Finite Group Databasebeta

← #29714065

Subgroups — #29714065

Order 15616.

1204 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840477C211yesmaximal, characteristic, complemented
2C2×#16840042C211yesmaximal, characteristic, complemented
3#21450716C211yesmaximal, characteristic, complemented
4#21446999C212yesmaximal, complemented
5#21446814C212yesmaximal, complemented
6#21446792C211yesmaximal, characteristic, complemented
7#21446617C211yesmaximal, characteristic, complemented
8#21446138C212yesmaximal, complemented
9#21446017C212yesmaximal, complemented
10#21445829C211yesmaximal, characteristic, complemented
11C2×C61×#442C211yesmaximal, characteristic, complemented
12#16840477—24no—
13#16840477—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519763—22no—
16C2×#14519751—21no—
17C2×#14519748C2212yes—
18C2×#14519748C2212yes—
19C2×C4×D244C2211yescharacteristic
20C2×#14519736—21no—
21C2×#14519735C2212yes—
22#16840042—24no—
23C2×#14519710C2212yes—
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519698C2212yes—
26C2×#14519698C2212yes—
27C2×#14519689C2212yes—
28C2×#14519689C2212yes—
29#16839687—21no—
30#16839671—21no—
31#16839641—22no—
32#16839628C2211yescharacteristic
33#16839619C2212yes—
34#16839525C2212yes—
35#16839525C2212yes—
36#16839524C2212yes—
37C22×D8×C61C2211yescharacteristic
38C2×C4×D4×C61C2211yescharacteristic
39C2×C4×D4×C61C2211yescharacteristic
40C61×#442—24no—
41C2×#106×C61C2211yescharacteristic
42C2×#102×C61C2211yescharacteristic
43C2×#102×C61C2211yescharacteristic
44C2×#97×C61C2211yescharacteristic
45C22×Dic122—21no—
46C22×D244D411yescharacteristic
47C22×C22⋊D61—22no—
48C22×C22⋊D61—22no—
49C22×C4×D61C2312yes—
50C22×#92418C2312yes—
51#14519777—22no—
52#14519777D414yescomplemented
53#14519763—216no—
54#14519751—28no—
55#14519748—24no—
56#14519748—24no—
57#14519748—24no—
58#14519748—42no—
59C4×D244—22no—
60C4×D244—22no—
61#14519736—28no—
62#14519735—24no—
63#14519735—24no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61D411yescharacteristic
66C2×C4⋊Dic61C2312yes—
67C2×C4⋊Dic61C2312yes—
68C2×C22⋊Dic61—22no—
69C2×C22⋊Dic61—22no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×#11758198D412yes—
73C2×C4×Dic61—22no—
74#14519710—24no—
75C4⋊#92418—22no—
76#14519698—24no—
77#14519698—42no—
78#14519689—24no—
79#14519689—42no—
80#14519684—21no—
81#14519684—21no—
82#14519684—22no—
83C22×D4×C61C2311yescharacteristic
84C22×D4×C61C2311yescharacteristic
85C23×C4×C61—21no—
86C23×C4×C61—21no—
87C2×D8×C61—22no—
88C2×D8×C61—22no—
89C2×D8×C61—22no—
90C22×C8×C61C2311yescharacteristic
91C22×C8×C61C2311yescharacteristic
92C4×D4×C61—22no—
93C4×D4×C61—22no—
94C4×D4×C61—22no—
95C4×D4×C61D414yescomplemented
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C22⋊C4×C61—21no—
99C2×C22⋊C4×C61—21no—
100C2×C42×C61C2311yescharacteristic
101#106×C61—22no—
102#102×C61—22no—
103#102×C61—22no—
104#97×C61—22no—
105C24×C61—21no—
106C24×C61—21no—
107C23×D61C2×D412yes—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61C2×D412yes—
119C2×D4×C61#4112yes—
120C22×C4×C61—21no—
121C22×C4×C61—24no—
122C22×C4×C61—21no—
123C22×C4×C61—44no—
124C22×C4×C61—42no—
125C22×C4×C61—82no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—28no—
130C2×C22⋊D61—44no—
131C2×C22⋊D61—44no—
132C2×C22⋊D61—82no—
133C2×C22⋊D61—82no—
134C2×C22⋊D61—82no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61C2×D412yes—
138C2×D244—22no—
139C2×D244—42no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×Dic122—24no—
147C2×Dic122—82no—
148D8×C61—28no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61#4112yes—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61C2×D412yes—
159C22⋊C4×C61—24no—
160C22⋊C4×C61—42no—
161C42×C61C2×D412yes—
162C42×C61C2×D412yes—
163C22⋊Dic61—28no—
164C22⋊Dic61—44no—
165#11758198—28no—
166#11758198—24no—
167#11758198—82no—
