Finite Group Databasebeta

← #29686910

Subgroups — #29686910

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×#16840033C211yesmaximal, characteristic
2#21454626C212yesmaximal, complemented
3C2×#16839873C211yesmaximal, characteristic, complemented
4#21454246C211yesmaximal, characteristic, complemented
5#21446729C211yesmaximal, characteristic, complemented
6#21446480C212yesmaximal, complemented
7#21446328C212yesmaximal, complemented
8#21445879C211yesmaximal, characteristic, complemented
9#21445727C211yesmaximal, characteristic, complemented
10#21445610C212yesmaximal, complemented
11C61×#1951C211yesmaximal, characteristic, complemented
12C2×#14519805C2212yes—
13C22×D488C2211yescharacteristic
14C2×#14519785C2211yescharacteristic
15C2×#14519783C2211yescharacteristic
16#16840230C2211yescharacteristic
17C2×#14519739C2212yes—
18#16840033C418yescomplemented
19#16839873—22no—
20#16839873—21no—
21#16839873—21no—
22#16839873—21no—
23#16839873—21no—
24#16839861—22no—
25#16839849—22no—
26#16839844—22no—
27C2×#14519713C2211yescharacteristic
28#16839766C2212yes—
29#16839761C2212yes—
30C2×#14519697C2212yes—
31C2×#14519693C2212yes—
32C2×#14519693C2212yes—
33#16839719—21no—
34#16839718C2211yescharacteristic
35#16839716—21no—
36M16×Dic61C2212yes—
37M16×Dic61C2211yescharacteristic
38C4×#11758189—21no—
39C2×#14519687C2211yescharacteristic
40#97×D61—21no—
41#16839531C2212yes—
42#16839526C2212yes—
43C2×#108×C61C2211yescharacteristic
44C61×#428C2212yes—
45C2×#106×C61C2211yescharacteristic
46C61×#405C2211yescharacteristic
47C61×#343C2212yes—
48C2×#11758222C2311yescharacteristic
49C22×D244C2312yes—
50C22×C4×D61—22no—
51#14519805—24no—
52#14519805C2×C418yescomplemented
53C2×#11758190—22no—
54C2×D488—22no—
55C2×D488—21no—
56C2×D488—21no—
57C2×D488—22no—
58C2×D488C2×C414yescomplemented
59#14519785—22no—
60#14519785—22no—
61#14519783—22no—
62#14519783C2×C414yescomplemented
63C2×#11758194C2312yes—
64C2×#11758194C2311yescharacteristic
65C2×#11758189—21no—
66C2×#11758189—21no—
67C22×#92418C2311yescharacteristic
68C2×C8×D61—21no—
69C2×C8×D61—21no—
70#14519742D412yes—
71#14519739—22no—
72#14519739—22no—
73#14519739—22no—
74#14519739—22no—
75#14519739D412yes—
76#14519737—22no—
77#14519734—21no—
78C2×#11758198—22no—
79C2×C4×Dic61C2311yescharacteristic
80C42×D61—21no—
81#14519713—22no—
82#14519700—22no—
83#14519697—22no—
84#14519697—22no—
85#14519697—22no—
86#14519697—22no—
87#14519697—22no—
88#14519693—24no—
89#14519693C2×C418yescomplemented
90#14519688—21no—
91#14519687—21no—
92#14519687—22no—
93#14519687D411yescharacteristic
94#14519687D411yescharacteristic
95#14519687D411yescharacteristic
96#14519687D411yescharacteristic
97C4×#92418—21no—
98C8×Dic61—21no—
99C8×Dic61—22no—
100C2×M16×C61C2312yes—
101C2×M16×C61C2311yescharacteristic
102C22×C8×C61C2311yescharacteristic
103#117×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2312yes—
105#108×C61C2×C414yescomplemented
106#107×C61—22no—
107#106×C61—21no—
108#106×C61—21no—
109#106×C61—22no—
110#97×C61—21no—
111C4×C8×C61—21no—
112C23×D61—22no—
113C22×C4×C61—22no—
114C22×C4×C61C2411yescharacteristic
115C2×C22⋊D61—21no—
116C22×Dic61#4112yes—
117#11758222—21no—
118#11758222—21no—
119#11758222C22×C412yes—
120#11758222#4112yes—
121C2×D244—22no—
122C2×D244—22no—
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—22no—
126C2×D244C22×C414yes—
127C2×D244C2×D411yescharacteristic
128C2×C4×D61—21no—
129C2×C4×D61—22no—
130C2×C4×D61—42no—
131C2×C4×D61—42no—
132C2×C4×D61—84no—
