Finite Group Databasebeta

← #29713318

Subgroups — #29713318

Order 15616.

1072 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2C2×#16840101C212yesmaximal
3#21448300C211yesmaximal, characteristic
4#21447153C211yesmaximal, characteristic, complemented
5#21447147C211yesmaximal, characteristic, complemented
6#21447062C211yesmaximal, characteristic, complemented
7#21446128C212yesmaximal
8#21446089C212yesmaximal, complemented
9#21446071C212yesmaximal, complemented
10#21445886C211yesmaximal, characteristic
11C61×#2043C211yesmaximal, characteristic, complemented
12#16840317—28no—
13C2×#14519777C2211yescharacteristic
14C2×#14519768—21no—
15C2×#14519757—21no—
16C2×#14519753C2212yes—
17C2×Q8×Dic61C2212yes—
18C2×#14519742—21no—
19C2×#14519735—21no—
20C2×#14519735C2211yescharacteristic
21C2×#14519735C2211yescharacteristic
22C2×C42×D61C2211yescharacteristic
23#16840101—28no—
24C2×#14519710C2212yes—
25C2×#14519709C2212yes—
26C2×#14519701C2212yes—
27C2×#14519701C2212yes—
28C2×#14519689C2212yes—
29#16839661C2211yescharacteristic
30#16839660C2211yescharacteristic
31#16839658C2211yescharacteristic
32#16839636—21no—
33#16839633—21no—
34C4⋊C4×Dic61—21no—
35#16839613—21no—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839517C2211yescharacteristic
39#16839516C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×#103×C61C2212yes—
42C2×#103×C61C2212yes—
43C61×#389C2211yescharacteristic
44C61×#336C2211yescharacteristic
45C22×Dic122—21no—
46C22×C4×D61D411yescharacteristic
47C22×C4×D61D411yescharacteristic
48C22×C4×D61C2311yescharacteristic
49C22×#92418C2312yes—
50#14519777—24no—
51#14519768—28no—
52#14519757—28no—
53#14519753—24no—
54#14519753D418yes—
55Q8×Dic61—24no—
56Q8×Dic61—24no—
57#14519742—28no—
58#14519735—28no—
59#14519735—22no—
60#14519735—22no—
61#14519735—24no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—22no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2312yes—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×#11758196—22no—
69C2×#11758196—21no—
70C2×#11758196—21no—
71C2×#11758198—22no—
72C2×#11758198—21no—
73C2×#11758198—21no—
74C2×C4×Dic61—22no—
75C2×C4×Dic61—21no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61D411yescharacteristic
78C2×C4×Dic61D411yescharacteristic
79C2×C4×Dic61C2311yescharacteristic
80C42×D61—24no—
81#14519710—24no—
82#14519709—24no—
83#14519701—24no—
84#14519701—24no—
85#14519689—24no—
86#14519684—21no—
87#14519684—21no—
88#14519684—21no—
89C22×Q8×C61C2312yes—
90C22×C8×C61C2312yes—
91C4⋊Q8×C61—24no—
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#95×C61—21no—
101C23×D61C2×D411yescharacteristic
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—21no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2411yescharacteristic
114C22×Dic61—22no—
115C22×Dic61—21no—
116C22×Dic61—21no—
117C22×Dic61C2×D411yescharacteristic
118C22×Dic61C2×D411yescharacteristic
119C22×Dic61C2×D411yescharacteristic
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—21no—
123C2×C4×D61—21no—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—41no—
129C2×C4×D61C2×D414yes—
130C2×Dic122—24no—
131C2×Dic122—42no—
132C2×Dic122—42no—
133C2×C8×C61—22no—
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C4⋊C4×C61—24no—
137C4⋊C4×C61—24no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—42no—
141C4⋊C4×C61#4112yes—
142C42×C61—22no—
143#11758198—24no—
144#11758198—28no—
145#11758198—42no—
146C4⋊Dic61—24no—
147C4⋊Dic61—28no—
148C4⋊Dic61—22no—
149C4⋊Dic61—21no—
150C4⋊Dic61—22no—
151C4⋊Dic61—42no—
152C4⋊Dic61C2×D414yes—
153C4⋊Dic61#4112yes—
154#11758196—28no—
155#11758196—42no—
156#11758196—42no—
