Finite Group Databasebeta

← #29686918

Subgroups — #29686918

Order 15616.

938 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840029C211yesmaximal, characteristic
2#21454623C212yesmaximal, complemented
3C2×#16839874C211yesmaximal, characteristic, complemented
4#21454246C211yesmaximal, characteristic, complemented
5#21446599C211yesmaximal, characteristic, complemented
6#21446484C212yesmaximal, complemented
7#21446324C212yesmaximal, complemented
8#21445875C211yesmaximal, characteristic, complemented
9#21445727C211yesmaximal, characteristic, complemented
10#100×Dic61C212yesmaximal, complemented
11C61×#1978C211yesmaximal, characteristic, complemented
12C2×#14519803C2212yes—
13C2×#14519785C2211yescharacteristic
14C2×#14519783C2211yescharacteristic
15#16840230C2211yescharacteristic
16C2×D4×Dic61C2212yes—
17#16840029C418yescomplemented
18#16839874—21no—
19#16839874—21no—
20#16839874—21no—
21#16839874—21no—
22#16839871—22no—
23#16839859—22no—
24#16839853—22no—
25#16839842—22no—
26C2×#14519713C2211yescharacteristic
27#16839805C2212yes—
28C2×#14519709C2211yescharacteristic
29#16839764C2212yes—
30C2×#14519698C2212yes—
31C2×#14519694C2212yes—
32#16839719—21no—
33#16839718C2211yescharacteristic
34#16839716—21no—
35M16×Dic61C2212yes—
36M16×Dic61C2211yescharacteristic
37C4×#11758189—21no—
38C2×#14519687C2211yescharacteristic
39#97×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#14519742D412yes—
66D4×Dic61—22no—
67D4×Dic61—22no—
68D4×Dic61—22no—
69D4×Dic61—22no—
70#14519739D412yes—
71#14519737—22no—
72#14519734—21no—
73C2×C4⋊Dic61C2312yes—
74C2×C22⋊Dic61—22no—
75C2×C4×Dic61C2311yescharacteristic
76C42×D61—21no—
77#14519713C2×C414yescomplemented
78#14519710—22no—
79#14519709—21no—
80#14519709—21no—
81#14519708—22no—
82#14519702—22no—
83#14519699—22no—
84#14519698—22no—
85#14519698—22no—
86#14519694C2×C418yescomplemented
87#14519688—21no—
88#14519687—21no—
89#14519687—22no—
90#14519687D411yescharacteristic
91#14519687D411yescharacteristic
92#14519687D411yescharacteristic
93#14519687D411yescharacteristic
94C4×#92418—21no—
95C8×Dic61—21no—
96C8×Dic61—22no—
97C22×D4×C61C2312yes—
98C2×D8×C61—21no—
99C2×D8×C61—21no—
100C2×D8×C61—22no—
101C2×D8×C61C2×C414yescomplemented
102C2×M16×C61C2312yes—
103C2×M16×C61C2311yescharacteristic
104C22×C8×C61C2311yescharacteristic
105#117×C61C2311yescharacteristic
106#108×C61—22no—
107#102×C61—22no—
108#102×C61—22no—
109#100×C61—24no—
110#97×C61—21no—
111C4×C8×C61—21no—
112C24×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61C22×C414yes—
119C22×C4×C61C2411yescharacteristic
120C2×C22⋊D61—21no—
121C22×Dic61—22no—
122C22×Dic61—42no—
123C22×Dic61—42no—
124C22×Dic61—84no—
125C22×Dic61#4112yes—
126#11758222—21no—
127#11758222—21no—
128#11758222C22×C412yes—
129#11758222#4112yes—
130C2×D244C2×D411yescharacteristic
131C2×C4×D61—21no—
132C2×C4×D61—22no—
133C2×Dic122C2×D411yescharacteristic
134D8×C61—24no—
135D8×C61—24no—
136M16×C61—21no—
137M16×C61—22no—
138M16×C61—22no—
139M16×C61#4112yes—
140C2×C8×C61—21no—
141C2×C8×C61—22no—
142C2×C8×C61C22×C412yes—
143C2×C8×C61C2×D411yescharacteristic
144C2×C8×C61C2×D411yescharacteristic
145C2×C8×C61#4112yes—
146C4⋊C4×C61C2×D412yes—
147C22⋊C4×C61—21no—
148C42×C61—21no—
149C22⋊Dic61—42no—
150C22⋊Dic61—42no—
151#11758200—28no—
152#11758199—28no—
153#11758198—22no—
154C4⋊Dic61—22no—
155C4⋊Dic61—22no—
156#11758196—22no—
157C4×Dic61—22no—
158C4×Dic61C2×D412yes—
159C4×Dic61C2×D411yescharacteristic
160C4×Dic61C2×D411yescharacteristic
161#11758194—21no—
