Finite Group Databasebeta

← #29714929

Subgroups — #29714929

Order 15616.

1084 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840487C211yesmaximal, characteristic, complemented
2C2×#16839964C211yesmaximal, characteristic
3C2×#16839864C211yesmaximal, characteristic
4#21451178C211yesmaximal, characteristic
5#21447064C212yesmaximal, complemented
6#21446903C212yesmaximal
7#21446759C211yesmaximal, characteristic, complemented
8#21446172C212yesmaximal
9#21446075C212yesmaximal, complemented
10#21445831C211yesmaximal, characteristic, complemented
11C61×#1960C211yesmaximal, characteristic, complemented
12C22×#11758190C2211yescharacteristic
13C22×#11758201C2212yes—
14#16840487—24no—
15#16840487—24no—
16C2×#14519775C2212yes—
17C2×#14519768—21no—
18C2×#14519767—22no—
19C2×Q8×Dic61C2212yes—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×#14519742—21no—
23C2×C4×D244C2211yescharacteristic
24#16839964—24no—
25#16839864—24no—
26C2×#14519711C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519701C2212yes—
30C2×#14519700C2212yes—
31C2×#14519699C2211yescharacteristic
32#16839699—21no—
33#16839675—21no—
34#16839663C2212yes—
35#16839661C2212yes—
36#16839636—22no—
37#16839524C2212yes—
38#16839516C2212yes—
39#16839516C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×#103×C61C2212yes—
42C2×#103×C61C2212yes—
43C2×#97×C61C2211yescharacteristic
44C61×#382C2211yescharacteristic
45C22×Dic122D411yescharacteristic
46C22×Dic122C2311yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C4×D61—21no—
49C2×#11758190—22no—
50C2×#11758190—22no—
51C2×#11758190—22no—
52C2×#11758201—24no—
53C2×#11758201—24no—
54C2×#11758201—42no—
55C22×#92418C2312yes—
56#14519775—24no—
57#14519775—24no—
58#14519768—28no—
59#14519767—216no—
60Q8×Dic61—24no—
61Q8×Dic61—24no—
62C4×Dic122—22no—
63C4×Dic122—22no—
64C4×Dic122—22no—
65C4×Dic122D414yes—
66#14519742—28no—
67C4×D244—22no—
68C4×D244—22no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—22no—
76C2×#11758196—22no—
77C2×#11758196D412yes—
78C2×#11758198—21no—
79C2×#11758198—22no—
80C2×#11758198—22no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61C2312yes—
84#14519711—22no—
85C4⋊#92418—22no—
86#14519702—22no—
87#14519701—24no—
88#14519700—42no—
89#14519699—22no—
90#14519684—22no—
91C22×Q8×C61C2312yes—
92C22×C8×C61C2311yescharacteristic
93C22×C8×C61C2311yescharacteristic
94C4⋊Q8×C61—22no—
95C4⋊Q8×C61D414yes—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61C2312yes—
99C2×C42×C61C2311yescharacteristic
100#103×C61—24no—
101#103×C61—42no—
102#97×C61—22no—
103#95×C61—21no—
104#95×C61—21no—
105C23×D61—21no—
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61C2×D414yes—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2411yescharacteristic
117C22×Dic61—21no—
118C22×Dic61—21no—
119C22×Dic61—21no—
120C22×Dic61C2×D412yes—
121C22×Dic61C2×D412yes—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244#4112yes—
128C2×C4×D61—42no—
129C2×C4×D61—82no—
130C2×Dic122—22no—
131C2×Dic122—24no—
132C2×Dic122—21no—
133C2×Dic122—21no—
134C2×Dic122—21no—
135C2×Dic122—21no—
136C2×Dic122—21no—
137C2×Dic122C2×D412yes—
138C2×C8×C61—21no—
139C2×C8×C61—21no—
140C2×C8×C61—21no—
141C2×C8×C61—21no—
142C2×C8×C61—21no—
143C2×C8×C61#4112yes—
144C4⋊C4×C61—24no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C42×C61C2×D412yes—
149C42×C61C2×D412yes—
150#11758201—48no—
151#11758198—42no—
152#11758198—44no—
153#11758198—82no—
154C4⋊Dic61—28no—
155C4⋊Dic61—22no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—82no—
160C4⋊Dic61C2×D412yes—
161#11758196—28no—
162#11758196—28no—
