Finite Group Databasebeta

← #29734125

Subgroups — #29734125

Order 15616.

1012 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840486C211yesmaximal, characteristic
2C2×#16840059C211yesmaximal, characteristic
3#21451208C211yesmaximal, characteristic, complemented
4#21447056C212yesmaximal
5#21447056C212yesmaximal
6#21446759C211yesmaximal, characteristic, complemented
7#21446743C211yesmaximal, characteristic, complemented
8#21446085C212yesmaximal, complemented
9#21446071C212yesmaximal, complemented
10#21445645C211yesmaximal, characteristic
11C2×C61×#491C211yesmaximal, characteristic, complemented
12#16840486—24no—
13#16840486—24no—
14C2×#14519772—22no—
15C2×#14519766C2212yes—
16C2×#14519757—21no—
17C2×Q8×Dic61C2212yes—
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×#14519735—21no—
21#16840059—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519701C2212yes—
26C2×#14519701C2212yes—
27C2×#14519699C2212yes—
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30#16839692—21no—
31#16839682—21no—
32#16839658C2212yes—
33#16839658C2212yes—
34#16839657—22no—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839525C2212yes—
37#16839516C2212yes—
38#16839516C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×#125×C61C2211yescharacteristic
41C61×#491—24no—
42C2×#103×C61C2212yes—
43C2×#103×C61C2212yes—
44C2×#97×C61C2211yescharacteristic
45C22×Dic122D411yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C22×#92418C2311yescharacteristic
49C22×#92418C2311yescharacteristic
50#14519772—216no—
51#14519766—24no—
52#14519766—24no—
53#14519757—28no—
54Q8×Dic61—24no—
55Q8×Dic61—24no—
56C4×Dic122—22no—
57C4×Dic122—22no—
58C4×D244—22no—
59C4×D244D414yes—
60#14519735—28no—
61C2×C4⋊Dic61—22no—
62C2×C4⋊Dic61D411yescharacteristic
63C2×C4⋊Dic61C2311yescharacteristic
64C2×C4⋊Dic61C2312yes—
65C2×#11758196—22no—
66C2×#11758196—22no—
67C2×#11758196D412yes—
68C2×#11758198—21no—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C2×C4×Dic61—22no—
72C2×C4×Dic61—21no—
73C2×C4×Dic61C2312yes—
74#14519710—22no—
75#14519708—22no—
76C4⋊#92418—22no—
77#14519701—24no—
78#14519701—24no—
79#14519699—24no—
80#14519697—22no—
81#14519697—22no—
82#14519684—21no—
83#14519684—22no—
84#14519684—22no—
85C22×Q8×C61C2312yes—
86C22×C8×C61C2312yes—
87C4⋊Q8×C61—22no—
88C4⋊Q8×C61D414yes—
89#125×C61—22no—
90#125×C61—22no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C42×C61C2311yescharacteristic
98#103×C61—24no—
99#103×C61—24no—
100#97×C61—22no—
101C23×D61—21no—
102C2×Q8×C61—22no—
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61C2×D414yes—
108C22×C4×C61—22no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2411yescharacteristic
114C22×Dic61—21no—
115C22×Dic61—22no—
116C22×Dic61C2×D412yes—
117C22×Dic61C2×D412yes—
118C2×D244—22no—
119C2×D244—22no—
120C2×D244—21no—
121C2×D244C2×D412yes—
122C2×C4×D61—24no—
123C2×C4×D61—44no—
124C2×Dic122—22no—
125C2×Dic122—42no—
126C2×C8×C61—22no—
127C2×C8×C61—22no—
128C2×C8×C61—22no—
129C4⋊C4×C61—24no—
130C4⋊C4×C61—24no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—42no—
136C4⋊C4×C61—42no—
137C42×C61C2×D412yes—
138C42×C61C2×D412yes—
139#11758198—28no—
140#11758198—28no—
141#11758198—24no—
142C4⋊Dic61—28no—
143C4⋊Dic61—22no—
144C4⋊Dic61—21no—
145C4⋊Dic61—22no—
146C4⋊Dic61C2×D412yes—
