Finite Group Databasebeta

← #29718418

Subgroups — #29718418

Order 15616.

958 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840098C211yesmaximal, characteristic
2C22×#14519702C211yesmaximal, characteristic
3#21449463C211yesmaximal, characteristic
4#21448694C211yesmaximal, characteristic
5#21447213C212yesmaximal, complemented
6#21447066C212yesmaximal, complemented
7#21446741C212yesmaximal, complemented
8#21446461C212yesmaximal, complemented
9#21446242C211yesmaximal, characteristic
10#21445755C211yesmaximal, characteristic
11C61×#1850C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C2×#14519777C2212yes—
14C2×#14519756C2211yescharacteristic
15C22×C4⋊Dic61C2211yescharacteristic
16#16840098—24no—
17C2×#14519711C2211yescharacteristic
18C2×#14519711C2211yescharacteristic
19C2×#14519702—24no—
20C2×#14519702C2212yes—
21C2×#14519702C2212yes—
22C2×#14519702C2212yes—
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2212yes—
26C2×#14519702C2212yes—
27#16839683—22no—
28#16839668C2212yes—
29#16839660C2212yes—
30#16839645—22no—
31#16839637—21no—
32#16839637—21no—
33#16839622C2211yescharacteristic
34#16839620—21no—
35C4×C4⋊Dic61C2212yes—
36C22⋊C4×Dic61—21no—
37#16839525C2212yes—
38C23×C8×C61C2211yescharacteristic
39C2×C4⋊C8×C61C2212yes—
40C2×C22⋊C8×C61C2211yescharacteristic
41C4×C22⋊C4×C61C2211yescharacteristic
42C61×#336C2212yes—
43C22×Dic122—22no—
44C22×Dic122—24no—
45C22×Dic122—22no—
46C22×Dic122—21no—
47C22×Dic122—42no—
48C22×Dic122C2311yescharacteristic
49C22×Dic122C2312yes—
50C23×Dic61—21no—
51C23×Dic61—21no—
52#14519777—24no—
53#14519777—42no—
54#14519756—22no—
55#14519756—22no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—22no—
58C2×C4⋊Dic61—22no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61D414yes—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C4⋊Dic61C2312yes—
63C2×C4⋊Dic61C2311yescharacteristic
64C2×C4⋊Dic61C2311yescharacteristic
65C2×#11758196—21no—
66C2×#11758196—22no—
67C2×C22⋊Dic61—21no—
68C2×C22⋊Dic61—21no—
69C2×C4×Dic61—22no—
70#14519711—22no—
71#14519711—22no—
72#14519702—24no—
73#14519702—24no—
74#14519702—24no—
75#14519702—22no—
76#14519702—22no—
77#14519702—42no—
78#14519702—42no—
79#14519684—22no—
80#14519684—22no—
81#14519684—22no—
82#14519684—22no—
83#14519684—21no—
84C23×C4×C61C2311yescharacteristic
85C22×C8×C61—22no—
86C22×C8×C61—42no—
87C22×C8×C61C2312yes—
88C22×C8×C61C2311yescharacteristic
89C22×C8×C61C2311yescharacteristic
90C2×C22⋊C4×C61D411yescharacteristic
91C2×C22⋊C4×C61D411yescharacteristic
92C2×C42×C61C2312yes—
93C4⋊C8×C61—42no—
94C22⋊C8×C61—22no—
95#95×C61D412yes—
96C24×C61C2×D411yescharacteristic
97C22×C4×C61—22no—
98C22×C4×C61—22no—
99C22×C4×C61C2×D412yes—
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D412yes—
102C22×C4×C61C2×D412yes—
103C22×C4×C61#4112yes—
104C22×C4×C61C2411yescharacteristic
105C22×Dic61—22no—
106C22×Dic61—21no—
107C22×Dic61—21no—
108C22×Dic61—24no—
109C22×Dic61—21no—
110C22×Dic61—21no—
111C22×Dic61—22no—
112C22×Dic61—44no—
113C22×Dic61—42no—
114C22×Dic61—82no—
115C2×Dic122—28no—
116C2×Dic122—22no—
117C2×Dic122—22no—
118C2×Dic122—22no—
119C2×Dic122—22no—
120C2×Dic122—22no—
121C2×Dic122—21no—
122C2×Dic122—21no—
123C2×Dic122—22no—
124C2×Dic122—22no—
125C2×Dic122—48no—
126C2×Dic122—44no—
127C2×Dic122—44no—
128C2×Dic122—42no—
129C2×Dic122#4114yes—
130C2×C8×C61—21no—
131C2×C8×C61—21no—
132C2×C8×C61—22no—
133C2×C8×C61—22no—
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C2×C8×C61—22no—
