Finite Group Databasebeta

← #29696282

Subgroups — #29696282

Order 15616.

1206 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840478C211yesmaximal, characteristic
2C2×#16839827C211yesmaximal, characteristic, complemented
3#21448766C211yesmaximal, characteristic, complemented
4#21446781C211yesmaximal, characteristic, complemented
5#21446246C212yesmaximal, complemented
6#21446140C212yesmaximal, complemented
7#21445913C212yesmaximal, complemented
8#21445820C211yesmaximal, characteristic, complemented
9C4×#14519697C211yesmaximal, characteristic, complemented
10#21445534C212yesmaximal, complemented
11C61×#1987C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13#16840478—24no—
14#16840478C418yescomplemented
15C2×#14519766—22no—
16C2×#14519752—21no—
17C2×#14519752C2212yes—
18C2×C4×D244C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×#14519735—21no—
21C2×#14519735C2212yes—
22C2×#14519735C2212yes—
23#16839827—24no—
24C2×#14519699C2212yes—
25C2×#14519697C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519689C2212yes—
28C2×C4×#92418C2211yescharacteristic
29#16839664—21no—
30#16839654—22no—
31#16839632C2211yescharacteristic
32#16839619C2212yes—
33C4×C4⋊Dic61—21no—
34#16839525C2212yes—
35#16839524C2211yescharacteristic
36#16839524C2211yescharacteristic
37#16839524C2212yes—
38#16839524C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4⋊C8×C61C2211yescharacteristic
41C4×C4⋊C4×C61C2211yescharacteristic
42C61×#340C2212yes—
43C61×#340C2212yes—
44C22×D244—22no—
45C22×D244D411yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C22×C4×D61C2312yes—
49C22×#92418C2311yescharacteristic
50C22×#92418C2311yescharacteristic
51C2×C8×D61—24no—
52C2×C8×D61—24no—
53C2×C8×D61—42no—
54#14519766—216no—
55#14519752—28no—
56#14519752—24no—
57#14519752C2×C418yes—
58C4×D244—22no—
59C4×D244C2×C414yes—
60C4×D244C2×C414yes—
61C4×D244D414yes—
62#14519735—24no—
63#14519735—28no—
64#14519735—24no—
65#14519735—42no—
66#14519735C2×C418yes—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61D411yescharacteristic
70C2×C4⋊Dic61C2312yes—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758198—22no—
73C2×#11758198—21no—
74C2×#11758198—22no—
75C2×#11758198—22no—
76C2×#11758198—22no—
77C2×#11758198D412yes—
78C2×C4×Dic61—22no—
79C2×C4×Dic61—22no—
80#14519699—42no—
81#14519697—22no—
82#14519697—22no—
83#14519689—24no—
84C4×#92418—22no—
85#14519684—22no—
86C22×Q8×C61—21no—
87C22×C8×C61C2312yes—
88C4⋊Q8×C61—22no—
89C4⋊Q8×C61C2×C414yes—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C4⋊C4×C61C2312yes—
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C42×C61—21no—
95C2×C42×C61C2311yescharacteristic
96C4⋊C8×C61—22no—
97#95×C61—21no—
98C23×D61—21no—
99C23×D61C2×D412yes—
100C2×Q8×C61—24no—
101C2×Q8×C61—81no—
102C2×Q8×C61—81no—
103C22×C4×C61—22no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2411yescharacteristic
109C22×Dic61—21no—
110C22×Dic61—22no—
111C22×Dic61C2×D412yes—
112C2×D244—22no—
