Finite Group Databasebeta

← #29685245

Subgroups — #29685245

Order 15616.

1434 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840307C211yesmaximal, characteristic
2C22×#14519687C211yesmaximal, characteristic, complemented
3#21448214C211yesmaximal, characteristic, complemented
4#21446790C212yesmaximal, complemented
5#21446641C212yesmaximal, complemented
6#21446248C211yesmaximal, characteristic, complemented
7#21446247C211yesmaximal, characteristic, complemented
8#21446245C211yesmaximal, characteristic, complemented
9#21445646C212yesmaximal, complemented
10#21445643C212yesmaximal, complemented
11C61×#1893C211yesmaximal, characteristic, complemented
12C22×#11758224—21no—
13C23×#92418C2211yescharacteristic
14#16840307—24no—
15#16840307C418yescomplemented
16C2×#14519776C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519763—21no—
19C2×#14519751C2211yescharacteristic
20C2×#14519749—21no—
21C2×D4×Dic61C2211yescharacteristic
22C2×#14519739C2211yescharacteristic
23C2×#14519737—21no—
24C22×C4×Dic61C2211yescharacteristic
25C2×#14519699C2212yes—
26C2×#14519698C2212yes—
27C2×#14519697C2212yes—
28C2×#14519687—24no—
29C2×#14519687C2212yes—
30C2×#14519687C2212yes—
31#16839629—21no—
32#16839627C2212yes—
33#16839623—21no—
34#16839622C2211yescharacteristic
35C4⋊C4×Dic61C2212yes—
36C22⋊C4×Dic61—21no—
37#16839607—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C23×C8×C61C2211yescharacteristic
42C2×#121×C61C2211yescharacteristic
43C2×#102×C61C2212yes—
44C61×#380C2211yescharacteristic
45C61×#340C2212yes—
46C2×#11758224—28no—
47C2×#11758224—44no—
48C2×#11758224—44no—
49C22×Dic122—21no—
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—21no—
52C22×C22⋊D61—21no—
53C22×C22⋊D61—21no—
54C23×Dic61—21no—
55C23×Dic61D411yescharacteristic
56C23×Dic61D411yescharacteristic
57C22×C4×D61—21no—
58C22×#92418—22no—
59C22×#92418—24no—
60C22×#92418C2312yes—
61#14519776—22no—
62#14519776C2×C414yes—
63#14519773—22no—
64#14519773C2×C414yes—
65#14519763—28no—
66#14519751—22no—
67#14519751C2×C414yes—
68#14519749—28no—
69D4×Dic61—22no—
70D4×Dic61C2×C414yes—
71#14519739—22no—
72#14519739C2×C414yes—
73#14519737—28no—
74C2×C4⋊Dic61C2312yes—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61C2×C414yes—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61D414yes—
86C2×C4×Dic61C2311yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88#14519699—24no—
89#14519698—24no—
90#14519697—24no—
91#14519687—24no—
92#14519687—24no—
93#14519684—21no—
94#14519684—22no—
95#14519684—22no—
96#14519684—22no—
97C22×D4×C61—21no—
98C22×D4×C61C2311yescharacteristic
99C23×C4×C61C2311yescharacteristic
100C22×C8×C61—22no—
101C22×C8×C61—24no—
102C22×C8×C61C2312yes—
103#121×C61—22no—
104#121×C61C2×C414yes—
105C2×C4⋊C4×C61C2312yes—
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61—21no—
109#102×C61—24no—
110#95×C61—21no—
111C24×C61—21no—
112C24×C61C2×D411yescharacteristic
113C23×D61—21no—
114C2×D4×C61—24no—
115C2×D4×C61—21no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—42no—
119C2×D4×C61—42no—
120C2×D4×C61C22×C412yes—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—21no—
124C22×C4×C61—22no—
125C22×C4×C61C22×C412yes—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—24no—
130C2×C22⋊D61—24no—
131C2×C22⋊D61—24no—
132C2×C22⋊D61—44no—
133C2×C22⋊D61—42no—
134C2×C22⋊D61—42no—
135C2×C22⋊D61—42no—
136C2×C22⋊D61—42no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—21no—
140C22×Dic61—24no—
141C22×Dic61—22no—
142C22×Dic61—42no—
143C22×Dic61—42no—
144C22×Dic61—42no—
145C22×Dic61C22⋊C414yes—
146C22×Dic61C22⋊C414yes—
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151#11758224—432no—
152C2×D244—22no—
153C2×D244—22no—
154C2×D244—21no—
155C2×D244C22×C412yes—
156C2×C4×D61—24no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×Dic122—24no—
160C2×Dic122—42no—
161C2×Dic122—42no—
162C2×C8×C61—28no—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C2×C8×C61#4114yescomplemented
166C4⋊C4×C61—21no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61C22×C412yes—
170C22⋊C4×C61—24no—
171C22⋊C4×C61—42no—
172C22⋊Dic61—24no—
173C22⋊Dic61—42no—
174#11758198—24no—
