Finite Group Databasebeta

← #29707333

Subgroups — #29707333

Order 15616.

1266 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×#16840317C211yesmaximal, characteristic
2C2×#16839769C212yesmaximal
3#391×D61C211yesmaximal, characteristic
4#21446969C212yesmaximal, complemented
5#21446968C212yesmaximal, complemented
6#21446961C211yesmaximal, characteristic, complemented
7#21446960C211yesmaximal, characteristic, complemented
8#21446957C211yesmaximal, characteristic, complemented
9#21445537C212yesmaximal
10#21445511C211yesmaximal, characteristic
11C61×#2024C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C22×#11758189C2212yes—
14#16840317—28no—
15C2×#14519777C2211yescharacteristic
16C2×#14519768C2212yes—
17C2×#14519757—21no—
18C2×#14519753—21no—
19C2×Q8×Dic61—21no—
20C2×#14519742—21no—
21C2×#14519735C2212yes—
22C2×C4⋊C4×D61C2211yescharacteristic
23C2×C42×D61C2211yescharacteristic
24#16839769—28no—
25C2×#14519702C2212yes—
26C2×#14519701C2212yes—
27C2×#14519700C2212yes—
28#16839669C2211yescharacteristic
29#16839668C2211yescharacteristic
30#16839667C2211yescharacteristic
31#16839666—21no—
32#16839666—21no—
33#95×D61—21no—
34#95×D61—21no—
35#16839525C2212yes—
36#16839524C2212yes—
37#16839524C2212yes—
38#16839517C2211yescharacteristic
39#16839516C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×#103×C61C2212yes—
42C61×#391C2211yescharacteristic
43C61×#340C2212yes—
44C61×#336C2211yescharacteristic
45C22×Dic122—21no—
46C22×Dic122—21no—
47C22×Dic122C2311yescharacteristic
48C2×Q8×D61—22no—
49C2×Q8×D61—24no—
50C2×Q8×D61—24no—
51C2×Q8×D61Q814yescomplemented
52C22×C4×D61—21no—
53C22×C4×D61—21no—
54C22×C4×D61D411yescharacteristic
55C22×C4×D61D411yescharacteristic
56C22×C4×D61C2311yescharacteristic
57C2×#11758189—24no—
58C2×#11758189—24no—
59C2×#11758189—24no—
60C22×#92418C2312yes—
61#14519777—24no—
62#14519768—24no—
63#14519768—24no—
64#14519757—28no—
65#14519753—28no—
66Q8×Dic61—28no—
67#14519742—28no—
68#14519735—24no—
69#14519735D418yes—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2312yes—
74C2×#11758196—22no—
75C2×#11758196—21no—
76C4⋊C4×D61—24no—
77C2×#11758198—22no—
78C2×C4×Dic61—22no—
79C2×C4×Dic61D411yescharacteristic
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61C2311yescharacteristic
82C42×D61—24no—
83#14519702—24no—
84#14519701—24no—
85#14519700—24no—
86#14519684—21no—
87#14519684—21no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92C22×Q8×C61—21no—
93C22×Q8×C61C2311yescharacteristic
94C22×C8×C61C2312yes—
95C4⋊Q8×C61—24no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C4⋊C4×C61C2312yes—
99C2×C42×C61C2311yescharacteristic
100#103×C61—24no—
101#95×C61—21no—
102#95×C61—21no—
103C23×D61C2×D411yescharacteristic
104C2×Q8×C61—24no—
105C2×Q8×C61—21no—
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61—42no—
109C2×Q8×C61—42no—
110C2×Q8×C61C2×Q812yes—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61—21no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2411yescharacteristic
117C22×Dic61—21no—
118C22×Dic61—22no—
119C22×Dic61—21no—
120C22×Dic61C2×D411yescharacteristic
121C22×Dic61C2×D411yescharacteristic
122C22×Dic61C2×D411yescharacteristic
123Q8×D61—216no—
124Q8×D61—48no—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—24no—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—41no—
131C2×C4×D61—41no—
132C2×C4×D61—42no—
133C2×C4×D61—42no—
134C2×C4×D61—42no—
135C2×C4×D61—42no—
136C2×C4×D61—42no—
137C2×C4×D61C2×D414yes—
138C2×C4×D61C2×Q812yes—
139C2×C4×D61#4114yes—
140C2×C4×D61#4112yes—
141C2×Dic122—22no—
142C2×Dic122—22no—
143C2×Dic122—24no—
144C2×Dic122—21no—
145C2×Dic122—24no—
146C2×Dic122—42no—
147C2×Dic122—42no—
148C2×Dic122—42no—
149C2×Dic122—42no—
150C2×Dic122C2×Q812yes—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C4⋊C4×C61—24no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61C2×D414yes—
158C42×C61—22no—
