Finite Group Databasebeta

← #29736617

Subgroups — #29736617

Order 15616.

1306 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D61C211yesmaximal, characteristic
2C2×#16840111C212yesmaximal, complemented
3#21454168C211yesmaximal, characteristic
4#21446961C211yesmaximal, characteristic, complemented
5#21446960C211yesmaximal, characteristic, complemented
6#21446957C211yesmaximal, characteristic, complemented
7#21446778C212yesmaximal, complemented
8#21446125C212yesmaximal
9#21446123C212yesmaximal
10#21445576C211yesmaximal, characteristic
11C61×#2024C211yesmaximal, characteristic, complemented
12C22×Q8×D61—21no—
13C2×#14519777C2211yescharacteristic
14C2×#14519776—21no—
15C2×#14519776C2212yes—
16C2×#14519775—21no—
17C2×#14519775C2211yescharacteristic
18C4⋊Q8×D61—28no—
19C2×#14519757C2211yescharacteristic
20C2×#14519753C2211yescharacteristic
21C2×C4⋊C4×D61—21no—
22C2×C4⋊C4×D61C2212yes—
23C2×C42×D61C2211yescharacteristic
24#16840111—28no—
25C2×#14519702C2212yes—
26C2×#14519701C2212yes—
27C2×#14519700C2212yes—
28C2×#14519689C2212yes—
29C2×C8×Dic61C2212yes—
30#16839668C2211yescharacteristic
31#16839659—21no—
32#16839659—21no—
33#16839631C2211yescharacteristic
34#16839621—21no—
35#16839621—21no—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39#16839517C2211yescharacteristic
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×#103×C61C2212yes—
42C61×#391C2211yescharacteristic
43C61×#340C2212yes—
44C61×#336C2211yescharacteristic
45C22×Dic122—21no—
46C22×Dic122—22no—
47C22×Dic122C2311yescharacteristic
48C2×Q8×D61—28no—
49C2×Q8×D61—44no—
50C2×Q8×D61—44no—
51C22×C4×D61—22no—
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C22×C4×D61C2311yescharacteristic
55C22×#92418C2312yes—
56#14519777—24no—
57#14519776—28no—
58#14519776—24no—
59#14519776D418yes—
60#14519775—28no—
61#14519775—24no—
62#14519757—24no—
63#14519753—24no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2312yes—
68C2×#11758196—22no—
69C2×#11758196—22no—
70C2×#11758196—21no—
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C4⋊C4×D61—28no—
76C4⋊C4×D61—24no—
77C4⋊C4×D61—24no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61C2311yescharacteristic
83C42×D61—24no—
84#14519702—24no—
85#14519701—24no—
86#14519700—24no—
87#14519689—24no—
88C8×Dic61—24no—
89#14519684—21no—
90#14519684—21no—
91C22×Q8×C61—21no—
92C22×Q8×C61C2311yescharacteristic
93C22×C8×C61C2312yes—
94C4⋊Q8×C61—24no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C4⋊C4×C61C2312yes—
98C2×C42×C61C2311yescharacteristic
99#103×C61—24no—
100#95×C61—21no—
101#95×C61—21no—
102C23×D61C2×D411yescharacteristic
103C2×Q8×C61—24no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61—21no—
107C2×Q8×C61—21no—
108C2×Q8×C61—42no—
109C2×Q8×C61—42no—
110C22×C4×C61—22no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2411yescharacteristic
116C22×Dic61—22no—
117C22×Dic61—21no—
118C22×Dic61—21no—
119C22×Dic61C2×D411yescharacteristic
120C22×Dic61C2×D411yescharacteristic
121C22×Dic61C2×D411yescharacteristic
122Q8×D61—432no—
123C2×C4×D61—22no—
124C2×C4×D61—21no—
125C2×C4×D61—21no—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—28no—
131C2×C4×D61—44no—
132C2×C4×D61—44no—
133C2×C4×D61SD1614yes—
134C2×C4×D61Q1614yes—
135C2×C4×D61#4114yes—
136C2×Dic122—24no—
137C2×Dic122—28no—
138C2×Dic122—22no—
139C2×Dic122—22no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122—42no—
143C2×Dic122—42no—
144C2×Dic122—44no—
145C2×Dic122—44no—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61C2×D414yes—
153C42×C61—22no—
154#11758198—42no—
155#11758198—42no—
156C4⋊Dic61—24no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61—42no—
160C4⋊Dic61C2×D414yes—
161#11758196—28no—
162#11758196—28no—
163#11758196—42no—
164#11758196—42no—
165#11758196—42no—
166#11758196—42no—