175#11758198—24no—
176C4⋊Dic61—21no—
177C4⋊Dic61—22no—
178C4⋊Dic61—22no—
179C4⋊Dic61C22×C412yes—
180#11758196—24no—
181C4×Dic61—44no—
182C4×Dic61C22⋊C414yes—
183C4×Dic61C22⋊C414yes—
184C4×Dic61C22×C412yes—
185C4×Dic61C22×C412yes—
186C4×Dic61C2×D412yes—
187C4×Dic61C2×D412yes—
188C2×#92418—28no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418#4114yescomplemented
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—42no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—42no—
199C23×C61C2×C22⋊C412yes—
200C23×C61AGL(1,ℤ/8)12yes—
201C23×C61AGL(1,ℤ/8)12yes—
202C23×C61C22×D411yescharacteristic
203C22×D61—22no—
204C22×D61—42no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—42no—
208D4×C61—24no—
209D4×C61—48no—
210D4×C61—41no—
211D4×C61—41no—
212C2×C4×C61—22no—
213C2×C4×C61—24no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61C2×C22⋊C412yes—
223C2×C4×C61C23×C411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C2×#4112yes—
226C2×C4×C61C2×#4112yes—
227C22⋊D61—48no—
228C22⋊D61—48no—
229C22⋊D61—48no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—28no—
237C2×Dic61—22no—
238C2×Dic61—44no—
239C2×Dic61—48no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61C2×C22⋊C412yes—
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61C2×C22⋊C412yes—
248C2×Dic61C2×C22⋊C412yes—
249D244—24no—
250D244—41no—
251D244—41no—
252C4×D61—48no—
253Dic122—48no—
254C8×C61—24no—
255#92418—24no—
256#21408—611nomaximal, Sylow
257C22×C61—22no—
258C22×C61—24no—
259C22×C61—24no—
260C22×C61—22no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—82no—
265C22×C61—82no—
266C22×C61#41814yes—
267C22×C61C22×C22⋊C411yescharacteristic
268C22×C61#53212yes—
269C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C61#57512yes—
272C2×D61—42no—
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—82no—
276C2×D61—82no—
277C4×C61—22no—
278C4×C61—42no—
279C4×C61—82no—
280C4×C61C4×#4112yes—
281C4×C61#52511yescharacteristic
282C4×C61#52511yescharacteristic
283Dic61—22no—
284Dic61—22no—
285Dic61—22no—
286Dic61—22no—
287Dic61—42no—
288Dic61—82no—
289C2×#532—611no—
290C22×#97—611no—
291#2283—611no—
292#1993—612no—
293#1961—612no—
294#1893—611no—
295#1893—611no—
296#1893—611no—
297#1893D6111yescharacteristic, complemented
298#1762—612no—
299#1762—612no—
300C2×C61—24no—
301C2×C61—82no—
302C2×C61#229711yescharacteristic
303C2×C61C2×#41811yescharacteristic
304C2×C61#289611yescharacteristic
305C2×C61#318012yes—
306C2×C61#318012yes—
307D61—82no—
308C22×#41—1221no—
309C23×C8—611no—
310C23×C8C2×D6111yescharacteristic
311#532—618no—
312#532—1224no—
313C2×C4⋊Q8—611no—
314C2×#127—611no—
315C2×#124—1221no—
316C2×#121—611no—
317C2×#121—1221no—
318C2×#121C2×D6111yescharacteristic
319C2×C4×D4—611no—
320C2×C4×D4—611no—
321C2×C4×D4—1221no—
322C22×C42—611no—
323C2×#102—612no—
324C2×#102—612no—
325C2×#102—612no—
326C2×#102C2×D6112yes—
327C2×#97—612no—
328C2×#97—612no—
329C2×#97—1224no—
330#383—1221no—
331#382—612no—
332#381—1221no—
333#380—611no—
334#380C2×D6111yescharacteristic
335C4×C4⋊C4—612no—
336C4×C22⋊C4—1221no—
337#376—1221no—
338#340—612no—
339#340—612no—
340#340—612no—
341#340C2×D6112yes—
342C61#2140811yescharacteristic, complemented, Sylow
343C2×#41—1228no—
344C2×#41—2444no—
345C2×#41—2444no—
346C22×Q8—1221no—
347C22×D4—21no—
348C22×D4—611no—
349C22×D4—1221no—
350C22×D4—1221no—
351C22×D4—1221no—
352C22×D4C22×D6111yescharacteristic
353C23×C4—611no—
354C23×C4—611no—
355C23×C4—1221no—
356C23×C4—1221no—
357C23×C4C22×D6111yescharacteristic
358C22×C8—22no—
359C22×C8—24no—
360C22×C8—612no—
361C22×C8—1224no—
362C22×C8—1222no—
363C22×C8C22×D6112yes—
364C4⋊Q8—614no—
365C4⋊Q8—1222no—
366#127—614no—
367#127—1222no—
368#124—1228no—
369#121—22no—
370#121—614no—
371#121—1228no—
372#121—1222no—
373#121C4×D6114yes—
374C4×D4—614no—
375C4×D4—614no—
376C4×D4—1228no—
377C4×D4—1222no—
378C4×D4—1222no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4C22×D6112yes—
383C2×C4⋊C4C22×D6111yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C42—614no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—614no—
394C2×C42—611no—
395C2×C42—611no—
396C2×C42—1222no—
