Finite Group Databasebeta

← #29734150

Subgroups — #29734150

Order 15616.

1112 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840497C211yesmaximal, characteristic
2C2×#16840124C211yesmaximal, characteristic, complemented
3C2×#16840049C211yesmaximal, characteristic, complemented
4#21447060C212yesmaximal, complemented
5#21447053C212yesmaximal
6#21446760C211yesmaximal, characteristic, complemented
7#21446751C211yesmaximal, characteristic, complemented
8#21446086C212yesmaximal, complemented
9#21446079C212yesmaximal, complemented
10#21445647C211yesmaximal, characteristic, complemented
11C61×#2546C211yesmaximal, characteristic, complemented
12#16840497—24no—
13#16840497—24no—
14C2×#14519774C2211yescharacteristic
15C2×#14519766—21no—
16C2×#14519765C2211yescharacteristic
17C2×#14519765C2211yescharacteristic
18C2×#14519762—21no—
19C2×#14519762—21no—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×#14519739—21no—
24#16840124—24no—
25#16840049—24no—
26C2×#14519708C2212yes—
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519700C2212yes—
29C2×#14519700C2212yes—
30C2×#14519697C2212yes—
31C2×#14519697C2212yes—
32C2×#14519697C2212yes—
33C2×#14519697C2212yes—
34C2×#14519688C2211yescharacteristic
35#16839524C2212yes—
36#16839517C2212yes—
37#16839517C2212yes—
38C2×C4⋊Q8×C61—21no—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4×Q8×C61C2211yescharacteristic
41C61×#398—21no—
42C61×#398—21no—
43C61×#391—21no—
44C61×#389C2212yes—
45C61×#389C2212yes—
46C4×C4⋊C4×C61C2211yescharacteristic
47C22×Dic122C2311yescharacteristic
48C22×D244—21no—
49C22×D244—21no—
50C22×D244C2311yescharacteristic
51C22×D244C2311yescharacteristic
52C22×C4×D61—21no—
53C22×C4×D61—21no—
54C22×#92418C2312yes—
55C22×#92418C2312yes—
56#14519774—22no—
57#14519774—22no—
58#14519766—28no—
59#14519765—22no—
60#14519765—22no—
61#14519765—22no—
62#14519765D414yes—
63#14519762—28no—
64#14519762—28no—
65C4×Dic122—22no—
66C4×Dic122—22no—
67C4×D244—22no—
68C4×D244—22no—
69C4×D244—22no—
70C4×D244D414yes—
71#14519739—28no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×C4×Dic61—21no—
81#14519708—24no—
82C4⋊#92418—22no—
83#14519700—24no—
84#14519700—24no—
85#14519697—24no—
86#14519697—24no—
87#14519697—24no—
88#14519697—24no—
89#14519688—22no—
90C22×Q8×C61—21no—
91C22×Q8×C61D411yescharacteristic
92C4⋊Q8×C61—22no—
93C4⋊Q8×C61—22no—
94C4⋊Q8×C61—44no—
95C4×Q8×C61—22no—
96C4×Q8×C61—22no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61—22no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C4⋊C4×C61C2312yes—
104C2×C4⋊C4×C61C2312yes—
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C2×C42×C61C2311yescharacteristic
108C2×C42×C61C2311yescharacteristic
109#95×C61—21no—
110#95×C61—22no—
111#95×C61—22no—
112C23×D61—21no—
113C23×D61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—41no—
117C2×Q8×C61—42no—
118C2×Q8×C61—42no—
119C2×Q8×C61—42no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2411yescharacteristic
129C22×Dic61—21no—
130C22×Dic61—21no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—22no—
135C2×D244—24no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—24no—
139C2×D244—42no—
140C2×D244—42no—
141C2×D244—82no—
142C2×D244C2×D412yes—
143C2×D244C2×D412yes—
144C2×C4×D61—24no—
145C2×C4×D61—24no—
146C2×C4×D61—82no—
147C2×C4×D61—82no—
148C2×Dic122—22no—
149C2×Dic122—22no—
150C2×Dic122—21no—
151C2×Dic122C2×D412yes—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—44no—
159C4⋊C4×C61—42no—
160C4⋊C4×C61—44no—
161C4⋊C4×C61#4114yes—
162C42×C61—21no—
163C42×C61—21no—
164C42×C61—21no—
165C42×C61—21no—
166C42×C61—42no—
167C42×C61C2×D412yes—
168C42×C61C2×D412yes—
169#11758198—24no—
170#11758198—24no—
171#11758198—24no—
172#11758198—24no—
173#11758198—24no—
