Finite Group Databasebeta

← #29688205

Subgroups — #29688205

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456162C211yesmaximal, characteristic
2#21454758C211yesmaximal, characteristic, complemented
3#21454534C211yesmaximal, characteristic
4#21446794C211yesmaximal, characteristic, complemented
5#21446645C211yesmaximal, characteristic
6#21446533C211yesmaximal, characteristic, complemented
7#21446529C211yesmaximal, characteristic, complemented
8#21446523C211yesmaximal, characteristic, complemented
9#21445958C211yesmaximal, characteristic, complemented
10#21445666C211yesmaximal, characteristic, complemented
11#21445662C211yesmaximal, characteristic, complemented
12#21445657C211yesmaximal, characteristic, complemented
13#21445653C211yesmaximal, characteristic, complemented
14C4×#14519687C211yesmaximal, characteristic, complemented
15C61×#1937C211yesmaximal, characteristic, complemented
16C2×#14519785C2211yescharacteristic
17C2×#14519784C2211yescharacteristic
18#16840315C2211yescharacteristic
19#16840308C2211yescharacteristic
20#16840129C412yescomplemented
21#16840125C412yescomplemented
22#16840123C412yescomplemented
23#16840120C412yescomplemented
24#16839891—22no—
25#16839889—22no—
26#16839836—22no—
27#16839830—22no—
28#16839788C2211yescharacteristic
29#16839785C2211yescharacteristic
30#16839785C2211yescharacteristic
31C2×#14519705C2211yescharacteristic
32C2×#14519704C2211yescharacteristic
33C2×#14519703C2211yescharacteristic
34#16839759C2211yescharacteristic
35#16839758C2211yescharacteristic
36#16839756C2211yescharacteristic
37#16839728C2211yescharacteristic
38#16839720C2211yescharacteristic
39#16839718C2211yescharacteristic
40C2×#14519687C2211yescharacteristic
41C2×#14519687C2211yescharacteristic
42C2×#14519687C2211yescharacteristic
43C2×#14519687C2211yescharacteristic
44C2×C4×#92418C2211yescharacteristic
45C42×Dic61C2211yescharacteristic
46#16839527C2211yescharacteristic
47#16839526C2211yescharacteristic
48#16839526C2211yescharacteristic
49#16839523C2211yescharacteristic
50#16839522C2211yescharacteristic
51#16839521C2211yescharacteristic
52C61×#424C2211yescharacteristic
53C2×#104×C61C2211yescharacteristic
54C61×#416C2211yescharacteristic
55C61×#408C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C61×#341C2211yescharacteristic
58C61×#339C2211yescharacteristic
59C2×#11758222C2311yescharacteristic
60#14519785—22no—
61#14519785—22no—
62#14519784—22no—
63#14519784—22no—
64C2×#11758194—21no—
65C2×#11758194C2311yescharacteristic
66C2×#11758194C2311yescharacteristic
67C2×#11758189—21no—
68C22×#92418—21no—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C2×C8×D61—21no—
72#14519776C2×C412yes—
73#14519775C2×C411yescharacteristic
74#14519773C2×C411yescharacteristic
75#14519768C2×C412yes—
76#14519766C2×C411yescharacteristic
77#14519764C2×C411yescharacteristic
78#14519754—22no—
79#14519749—22no—
80C4×Dic122—22no—
81C4×C22⋊D61—22no—
82#14519736C2311yescharacteristic
83C2×C4×Dic61—22no—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519706—22no—
87#14519706—22no—
88#14519706—22no—
89#14519705—22no—
90#14519704—22no—
91#14519703—22no—
92#14519702C2×C412yes—
93#14519701C2×C412yes—
