Finite Group Databasebeta

← #29734172

Subgroups — #29734172

Order 15616.

1048 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840485C211yesmaximal, characteristic
2C2×#16840124C211yesmaximal, characteristic
3C2×#16840054C211yesmaximal, characteristic, complemented
4#21447065C212yesmaximal, complemented
5#21447060C212yesmaximal
6#21446760C211yesmaximal, characteristic, complemented
7#21446751C211yesmaximal, characteristic
8#21446072C212yesmaximal, complemented
9#21446065C212yesmaximal
10#21445652C211yesmaximal, characteristic, complemented
11C61×#2546C211yesmaximal, characteristic, complemented
12#16840485—24no—
13#16840485—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519772—21no—
16C2×#14519772—21no—
17C2×#14519765C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×#14519753—21no—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×#14519735—21no—
24#16840124—24no—
25#16840054—24no—
26C2×#14519710C2212yes—
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519700C2212yes—
29C2×#14519700C2212yes—
30C2×#14519700C2212yes—
31C2×#14519700C2212yes—
32C2×#14519697C2212yes—
33C2×#14519697C2212yes—
34C2×#14519688C2211yescharacteristic
35#16839524C2212yes—
36#16839517C2212yes—
37#16839517C2212yes—
38C2×C4⋊Q8×C61—21no—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4×Q8×C61C2211yescharacteristic
41C61×#398—21no—
42C61×#398—21no—
43C61×#391—21no—
44C61×#389C2212yes—
45C61×#389C2212yes—
46C4×C4⋊C4×C61C2211yescharacteristic
47C22×Dic122C2311yescharacteristic
48C22×Dic122C2311yescharacteristic
49C22×D244C2311yescharacteristic
50C22×C4×D61—21no—
51C22×#92418C2312yes—
52C22×#92418C2312yes—
53#14519777—22no—
54#14519777—22no—
55#14519772—28no—
56#14519772—28no—
57#14519765—22no—
58#14519765—22no—
59#14519765—22no—
60#14519765D414yes—
61#14519753—28no—
62C4×Dic122—22no—
63C4×Dic122—22no—
64C4×Dic122—22no—
65C4×Dic122D414yes—
66C4×D244—22no—
67C4×D244—22no—
68#14519735—28no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81#14519710—24no—
82C4⋊#92418—22no—
83#14519700—24no—
84#14519700—24no—
85#14519700—24no—
86#14519700—24no—
87#14519697—24no—
88#14519697—24no—
89#14519688—22no—
90C22×Q8×C61—21no—
91C22×Q8×C61D411yescharacteristic
92C4⋊Q8×C61—22no—
93C4⋊Q8×C61—22no—
94C4⋊Q8×C61—44no—
95C4×Q8×C61—22no—
96C4×Q8×C61—22no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61—22no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C4⋊C4×C61C2312yes—
104C2×C4⋊C4×C61C2312yes—
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C2×C42×C61C2311yescharacteristic
108C2×C42×C61C2311yescharacteristic
109#95×C61—21no—
110#95×C61—22no—
111#95×C61—22no—
112C23×D61—21no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—41no—
116C2×Q8×C61—42no—
117C2×Q8×C61—42no—
118C2×Q8×C61—42no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—21no—
134C2×D244C2×D412yes—
135C2×C4×D61—24no—
136C2×C4×D61—42no—
137C2×C4×D61—42no—
138C2×Dic122—22no—
139C2×Dic122—22no—
140C2×Dic122—22no—
141C2×Dic122—22no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122C2×D412yes—
145C2×Dic122C2×D412yes—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—44no—
153C4⋊C4×C61—42no—
154C4⋊C4×C61—44no—
155C4⋊C4×C61#4114yes—
156C42×C61—21no—
157C42×C61—21no—
158C42×C61—21no—
159C42×C61—21no—
160C42×C61—42no—
161C42×C61C2×D412yes—
162C42×C61C2×D412yes—
163#11758198—24no—
164#11758198—24no—
165#11758198—24no—
166#11758198—24no—
167#11758198—24no—
168C4⋊Dic61—24no—
169C4⋊Dic61—24no—
170C4⋊Dic61—21no—
171C4⋊Dic61C2×D412yes—
