Finite Group Databasebeta

← #29737039

Subgroups — #29737039

Order 15616.

1176 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840472C211yesmaximal, characteristic, complemented
2C2×#16840126C211yesmaximal, characteristic, complemented
3C2×#16839954C211yesmaximal, characteristic
4#21446965C212yesmaximal, complemented
5#21446955C212yesmaximal, complemented
6#21446776C211yesmaximal, characteristic, complemented
7#21446760C211yesmaximal, characteristic, complemented
8#21446170C212yesmaximal, complemented
9#21446170C212yesmaximal, complemented
10#21445647C211yesmaximal, characteristic, complemented
11C61×#2548C211yesmaximal, characteristic, complemented
12C22×#11758199C2212yes—
13#16840472—24no—
14#16840472—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519769C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×#14519755—21no—
19C2×#14519755—21no—
20C2×#14519753—21no—
21C2×C4×D244C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×C4×D244C2211yescharacteristic
24C2×C4⋊C4×D61—21no—
25#16840126—24no—
26#16839954—24no—
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519698C2212yes—
29C2×#14519698C2212yes—
30C2×#14519698C2212yes—
31C2×#14519697C2212yes—
32C2×#14519688C2211yescharacteristic
33#16839524C2212yes—
34#16839524C2212yes—
35#16839524C2212yes—
36#16839517C2212yes—
37#16839517C2212yes—
38C2×#127×C61C2211yescharacteristic
39C2×#124×C61—21no—
40C2×C4×Q8×C61C2211yescharacteristic
41C61×#394—21no—
42C61×#394—21no—
43C61×#390—21no—
44C61×#390C2212yes—
45C61×#390C2212yes—
46C4×C4⋊C4×C61C2211yescharacteristic
47C22×Dic122—21no—
48C22×D244C2311yescharacteristic
49C22×C4×D61—21no—
50C22×C4×D61—21no—
51C22×C4×D61—21no—
52C2×#11758199—24no—
53C2×#11758199—24no—
54C2×#11758199—24no—
55C22×#92418C2312yes—
56C22×#92418C2312yes—
57#14519777—22no—
58#14519777—22no—
59#14519769—22no—
60#14519769—22no—
61#14519769—22no—
62#14519769D414yes—
63#14519755—28no—
64#14519755—28no—
65#14519753—28no—
66C4×D244—22no—
67C4×D244—22no—
68C4×D244—22no—
69C4×D244—22no—
70C4×D244—22no—
71C4×D244D414yes—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C4⋊C4×D61—28no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C4⋊#92418—22no—
85#14519698—24no—
86#14519698—24no—
87#14519698—24no—
88#14519697—24no—
89#14519688—22no—
90C22×Q8×C61D411yescharacteristic
91C22×D4×C61—21no—
92C22×D4×C61—21no—
93C22×D4×C61C2312yes—
94#127×C61—22no—
95#127×C61—22no—
96#124×C61—44no—
97C4×Q8×C61—22no—
98C4×Q8×C61—22no—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C4⋊C4×C61C2312yes—
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61—22no—
105C2×C22⋊C4×C61—22no—
106C2×C22⋊C4×C61—22no—
107C2×C42×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C2×C42×C61C2311yescharacteristic
110C2×C42×C61C2311yescharacteristic
111#95×C61—21no—
112C24×C61—22no—
113C23×D61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—41no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—42no—
124C2×D4×C61—42no—
125C2×D4×C61—42no—
126C2×D4×C61—42no—
127C2×D4×C61—82no—
128C22×C4×C61—22no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2411yescharacteristic
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C2×D244—21no—
140C2×D244—22no—
141C2×D244C2×D412yes—
142C2×D244#4114yes—
143C2×C4×D61—24no—
144C2×C4×D61—24no—
145C2×C4×D61—24no—
146C2×C4×D61—42no—
147C2×C4×D61—42no—
148C2×C4×D61—82no—
149C2×C4×D61—82no—
150C2×Dic122—24no—
151C2×Dic122—82no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C22⋊C4×C61—44no—
158C22⋊C4×C61—44no—
159C22⋊C4×C61—82no—
160C22⋊C4×C61—82no—
161C42×C61—21no—
162C42×C61—21no—
163C42×C61—21no—
164C42×C61—21no—
165C42×C61—42no—
166C42×C61C2×D412yes—