168C4⋊Dic61—24no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61C2×D414yes—
174C4×Dic61—28no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418—22no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61C22×D411yescharacteristic
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—22no—
196D4×C61—22no—
197D4×C61—22no—
198D4×C61—22no—
199D4×C61—22no—
200D4×C61—41no—
201D4×C61—41no—
202D4×C61—41no—
203D4×C61—41no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—48no—
208C2×C4×C61—42no—
209C2×C4×C61—41no—
210C2×C4×C61—42no—
211C2×C4×C61—84no—
212C2×C4×C61#12012yes—
213C2×C4×C61#12012yes—
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61H(5,𝔽2)11yescharacteristic
219C2×C4×C612−1+411yescharacteristic
220C22⋊D61—88no—
221C22⋊D61—88no—
222C2×Dic61—22no—
223C2×Dic61—24no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231D244—44no—
232C4×D61—42no—
233C4×D61—42no—
234C4×D61—42no—
235C4×D61—42no—
236Dic122—84no—
237C8×C61—22no—
238C8×C61—41no—
239#92418—42no—
240#24129—611nomaximal, Sylow
241C22×C61—42no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—44no—
246C22×C61—82no—
247C22×C61—82no—
248C22×C61—81no—
249C22×C61—81no—
250C22×C61—82no—
251C22×C61C2×#12011yescharacteristic
252C22×C61#54611yescharacteristic
253C22×C61#54711yescharacteristic
254C22×C61#57512yes—
255C22×C61#57612yes—
256C2×D61—42no—
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—42no—
263C4×C61—22no—
264C4×C61—22no—
265C4×C61—42no—
266C4×C61—81no—
267C4×C61#52511yescharacteristic
268C4×C61#53711yescharacteristic
269C4×C61#54811yescharacteristic
270C4×C61#55011yescharacteristic
271Dic61—42no—
272Dic61—44no—
273Dic61—82no—
274C2×#555—611no—
275C2×#474—611no—
276C2×#442D6111yescharacteristic, complemented
277#2535—611no—
278#2033—612no—
279#1999—612no—
280#1993—611no—
281#1955—611no—
282#1875—612no—
283#1854—612no—
284#1804—611no—
285C2×C61—82no—
286C2×C61—82no—
287C2×C61#240711yescharacteristic
288C2×C61#300511yescharacteristic
289C2×C61#300811yescharacteristic
290C2×C61#331912yes—
291C2×C61#334712yes—
292D61—82no—
293C22×D8C2×D6111yescharacteristic
294#555—1224no—
295#555—1224no—
296C2×C4⋊Q8—611no—
297C2×#124—1222no—
298C2×#121—612no—
299C2×#121—612no—
300C2×#121—1221no—
301C2×C4×D4—611no—
302C2×C4×D4C2×D6111yescharacteristic
303C2×C4×D4C2×D6111yescharacteristic
304C2×#117—612no—
305C2×#117—1221no—
306#474—1224no—
307#442—24no—
308C2×#107—612no—
309C2×#106C2×D6111yescharacteristic
310C2×C4⋊C8—611no—
311C2×#102—612no—
312C2×#102—612no—
313C2×#102C2×D6111yescharacteristic
314C2×#102C2×D6111yescharacteristic
315C2×C22⋊C8—612no—
316C2×C22⋊C8—612no—
317C2×#97C2×D6111yescharacteristic
318#397—1221no—
319#392—1221no—
320#386—1222no—
321#382—611no—
322#380—612no—
323#340—612no—
324#340—612no—
325#340—612no—
326C61#2412911yescharacteristic, complemented, Sylow
327C22×Q8—1221no—
328C22×D4—611no—
329C22×D4—1222no—
330C22×D4—1222no—
331C22×D4C22×D6111yescharacteristic
332C22×D4C22×D6111yescharacteristic
333C23×C4—21no—
334C23×C4—21no—
335C23×C4—612no—
336C2×D8—22no—
337C2×D8—22no—
338C2×D8—22no—
339C22×C8—612no—
340C22×C8C22×D6111yescharacteristic
341C22×C8C22×D6111yescharacteristic
342C4⋊Q8—614no—
343C4⋊Q8—1222no—
344#124—12216no—
345#121—1224no—
346#121—1228no—
347#121—1224no—
348#121—1224no—
349#121—2442no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—22no—
353C4×D4—1222no—
354C4×D4—1222no—
355C4×D4C22⋊D6114yescomplemented
356#117—1224no—
357#117—1228no—
358#117—1224no—
359C2×C4⋊C4—612no—
360C2×C4⋊C4—611no—
361C2×C4⋊C4—612no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4C22×D6111yescharacteristic
364C2×C4⋊C4C22×D6111yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—612no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4—1222no—
370C2×C22⋊C4—1222no—
371C2×C22⋊C4—1222no—
372C2×C42—1222no—
373C2×C42C22×D6111yescharacteristic