174#11758198—24no—
175C4⋊Dic61—21no—
176C4⋊Dic61C2×D412yes—
177#11758196—24no—
178C4×Dic61—24no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C23×C61C22×D411yescharacteristic
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—42no—
195C22×D61—42no—
196Q8×C61—44no—
197Q8×C61—84no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—22no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—24no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61#12012yes—
217C2×C4×C61#12012yes—
218C2×C4×C61AGL(1,ℤ/8)12yes—
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C2×#4112yes—
222C2×C4×C612−1+412yes—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227D244—22no—
228D244—22no—
229D244—24no—
230D244—48no—
231D244—41no—
232D244—41no—
233D244—42no—
234D244—84no—
235C4×D61—84no—
236C4×D61—84no—
237Dic122—24no—
238Dic122—41no—
239Dic122—41no—
240#92418—42no—
241#92418—42no—
242#25659—611nomaximal, Sylow
243C22×C61C2×#12011yescharacteristic
244C22×C61#54812yes—
245C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
246C22×C61#57511yescharacteristic
247C22×C61#57712yes—
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—42no—
251C2×D61—42no—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—82no—
255C2×D61—82no—
256C2×D61—82no—
257C2×D61—82no—
258C4×C61—22no—
259C4×C61—22no—
260C4×C61—22no—
261C4×C61—44no—
262C4×C61—41no—
263C4×C61—41no—
264C4×C61—41no—
265C4×C61—82no—
266C4×C61#53211yescharacteristic
267C4×C61#53611yescharacteristic
268C4×C61#54712yes—
269Dic61—42no—
270Dic61—42no—
271C2×#559—611no—
272C2×#489—611no—
273C2×#476—611no—
274#2546D6111yescharacteristic, complemented
275#2042—612no—
276#2041—612no—
277#1985—611no—
278#1983—611no—
279#1865—612no—
280#1864—612no—
281#1762—611no—
282C2×C61#243211yescharacteristic
283C2×C61#300611yescharacteristic
284C2×C61#300911yescharacteristic
285C2×C61#337112yes—
286C2×C61#337312yes—
287D61—82no—
288D61—82no—
289#559—1224no—
290#559—1224no—
291C2×C4⋊Q8—21no—
292C2×C4⋊Q8C2×D6111yescharacteristic
293C2×#127—611no—
294C2×#124—611no—
295C2×#124—611no—
296C2×#124—1221no—
297C2×#123—1221no—
298C2×#123—1221no—
299C2×C4×Q8—611no—
300C2×C4×Q8C2×D6111yescharacteristic
301C2×C4×D4—611no—
302C2×C4×D4—611no—
303C2×C4×D4—1221no—
304#489—1224no—
305#476—1224no—
306C2×#106—612no—
307C2×C4⋊C8—611no—
308C2×#103—612no—
309C2×#103—612no—
310C2×#102—612no—
311C2×#102—612no—
312C2×#102—612no—
313C2×#102—612no—
314C2×#97—611no—
315#398—21no—
316#398—21no—
317#391—21no—
318#389C2×D6112yes—
319#389C2×D6112yes—
320C4×C4⋊C4C2×D6111yescharacteristic
321#340—612no—
322#336—612no—
323#336—612no—
324C61#2565911yescharacteristic, complemented, Sylow
325C22×Q8—21no—
326C22×Q8—611no—
327C22×Q8C22⋊D6111yescharacteristic
328C22×D4—611no—
329C22×D4—611no—
330C22×D4—1221no—
331C22×D4—1221no—
332C23×C4—1221no—
333C23×C4—1221no—
334C22×C8—612no—
335C22×C8—612no—
336C4⋊Q8—22no—
337C4⋊Q8—22no—
338C4⋊Q8—44no—
339#127—1222no—
340#127—1222no—
341#124—614no—
342#124—1222no—
343#124—1222no—
344#124—1228no—
345#124—1222no—
346#123—1228no—
347#123—1228no—
348C4×Q8—22no—
349C4×Q8—22no—
350C4×Q8—1222no—
351C4×Q8—1222no—
352C4×D4—614no—
353C4×D4—1222no—
354C4×D4—1222no—
355C4×D4—1228no—
356C4×D4—1222no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—611no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4C22⋊D6111yescharacteristic
363C2×C4⋊C4C22⋊D6111yescharacteristic
364C2×C4⋊C4C22⋊D6111yescharacteristic
365C2×C4⋊C4C22×D6112yes—
366C2×C4⋊C4C22×D6112yes—
367C2×C22⋊C4—1221no—
368C2×C22⋊C4—1221no—
369C2×C22⋊C4—1221no—
370C2×C22⋊C4—1221no—
371C2×C22⋊C4—1221no—
372C2×C22⋊C4—1221no—
373C2×C42—21no—
374C2×C42—1221no—
375C2×C42C22×D6111yescharacteristic
376C2×C42C22×D6111yescharacteristic
377C2×C42C22×D6111yescharacteristic
378#106—1224no—
