximal, complemented
7#21446324C212yesmaximal, complemented
8#21445880C211yesmaximal, characteristic, complemented
9#108×Dic61C211yesmaximal, characteristic, complemented
10#100×Dic61C212yesmaximal, complemented
11C61×#1978C211yesmaximal, characteristic, complemented
12C2×#14519803C2212yes—
13C2×#14519785C2211yescharacteristic
14C2×#14519783C2211yescharacteristic
15#16840230C2211yescharacteristic
16C2×D4×Dic61C2212yes—
17#16840029C418yescomplemented
18#16839871—22no—
19#16839859—22no—
20#16839853—22no—
21D8×Dic61—21no—
22D8×Dic61—21no—
23D8×Dic61—21no—
24D8×Dic61—21no—
25#16839842—22no—
26C2×#14519713C2211yescharacteristic
27#16839805C2212yes—
28C2×#14519711C2211yescharacteristic
29#16839764C2212yes—
30C2×#14519698C2212yes—
31C2×#14519694C2212yes—
32#16839719—21no—
33#16839718C2211yescharacteristic
34#16839716—21no—
35#16839714—21no—
36M16×Dic61C2212yes—
37M16×Dic61C2211yescharacteristic
38C2×C8×Dic61C2211yescharacteristic
39C4×C8×D61—21no—
40#16839530C2212yes—
41#16839530C2212yes—
42#16839526C2212yes—
43C22×D8×C61C2211yescharacteristic
44C2×#108×C61C2211yescharacteristic
45C61×#418C2212yes—
46C2×#100×C61C2212yes—
47C61×#405C2211yescharacteristic
48C2×#11758222C2311yescharacteristic
49C23×Dic61—22no—
50#14519803—24no—
51#14519803C2×C418yescomplemented
52C2×#11758200—22no—
53C2×#11758199—22no—
54#14519785—22no—
55#14519785—22no—
56#14519783—22no—
57#14519783C2×C414yescomplemented
58C2×#11758194C2311yescharacteristic
59C2×#11758194C2312yes—
60C2×#11758189—21no—
61C2×#11758189—21no—
62C22×#92418C2311yescharacteristic
63C2×C8×D61—21no—
64C2×C8×D61—21no—
65#14519742D411yescharacteristic
66#14519742D411yescharacteristic
67D4×Dic61—22no—
68D4×Dic61—22no—
69D4×Dic61—22no—
70D4×Dic61—22no—
71#14519739D411yescharacteristic
72#14519739D411yescharacteristic
73#14519737—21no—
74#14519737—21no—
75#14519734—21no—
76C2×C4⋊Dic61C2312yes—
77C2×C22⋊Dic61—22no—
78C2×C4×Dic61C2311yescharacteristic
79C42×D61—21no—
80#14519713C2×C414yescomplemented
81#14519711—21no—
82#14519711—21no—
83#14519710—22no—
84#14519708—22no—
85#14519702—22no—
86#14519699—22no—
87#14519698—22no—
88#14519698—22no—
89#14519694C2×C418yescomplemented
90#14519688—21no—
91#14519687—21no—
92#14519687—22no—
93C4×#92418—21no—
94C8×Dic61—21no—
95C8×Dic61—22no—
96C8×Dic61D411yescharacteristic
97C8×Dic61D411yescharacteristic
98C8×Dic61D411yescharacteristic
99C8×Dic61D411yescharacteristic
100C22×D4×C61C2312yes—
101C2×D8×C61—21no—
102C2×D8×C61—21no—
103C2×D8×C61—22no—
104C2×D8×C61C2×C414yescomplemented
105C2×M16×C61C2312yes—
106C2×M16×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108#117×C61C2311yescharacteristic
109#108×C61—22no—
110#102×C61—22no—
111#102×C61—22no—
112#100×C61—24no—
113#97×C61—21no—
114C4×C8×C61—21no—
115C24×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61C22×C414yes—
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—21no—