374#107—1224no—
375#106—22no—
376C4⋊C8—1222no—
377#102—22no—
378#102—22no—
379#102—1224no—
380#102—2442no—
381C22⋊C8—1224no—
382C22⋊C8—2442no—
383#97—22no—
384#95—1222no—
385#95—1221no—
386#95—1221no—
387C24—21no—
388C24—21no—
389C24—612no—
390C2×Q8—1224no—
391C2×Q8—4882no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—1228no—
403C2×D4—1222no—
404C2×D4—4882no—
405C2×D4—4882no—
406C2×D4—4882no—
407C2×D4—2442no—
408C2×D4—2444no—
409C2×D4—2444no—
410C2×D4#1175822212yes—
411C2×D4C2×C22⋊D6112yes—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—44no—
416C22×C4—42no—
417C22×C4—82no—
418C22×C4—612no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4D4×D6111yescharacteristic
428C22×C4D4×D6111yescharacteristic
429C22×C4C23×D6111yescharacteristic
430D8—28no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—1222no—
437C2×C8—1222no—
438C2×C8—1222no—
439C2×C8#1175822212yes—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—614no—
444C4⋊C4—1222no—
445C4⋊C4—1222no—
446C4⋊C4—1224no—
447C4⋊C4—1222no—
448C4⋊C4—1222no—
449C4⋊C4C2×C22⋊D6112yes—
450C22⋊C4—24no—
451C22⋊C4—42no—
452C22⋊C4—1228no—
453C22⋊C4—1228no—
454C22⋊C4—1224no—
455C22⋊C4—4882no—
456C22⋊C4—2444no—
457C42—1228no—
458C42D4×D6112yes—
459C42C2×C22⋊D6112yes—
460C23—22no—
461C23—22no—
462C23—41no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—1222no—
471C23—1222no—
472C23—1222no—
473C23—1222no—
474C23—1222no—
475C23—1222no—
476C23—1222no—
477C23C2×D4×D6111yescharacteristic
478Q8—4884no—
479D4—22no—
480D4—22no—
481D4—22no—
482D4—22no—
483D4—41no—
484D4—41no—
485D4—41no—
486D4—41no—
487D4—4888no—
488D4—4888no—
489D4—2444no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—48no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—84no—
498C2×C4—1222no—
499C2×C4—1224no—
500C2×C4—1222no—
501C2×C4—1222no—
502C2×C4—2442no—
503C2×C4—2442no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4#1451974512yes—
512C2×C4#1451974512yes—
513C2×C4C2×D4×D6111yescharacteristic
514C2×C4C22×C22⋊D6111yescharacteristic
515C2×C4C2×#1175822211yescharacteristic
516C2×C4C2×#1175822211yescharacteristic
517C2×C4#1451982311yescharacteristic
518C2×C4#1451982411yescharacteristic
519C8—22no—
520C8—41no—
521C8—2442no—
522C22—44no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—81no—
531C22—81no—
532C22—2442no—
533C22—2442no—
534C22—2442no—
535C22—2442no—
536C22—2442no—
537C22—2442no—
538C22—2442no—
539C22C2×#1451974511yescharacteristic
540C22#1684040811yescharacteristic
541C22#1684041711yescharacteristic
542C22#1684055812yes—
543C22#1684056612yes—
544C4—22no—
545C4—22no—
546C4—42no—
547C4—81no—
548C4—4882no—
549C4—2442no—
550C4—2444no—
551C4#1684027311yescharacteristic
552C4#1684034011yescharacteristic
553C4#1684042111yescharacteristic
554C4#1684043311yescharacteristic
555C2—82no—
556C2—82no—
557C2—4882no—
558C2#2144916411yescharacteristic
559C2#2145505811yescharacteristic
560C2#2145508611yescharacteristic
561C2#2145798012yes—
562C2#2145832412yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#442C2
socleC23×C61C22×D4
hypercenterC2×#442D61
hypocenterC61#24129
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#442

Complements

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

NH ≅ G/Ntype
1. C2×#16840477C2semidirect
2. C2×#16840042C2semidirect
3. #21450716C2semidirect
4. #21446999C2semidirect
5. #21446814C2semidirect
6. #21446792C2semidirect
7. #21446617C2semidirect
8. #21446138C2semidirect
9. #21446017C2semidirect
10. #21445829C2semidirect
11. C2×C61×#442C2semidirect
52. #14519777D4semidirect
95. C4×D4×C61D4semidirect
276. C2×#442D61semidirect
326. C61#24129semidirect
355. C4×D4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168404772
C2×#168400422
#214507162
#214469992
#214468142
#214467922
#214466172
#214461382
#214460172
#214458292
C2×C61×#4422
#2412961