Finite Group Databasebeta

← #29736975

Subgroups — #29736975

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×#16840124C211yesmaximal, characteristic
2C2×#16839984C211yesmaximal, characteristic
3#21451307C211yesmaximal, characteristic
4#21446967C212yesmaximal, complemented
5#21446962C212yesmaximal
6#21446768C211yesmaximal, characteristic, complemented
7#21446760C211yesmaximal, characteristic, complemented
8#21446174C212yesmaximal
9#21446160C212yesmaximal, complemented
10#21445649C211yesmaximal, characteristic, complemented
11C2×C61×#557C211yesmaximal, characteristic, complemented
12C22×#11758202C2211yescharacteristic
13C22×#11758201C2211yescharacteristic
14C2×#14519765C2211yescharacteristic
15#16840124—24no—
16#16839984—24no—
17C2×C4⋊#92418C2211yescharacteristic
18C2×#14519701C2212yes—
19C2×#14519700C2212yes—
20C2×#14519700C2211yescharacteristic
21C2×#14519697C2212yes—
22C2×#14519697C2211yescharacteristic
23C2×#14519688C2211yescharacteristic
24#16839688—21no—
25#16839673—21no—
26#16839668C2212yes—
27#16839664C2212yes—
28#16839652—22no—
29C4×C4⋊Dic61C2211yescharacteristic
30#16839524C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34#16839517C2212yes—
35C61×#557—24no—
36C61×#557—24no—
37C2×C4⋊Q8×C61—21no—
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×#125×C61—22no—
40C2×#125×C61C2212yes—
41C2×C4×Q8×C61—21no—
42C2×C4×Q8×C61C2212yes—
43C2×C4×Q8×C61C2211yescharacteristic
44C2×C4×Q8×C61C2211yescharacteristic
45C22×Dic122C2311yescharacteristic
46C22×D244C2311yescharacteristic
47C2×#11758202—22no—
48C2×#11758202—24no—
49C2×#11758201—22no—
50C2×#11758201—24no—
51C22×#92418C2312yes—
52C22×#92418C2311yescharacteristic
53C22×#92418C2311yescharacteristic
54#14519765—22no—
55#14519765—22no—
56C2×C4⋊Dic61C2312yes—
57C2×#11758198—21no—
58C2×#11758198—21no—
59C2×#11758198—22no—
60C2×#11758198—22no—
61C2×C4×Dic61—21no—
62C4⋊#92418—22no—
63#14519701—24no—
64#14519700—22no—
65#14519700—24no—
66#14519697—22no—
67#14519697—24no—
68#14519688—22no—
69#14519684—21no—
70#14519684—22no—
71#14519684—22no—
72C22×Q8×C61—21no—
73C22×Q8×C61D411yescharacteristic
74C22×Q8×C61C2311yescharacteristic
75C4⋊Q8×C61—28no—
76C4⋊Q8×C61—22no—
77C4⋊Q8×C61D414yes—
78#125×C61—216no—
79#125×C61—24no—
80#125×C61—24no—
81C4×Q8×C61—28no—
82C4×Q8×C61—24no—
83C4×Q8×C61—24no—
84C4×Q8×C61—22no—
85C4×Q8×C61—22no—
86C4×Q8×C61—22no—
87C4×Q8×C61D414yes—
88C2×C4⋊C4×C61—22no—
89C2×C4⋊C4×C61—22no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—22no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61D411yescharacteristic
94C2×C4⋊C4×C61D412yes—
95C2×C4⋊C4×C61C2312yes—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C42×C61—21no—
98C2×C42×C61—22no—
99C2×C42×C61C2312yes—
100C2×C42×C61C2311yescharacteristic
101C23×D61—21no—
102C2×Q8×C61—22no—
103C2×Q8×C61—24no—
104C2×Q8×C61—21no—
105C2×Q8×C61—22no—
106C2×Q8×C61—42no—
107C2×Q8×C61—44no—
108C2×Q8×C61C2×D412yes—
109C2×Q8×C61#4114yes—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2411yescharacteristic
118C22×Dic61—21no—
119C22×Dic61—22no—
120C2×D244—21no—
121C2×D244—21no—
122C2×D244—22no—
123C2×D244—22no—
124C2×Dic122—21no—
125C2×Dic122—21no—
126C2×Dic122—22no—
127C2×Dic122—22no—
128C4⋊C4×C61—28no—
129C4⋊C4×C61—28no—