94#14519700C2×C412yes—
95#14519699C2×C412yes—
96#14519698C2×C412yes—
97#14519697C2×C412yes—
98#14519689—22no—
99#14519688—21no—
100#14519687—22no—
101#14519687—21no—
102#14519687—22no—
103#14519687C2×C412yes—
104#14519687C2×C412yes—
105#14519687D412yes—
106#14519687D412yes—
107C4×#92418—21no—
108C4×#92418—22no—
109C8×Dic61—21no—
110C2×#41×C61C2311yescharacteristic
111C2×M16×C61C2311yescharacteristic
112C2×M16×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115#117×C61C2311yescharacteristic
116C2×C42×C61C2311yescharacteristic
117C4⋊C8×C61—22no—
118#104×C61—22no—
119#103×C61C2×C412yes—
120#102×C61C2×C412yes—
121C22⋊C8×C61—22no—
122C4×C8×C61—22no—
123#41×C61—21no—
124#41×C61—21no—
125#41×C61—21no—
126#41×C61—21no—
127C2×Q8×C61C22×C411yescharacteristic
128C2×D4×C61—21no—
129C2×D4×C61C22×C411yescharacteristic
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61C2411yescharacteristic
133C2×C22⋊D61—21no—
134C2×C22⋊D61—22no—
135C22×Dic61—21no—
136C22×Dic61#4112yes—
137#11758222—21no—
138#11758222—21no—
139#11758222—21no—
140#11758222—21no—
141C2×D244C22×C411yescharacteristic
142C2×C4×D61—21no—
143C2×Dic122—22no—
144C2×Dic122C22×C411yescharacteristic
145M16×C61—21no—
146M16×C61—21no—
147M16×C61—21no—
148M16×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61C22×C412yes—
156C2×C8×C61C2×D412yes—
157C4⋊C4×C61C22×C412yes—
158C22⋊C4×C61—21no—
159C42×C61—21no—
160C42×C61—21no—
161C42×C61#4112yes—
162C22⋊Dic61—21no—
163C22⋊Dic61—22no—
164#11758198—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61C22×C412yes—
167#11758196—22no—
168#11758196—22no—
169#11758196—22no—
170C4×Dic61—21no—
171C4×Dic61—21no—
172C4×Dic61—44no—
173C4×Dic61C22×C411yescharacteristic
174C4×Dic61C22×C411yescharacteristic
175C4×Dic61C2×D412yes—
176C4×Dic61#4112yes—
177#11758194—21no—
178#11758194—21no—
179#11758194—21no—
180#11758194—21no—
181#11758194—42no—
182C2×#92418—21no—
183C2×#92418—21no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—42no—
187C2×#92418C22×C412yes—
188C2×#92418C2×D411yescharacteristic
189C2×#92418C2×D411yescharacteristic
190C2×#92418#4112yes—
191C2×#92418#4112yes—
192#11758189—42no—
193C8×D61—42no—
194C23×C61—21no—
195C23×C61C2×#4111yescharacteristic
196C22×D61—21no—
197Q8×C61—21no—
198Q8×C61—21no—
199D4×C61—21no—
200D4×C61—21no—
201D4×C61—41no—
202D4×C61—41no—
203C2×C4×C61—24no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—21no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61C23×C411yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C61C2×#4111yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C2×C4×C61C2×#4111yescharacteristic
215C22⋊D61—41no—
216C22⋊D61—41no—
217C22⋊D61—82no—
218C2×Dic61—22no—
219C2×Dic61—21no—
220C2×Dic61—22no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61C4×D412yes—
224C2×Dic61C4×D412yes—
225D244—21no—
226D244—21no—
227C4×D61—41no—
228C4×D61—41no—
229Dic122—21no—
230Dic122—21no—
231Dic122—82no—
232C8×C61—22no—
233C8×C61—22no—
234C8×C61—41no—
235C8×C61—41no—
236#92418—22no—
237#92418—41no—
238#92418—41no—
239#92418#12412yes—
240#92418#12412yes—
241#21662—611nomaximal, Sylow
242C22×C61—22no—
243C22×C61—42no—
244C22×C61—41no—
245C22×C61C2×C4×D411yescharacteristic
