Finite Group Databasebeta

← #29685260

Subgroups — #29685260

Order 15616.

1466 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840309C211yesmaximal, characteristic
2C2×#16839714C211yesmaximal, characteristic, complemented
3#21448223C211yesmaximal, characteristic, complemented
4#21446778C212yesmaximal, complemented
5#21446623C212yesmaximal, complemented
6#21446256C212yesmaximal, complemented
7#21446254C212yesmaximal, complemented
8#21445647C211yesmaximal, characteristic, complemented
9#21445645C211yesmaximal, characteristic, complemented
10#21445644C211yesmaximal, characteristic, complemented
11C61×#1762C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13C22×#11758189C2212yes—
14#16840309—24no—
15#16840309C418yescomplemented
16C2×#14519776C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519763—21no—
19C2×#14519752C2211yescharacteristic
20C2×#14519750—21no—
21C2×#14519748C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×C4×C22⋊D61—21no—
24C2×#14519734—21no—
25C2×C42×D61C2211yescharacteristic
26C2×#14519699C2212yes—
27C2×#14519698C2212yes—
28C2×#14519697C2212yes—
29#16839714—24no—
30C2×#14519688C2211yescharacteristic
31C2×C8×Dic61C2212yes—
32#16839631C2212yes—
33#16839626—21no—
34#16839621—21no—
35#16839619C2212yes—
36C4×#11758196—21no—
37C4×C4⋊Dic61C2211yescharacteristic
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×D4×C61C2211yescharacteristic
42C2×#102×C61C2212yes—
43C2×#97×C61C2211yescharacteristic
44C4×C4⋊C4×C61C2211yescharacteristic
45C61×#340C2212yes—
46C2×#11758222—28no—
47C2×#11758222—48no—
48C22×Dic122—21no—
49C22×D244—21no—
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—21no—
52C22×C22⋊D61—21no—
53C22×C22⋊D61—21no—
54C22×C4×D61—21no—
55C22×C4×D61D411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61C2311yescharacteristic
58C2×#11758189—24no—
59C2×#11758189—24no—
60C2×#11758189—24no—
61C22×#92418C2312yes—
62#14519776—22no—
63#14519776C2×C414yes—
64#14519773—22no—
65#14519773C2×C414yes—
66#14519763—28no—
67#14519752—22no—
68#14519752C2×C414yes—
69#14519750—28no—
70#14519748—22no—
71#14519748C2×C414yes—
72C4×D244—22no—
73C4×D244C2×C414yes—
74C4×C22⋊D61—28no—
75#14519734—44no—
76C2×C4⋊Dic61C2312yes—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—22no—
80C2×#11758196—22no—
81C2×C22⋊Dic61—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—22no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89C42×D61C2×C414yes—
90C42×D61D414yes—
91#14519699—24no—
92#14519698—24no—
93#14519697—24no—
94#14519688—22no—
95C8×Dic61—24no—
96#14519684—21no—
97#14519684—22no—
98C22×D4×C61C2311yescharacteristic
99C23×C4×C61—21no—
100C22×C8×C61C2312yes—
101C4×D4×C61—22no—
102C4×D4×C61C2×C414yes—
103C2×C4⋊C4×C61C2312yes—
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C22⋊C4×C61—21no—
106C2×C42×C61—21no—
107C2×C42×C61C2311yescharacteristic
108#102×C61—24no—
109#97×C61—22no—
110#95×C61—21no—
111C24×C61—21no—
112C23×D61—21no—
113C23×D61C2×D411yescharacteristic
114C2×D4×C61—21no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61C22×C412yes—
118C22×C4×C61—21no—
119C22×C4×C61—24no—
120C22×C4×C61—22no—