397#102—24no—
398#102—1224no—
399#102—1224no—
400#102—1224no—
401#97—1224no—
402#97—1224no—
403#95—21no—
404#95—1222no—
405#95—1222no—
406#95—1222no—
407#95—1221no—
408C24—21no—
409C24—1221no—
410C24D4×D6111yescharacteristic
411#41—24432no—
412C2×Q8—1224no—
413C2×Q8—2442no—
414C2×Q8—2442no—
415C2×D4—24no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—612no—
422C2×D4—1224no—
423C2×D4—1224no—
424C2×D4—1224no—
425C2×D4—1221no—
426C2×D4—1222no—
427C2×D4—1222no—
428C2×D4—2442no—
429C2×D4—2442no—
430C2×D4—2444no—
431C2×D4—2442no—
432C2×D4—2442no—
433C2×D4C2×C4×D6112yes—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—614no—
443C22×C4—614no—
444C22×C4—1224no—
445C22×C4—1224no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1221no—
449C22×C4—1222no—
450C22×C4—2442no—
451C22×C4—2442no—
452C22×C4—2442no—
453C22×C4—2442no—
454C22×C4—2442no—
455C22×C4C2×C4×D6112yes—
456C22×C4D4×D6112yes—
457C22×C4D4×D6111yescharacteristic
458C22×C4C23×D6111yescharacteristic
459C2×C8—28no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—614no—
463C2×C8—1228no—
464C2×C8—1222no—
465C2×C8—1222no—
466C2×C8#1175822414yescomplemented
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—612no—
471C4⋊C4—1224no—
472C4⋊C4—1221no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4C2×C4×D6112yes—
476C22⋊C4—24no—
477C22⋊C4—42no—
478C22⋊C4—1224no—
479C22⋊C4—1224no—
480C22⋊C4—1224no—
481C22⋊C4—2442no—
482C42—612no—
483C42—612no—
484C42—612no—
485C42—612no—
486C42—614no—
487C42—614no—
488C42—2444no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23—1222no—
497C23—2441no—
498C23—2441no—
499C23—2442no—
500C23—2442no—
501C23C22⋊C4×D6112yes—
502C23#1451980112yes—
503C23#1451980412yes—
504C23C2×D4×D6111yescharacteristic
505Q8—2448no—
506D4—24no—
507D4—48no—
508D4—41no—
509D4—41no—
510D4—1224no—
511D4—2448no—
512D4—2448no—
513D4—2448no—
514D4—2441no—
515D4—2441no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—612no—
527C2×C4—612no—
528C2×C4—612no—
529C2×C4—612no—
530C2×C4—1228no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1222no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—2448no—
539C2×C4—2444no—
540C2×C4—2448no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4C22⋊C4×D6112yes—
547C2×C4C22×C4×D6111yescharacteristic
548C2×C4C2×D4×D6111yescharacteristic
549C2×C4#41×D6112yes—
550C2×C4C2×#1175822412yes—
551C8—24no—
552C8—1224no—
553C22—22no—
554C22—24no—
555C22—24no—
556C22—22no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—82no—
561C22—82no—
562C22—4882no—
563C22—4882no—
564C22—2442no—
565C22—2442no—
566C22—2442no—
567C22#1683976514yes—
568C22C2×C22⋊C4×D6111yescharacteristic
569C22#1684030712yes—
570C22C2×#1451980111yescharacteristic
571C22C2×#1451980411yescharacteristic
572C22#1684055912yes—
573C4—22no—
574C4—42no—
575C4—82no—
576C4—1222no—
577C4—1222no—
578C4—1222no—
579C4—1222no—
580C4—4882no—
581C4—2442no—
582C4C4×#1175822412yes—
583C4#1684026811yescharacteristic
584C4#1684027011yescharacteristic
585C2—24no—
586C2—82no—
587C2—4882no—
588C2#2144830411yescharacteristic
589C2C2×#1683976511yescharacteristic
590C2#2145433411yescharacteristic
591C2#2145651012yes—
592C2#2145651212yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1893C2
socleC23×C61C22×D4
prime radicalC2×#16840307C2
cosocleC22×C4×C61C24
hypercenter#1893D61
hypocenterC61#21408
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1893

Complements

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

NH ≅ G/Ntype
2. C22×#14519687C2semidirect
3. #21448214C2semidirect
4. #21446790C2semidirect
5. #21446641C2semidirect
6. #21446248C2semidirect
7. #21446247C2semidirect
8. #21446245C2semidirect
9. #21445646C2semidirect
10. #21445643C2semidirect
11. C61×#1893C2semidirect
15. #16840307C4semidirect
165. C2×C8×C61#41semidirect
191. C2×#92418#41semidirect
297. #1893D61semidirect
342. C61#21408semidirect
466. C2×C8#11758224semidirect

Maximal subgroups

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

subgroupindex
C2×#168403072
C22×#145196872
#214482142
#214467902
#214466412
#214462482
#214462472
#214462452
#214456462
#214456432
C61×#18932
#2140861