Finite Group Databasebeta

← #29716380

Subgroups — #29716380

Order 15616.

1330 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840427C211yesmaximal, characteristic, complemented
2C2×C4×#11758199C211yesmaximal, characteristic
3#21450741C211yesmaximal, characteristic, complemented
4#21447145C212yesmaximal, complemented
5#21446953C212yesmaximal, complemented
6#21446781C211yesmaximal, characteristic, complemented
7#21446754C211yesmaximal, characteristic, complemented
8#21446164C212yesmaximal
9#21446063C212yesmaximal, complemented
10#21445820C211yesmaximal, characteristic, complemented
11C2×C61×#474C211yesmaximal, characteristic, complemented
12C22×Q8×D61—21no—
13C22×#11758199C2211yescharacteristic
14#16840427—24no—
15#16840427—24no—
16C2×#14519776C2212yes—
17C2×#14519756—21no—
18C2×#14519755—22no—
19C2×#14519753—21no—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×#14519739C2212yes—
23C4×#11758199—24no—
24C2×#14519710C2211yescharacteristic
25C2×#14519699C2212yes—
26C2×#14519698C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×C4×#92418C2211yescharacteristic
29#16839685—22no—
30#16839665C2212yes—
31#16839658C2212yes—
32#16839632C2211yescharacteristic
33#16839622—21no—
34#16839525C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839516C2212yes—
38#16839516C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4×D4×C61C2211yescharacteristic
41C61×#474—24no—
42C2×#107×C61C2212yes—
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#102×C61C2212yes—
45C22×Dic122—22no—
46C22×Dic122D411yescharacteristic
47C2×Q8×D61—28no—
48C2×Q8×D61—44no—
49C2×Q8×D61—44no—
50C22×D244C2311yescharacteristic
51C22×C4×D61—21no—
52C22×C4×D61—22no—
53C2×#11758199—22no—
54C2×#11758199—22no—
55C2×#11758199Q814yescomplemented
56C22×#92418C2311yescharacteristic
57C22×#92418C2311yescharacteristic
58#14519776—24no—
59#14519776—24no—
60#14519756—44no—
61#14519755—216no—
62#14519753—28no—
63C4×Dic122—22no—
64C4×Dic122—22no—
65C4×D244—22no—
66C4×D244D414yes—
67#14519739—24no—
68#14519739—24no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—22no—
75C2×#11758196—22no—
76C2×#11758196D412yes—
77C2×C22⋊Dic61—22no—
78C2×C22⋊Dic61—22no—
79C2×#11758198—21no—
80C2×#11758198—22no—
81C2×C4×Dic61C2312yes—
82#14519710—22no—
83#14519699—24no—
84#14519698—22no—
85#14519697—22no—
86C4×#92418—22no—
87#14519684—22no—
88C22×Q8×C61—21no—
89C22×D4×C61C2311yescharacteristic
90C23×C4×C61—21no—
91C22×C8×C61C2312yes—
92C4⋊Q8×C61—22no—
93C4⋊Q8×C61D414yes—
94C4×D4×C61—22no—
95C4×D4×C61—22no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C4⋊C4×C61C2312yes—
99C2×C22⋊C4×C61—21no—
100C2×C42×C61C2311yescharacteristic
101#107×C61—24no—
102C4⋊C8×C61—22no—
103#102×C61—24no—
104C24×C61—21no—
105C23×D61—21no—
106C2×Q8×C61—24no—
107C2×Q8×C61—42no—
108C2×Q8×C61—42no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61C2×Q812yescomplemented
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—42no—
118C22×C4×C61—42no—
119C22×C4×C61—42no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C22×Dic61—21no—
124C22×Dic61—22no—
125C22×Dic61C2×D412yes—
126C22×Dic61C2×D412yes—
127Q8×D61—432no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244C2×D412yes—
132C2×D244C2×Q812yes—
133C2×D244#4112yes—
134C2×C4×D61—28no—
135C2×C4×D61—24no—
136C2×C4×D61—44no—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×C4×D61—44no—
140C2×Dic122—22no—
141C2×Dic122—22no—
142C2×Dic122—28no—
143C2×Dic122—44no—
144C2×Dic122—44no—
145C2×Dic122D814yescomplemented
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61C2×D414yes—
154C22⋊C4×C61—42no—
155C42×C61C2×D412yes—
156C42×C61C2×D412yes—
157C22⋊Dic61—44no—
158C22⋊Dic61—82no—