137C2×C8×C61—44no—
138C22⋊C4×C61—22no—
139C22⋊C4×C61Q1614yes—
140C42×C61—22no—
141C42×C61—22no—
142C22⋊Dic61—42no—
143C22⋊Dic61—42no—
144C4⋊Dic61—24no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—22no—
149C4⋊Dic61—21no—
150C4⋊Dic61—22no—
151C4⋊Dic61—44no—
152C4⋊Dic61—44no—
153C4⋊Dic61C2×D412yes—
154C4⋊Dic61C2×D412yes—
155#11758196—42no—
156#11758196—82no—
157C4×Dic61—82no—
158C23×C61—22no—
159C23×C61—22no—
160C23×C61#12112yes—
161C23×C61C2×Q1612yes—
162C23×C61#13712yes—
163C23×C61C22×D411yescharacteristic
164C2×C4×C61—22no—
165C2×C4×C61—22no—
166C2×C4×C61—22no—
167C2×C4×C61—22no—
168C2×C4×C61—22no—
169C2×C4×C61—24no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61#12112yes—
174C2×C4×C61C2×Q1614yes—
175C2×C4×C61C22×D411yescharacteristic
176C2×C4×C61C2×#4112yes—
177C2×C4×C61C2×#4111yescharacteristic
178C2×C4×C61H(5,𝔽2)12yes—
179C2×Dic61—22no—
180C2×Dic61—22no—
181C2×Dic61—24no—
182C2×Dic61—48no—
183C2×Dic61—42no—
184C2×Dic61—42no—
185C2×Dic61—42no—
186C2×Dic61—42no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—42no—
190C2×Dic61—42no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193C2×Dic61—42no—
194C2×Dic61—84no—
195Dic122—24no—
196Dic122—24no—
197Dic122—48no—
198Dic122—42no—
199Dic122—42no—
200Dic122—41no—
201Dic122—41no—
202Dic122—42no—
203C8×C61—41no—
204C8×C61—41no—
205C8×C61—42no—
206#24482—611nomaximal, Sylow
207C22×C61—22no—
208C22×C61—22no—
209C22×C61—24no—
210C22×C61—24no—
211C22×C61#46214yes—
212C22×C61C2×#12111yescharacteristic
213C22×C61#54612yes—
214C22×C61C22×Q1611yescharacteristic
215C22×C61C2×#13711yescharacteristic
216C22×C61#57712yes—
217C4×C61—22no—
218C4×C61—24no—
219C4×C61—42no—
220C4×C61#52511yescharacteristic
221C4×C61#53511yescharacteristic
222C4×C61#55112yes—
223Dic61—44no—
224Dic61—42no—
225Dic61—42no—
226Dic61—81no—
227Dic61—81no—
228Dic61—82no—
229C2×#484—611no—
230C22×#103—611no—
231#2434—611no—
232#2345—611no—
233#2068—612no—
234#2043—612no—
235#1981—612no—
236#1928—612no—
237#1892—611no—
238#1850D6111yescharacteristic
239#1787—611no—
240C2×C61—24no—
241C2×C61#241111yescharacteristic
242C2×C61C2×#46211yescharacteristic
243C2×C61#304211yescharacteristic
244C2×C61#329912yes—
245C2×C61#332012yes—
246C23×Q8—611no—
247C23×C8C2×D6111yescharacteristic
248C2×C4⋊Q8—612no—
249C2×C22⋊Q8—611no—
250C22×C4⋊C4—611no—
251#484—1224no—
252C2×#107—611no—
253C2×#107—611no—
254C2×C4⋊C8C2×D6112yes—
255C2×#103—611no—
256C2×#103—611no—
257C2×#103—612no—
258C2×#103—612no—
259C2×#103—612no—
260C2×#103—612no—
261C2×#103—612no—
262C2×#103—1224no—
263C2×C22⋊C8C2×D6111yescharacteristic
264#395—1222no—
265#391—612no—
266#389—612no—
267#387—1222no—
268#385—1221no—
269#385—1221no—
270#380—611no—
271#380—1221no—
272C4×C4⋊C4—612no—
273C4×C22⋊C4—1221no—
274C4×C22⋊C4C2×D6111yescharacteristic
275#340—612no—
276#336C2×D6112yes—
277C61#2448211yescharacteristic, complemented, Sylow
278C22×Q8—612no—
279C22×Q8—611no—
280C22×Q8—1224no—
281C22×Q8—1221no—
282C22×Q8—1222no—
283C22×Q8—1222no—
284C22×Q8—2442no—
285C23×C4—1221no—
286C23×C4—1221no—
287C23×C4C22×D6111yescharacteristic
288C22×C8—22no—
289C22×C8—42no—
290C22×C8C22×D6112yes—
291C22×C8C22×D6111yescharacteristic
292C22×C8C22×D6111yescharacteristic
293C4⋊Q8—1224no—
294C4⋊Q8—2442no—
295C22⋊Q8—1222no—
296C22⋊Q8—1222no—
297C2×C4⋊C4—614no—
298C2×C4⋊C4—611no—
299C2×C4⋊C4—611no—
300C2×C4⋊C4—611no—
301C2×C4⋊C4—612no—
302C2×C4⋊C4—1221no—
303C2×C4⋊C4—1222no—
304C2×C4⋊C4—1222no—
305C2×C4⋊C4—1221no—
