Finite Group Databasebeta

← #29714949

Subgroups — #29714949

Order 15616.

1068 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840486C211yesmaximal, characteristic
2C2×#16839860C211yesmaximal, characteristic
3#21451181C211yesmaximal, characteristic, complemented
4#21447056C212yesmaximal
5#21446910C212yesmaximal
6#21446758C211yesmaximal, characteristic, complemented
7#21446640C211yesmaximal, characteristic, complemented
8#21446162C212yesmaximal, complemented
9#21446071C212yesmaximal, complemented
10#21445830C211yesmaximal, characteristic
11C2×C61×#462C211yesmaximal, characteristic, complemented
12C22×#11758201C2211yescharacteristic
13#16840486—24no—
14#16840486—24no—
15C2×#14519772—22no—
16C2×#14519766C2212yes—
17C2×#14519757—21no—
18C2×Q8×Dic61C2212yes—
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×#14519735—21no—
22#16839860—24no—
23C2×#14519710C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519701C2212yes—
26C2×#14519701C2212yes—
27C2×#14519699C2212yes—
28C2×#14519697C2211yescharacteristic
29C2×#14519688C2211yescharacteristic
30#16839701—21no—
31#16839682—21no—
32#16839669C2212yes—
33#16839658C2212yes—
34#16839646—22no—
35#16839628C2211yescharacteristic
36#16839525C2212yes—
37#16839516C2212yes—
38#16839516C2212yes—
39C22×Q16×C61C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×C4×Q8×C61C2211yescharacteristic
42C61×#462—24no—
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#103×C61C2212yes—
45C22×Dic122D411yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C2×#11758201—22no—
49C2×#11758201—22no—
50C2×#11758201—22no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2311yescharacteristic
53#14519772—216no—
54#14519766—24no—
55#14519766—24no—
56#14519757—28no—
57Q8×Dic61—24no—
58Q8×Dic61—24no—
59C4×Dic122—22no—
60C4×Dic122—22no—
61C4×D244—22no—
62C4×D244D414yes—
63#14519735—28no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61D411yescharacteristic
66C2×C4⋊Dic61C2312yes—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×#11758196—22no—
69C2×#11758196D412yes—
70C2×#11758198—21no—
71C2×#11758198—22no—
72C2×#11758198—22no—
73C2×C4×Dic61—22no—
74C2×C4×Dic61C2312yes—
75#14519710—22no—
76#14519701—22no—
77#14519701—24no—
78#14519701—42no—
79#14519699—42no—
80#14519697—22no—
81#14519688—22no—
82#14519684—21no—
83#14519684—21no—
84#14519684—22no—
85#14519684—22no—
86#14519684—22no—
87C22×Q8×C61C2312yes—
88C22×Q8×C61C2311yescharacteristic
89C2×Q16×C61—24no—
90C2×Q16×C61—24no—
91C2×Q16×C61—42no—
92C22×C8×C61C2312yes—
93C4⋊Q8×C61—22no—
94C4⋊Q8×C61D414yes—
95C4×Q8×C61—22no—
96C4×Q8×C61—22no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C42×C61—21no—
102C2×C42×C61C2311yescharacteristic
103C4⋊C8×C61—22no—
104#103×C61—24no—
105C23×D61—21no—
106C2×Q8×C61—22no—
107C2×Q8×C61—21no—
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61C2×D414yes—
117C2×Q8×C61#4112yes—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61—21no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C22×Dic61—21no—
125C22×Dic61—22no—
126C22×Dic61C2×D412yes—
127C22×Dic61C2×D412yes—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244C2×D412yes—
134C2×C4×D61—24no—
135C2×C4×D61—44no—
136C2×Dic122—22no—
137C2×Dic122—42no—
138Q16×C61—48no—
139C2×C8×C61—22no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—24no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—42no—
147C42×C61—42no—
148C42×C61C2×D412yes—
149C42×C61C2×D412yes—
150#11758201—28no—
151#11758198—28no—
152#11758198—28no—
153#11758198—24no—
154C4⋊Dic61—28no—
155C4⋊Dic61—22no—
