Finite Group Databasebeta

← #29718366

Subgroups — #29718366

Order 15616.

1130 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519697C211yesmaximal, characteristic
2#21449471C211yesmaximal, characteristic
3#21448688C211yesmaximal, characteristic, complemented
4#21447188C212yesmaximal, complemented
5#21447057C212yesmaximal, complemented
6#21446748C212yesmaximal, complemented
7#21446445C212yesmaximal, complemented
8#21446243C211yesmaximal, characteristic, complemented
9#21445996C211yesmaximal, characteristic, complemented
10#21445751C211yesmaximal, characteristic
11C2×C61×#483C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14C2×#14519766C2212yes—
15C2×#14519736—21no—
16C2×#14519706C2212yescomplemented
17C2×#14519701C2212yescomplemented
18C2×#14519699C2212yes—
19C2×#14519699C2212yes—
20C2×#14519697—24no—
21C2×#14519697C2212yes—
22C2×#14519697C2212yes—
23#16839679—22no—
24#16839663C2212yes—
25#16839661C2212yescomplemented
26#16839648—22no—
27#16839625—21no—
28#16839622C2211yescharacteristic
29#16839617—21no—
30#16839617—21no—
31C4⋊C4×Dic61C2212yescomplemented
32C4×C22⋊Dic61C2211yescharacteristic
33#16839524C2212yescomplemented
34#16839516C2212yescomplemented
35C2×C22⋊Q8×C61C2211yescharacteristic
36C22×C4⋊C4×C61C2211yescharacteristic
37C61×#483—24no—
38C2×#106×C61C2211yescharacteristic, complemented
39C2×#106×C61C2211yescharacteristic, complemented
40C2×#103×C61C2211yescharacteristic, complemented
41C2×#103×C61C2211yescharacteristic, complemented
42C2×C22⋊C8×C61C2211yescharacteristic
43C24×D61—21no—
44C22×D244—22no—
45C22×D244—24no—
46C22×D244—22no—
47C22×D244—21no—
48C22×D244—42no—
49C22×D244C2312yes—
50C22×#92418—22no—
51C22×#92418—42no—
52C22×#92418C2312yes—
53#14519766—24no—
54#14519766—42no—
55#14519736—44no—
56C2×C4⋊Dic61—22no—
57C2×C4⋊Dic61C2312yes—
58C2×C22⋊Dic61—21no—
59C2×C22⋊Dic61D411yescharacteristic
60C2×C22⋊Dic61D411yescharacteristic
61C2×#11758198—22no—
62C2×#11758198—22no—
63C2×#11758198—22no—
64C2×#11758198—22no—
65C2×C4×Dic61—22no—
66C2×C4×Dic61C2312yes—
67#14519706—42no—
68#14519701—42no—
69#14519699—24no—
70#14519699—42no—
71#14519697—24no—
72#14519697—24no—
73#14519684—21no—
74#14519684—22no—
75#14519684—22no—
76#14519684D412yes—
77C22×Q8×C61C2311yescharacteristic
78C23×C4×C61—21no—
79C23×C4×C61C2311yescharacteristic
80C22×C8×C61C2311yescharacteristic
81C22×C8×C61C2311yescharacteristic
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—22no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—22no—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61D414yes—
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C22⋊C4×C61—21no—
92#106×C61—22no—
93#106×C61—22no—
94#103×C61—22no—
95#103×C61—22no—
96C22⋊C8×C61—22no—
97C24×C61C2×D411yescharacteristic
98C23×D61—22no—
99C23×D61—44no—
100C23×D61—42no—
101C23×D61—42no—
102C23×D61—42no—
103C23×D61—82no—
104C2×Q8×C61—21no—
105C2×Q8×C61—21no—
