Finite Group Databasebeta

← #29736983

Subgroups — #29736983

Order 15616.

1022 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840490C211yesmaximal, characteristic, complemented
2C2×#16840122C211yesmaximal, characteristic
3C2×#16839985C211yesmaximal, characteristic
4#21451307C211yesmaximal, characteristic
5#21446971C212yesmaximal, complemented
6#21446966C212yesmaximal
7#21446766C211yesmaximal, characteristic, complemented
8#21446173C212yesmaximal
9#21446159C212yesmaximal, complemented
10#21445650C211yesmaximal, characteristic, complemented
11C61×#1985C211yesmaximal, characteristic, complemented
12C22×Dic244C2211yescharacteristic
13C22×#11758190C2211yescharacteristic
14#16840490—24no—
15#16840490—24no—
16C2×#14519777C2211yescharacteristic
17C2×#14519775—21no—
18C2×#14519769C2212yes—
19C2×#14519767—22no—
20C2×#14519765C2211yescharacteristic
21C2×C4×Dic122C2212yes—
22C2×C4×Dic122C2211yescharacteristic
23C2×#14519742—21no—
24#16840122—24no—
25#16839985—24no—
26C2×#14519702C2212yes—
27C2×#14519702C2212yes—
28C2×#14519702C2211yescharacteristic
29C2×#14519699C2212yes—
30C2×#14519699C2211yescharacteristic
31#16839688—21no—
32#16839673—21no—
33#16839668C2212yes—
34#16839664C2212yes—
35#16839652—22no—
36C4×C4⋊Dic61C2211yescharacteristic
37#16839525C2212yes—
38#16839525C2212yes—
39#16839525C2212yes—
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×#97×C61C2211yescharacteristic
43C61×#336C2212yes—
44C61×#336C2212yes—
45C22×Dic122—21no—
46C22×Dic122C2311yescharacteristic
47C22×Dic122C2311yescharacteristic
48C22×D244C2311yescharacteristic
49C2×Dic244—22no—
50C2×Dic244—24no—
51C2×#11758190—22no—
52C2×#11758190—24no—
53#14519777—22no—
54#14519777D414yes—
55#14519775—28no—
56#14519769—24no—
57#14519769—24no—
58#14519767—216no—
59#14519765—22no—
60#14519765—22no—
61C4×Dic122—24no—
62C4×Dic122—22no—
63C4×Dic122—24no—
64C4×Dic122D414yes—
65#14519742—28no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2312yes—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2312yes—
70C2×#11758196—22no—
71C2×#11758196—21no—
72C2×#11758196—22no—
73C2×#11758196—22no—
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×#11758198—22no—
77C2×#11758198—22no—
78C2×C4×Dic61—22no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81#14519702—24no—
82#14519702—22no—
83#14519702—24no—
84#14519699—22no—
85#14519699—24no—
86#14519684—21no—
87#14519684—22no—
88#14519684—22no—
89C22×Q8×C61D411yescharacteristic
90C22×C8×C61C2312yes—
91C22×C8×C61C2311yescharacteristic
92C22×C8×C61C2311yescharacteristic
93C4×Q8×C61—22no—
94C4×Q8×C61—22no—
95C2×C4⋊C4×C61D411yescharacteristic
96C2×C4⋊C4×C61D412yes—
97C2×C42×C61C2312yes—
98C2×C42×C61C2311yescharacteristic
99C4⋊C8×C61—22no—
100#97×C61—22no—
101C23×D61—21no—
102C2×Q8×C61—22no—
103C2×Q8×C61—42no—
104C22×C4×C61C2×D412yes—
105C22×C4×C61C2×D412yes—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2411yescharacteristic
109C22×Dic61—21no—
110C22×Dic61—22no—
111C22×Dic61—21no—
112C22×Dic61—21no—
113C22×Dic61—22no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244—22no—
117C2×D244—22no—
118C2×Dic122—21no—
119C2×Dic122—22no—
120C2×Dic122—24no—
121C2×Dic122—21no—
122C2×Dic122—21no—
123C2×Dic122—22no—
124C2×Dic122—22no—
125C2×Dic122—44no—
126C2×Dic122C2×D412yes—
127C2×Dic122#4114yes—
128C2×C8×C61—22no—
129C2×C8×C61—21no—
130C2×C8×C61—21no—
131C2×C8×C61—22no—
132C2×C8×C61—22no—
133C2×C8×C61—22no—
134C2×C8×C61—22no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—24no—