113C2×D244—22no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—28no—
120C2×D244—82no—
121C2×D244—82no—
122C2×D244C22⋊C414yes—
123C2×D244C22×C412yes—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—24no—
129C2×C4×D61—42no—
130C2×C4×D61—42no—
131C2×C4×D61C22×C414yes—
132C2×C4×D61C2×D414yes—
133C2×C8×C61—22no—
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C4⋊C4×C61—24no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61C22×C414yes—
142C4⋊C4×C61#4112yes—
143C42×C61—42no—
144C42×C61C22×C412yes—
145C42×C61C2×D412yes—
146#11758198—28no—
147#11758198—24no—
148#11758198—28no—
149#11758198—44no—
150#11758198—82no—
151#11758198C22⋊C418yes—
152C4⋊Dic61—21no—
153C4⋊Dic61—22no—
154C4⋊Dic61—22no—
155C4⋊Dic61—42no—
156C4⋊Dic61—81no—
157C4⋊Dic61C22⋊C414yes—
158C4⋊Dic61C22×C412yes—
159C4×Dic61—28no—
160C4×Dic61—44no—
161C2×#92418—21no—
162C2×#92418—21no—
163C2×#92418—21no—
164C2×#92418—21no—
165C2×#92418—21no—
166C2×#92418#4112yes—
167C8×D61—48no—
168C23×C61C22×D411yescharacteristic
169C22×D61—22no—
170C22×D61—22no—
171C22×D61—22no—
172C22×D61—22no—
173C22×D61—22no—
174C22×D61—22no—
175C22×D61—22no—
176C22×D61—41no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61C2×C22⋊C414yes—
183Q8×C61—84no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—24no—
187C2×C4×C61—22no—
188C2×C4×C61—41no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61C2×C22⋊C412yes—
196C2×C4×C61C2×C22⋊C412yes—
197C2×C4×C61#13512yes—
198C2×C4×C61#13512yes—
199C2×C4×C61C23×C411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C2×#4111yescharacteristic
202C2×C4×C61C2×#4111yescharacteristic
203C2×C4×C61H(5,𝔽2)11yescharacteristic
204C2×C4×C612−1+411yescharacteristic
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61C2×C22⋊C414yes—
214D244—22no—
215D244—22no—
216D244—28no—
217D244—41no—
218D244—41no—
219D244—88no—
220C4×D61—24no—
221C4×D61—24no—
222C4×D61—24no—
223C4×D61—24no—
224C4×D61—48no—
225C8×C61—42no—
226#92418—22no—
227#92418—41no—
228#22489—611nomaximal, Sylow
229C22×C61C22×C22⋊C411yescharacteristic
230C22×C61#54611yescharacteristic
231C22×C61#54711yescharacteristic
232C22×C61C2×#13511yescharacteristic
233C22×C61C2×#13511yescharacteristic
234C22×C61#57611yescharacteristic
235C22×C61#57711yescharacteristic
236C2×D61—24no—
237C2×D61—24no—
238C2×D61—24no—
239C2×D61—42no—
240C2×D61—42no—
241C2×D61—42no—
242C2×D61—42no—
243C2×D61—42no—
244C2×D61—42no—
245C2×D61—42no—
246C2×D61—81no—
247C2×D61—81no—
248C2×D61—81no—
249C2×D61—81no—
250C4×C61—22no—
251C4×C61—44no—
252C4×C61—42no—
253C4×C61—81no—
254C4×C61#41614yes—
255C4×C61C4×#4111yescharacteristic
256C4×C61#52011yescharacteristic
257C4×C61#54611yescharacteristic
258C4×C61#54711yescharacteristic
259Dic61—24no—
260Dic61—42no—
261Dic61—82no—
262C2×#555—611no—
263C2×C8×D4—611no—
264#2354—611no—
265#1990—611no—
266#1987D6111yescharacteristic, complemented
267#1893—612no—
268#1875—612no—
269#1825—612no—
270#1802—611no—
271C4×#102—611no—
272#1735—612no—
273C2×C61#233911yescharacteristic
274C2×C61C2×#41611yescharacteristic
