Finite Group Databasebeta

← #29686909

Subgroups — #29686909

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
2C4×#14519803C212yesmaximal, complemented
3#21454244C211yesmaximal, characteristic, complemented
4#21446730C211yesmaximal, characteristic, complemented
5#21446597C211yesmaximal, characteristic, complemented
6#21446478C212yesmaximal, complemented
7#21446330C212yesmaximal, complemented
8#21445879C211yesmaximal, characteristic, complemented
9#21445725C211yesmaximal, characteristic, complemented
10C4×#14519694C212yesmaximal, complemented
11C2×C61×#442C211yesmaximal, characteristic, complemented
12C2×#14519803C2212yes—
13C2×#14519783C2211yescharacteristic
14C2×#14519783C2211yescharacteristic
15C4×#11758222C2211yescharacteristic
16#16840029C418yescomplemented
17#16839872—22no—
18#16839862—22no—
19C4×#11758200—22no—
20C4×#11758199—22no—
21C2×#14519713C2211yescharacteristic
22C2×#14519713C2211yescharacteristic
23#16839802C2212yes—
24#16839767C2212yes—
25#16839761C2212yes—
26C2×#14519694C2212yes—
27C2×#14519694C2212yes—
28#16839719—21no—
29#16839719—21no—
30#16839717C2211yescharacteristic
31#16839717C2211yescharacteristic
32C4×#11758194C2212yes—
33C4×#11758194C2211yescharacteristic
34#16839714—21no—
35#97×D61—21no—
36#16839531C2212yes—
37#16839531C2212yes—
38#16839527C2212yes—
39C22×D8×C61C2211yescharacteristic
40C2×C4×D4×C61C2212yes—
41C61×#442—21no—
42C61×#442—21no—
43C61×#442—21no—
44C61×#442—21no—
45C2×#106×C61C2211yescharacteristic
46C2×#102×C61C2212yes—
47C2×#97×C61C2211yescharacteristic
48C2×#11758222C2311yescharacteristic
49#14519803—24no—
50#14519803C2×C418yescomplemented
51C2×#11758200—22no—
52C2×#11758199—22no—
53#14519783—22no—
54#14519783—22no—
55#14519783—22no—
56#14519783C2×C414yescomplemented
57C2×#11758194C2312yes—
58C2×#11758194C2311yescharacteristic
59C2×#11758194C2312yes—
60C2×#11758194C2311yescharacteristic
61C2×#11758189—21no—
62C2×#11758189—21no—
63C2×C8×D61—21no—
64C2×C8×D61—21no—
65C4×Dic122D412yes—
66C4×D244D412yes—
67C4×C22⋊D61—22no—
68#14519736C2311yescharacteristic
69#14519734—21no—
70C42×D61—21no—
71#14519713—22no—
72#14519713C2×C414yescomplemented
73#14519710—22no—
74#14519708—22no—
75#14519700—22no—
76#14519698—22no—
77#14519698—22no—
78#14519697—22no—
79#14519694—24no—
80#14519694C2×C418yescomplemented
81#14519688—21no—
82#14519688—22no—
83#14519687—21no—
84#14519687—21no—
85C4×#92418—21no—
86C4×#92418—22no—
87C8×Dic61—21no—
88C8×Dic61—21no—
89C22×D4×C61C2312yes—
90C23×C4×C61—22no—
91C2×D8×C61—21no—
92C2×D8×C61—21no—
93C2×D8×C61—22no—
94C2×D8×C61C2×C414yescomplemented
95C22×C8×C61C2311yescharacteristic
96C22×C8×C61C2311yescharacteristic
97C4×D4×C61—22no—
98C4×D4×C61—22no—
99C4×D4×C61—22no—
100C4×D4×C61—22no—
101C2×C4⋊C4×C61C2312yes—
102C2×C22⋊C4×C61—22no—
103C2×C42×C61C2311yescharacteristic
104#106×C61—21no—
105#106×C61—21no—
106#102×C61—22no—
107#102×C61—22no—
108#97×C61D411yescharacteristic
109#97×C61D411yescharacteristic
110#97×C61D411yescharacteristic
111#97×C61D411yescharacteristic
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×C61—22no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61—84no—
123C22×C4×C61#4112yes—
124C22×C4×C61C2411yescharacteristic
125C2×C22⋊D61—21no—
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—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C2×C8×C61C22×C412yes—
139C2×C8×C61C2×D411yescharacteristic
140C2×C8×C61C2×D411yescharacteristic
141C2×C8×C61C2×D411yescharacteristic
