Finite Group Databasebeta

← #29714916

Subgroups — #29714916

Order 15616.

1180 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840498C211yesmaximal, characteristic, complemented
2C2×#16839862C211yesmaximal, characteristic, complemented
3#21451173C211yesmaximal, characteristic, complemented
4#21447058C212yesmaximal, complemented
5#21446894C212yesmaximal, complemented
6#21446758C211yesmaximal, characteristic, complemented
7#21446644C211yesmaximal, characteristic, complemented
8#21446169C212yesmaximal, complemented
9#21446064C212yesmaximal, complemented
10#21445835C211yesmaximal, characteristic, complemented
11C2×C61×#459C211yesmaximal, characteristic
12C22×#11758200C2211yescharacteristic
13#16840498—24no—
14#16840498—24no—
15C2×#14519764—21no—
16C2×#14519764C2212yes—
17C2×#14519758—22no—
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×D4×Dic61—21no—
21C2×D4×Dic61C2212yes—
22#16839862—24no—
23C2×#14519710C2211yescharacteristic
24C2×#14519702C2212yes—
25C2×#14519700C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2212yes—
28C2×#14519698C2212yes—
29C2×#14519688C2211yescharacteristic
30#16839698—21no—
31#16839674—21no—
32#16839665C2212yes—
33#16839658C2212yes—
34#16839635—22no—
35#16839628C2211yescharacteristic
36#16839525C2212yes—
37#16839516C2212yes—
38#16839516C2212yes—
39C22×D8×C61C2212yes—
40C2×#127×C61C2211yescharacteristic
41C2×C4×D4×C61C2211yescharacteristic
42C61×#459—24no—
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#102×C61C2212yes—
45C22×Dic122D411yescharacteristic
46C22×Dic122C2311yescharacteristic
47C23×Dic61—22no—
48C2×#11758200—22no—
49C2×#11758200—22no—
50C2×#11758200—22no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2311yescharacteristic
53#14519764—28no—
54#14519764—24no—
55#14519764—24no—
56#14519758—216no—
57C4×Dic122—22no—
58C4×Dic122—22no—
59C4×Dic122—22no—
60C4×Dic122D414yes—
61D4×Dic61—28no—
62D4×Dic61—24no—
63D4×Dic61—24no—
64C2×C4⋊Dic61D411yescharacteristic
65C2×C4⋊Dic61C2312yes—
66C2×C4⋊Dic61C2311yescharacteristic
67C2×#11758196—21no—
68C2×#11758196—22no—
69C2×#11758196D412yes—
70C2×C22⋊Dic61—22no—
71C2×C22⋊Dic61—22no—
72C2×C22⋊Dic61—22no—
73C2×C22⋊Dic61—22no—
74C2×C22⋊Dic61—22no—
75C2×C4×Dic61—21no—
76C2×C4×Dic61C2312yes—
77#14519710—22no—
78#14519702—42no—
79#14519700—22no—
80#14519698—22no—
81#14519698—24no—
82#14519698—42no—
83#14519688—22no—
84#14519684—21no—
85#14519684—21no—
86#14519684—22no—
87C22×D4×C61—21no—
88C22×D4×C61—21no—
89C22×D4×C61C2312yes—
90C22×D4×C61C2311yescharacteristic
91C23×C4×C61—21no—
92C2×D8×C61—24no—
93C2×D8×C61—24no—
94C2×D8×C61—42no—
95C22×C8×C61C2312yes—
96#127×C61—22no—
97#127×C61D414yes—
98C4×D4×C61—22no—
99C4×D4×C61—22no—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C22⋊C4×C61—21no—
102C2×C42×C61C2311yescharacteristic
103C4⋊C8×C61—22no—
104#102×C61—24no—
105C24×C61—22no—
106C24×C61—21no—
107C2×D4×C61—24no—
108C2×D4×C61—24no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—21no—
119C2×D4×C61—44no—
120C2×D4×C61—82no—
121C2×D4×C61C2×D414yes—
122C2×D4×C61#4112yes—
123C22×C4×C61—21no—
124C22×C4×C61—42no—
125C22×C4×C61—82no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2411yescharacteristic
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—28no—
132C22×Dic61—22no—
133C22×Dic61—82no—
134C22×Dic61—82no—
135C22×Dic61C2×D412yes—
136C22×Dic61C2×D412yes—
137C2×Dic122—21no—
138C2×Dic122—21no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122—22no—
142C2×Dic122—21no—
143C2×Dic122—42no—
144C2×Dic122C2×D412yes—
145D8×C61—48no—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C22⋊C4×C61—42no—
152C42×C61C2×D412yes—
153C42×C61C2×D412yes—
154C22⋊Dic61—28no—
155C22⋊Dic61—28no—
156C22⋊Dic61—28no—
157C22⋊Dic61—44no—
158C22⋊Dic61—82no—
159#11758200—28no—
160C4⋊Dic61—22no—