121C22×C4×C61—44no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2411yescharacteristic
125C2×C22⋊D61—24no—
126C2×C22⋊D61—24no—
127C2×C22⋊D61—24no—
128C2×C22⋊D61—44no—
129C2×C22⋊D61—42no—
130C2×C22⋊D61—44no—
131C2×C22⋊D61—42no—
132C22×Dic61—21no—
133C22×Dic61—22no—
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137#11758222—432no—
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—24no—
141C2×D244—21no—
142C2×D244—44no—
143C2×D244C22×C412yes—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—24no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—24no—
150C2×C4×D61—44no—
151C2×C4×D61C22⋊C414yes—
152C2×C4×D61C22⋊C414yes—
153C2×C4×D61C22×C412yes—
154C2×C4×D61C2×D412yes—
155C2×Dic122—24no—
156C2×Dic122—44no—
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C2×C8×C61#4114yescomplemented
160C4⋊C4×C61—21no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61C22×C412yes—
164C22⋊C4×C61—24no—
165C42×C61—42no—
166C42×C61C22×C412yes—
167C42×C61C2×D412yes—
168C22⋊Dic61—24no—
169#11758198—24no—
170#11758198—24no—
171#11758198—44no—
172C4⋊Dic61—21no—
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C4⋊Dic61C22×C412yes—
176#11758196—24no—
177#11758196—44no—
178#11758196—44no—
179C4×Dic61—42no—
180C4×Dic61C22⋊C414yes—
181C4×Dic61C22⋊C414yes—
182C4×Dic61C22×C412yes—
183C4×Dic61C2×D412yes—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418#4114yescomplemented
187#11758189—216no—
188C23×C61—22no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61C22×D411yescharacteristic
194C22×D61—22no—
195C22×D61—21no—
196C22×D61—22no—
197C22×D61—21no—
198C22×D61—22no—
199C22×D61—42no—
200C22×D61—42no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61C2×C22⋊C412yes—
204D4×C61—22no—
205D4×C61—22no—
206D4×C61—42no—
207C2×C4×C61—21no—
208C2×C4×C61—22no—
209C2×C4×C61—21no—
210C2×C4×C61—22no—
211C2×C4×C61—48no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61C2×C22⋊C412yes—
217C2×C4×C61C2×C22⋊C412yes—
218C2×C4×C61C23×C411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C2×#4112yes—
221C2×C4×C61C2×#4112yes—
222C22⋊D61—48no—
223C22⋊D61—48no—
224C22⋊D61—48no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61C2×C22⋊C412yes—
232C2×Dic61C2×C22⋊C412yes—
233C2×Dic61C2×C22⋊C412yes—
234C2×Dic61AGL(1,ℤ/8)12yes—
235C2×Dic61AGL(1,ℤ/8)14yes—
236C2×Dic61#13712yes—
237D244—22no—
238D244—22no—
239D244—48no—
240D244—42no—
241C4×D61—22no—
242C4×D61—22no—
243C4×D61—24no—
244C4×D61—28no—
245C4×D61—28no—
246C4×D61—48no—
247Dic122—48no—
248C8×C61—24no—
249#92418—24no—
250#21410—611nomaximal, Sylow
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—82no—
255C22×C61—82no—
256C22×C61C22×C22⋊C411yescharacteristic
257C22×C61#53212yes—
258C22×C61C2×AGL(1,ℤ/8)12yes—
259C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C61C2×#13711yescharacteristic
261C2×D61—22no—
262C2×D61—22no—
263C2×D61—22no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—82no—
270C2×D61—82no—
271C4×C61—22no—
272C4×C61—22no—
273C4×C61—42no—
274C4×C61—82no—
275C4×C61C4×#4111yescharacteristic
276C4×C61C4×#4111yescharacteristic
277C4×C61#52512yes—
278Dic61—22no—
279Dic61—42no—
280Dic61—82no—
281Dic61#41714yes—
282Dic61#41814yes—
283C2×#532—611no—