167#11758196—42no—
168C4×Dic61—22no—
169C4×Dic61—22no—
170C4×Dic61C2×D414yes—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418—22no—
174C23×C61C22×D411yescharacteristic
175C22×D61—21no—
176C22×D61—21no—
177C22×D61—21no—
178C22×D61—21no—
179C22×D61#12412yes—
180C22×D61C2×SD1612yes—
181C22×D61C2×Q1612yes—
182Q8×C61—48no—
183Q8×C61—42no—
184Q8×C61—42no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—24no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61C2×SD1612yes—
199C2×C4×C61C2×Q1612yes—
200C2×C4×C61C22×D412yes—
201C2×C4×C61C2×#4112yes—
202C2×C4×C612−1+411yescharacteristic
203C2×C4×C612−1+411yescharacteristic
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—24no—
207C2×Dic61—42no—
208C2×Dic61—42no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61#12012yes—
216C2×Dic61#12012yes—
217C2×Dic61#12412yes—
218C2×Dic61C2×SD1612yes—
219C2×Dic61C2×Q1612yes—
220C2×Dic61AGL(1,ℤ/8)12yes—
221C2×Dic61#13712yes—
222C4×D61—24no—
223C4×D61—24no—
224C4×D61—28no—
225C4×D61—28no—
226C4×D61—416no—
227Dic122—48no—
228Dic122—416no—
229Dic122—42no—
230Dic122—42no—
231C8×C61—42no—
232#92418—42no—
233#25902—611nomaximal, Sylow
234C22×C61C2×#12011yescharacteristic
235C22×C61C2×#12411yescharacteristic
236C22×C61#53611yescharacteristic
237C22×C61C22×SD1611yescharacteristic
238C22×C61C22×Q1611yescharacteristic
239C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
240C22×C61C2×#13711yescharacteristic
241C2×D61—22no—
242C2×D61—22no—
243C2×D61—22no—
244C2×D61—22no—
245C2×D61—22no—
246C2×D61—22no—
247C2×D61#48714yescomplemented
248C4×C61—22no—
249C4×C61—22no—
250C4×C61—44no—
251C4×C61—81no—
252C4×C61—81no—
253C4×C61#54812yes—
254C4×C61#54812yes—
255Dic61—22no—
256Dic61—44no—
257Dic61—81no—
258Dic61—81no—
259Dic61#45014yes—
260Dic61#45114yes—
261C22×C4⋊Q8—611no—
262C2×#487—612no—
263#2857—611no—
264#2024—611no—
265#2024—611no—
266#2024—611no—
267#2024D6111yescharacteristic, complemented
268#1990—612no—
269#1872—612no—
270#1872—612no—
271C4×C22⋊C8—611no—
272C2×C61#243611yescharacteristic
273C2×C61C2×#45011yescharacteristic
274C2×C61C2×#45111yescharacteristic
275C2×C61C2×#48711yescharacteristic
276C2×C61#314111yescharacteristic
277C2×C61#322911yescharacteristic
278C2×C61#323011yescharacteristic
279D61—24no—
280C23×Q8—1221no—
281C2×C4⋊Q8—612no—
282C2×C4⋊Q8—611no—
283C2×C4⋊Q8—611no—
284C2×C4⋊Q8—1221no—
285C2×C4⋊Q8—1221no—
286C2×C4⋊Q8—1228no—
287C2×C4⋊Q8C2×D6111yescharacteristic
288C2×C22⋊Q8—611no—
289C2×C22⋊Q8—611no—
290C22×C4⋊C4—612no—
291C22×C4⋊C4—1221no—
292C22×C42—611no—
293#487—1228no—
294C2×#103—612no—
295C2×#103—612no—
296C2×#103—612no—
297C2×#103C2×D6112yes—
298C2×C22⋊C8—612no—
299C2×C4×C8—612no—
300#391—611no—
301#391C2×D6111yescharacteristic
302#389—1221no—
303#389—1221no—
304#384—611no—
305#380—1221no—
306#380—1221no—
307#340—612no—
308#340—612no—
309#340—612no—
310#340C2×D6112yes—
311#336—611no—
312#336C2×D6111yescharacteristic
313C61#2590211yescharacteristic, complemented, Sylow
314C22×Q8—21no—
315C22×Q8—611no—
316C22×Q8—1222no—
317C22×Q8—1228no—
318C22×Q8—1221no—
319C22×Q8—2444no—
320C22×Q8—2444no—
321C22×Q8C22×D6111yescharacteristic
322C23×C4—611no—
323C23×C4—611no—
324C23×C4—611no—
325C23×C4—1222no—
326C22×C8—612no—
327C22×C8C22×D6112yes—
328C4⋊Q8—24no—
329C4⋊Q8—618no—
330C4⋊Q8—1224no—
331C4⋊Q8—1224no—
332C4⋊Q8—1224no—
333C4⋊Q8—1228no—
334C4⋊Q8—1228no—
335C22⋊Q8—1224no—
336C22⋊Q8—1224no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—612no—
339C2×C4⋊C4—611no—
340C2×C4⋊C4—1224no—
341C2×C4⋊C4—1228no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—1222no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1224no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4C22×D6111yescharacteristic
353C2×C4⋊C4C22×D6112yes—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1221no—