163#11758196—24no—
164#11758196—24no—
165C4×Dic61—28no—
166C4×Dic61—24no—
167C4×Dic61—22no—
168C4×Dic61—22no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—22no—
172#11758190—28no—
173C23×C61C22×D411yescharacteristic
174C22×D61—22no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—42no—
178C22×D61—42no—
179Q8×C61—24no—
180Q8×C61—24no—
181Q8×C61—42no—
182Q8×C61—42no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—24no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61#12012yes—
192C2×C4×C61#12012yes—
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C2×C4×C61C2×#4111yescharacteristic
197C2×C4×C61H(5,𝔽2)11yescharacteristic
198C2×C4×C612−1+411yescharacteristic
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—22no—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—42no—
209C2×Dic61—41no—
210C2×Dic61—42no—
211C2×Dic61#13714yes—
212D244—22no—
213D244—22no—
214D244—41no—
215D244—41no—
216C4×D61—84no—
217Dic122—28no—
218Dic122—22no—
219Dic122—22no—
220Dic122—41no—
221Dic122—41no—
222C8×C61—22no—
223C8×C61—41no—
224#92418—42no—
225#24199—611nomaximal, Sylow
226C22×C61C2×#12011yescharacteristic
227C22×C61#54811yescharacteristic
228C22×C61#55011yescharacteristic
229C22×C61C2×#13712yes—
230C22×C61#57712yes—
231C2×D61—44no—
232C2×D61—42no—
233C2×D61—82no—
234C2×D61—81no—
235C2×D61—81no—
236C4×C61—22no—
237C4×C61—22no—
238C4×C61—44no—
239C4×C61—82no—
240C4×C61#52511yescharacteristic
241C4×C61#53711yescharacteristic
242C4×C61#54611yescharacteristic
243C4×C61#54711yescharacteristic
244Dic61—24no—
245Dic61—42no—
246Dic61—42no—
247Dic61—42no—
248Dic61—81no—
249C2×#557—611no—
250C2×#461—611no—
251C2×#440—611no—
252#2580—611no—
253#2043—612no—
254#2013—612no—
255#1985—611no—
256#1960D6111yescharacteristic, complemented
257#1880—612no—
258#1863—612no—
259#1805—611no—
260C2×C61#240711yescharacteristic
261C2×C61#300411yescharacteristic
262C2×C61#301011yescharacteristic
263C2×C61#337212yes—
264C2×C61#340012yes—
265D61—82no—
266C22×SD16—612no—
267C22×SD16—611no—
268#557—1224no—
269#557—1224no—
270C2×C4⋊Q8—612no—
271C2×C4⋊Q8C2×D6111yescharacteristic
272C2×#125—1222no—
273C2×#125—1221no—
274C2×C4×Q8—611no—
275C2×C4×Q8—612no—
276C2×C4×Q8—611no—
277C2×C4×Q8—1221no—
278C2×C4×D4—611no—
279#461—1224no—
280#440—1224no—
281C2×#107—611no—
282C2×C4⋊C8—611no—
283C2×#103—612no—
284C2×#103—611no—
285C2×#103—612no—
286C2×#103C2×D6112yes—
287C2×#103C2×D6112yes—
288C2×#102—611no—
289C2×#97C2×D6111yescharacteristic
290#399—1221no—
291#393—1221no—
292#390—612no—
293#389—612no—
294#385—1222no—
295#382C2×D6111yescharacteristic
296#340—612no—
297#336—612no—
298#336—612no—
299C61#2419911yescharacteristic, complemented, Sylow
300C22×Q8—611no—
301C22×Q8—611no—
302C22×Q8C22×D6112yes—
303C22×D4—611no—
304C23×C4—1221no—
305C2×SD16—1222no—
306C2×SD16—1222no—
307C2×SD16—1224no—
308C2×SD16—1224no—
309C2×SD16—1222no—
310C2×SD16—2442no—
311C22×C8—612no—
312C22×C8C22×D6111yescharacteristic
313C22×C8C22×D6111yescharacteristic
314C4⋊Q8—22no—
315C4⋊Q8—1224no—
316C4⋊Q8—1224no—
317C4⋊Q8D24414yes—
318#125—12216no—
319#125—1228no—
320C4×Q8—614no—
321C4×Q8—1228no—
322C4×Q8—1224no—
323C4×Q8—1224no—
324C4×Q8—1222no—
325C4×Q8—1222no—
326C4×Q8—1222no—
327C4×D4—1222no—
328C4×D4—1222no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—611no—
332C2×C4⋊C4—612no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4—1222no—
336C2×C4⋊C4—1222no—
337C2×C4⋊C4—1222no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4—1222no—
340C2×C4⋊C4C22×D6112yes—
341C2×C22⋊C4—1222no—
342C2×C22⋊C4—1222no—
343C2×C22⋊C4—1221no—