137C42×C61—22no—
138C42×C61—22no—
139C42×C61C2×D412yes—
140C42×C61C2×D412yes—
141#11758198—44no—
142#11758198—42no—
143#11758198—42no—
144#11758198—82no—
145C4⋊Dic61—24no—
146C4⋊Dic61—21no—
147C4⋊Dic61—22no—
148C4⋊Dic61—22no—
149C4⋊Dic61—22no—
150C4⋊Dic61C2×D414yes—
151C4⋊Dic61C2×D412yes—
152#11758196—28no—
153#11758196—28no—
154#11758196—28no—
155#11758196—24no—
156C4×Dic61—28no—
157C4×Dic61—24no—
158C4×Dic61—81no—
159Dic244—44no—
160#11758190—44no—
161C23×C61C22×D411yescharacteristic
162C22×D61—41no—
163C22×D61—42no—
164C22×D61—42no—
165C22×D61—41no—
166C22×D61—41no—
167Q8×C61—44no—
168C2×C4×C61—22no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61#12012yes—
178C2×C4×C61#12012yes—
179C2×C4×C61C22×D411yescharacteristic
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C61C2×#4112yes—
182C2×C4×C612−1+412yes—
183C2×Dic61—22no—
184C2×Dic61—24no—
185C2×Dic61—22no—
186C2×Dic61—24no—
187C2×Dic61—44no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193C2×Dic61—42no—
194C2×Dic61—42no—
195D244—41no—
196D244—41no—
197D244—42no—
198Dic122—22no—
199Dic122—24no—
200Dic122—48no—
201Dic122—41no—
202Dic122—41no—
203Dic122—42no—
204Dic122#12414yes—
205C8×C61—41no—
206C8×C61—41no—
207C8×C61—42no—
208#25941—611nomaximal, Sylow
209C22×C61C2×#12011yescharacteristic
210C22×C61#54812yes—
211C22×C61#57512yes—
212C22×C61#57711yescharacteristic
213C22×C61#57811yescharacteristic
214C2×D61—82no—
215C2×D61—82no—
216C2×D61—81no—
217C2×D61—81no—
218C2×D61—81no—
219C4×C61—22no—
220C4×C61—22no—
221C4×C61—42no—
222C4×C61—42no—
223C4×C61C2×#12411yescharacteristic
224C4×C61#53611yescharacteristic
225C4×C61#54812yes—
226Dic61—24no—
227Dic61—44no—
228Dic61—42no—
229Dic61—82no—
230Dic61—81no—
231C2×#557—611no—
232C2×#489—611no—
233C2×#464—611no—
234#2592—611no—
235#2026—612no—
236#2025—612no—
237#1987—611no—
238#1985D6111yescharacteristic, complemented
239#1880—612no—
240#1878—612no—
241#1763—611no—
242C2×C61#243611yescharacteristic
243C2×C61#300511yescharacteristic
244C2×C61#301311yescharacteristic
245C2×C61#337112yes—
246C2×C61#337712yes—
247D61—161no—
248C22×Q16—611no—
249C22×SD16—611no—
250#557—1224no—
251#557—1224no—
252C2×C4⋊Q8—611no—
253C2×C4⋊Q8—1221no—
254C2×#125—612no—
255C2×#125—1222no—
256C2×#124—611no—
257C2×C4×Q8—611no—
258C2×C4×Q8—612no—
259C2×C4×Q8—1221no—
260C2×C4×Q8C2×D6111yescharacteristic
261#489—1224no—
262#464—1224no—
263C2×C4⋊C8C2×D6111yescharacteristic
264C2×#103—612no—
265C2×#103—612no—
266C2×#103—611no—
267C2×#102—612no—
268C2×#102—611no—
269C2×#97C2×D6111yescharacteristic
270#397—1221no—
271#393—1221no—
272#391—612no—
273#390—612no—
274#388—1222no—
275C4×C4⋊C4—611no—
276#340—612no—
277#340—612no—
278#340—612no—
279#336C2×D6112yes—
280#336C2×D6112yes—
281C61#2594111yescharacteristic, complemented, Sylow
282C22×Q8—611no—
283C22×Q8—611no—
284C22×Q8—1221no—
285C22×Q8D24411yescharacteristic
286C22×D4—611no—
287C2×Q16—1222no—
288C2×Q16—1224no—
289C2×SD16—1222no—
290C2×SD16—1224no—
291C22×C8C22×D6111yescharacteristic
292C22×C8C22×D6111yescharacteristic
293C22×C8C22×D6112yes—
294C4⋊Q8—614no—
295C4⋊Q8—1228no—
296C4⋊Q8—1222no—
297#125—1224no—
298#125—1224no—
299#125—12216no—
300#124—1222no—
301#124—1222no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×Q8—614no—
305C4×Q8—1224no—
306C4×Q8—1224no—
307C4×Q8—1228no—
308C4×Q8—1222no—
309C2×C4⋊C4—611no—
310C2×C4⋊C4—612no—
311C2×C4⋊C4—612no—
312C2×C4⋊C4—1221no—
313C2×C4⋊C4—1222no—
314C2×C4⋊C4—1222no—