159#11758198—28no—
160C4⋊Dic61—24no—
161C4⋊Dic61—28no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61C2×D414yes—
165#11758196—28no—
166#11758196—42no—
167C4×Dic61—22no—
168C4×Dic61—22no—
169C4×Dic61—22no—
170C4×Dic61—28no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418—22no—
174#11758189—48no—
175C23×C61C22×D411yescharacteristic
176C22×D61—21no—
177C22×D61—21no—
178C22×D61—21no—
179C22×D61—21no—
180C22×D61#12012yes—
181C22×D61C22⋊Q812yes—
182C22×D61#12412yes—
183Q8×C61—24no—
184Q8×C61—48no—
185Q8×C61—42no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—24no—
191C2×C4×C61—22no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61#13712yes—
200C2×C4×C61#13712yes—
201C2×C4×C61C22×D412yes—
202C2×C4×C61C22×Q811yescharacteristic
203C2×C4×C61C2×#4112yes—
204C2×C4×C61C2×#4111yescharacteristic
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—24no—
208C2×Dic61—22no—
209C2×Dic61—21no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—21no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61#12012yes—
221C2×Dic61C22⋊Q812yes—
222C2×Dic61#12412yes—
223C4×D61—44no—
224C4×D61—44no—
225C4×D61—48no—
226C4×D61—84no—
227C4×D61C22⋊Q818yes—
228C4×D61#12418yes—
229Dic122—24no—
230Dic122—48no—
231Dic122—42no—
232Dic122—48no—
233C8×C61—42no—
234#92418—42no—
235#23455—611nomaximal, Sylow
236C22×C61C2×#12011yescharacteristic
237C22×C61C2×C22⋊Q811yescharacteristic
238C22×C61C2×#12411yescharacteristic
239C22×C61C2×#13711yescharacteristic
240C22×C61C2×#13711yescharacteristic
241C22×C61#57511yescharacteristic
242C22×C61#57511yescharacteristic
243C2×D61—22no—
244C2×D61—22no—
245C2×D61—22no—
246C2×D61—22no—
247C2×D61—22no—
248C2×D61—22no—
249C2×D61#39314yes—
250C4×C61—22no—
251C4×C61—22no—
252C4×C61—42no—
253C4×C61—44no—
254C4×C61—81no—
255C4×C61C2×C22⋊Q812yes—
256C4×C61C2×#12412yes—
257Dic61—42no—
258Dic61—44no—
259Dic61—41no—
260Dic61—41no—
261Dic61—81no—
262Dic61#39314yes—
263C2×#533—611no—
264C2×#419—612no—
265C2×#391—611no—
266#2026—612no—
267#2026—612no—
268#2024—611no—
269#2024—611no—
270#2024—611no—
271#2024D6111yescharacteristic, complemented
272#1736—612no—
273#1728—611no—
274C2×C61C2×#39311yescharacteristic
275C2×C61#300711yescharacteristic
276C2×C61#300711yescharacteristic
277C2×C61#307811yescharacteristic
278C2×C61#307811yescharacteristic
279C2×C61#313911yescharacteristic
280C2×C61#314011yescharacteristic
281D61—24no—
282C23×Q8—611no—
283C22×M16—612no—
284#533—1228no—
285C2×C4⋊Q8—611no—
286C2×C4⋊Q8C2×D6111yescharacteristic
287C2×#125—612no—
288C2×C22⋊Q8—1221no—
289C2×C22⋊Q8—1221no—
290C2×C4×Q8—1221no—
291C2×C4×Q8—1221no—
292C2×#117—612no—
293C22×C4⋊C4—611no—
294C22×C42—611no—
295#419—1228no—
296C2×#103—612no—
297C2×#103—612no—
298C2×#103—612no—
299C2×#103C2×D6112yes—
300#391—611no—
301#391—611no—
302#391—611no—
303#391—1221no—
304#391—1221no—
305#391C2×D6111yescharacteristic
306C2×#95—1221no—
307C2×#95—1221no—
308#340—612no—
309#340—612no—
310#340—612no—
311#340C2×D6112yes—
312#336—612no—
313#336—611no—
314#336C2×D6111yescharacteristic
315C61#2345511yescharacteristic, complemented, Sylow
316C22×Q8—21no—
317C22×Q8—614no—
318C22×Q8—611no—
319C22×Q8—1221no—
320C22×Q8—1222no—
321C22×Q8—1224no—
322C22×Q8—1224no—
323C22×Q8—1221no—
324C22×Q8C22×D6111yescharacteristic
325C23×C4—611no—
326C23×C4—611no—
327C23×C4—611no—
328C23×C4—1221no—
329C23×C4—1221no—
330C2×M16—1224no—
331C2×M16—1224no—
332C2×M16—1224no—
333C22×C8—612no—
334C22×C8C22×D6112yes—
335C4⋊Q8—24no—
336C4⋊Q8—1224no—
337#125—1224no—
338#125—1224no—
339C22⋊Q8—1228no—
340C22⋊Q8—1228no—
341C4×Q8—1228no—
342C4×Q8—1228no—
343#117—618no—
344#117—1224no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—612no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—1224no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4—1222no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4C22×D6111yescharacteristic
354C2×C4⋊C4C22×D6112yes—