379C4⋊C8—1222no—
380#103—1224no—
381#103—1224no—
382#102—1224no—
383#102—1224no—
384#102—1224no—
385#102—1224no—
386#97—1222no—
387#95—21no—
388#95—22no—
389#95—22no—
390C24—1221no—
391C24—1221no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—41no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—612no—
399C2×Q8—1221no—
400C2×Q8—1222no—
401C2×Q8—1222no—
402C2×D4—612no—
403C2×D4—612no—
404C2×D4—1224no—
405C2×D4—1224no—
406C2×D4—1221no—
407C2×D4—1221no—
408C2×D4—1222no—
409C2×D4—1222no—
410C2×D4—1222no—
411C2×D4—1222no—
412C2×D4—4882no—
413C2×D4—2442no—
414C2×D4—2442no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—1224no—
418C22×C4—1221no—
419C22×C4—1221no—
420C22×C4—1224no—
421C22×C4—4882no—
422C22×C4—4882no—
423C22×C4C2×C22⋊D6111yescharacteristic
424C22×C4C2×C22⋊D6111yescharacteristic
425C22×C4C2×C22⋊D6111yescharacteristic
426C22×C4C2×C22⋊D6111yescharacteristic
427C22×C4C2×C22⋊D6111yescharacteristic
428C22×C4C2×C22⋊D6111yescharacteristic
429C22×C4C23×D6111yescharacteristic
430C2×C8—1222no—
431C2×C8—1222no—
432C2×C8—1222no—
433C2×C8—1222no—
434C2×C8—1222no—
435C2×C8—1222no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—44no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—612no—
446C4⋊C4—1224no—
447C4⋊C4—1221no—
448C4⋊C4#1175822614yes—
449C22⋊C4—1224no—
450C22⋊C4—1224no—
451C22⋊C4—1224no—
452C22⋊C4—1224no—
453C22⋊C4—1224no—
454C22⋊C4—1224no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—42no—
460C42—1224no—
461C42D4×D6112yes—
462C42D4×D6112yes—
463C23—1222no—
464C23—1222no—
465C23—2441no—
466C23—2441no—
467C23—2441no—
468C23—2441no—
469C23—2442no—
470C23—2442no—
471C23—2442no—
472C23—2442no—
473C23C22×C22⋊D6111yescharacteristic
474Q8—44no—
475Q8—84no—
476Q8—1224no—
477Q8—2441no—
478Q8—2441no—
479D4—1222no—
480D4—1222no—
481D4—1224no—
482D4—4884no—
483D4—2448no—
484D4—2441no—
485D4—2441no—
486D4—2442no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—4884no—
508C2×C4—4884no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4#1451974512yes—
512C2×C4#1451974512yes—
513C2×C4#1451980312yes—
514C2×C4C2×D4×D6111yescharacteristic
515C2×C4C2×D4×D6111yescharacteristic
516C2×C4C2×#1175822612yes—
517C2×C4#1451982512yes—
518C8—2442no—
519C8—2442no—
520C22—4882no—
521C22—4882no—
522C22—4882no—
523C22—4882no—
524C22—2442no—
525C22—2442no—
526C22—2442no—
527C22—2442no—
528C22—2442no—
529C22—2442no—
530C22C2×#1451974511yescharacteristic
531C22#1684042312yes—
532C22C2×#1451980311yescharacteristic
533C22#1684056011yescharacteristic
534C22#1684057012yes—
535C4—22no—
536C4—22no—
537C4—22no—
538C4—44no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—82no—
543C4—2442no—
544C4—2442no—
545C4#1684031311yescharacteristic
546C4#1684033711yescharacteristic
547C4#1684041512yes—
548C2—4882no—
549C2—4882no—
550C2#2144944411yescharacteristic
551C2#2145506911yescharacteristic
552C2#2145509411yescharacteristic
553C2#2145858712yes—
554C2#2145861112yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2546C2
socleC23×C61C22×D4
prime radicalC2×#14519774C22
hypercenter#2546D61
hypocenterC61#25659
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2546

Complements

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

NH ≅ G/Ntype
2. C2×#16840124C2semidirect
3. C2×#16840049C2semidirect
4. #21447060C2semidirect
6. #21446760C2semidirect
7. #21446751C2semidirect
8. #21446086C2semidirect
9. #21446079C2semidirect
10. #21445647C2semidirect
11. C61×#2546C2semidirect
274. #2546D61semidirect
324. C61#25659semidirect

Maximal subgroups

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

subgroupindex
C2×#168404972
C2×#168401242
C2×#168400492
#214470602
#214470532
#214467602
#214467512
#214460862
#214460792
#214456472
C61×#25462
#2565961