157C4×Dic61—24no—
158C4×Dic61—28no—
159C4×Dic61—22no—
160C4×Dic61—22no—
161C4×Dic61—22no—
162C4×Dic61—42no—
163C2×#92418—22no—
164C2×#92418—22no—
165C2×#92418—22no—
166C23×C61C22×D411yescharacteristic
167C22×D61—21no—
168C22×D61—21no—
169C22×D61—21no—
170C22×D61—21no—
171C22×D61—21no—
172C22×D61—21no—
173C22×D61#12012yes—
174Q8×C61—24no—
175Q8×C61—24no—
176Q8×C61—42no—
177Q8×C61—42no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—24no—
183C2×C4×C61—22no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61#13512yes—
192C2×C4×C61#13712yes—
193C2×C4×C61C22×D412yes—
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C2×C4×C612−1+412yes—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—24no—
203C2×Dic61—22no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—42no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—42no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61#12012yes—
214C2×Dic61#13712yes—
215C4×D61—28no—
216C4×D61—24no—
217C4×D61—24no—
218C4×D61—44no—
219Dic122—48no—
220C8×C61—42no—
221#92418—42no—
222#24065—611nomaximal, Sylow
223C22×C61C2×#12011yescharacteristic
224C22×C61#52511yescharacteristic
225C22×C61#53611yescharacteristic
226C22×C61C2×#13511yescharacteristic
227C22×C61C2×#13711yescharacteristic
228C22×C61C2×#13711yescharacteristic
229C22×C61#57511yescharacteristic
230C2×D61—22no—
231C2×D61—22no—
232C2×D61—22no—
233C2×D61—41no—
234C2×D61—41no—
235C2×D61—41no—
236C2×D61—41no—
237C4×C61—22no—
238C4×C61—22no—
239C4×C61—42no—
240C4×C61—44no—
241C4×C61—81no—
242C4×C61#54712yes—
243C4×C61#54712yes—
244Dic61—22no—
245Dic61—22no—
246Dic61—42no—
247Dic61—44no—
248Dic61—41no—
249Dic61—81no—
250C2×#533—611no—
251C2×#485—612no—
252#2296—611no—
253#2058—611no—
254#2057—611no—
255#2043—611no—
256#2043D6111yescharacteristic, complemented
257#1873—612no—
258#1865—612no—
259#1863—612no—
260#1817—611no—
261C2×C61#240611yescharacteristic
262C2×C61#300211yescharacteristic
263C2×C61#300411yescharacteristic
264C2×C61#310711yescharacteristic
265C2×C61#311611yescharacteristic
266C2×C61#323111yescharacteristic
267C2×C61#323711yescharacteristic
268D61—42no—
269#533—1228no—
270C2×C4⋊Q8—611no—
271C2×C4⋊Q8C2×D6111yescharacteristic
272C2×#125—1221no—
273C2×C22⋊Q8—612no—
274C2×C22⋊Q8—1221no—
275C2×C4×Q8—612no—
276C2×C4×Q8—1221no—
277C2×#117—611no—
278C2×#117—611no—
279C2×#117—1221no—
280C22×C42—611no—
281#485—1228no—
282C2×#107—612no—
283C2×#106—612no—
284C2×#103—612no—
285C2×#103—612no—
286C2×#103C2×D6112yes—
287C2×#103C2×D6112yes—
288C2×C22⋊C8—612no—
289#389—611no—
290#389—611no—
291#389—611no—
292#389C2×D6111yescharacteristic
293#385—1221no—
294#385—1221no—
295C4×C4⋊C4—1221no—
296C4×C4⋊C4—1221no—
297#340—612no—
298#340—612no—
299#336—612no—
300#336—611no—
301#336C2×D6111yescharacteristic
302C61#2406511yescharacteristic, complemented, Sylow
303C22×Q8—1221no—
304C22×Q8C22×D6112yes—
305C23×C4—611no—
306C23×C4—611no—
307C23×C4—611no—
308C22×C8—612no—
309C22×C8C22×D6112yes—
310C4⋊Q8—24no—
311C4⋊Q8—1224no—
312#125—1228no—
313C22⋊Q8—618no—
314C22⋊Q8—1228no—
315C22⋊Q8—1224no—
316C4×Q8—1228no—
317C4×Q8—1224no—
318C4×Q8—1224no—
319#117—1222no—
320#117—1228no—
321#117—1222no—
322#117—1224no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—611no—
327C2×C4⋊C4—612no—
328C2×C4⋊C4—611no—
329C2×C4⋊C4—1222no—
330C2×C4⋊C4—1222no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4—1221no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4C22×D6111yescharacteristic
336C2×C4⋊C4C22×D6111yescharacteristic
337C2×C22⋊C4—1221no—
338C2×C22⋊C4—1222no—
339C2×C22⋊C4—1221no—
340C2×C42—611no—
341C2×C42—611no—
342C2×C42—611no—
343C2×C42—1224no—