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—24no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—42no—
116C22×C4×C61—42no—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D412yes—
119C22×C4×C61#4112yes—
120C22×C4×C61C2411yescharacteristic
121C22×Dic61—22no—
122C22×Dic61C2×D412yes—
123C22×Dic61C2×D412yes—
124C2×D244—28no—
125C2×D244—22no—
126C2×D244—22no—
127C2×D244—22no—
128C2×D244—22no—
129C2×D244—22no—
130C2×D244—48no—
131C2×D244—44no—
132C2×D244—44no—
133C2×D244—42no—
134C2×D244#4114yes—
135C2×C8×C61—21no—
136C2×C8×C61—21no—
137C2×C8×C61—22no—
138C2×C8×C61—22no—
139C4⋊C4×C61—24no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—42no—
145C4⋊C4×C61C2×D412yes—
146C4⋊C4×C61C2×D412yes—
147C22⋊C4×C61—42no—
148C22⋊Dic61—22no—
149C22⋊Dic61—22no—
150C22⋊Dic61—42no—
151#11758198—44no—
152#11758198—44no—
153#11758198—82no—
154#11758198—82no—
155C4⋊Dic61—22no—
156C4⋊Dic61—22no—
157C4⋊Dic61—44no—
158C4×Dic61—22no—
159C4×Dic61—22no—
160C4×Dic61—44no—
161C2×#92418—22no—
162C2×#92418—22no—
163C2×#92418—22no—
164C2×#92418—44no—
165C23×C61—22no—
166C23×C61—22no—
167C23×C61#12112yes—
168C23×C61#13512yes—
169C23×C61AGL(1,ℤ/8)12yes—
170C23×C61C22×D411yescharacteristic
171C22×D61—24no—
172C22×D61—48no—
173C22×D61—42no—
174C22×D61—42no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—42no—
178C22×D61—42no—
179C22×D61—88no—
180C22×D61—88no—
181C22×D61—84no—
182C22×D61—84no—
183Q8×C61—41no—
184Q8×C61—41no—
185Q8×C61—42no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—24no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—48no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61#12112yes—
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C2×#4112yes—
204C2×C4×C61C2×#4111yescharacteristic
205C2×C4×C61H(5,𝔽2)12yes—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—22no—
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215D244—24no—
216D244—24no—
217D244—48no—
218D244—42no—
219D244—42no—
220C8×C61—41no—
221C8×C61—41no—
222#92418—42no—
223#24477—611nomaximal, Sylow
224C22×C61—22no—
225C22×C61—24no—
226C22×C61—22no—
227C22×C61—24no—
228C22×C61#46314yes—
229C22×C61C2×#12111yescharacteristic
230C22×C61#54612yes—
231C22×C61C2×#13511yescharacteristic
232C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
233C22×C61#57712yes—
234C2×D61—44no—
235C2×D61—44no—
236C2×D61—44no—
237C2×D61—816no—
238C2×D61—82no—
239C2×D61—82no—
240C2×D61—82no—
241C2×D61—82no—
242C4×C61—22no—
243C4×C61—42no—
244C4×C61—42no—
245C4×C61—81no—
246C4×C61—81no—
247C4×C61#52511yescharacteristic
248C4×C61#53511yescharacteristic
249C4×C61#55112yes—
250Dic61—42no—
251Dic61—42no—
252Dic61—82no—
253C2×#483D6111yescharacteristic, complemented
254C22×#102—611no—
255#2435—611no—
256#2344—611no—
257#2065—612no—
258#2042—612no—
259#1983—612no—
260#1926—612no—
261#1892—611no—
262#1850—611no—
263#1786—611no—
264C2×C61—24no—
265C2×C61#241111yescharacteristic
266C2×C61C2×#46311yescharacteristic