275C2×C61#289711yescharacteristic
276C2×C61#330311yescharacteristic
277C2×C61#330511yescharacteristic
278C2×C61#334011yescharacteristic
279C2×C61#334111yescharacteristic
280D61—44no—
281D61—82no—
282C23×C8—612no—
283#555—618no—
284#555—1224no—
285C2×C4⋊Q8C2×D6111yescharacteristic
286C2×#124—1222no—
287C2×#121—612no—
288C2×#121—1221no—
289C2×C4×D4—611no—
290C2×C4×D4—611no—
291C2×#117—612no—
292C2×#117—612no—
293C2×#117—1221no—
294C8×D4—1224no—
295C2×C4⋊C8C2×D6111yescharacteristic
296C2×#102—611no—
297C2×#102—611no—
298C2×#102—612no—
299C2×C22⋊C8—612no—
300C2×C4×C8—611no—
301#390—1221no—
302#388—1222no—
303#384—611no—
304#380—612no—
305C4×C4⋊C4—1221no—
306C4×C4⋊C4C2×D6111yescharacteristic
307#340—612no—
308#340—611no—
309#340—612no—
310#340—612no—
311#340—611no—
312#340C2×D6112yes—
313#340C2×D6112yes—
314C61#2248911yescharacteristic, complemented, Sylow
315C22×Q8—21no—
316C22×D4—611no—
317C22×D4—611no—
318C22×D4—1222no—
319C23×C4—612no—
320C23×C4—1221no—
321C22×C8—611no—
322C22×C8—611no—
323C22×C8—1224no—
324C22×C8—1224no—
325C22×C8—2442no—
326C22×C8C22×D6112yes—
327C4⋊Q8—22no—
328C4⋊Q8C4×D6114yes—
329#124—12216no—
330#121—618no—
331#121—1224no—
332#121—1228no—
333C4×D4—614no—
334C4×D4—614no—
335C4×D4—614no—
336C4×D4—1222no—
337#117—618no—
338#117—1228no—
339#117—1224no—
340#117—1224no—
341#117—2442no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—611no—
344C2×C4⋊C4—611no—
345C2×C4⋊C4—612no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4C22×D6111yescharacteristic
349C2×C4⋊C4C22×D6112yes—
350C2×C4⋊C4C22×D6111yescharacteristic
351C2×C22⋊C4—612no—
352C2×C22⋊C4—1222no—
353C2×C22⋊C4—1222no—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1222no—
356C2×C22⋊C4—1222no—
357C2×C42—21no—
358C2×C42—1222no—
359C2×C42—1222no—
360C2×C42C22×D6111yescharacteristic
361C4⋊C8—22no—
362#102—1222no—
363#102—1222no—
364#102—2442no—
365C22⋊C8—1224no—
366C4×C8—1222no—
367#95—21no—
368#95—1222no—
369C24—612no—
370C24—1221no—
371C2×Q8—24no—
372C2×Q8—81no—
373C2×Q8—81no—
374C2×D4—612no—
375C2×D4—614no—
376C2×D4—1228no—
377C2×D4—1221no—
378C2×D4—1221no—
379C2×D4—1221no—
380C2×D4—1221no—
381C2×D4—1221no—
382C2×D4—1222no—
383C2×D4—1222no—
384C2×D4—4882no—
385C2×D4—4882no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—612no—
390C22×C4—614no—
391C22×C4—614no—
392C22×C4—1222no—
393C22×C4—1222no—
394C22×C4—1222no—
395C22×C4—1222no—
396C22×C4—1221no—
397C22×C4—1224no—
398C22×C4—1222no—
399C22×C4—2442no—
400C22×C4—2442no—
401C22×C4D4×D6111yescharacteristic
402C22×C4D4×D6111yescharacteristic
403C22×C4C23×D6111yescharacteristic
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—612no—
408C2×C8—1221no—
409C2×C8—1221no—
410C2×C8—1221no—
411C2×C8—1221no—
412C2×C8—1221no—
413C2×C8—2448no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—24no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—614no—
420C4⋊C4—612no—
421C4⋊C4—1221no—
422C4⋊C4—1222no—
423C4⋊C4—1222no—
424C4⋊C4—4881no—
425C4⋊C4—2442no—