161C4⋊Dic61—21no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61C2×D412yes—
165#11758196—24no—
166#11758196—24no—
167#11758196—82no—
168C4×Dic61—24no—
169C4×Dic61—22no—
170C4×Dic61—22no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—21no—
176C2×#92418#4112yes—
177C23×C61—24no—
178C23×C61—22no—
179C23×C61—41no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—41no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61C22×D411yescharacteristic
190D4×C61—22no—
191D4×C61—22no—
192D4×C61—24no—
193D4×C61—24no—
194D4×C61—48no—
195D4×C61—42no—
196D4×C61—41no—
197D4×C61—42no—
198D4×C61—41no—
199D4×C61—84no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—41no—
203C2×C4×C61—42no—
204C2×C4×C61—84no—
205C2×C4×C61#12012yes—
206C2×C4×C61#12012yes—
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C61H(5,𝔽2)11yescharacteristic
212C2×C4×C612−1+411yescharacteristic
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—88no—
228Dic122—22no—
229Dic122—22no—
230Dic122—44no—
231Dic122—41no—
232Dic122—41no—
233C8×C61—42no—
234#92418—22no—
235#92418—41no—
236#24198—611nomaximal, Sylow
237C22×C61—44no—
238C22×C61—42no—
239C22×C61—44no—
240C22×C61—44no—
241C22×C61—44no—
242C22×C61—82no—
243C22×C61—82no—
244C22×C61—82no—
245C22×C61—82no—
246C22×C61—82no—
247C22×C61—81no—
248C22×C61—81no—
249C22×C61C2×#12011yescharacteristic
250C22×C61#54811yescharacteristic
251C22×C61#55011yescharacteristic
252C22×C61#57512yes—
253C22×C61#57612yes—
254C4×C61—22no—
255C4×C61—22no—
256C4×C61—81no—
257C4×C61#52511yescharacteristic
258C4×C61#53711yescharacteristic
259C4×C61#54611yescharacteristic
260C4×C61#54711yescharacteristic
261Dic61—42no—
262Dic61—42no—
263Dic61—42no—
264Dic61—42no—
265Dic61—82no—
266C2×#559—611no—
267C2×#459D6111yescharacteristic
268C2×#440—611no—
269#2580—611no—
270#2042—612no—
271#2012—612no—
272#1985—611no—
273#1961—611no—
274#1879—612no—
275#1862—612no—
276#1805—611no—
277C2×C61—82no—
278C2×C61—84no—
279C2×C61#240711yescharacteristic
280C2×C61#300511yescharacteristic
281C2×C61#300811yescharacteristic
282C2×C61#337012yes—
283C2×C61#340312yes—
284C22×SD16—611no—
285C22×D8C2×D6112yes—
286#559—1224no—
287#559—1224no—
288C2×#127C2×D6111yescharacteristic
289C2×#124—612no—
290C2×#124—1221no—
291C2×#123—1222no—
292C2×C4×Q8—611no—
293C2×C4×Q8—611no—
294C2×C4×D4—612no—
295C2×C4×D4—1221no—
296C2×C4×D4C2×D6111yescharacteristic
297#459—24no—
298#440—1224no—
299C2×#107—611no—
300C2×C4⋊C8C2×D6111yescharacteristic
301C2×#103—611no—
302C2×#103—612no—
303C2×#102—612no—
304C2×#102—611no—
305C2×#102—612no—
306C2×#102C2×D6112yes—
307C2×#97—611no—
308#399—1221no—
309#393—1221no—
310#390—612no—
311#389—612no—
312#385—1222no—
313#382—611no—
314#340—612no—
315#336—612no—
316#336—612no—
317C61#2419811yescharacteristic, complemented, Sylow
318C22×Q8—611no—
319C22×Q8—611no—
320C22×D4—21no—
321C22×D4—21no—
322C22×D4C22×D6111yescharacteristic
323C22×D4C22×D6112yes—
324C23×C4—21no—
325C23×C4—1222no—
326C2×SD16—1222no—
327C2×SD16—1222no—
328C2×SD16—1222no—
329C2×D8—24no—
330C2×D8—24no—
331C2×D8—42no—
332C22×C8—611no—
333C22×C8—611no—
334C22×C8C22×D6112yes—
335#127—22no—
336#127C22⋊D6114yes—
337#124—1224no—
338#124—1228no—
339#124—1224no—
340#123—12216no—
341C4×Q8—614no—
342C4×Q8—1222no—
343C4×Q8—1222no—
344C4×Q8—1222no—
345C4×D4—22no—
346C4×D4—22no—
347C4×D4—1224no—
348C4×D4—1228no—
349C4×D4—1224no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—612no—
353C2×C4⋊C4—612no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1222no—
356C2×C4⋊C4C22×D6111yescharacteristic
357C2×C22⋊C4—21no—
358C2×C22⋊C4—1222no—
359C2×C22⋊C4—1222no—
360C2×C22⋊C4—1222no—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4—1222no—
363C2×C42—612no—
364C2×C42—1221no—
365C2×C42C22×D6111yescharacteristic
366#107—1222no—