130C4⋊C4×C61—28no—
131C4⋊C4×C61—24no—
132C4⋊C4×C61—24no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—24no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61C2×D414yes—
138C4⋊C4×C61C2×D412yes—
139C42×C61—28no—
140C42×C61—24no—
141C42×C61—22no—
142C42×C61—22no—
143C42×C61C2×D412yes—
144C42×C61C2×D412yes—
145#11758202—44no—
146#11758201—44no—
147#11758198—44no—
148#11758198—42no—
149#11758198—42no—
150#11758198—82no—
151C4⋊Dic61—22no—
152C4⋊Dic61—22no—
153C4×Dic61—81no—
154C2×#92418—22no—
155C2×#92418—21no—
156C2×#92418—21no—
157C2×#92418—22no—
158C2×#92418—22no—
159C2×#92418—22no—
160C2×#92418—22no—
161C23×C61C22×D411yescharacteristic
162C22×D61—41no—
163C22×D61—42no—
164C22×D61—42no—
165C22×D61—41no—
166C22×D61—41no—
167Q8×C61—22no—
168Q8×C61—24no—
169Q8×C61—44no—
170Q8×C61—48no—
171Q8×C61#12414yes—
172C2×C4×C61—22no—
173C2×C4×C61—24no—
174C2×C4×C61—22no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—24no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—44no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61#12012yes—
189C2×C4×C61#12012yes—
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C2×#4112yes—
193C2×C4×C612−1+412yes—
194C2×Dic61—41no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199D244—41no—
200D244—41no—
201D244—42no—
202Dic122—41no—
203Dic122—41no—
204Dic122—42no—
205#92418—41no—
206#92418—41no—
207#92418—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—24no—
220C4×C61—22no—
221C4×C61—22no—
222C4×C61—42no—
223C4×C61—44no—
224C4×C61—42no—
225C4×C61—42no—
226C4×C61C2×#12411yescharacteristic
227C4×C61#53611yescharacteristic
228C4×C61#54812yes—
229Dic61—82no—
230Dic61—81no—
231C2×#557D6111yescharacteristic, complemented
232C2×#489—611no—
233C2×#464—611no—
234#2592—611no—
235#2026—612no—
236#2025—612no—
237#1987—611no—
238#1985—611no—
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—24no—
251#557—24no—
252C2×C4⋊Q8—21no—
253C2×C4⋊Q8C2×D6111yescharacteristic
254C2×#125—22no—
255C2×#125C2×D6112yes—
256C2×#124—611no—
257C2×C4×Q8—21no—
258C2×C4×Q8C2×D6112yes—
259C2×C4×Q8C2×D6111yescharacteristic
260C2×C4×Q8C2×D6111yescharacteristic
261#489—1224no—
262#464—1224no—
263C2×C4⋊C8—611no—
264C2×#103—612no—
265C2×#103—612no—
266C2×#103—611no—
267C2×#102—612no—
268C2×#102—611no—
269C2×#97—611no—
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#336—612no—
280#336—612no—
281C61#2594111yescharacteristic, complemented, Sylow
282C22×Q8—21no—
283C22×Q8—611no—
284C22×Q8C22⋊D6111yescharacteristic
285C22×Q8C22×D6111yescharacteristic
286C22×D4—611no—
287C2×Q16—1224no—
288C2×Q16—1222no—
289C2×SD16—1224no—
290C2×SD16—1222no—
291C22×C8—611no—
292C22×C8—611no—
293C22×C8—612no—
294C4⋊Q8—28no—
295C4⋊Q8—22no—
296C4⋊Q8C22⋊D6114yes—
297#125—216no—
298#125—24no—
299#125—24no—
300#124—1222no—
301#124—1222no—
302C4×Q8—28no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8—22no—
306C4×Q8—22no—
307C4×Q8—22no—
308C4×Q8C22⋊D6114yes—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—612no—
315C2×C4⋊C4C22⋊D6111yescharacteristic
316C2×C4⋊C4C22⋊D6112yes—
317C2×C4⋊C4C22×D6111yescharacteristic
318C2×C4⋊C4C22×D6112yes—
319C2×C22⋊C4—1222no—
320C2×C22⋊C4—1221no—
321C2×C22⋊C4—1221no—