356C2×C42—611no—
357C2×C42—611no—
358C2×C42—611no—
359C2×C42—1224no—
360C2×C42C22×D6111yescharacteristic
361#103—24no—
362#103—1224no—
363#103—1224no—
364#103—1224no—
365C22⋊C8—1224no—
366C4×C8—1224no—
367#95—21no—
368#95—21no—
369#95—1221no—
370#95—1221no—
371C24—611no—
372C2×Q8—24no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—1228no—
380C2×Q8—1224no—
381C2×Q8—1222no—
382C2×Q8—1222no—
383C2×Q8—1221no—
384C2×Q8—1221no—
385C2×Q8—2442no—
386C2×Q8—2444no—
387C2×Q8—2444no—
388C2×Q8—24432no—
389C2×Q8—2442no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—611no—
394C22×C4—611no—
395C22×C4—611no—
396C22×C4—614no—
397C22×C4—614no—
398C22×C4—614no—
399C22×C4—1222no—
400C22×C4—1222no—
401C22×C4—1222no—
402C22×C4—1222no—
403C22×C4—1228no—
404C22×C4—1222no—
405C22×C4—1221no—
406C22×C4—1221no—
407C22×C4—1222no—
408C22×C4—1221no—
409C22×C4—1221no—
410C22×C4—2444no—
411C22×C4—2444no—
412C22×C4D4×D6111yescharacteristic
413C22×C4D4×D6111yescharacteristic
414C22×C4C23×D6111yescharacteristic
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—1222no—
419C2×C8—1222no—
420C2×C8—1222no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—614no—
425C4⋊C4—1228no—
426C4⋊C4—1228no—
427C4⋊C4—1224no—
428C4⋊C4—1222no—
429C4⋊C4—1222no—
430C4⋊C4—2442no—
431C4⋊C4—2442no—
432C4⋊C4—2442no—
433C4⋊C4—2442no—
434C4⋊C4—2442no—
435C4⋊C4—2442no—
436C4⋊C4D4×D6114yes—
437C22⋊C4—2442no—
438C22⋊C4—2442no—
439C42—22no—
440C42—614no—
441C42—1222no—
442C42—1222no—
443C23—612no—
444C23—612no—
445C23—612no—
446C23—1221no—
447C23—1221no—
448C23—1221no—
449C23—1221no—
450C23C2×D4×D6111yescharacteristic
451Q8—48no—
452Q8—42no—
453Q8—42no—
454Q8—2448no—
455Q8—24416no—
456Q8—2442no—
457Q8—2442no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—24no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—612no—
472C2×C4—612no—
473C2×C4—612no—
474C2×C4—612no—
475C2×C4—612no—
476C2×C4—612no—
477C2×C4—612no—
478C2×C4—1228no—
479C2×C4—1228no—
480C2×C4—1221no—
481C2×C4—1221no—
482C2×C4—1224no—
483C2×C4—1224no—
484C2×C4—1224no—
485C2×C4—24416no—
486C2×C4—2441no—
487C2×C4—2441no—
488C2×C4—2441no—
489C2×C4—2441no—
490C2×C4—2441no—
491C2×C4—2441no—
492C2×C4—2442no—
493C2×C4—2442no—
494C2×C4SD16×D6112yes—
495C2×C4Q16×D6112yes—
496C2×C4C2×D4×D6112yes—
497C2×C4#41×D6112yes—
498C2×C4#1451982411yescharacteristic
499C2×C4#1451982511yescharacteristic
500C8—42no—
501C8—2442no—
502C22—614no—
503C22—1222no—
504C22—1222no—
505C22—1222no—
506C22—1222no—
507C22—1222no—
508C22—1222no—
509C22#120×D6111yescharacteristic
510C22#124×D6111yescharacteristic
511C22#1684033511yescharacteristic
512C22C2×SD16×D6111yescharacteristic
513C22C2×Q16×D6111yescharacteristic
514C22AGL(1,ℤ/8)×D6111yescharacteristic
515C22#137×D6111yescharacteristic
516C4—22no—
517C4—22no—
518C4—44no—
519C4—81no—
520C4—81no—
521C4—614no—
522C4—614no—
523C4—1222no—
524C4—4881no—
525C4—4881no—
526C4—2444no—
527C4#1684042012yes—
528C4#1684042212yes—
529C2—1224no—
530C2#2144947211yescharacteristic
531C2#450×D6111yescharacteristic
532C2#451×D6111yescharacteristic
533C2#487×D6111yescharacteristic
534C2#2145621111yescharacteristic
535C2#2145696211yescharacteristic
536C2#2145696911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2024C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2024D61
hypocenterC61#25902
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
2. C2×#16840111C2semidirect
4. #21446961C2semidirect
5. #21446960C2semidirect
6. #21446957C2semidirect
7. #21446778C2semidirect
11. C61×#2024C2semidirect
247. C2×D61#487semidirect
267. #2024D61semidirect
313. C61#25902semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D612
C2×#168401112
#214541682
#214469612
#214469602
#214469572
#214467782
#214461252
#214461232
#214455762
C61×#20242
#2590261