156C4⋊Dic61—21no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61C2×D412yes—
160#11758196—24no—
161#11758196—44no—
162C4×Dic61—28no—
163C4×Dic61—22no—
164C4×Dic61—22no—
165C2×#92418—21no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418—21no—
170C2×#92418#4112yes—
171C23×C61C22×D411yescharacteristic
172C22×D61—22no—
173C22×D61—41no—
174C22×D61—41no—
175C22×D61—42no—
176C22×D61—42no—
177Q8×C61—22no—
178Q8×C61—22no—
179Q8×C61—24no—
180Q8×C61—24no—
181Q8×C61—42no—
182Q8×C61—42no—
183Q8×C61—41no—
184Q8×C61—41no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—24no—
188C2×C4×C61—22no—
189C2×C4×C61—41no—
190C2×C4×C61—42no—
191C2×C4×C61—41no—
192C2×C4×C61—42no—
193C2×C4×C61—41no—
194C2×C4×C61—42no—
195C2×C4×C61#12012yes—
196C2×C4×C61#12012yes—
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C2×#4111yescharacteristic
200C2×C4×C61C2×#4111yescharacteristic
201C2×C4×C61H(5,𝔽2)11yescharacteristic
202C2×C4×C612−1+411yescharacteristic
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—24no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213D244—22no—
214D244—22no—
215D244—41no—
216D244—41no—
217C4×D61—48no—
218Dic122—44no—
219C8×C61—42no—
220#92418—22no—
221#92418—41no—
222#24201—611nomaximal, Sylow
223C22×C61C2×#12011yescharacteristic
224C22×C61#54811yescharacteristic
225C22×C61#55011yescharacteristic
226C22×C61#57512yes—
227C22×C61#57812yes—
228C2×D61—42no—
229C2×D61—42no—
230C2×D61—42no—
231C2×D61—82no—
232C2×D61—82no—
233C4×C61—22no—
234C4×C61—22no—
235C4×C61—44no—
236C4×C61—42no—
237C4×C61—81no—
238C4×C61#52511yescharacteristic
239C4×C61#53711yescharacteristic
240C4×C61#54611yescharacteristic
241C4×C61#54711yescharacteristic
242Dic61—42no—
243Dic61—42no—
244Dic61—42no—
245Dic61—82no—
246C2×#556—611no—
247C2×#462D6111yescharacteristic, complemented
248C2×#440—611no—
249#2581—611no—
250#2042—612no—
251#2015—612no—
252#1985—611no—
253#1960—611no—
254#1878—612no—
255#1863—612no—
256#1804—611no—
257C2×C61#240711yescharacteristic
258C2×C61#300511yescharacteristic
259C2×C61#301111yescharacteristic
260C2×C61#337712yes—
261C2×C61#340212yes—
262D61—82no—
263C22×Q16C2×D6112yes—
264C22×SD16—611no—
265#556—1224no—
266#556—1224no—
267C2×C4⋊Q8C2×D6111yescharacteristic
268C2×#126—1222no—
269C2×#124—612no—
270C2×C22⋊Q8—1221no—
271C2×C4×Q8—611no—
272C2×C4×Q8—612no—
273C2×C4×Q8C2×D6111yescharacteristic
274C2×C4×D4—611no—
275C2×#117—1221no—
276#462—24no—
277#440—1224no—
278C2×#107—611no—
279C2×C4⋊C8C2×D6111yescharacteristic
280C2×#103—612no—
281C2×#103—611no—
282C2×#103—612no—
283C2×#103C2×D6112yes—
284C2×#102—611no—
285C2×#102—612no—
286C2×#97—611no—
287#400—1221no—
288#395—1221no—
289#391—612no—
290#389—612no—
291#387—1222no—
292#382—611no—
293#340—612no—
294#336—612no—
295#336—612no—
296C61#2420111yescharacteristic, complemented, Sylow
297C22×Q8—611no—
298C22×Q8C22×D6111yescharacteristic
299C22×Q8C22×D6112yes—
300C22×D4—611no—
301C23×C4—1221no—
302C2×Q16—24no—
303C2×Q16—24no—
304C2×Q16—42no—
305C2×SD16—1222no—
306C2×SD16—1222no—
307C2×SD16—1222no—
308C22×C8—611no—
309C22×C8—611no—
310C22×C8C22×D6112yes—
311C4⋊Q8—22no—
312C4⋊Q8C22⋊D6114yes—
313#126—12216no—
314#124—1224no—
315#124—1224no—
316C22⋊Q8—1228no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8—1224no—
320C4×Q8—1224no—
321C4×Q8—1222no—
322C4×Q8—1222no—
323C4×D4—614no—
324C4×D4—1222no—
325#117—1228no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—611no—
330C2×C4⋊C4—611no—
331C2×C4⋊C4—612no—
332C2×C4⋊C4—612no—
333C2×C4⋊C4—1222no—
334C2×C4⋊C4—1222no—
335C2×C4⋊C4C22×D6111yescharacteristic
336C2×C22⋊C4—1222no—
337C2×C22⋊C4—1222no—