133C2×Dic122C2×D411yescharacteristic
134M16×C61—21no—
135M16×C61—22no—
136M16×C61C22×C414yes—
137M16×C61#4112yes—
138C2×C8×C61—21no—
139C2×C8×C61—22no—
140C2×C8×C61C22×C412yes—
141C2×C8×C61C2×D411yescharacteristic
142C2×C8×C61C2×D411yescharacteristic
143C2×C8×C61#4112yes—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61C2×D412yes—
147C22⋊C4×C61—21no—
148C42×C61—21no—
149#11758198—22no—
150#11758198—42no—
151#11758198—42no—
152#11758196—22no—
153C4×Dic61—22no—
154C4×Dic61C2×D412yes—
155C4×Dic61C2×D411yescharacteristic
156C4×Dic61C2×D411yescharacteristic
157#11758194—21no—
158#11758194—22no—
159#11758194—22no—
160#11758194C22×C412yes—
161C2×#92418—21no—
162C2×#92418—21no—
163C2×#92418—21no—
164C2×#92418—22no—
165C2×#92418C2×D411yescharacteristic
166C2×#92418C2×D411yescharacteristic
167D488—28no—
168D488—24no—
169D488—24no—
170#11758190—28no—
171#11758189—24no—
172#11758189—24no—
173C8×D61—24no—
174C8×D61—24no—
175C23×C61C2×#4111yescharacteristic
176C22×D61—21no—
177C22×D61—24no—
178C22×D61—42no—
179C22×D61—42no—
180C22×D61—82no—
181C22×D61—82no—
182C2×C4×C61—22no—
183C2×C4×C61—42no—
184C2×C4×C61—82no—
185C2×C4×C61C23×C411yescharacteristic
186C2×C4×C61C22×D411yescharacteristic
187C2×C4×C61C22×D411yescharacteristic
188C2×C4×C61C2×#4111yescharacteristic
189C2×C4×C61C2×#4111yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C22⋊D61—22no—
192C22⋊D61—22no—
193C2×Dic61—21no—
194C2×Dic61—42no—
195C2×Dic61#12112yes—
196C2×Dic61#12112yes—
197D244—24no—
198D244—24no—
199D244—21no—
200D244—42no—
201D244—42no—
202D244C4×D412yes—
203C4×D61—22no—
204C4×D61—22no—
205C4×D61—82no—
206C4×D61—84no—
207C4×D61—84no—
208Dic122—21no—
209Dic122C4×D412yes—
210C8×C61—24no—
211C8×C61—22no—
212C8×C61C4×D412yes—
213C8×C61C4×D412yes—
214#92418—22no—
215#92418—21no—
216#92418—21no—
217#92418—42no—
218#21555—611nomaximal, Sylow
219C22×C61—22no—
220C22×C61C4×#4111yescharacteristic
221C22×C61C2×#12111yescharacteristic
222C22×C61#52911yescharacteristic
223C2×D61—22no—
224C2×D61—42no—
225C2×D61—44no—
226C2×D61—84no—
227C2×D61—82no—
228C2×D61—82no—
229C2×D61—84no—
230C4×C61—42no—
231C4×C61—82no—
232C4×C61C2×C4×D411yescharacteristic
233C4×C61C2×C4×D411yescharacteristic
234C4×C61#52511yescharacteristic
235C4×C61#53211yescharacteristic
236Dic61—22no—
237Dic61—22no—
238Dic61—22no—
239C2×#472—611no—
240C4×AGL(1,ℤ/8)—612no—
241C2×#442—611no—
242#2875—611no—
243#1978—611no—
244#1951D6111yescharacteristic, complemented
245#1931—612no—
246#1907—612no—
247#1815—611no—
248#1778—611no—
249C4×#100—612no—
250C2×C61—21no—
251C2×C61—21no—
252C2×C61C4×#12111yescharacteristic
253C2×C61#306812yes—
254D61—42no—
255D61—84no—
256C2×AGL(1,ℤ/8)—612no—
257C22×D8—611no—
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#428C2×D6112yes—
274C2×#106C2×D6111yescharacteristic
275#418—612no—
276#417—612no—
277C2×#102—612no—
278C2×#100—612no—
279C2×#100—612no—
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#343C2×D6112yes—
291#341—612no—
292C61#2155511yescharacteristic, complemented, Sylow
293C2×#41—611no—
294C22×D4—612no—
295C23×C4—1222no—
296AGL(1,ℤ/8)—618no—
297AGL(1,ℤ/8)—1224no—
298C2×SD16—1222no—
299C2×D8—614no—
300C2×D8—1222no—
301C2×D8—1221no—
302C2×D8—1221no—
303C2×D8—1222no—
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⋊C4C22×D6112yes—
328C2×C22⋊C4—1222no—
329C2×C42—611no—
330C2×C42—1221no—
331#108—1222no—
332#108C4×D6114yescomplemented
333#107—22no—
334#106—21no—
335#106—21no—
336#106—22no—
337#103—1222no—