284C2×C4×M16—611no—
285#2285—611no—
286#1990—612no—
287#1957—612no—
288#1895—612no—
289#1895—612no—
290#1762—611no—
291#1762—611no—
292#1762—611no—
293#1762D6111yescharacteristic, complemented
294C2×C61—82no—
295C2×C61#229711yescharacteristic
296C2×C61C2×#41811yescharacteristic
297C2×C61C2×#41711yescharacteristic
298C2×C61#318212yes—
299C2×C61#318412yes—
300D61—42no—
301D61—82no—
302C22×#41—1221no—
303C22×M16—612no—
304#532—618no—
305#532—1224no—
306C2×C4⋊Q8—611no—
307C2×#127—611no—
308C2×#124—1221no—
309C2×#121—611no—
310C2×#121—611no—
311C2×#121—1221no—
312C2×C4×D4—611no—
313C2×C4×D4—1221no—
314C2×C4×D4C2×D6111yescharacteristic
315C2×#117—1221no—
316C22×C42—611no—
317C2×#102—612no—
318C2×#102—612no—
319C2×#102—612no—
320C2×#102C2×D6112yes—
321C4×M16—1224no—
322C2×#97—611no—
323C2×#97C2×D6111yescharacteristic
324C2×C4×C8—612no—
325#384—612no—
326#382—1221no—
327#380—612no—
328#380—1221no—
329C4×C4⋊C4—611no—
330C4×C4⋊C4—1221no—
331C4×C4⋊C4C2×D6111yescharacteristic
332#340—612no—
333#340—612no—
334#340—612no—
335#340C2×D6112yes—
336C61#2141011yescharacteristic, complemented, Sylow
337C2×#41—1228no—
338C2×#41—2448no—
339C22×Q8—1221no—
340C22×D4—611no—
341C22×D4—1221no—
342C22×D4—1221no—
343C22×D4—1221no—
344C22×D4—1221no—
345C22×D4C22×D6111yescharacteristic
346C23×C4—21no—
347C23×C4—611no—
348C23×C4—611no—
349C23×C4—611no—
350C23×C4—1221no—
351C2×M16—1224no—
352C2×M16—1224no—
353C2×M16—1224no—
354C22×C8—612no—
355C22×C8C22×D6112yes—
356C4⋊Q8—614no—
357C4⋊Q8—1222no—
358#127—614no—
359#127—1222no—
360#124—1228no—
361#121—614no—
362#121—614no—
363#121—1228no—
364#121—1222no—
365#121—1222no—
366C4×D4—22no—
367C4×D4—614no—
368C4×D4—1228no—
369C4×D4—1222no—
370C4×D4C4×D6114yes—
371#117—2444no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—612no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4C22×D6112yes—
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—1221no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1222no—
384C2×C42—21no—
385C2×C42—611no—
386C2×C42—611no—
387C2×C42—611no—
388C2×C42—614no—
389C2×C42—614no—
390C2×C42—1221no—
391C2×C42C22×D6111yescharacteristic
392#102—24no—
393#102—1224no—
394#102—1224no—
395#102—1224no—
396#97—22no—
397#97—1222no—
398C4×C8—1224no—
399#95—21no—
400#95—1222no—
401#95—1221no—
402C24—21no—
403C24—611no—
404C24—1221no—
405#41—24432no—
406C2×Q8—1224no—
407C2×Q8—2444no—
408C2×D4—21no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—612no—
412C2×D4—1224no—
413C2×D4—1224no—
414C2×D4—1224no—
415C2×D4—1224no—
416C2×D4—1221no—
417C2×D4—1222no—
418C2×D4—1222no—
419C2×D4—2444no—
420C2×D4—2444no—
421C2×D4—2442no—
422C2×D4—2442no—
423C2×D4—2444no—
424C2×D4C2×C4×D6112yes—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—44no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—612no—
433C22×C4—612no—
434C22×C4—614no—
435C22×C4—614no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1224no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1221no—
442C22×C4—1224no—
443C22×C4—1222no—
444C22×C4—2444no—
445C22×C4D4×D6111yescharacteristic
446C22×C4D4×D6111yescharacteristic
447C22×C4C23×D6111yescharacteristic