167C42×C61C2×D412yes—
168#11758199—48no—
169#11758198—24no—
170#11758198—24no—
171#11758198—24no—
172C4⋊Dic61—24no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61C2×D412yes—
177C4⋊Dic61C2×D412yes—
178C4⋊Dic61C2×D412yes—
179#11758196—24no—
180#11758196—24no—
181#11758196—24no—
182#11758196—24no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—22no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61C22×D411yescharacteristic
197C22×D61—22no—
198C22×D61—24no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—42no—
202Q8×C61—44no—
203D4×C61—42no—
204D4×C61—42no—
205D4×C61—42no—
206D4×C61—42no—
207D4×C61—84no—
208D4×C61—84no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—22no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61#12012yes—
225C2×C4×C61#12012yes—
226C2×C4×C61AGL(1,ℤ/8)12yes—
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C2×#4112yes—
230C2×C4×C612−1+412yes—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238D244—22no—
239D244—24no—
240D244#12414yescomplemented
241C4×D61—48no—
242C4×D61—84no—
243C4×D61—84no—
244Dic122—84no—
245#92418—42no—
246#92418—42no—
247#25946—611nomaximal, Sylow
248C22×C61—82no—
249C22×C61—82no—
250C22×C61—82no—
251C22×C61—82no—
252C22×C61—82no—
253C22×C61—82no—
254C22×C61—82no—
255C22×C61C2×#12011yescharacteristic
256C22×C61#54812yes—
257C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
258C22×C61#57511yescharacteristic
259C22×C61#57612yes—
260C2×D61—24no—
261C2×D61—44no—
262C2×D61—44no—
263C2×D61—42no—
264C2×D61—42no—
265C4×C61—22no—
266C4×C61—22no—
267C4×C61—22no—
268C4×C61—41no—
269C4×C61—41no—
270C4×C61—41no—
271C4×C61—82no—
272C4×C61C2×#12411yescharacteristic
273C4×C61#53611yescharacteristic
274C4×C61#54812yes—
275Dic61—42no—
276Dic61—42no—
277Dic61—42no—
278C2×#554—611no—
279C2×#490—611no—
280C2×#459—611no—
281#2548D6111yescharacteristic, complemented
282#2025—612no—
283#2023—612no—
284#1989—611no—
285#1985—611no—
286#1879—612no—
287#1879—612no—
288#1762—611no—
289C2×C61—162no—
290C2×C61#243611yescharacteristic
291C2×C61#300611yescharacteristic
292C2×C61#300811yescharacteristic
293C2×C61#336612yes—
294C2×C61#337012yes—
295D61—44no—
296C22×D8—612no—
297#554—1224no—
298#554—1224no—
299C2×C4⋊Q8—611no—
300C2×#127C2×D6111yescharacteristic
301C2×#125—611no—
302C2×#125—611no—
303C2×#124—21no—
304C2×C22⋊Q8—1221no—
305C2×C22⋊Q8—1221no—
306C2×C22⋊Q8—1221no—
307C2×C4×Q8C2×D6111yescharacteristic
308C2×C4×D4—611no—
309C2×C4×D4—611no—
310C2×C4×D4—611no—
311C22×C4⋊C4—1221no—
312#490—1224no—
313#459—1224no—
314C2×C4⋊C8—611no—
315C2×#102—612no—
316C2×#102—612no—
317C2×#102—612no—
318C2×#102—612no—
319C2×#97—611no—
320#394—21no—
321#394—21no—
322#390—21no—
323#390C2×D6112yes—
324#390C2×D6112yes—
325C4×C4⋊C4C2×D6111yescharacteristic
326#340—612no—
327#340—612no—
328#340—612no—
329#336—612no—
330#336—612no—
331C61#2594611yescharacteristic, complemented, Sylow
332C22×Q8—1221no—
333C22×Q8C22⋊D6111yescharacteristic
334C22×D4—21no—
335C22×D4—21no—
336C22×D4—611no—
337C22×D4C22×D6112yes—
338C23×C4—1221no—
339C23×C4—1221no—
340C23×C4—1221no—
341C2×D8—1224no—
342C2×D8—1224no—
343C2×D8—1224no—
344C22×C8—612no—
345C22×C8—612no—
346C4⋊Q8—1222no—
347C4⋊Q8—1222no—
348#127—22no—
349#127—22no—
350#125—614no—
351#125—1222no—
352#125—1222no—
353#125—1222no—
354#124—44no—
355C22⋊Q8—1228no—
356C22⋊Q8—1228no—
357C22⋊Q8—1228no—
358C4×Q8—22no—
359C4×Q8—22no—
360C4×D4—614no—
361C4×D4—1222no—
362C4×D4—1222no—
363C4×D4—1222no—
364C4×D4—1222no—
365C4×D4—1222no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—1228no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4C22⋊D6111yescharacteristic
376C2×C4⋊C4C22⋊D6111yescharacteristic