162#11758194—22no—
163#11758194C22×C412yes—
164#11758194C22×C414yes—
165C2×#92418—21no—
166C2×#92418—22no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418C2×D411yescharacteristic
170C2×#92418C2×D411yescharacteristic
171#11758189—24no—
172#11758189—24no—
173C8×D61—24no—
174C8×D61—24no—
175C23×C61—24no—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61—82no—
179C23×C61—82no—
180C23×C61C2×#4111yescharacteristic
181C22×D61—21no—
182D4×C61—24no—
183D4×C61—24no—
184D4×C61—42no—
185D4×C61—42no—
186C2×C4×C61—22no—
187C2×C4×C61C23×C411yescharacteristic
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C2×C4×C61C2×#4111yescharacteristic
192C2×C4×C61C2×#4111yescharacteristic
193C22⋊D61—22no—
194C22⋊D61—22no—
195C2×Dic61—21no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—84no—
199C2×Dic61—84no—
200C2×Dic61—82no—
201C2×Dic61#12112yes—
202C2×Dic61#12112yes—
203D244—21no—
204D244C4×D412yes—
205C4×D61—22no—
206C4×D61—22no—
207C4×D61—82no—
208Dic122—21no—
209Dic122C4×D412yes—
210C8×C61—22no—
211C8×C61—42no—
212C8×C61C4×D412yes—
213C8×C61C4×D412yes—
214#92418—22no—
215#92418—24no—
216#92418—21no—
217#92418—21no—
218#21555—611nomaximal, Sylow
219C22×C61—22no—
220C22×C61—44no—
221C22×C61—84no—
222C22×C61—84no—
223C22×C61—82no—
224C22×C61—82no—
225C22×C61C4×#4111yescharacteristic
226C22×C61C2×#12111yescharacteristic
227C22×C61#52911yescharacteristic
228C2×D61—22no—
229C2×D61—42no—
230C4×C61—42no—
231C4×C61C2×C4×D411yescharacteristic
232C4×C61C2×C4×D411yescharacteristic
233C4×C61#52511yescharacteristic
234C4×C61#53211yescharacteristic
235Dic61—22no—
236Dic61—22no—
237Dic61—22no—
238Dic61—82no—
239C2×#472—611no—
240C4×AGL(1,ℤ/8)—612no—
241C2×#442—611no—
242#2875—611no—
243#1978D6111yescharacteristic, complemented
244#1951—611no—
245#1931—612no—
246#1907—612no—
247#1815—611no—
248#1778—611no—
249C4×#100—612no—
250C2×C61—21no—
251C2×C61—21no—
252C2×C61—84no—
253C2×C61C4×#12111yescharacteristic
254C2×C61#306812yes—
255D61—42no—
256C2×AGL(1,ℤ/8)—612no—
257C22×D8C2×D6111yescharacteristic
258C2×#131—611no—
259C2×#131—611no—
260C4×#41—611no—
261C2×C4×D4—612no—
262#472—618no—
263#442—1221no—
264#442—1221no—
265#442—1221no—
266#442—1221no—
267#442—1222no—
268#440—1222no—
269C4×SD16—1222no—
270C4×D8—1222no—
271C2×#108—611no—
272C2×#108C2×D6111yescharacteristic
273#428—612no—
274C2×#106—611no—
275#418C2×D6112yes—
276#417—612no—
277C2×#102—612no—
278C2×#100—612no—
279C2×#100C2×D6112yes—
280#405—611no—
281#405—1221no—
282#405—1221no—
283#405C2×D6111yescharacteristic
284C4×M16—611no—
285C4×M16—612no—
286C4×M16—1221no—
287C2×#97—611no—
288C2×#97—1221no—
289#343—612no—
290#343—612no—
291#341—612no—
292C61#2155511yescharacteristic, complemented, Sylow
293C2×#41—611no—
294C22×D4C22×D6112yes—
295C23×C4—1222no—
296AGL(1,ℤ/8)—618no—
297AGL(1,ℤ/8)—1224no—
298C2×SD16—1222no—
299C2×D8—21no—
300C2×D8—21no—
301C2×D8—22no—
302C2×D8—1222no—
303C2×D8C4×D6114yescomplemented
304#131—614no—
305#131—1222no—
306#131—1222no—
307#131—1222no—
308C2×M16—611no—
309C2×M16—612no—
310C2×M16—1221no—
311C2×M16—1221no—
312C2×M16C22×D6112yes—
313C2×M16C22×D6111yescharacteristic
314C22×C8—611no—
315C22×C8—1221no—
316C22×C8—1221no—
317C22×C8C22×D6111yescharacteristic
318C4×Q8—612no—
319C4×D4—612no—
320C4×D4—1222no—
321C4×D4—1222no—
322C4×D4—1222no—
323C4×D4—1222no—
324C4×D4—1222no—
325#117—1221no—
326#117C22×D6111yescharacteristic
327C2×C4⋊C4—612no—
328C2×C22⋊C4—1222no—
329C2×C42—611no—
330C2×C42—1221no—
331#108—22no—
332#108—614no—