147C4⋊Dic61#4114yes—
148#11758196—24no—
149#11758196—44no—
150#11758196—82no—
151C4×Dic61—28no—
152C4×Dic61—22no—
153C4×Dic61—22no—
154C4×Dic61—81no—
155C2×#92418—21no—
156C2×#92418—21no—
157C2×#92418—22no—
158C2×#92418—22no—
159C23×C61C22×D411yescharacteristic
160C22×D61—22no—
161C22×D61—42no—
162C22×D61—42no—
163C22×D61—41no—
164C22×D61—41no—
165Q8×C61—24no—
166Q8×C61—24no—
167Q8×C61—42no—
168Q8×C61—42no—
169C2×C4×C61—22no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—24no—
173C2×C4×C61—42no—
174C2×C4×C61—42no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—42no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61#12012yes—
181C2×C4×C61#12012yes—
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C22×D411yescharacteristic
184C2×C4×C61C2×#4112yes—
185C2×C4×C612−1+412yes—
186C2×Dic61—22no—
187C2×Dic61—22no—
188C2×Dic61—22no—
189C2×Dic61—24no—
190C2×Dic61—22no—
191C2×Dic61—22no—
192C2×Dic61—22no—
193C2×Dic61—22no—
194C2×Dic61—42no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197D244—22no—
198D244—22no—
199D244—42no—
200C4×D61—48no—
201Dic122—44no—
202C8×C61—42no—
203#92418—41no—
204#92418—41no—
205#25656—611nomaximal, Sylow
206C22×C61C2×#12011yescharacteristic
207C22×C61#54812yes—
208C22×C61#57512yes—
209C22×C61#57711yescharacteristic
210C22×C61#57811yescharacteristic
211C2×D61—42no—
212C2×D61—42no—
213C2×D61—42no—
214C2×D61—82no—
215C2×D61—82no—
216C4×C61—22no—
217C4×C61—22no—
218C4×C61—44no—
219C4×C61—81no—
220C4×C61—81no—
221C4×C61#53211yescharacteristic
222C4×C61#53611yescharacteristic
223C4×C61#54712yes—
224Dic61—44no—
225Dic61—42no—
226Dic61—42no—
227Dic61—42no—
228C2×#556—611no—
229C2×#491D6111yescharacteristic, complemented
230C2×#478—611no—
231#2584—611no—
232#2042—612no—
233#2042—612no—
234#1985—611no—
235#1981—611no—
236#1865—612no—
237#1863—612no—
238#1762—611no—
239C2×C61#243211yescharacteristic
240C2×C61#300511yescharacteristic
241C2×C61#301311yescharacteristic
242C2×C61#337612yes—
243C2×C61#337712yes—
244D61—82no—
245#556—1224no—
246#556—1224no—
247C2×C4⋊Q8C2×D6111yescharacteristic
248C2×#126—1222no—
249C2×#125C2×D6111yescharacteristic
250C2×#124—612no—
251C2×C22⋊Q8—1221no—
252C2×C4×Q8—611no—
253C2×C4×Q8—612no—
254C2×C4×D4—611no—
255C2×#117—1221no—
256#491—24no—
257#478—1224no—
258C2×#107—611no—
259C2×#106—611no—
260C2×C4⋊C8—611no—
261C2×#103—612no—
262C2×#103—612no—
263C2×#103C2×D6112yes—
264C2×#103C2×D6112yes—
265C2×#102—612no—
266C2×#102—611no—
267C2×#102—611no—
268C2×#97C2×D6111yescharacteristic
269#398—1221no—
270#395—1221no—
271#389—612no—
272#389—612no—
273#389—1222no—
274C4×C4⋊C4—611no—
275#340—612no—
276#336—612no—
277#336—612no—
278C61#2565611yescharacteristic, complemented, Sylow
279C22×Q8—611no—
280C22×Q8C22×D6112yes—
281C22×D4—611no—
282C23×C4—1221no—
283C22×C8—611no—
284C22×C8—611no—
285C22×C8C22×D6112yes—
286C4⋊Q8—22no—
287C4⋊Q8C22⋊D6114yes—
288#126—12216no—
289#125—22no—
290#125—22no—
291#124—1224no—
292#124—1224no—
293C22⋊Q8—1228no—
294C4×Q8—1224no—
295C4×Q8—1224no—
296C4×Q8—1222no—
297C4×Q8—1222no—
298C4×D4—614no—
299C4×D4—1222no—
300#117—1228no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—611no—
306C2×C4⋊C4—612no—
307C2×C4⋊C4—611no—
308C2×C4⋊C4—612no—
309C2×C4⋊C4—1222no—
310C2×C4⋊C4—1222no—
311C2×C4⋊C4—1222no—
312C2×C4⋊C4C22×D6111yescharacteristic