267C2×C61#304311yescharacteristic
268C2×C61#329612yes—
269C2×C61#330612yes—
270D61—84no—
271C23×D4—611no—
272C23×C8—611no—
273C2×#124—612no—
274C2×C22⋊Q8C2×D6111yescharacteristic
275C2×#117—1221no—
276C22×C4⋊C4C2×D6111yescharacteristic
277#483—24no—
278C2×#106C2×D6111yescharacteristic, complemented
279C2×#106C2×D6111yescharacteristic, complemented
280C2×C4⋊C8—612no—
281C2×#103—612no—
282C2×#103C2×D6111yescharacteristic, complemented
283C2×#103C2×D6111yescharacteristic, complemented
284C2×#102—612no—
285C2×#102—612no—
286C2×#102—612no—
287C2×#102—612no—
288C2×#102—1224no—
289C2×C22⋊C8C2×D6111yescharacteristic
290#394—1222no—
291#390—612no—
292#389—612no—
293#387—1222no—
294#381—1221no—
295#380—611no—
296#379—1221no—
297#379—1221no—
298C4×C4⋊C4—612no—
299C4×C22⋊C4—611no—
300#340—612no—
301#336—612no—
302C61#2447711yescharacteristic, complemented, Sylow
303C25—1221no—
304C22×Q8C22×D6111yescharacteristic
305C22×D4—612no—
306C22×D4—1224no—
307C22×D4—1221no—
308C22×D4—1222no—
309C22×D4—1222no—
310C22×D4—2442no—
311C23×C4—21no—
312C23×C4C22×D6111yescharacteristic
313C22×C8—612no—
314C22×C8—1222no—
315C22×C8—2442no—
316C22×C8C22×D6111yescharacteristic
317C22×C8C22×D6111yescharacteristic
318#124—1224no—
319#124—2442no—
320C22⋊Q8—22no—
321C22⋊Q8—22no—
322#117—2444no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—612no—
327C2×C4⋊C4—1222no—
328C2×C4⋊C4C22⋊D6114yes—
329C2×C4⋊C4C22×D6111yescharacteristic
330C2×C4⋊C4C22×D6111yescharacteristic
331C2×C4⋊C4C22×D6111yescharacteristic
332C2×C22⋊C4—21no—
333C2×C22⋊C4—611no—
334C2×C22⋊C4—611no—
335C2×C22⋊C4—1222no—
336C2×C22⋊C4—1222no—
337C2×C22⋊C4—1222no—
338C2×C22⋊C4—1222no—
339C2×C22⋊C4—1221no—
340C2×C42—612no—
341C2×C42—1222no—
342#106—22no—
343#106—22no—
344C4⋊C8—2442no—
345#103—22no—
346#103—22no—
347#103—2442no—
348#102—1224no—
349#102—1224no—
350#102—1224no—
351#102—2442no—
352C22⋊C8—22no—
353#95—612no—
354#95—1222no—
355#95—1222no—
356#95—1221no—
357C24—1222no—
358C24—4882no—
359C24—2444no—
360C24—2442no—
361C24—2442no—
362C24—2442no—
363C24D4×D6111yescharacteristic
364C2×Q8—21no—
365C2×Q8—22no—
366C2×Q8—21no—
367C2×Q8—22no—
368C2×D4—614no—
369C2×D4—1228no—
370C2×D4—1222no—
371C2×D4—1222no—
372C2×D4—1222no—
373C2×D4—1222no—
374C2×D4—1222no—
375C2×D4—2448no—
376C2×D4—2444no—
377C2×D4—2444no—
378C2×D4—2442no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—612no—
389C22×C4—612no—
390C22×C4—1222no—
391C22×C4D4×D6111yescharacteristic
392C22×C4#1175822412yes—
393C22×C4C2×C22⋊D6112yes—
394C22×C4C23×D6111yescharacteristic
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—1222no—
400C2×C8—1222no—
401C2×C8—1222no—
402C2×C8—2444no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—24no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—42no—
409C4⋊C4—1222no—
410C4⋊C4—1222no—
411C4⋊C4—2444no—
412C4⋊C4C2×C22⋊D6112yes—
413C4⋊C4C2×C22⋊D6112yes—
414C22⋊C4—42no—
415C22⋊C4—1222no—