426C4⋊C4C2×C4×D6114yes—
427C4⋊C4#1175822612yes—
428C22⋊C4—618no—
429C22⋊C4—1228no—
430C22⋊C4—1228no—
431C22⋊C4—1224no—
432C22⋊C4—4882no—
433C22⋊C4—2444no—
434C42—42no—
435C42—1228no—
436C42—2444no—
437C42C2×C4×D6112yes—
438C42D4×D6112yes—
439C23—614no—
440C23—1222no—
441C23—1222no—
442C23—1222no—
443C23—1222no—
444C23—1222no—
445C23—1222no—
446C23—1222no—
447C23—2441no—
448C23—2441no—
449C23—2441no—
450C23—2441no—
451C23—2441no—
452C23—2441no—
453C23C2×D4×D6111yescharacteristic
454Q8—84no—
455D4—1222no—
456D4—1222no—
457D4—1228no—
458D4—4888no—
459D4—2441no—
460D4—2441no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—614no—
473C2×C4—1222no—
474C2×C4—1222no—
475C2×C4—1224no—
476C2×C4—1224no—
477C2×C4—1224no—
478C2×C4—1224no—
479C2×C4—1222no—
480C2×C4—2442no—
481C2×C4—2442no—
482C2×C4—2442no—
483C2×C4—2441no—
484C2×C4—2441no—
485C2×C4—2448no—
486C2×C4C22⋊C4×D6112yes—
487C2×C4C22⋊C4×D6112yes—
488C2×C4#1451979812yes—
489C2×C4#1451980012yes—
490C2×C4C22×C4×D6111yescharacteristic
491C2×C4C2×D4×D6111yescharacteristic
492C2×C4#41×D6111yescharacteristic
493C2×C4C2×#1175822611yescharacteristic
494C2×C4#1451982311yescharacteristic
495C2×C4#1451982511yescharacteristic
496C8—42no—
497C8—1222no—
498C8—2441no—
499C22—1224no—
500C22—1224no—
501C22—1224no—
502C22—4881no—
503C22—4881no—
504C22—4881no—
505C22—4881no—
506C22—2442no—
507C22—2442no—
508C22—2442no—
509C22—2442no—
510C22—2442no—
511C22—2442no—
512C22—2442no—
513C22C2×C22⋊C4×D6111yescharacteristic
514C22#1684040311yescharacteristic
515C22#1684041411yescharacteristic
516C22C2×#1451979811yescharacteristic
517C22C2×#1451980011yescharacteristic
518C22#1684056611yescharacteristic
519C22#1684057211yescharacteristic
520C4—22no—
521C4—44no—
522C4—42no—
523C4—81no—
524C4—1224no—
525C4—4882no—
526C4—2442no—
527C4#1683975714yes—
528C4C4×#1175822611yescharacteristic
529C4#1684024411yescharacteristic
530C4#1684040511yescharacteristic
531C4#1684041511yescharacteristic
532C2—4882no—
533C2—2444no—
534C2#2144864811yescharacteristic
535C2C2×#1683975711yescharacteristic
536C2#2145434411yescharacteristic
537C2#2145774411yescharacteristic
538C2#2145776611yescharacteristic
539C2#2145823511yescharacteristic
540C2#2145824011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1987C2
socleC23×C61C22×D4
prime radicalC2×#16840478C2
cosocleC22×C4×C61C24
hypercenter#1987D61
hypocenterC61#22489
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
2. C2×#16839827C2semidirect
3. #21448766C2semidirect
4. #21446781C2semidirect
5. #21446246C2semidirect
6. #21446140C2semidirect
7. #21445913C2semidirect
8. #21445820C2semidirect
9. C4×#14519697C2semidirect
10. #21445534C2semidirect
11. C61×#1987C2semidirect
14. #16840478C4semidirect
266. #1987D61semidirect
314. C61#22489semidirect

Maximal subgroups

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

subgroupindex
C2×#168404782
C2×#168398272
#214487662
#214467812
#214462462
#214461402
#214459132
#214458202
C4×#145196972
#214455342
C61×#19872
#2248961