124C22×Dic61—22no—
125C22×Dic61—42no—
126C22×Dic61—42no—
127C22×Dic61—84no—
128C22×Dic61#4111yescharacteristic
129C22×Dic61#4111yescharacteristic
130#11758222—21no—
131#11758222—21no—
132#11758222C22×C412yes—
133#11758222#4112yes—
134C2×D244C2×D411yescharacteristic
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×Dic122C2×D411yescharacteristic
139D8×C61—24no—
140D8×C61—24no—
141M16×C61—21no—
142M16×C61—22no—
143M16×C61—22no—
144M16×C61#4112yes—
145C2×C8×C61—21no—
146C2×C8×C61—22no—
147C2×C8×C61C22×C412yes—
148C2×C8×C61C2×D411yescharacteristic
149C2×C8×C61C2×D411yescharacteristic
150C2×C8×C61#4112yes—
151C4⋊C4×C61C2×D411yescharacteristic
152C4⋊C4×C61C2×D411yescharacteristic
153C22⋊C4×C61—21no—
154C42×C61—21no—
155C22⋊Dic61—42no—
156C22⋊Dic61—42no—
157#11758200—28no—
158#11758199—28no—
159#11758198—21no—
160#11758198—21no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163#11758196—21no—
164#11758196—21no—
165C4×Dic61—21no—
166C4×Dic61—21no—
167C4×Dic61C2×D411yescharacteristic
168C4×Dic61C2×D411yescharacteristic
169C4×Dic61C2×D411yescharacteristic
170C4×Dic61C2×D411yescharacteristic
171#11758194—21no—
172#11758194—22no—
173#11758194C22×C412yes—
174#11758194C22×C414yes—
175C2×#92418—21no—
176C2×#92418—22no—
177C2×#92418—21no—
178C2×#92418—21no—
179C2×#92418C2×D411yescharacteristic
180C2×#92418C2×D411yescharacteristic
181#11758189—24no—
182#11758189—24no—
183C8×D61—24no—
184C8×D61—24no—
185C23×C61—24no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—82no—
189C23×C61—82no—
190C23×C61C2×#4111yescharacteristic
191C22×D61—21no—
192D4×C61—24no—
193D4×C61—24no—
194D4×C61—42no—
195D4×C61—42no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61C23×C411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C2×#4111yescharacteristic
202C2×C4×C61C2×#4111yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C22⋊D61—22no—
205C22⋊D61—22no—
206C2×Dic61—21no—
207C2×Dic61—22no—
208C2×Dic61—41no—
209C2×Dic61—42no—
210C2×Dic61—84no—
211C2×Dic61—84no—
212C2×Dic61—82no—
213C2×Dic61#12111yescharacteristic
214C2×Dic61#12111yescharacteristic
215C2×Dic61#12111yescharacteristic
216C2×Dic61#12111yescharacteristic
217D244—21no—
218D244C4×D412yes—
219C4×D61—22no—
220C4×D61—22no—
221C4×D61—42no—
222C4×D61—42no—
223Dic122—21no—
224Dic122C4×D412yes—
225C8×C61—22no—
226C8×C61—42no—
227C8×C61C4×D412yes—
228C8×C61C4×D412yes—
229#92418—22no—
230#92418—24no—
231#92418—21no—
232#92418—21no—
233C4×#472—611nomaximal, Sylow
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—44no—
237C22×C61—84no—
238C22×C61—84no—
239C22×C61—82no—
240C22×C61—82no—
241C22×C61C4×#4111yescharacteristic
242C22×C61C2×#12111yescharacteristic
243C22×C61#52911yescharacteristic
244C2×D61—22no—
245C2×D61—41no—
246C2×D61—41no—
247C4×C61—41no—
248C4×C61—41no—
249C4×C61C2×C4×D411yescharacteristic
250C4×C61C2×C4×D411yescharacteristic