142C2×C8×C61C2×D411yescharacteristic
143C2×C8×C61#4112yes—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C22⋊C4×C61—42no—
147C22⋊C4×C61—42no—
148C42×C61C2×D412yes—
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151C22⋊Dic61—21no—
152#11758200—28no—
153#11758199—28no—
154#11758198—22no—
155C4⋊Dic61C2×D412yes—
156#11758196—22no—
157C4×Dic61—22no—
158C4×Dic61—21no—
159#11758194—21no—
160#11758194—21no—
161#11758194—22no—
162#11758194—22no—
163#11758194—22no—
164#11758194C22×C412yes—
165#11758194C22×C414yes—
166#11758194#4112yes—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418—22no—
170C2×#92418—22no—
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—42no—
187C2×C4×C61—42no—
188C2×C4×C61—84no—
189C2×C4×C61—84no—
190C2×C4×C61—82no—
191C2×C4×C61#12112yes—
192C2×C4×C61#12112yes—
193C2×C4×C61C23×C411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C2×#4111yescharacteristic
197C2×C4×C61C2×#4111yescharacteristic
198C2×C4×C61C2×#4111yescharacteristic
199C22⋊D61—22no—
200C22⋊D61—22no—
201C2×Dic61—21no—
202C2×Dic61—22no—
203D244—21no—
204D244C4×D412yes—
205C4×D61—22no—
206C4×D61—22no—
207C4×D61—82no—
208Dic122—21no—
209Dic122C4×D412yes—
210C8×C61—21no—
211C8×C61—21no—
212C8×C61C4×D412yes—
213C8×C61C4×D412yes—
214#92418—22no—
215#92418—24no—
216#92418—22no—
217#92418—42no—
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—22no—
231C4×C61—22no—
232C4×C61—82no—
233C4×C61C2×C4×D411yescharacteristic
234C4×C61C2×C4×D411yescharacteristic
235C4×C61#52511yescharacteristic
236C4×C61#53211yescharacteristic
237Dic61—22no—
238Dic61—42no—
239C2×#472—611no—
240C4×AGL(1,ℤ/8)—612no—
241C2×#442D6111yescharacteristic, complemented
242#2875—611no—
243#1978—611no—
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×D4C2×D6112yes—
262#472—618no—
263#442—21no—
264#442—21no—
265#442—21no—
266#442—21no—
267#442—1222no—
268#440—1222no—
269C4×SD16—1222no—
270C4×D8—1222no—
271C2×#108—611no—
272C2×#108—611no—
273#428—612no—
274C2×#106C2×D6111yescharacteristic
275#418—612no—
276#417—612no—
277C2×#102C2×D6112yes—
278C2×#100—612no—
279C2×#100—612no—
280#405—611no—
281#405—611no—
282#405—1221no—
283#405—1221no—
284C4×M16—611no—
285C4×M16—612no—
286C4×M16—1221no—
287C2×#97—1221no—
288C2×#97C2×D6111yescharacteristic
289#343—612no—
290#343—612no—
291#341—612no—
292C61#2155511yescharacteristic, complemented, Sylow
293C2×#41—611no—
294C22×D4C22×D6112yes—
295C23×C4—22no—
296AGL(1,ℤ/8)—618no—
297AGL(1,ℤ/8)—1224no—
298C2×SD16—1222no—
299C2×D8—22no—
300C2×D8—21no—
301C2×D8—21no—
302C2×D8—1222no—
303C2×D8C4×D6114yescomplemented
304#131—614no—
305#131—1222no—
306#131—1222no—
307#131—1222no—
308C2×M16—611no—
309C2×M16—611no—
310C2×M16—612no—
311C2×M16—612no—
312C2×M16—1221no—
313C2×M16—1221no—
314C22×C8—1221no—
315C22×C8—1221no—
316C22×C8C22×D6111yescharacteristic
317C22×C8C22×D6111yescharacteristic
318C4×Q8—612no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—612no—
324C4×D4—1222no—
325#117—611no—
326#117—1221no—
327C2×C4⋊C4C22×D6112yes—
328C2×C22⋊C4—22no—
329C2×C42—1221no—
330C2×C42C22×D6111yescharacteristic
331#108—614no—
332#108—1222no—
333#107—1222no—
334#106—21no—
335#106—21no—
336#106—1222no—
337#103—1222no—
338#102—22no—
339#102—22no—
340#102—1222no—
341#102—1222no—
342#102—1222no—
343#100—618no—
344#100—1224no—
345#97—1221no—
346#97—1221no—
347#97—1222no—
348#97—1221no—
349#97C22⋊D6111yescharacteristic
350#97C22⋊D6111yescharacteristic
351#97C22⋊D6111yescharacteristic
352#97C22⋊D6111yescharacteristic