306C2×C4⋊C4—1222no—
307C2×C4⋊C4—1222no—
308C2×C22⋊C4—1221no—
309C2×C22⋊C4—1221no—
310C2×C22⋊C4D24411yescharacteristic
311C2×C22⋊C4D24411yescharacteristic
312C2×C42—1222no—
313C2×C42C22×D6112yes—
314#107—1222no—
315#107—1222no—
316C4⋊C8—42no—
317#103—1224no—
318#103—1224no—
319#103—1222no—
320#103—1222no—
321#103—1224no—
322#103—2442no—
323#103—2442no—
324C22⋊C8—22no—
325#95—1222no—
326#95—1222no—
327#95—1221no—
328#95—1222no—
329#95—1222no—
330#95D24412yes—
331C24C2×D24411yescharacteristic
332C2×Q8—614no—
333C2×Q8—1228no—
334C2×Q8—1222no—
335C2×Q8—1222no—
336C2×Q8—1221no—
337C2×Q8—1222no—
338C2×Q8—1222no—
339C2×Q8—1222no—
340C2×Q8—1221no—
341C2×Q8—1222no—
342C2×Q8—1222no—
343C2×Q8—2444no—
344C2×Q8—2448no—
345C2×Q8—2444no—
346C2×Q8—2442no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—1224no—
350C22×C4—1221no—
351C22×C4—1221no—
352C22×C4—1222no—
353C22×C4—1221no—
354C22×C4—1222no—
355C22×C4—1221no—
356C22×C4—4882no—
357C22×C4—2444no—
358C22×C4—2442no—
359C22×C4C2×D24412yes—
360C22×C4C2×D24411yescharacteristic
361C22×C4C2×D24412yes—
362C22×C4D4×D6112yes—
363C22×C4#1175822612yes—
364C22×C4C23×D6111yescharacteristic
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—44no—
373C4⋊C4—612no—
374C4⋊C4—612no—
375C4⋊C4—1221no—
376C4⋊C4—1221no—
377C4⋊C4—1224no—
378C4⋊C4—1221no—
379C4⋊C4—1222no—
380C4⋊C4—1221no—
381C4⋊C4—1222no—
382C4⋊C4—4882no—
383C4⋊C4—2444no—
384C4⋊C4—2444no—
385C4⋊C4—2442no—
386C22⋊C4—22no—
387C22⋊C4—2442no—
388C22⋊C4—2442no—
389C22⋊C4Dic24414yes—
390C42—22no—
391C42—22no—
392C42—4882no—
393C23—22no—
394C23—22no—
395C23#1451975212yes—
396C23C2×Dic24412yes—
397C23#1451981012yes—
398C23C22×D24411yescharacteristic
399Q8—1224no—
400Q8—1224no—
401Q8—2448no—
402Q8—2441no—
403Q8—2441no—
404Q8—2442no—
405Q8—2442no—
406Q8—2442no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—1222no—
417C2×C4—1222no—
418C2×C4—1224no—
419C2×C4—4884no—
420C2×C4—2441no—
421C2×C4—2441no—
422C2×C4—2442no—
423C2×C4—2442no—
424C2×C4—2442no—
425C2×C4—2442no—
426C2×C4—2442no—
427C2×C4—2448no—
428C2×C4—2442no—
429C2×C4—2442no—
430C2×C4—2442no—
431C2×C4—2442no—
432C2×C4#1451975212yes—
433C2×C4C2×Dic24414yes—
434C2×C4C2×D4×D6111yescharacteristic
435C2×C4#41×D6112yes—
436C2×C4C2×#1175822611yescharacteristic
437C2×C4#1451982212yes—
438C8—41no—
439C8—41no—
440C8—42no—
441C22—24no—
442C22—24no—
443C22—22no—
444C22—22no—
445C22#1683997314yes—
446C22C2×#1451975211yescharacteristic
447C22#1684040712yes—
448C22C22×Dic24411yescharacteristic
449C22C2×#1451981011yescharacteristic
450C22#1684057412yes—
451C4—24no—
452C4—22no—
453C4—42no—
454C4—4882no—
455C4—4881no—
456C4—4881no—
457C4—2442no—
458C4—2442no—
459C4—2444no—
460C4#1684027011yescharacteristic
461C4#1684032711yescharacteristic
462C4#1684043912yes—
463C2—24no—
464C2#2144920211yescharacteristic
465C2C2×#1683997311yescharacteristic
466C2#2145533211yescharacteristic
467C2#2145769612yes—
468C2#2145799212yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1850C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1850D61
hypocenterC61#24482
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
5. #21447213C2semidirect
6. #21447066C2semidirect
7. #21446741C2semidirect
8. #21446461C2semidirect
277. C61#24482semidirect

Maximal subgroups

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

subgroupindex
C2×#168400982
C22×#145197022
#214494632
#214486942
#214472132
#214470662
#214467412
#214464612
#214462422
#214457552
C61×#18502
#2448261