Finite Group Databasebeta

← #29696015

Subgroups — #29696015

Order 15616.

1312 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840576C211yesmaximal, characteristic
2#21454703C211yesmaximal, characteristic, complemented
3#21454353C212yesmaximal, complemented
4#21446732C211yesmaximal, characteristic, complemented
5#21446690C211yesmaximal, characteristic, complemented
6#21446334C212yesmaximal, complemented
7#21446186C212yesmaximal, complemented
8#21445875C211yesmaximal, characteristic, complemented
9#21445779C212yesmaximal, complemented
10#21445663C211yesmaximal, characteristic, complemented
11C2×C61×#444C211yesmaximal, characteristic, complemented
12C2×#14519824C2212yes—
13#16840576—24no—
14#16840576C418yescomplemented
15C2×#14519807C2212yes—
16C2×#14519806—22no—
17C2×#14519796—21no—
18C22×Dic244C2211yescharacteristic
19C2×Q16×D61—21no—
20C2×#14519783C2211yescharacteristic
21#16840245C2212yes—
22#16839874—21no—
23#16839869—21no—
24SD16×Dic61—22no—
25C2×#14519709C2211yescharacteristic
26C2×#14519705C2211yescharacteristic
27#16839772—22no—
28#16839772C2212yes—
29#16839764—22no—
30#16839764C2212yes—
31#16839763C2212yes—
32#16839759—22no—
33#16839759C2211yescharacteristic
34C2×#14519702—22no—
35C2×#14519702C2212yes—
36C2×#14519695C2212yes—
37#16839717C2211yescharacteristic
38#16839668C2212yes—
39#16839530C2212yes—
40#16839527C2211yescharacteristic
41#16839522C2211yescharacteristic
42C2×#135×C61C2211yescharacteristic
43C2×#131×C61C2212yes—
44C61×#444—24no—
45C2×#108×C61C2211yescharacteristic
46C2×#104×C61C2212yes—
47C2×#97×C61C2211yescharacteristic
48#14519824—24no—
49#14519824—24no—
50#14519824—42no—
51#14519824C2×C418yescomplemented
52C2×#11758222—22no—
53C2×#11758222C2311yescharacteristic
54C2×#11758224—22no—
55C2×#11758224—42no—
56C22×Dic122—22no—
57C22×Dic122C2312yes—
58C2×Q8×D61—22no—
59#14519807—24no—
60#14519807C2×C418yescomplemented
61#14519806—216no—
62#14519796—28no—
63C2×Dic244—21no—
64C2×Dic244—21no—
65C2×Dic244—22no—
66C2×Dic244C2×C414yescomplemented
67C2×#11758202—22no—
68Q16×D61—28no—
69C2×#11758200—22no—
70#14519783—22no—
71#14519783C2×C414yescomplemented
72C2×#11758194C2311yescharacteristic
73C2×#11758194C2311yescharacteristic
74C2×#11758189—21no—
75C2×C8×D61—21no—
76Q8×Dic61—22no—
77D4×Dic61—22no—
78D4×Dic61—42no—
79#14519736—22no—
80#14519736C2311yescharacteristic
81C2×C4⋊Dic61—22no—
82C2×C4⋊Dic61C2312yes—
83#14519709—21no—
84#14519709—21no—
85#14519705—22no—
86#14519702—22no—
87#14519702—22no—
88#14519702—22no—
89#14519702—22no—
90#14519702—22no—
91#14519702D412yes—
92#14519701—22no—
93#14519699—22no—
94#14519699D412yes—
95#14519698—22no—
96#14519695—42no—
97#14519687—21no—
98C8×Dic61—21no—
99#14519684—42no—
100C2×#41×C61C2312yes—
101#135×C61C2×C414yes—
102#135×C61D414yes—
103C2×Q16×C61D411yescharacteristic
104C2×SD16×C61D412yes—
105C2×D8×C61D411yescharacteristic
106#131×C61—24no—
107#131×C61—42no—
108C2×M16×C61—22no—
109C2×M16×C61C2312yes—
110C22×C8×C61—22no—
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C2×C42×C61C2311yescharacteristic
114#108×C61—22no—
