Finite Group Databasebeta

← #29714922

Subgroups — #29714922

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×#16840497C211yesmaximal, characteristic, complemented
2C2×#16839957C211yesmaximal, characteristic
3C2×#16839862C211yesmaximal, characteristic, complemented
4#21447060C212yesmaximal, complemented
5#21446895C212yesmaximal, complemented
6#21446760C211yesmaximal, characteristic, complemented
7#21446642C211yesmaximal, characteristic, complemented
8#21446170C212yesmaximal, complemented
9#21446065C212yesmaximal, complemented
10#21445831C211yesmaximal, characteristic, complemented
11C61×#2580C211yesmaximal, characteristic, complemented
12C22×#11758200C2211yescharacteristic
13C22×#11758199C2212yes—
14#16840497—24no—
15#16840497—24no—
16C2×#14519774C2211yescharacteristic
17C2×#14519766—21no—
18C2×#14519765C2212yes—
19C2×#14519762—22no—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×D244C2212yes—
22C2×#14519739—21no—
23#16839957—24no—
24#16839862—24no—
25C2×#14519710C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519700C2212yes—
29C2×#14519698C2211yescharacteristic
30C2×#14519697C2212yes—
31C2×#14519697C2212yes—
32C2×#14519697C2212yes—
33C2×#14519688C2211yescharacteristic
34#16839524C2212yes—
35#16839517C2212yes—
36#16839517C2212yes—
37C2×C4×Q8×C61C2211yescharacteristic
38C2×C4×D4×C61C2211yescharacteristic
39C61×#399—21no—
40C61×#393—21no—
41C61×#390C2212yes—
42C61×#389C2212yes—
43C61×#385—22no—
44C61×#382C2211yescharacteristic
45C22×Dic122C2311yescharacteristic
46C22×D244—21no—
47C22×D244—21no—
48C22×D244C2312yes—
49C22×C4×D61—22no—
50C2×#11758200—22no—
51C2×#11758200—22no—
52C2×#11758200—22no—
53C2×#11758199—24no—
54C2×#11758199—24no—
55C2×#11758199—42no—
56C22×#92418C2311yescharacteristic
57C22×#92418C2312yes—
58C22×#92418C2311yescharacteristic
59#14519774—22no—
60#14519774D414yes—
61#14519766—28no—
62#14519765—24no—
63#14519765—24no—
64#14519762—216no—
65C4×Dic122—22no—
66C4×Dic122D414yes—
67C4×D244—24no—
68C4×D244—24no—
69#14519739—28no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758198—22no—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×C4×Dic61—21no—
76#14519710—22no—
77C4⋊#92418—22no—
78#14519700—22no—
79#14519700—42no—
80#14519698—22no—
81#14519697—24no—
82#14519697—24no—
83#14519697—42no—
84#14519688—22no—
85C22×Q8×C61D411yescharacteristic
86C22×D4×C61C2311yescharacteristic
87C23×C4×C61—21no—
88C4×Q8×C61—22no—
89C4×Q8×C61—22no—
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92C2×C4⋊C4×C61—22no—
93C2×C4⋊C4×C61D411yescharacteristic
94C2×C4⋊C4×C61D412yes—
95C2×C4⋊C4×C61C2312yes—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C22⋊C4×C61—22no—
98C2×C22⋊C4×C61—21no—
99C2×C22⋊C4×C61—22no—
100C2×C42×C61C2312yes—
101C2×C42×C61C2311yescharacteristic
102#95×C61—21no—
103#95×C61—21no—
104#95×C61—22no—
105C24×C61—21no—
106C23×D61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61—42no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61#4112yes—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—42no—
118C22×C4×C61—82no—
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C22×Dic61—21no—
125C22×Dic61—21no—
126C2×D244—22no—
127C2×D244—22no—
128C2×D244—22no—
129C2×D244—22no—
130C2×D244—24no—
131C2×D244—22no—
132C2×D244—24no—
133C2×D244—44no—
134C2×D244—82no—
135C2×D244C2×D414yes—
136C2×C4×D61—28no—
137C2×C4×D61—82no—
138C2×C4×D61—82no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122C2×D412yes—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—82no—
152C22⋊C4×C61—44no—
153C22⋊C4×C61—42no—
154C22⋊C4×C61—82no—
155C42×C61—22no—
156C42×C61—22no—
157C42×C61C2×D412yes—
158C42×C61C2×D412yes—
159#11758200—28no—
160#11758199—48no—
161#11758198—28no—
162#11758198—28no—
163#11758198—28no—
164C4⋊Dic61—21no—