353C4×C8—1221no—
354C4×C8—1221no—
355C4×C8—1222no—
356C4×C8—1221no—
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—22no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—84no—
375C22×C4—1221no—
376C22×C4—1222no—
377C22×C4#1175822412yes—
378C22×C4C23×D6111yescharacteristic
379SD16—1228no—
380D8—24no—
381D8—24no—
382D8—1228no—
383M16—612no—
384M16—614no—
385M16—612no—
386M16—1224no—
387M16—1221no—
388M16—1222no—
389M16—1221no—
390M16—1224no—
391M16—1222no—
392M16—1222no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—1224no—
396C2×C8—1222no—
397C2×C8—1221no—
398C2×C8—1221no—
399C2×C8—1224no—
400C2×C8—1222no—
401C2×C8C2×C4×D6112yes—
402C2×C8#1175822212yes—
403C2×C8C2×C22⋊D6111yescharacteristic
404C2×C8C2×C22⋊D6111yescharacteristic
405C2×C8C2×C22⋊D6111yescharacteristic
406C2×C8C2×C22⋊D6111yescharacteristic
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—612no—
410C4⋊C4—1222no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—1221no—
414C22⋊C4—1222no—
415C42—1221no—
416C42—1222no—
417C42D4×D6112yes—
418C42C2×C22⋊D6111yescharacteristic
419C42C2×C22⋊D6111yescharacteristic
420C23—24no—
421C23—42no—
422C23—42no—
423C23—82no—
424C23—82no—
425C23—1221no—
426C23C2×#1175822411yescharacteristic
427Q8—612no—
428Q8—1221no—
429D4—24no—
430D4—24no—
431D4—42no—
432D4—42no—
433D4—612no—
434D4—1222no—
435D4—1222no—
436D4—1221no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—84no—
440C2×C4—84no—
441C2×C4—82no—
442C2×C4—1221no—
443C2×C4—1222no—
444C2×C4—1222no—
445C2×C4—1222no—
446C2×C4—4882no—
447C2×C4#1451974812yes—
448C2×C4#1451974812yes—
449C2×C4C22×C4×D6111yescharacteristic
450C2×C4C2×D4×D6111yescharacteristic
451C2×C4C22×C22⋊D6111yescharacteristic
452C2×C4#41×D6111yescharacteristic
453C2×C4#41×D6111yescharacteristic
454C2×C4C2×#1175822211yescharacteristic
455C8—21no—
456C8—21no—
457C8—1224no—
458C8—1222no—
459C8—1222no—
460C8—2442no—
461C8C4×C22⋊D6112yes—
462C8C4×C22⋊D6112yes—
463C22—22no—
464C22—44no—
465C22—84no—
466C22—84no—
467C22—82no—
468C22—82no—
469C22—1222no—
470C22—2442no—
471C22C4×#1175822411yescharacteristic
472C22C2×#1451974811yescharacteristic
473C22#1684029411yescharacteristic
474C4—22no—
475C4—22no—
476C4—82no—
477C4—1222no—
478C4—2442no—
479C4C4×D4×D6111yescharacteristic
480C4C2×C4×C22⋊D6111yescharacteristic
481C4#1684026911yescharacteristic
482C4#1684030911yescharacteristic
483C2—21no—
484C2—21no—
485C2—84no—
486C2—2442no—
487C2C4×#1451974811yescharacteristic
488C2#2145554312yes—
subgroupG/H
centerC22C2×#14519748
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#442C2
socleC22×C61C2×#121
prime radicalC2×#16840029C2
cosocleC22×C4×C61C24
hypercenterC2×#442D61
hypocenterC61#21555
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#442

Complements

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

NH ≅ G/Ntype
2. C4×#14519803C2semidirect
3. #21454244C2semidirect
4. #21446730C2semidirect
5. #21446597C2semidirect
6. #21446478C2semidirect
7. #21446330C2semidirect
8. #21445879C2semidirect
9. #21445725C2semidirect
10. C4×#14519694C2semidirect
11. C2×C61×#442C2semidirect
16. #16840029C4semidirect
50. #14519803C2×C4semidirect
56. #14519783C2×C4semidirect
72. #14519713C2×C4semidirect
80. #14519694C2×C4semidirect
94. C2×D8×C61C2×C4semidirect
241. C2×#442D61semidirect
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
C4×#145198032
#214542442
#214467302
#214465972
#214464782
#214463302
#214458792
#214457252
C4×#145196942
C2×C61×#4422
#2155561