115#104×C61—24no—
116#97×C61—22no—
117#41×C61—22no—
118#41×C61—22no—
119#41×C61C22×C414yes—
120#41×C61C2×D414yes—
121C2×Q8×C61C2×D412yes—
122C2×D4×C61—22no—
123C2×D4×C61C2×D412yes—
124C22×C4×C61—22no—
125C22×C4×C61—21no—
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—21no—
128C2×C22⋊D61—42no—
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—42no—
132C22×Dic61—42no—
133Q8×D61—28no—
134Q8×D61—44no—
135#11758224—28no—
136#11758224—44no—
137#11758224—48no—
138#11758224—48no—
139#11758222—21no—
140#11758222—21no—
141#11758222—28no—
142#11758222—44no—
143#11758222C22×C412yes—
144#11758222#4112yes—
145C2×D244C2×D411yescharacteristic
146C2×C4×D61—21no—
147C2×C4×D61—42no—
148C2×Dic122—22no—
149C2×Dic122—22no—
150C2×Dic122—22no—
151C2×Dic122—22no—
152C2×Dic122—28no—
153C2×Dic122—22no—
154C2×Dic122—44no—
155C2×Dic122—42no—
156C2×Dic122C22×C414yes—
157C2×Dic122C2×D411yescharacteristic
158Q16×C61C22⋊C414yes—
159SD16×C61C22⋊C418yes—
160D8×C61C22⋊C414yes—
161M16×C61—22no—
162M16×C61—22no—
163M16×C61—44no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—22no—
167C2×C8×C61—44no—
168C2×C8×C61C22×C412yes—
169C2×C8×C61C2×D412yes—
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2×D412yes—
173C42×C61—21no—
174C42×C61—21no—
175C22⋊Dic61—21no—
176C22⋊Dic61—42no—
177#11758202—28no—
178#11758200—28no—
179C4⋊Dic61—22no—
180C4⋊Dic61—22no—
181C4⋊Dic61—42no—
182C4⋊Dic61C2×D412yes—
183C4×Dic61—21no—
184C4×Dic61—42no—
185#11758194—21no—
186#11758194—21no—
187#11758194—21no—
188#11758194C22×C412yes—
189C2×#92418—21no—
190C2×#92418—21no—
191Dic244—24no—
192Dic244—24no—
193#11758189—24no—
194C8×D61—24no—
195C23×C61—22no—
196C23×C61H(5,𝔽2)11yescharacteristic
197C22×D61—21no—
198Q8×C61—22no—
199Q8×C61C2×C22⋊C414yes—
200D4×C61—24no—
201D4×C61—24no—
202D4×C61—22no—
203D4×C61C2×C22⋊C414yes—
204C2×C4×C61—22no—
205C2×C4×C61—24no—
206C2×C4×C61—24no—
207C2×C4×C61—42no—
208C2×C4×C61—41no—
209C2×C4×C61C23×C411yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C2×C4×C61H(5,𝔽2)11yescharacteristic
215C22⋊D61—22no—
216C22⋊D61—22no—
217C22⋊D61—48no—
218C2×Dic61—21no—
219C2×Dic61—24no—
220C2×Dic61—22no—
221C2×Dic61—48no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—82no—
226C2×Dic61—162no—
227D244—21no—
228D244C4×D412yes—
229C4×D61—22no—
230C4×D61—22no—
231C4×D61—48no—
232Dic122—21no—
233Dic122—24no—
234Dic122—24no—
235Dic122—48no—
236Dic122—42no—
237Dic122—42no—
238Dic122C4×D412yes—
239C8×C61—22no—
240C8×C61—42no—
241C8×C61C2×C22⋊C412yes—
242C8×C61C2×C22⋊C412yes—
243#92418—22no—
244#92418—41no—
245#22465—611nomaximal, Sylow
246C22×C61—22no—
247C22×C61—24no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61#51811yescharacteristic
251C22×C61D4211yescharacteristic
252C22×C61#54611yescharacteristic
253C2×D61—22no—
254C2×D61—42no—
255C4×C61—24no—
256C4×C61—81no—
257C4×C61C22×C22⋊C411yescharacteristic
258C4×C61C2×C4×D411yescharacteristic
259C4×C61D4211yescharacteristic
260C4×C61#54611yescharacteristic
261Dic61—22no—
262Dic61—44no—
263Dic61—42no—
264Dic61—82no—
265C2×#578—611no—