416C22⋊C4—1222no—
417C22⋊C4—4882no—
418C22⋊C4—4882no—
419C22⋊C4—2444no—
420C22⋊C4—2444no—
421C22⋊C4—2442no—
422C42—1222no—
423C42—1222no—
424C42—2444no—
425C23—22no—
426C23—22no—
427C23—1224no—
428C23—4884no—
429C23—4888no—
430C23—4888no—
431C23—4884no—
432C23—2448no—
433C23—2442no—
434C23—2442no—
435C23—2442no—
436C23—2442no—
437C23—2442no—
438C23—2442no—
439C23#1451974912yes—
440C23#1451979812yes—
441C23#1451980412yes—
442C23C2×D4×D6111yescharacteristic
443Q8—41no—
444Q8—41no—
445Q8—42no—
446D4—1224no—
447D4—1224no—
448D4—2448no—
449D4—2442no—
450D4—2442no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—48no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—1222no—
467C2×C4—1222no—
468C2×C4—1222no—
469C2×C4—1222no—
470C2×C4—1222no—
471C2×C4—1222no—
472C2×C4—2442no—
473C2×C4—2442no—
474C2×C4—2442no—
475C2×C4#1451974912yes—
476C2×C4C22×C22⋊D6111yescharacteristic
477C2×C4#41×D6112yes—
478C2×C4C2×#1175822411yescharacteristic
479C2×C4#1451982312yes—
480C8—41no—
481C8—41no—
482C8—2442no—
483C22—24no—
484C22—22no—
485C22—24no—
486C22—22no—
487C22—48816no—
488C22—4882no—
489C22—4882no—
490C22—4882no—
491C22—4882no—
492C22—2444no—
493C22—2444no—
494C22—2444no—
495C22#1683997914yes—
496C22C2×#1451974911yescharacteristic
497C22#1684040412yes—
498C22C2×#1451979811yescharacteristic
499C22C2×#1451980411yescharacteristic
500C22#1684057112yes—
501C4—22no—
502C4—42no—
503C4—42no—
504C4—81no—
505C4—81no—
506C4—4882no—
507C4—2442no—
508C4—2442no—
509C4#1684026911yescharacteristic
510C4#1684032611yescharacteristic
511C4#1684043512yes—
512C2—24no—
513C2—4884no—
514C2#2144920511yescharacteristic
515C2C2×#1683997911yescharacteristic
516C2#2145534211yescharacteristic
517C2#2145765212yes—
518C2#2145777812yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#483C2
socleC23×C61C22×D4
prime radicalC23×D244C22
hypercenterC2×#483D61
hypocenterC61#24477
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#483

Complements

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

NH ≅ G/Ntype
3. #21448688C2semidirect
4. #21447188C2semidirect
5. #21447057C2semidirect
6. #21446748C2semidirect
7. #21446445C2semidirect
8. #21446243C2semidirect
9. #21445996C2semidirect
11. C2×C61×#483C2semidirect
16. C2×#14519706C22semidirect
17. C2×#14519701C22semidirect
25. #16839661C22semidirect
31. C4⋊C4×Dic61C22semidirect
33. #16839524C22semidirect
34. #16839516C22semidirect
38. C2×#106×C61C22semidirect
39. C2×#106×C61C22semidirect
40. C2×#103×C61C22semidirect
41. C2×#103×C61C22semidirect
253. C2×#483D61semidirect
278. C2×#106C2×D61semidirect
279. C2×#106C2×D61semidirect
282. C2×#103C2×D61semidirect
283. C2×#103C2×D61semidirect
302. C61#24477semidirect

Maximal subgroups

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

subgroupindex
C22×#145196972
#214494712
#214486882
#214471882
#214470572
#214467482
#214464452
#214462432
#214459962
#214457512
C2×C61×#4832
#2447761