315C2×C4⋊C4—1222no—
316C2×C4⋊C4—1221no—
317C2×C4⋊C4D24411yescharacteristic
318C2×C4⋊C4D24412yes—
319C2×C22⋊C4—1222no—
320C2×C22⋊C4—1221no—
321C2×C22⋊C4—1221no—
322C2×C22⋊C4—1222no—
323C2×C42—1222no—
324C2×C42—1221no—
325C2×C42—1221no—
326C2×C42C22×D6112yes—
327C2×C42C22×D6111yescharacteristic
328C4⋊C8—22no—
329#103—1222no—
330#103—1224no—
331#103—1224no—
332#102—1222no—
333#102—1224no—
334#97—22no—
335#95—1221no—
336#95—1222no—
337#95—1222no—
338C24—1221no—
339C2×Q8—22no—
340C2×Q8—42no—
341C2×Q8—612no—
342C2×Q8—614no—
343C2×Q8—1221no—
344C2×Q8—1222no—
345C2×Q8—1224no—
346C2×Q8—1221no—
347C2×Q8—1221no—
348C2×Q8—1222no—
349C2×Q8—1222no—
350C2×Q8—2444no—
351C2×D4—1221no—
352C2×D4—1221no—
353C2×D4—1222no—
354C2×D4—1222no—
355C22×C4—1221no—
356C22×C4—1221no—
357C22×C4—1222no—
358C22×C4—1221no—
359C22×C4—1222no—
360C22×C4C2×D24412yes—
361C22×C4C2×D24412yes—
362C22×C4C2×D24411yescharacteristic
363C22×C4C2×D24411yescharacteristic
364C22×C4C23×D6111yescharacteristic
365Q16—2444no—
366SD16—2444no—
367C2×C8—22no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—612no—
377C4⋊C4—614no—
378C4⋊C4—1228no—
379C4⋊C4—1228no—
380C4⋊C4—1228no—
381C4⋊C4—1221no—
382C4⋊C4—1222no—
383C4⋊C4—1224no—
384C4⋊C4—1224no—
385C4⋊C4—1222no—
386C4⋊C4—1222no—
387C22⋊C4—4882no—
388C22⋊C4—2444no—
389C22⋊C4—2442no—
390C22⋊C4—2442no—
391C42—22no—
392C42—22no—
393C42—1228no—
394C42—1224no—
395C42—4881no—
396C42D4×D6112yes—
397C42D4×D6112yes—
398C23—2441no—
399C23—2442no—
400C23—2441no—
401C23—2441no—
402C23—2442no—
403C23C22×D24411yescharacteristic
404Q8—44no—
405Q8—614no—
406Q8—1222no—
407Q8—1224no—
408Q8—2448no—
409Q8—2441no—
410Q8—2441no—
411Q8—2442no—
412D4—2441no—
413D4—2441no—
414D4—2442no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—1222no—
425C2×C4—1224no—
426C2×C4—1222no—
427C2×C4—1224no—
428C2×C4—2441no—
429C2×C4—2442no—
430C2×C4—2442no—
431C2×C4—2442no—
432C2×C4—2442no—
433C2×C4—2444no—
434C2×C4—2441no—
435C2×C4—2441no—
436C2×C4#1451974712yes—
437C2×C4#1451974712yes—
438C2×C4C2×D4×D6111yescharacteristic
439C2×C4C2×D4×D6111yescharacteristic
440C2×C4#41×D6112yes—
441C2×C4#1451982512yes—
442C8—41no—
443C8—41no—
444C8—42no—
445C22—4881no—
446C22—4881no—
447C22—4882no—
448C22—4881no—
449C22—4882no—
450C22C2×#1451974711yescharacteristic
451C22#1684042412yes—
452C22#1684056212yes—
453C22#1684057411yescharacteristic
454C22#1684057911yescharacteristic
455C4—22no—
456C4—22no—
457C4—42no—
458C4—42no—
459C4—1224no—
460C4—4881no—
461C4—4882no—
462C4—2442no—
463C4—2444no—
464C4#124×D6111yescharacteristic
465C4#1684033711yescharacteristic
466C4#1684042212yes—
467C2—9761no—
468C2#2144947611yescharacteristic
469C2#2145506411yescharacteristic
470C2#2145513711yescharacteristic
471C2#2145858912yes—
472C2#2145866312yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1985C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1985D61
hypocenterC61#25941
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1985

Complements

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

NH ≅ G/Ntype
1. C2×#16840490C2semidirect
5. #21446971C2semidirect
7. #21446766C2semidirect
9. #21446159C2semidirect
10. #21445650C2semidirect
11. C61×#1985C2semidirect
238. #1985D61semidirect
281. C61#25941semidirect

Maximal subgroups

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

subgroupindex
C2×#168404902
C2×#168401222
C2×#168399852
#214513072
#214469712
#214469662
#214467662
#214461732
#214461592
#214456502
C61×#19852
#2594161