165C4⋊Dic61C2×D412yes—
166#11758196—24no—
167C4×Dic61—24no—
168C2×#92418—21no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418—21no—
174C2×#92418—22no—
175C2×#92418—21no—
176C2×#92418#4112yes—
177C23×C61—22no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61C22×D411yescharacteristic
183C22×D61—24no—
184C22×D61—42no—
185C22×D61—42no—
186C22×D61—42no—
187C22×D61—42no—
188C22×D61—42no—
189C22×D61—42no—
190Q8×C61—44no—
191D4×C61—22no—
192D4×C61—22no—
193D4×C61—41no—
194D4×C61—41no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—41no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—84no—
209C2×C4×C61#12012yes—
210C2×C4×C61#12012yes—
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C2×C4×C61C2×#4111yescharacteristic
215C2×C4×C61H(5,𝔽2)11yescharacteristic
216C2×C4×C612−1+411yescharacteristic
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—42no—
222D244—24no—
223D244—24no—
224D244—48no—
225D244—42no—
226D244—42no—
227D244—84no—
228C4×D61—88no—
229Dic122—22no—
230Dic122—22no—
231Dic122—41no—
232Dic122—41no—
233#92418—22no—
234#92418—41no—
235#92418—42no—
236#24198—611nomaximal, Sylow
237C22×C61—44no—
238C22×C61—42no—
239C22×C61—82no—
240C22×C61—81no—
241C22×C61—81no—
242C22×C61C2×#12011yescharacteristic
243C22×C61#54811yescharacteristic
244C22×C61#55011yescharacteristic
245C22×C61#57512yes—
246C22×C61#57612yes—
247C2×D61—44no—
248C2×D61—44no—
249C2×D61—44no—
250C2×D61—82no—
251C2×D61—82no—
252C2×D61—82no—
253C2×D61—82no—
254C4×C61—22no—
255C4×C61—22no—
256C4×C61—42no—
257C4×C61—42no—
258C4×C61—82no—
259C4×C61—81no—
260C4×C61#52511yescharacteristic
261C4×C61#53711yescharacteristic
262C4×C61#54611yescharacteristic
263C4×C61#54711yescharacteristic
264Dic61—42no—
265Dic61—42no—
266C2×#559—611no—
267C2×#459—611no—
268C2×#440—611no—
269#2580D6111yescharacteristic, complemented
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#240711yescharacteristic
279C2×C61#300511yescharacteristic
280C2×C61#300811yescharacteristic
281C2×C61#337012yes—
282C2×C61#340312yes—
283D61—84no—
284C22×SD16—611no—
285C22×D8—612no—
286#559—1224no—
287#559—1224no—
288C2×#127—611no—
289C2×#124—612no—
290C2×#124—1221no—
291C2×#123—1222no—
292C2×C4×Q8—611no—
293C2×C4×Q8C2×D6111yescharacteristic
294C2×C4×D4—612no—
295C2×C4×D4—1221no—
296C2×C4×D4C2×D6111yescharacteristic
297#459—1224no—
298#440—1224no—
299C2×#107—611no—
300C2×C4⋊C8—611no—
301C2×#103—611no—
302C2×#103—612no—
303C2×#102—611no—
304C2×#102—612no—
305C2×#102—612no—
306C2×#102—612no—
307C2×#97—611no—
308#399—21no—
309#393—21no—
310#390C2×D6112yes—
311#389C2×D6112yes—
312#385—22no—
313#382C2×D6111yescharacteristic
314#340—612no—
315#336—612no—
316#336—612no—
317C61#2419811yescharacteristic, complemented, Sylow
318C22×Q8—611no—
319C22×Q8C22⋊D6111yescharacteristic
320C22×D4—612no—
321C22×D4—1221no—
322C22×D4—1221no—
323C22×D4C22×D6111yescharacteristic
324C23×C4—21no—
325C23×C4—1222no—
326C2×SD16—1222no—
327C2×SD16—1222no—
328C2×SD16—1222no—
329C2×D8—1224no—
330C2×D8—1224no—
331C2×D8—2442no—
332C22×C8—611no—
333C22×C8—612no—
334C22×C8—611no—
335#127—614no—
336#127—1222no—
337#124—1224no—
338#124—1224no—
339#124—1228no—
340#123—12216no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×Q8—614no—
344C4×Q8—1222no—
345C4×D4—22no—
346C4×D4—22no—
347C4×D4—1224no—
348C4×D4—1224no—
349C4×D4—1228no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4C22⋊D6111yescharacteristic
354C2×C4⋊C4C22⋊D6112yes—
355C2×C4⋊C4C22×D6111yescharacteristic
356C2×C4⋊C4C22×D6112yes—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—1222no—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4—1222no—
363C2×C42—1221no—
364C2×C42C22×D6112yes—
365C2×C42C22×D6111yescharacteristic
366#107—1222no—
367C4⋊C8—1222no—
368#103—1222no—
369#103—2442no—