322C2×C22⋊C4—1222no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42—1221no—
326C2×C42C22×D6112yes—
327C2×C42C22×D6111yescharacteristic
328C4⋊C8—1222no—
329#103—1224no—
330#103—1222no—
331#103—1224no—
332#102—1222no—
333#102—1224no—
334#97—1222no—
335#95—1221no—
336#95—1222no—
337#95—1222no—
338C24—1221no—
339C2×Q8—21no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—24no—
343C2×Q8—42no—
344C2×Q8—44no—
345C2×Q8—1221no—
346C2×Q8—1221no—
347C2×Q8—1222no—
348C2×Q8—1222no—
349C2×Q8#1175822614yes—
350C2×Q8C2×C22⋊D6112yes—
351C2×D4—1221no—
352C2×D4—1221no—
353C2×D4—1222no—
354C2×D4—1222no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—1221no—
359C22×C4—1222no—
360C22×C4C2×C22⋊D6112yes—
361C22×C4C2×C22⋊D6112yes—
362C22×C4C2×C22⋊D6111yescharacteristic
363C22×C4C2×C22⋊D6111yescharacteristic
364C22×C4C23×D6111yescharacteristic
365Q16—2444no—
366SD16—2444no—
367C2×C8—1222no—
368C2×C8—1221no—
369C2×C8—1221no—
370C2×C8—1222no—
371C2×C8—1222no—
372C2×C8—1222no—
373C2×C8—1222no—
374C4⋊C4—28no—
375C4⋊C4—28no—
376C4⋊C4—28no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—1222no—
384C4⋊C4—1222no—
385C4⋊C4C2×C22⋊D6114yes—
386C4⋊C4C2×C22⋊D6112yes—
387C22⋊C4—4882no—
388C22⋊C4—2444no—
389C22⋊C4—2442no—
390C22⋊C4—2442no—
391C42—28no—
392C42—22no—
393C42—22no—
394C42—24no—
395C42—4881no—
396C42C2×C22⋊D6112yes—
397C42C2×C22⋊D6112yes—
398C23—2441no—
399C23—2442no—
400C23—2441no—
401C23—2441no—
402C23—2442no—
403C23C22×C22⋊D6111yescharacteristic
404Q8—22no—
405Q8—24no—
406Q8—44no—
407Q8—48no—
408Q8—2441no—
409Q8—2441no—
410Q8—2442no—
411Q8#1451976614yes—
412D4—2441no—
413D4—2441no—
414D4—2442no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—44no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—2441no—
432C2×C4—2442no—
433C2×C4—2442no—
434C2×C4—2442no—
435C2×C4—2442no—
436C2×C4#1451974612yes—
437C2×C4#1451974612yes—
438C2×C4C22×C22⋊D6111yescharacteristic
439C2×C4C22×C22⋊D6111yescharacteristic
440C2×C4C2×#1175822612yes—
441C2×C4#1451982512yes—
442C8—2441no—
443C8—2441no—
444C8—2442no—
445C22—4881no—
446C22—4881no—
447C22—4882no—
448C22—4881no—
449C22—4882no—
450C22C2×#1451974611yescharacteristic
451C22#1684042312yes—
452C22#1684056012yes—
453C22#1684057011yescharacteristic
454C22#1684057711yescharacteristic
455C4—24no—
456C4—22no—
457C4—22no—
458C4—42no—
459C4—42no—
460C4—42no—
461C4—44no—
462C4—4881no—
463C4—4882no—
464C4C2×#1451976611yescharacteristic
465C4#1684033611yescharacteristic
466C4#1684042312yes—
467C2—9761no—
468C2#2144947311yescharacteristic
469C2#2145506111yescharacteristic
470C2#2145513411yescharacteristic
471C2#2145858712yes—
472C2#2145866112yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#557C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC2×#557D61
hypocenterC61#25941
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
4. #21446967C2semidirect
6. #21446768C2semidirect
7. #21446760C2semidirect
9. #21446160C2semidirect
10. #21445649C2semidirect
11. C2×C61×#557C2semidirect
231. C2×#557D61semidirect
281. C61#25941semidirect

Maximal subgroups

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

subgroupindex
C2×#168401242
C2×#168399842
#214513072
#214469672
#214469622
#214467682
#214467602
#214461742
#214461602
#214456492
C2×C61×#5572
#2594161