246C22×C61#52911yescharacteristic
247C22×C61#52911yescharacteristic
248C2×D61—42no—
249C2×D61—41no—
250C4×C61—24no—
251C4×C61—42no—
252C4×C61—41no—
253C4×C61C4×#4111yescharacteristic
254C4×C61C4×#4111yescharacteristic
255C4×C61C2×#12411yescharacteristic
256C4×C61#53211yescharacteristic
257Dic61—22no—
258Dic61—22no—
259Dic61—42no—
260Dic61—42no—
261Dic61—41no—
262#3135—611no—
263#2961—611no—
264#2932—611no—
265#1994—611no—
266#1962—611no—
267#1938—611no—
268#1938—611no—
269#1937—611no—
270#1937D6111yescharacteristic, complemented
271#1838—611no—
272#1765—611no—
273#1765—611no—
274#1764—611no—
275#1764—611no—
276C4×#97—611no—
277C2×C61—21no—
278C2×C61—21no—
279C2×C61—81no—
280C2×C61C4×#12411yescharacteristic
281C2×C61#295612yes—
282D61—81no—
283C2×#131—611no—
284C2×#131—611no—
285#533—611no—
286#532—611no—
287#491—612no—
288#490—612no—
289#489—612no—
290#489—612no—
291#446—1222no—
292#446—1222no—
293#435—1222no—
294#435—1222no—
295#424—611no—
296#424C2×D6111yescharacteristic
297#423—611no—
298#423—611no—
299C2×#104—611no—
300C2×#104—611no—
301C2×#104—611no—
302C2×#104C2×D6111yescharacteristic
303#416—611no—
304#416—611no—
305#416—611no—
306#416C2×D6111yescharacteristic
307#408—611no—
308#408C2×D6111yescharacteristic
309#405—611no—
310#405—611no—
311C2×#97—611no—
312C2×#97—611no—
313C2×#97—611no—
314C2×#97—611no—
315C2×C4×C8—611no—
316C2×C4×C8C2×D6111yescharacteristic
317C43—611no—
318#341—611no—
319#341—611no—
320#341—611no—
321#341C2×D6111yescharacteristic
322#339—611no—
323#339—611no—
324#339—611no—
325#339C2×D6111yescharacteristic
326C61#2166211yescharacteristic, complemented, Sylow
327C2×#41—611no—
328C2×#41C22×D6111yescharacteristic
329#131—1222no—
330#131—1222no—
331#131—1222no—
332#131—1222no—
333C2×M16—611no—
334C2×M16—611no—
335C2×M16—1221no—
336C2×M16—1221no—
337C2×M16C22×D6111yescharacteristic
338C2×M16C22×D6111yescharacteristic
339C22×C8—611no—
340C22×C8—611no—
341C22×C8—1221no—
342C22×C8—1221no—
343C22×C8C22×D6111yescharacteristic
344C22×C8C22×D6111yescharacteristic
345C4⋊Q8—612no—
346C4⋊Q8—611no—
347#127—611no—
348#125—612no—
349#124—611no—
350#124—611no—
351C22⋊Q8—1222no—
352#121—1222no—
353C4×Q8—1222no—
354C4×D4—1222no—
355#117—611no—
356#117C22×D6111yescharacteristic
357C2×C42—611no—
358C2×C42—611no—
359C2×C42—1222no—
360C2×C42C22×D6111yescharacteristic
361C4⋊C8—22no—
362C4⋊C8—1222no—
363C4⋊C8—1222no—
364C4⋊C8—1222no—
365#104—22no—
366#104—1222no—
367#104—1222no—
368#104—1222no—
369#103—612no—
370#103—612no—
371#103—612no—
372#103C4×D6112yes—
373#102—612no—
374#102—612no—
375#102—612no—
376#102C4×D6112yes—
377C22⋊C8—22no—
378C22⋊C8—1222no—
379#97—612no—
380#97—612no—
381#97—612no—
382#97—612no—
383#97—1221no—
384#97—1222no—
385#97—1222no—
386#97—1221no—
387C4×C8—22no—
388C4×C8—1222no—
389C4×C8—1221no—
390C4×C8—1221no—
391#41—21no—
392#41—21no—
393#41—21no—
394#41—21no—
395#41—1221no—
396#41—1221no—
397#41—1221no—
398#41—1221no—
399C2×Q8—611no—
400C2×Q8—1222no—
401C2×Q8C2×C4×D6111yescharacteristic
402C2×D4—21no—
403C2×D4—611no—
404C2×D4—1222no—
405C2×D4—1221no—
406C2×D4C2×C4×D6111yescharacteristic
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—612no—