344C2×C42—1221no—
345C2×C42—1222no—
346C2×C42—1221no—
347C2×C42C22×D6111yescharacteristic
348#107—1224no—
349#106—1224no—
350#103—24no—
351#103—24no—
352#103—1224no—
353#103—1224no—
354C22⋊C8—1224no—
355#95—21no—
356#95—1221no—
357#95—1221no—
358#95—1221no—
359C24—611no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—1224no—
366C2×Q8—2442no—
367C2×Q8—2442no—
368C2×Q8D4×D6114yes—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—611no—
373C22×C4—611no—
374C22×C4—614no—
375C22×C4—611no—
376C22×C4—1222no—
377C22×C4—1222no—
378C22×C4—1222no—
379C22×C4—1222no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4—1222no—
383C22×C4—1221no—
384C22×C4—1222no—
385C22×C4—1222no—
386C22×C4—1221no—
387C22×C4—2441no—
388C22×C4D4×D6111yescharacteristic
389C22×C4D4×D6111yescharacteristic
390C22×C4C23×D6111yescharacteristic
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—1222no—
395C2×C8—1222no—
396C2×C8—1222no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—42no—
402C4⋊C4—614no—
403C4⋊C4—612no—
404C4⋊C4—1228no—
405C4⋊C4—1224no—
406C4⋊C4—1228no—
407C4⋊C4—1222no—
408C4⋊C4—1221no—
409C4⋊C4—1222no—
410C4⋊C4—2442no—
411C4⋊C4—2442no—
412C4⋊C4—2442no—
413C4⋊C4#1175822412yes—
414C22⋊C4—1228no—
415C22⋊C4—1224no—
416C22⋊C4—2442no—
417C42—22no—
418C42—1222no—
419C42—1222no—
420C42—1224no—
421C42—1228no—
422C42—1222no—
423C42—2442no—
424C23—612no—
425C23—1221no—
426C23—1221no—
427C23—1221no—
428C23—1221no—
429C23—1221no—
430C23—1221no—
431C23C2×D4×D6111yescharacteristic
432Q8—24no—
433Q8—24no—
434Q8—42no—
435Q8—42no—
436Q8—2448no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—612no—
451C2×C4—612no—
452C2×C4—1228no—
453C2×C4—1221no—
454C2×C4—1221no—
455C2×C4—1221no—
456C2×C4—1221no—
457C2×C4—1224no—
458C2×C4—1224no—
459C2×C4—1221no—
460C2×C4—1224no—
461C2×C4—1222no—
462C2×C4—1221no—
463C2×C4—1221no—
464C2×C4—2444no—
465C2×C4—2441no—
466C2×C4—2441no—
467C2×C4—2441no—
468C2×C4—2442no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4—2442no—
472C2×C4#1451979812yes—
473C2×C4#1451980912yes—
474C2×C4C2×D4×D6112yes—
475C2×C4#41×D6111yescharacteristic
476C2×C4C2×#1175822411yescharacteristic
477C2×C4#1451982412yes—
478C8—42no—
479C8—2442no—
480C22—1222no—
481C22—1222no—
482C22—1222no—
483C22—2441no—
484C22—2441no—
485C22—2441no—
486C22—2441no—
487C22#120×D6111yescharacteristic
488C22#1684027211yescharacteristic
489C22#1684033211yescharacteristic
490C22C2×#1451979811yescharacteristic
491C22#137×D6111yescharacteristic
492C22C2×#1451980911yescharacteristic
493C22#1684056111yescharacteristic
494C4—22no—
495C4—22no—
496C4—42no—
497C4—44no—
498C4—81no—
499C4—1222no—
500C4—1222no—
501C4—4881no—
502C4—2442no—
503C4—2441no—
504C4—2444no—
505C4#1684041012yes—
506C4#1684041212yes—
507C2—2442no—
508C2#2144914911yescharacteristic
509C2#2145503611yescharacteristic
510C2#2145505411yescharacteristic
511C2#2145588711yescharacteristic
512C2#2145598011yescharacteristic
513C2#2145697311yescharacteristic
514C2#2145702811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2043C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2043D61
hypocenterC61#24065
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2043

Complements

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

NH ≅ G/Ntype
4. #21447153C2semidirect
5. #21447147C2semidirect
6. #21447062C2semidirect
8. #21446089C2semidirect
9. #21446071C2semidirect
11. C61×#2043C2semidirect
256. #2043D61semidirect
302. C61#24065semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168401012
#214483002
#214471532
#214471472
#214470622
#214461282
#214460892
#214460712
#214458862
C61×#20432
#2406561