251C4×C61#52511yescharacteristic
252C4×C61#53211yescharacteristic
253Dic61—21no—
254Dic61—21no—
255Dic61—21no—
256Dic61—22no—
257Dic61—82no—
258Dic61#47212yescomplemented
259C2×#472—611no—
260C4×AGL(1,ℤ/8)—612no—
261C2×C4×D8—611no—
262C4×#131—611no—
263#1978D6111yescharacteristic, complemented
264#1952—611no—
265#1931—612no—
266#1907—612no—
267#1816—611no—
268C4×#108—611no—
269C4×#100—612no—
270C2×C61—21no—
271C2×C61—21no—
272C2×C61—84no—
273C2×C61C4×#12111yescharacteristic
274C2×C61C2×#47211yescharacteristic
275C2×C61#306711yescharacteristic
276D61—42no—
277C2×AGL(1,ℤ/8)—612no—
278C22×D8C2×D6111yescharacteristic
279C2×#131—611no—
280C2×#131—611no—
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×#107—611no—
296#418C2×D6112yes—
297#417—612no—
298C2×#102—612no—
299C2×#100—612no—
300C2×#100C2×D6112yes—
301#405—611no—
302#405—1221no—
303#405—1221no—
304#405C2×D6111yescharacteristic
305C4×M16—611no—
306C4×M16—612no—
307C4×M16—1221no—
308C2×C4×C8—611no—
309C2×C4×C8—1221no—
310#343—612no—
311#343—612no—
312#341—612no—
313C61C4×#47211yescharacteristic, complemented, Sylow
314C2×#41—611no—
315C22×D4C22×D6112yes—
316C23×C4—1222no—
317AGL(1,ℤ/8)—618no—
318AGL(1,ℤ/8)—1224no—
319C2×SD16—1222no—
320C2×D8—21no—
321C2×D8—21no—
322C2×D8—22no—
323C2×D8—1222no—
324C2×D8C4×D6114yescomplemented
325#131—614no—
326#131—1222no—
327#131—1222no—
328#131—1222no—
329C2×M16—611no—
330C2×M16—612no—
331C2×M16—1221no—
332C2×M16—1221no—
333C2×M16C22×D6112yes—
334C2×M16C22×D6111yescharacteristic
335C22×C8—611no—
336C22×C8—1221no—
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—1221no—
350#117C22×D6111yescharacteristic
351C2×C4⋊C4—612no—
352C2×C22⋊C4—1222no—
353C2×C42—611no—
354C2×C42—1221no—
355#108—22no—
356#108—614no—
357#107—1221no—
358#107—1221no—
359#107—1222no—
360#106—1222no—
361#103—1222no—
362#102—22no—
363#102—22no—
364#102—1222no—
365#102—1222no—
366#102—1222no—
367#100—24no—
368#100—618no—
369#97—21no—
370#97—1221no—
371#97—1221no—
372#97—1222no—
373C4×C8—21no—
374C4×C8—611no—
375C4×C8—611no—
376C4×C8—611no—
377C4×C8—611no—
378C4×C8—1221no—
379C4×C8—1221no—
380C4×C8—1222no—
381C24—22no—
382#41—612no—
383#41—612no—
384#41—1221no—
385#41—1221no—
386C2×Q8—611no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—611no—
393C2×D4—1221no—
394C2×D4C2×C4×D6114yes—
395C22×C4—611no—
396C22×C4—611no—
397C22×C4—1221no—
398C22×C4—1221no—
399C22×C4—1221no—
400C22×C4—1222no—
401C22×C4—4884no—
402C22×C4—2442no—
403C22×C4—2442no—
404C22×C4C23×D6111yescharacteristic
405SD16—1228no—
406D8—24no—
407D8—24no—
408D8—1228no—
409M16—21no—
410M16—22no—
411M16—22no—
412M16—612no—
413M16—614no—
414M16—1224no—
415M16—1222no—
416M16—1221no—
417M16—1224no—
418M16#1175822412yes—
419C2×C8—21no—
420C2×C8—22no—
421C2×C8—611no—
422C2×C8—611no—
423C2×C8—1224no—
424C2×C8—1221no—
425C2×C8—1222no—
426C2×C8—1221no—