338#102—1222no—
339#102—1222no—
340#102—1222no—
341#102—1222no—
342#102—1222no—
343#100—618no—
344#100—1224no—
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—1222no—
358#41—612no—
359#41—612no—
360#41—1221no—
361#41—1221no—
362C2×Q8—611no—
363C2×D4—614no—
364C2×D4—611no—
365C2×D4—1221no—
366C2×D4—1222no—
367C2×D4—1222no—
368C2×D4—1222no—
369C2×D4—1222no—
370C2×D4—1222no—
371C22×C4—22no—
372C22×C4—612no—
373C22×C4—1222no—
374C22×C4—1221no—
375C22×C4—4884no—
376C22×C4—2442no—
377C22×C4—2442no—
378C22×C4C23×D6111yescharacteristic
379SD16—1228no—
380D8—1228no—
381D8—1224no—
382D8—1224no—
383M16—21no—
384M16—22no—
385M16—612no—
386M16—1224no—
387M16—1221no—
388M16—1224no—
389M16—1222no—
390M16—1222no—
391M16C2×C4×D6114yes—
392M16#1175822612yes—
393C2×C8—21no—
394C2×C8—22no—
395C2×C8—611no—
396C2×C8—611no—
397C2×C8—1224no—
398C2×C8—1221no—
399C2×C8—1221no—
400C2×C8—1221no—
401C2×C8—1224no—
402C2×C8—1222no—
403C2×C8C2×C4×D6112yes—
404C2×C8D4×D6111yescharacteristic
405C2×C8D4×D6111yescharacteristic
406C2×C8#1175822612yes—
407C4⋊C4—22no—
408C4⋊C4—22no—
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—1221no—
421C23—1224no—
422C23—4882no—
423C23—4882no—
424C23—2442no—
425C23—2442no—
426C23#41×D6111yescharacteristic
427Q8—612no—
428Q8—1221no—
429D4—612no—
430D4—1222no—
431D4—1222no—
432D4—1224no—
433D4—1224no—
434D4—1221no—
435D4—2442no—
436D4—2442no—
437C2×C4—22no—
438C2×C4—42no—
439C2×C4—82no—
440C2×C4—612no—
441C2×C4—612no—
442C2×C4—1221no—
443C2×C4—1222no—
444C2×C4—1222no—
445C2×C4—4882no—
446C2×C4—4884no—
447C2×C4—4884no—
448C2×C4—2442no—
449C2×C4C22×C4×D6111yescharacteristic
450C2×C4C2×D4×D6111yescharacteristic
451C2×C4C2×D4×D6111yescharacteristic
452C2×C4#41×D6111yescharacteristic
453C2×C4C2×#1175822611yescharacteristic
454C2×C4C2×#1175822611yescharacteristic
455C8—24no—
456C8—22no—
457C8—1222no—
458C8—1221no—
459C8—1221no—
460C8—2442no—
461C8#1451973912yes—
462C8#1451973912yes—
463C22—22no—
464C22—1222no—
465C22—4884no—
466C22—4884no—
467C22—4882no—
468C22—4882no—
469C22—2444no—
470C22—2442no—
471C22C4×#1175822611yescharacteristic
472C22#121×D6111yescharacteristic
473C22#1684029711yescharacteristic
474C4—42no—
475C4—82no—
476C4—1222no—
477C4—1222no—
478C4—1222no—
479C4C4×D4×D6111yescharacteristic
480C4C2×#1451973911yescharacteristic
481C4#1684027511yescharacteristic
482C4#1684031311yescharacteristic
483C2—21no—
484C2—21no—
485C2—4884no—
486C2—2442no—
487C2#2144832711yescharacteristic
488C2#2145554212yes—
subgroupG/H
centerC22#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1951C2
socleC22×C61C2×#121
prime radicalC2×#16840033C2
cosocleC22×C4×C61C24
hypercenter#1951D61
hypocenterC61#21555
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1951

Complements

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

NH ≅ G/Ntype
2. #21454626C2semidirect
3. C2×#16839873C2semidirect
4. #21454246C2semidirect
5. #21446729C2semidirect
6. #21446480C2semidirect
7. #21446328C2semidirect
8. #21445879C2semidirect
9. #21445727C2semidirect
10. #21445610C2semidirect
11. C61×#1951C2semidirect
18. #16840033C4semidirect
52. #14519805C2×C4semidirect
58. C2×D488C2×C4semidirect
62. #14519783C2×C4semidirect
89. #14519693C2×C4semidirect
105. #108×C61C2×C4semidirect
244. #1951D61semidirect
292. C61#21555semidirect
332. #108C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400332
#214546262
C2×#168398732
#214542462
#214467292
#214464802
#214463282
#214458792
#214457272
#214456102
C61×#19512
#2155561