367C4⋊C8—22no—
368#103—1222no—
369#103—2442no—
370#102—24no—
371#102—1222no—
372#102—1224no—
373#102—2442no—
374#97—1222no—
375#95—1222no—
376#95—1221no—
377#95—1221no—
378C24—21no—
379C24—22no—
380C2×Q8—612no—
381C2×Q8—1221no—
382C2×Q8—1222no—
383C2×Q8—1221no—
384C2×Q8—1221no—
385C2×Q8—1221no—
386C2×Q8—1221no—
387C2×Q8—2442no—
388C2×D4—21no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—21no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4—21no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—44no—
401C2×D4—82no—
402C2×D4#1175822412yes—
403C2×D4C2×C22⋊D6114yes—
404C22×C4—21no—
405C22×C4—42no—
406C22×C4—82no—
407C22×C4—612no—
408C22×C4—612no—
409C22×C4—1228no—
410C22×C4—1221no—
411C22×C4—1221no—
412C22×C4—1222no—
413C22×C4—4882no—
414C22×C4—4882no—
415C22×C4D4×D6111yescharacteristic
416C22×C4D4×D6111yescharacteristic
417C22×C4C23×D6111yescharacteristic
418SD16—1228no—
419D8—48no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—612no—
424C2×C8—1221no—
425C2×C8—1221no—
426C2×C8—1221no—
427C2×C8—1221no—
428C2×C8—1221no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—612no—
432C4⋊C4—1221no—
433C4⋊C4—1224no—
434C4⋊C4—1222no—
435C4⋊C4—1224no—
436C4⋊C4—1222no—
437C4⋊C4—1222no—
438C4⋊C4—4882no—
439C22⋊C4—42no—
440C22⋊C4—1228no—
441C22⋊C4—1228no—
442C22⋊C4—1228no—
443C22⋊C4—4882no—
444C22⋊C4—2444no—
445C42—1222no—
446C42—1222no—
447C42—1224no—
448C42D4×D6112yes—
449C42C2×C22⋊D6112yes—
450C23—24no—
451C23—22no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—41no—
462C23C2×D4×D6111yescharacteristic
463Q8—1222no—
464Q8—1222no—
465Q8—2444no—
466Q8—2441no—
467Q8—2441no—
468D4—22no—
469D4—22no—
470D4—24no—
471D4—24no—
472D4—48no—
473D4—42no—
474D4—41no—
475D4—42no—
476D4—41no—
477D4—84no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—84no—
483C2×C4—1222no—
484C2×C4—1222no—
485C2×C4—1222no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—4888no—
492C2×C4—2442no—
493C2×C4—2442no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2441no—
497C2×C4—2441no—
498C2×C4#1451974512yes—
499C2×C4#1451974512yes—
500C2×C4C2×D4×D6111yescharacteristic
501C2×C4C22×C22⋊D6111yescharacteristic
502C2×C4#41×D6111yescharacteristic
503C2×C4C2×#1175822411yescharacteristic
504C2×C4#1451982211yescharacteristic
505C2×C4#1451982411yescharacteristic
506C8—42no—
507C8—1222no—
508C8—2441no—
509C22—44no—
510C22—42no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22—82no—
515C22—82no—
516C22—81no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—81no—
521C22C2×#1451974511yescharacteristic
522C22#1684041811yescharacteristic
523C22#1684043011yescharacteristic
524C22#1684055812yes—
525C22#1684056312yes—
526C4—22no—
527C4—22no—
528C4—81no—
529C4—4882no—
530C4—2442no—
531C4—2442no—
532C4—2442no—
533C4—2442no—
534C4#1684027211yescharacteristic
535C4#1684034111yescharacteristic
536C4#1684040611yescharacteristic
537C4#1684041111yescharacteristic
538C2—82no—
539C2—84no—
540C2#2144916011yescharacteristic
541C2#2145505811yescharacteristic
542C2#2145508311yescharacteristic
543C2#2145856512yes—
544C2#2145890212yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#459C2
socleC23×C61C22×D4
hypercenterC2×#459D61
hypocenterC61#24198
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. C2×#16840498C2semidirect
2. C2×#16839862C2semidirect
3. #21451173C2semidirect
4. #21447058C2semidirect
5. #21446894C2semidirect
6. #21446758C2semidirect
7. #21446644C2semidirect
8. #21446169C2semidirect
9. #21446064C2semidirect
10. #21445835C2semidirect
317. C61#24198semidirect

Maximal subgroups

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

subgroupindex
C2×#168404982
C2×#168398622
#214511732
#214470582
#214468942
#214467582
#214466442
#214461692
#214460642
#214458352
C2×C61×#4592
#2419861