338C2×C22⋊C4—1221no—
339C2×C42—21no—
340C2×C42—612no—
341C2×C42—1222no—
342C2×C42C22×D6111yescharacteristic
343#107—1222no—
344C4⋊C8—22no—
345#103—24no—
346#103—1222no—
347#103—1224no—
348#103—2442no—
349#102—1222no—
350#102—2442no—
351#97—1222no—
352#95—1222no—
353#95—1222no—
354#95—1222no—
355#95—1221no—
356#95—1221no—
357C24—1221no—
358C2×Q8—22no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—22no—
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—21no—
367C2×Q8—22no—
368C2×Q8—1222no—
369C2×Q8—2442no—
370C2×Q8#1175822412yes—
371C2×Q8C2×C22⋊D6114yes—
372C2×D4—612no—
373C2×D4—1221no—
374C2×D4—1221no—
375C2×D4—1221no—
376C2×D4—1221no—
377C2×D4—1221no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—612no—
382C22×C4—612no—
383C22×C4—1221no—
384C22×C4—1224no—
385C22×C4—1222no—
386C22×C4—2444no—
387C22×C4D4×D6111yescharacteristic
388C22×C4D4×D6111yescharacteristic
389C22×C4C23×D6111yescharacteristic
390Q16—48no—
391SD16—1228no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—612no—
396C2×C8—1221no—
397C2×C8—1221no—
398C2×C8—1221no—
399C2×C8—1221no—
400C2×C8—1221no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—42no—
406C4⋊C4—612no—
407C4⋊C4—1228no—
408C4⋊C4—1221no—
409C4⋊C4—1224no—
410C4⋊C4—1222no—
411C4⋊C4—1222no—
412C4⋊C4—1222no—
413C4⋊C4—2444no—
414C22⋊C4—1228no—
415C22⋊C4—1228no—
416C22⋊C4—1224no—
417C42—42no—
418C42—1228no—
419C42—1222no—
420C42—1222no—
421C42D4×D6112yes—
422C42C2×C22⋊D6112yes—
423C23—1222no—
424C23—2442no—
425C23—2442no—
426C23—2441no—
427C23—2441no—
428C23C2×D4×D6111yescharacteristic
429Q8—22no—
430Q8—22no—
431Q8—24no—
432Q8—24no—
433Q8—42no—
434Q8—42no—
435Q8—41no—
436Q8—41no—
437Q8—2444no—
438D4—1222no—
439D4—1222no—
440D4—2441no—
441D4—2441no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—1222no—
453C2×C4—1222no—
454C2×C4—1222no—
455C2×C4—1222no—
456C2×C4—1222no—
457C2×C4—1224no—
458C2×C4—2448no—
459C2×C4—2442no—
460C2×C4—2442no—
461C2×C4—2442no—
462C2×C4—2442no—
463C2×C4#1451974512yes—
464C2×C4#1451974512yes—
465C2×C4C2×D4×D6111yescharacteristic
466C2×C4C22×C22⋊D6111yescharacteristic
467C2×C4#41×D6111yescharacteristic
468C2×C4C2×#1175822411yescharacteristic
469C2×C4#1451982211yescharacteristic
470C2×C4#1451982411yescharacteristic
471C8—42no—
472C8—1222no—
473C8—2441no—
474C22—4882no—
475C22—4882no—
476C22—2442no—
477C22—2442no—
478C22—2442no—
479C22C2×#1451974511yescharacteristic
480C22#1684041811yescharacteristic
481C22#1684043011yescharacteristic
482C22#1684056112yes—
483C22#1684057512yes—
484C4—22no—
485C4—22no—
486C4—44no—
487C4—42no—
488C4—81no—
489C4—4882no—
490C4—2442no—
491C4—2442no—
492C4—2442no—
493C4#1684027211yescharacteristic
494C4#1684034111yescharacteristic
495C4#1684040611yescharacteristic
496C4#1684041111yescharacteristic
497C2—4882no—
498C2#2144916011yescharacteristic
499C2#2145506011yescharacteristic
500C2#2145511011yescharacteristic
501C2#2145865412yes—
502C2#2145889312yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#462C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC2×#462D61
hypocenterC61#24201
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
3. #21451181C2semidirect
6. #21446758C2semidirect
7. #21446640C2semidirect
8. #21446162C2semidirect
9. #21446071C2semidirect
11. C2×C61×#462C2semidirect
247. C2×#462D61semidirect
296. C61#24201semidirect

Maximal subgroups

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

subgroupindex
C2×#168404862
C2×#168398602
#214511812
#214470562
#214469102
#214467582
#214466402
#214461622
#214460712
#214458302
C2×C61×#4622
#2420161