410C22×C4—1221no—
411C22×C4—1221no—
412C22×C4C23×D6111yescharacteristic
413M16—21no—
414M16—21no—
415M16—21no—
416M16—21no—
417M16—1221no—
418M16—1221no—
419M16—1221no—
420M16—1221no—
421M16—2442no—
422M16—2442no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—611no—
430C2×C8—611no—
431C2×C8—612no—
432C2×C8—612no—
433C2×C8—612no—
434C2×C8—1221no—
435C2×C8—1221no—
436C2×C8—1221no—
437C2×C8—1221no—
438C2×C8—2442no—
439C2×C8—2442no—
440C2×C8C2×C4×D6112yes—
441C2×C8D4×D6112yes—
442C4⋊C4—612no—
443C4⋊C4—1222no—
444C4⋊C4—1222no—
445C4⋊C4—1222no—
446C4⋊C4—1222no—
447C4⋊C4C2×C4×D6112yes—
448C22⋊C4—21no—
449C22⋊C4—1222no—
450C22⋊C4—1222no—
451C22⋊C4—1221no—
452C42—21no—
453C42—21no—
454C42—611no—
455C42—611no—
456C42—612no—
457C42—612no—
458C42—1221no—
459C42—1221no—
460C42—2444no—
461C42#1175822412yes—
462C23—21no—
463C23—1221no—
464C23#41×D6111yescharacteristic
465Q8—21no—
466Q8—21no—
467Q8—1221no—
468Q8—1221no—
469Q8—4882no—
470D4—21no—
471D4—21no—
472D4—41no—
473D4—41no—
474D4—1221no—
475D4—1221no—
476D4—4882no—
477D4—2441no—
478D4—2441no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—612no—
486C2×C4—612no—
487C2×C4—1222no—
488C2×C4—1221no—
489C2×C4—1222no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4—2441no—
493C2×C4—2441no—
494C2×C4C22×C4×D6111yescharacteristic
495C2×C4C2×D4×D6111yescharacteristic
496C2×C4#41×D6111yescharacteristic
497C2×C4#41×D6111yescharacteristic
498C2×C4#41×D6111yescharacteristic
499C2×C4C2×#1175822411yescharacteristic
500C8—22no—
501C8—22no—
502C8—41no—
503C8—41no—
504C8—612no—
505C8—612no—
506C8—1222no—
507C8—2441no—
508C8—2441no—
509C22—22no—
510C22—42no—
511C22—41no—
512C22—2442no—
513C22—2441no—
514C22C4×D4×D6111yescharacteristic
515C22#1684028911yescharacteristic
516C22#1684029511yescharacteristic
517C4—24no—
518C4—42no—
519C4—41no—
520C4—1222no—
521C4—1222no—
522C4—2442no—
523C4—2442no—
524C4—2441no—
525C4C4×#1175822411yescharacteristic
526C4#1684023011yescharacteristic
527C4#124×D6111yescharacteristic
528C4#1684030711yescharacteristic
529C2—21no—
530C2—21no—
531C2—81no—
532C2—4881no—
533C2#2144836011yescharacteristic
534C2#2145472412yes—
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1937C2
socleC22×C61C2×C4×D4
prime radical#16840308C22
cosocleC22×C4×C61C24
hypercenter#1937D61
hypocenterC61#21662
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1937

Complements

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

NH ≅ G/Ntype
2. #21454758C2semidirect
4. #21446794C2semidirect
6. #21446533C2semidirect
7. #21446529C2semidirect
8. #21446523C2semidirect
9. #21445958C2semidirect
10. #21445666C2semidirect
11. #21445662C2semidirect
12. #21445657C2semidirect
13. #21445653C2semidirect
14. C4×#14519687C2semidirect
15. C61×#1937C2semidirect
20. #16840129C4semidirect
21. #16840125C4semidirect
22. #16840123C4semidirect
23. #16840120C4semidirect
270. #1937D61semidirect
326. C61#21662semidirect

Maximal subgroups

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

subgroupindex
#214561622
#214547582
#214545342
#214467942
#214466452
#214465332
#214465292
#214465232
#214459582
#214456662
#214456622
#214456572
#214456532
C4×#145196872
C61×#19372
#2166261