266C2×#444D6111yescharacteristic, complemented
267#2952—611no—
268#2898—612no—
269#1979—611no—
270#1971—611no—
271#1908—612no—
272#1882—612no—
273#1815—611no—
274#1793—612no—
275#1765—611no—
276C2×C61—21no—
277C2×C61—21no—
278C2×C61—44no—
279C2×C61D4×C22⋊C411yescharacteristic
280C2×C61#328912yes—
281D61—42no—
282C2×2−1+4—612no—
283#578—618no—
284#578—1224no—
285C2×#137—612no—
286C2×#137—1222no—
287C2×#135—1221no—
288C2×#135C2×D6111yescharacteristic
289C22×Q16—611no—
290C22×Q16—1221no—
291C2×#131—611no—
292C2×#131C2×D6112yes—
293#520—612no—
294#444—24no—
295#442—1221no—
296#441—1221no—
297C4×SD16—1222no—
298C2×#108C2×D6111yescharacteristic
299C2×#106—611no—
300C2×#104—611no—
301C2×#104C2×D6112yes—
302#419—612no—
303#419—1222no—
304#417—612no—
305#417—612no—
306#417—1222no—
307#416—611no—
308#416—1222no—
309C2×#103—612no—
310C2×#103—1222no—
311C2×#101—612no—
312#405—611no—
313C2×#97C2×D6111yescharacteristic
314#391—612no—
315#343—612no—
316#341—611no—
317#339—611no—
318C61#2246511yescharacteristic, complemented, Sylow
3192−1+4—618no—
3202−1+4—1224no—
3212−1+4—1224no—
3222−1+4—2442no—
323C2×#41—611no—
324C2×#41—1222no—
325C2×#41—1222no—
326C2×#41—2442no—
327C2×#41C22×D6112yes—
328C22×Q8—612no—
329C22×Q8—1222no—
330C22×Q8—1222no—
331#137—618no—
332#137—1224no—
333#137—12216no—
334#135—1228no—
335#135C4×D6114yes—
336#135C22⋊D6114yes—
337C2×Q16—614no—
338C2×Q16—1222no—
339C2×Q16—1228no—
340C2×Q16—1221no—
341C2×Q16—1221no—
342C2×Q16—1222no—
343C2×Q16D24411yescharacteristic
344C2×SD16—1222no—
345C2×SD16D24412yes—
346C2×D8D24411yescharacteristic
347#131—24no—
348#131—42no—
349#131—614no—
350#131—1222no—
351C2×M16—22no—
352C2×M16—611no—
353C2×M16—611no—
354C2×M16—1221no—
355C2×M16C22×D6112yes—
356C22×C8—22no—
357C22×C8—1221no—
358C22×C8C22×D6111yescharacteristic
359C22×C8C22×D6111yescharacteristic
360C4×Q8—1222no—
361C4×D4—1222no—
362C4×D4—2442no—
363#117—611no—
364#117—1222no—
365C2×C4⋊C4—612no—
366C2×C4⋊C4—1222no—
367C2×C42C22×D6111yescharacteristic
368#108—22no—
369#106—1221no—
370#106—1221no—
371#104—24no—
372#104—1222no—
373#103—612no—
374#103—1222no—
375#103—1222no—
376#103—1222no—
377#103—1222no—
378#103—1222no—
379#103—1222no—
380#102—612no—
381#102—1222no—
382#102—1222no—
383#101—2442no—
384#97—22no—
385#97—1221no—
386C4×C8—1221no—
387#95—2442no—
388#41—22no—
389#41—22no—
390#41—612no—
391#41—612no—
392#41—1228no—
393#41—1221no—
394#41—1221no—
395#41—1228no—
396#41—2444no—
397#41—2448no—
398#41—2444no—
399#41—2448no—
400#41C2×C4×D6114yes—
401#41C2×C22⋊D6114yes—
402C2×Q8—614no—
403C2×Q8—611no—
404C2×Q8—1228no—
405C2×Q8—1222no—
406C2×Q8—1228no—
407C2×Q8—1222no—
408C2×Q8—1222no—
409C2×Q8—1222no—
410C2×Q8—1222no—
411C2×Q8—2442no—
412C2×Q8—2444no—
413C2×Q8—2444no—
414C2×Q8C2×D24412yes—
415C2×D4—22no—
416C2×D4—611no—
417C2×D4—1221no—
418C2×D4—2442no—
419C2×D4C2×D24412yes—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—1221no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—2442no—
426C22×C4—2442no—
427C22×C4—2442no—
428C22×C4C23×D6111yescharacteristic
429Q16—1228no—
430Q16—1224no—
431Q16—1224no—