313C2×C4⋊C4C22×D6111yescharacteristic
314C2×C22⋊C4—1222no—
315C2×C22⋊C4—1222no—
316C2×C22⋊C4—1221no—
317C2×C42—612no—
318C2×C42—1222no—
319C2×C42—1221no—
320C2×C42C22×D6111yescharacteristic
321#107—1222no—
322#106—1222no—
323C4⋊C8—1222no—
324#103—24no—
325#103—24no—
326#103—1224no—
327#103—1224no—
328#102—1222no—
329#102—1222no—
330#102—1224no—
331#97—22no—
332#95—1222no—
333#95—1222no—
334#95—1221no—
335C24—1221no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—1222no—
342C2×Q8—2442no—
343C2×Q8C2×C22⋊D6114yes—
344C2×D4—612no—
345C2×D4—1221no—
346C2×D4—1222no—
347C2×D4—1222no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—612no—
352C22×C4—612no—
353C22×C4—1222no—
354C22×C4—1221no—
355C22×C4—1224no—
356C22×C4—2444no—
357C22×C4D4×D6111yescharacteristic
358C22×C4D4×D6111yescharacteristic
359C22×C4C23×D6111yescharacteristic
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—1221no—
364C2×C8—1221no—
365C2×C8—1222no—
366C2×C8—1222no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—612no—
376C4⋊C4—614no—
377C4⋊C4—1228no—
378C4⋊C4—1221no—
379C4⋊C4—1222no—
380C4⋊C4—1224no—
381C4⋊C4—1222no—
382C4⋊C4—4882no—
383C4⋊C4—2444no—
384C22⋊C4—1228no—
385C22⋊C4—1228no—
386C22⋊C4—1224no—
387C42—1228no—
388C42—1222no—
389C42—1222no—
390C42—4881no—
391C42D4×D6112yes—
392C42C2×C22⋊D6112yes—
393C23—1222no—
394C23—2442no—
395C23—2441no—
396C23—2441no—
397C23—2442no—
398C23C2×D4×D6111yescharacteristic
399Q8—24no—
400Q8—24no—
401Q8—42no—
402Q8—42no—
403Q8—2444no—
404D4—1222no—
405D4—1222no—
406D4—2442no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—24no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—1222no—
419C2×C4—1222no—
420C2×C4—1222no—
421C2×C4—1222no—
422C2×C4—1222no—
423C2×C4—1222no—
424C2×C4—1222no—
425C2×C4—1224no—
426C2×C4—2448no—
427C2×C4—2442no—
428C2×C4—2442no—
429C2×C4—2442no—
430C2×C4#1451974512yes—
431C2×C4#1451974512yes—
432C2×C4C2×D4×D6111yescharacteristic
433C2×C4C22×C22⋊D6111yescharacteristic
434C2×C4#41×D6112yes—
435C2×C4#1451982412yes—
436C8—42no—
437C8—2441no—
438C8—2441no—
439C22—4882no—
440C22—4882no—
441C22—2442no—
442C22—2442no—
443C22—2442no—
444C22C2×#1451974511yescharacteristic
445C22#1684042012yes—
446C22#1684056112yes—
447C22#1684057111yescharacteristic
448C22#1684057811yescharacteristic
449C4—22no—
450C4—22no—
451C4—44no—
452C4—81no—
453C4—81no—
454C4—2442no—
455C4—2444no—
456C4—2442no—
457C4—2442no—
458C4#1684031111yescharacteristic
459C4#1684033211yescharacteristic
460C4#1684041112yes—
461C2—4882no—
462C2#2144945011yescharacteristic
463C2#2145506011yescharacteristic
464C2#2145513311yescharacteristic
465C2#2145864112yes—
466C2#2145865612yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#491C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC2×#491D61
hypocenterC61#25656
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
3. #21451208C2semidirect
6. #21446759C2semidirect
7. #21446743C2semidirect
8. #21446085C2semidirect
9. #21446071C2semidirect
11. C2×C61×#491C2semidirect
229. C2×#491D61semidirect
278. C61#25656semidirect

Maximal subgroups

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

subgroupindex
C2×#168404862
C2×#168400592
#214512082
#214470562
#214467592
#214467432
#214460852
#214460712
#214456452
C2×C61×#4912
#2565661