355C2×C22⋊C4—1222no—
356C2×C42—611no—
357C2×C42—611no—
358C2×C42—611no—
359C2×C42—1224no—
360C2×C42—1222no—
361C2×C42C22×D6111yescharacteristic
362#103—24no—
363#103—1224no—
364#103—1224no—
365#103—1224no—
366#95—21no—
367#95—21no—
368#95—1221no—
369#95—1221no—
370#95—1221no—
371#95—1221no—
372#95—1221no—
373#95—1221no—
374C24—611no—
375C2×Q8—24no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—612no—
382C2×Q8—1224no—
383C2×Q8—1224no—
384C2×Q8—12216no—
385C2×Q8—1221no—
386C2×Q8—1222no—
387C2×Q8—1222no—
388C2×Q8—2442no—
389C2×Q8—2442no—
390C2×Q8—2442no—
391C2×Q8—2448no—
392C2×Q8—2442no—
393C2×Q8Q8×D6112yes—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—611no—
398C22×C4—614no—
399C22×C4—612no—
400C22×C4—614no—
401C22×C4—612no—
402C22×C4—611no—
403C22×C4—611no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1224no—
407C22×C4—1221no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1221no—
412C22×C4—2441no—
413C22×C4—2441no—
414C22×C4—2442no—
415C22×C4—2442no—
416C22×C4—2442no—
417C22×C4—2442no—
418C22×C4—2442no—
419C22×C4D4×D6111yescharacteristic
420C22×C4D4×D6111yescharacteristic
421C22×C4C23×D6111yescharacteristic
422M16—2448no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—1222no—
427C2×C8—1222no—
428C2×C8—1222no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—614no—
433C4⋊C4—1228no—
434C4⋊C4—1228no—
435C4⋊C4—1224no—
436C4⋊C4—1222no—
437C4⋊C4—1222no—
438C4⋊C4—2442no—
439C4⋊C4D4×D6114yes—
440C22⋊C4—1228no—
441C42—22no—
442C42—1222no—
443C42—1222no—
444C42—1228no—
445C42—1222no—
446C23—612no—
447C23—612no—
448C23—612no—
449C23—1221no—
450C23—1221no—
451C23—1221no—
452C23—1221no—
453C23C2×D4×D6111yescharacteristic
454Q8—24no—
455Q8—48no—
456Q8—42no—
457Q8—1224no—
458Q8—2448no—
459Q8—2448no—
460Q8—2442no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—612no—
475C2×C4—612no—
476C2×C4—612no—
477C2×C4—618no—
478C2×C4—618no—
479C2×C4—1221no—
480C2×C4—1221no—
481C2×C4—1224no—
482C2×C4—1222no—
483C2×C4—1221no—
484C2×C4—1221no—
485C2×C4—1221no—
486C2×C4—1221no—
487C2×C4—4884no—
488C2×C4—2444no—
489C2×C4—2444no—
490C2×C4—2448no—
491C2×C4—2441no—
492C2×C4—2441no—
493C2×C4—2441no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4—2442no—
497C2×C4—2442no—
498C2×C4#1451980612yes—
499C2×C4#1451980912yes—
500C2×C4C2×D4×D6112yes—
501C2×C4C2×Q8×D6111yescharacteristic
502C2×C4#41×D6112yes—
503C2×C4#41×D6111yescharacteristic
504C8—42no—
505C8—2442no—
506C22—614no—
507C22—1222no—
508C22—1222no—
509C22—1222no—
510C22—1222no—
511C22—1222no—
512C22—1222no—
513C22#120×D6111yescharacteristic
514C22C22⋊Q8×D6111yescharacteristic
515C22#124×D6111yescharacteristic
516C22C2×#1451980611yescharacteristic
517C22C2×#1451980911yescharacteristic
518C22#1684055811yescharacteristic
519C22#1684056111yescharacteristic
520C4—22no—
521C4—22no—
522C4—42no—
523C4—44no—
524C4—81no—
525C4—614no—
526C4—4881no—
527C4—2442no—
528C4—2441no—
529C4—2441no—
530C4—2444no—
531C4C22⋊Q8×D6112yes—
532C4#124×D6112yes—
533C2—1224no—
534C2#393×D6111yescharacteristic
535C2#2145507411yescharacteristic
536C2#2145507811yescharacteristic
537C2#2145561911yescharacteristic
538C2#2145562511yescharacteristic
539C2#2145619611yescharacteristic
540C2#2145620411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2024C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2024D61
hypocenterC61#23455
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2024

Complements

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

NH ≅ G/Ntype
4. #21446969C2semidirect
5. #21446968C2semidirect
6. #21446961C2semidirect
7. #21446960C2semidirect
8. #21446957C2semidirect
11. C61×#2024C2semidirect
51. C2×Q8×D61Q8semidirect
271. #2024D61semidirect
315. C61#23455semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168397692
#391×D612
#214469692
#214469682
#214469612
#214469602
#214469572
#214455372
#214455112
C61×#20242
#2345561