172#11758196—24no—
173#11758196—24no—
174C4×Dic61—24no—
175C4×Dic61—24no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C23×C61C22×D411yescharacteristic
183C22×D61—22no—
184C22×D61—42no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—42no—
188Q8×C61—44no—
189Q8×C61—84no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—22no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—24no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61#12012yes—
209C2×C4×C61#12012yes—
210C2×C4×C61AGL(1,ℤ/8)12yes—
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C2×#4112yes—
214C2×C4×C612−1+412yes—
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222D244—24no—
223D244—41no—
224D244—41no—
225C4×D61—48no—
226Dic122—22no—
227Dic122—22no—
228Dic122—24no—
229Dic122—41no—
230Dic122—41no—
231Dic122—42no—
232#92418—42no—
233#92418—42no—
234#25661—611nomaximal, Sylow
235C22×C61C2×#12011yescharacteristic
236C22×C61#54812yes—
237C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
238C22×C61#57511yescharacteristic
239C22×C61#57812yes—
240C2×D61—42no—
241C2×D61—42no—
242C2×D61—42no—
243C2×D61—82no—
244C2×D61—82no—
245C4×C61—22no—
246C4×C61—22no—
247C4×C61—22no—
248C4×C61—44no—
249C4×C61—41no—
250C4×C61—41no—
251C4×C61—41no—
252C4×C61—82no—
253C4×C61#53211yescharacteristic
254C4×C61#53611yescharacteristic
255C4×C61#54712yes—
256Dic61—42no—
257Dic61—42no—
258Dic61—42no—
259C2×#556—611no—
260C2×#489—611no—
261C2×#477—611no—
262#2546D6111yescharacteristic, complemented
263#2043—612no—
264#2042—612no—
265#1985—611no—
266#1983—611no—
267#1863—612no—
268#1862—612no—
269#1763—611no—
270C2×C61#243211yescharacteristic
271C2×C61#300611yescharacteristic
272C2×C61#301111yescharacteristic
273C2×C61#336712yes—
274C2×C61#337712yes—
275D61—82no—
276#556—1224no—
277#556—1224no—
278C2×C4⋊Q8—21no—
279C2×C4⋊Q8—611no—
280C2×C4⋊Q8C2×D6111yescharacteristic
281C2×#126—1221no—
282C2×#126—1221no—
283C2×#124—611no—
284C2×#124—611no—
285C2×C22⋊Q8—1221no—
286C2×C4×Q8—611no—
287C2×C4×Q8—611no—
288C2×C4×Q8C2×D6111yescharacteristic
289C2×C4×D4—611no—
290C2×#117—1221no—
291#489—1224no—
292#477—1224no—
293C2×#107—612no—
294C2×C4⋊C8—611no—
295C2×#103—612no—
296C2×#103—612no—
297C2×#103—612no—
298C2×#103—612no—
299C2×#102—612no—
300C2×#102—612no—
301C2×#97—611no—
302#398—21no—
303#398—21no—
304#391—21no—
305#389C2×D6112yes—
306#389C2×D6112yes—
307C4×C4⋊C4C2×D6111yescharacteristic
308#340—612no—
309#336—612no—
310#336—612no—
311C61#2566111yescharacteristic, complemented, Sylow
312C22×Q8—21no—
313C22×Q8—611no—
314C22×Q8—611no—
315C22×Q8C22⋊D6111yescharacteristic
316C22×D4—611no—
317C23×C4—1221no—
318C22×C8—612no—
319C22×C8—612no—
320C4⋊Q8—22no—
321C4⋊Q8—22no—
322C4⋊Q8—44no—
323C4⋊Q8—1222no—
324C4⋊Q8—1222no—
325#126—1228no—
326#126—1228no—
327#124—614no—
328#124—1222no—
329#124—1222no—
330#124—1222no—
331C22⋊Q8—1228no—
332C4×Q8—22no—
333C4×Q8—22no—
334C4×Q8—614no—
335C4×Q8—1222no—
336C4×Q8—1222no—
337C4×Q8—1222no—
338C4×D4—1222no—
339C4×D4—1222no—
340#117—1228no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—611no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4C22⋊D6111yescharacteristic
350C2×C4⋊C4C22⋊D6111yescharacteristic
351C2×C4⋊C4C22⋊D6111yescharacteristic
352C2×C4⋊C4C22×D6112yes—
353C2×C4⋊C4C22×D6112yes—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1221no—
357C2×C22⋊C4—1221no—
358C2×C22⋊C4—1221no—
359C2×C42—21no—
360C2×C42—1221no—
361C2×C42—1221no—
362C2×C42C22×D6111yescharacteristic
363C2×C42C22×D6111yescharacteristic