377C2×C4⋊C4C22⋊D6111yescharacteristic
378C2×C4⋊C4C22×D6112yes—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C42—21no—
387C2×C42C22×D6111yescharacteristic
388C2×C42C22×D6111yescharacteristic
389C2×C42C22×D6111yescharacteristic
390C4⋊C8—1222no—
391#102—1224no—
392#102—1224no—
393#102—1224no—
394#102—1224no—
395#97—1222no—
396#95—21no—
397C24—22no—
398C24—1221no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—41no—
402C2×Q8—1224no—
403C2×Q8—4882no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—82no—
415C2×D4—612no—
416C2×D4—614no—
417C2×D4—1221no—
418C2×D4—1222no—
419C22×C4—22no—
420C22×C4—1224no—
421C22×C4—1224no—
422C22×C4—1221no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—1224no—
426C22×C4—4882no—
427C22×C4—4882no—
428C22×C4—2442no—
429C22×C4—2442no—
430C22×C4C2×C22⋊D6111yescharacteristic
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C2×C22⋊D6111yescharacteristic
433C22×C4C2×C22⋊D6111yescharacteristic
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C23×D6111yescharacteristic
437D8—2448no—
438C2×C8—1222no—
439C2×C8—1222no—
440C2×C8—1222no—
441C2×C8—1222no—
442C2×C8—1222no—
443C2×C8—1222no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—612no—
450C4⋊C4—612no—
451C4⋊C4—612no—
452C4⋊C4—1224no—
453C4⋊C4—1224no—
454C4⋊C4—1224no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—1224no—
459C4⋊C4—1224no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—82no—
463C22⋊C4—82no—
464C22⋊C4—1224no—
465C22⋊C4—1224no—
466C22⋊C4—1224no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—42no—
472C42D4×D6112yes—
473C42D4×D6112yes—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—1222no—
482C23—1224no—
483C23—2441no—
484C23—2441no—
485C23—2442no—
486C23C22×C22⋊D6111yescharacteristic
487Q8—44no—
488Q8—4884no—
489D4—42no—
490D4—42no—
491D4—42no—
492D4—42no—
493D4—84no—
494D4—84no—
495D4—614no—
496D4—1222no—
497D4—1224no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1222no—
516C2×C4—4884no—
517C2×C4—4884no—
518C2×C4—2448no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4#1451974512yes—
524C2×C4#1451974512yes—
525C2×C4#1451980312yes—
526C2×C4C2×D4×D6111yescharacteristic
527C2×C4C2×D4×D6111yescharacteristic
528C2×C4#41×D6112yes—
529C2×C4#1451982412yes—
530C8—2442no—
531C8—2442no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—1224no—
540C22—2444no—
541C22—2444no—
542C22—2442no—
543C22—2442no—
544C22C2×#1451974511yescharacteristic
545C22#1684041912yes—
546C22C2×#1451980311yescharacteristic
547C22#1684056011yescharacteristic
548C22#1684056512yes—
549C4—22no—
550C4—22no—
551C4—22no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—82no—
556C4—2442no—
557C4—2442no—
558C4—2442no—
559C4#124×D6111yescharacteristic
560C4#1684033211yescharacteristic
561C4#1684042012yes—
562C2—162no—
563C2—2444no—
564C2#2144947411yescharacteristic
565C2#2145506911yescharacteristic
566C2#2145508511yescharacteristic
567C2#2145852212yes—
568C2#2145856612yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2548C2
socleC23×C61C22×D4
prime radicalC2×#16839954C2
hypercenter#2548D61
hypocenterC61#25946
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. C2×#16840472C2semidirect
2. C2×#16840126C2semidirect
4. #21446965C2semidirect
5. #21446955C2semidirect
6. #21446776C2semidirect
7. #21446760C2semidirect
8. #21446170C2semidirect
9. #21446170C2semidirect
10. #21445647C2semidirect
11. C61×#2548C2semidirect
240. D244#124semidirect
281. #2548D61semidirect
331. C61#25946semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
C2×#168401262
C2×#168399542
#214469652
#214469552
#214467762
#214467602
#214461702
#214456472
C61×#25482
#2594661