427C2×C8—1221no—
428C2×C8—1224no—
429C2×C8C2×C4×D6112yes—
430C2×C8D4×D6111yescharacteristic
431C2×C8D4×D6111yescharacteristic
432C2×C8#1175822412yes—
433C4⋊C4—1221no—
434C4⋊C4—1221no—
435C4⋊C4—1222no—
436C4⋊C4—1222no—
437C4⋊C4D4×D6111yescharacteristic
438C4⋊C4D4×D6111yescharacteristic
439C22⋊C4—21no—
440C22⋊C4—1221no—
441C22⋊C4—1221no—
442C22⋊C4—2442no—
443C22⋊C4—2442no—
444C42—21no—
445C42—611no—
446C42—611no—
447C42—611no—
448C42—611no—
449C42—1221no—
450C42—1221no—
451C23—24no—
452C23—42no—
453C23—42no—
454C23—82no—
455C23—82no—
456C23—1221no—
457C23#41×D6111yescharacteristic
458Q8—612no—
459Q8—1221no—
460D4—24no—
461D4—24no—
462D4—42no—
463D4—42no—
464D4—612no—
465D4—1222no—
466D4—1222no—
467D4—1221no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—611no—
471C2×C4—611no—
472C2×C4—611no—
473C2×C4—611no—
474C2×C4—1222no—
475C2×C4—1221no—
476C2×C4—1222no—
477C2×C4—1222no—
478C2×C4—4884no—
479C2×C4—4884no—
480C2×C4—4882no—
481C2×C4—2442no—
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×#1175822411yescharacteristic
491C8—22no—
492C8—42no—
493C8—1224no—
494C8—1222no—
495C8—1221no—
496C8—1221no—
497C8#1451973712yes—
498C8#1451973712yes—
499C22—21no—
500C22—21no—
501C22—44no—
502C22—84no—
503C22—84no—
504C22—82no—
505C22—82no—
506C22—1222no—
507C22—2441no—
508C22—2441no—
509C22C4×#1175822411yescharacteristic
510C22#121×D6111yescharacteristic
511C22#1684029111yescharacteristic
512C4—41no—
513C4—41no—
514C4—612no—
515C4—1222no—
516C4—1221no—
517C4—1221no—
518C4—1221no—
519C4—4882no—
520C4C4×D4×D6111yescharacteristic
521C4C2×#1451973711yescharacteristic
522C4#1684027211yescharacteristic
523C4#1684031011yescharacteristic
524C2—21no—
525C2—21no—
526C2—84no—
527C2—2442no—
528C2#2144833511yescharacteristic
529C2#472×D6111yescharacteristic
530C2#2145553711yescharacteristic
subgroupG/H
centerC22#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1978C2
socleC22×C61C2×#121
prime radicalC2×#16840029C2
cosocleC22×C4×C61C24
hypercenter#1978D61
hypocenterC61C4×#472
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1978

Complements

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

NH ≅ G/Ntype
2. #21454623C2semidirect
3. C2×D8×Dic61C2semidirect
4. #21454241C2semidirect
5. #21446604C2semidirect
6. #21446484C2semidirect
7. #21446324C2semidirect
8. #21445880C2semidirect
9. #108×Dic61C2semidirect
10. #100×Dic61C2semidirect
11. C61×#1978C2semidirect
17. #16840029C4semidirect
51. #14519803C2×C4semidirect
57. #14519783C2×C4semidirect
80. #14519713C2×C4semidirect
89. #14519694C2×C4semidirect
104. C2×D8×C61C2×C4semidirect
258. Dic61#472semidirect
263. #1978D61semidirect
313. C61C4×#472semidirect
324. C2×D8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400292
#214546232
C2×D8×Dic612
#214542412
#214466042
#214464842
#214463242
#214458802
#108×Dic612
#100×Dic612
C61×#19782
C4×#47261