333#107—1222no—
334#106—1221no—
335#106—1221no—
336#106—1222no—
337#103—1222no—
338#102—22no—
339#102—22no—
340#102—1222no—
341#102—1222no—
342#102—1222no—
343#100—24no—
344#100—618no—
345#97—21no—
346#97—611no—
347#97—611no—
348#97—611no—
349#97—611no—
350#97—1221no—
351#97—1221no—
352#97—1222no—
353C4×C8—21no—
354C4×C8—1221no—
355C4×C8—1221no—
356C4×C8—1222no—
357C24—22no—
358#41—612no—
359#41—612no—
360#41—1221no—
361#41—1221no—
362C2×Q8—611no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—611no—
369C2×D4—1221no—
370C2×D4C2×C4×D6114yes—
371C22×C4—612no—
372C22×C4—1222no—
373C22×C4—1221no—
374C22×C4—1222no—
375C22×C4—4884no—
376C22×C4—2442no—
377C22×C4—2442no—
378C22×C4C23×D6111yescharacteristic
379SD16—1228no—
380D8—24no—
381D8—24no—
382D8—1228no—
383M16—21no—
384M16—22no—
385M16—22no—
386M16—612no—
387M16—614no—
388M16—1224no—
389M16—1222no—
390M16—1221no—
391M16—1224no—
392M16#1175822412yes—
393C2×C8—21no—
394C2×C8—22no—
395C2×C8—611no—
396C2×C8—611no—
397C2×C8—1224no—
398C2×C8—1221no—
399C2×C8—1222no—
400C2×C8—1221no—
401C2×C8—1221no—
402C2×C8—1224no—
403C2×C8C2×C4×D6112yes—
404C2×C8D4×D6111yescharacteristic
405C2×C8D4×D6111yescharacteristic
406C2×C8#1175822412yes—
407C4⋊C4—1222no—
408C4⋊C4—1222no—
409C4⋊C4—1222no—
410C4⋊C4D4×D6112yes—
411C22⋊C4—21no—
412C22⋊C4—1222no—
413C22⋊C4—2442no—
414C22⋊C4—2442no—
415C42—21no—
416C42—611no—
417C42—612no—
418C42—611no—
419C42—1222no—
420C23—24no—
421C23—42no—
422C23—42no—
423C23—82no—
424C23—82no—
425C23—1221no—
426C23#41×D6111yescharacteristic
427Q8—612no—
428Q8—1221no—
429D4—24no—
430D4—24no—
431D4—42no—
432D4—42no—
433D4—612no—
434D4—1222no—
435D4—1222no—
436D4—1221no—
437C2×C4—22no—
438C2×C4—612no—
439C2×C4—612no—
440C2×C4—1221no—
441C2×C4—1222no—
442C2×C4—1222no—
443C2×C4—4882no—
444C2×C4—4884no—
445C2×C4—4884no—
446C2×C4—4882no—
447C2×C4—2442no—
448C2×C4—2442no—
449C2×C4C22×C4×D6111yescharacteristic
450C2×C4C2×D4×D6111yescharacteristic
451C2×C4C2×D4×D6111yescharacteristic
452C2×C4#41×D6111yescharacteristic
453C2×C4C2×#1175822411yescharacteristic
454C2×C4C2×#1175822411yescharacteristic
455C8—22no—
456C8—42no—
457C8—1224no—
458C8—1222no—
459C8—1221no—
460C8—1221no—
461C8#1451973712yes—
462C8#1451973712yes—
463C22—22no—
464C22—44no—
465C22—84no—
466C22—84no—
467C22—82no—
468C22—82no—
469C22—1222no—
470C22—2442no—
471C22C4×#1175822411yescharacteristic
472C22#121×D6111yescharacteristic
473C22#1684029111yescharacteristic
474C4—42no—
475C4—1222no—
476C4—1222no—
477C4—1222no—
478C4—4882no—
479C4C4×D4×D6111yescharacteristic
480C4C2×#1451973711yescharacteristic
481C4#1684027211yescharacteristic
482C4#1684031011yescharacteristic
483C2—21no—
484C2—21no—
485C2—84no—
486C2—2442no—
487C2#2144833511yescharacteristic
488C2#2145554512yes—
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
hypocenterC61#21555
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×#16839874C2semidirect
4. #21454246C2semidirect
5. #21446599C2semidirect
6. #21446484C2semidirect
7. #21446324C2semidirect
8. #21445875C2semidirect
9. #21445727C2semidirect
10. #100×Dic61C2semidirect
11. C61×#1978C2semidirect
17. #16840029C4semidirect
51. #14519803C2×C4semidirect
57. #14519783C2×C4semidirect
77. #14519713C2×C4semidirect
86. #14519694C2×C4semidirect
101. C2×D8×C61C2×C4semidirect
243. #1978D61semidirect
292. C61#21555semidirect
303. 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×#168398742
#214542462
#214465992
#214464842
#214463242
#214458752
#214457272
#100×Dic612
C61×#19782
#2155561