364C2×C42C22×D6111yescharacteristic
365#107—1224no—
366C4⋊C8—1222no—
367#103—1224no—
368#103—1224no—
369#103—1224no—
370#103—1224no—
371#102—1224no—
372#102—1224no—
373#97—1222no—
374#95—21no—
375#95—22no—
376#95—22no—
377C24—1221no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—41no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—612no—
385C2×Q8—612no—
386C2×Q8—1221no—
387C2×Q8—1221no—
388C2×Q8—1222no—
389C2×Q8—1222no—
390C2×Q8—1222no—
391C2×Q8—1222no—
392C2×D4—612no—
393C2×D4—1221no—
394C2×D4—1222no—
395C2×D4—1222no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—1224no—
399C22×C4—1221no—
400C22×C4—1221no—
401C22×C4—1221no—
402C22×C4—2442no—
403C22×C4—2442no—
404C22×C4C2×C22⋊D6111yescharacteristic
405C22×C4C2×C22⋊D6111yescharacteristic
406C22×C4C2×C22⋊D6111yescharacteristic
407C22×C4C2×C22⋊D6111yescharacteristic
408C22×C4C2×C22⋊D6111yescharacteristic
409C22×C4C2×C22⋊D6111yescharacteristic
410C22×C4C23×D6111yescharacteristic
411C2×C8—1222no—
412C2×C8—1222no—
413C2×C8—1222no—
414C2×C8—1222no—
415C2×C8—1222no—
416C2×C8—1222no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—44no—
424C4⋊C4—42no—
425C4⋊C4—44no—
426C4⋊C4—612no—
427C4⋊C4—1224no—
428C4⋊C4—1224no—
429C4⋊C4—1224no—
430C4⋊C4—1221no—
431C4⋊C4—1224no—
432C4⋊C4#1175822614yes—
433C22⋊C4—1224no—
434C22⋊C4—1224no—
435C22⋊C4—1224no—
436C22⋊C4—1224no—
437C22⋊C4—1224no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—42no—
443C42—1224no—
444C42—1224no—
445C42D4×D6112yes—
446C42D4×D6112yes—
447C23—1222no—
448C23—2441no—
449C23—2441no—
450C23—2442no—
451C23—2442no—
452C23C22×C22⋊D6111yescharacteristic
453Q8—44no—
454Q8—84no—
455Q8—1222no—
456Q8—1222no—
457Q8—1224no—
458Q8—2441no—
459Q8—2441no—
460Q8—2442no—
461D4—1224no—
462D4—2441no—
463D4—2441no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—1222no—
483C2×C4—1222no—
484C2×C4—1222no—
485C2×C4—2448no—
486C2×C4—2441no—
487C2×C4—2441no—
488C2×C4—2442no—
489C2×C4—2442no—
490C2×C4#1451974512yes—
491C2×C4#1451974512yes—
492C2×C4#1451980312yes—
493C2×C4C2×D4×D6111yescharacteristic
494C2×C4C2×D4×D6111yescharacteristic
495C2×C4C2×#1175822612yes—
496C2×C4#1451982512yes—
497C8—2442no—
498C8—2442no—
499C22—4882no—
500C22—4882no—
501C22—2442no—
502C22—2442no—
503C22—2442no—
504C22C2×#1451974511yescharacteristic
505C22#1684042312yes—
506C22C2×#1451980311yescharacteristic
507C22#1684056011yescharacteristic
508C22#1684057712yes—
509C4—22no—
510C4—22no—
511C4—22no—
512C4—41no—
513C4—41no—
514C4—44no—
515C4—41no—
516C4—82no—
517C4—2442no—
518C4—2442no—
519C4—2442no—
520C4#1684031311yescharacteristic
521C4#1684033711yescharacteristic
522C4#1684041512yes—
523C2—4882no—
524C2#2144944411yescharacteristic
525C2#2145506911yescharacteristic
526C2#2145511211yescharacteristic
527C2#2145853712yes—
528C2#2145866112yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2546C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2546D61
hypocenterC61#25661
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2546

Complements

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

NH ≅ G/Ntype
3. C2×#16840054C2semidirect
4. #21447065C2semidirect
6. #21446760C2semidirect
8. #21446072C2semidirect
10. #21445652C2semidirect
11. C61×#2546C2semidirect
262. #2546D61semidirect
311. C61#25661semidirect

Maximal subgroups

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

subgroupindex
C2×#168404852
C2×#168401242
C2×#168400542
#214470652
#214470602
#214467602
#214467512
#214460722
#214460652
#214456522
C61×#25462
#2566161