432Q16#1175819814yes—
433SD16—1228no—
434SD16#1175819818yes—
435D8#1175819814yes—
436M16—22no—
437M16—22no—
438M16—44no—
439M16—612no—
440M16—1221no—
441M16—1224no—
442M16—1221no—
443M16—1221no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—22no—
447C2×C8—44no—
448C2×C8—1221no—
449C2×C8—1224no—
450C2×C8—1221no—
451C2×C8C2×C4×D6112yes—
452C2×C8C2×D24411yescharacteristic
453C2×C8C2×D24411yescharacteristic
454C2×C8D4×D6112yes—
455C2×C8C2×C22⋊D6112yes—
456C4⋊C4—612no—
457C4⋊C4—1222no—
458C4⋊C4—1222no—
459C4⋊C4—2442no—
460C22⋊C4—1221no—
461C22⋊C4—2442no—
462C42—21no—
463C42—21no—
464C42—1221no—
465C42—2442no—
466C23—22no—
467C23—1221no—
468C23#1451982311yescharacteristic
469Q8—22no—
470Q8—612no—
471Q8—1221no—
472Q8—1224no—
473Q8—1224no—
474Q8—2448no—
475Q8—2442no—
476Q8—2442no—
477Q8C2×#1175819814yes—
478D4—24no—
479D4—24no—
480D4—22no—
481D4—612no—
482D4—1222no—
483D4—1222no—
484D4—1221no—
485D4—2448no—
486D4C2×#1175819814yes—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—1221no—
493C2×C4—1222no—
494C2×C4—1222no—
495C2×C4—1224no—
496C2×C4—1222no—
497C2×C4—9762no—
498C2×C4—4882no—
499C2×C4—2448no—
500C2×C4—2448no—
501C2×C4—2442no—
502C2×C4—2442no—
503C2×C4—2442no—
504C2×C4C22×C4×D6111yescharacteristic
505C2×C4C2×D4×D6111yescharacteristic
506C2×C4C22×C22⋊D6111yescharacteristic
507C2×C4C22×D24411yescharacteristic
508C2×C4#41×D6111yescharacteristic
509C2×C4#1451982211yescharacteristic
510C8—22no—
511C8—42no—
512C8—1222no—
513C8—2441no—
514C8C2×#1175819812yes—
515C8C2×#1175819812yes—
516C22—22no—
517C22—24no—
518C22—42no—
519C22—42no—
520C22—1222no—
521C22—2442no—
522C22#1684023411yescharacteristic
523C22D4×D24411yescharacteristic
524C22#1684040011yescharacteristic
525C4—24no—
526C4—81no—
527C4—1222no—
528C4—4882no—
529C4—2442no—
530C4—2444no—
531C4C22×#1175819811yescharacteristic
532C4C4×D4×D6111yescharacteristic
533C4D4×C22⋊D6111yescharacteristic
534C4#1684040711yescharacteristic
535C2—21no—
536C2—21no—
537C2—44no—
538C2—2442no—
539C2D4×#1175819811yescharacteristic
540C2#2145759512yes—
subgroupG/H
centerC22D4×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#444C2
socleC22×C61D42
prime radicalC2×#16840576C2
cosocleC22×C4×C61C24
hypercenterC2×#444D61
hypocenterC61#22465
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#444

Complements

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

NH ≅ G/Ntype
2. #21454703C2semidirect
3. #21454353C2semidirect
4. #21446732C2semidirect
5. #21446690C2semidirect
6. #21446334C2semidirect
7. #21446186C2semidirect
8. #21445875C2semidirect
9. #21445779C2semidirect
10. #21445663C2semidirect
11. C2×C61×#444C2semidirect
14. #16840576C4semidirect
51. #14519824C2×C4semidirect
60. #14519807C2×C4semidirect
66. C2×Dic244C2×C4semidirect
71. #14519783C2×C4semidirect
266. C2×#444D61semidirect
318. C61#22465semidirect

Maximal subgroups

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

subgroupindex
C2×#168405762
#214547032
#214543532
#214467322
#214466902
#214463342
#214461862
#214458752
#214457792
#214456632
C2×C61×#4442
#2246561