448M16—12216no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—614no—
452C2×C8—1222no—
453C2×C8—1222no—
454C2×C8#1175822214yescomplemented
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—612no—
459C4⋊C4—1224no—
460C4⋊C4—1221no—
461C4⋊C4—1222no—
462C4⋊C4—1222no—
463C4⋊C4—2444no—
464C4⋊C4—2444no—
465C4⋊C4C2×C4×D6112yes—
466C22⋊C4—24no—
467C22⋊C4—1224no—
468C22⋊C4—1224no—
469C22⋊C4—1224no—
470C22⋊C4—2444no—
471C42—42no—
472C42—612no—
473C42—612no—
474C42—614no—
475C42—614no—
476C42—2442no—
477C42C2×C4×D6112yes—
478C42D4×D6112yes—
479C23—22no—
480C23—42no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—612no—
485C23—1221no—
486C23—1222no—
487C23—1221no—
488C23—1222no—
489C23—1222no—
490C23—2442no—
491C23—2441no—
492C23—2441no—
493C23—2442no—
494C23C2×D4×D6111yescharacteristic
495Q8—2448no—
496D4—22no—
497D4—22no—
498D4—42no—
499D4—1222no—
500D4—1222no—
501D4—2448no—
502D4—2448no—
503D4—2448no—
504D4—2448no—
505D4—2442no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—48no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—612no—
516C2×C4—612no—
517C2×C4—612no—
518C2×C4—614no—
519C2×C4—612no—
520C2×C4—612no—
521C2×C4—1228no—
522C2×C4—1228no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—1224no—
527C2×C4—1222no—
528C2×C4—2448no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4C22⋊C4×D6112yes—
534C2×C4C22⋊C4×D6112yes—
535C2×C4C22×C4×D6111yescharacteristic
536C2×C4C2×D4×D6111yescharacteristic
537C2×C4#41×D6112yes—
538C2×C4C2×#1175822212yes—
539C8—24no—
540C8—1224no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—82no—
545C22—82no—
546C22—1222no—
547C22—1222no—
548C22—1222no—
549C22—4882no—
550C22—4882no—
551C22—2442no—
552C22—2442no—
553C22—2442no—
554C22—2442no—
555C22—2442no—
556C22C2×C22⋊C4×D6111yescharacteristic
557C22#1684030912yes—
558C22AGL(1,ℤ/8)×D6112yes—
559C22AGL(1,ℤ/8)×D6111yescharacteristic
560C22#137×D6111yescharacteristic
561C4—22no—
562C4—22no—
563C4—42no—
564C4—82no—
565C4—614no—
566C4—614no—
567C4—1222no—
568C4—4882no—
569C4—2442no—
570C4#1684023011yescharacteristic
571C4C4×#1175822211yescharacteristic
572C4#1684027112yes—
573C2—82no—
574C2—4882no—
575C2—2442no—
576C2#2144830511yescharacteristic
577C2#418×D6111yescharacteristic
578C2#417×D6111yescharacteristic
579C2#2145653412yes—
580C2#2145655412yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1762C2
socleC23×C61C22×D4
prime radicalC2×#16840309C2
cosocleC22×C4×C61C24
hypercenter#1762D61
hypocenterC61#21410
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C2×#16839714C2semidirect
3. #21448223C2semidirect
4. #21446778C2semidirect
5. #21446623C2semidirect
6. #21446256C2semidirect
7. #21446254C2semidirect
8. #21445647C2semidirect
9. #21445645C2semidirect
10. #21445644C2semidirect
11. C61×#1762C2semidirect
15. #16840309C4semidirect
159. C2×C8×C61#41semidirect
186. C2×#92418#41semidirect
293. #1762D61semidirect
336. C61#21410semidirect
454. C2×C8#11758222semidirect

Maximal subgroups

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

subgroupindex
C2×#168403092
C2×#168397142
#214482232
#214467782
#214466232
#214462562
#214462542
#214456472
#214456452
#214456442
C61×#17622
#2141061