344C2×C42—612no—
345C2×C42—1222no—
346C2×C42—1221no—
347C2×C42C22×D6111yescharacteristic
348#107—1222no—
349C4⋊C8—1222no—
350#103—24no—
351#103—42no—
352#103—1222no—
353#103—1224no—
354#103—2442no—
355#102—1222no—
356#97—22no—
357#95—21no—
358#95—21no—
359#95—1222no—
360C24—1221no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—612no—
367C2×Q8—1224no—
368C2×Q8—1221no—
369C2×Q8—1222no—
370C2×Q8—1221no—
371C2×Q8—1221no—
372C2×Q8—1221no—
373C2×Q8—1221no—
374C2×Q8C2×D24414yes—
375C2×D4—612no—
376C2×D4—1221no—
377C2×D4—1221no—
378C2×D4—1221no—
379C2×D4—1221no—
380C2×D4—1221no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—612no—
384C22×C4—612no—
385C22×C4—1221no—
386C22×C4—1221no—
387C22×C4—1221no—
388C22×C4—4882no—
389C22×C4—2442no—
390C22×C4D4×D6111yescharacteristic
391C22×C4D4×D6111yescharacteristic
392C22×C4C23×D6111yescharacteristic
393SD16—1228no—
394SD16—2448no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—1222no—
401C2×C8—1222no—
402C2×C8—1222no—
403C2×C8#1175822412yes—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—612no—
409C4⋊C4—1228no—
410C4⋊C4—1228no—
411C4⋊C4—1228no—
412C4⋊C4—1221no—
413C4⋊C4—1224no—
414C4⋊C4—1222no—
415C4⋊C4—1224no—
416C4⋊C4—1221no—
417C4⋊C4—1221no—
418C4⋊C4—4882no—
419C22⋊C4—4882no—
420C22⋊C4—2444no—
421C22⋊C4—2442no—
422C42—1228no—
423C42—1222no—
424C42—1222no—
425C42—1224no—
426C42C2×D24412yes—
427C42D4×D6112yes—
428C23—1222no—
429C23—2441no—
430C23—2442no—
431C23—2442no—
432C23—2441no—
433C23C2×D4×D6111yescharacteristic
434Q8—24no—
435Q8—24no—
436Q8—42no—
437Q8—42no—
438Q8—1228no—
439Q8—1222no—
440Q8—1222no—
441Q8—2441no—
442Q8—2441no—
443D4—1222no—
444D4—1222no—
445D4—2441no—
446D4—2441no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—614no—
456C2×C4—1222no—
457C2×C4—1222no—
458C2×C4—1222no—
459C2×C4—1222no—
460C2×C4—1222no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—4884no—
464C2×C4—2441no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4—2441no—
468C2×C4—2441no—
469C2×C4#1451974712yes—
470C2×C4#1451974712yes—
471C2×C4C2×D4×D6111yescharacteristic
472C2×C4C22×D24411yescharacteristic
473C2×C4#41×D6111yescharacteristic
474C2×C4C2×#1175822411yescharacteristic
475C2×C4#1451982211yescharacteristic
476C2×C4#1451982411yescharacteristic
477C8—22no—
478C8—41no—
479C8—2442no—
480C22—4881no—
481C22—4882no—
482C22—4881no—
483C22—2444no—
484C22—2442no—
485C22C2×#1451974711yescharacteristic
486C22#1684042011yescharacteristic
487C22#1684043211yescharacteristic
488C22#137×D6112yes—
489C22#1684057112yes—
490C4—22no—
491C4—22no—
492C4—44no—
493C4—82no—
494C4—1224no—
495C4—4881no—
496C4—2442no—
497C4—2442no—
498C4—2442no—
499C4#1684027211yescharacteristic
500C4#1684034111yescharacteristic
501C4#1684040711yescharacteristic
502C4#1684041611yescharacteristic
503C2—4882no—
504C2#2144916211yescharacteristic
505C2#2145505511yescharacteristic
506C2#2145510611yescharacteristic
507C2#2145859712yes—
508C2#2145886912yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1960C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1960D61
hypocenterC61#24199
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
1. C2×#16840487C2semidirect
5. #21447064C2semidirect
7. #21446759C2semidirect
9. #21446075C2semidirect
10. #21445831C2semidirect
11. C61×#1960C2semidirect
256. #1960D61semidirect
299. C61#24199semidirect

Maximal subgroups

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

subgroupindex
C2×#168404872
C2×#168399642
C2×#168398642
#214511782
#214470642
#214469032
#214467592
#214461722
#214460752
#214458312
C61×#19602
#2419961