370#102—1224no—
371#102—1224no—
372#102—1222no—
373#102—2442no—
374#97—1222no—
375#95—21no—
376#95—21no—
377#95—22no—
378C24—21no—
379C24—1222no—
380C2×Q8—22no—
381C2×Q8—42no—
382C2×Q8—612no—
383C2×Q8—1221no—
384C2×Q8—1221no—
385C2×Q8—1221no—
386C2×Q8—1221no—
387C2×Q8—1221no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—614no—
394C2×D4—1222no—
395C2×D4—1222no—
396C2×D4—1222no—
397C2×D4—1222no—
398C2×D4—1222no—
399C2×D4—1224no—
400C2×D4—1224no—
401C2×D4—4882no—
402C2×D4—2444no—
403C2×D4#1175822612yes—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—42no—
407C22×C4—82no—
408C22×C4—1228no—
409C22×C4—1221no—
410C22×C4—1221no—
411C22×C4—4882no—
412C22×C4—4882no—
413C22×C4C2×C22⋊D6112yes—
414C22×C4C2×C22⋊D6112yes—
415C22×C4C2×C22⋊D6111yescharacteristic
416C22×C4C2×C22⋊D6111yescharacteristic
417C22×C4C23×D6111yescharacteristic
418SD16—1228no—
419D8—2448no—
420C2×C8—612no—
421C2×C8—1221no—
422C2×C8—1221no—
423C2×C8—1221no—
424C2×C8—1222no—
425C2×C8—1222no—
426C2×C8—1221no—
427C2×C8—1222no—
428C2×C8—1221no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—82no—
436C4⋊C4—612no—
437C4⋊C4—1221no—
438C4⋊C4—1224no—
439C22⋊C4—44no—
440C22⋊C4—42no—
441C22⋊C4—82no—
442C22⋊C4—1228no—
443C22⋊C4—1228no—
444C22⋊C4—1228no—
445C42—22no—
446C42—22no—
447C42—1224no—
448C42D4×D6112yes—
449C42D4×D6112yes—
450C23—22no—
451C23—41no—
452C23—42no—
453C23—42no—
454C23—41no—
455C23—1224no—
456C23—2442no—
457C23—2442no—
458C23—2442no—
459C23—2442no—
460C23—2442no—
461C23—2442no—
462C23C22×C22⋊D6111yescharacteristic
463Q8—44no—
464Q8—1222no—
465Q8—1222no—
466Q8—2441no—
467Q8—2441no—
468D4—22no—
469D4—22no—
470D4—41no—
471D4—41no—
472D4—1224no—
473D4—1224no—
474D4—4884no—
475D4—2448no—
476D4—2442no—
477D4—2442no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—84no—
492C2×C4—1222no—
493C2×C4—1222no—
494C2×C4—4888no—
495C2×C4—2442no—
496C2×C4—2441no—
497C2×C4—2441no—
498C2×C4#1451974512yes—
499C2×C4#1451974512yes—
500C2×C4C2×D4×D6111yescharacteristic
501C2×C4C2×D4×D6111yescharacteristic
502C2×C4#41×D6111yescharacteristic
503C2×C4C2×#1175822611yescharacteristic
504C2×C4#1451982311yescharacteristic
505C2×C4#1451982511yescharacteristic
506C8—1222no—
507C8—2441no—
508C8—2442no—
509C22—44no—
510C22—42no—
511C22—81no—
512C22—82no—
513C22—81no—
514C22—4882no—
515C22—4882no—
516C22—4882no—
517C22—4882no—
518C22—2444no—
519C22—2444no—
520C22—2444no—
521C22C2×#1451974511yescharacteristic
522C22#1684042311yescharacteristic
523C22#1684043111yescharacteristic
524C22#1684056012yes—
525C22#1684056512yes—
526C4—22no—
527C4—22no—
528C4—42no—
529C4—42no—
530C4—82no—
531C4—81no—
532C4—2442no—
533C4—2442no—
534C4#1684027511yescharacteristic
535C4#1684034211yescharacteristic
536C4#1684040311yescharacteristic
537C4#1684041411yescharacteristic
538C2—82no—
539C2—4884no—
540C2#2144916611yescharacteristic
541C2#2145505911yescharacteristic
542C2#2145508511yescharacteristic
543C2#2145857212yes—
544C2#2145890312yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2580C2
socleC23×C61C22×D4
prime radicalC2×#16839957C2
hypercenter#2580D61
hypocenterC61#24198
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

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

NH ≅ G/Ntype
1. C2×#16840497C2semidirect
3. C2×#16839862C2semidirect
4. #21447060C2semidirect
5. #21446895C2semidirect
6. #21446760C2semidirect
7. #21446642C2semidirect
8. #21446170C2semidirect
9. #21446065C2semidirect
10. #21445831C2semidirect
11. C61×#2580C2semidirect
269. #2580D61semidirect
317. C61#24198semidirect

Maximal subgroups

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

subgroupindex
C2×#168404972
C2×#168399572
C2×#168398622
#214470602
#214468952
#214467602
#214466422
#214461702
#214460652
#214458312
C61×#25802
#2419861