159#11758199—28no—
160#11758198—28no—
161#11758198—24no—
162C4⋊Dic61—22no—
163C4⋊Dic61—21no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—42no—
167C4⋊Dic61—81no—
168C4⋊Dic61C2×D412yes—
169#11758196—28no—
170#11758196—24no—
171#11758196—44no—
172C4×Dic61—22no—
173C4×Dic61—22no—
174C2×#92418—21no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—21no—
179C2×#92418C2×Q812yescomplemented
180C23×C61—22no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61C22×D411yescharacteristic
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195Q8×C61—48no—
196D4×C61—22no—
197D4×C61—22no—
198D4×C61—41no—
199D4×C61—41no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—24no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—84no—
209C2×C4×C61#12012yes—
210C2×C4×C61#12012yes—
211C2×C4×C61C2×D812yes—
212C2×C4×C61AGL(1,ℤ/8)12yes—
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×Q811yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C612−1+411yescharacteristic
218C2×C4×C612−1+411yescharacteristic
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61C2×D814yes—
231D244—22no—
232D244—22no—
233D244—22no—
234D244C22⋊Q814yescomplemented
235C4×D61—416no—
236C4×D61—48no—
237Dic122—28no—
238Dic122—416no—
239C8×C61—42no—
240#92418—22no—
241#92418—41no—
242#24326—611nomaximal, Sylow
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—81no—
247C22×C61—81no—
248C22×C61—81no—
249C22×C61—81no—
250C22×C61C2×#12011yescharacteristic
251C22×C61#54811yescharacteristic
252C22×C61D4×Q811yescharacteristic
253C22×C61C22×D811yescharacteristic
254C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C61#57711yescharacteristic
256C22×C61#57811yescharacteristic
257C2×D61—24no—
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—42no—
263C2×D61—42no—
264C4×C61—22no—
265C4×C61—44no—
266C4×C61—81no—
267C4×C61#44714yes—
268C4×C61C2×C22⋊Q811yescharacteristic
269C4×C61#53611yescharacteristic
270C4×C61#54811yescharacteristic
271C4×C61D4×Q811yescharacteristic
272Dic61—24no—
273Dic61—42no—
274Dic61—42no—
275Dic61—82no—
276C2×D4×Q8—611no—
277C2×#474D6111yescharacteristic, complemented
278C2×C4×D8—611no—
279#2538—611no—
280#2057—612no—
281#2023—612no—
282#1990—611no—
283#1984—611no—
284#1879—612no—
285#1862—612no—
286#1802—611no—
287C2×C61—82no—
288C2×C61#240911yescharacteristic
289C2×C61C2×#44711yescharacteristic
290C2×C61#301311yescharacteristic
291C2×C61#335811yescharacteristic
292C2×C61#336711yescharacteristic
293C2×C61Q8×D811yescharacteristic
294C2×C61#338411yescharacteristic
295D61—44no—
296C23×Q8—1221no—
297C22×D8—611no—
298D4×Q8—1224no—
299D4×Q8—1224no—
300C2×C4⋊Q8—612no—
301C2×C4⋊Q8C2×D6111yescharacteristic
302C2×C22⋊Q8—1222no—
303C2×C22⋊Q8—1221no—
304C2×C22⋊Q8—1221no—
305C2×C4×Q8—611no—
306C2×C4×D4—612no—
307C2×C4×D4—611no—
308C2×C4×D4C2×D6111yescharacteristic
309#474—24no—
310C4×D8—1224no—
311C2×#107—611no—
312C2×#107C2×D6112yes—
313C2×C4⋊C8C2×D6111yescharacteristic
314C2×#102—611no—
315C2×#102—612no—
316C2×#102—611no—
317C2×#102C2×D6112yes—
318C2×C4×C8—611no—
319#396—1222no—
320#390—612no—
321#389—612no—
322#384—611no—
323#380—1221no—
324#340—612no—
325#340—612no—
326#340—612no—
327#336—612no—
328#336—612no—
329C61#2432611yescharacteristic, complemented, Sylow
330C22×Q8—21no—
331C22×Q8—611no—
332C22×Q8—1222no—
333C22×Q8—1228no—
334C22×Q8—2444no—
335C22×Q8—2444no—
336C22×D4—611no—
337C22×D4C22×D6111yescharacteristic
338C23×C4—21no—
339C23×C4—1222no—
340C23×C4—1221no—
341C2×D8—614no—
342C2×D8—1222no—
343C2×D8—1222no—
344C22×C8—611no—
345C22×C8—611no—
346C22×C8C22×D6112yes—
347C4⋊Q8—22no—
348C4⋊Q8—1224no—
349C4⋊Q8—1224no—
350C4⋊Q8C22⋊D6114yes—
351C22⋊Q8—12216no—
352C22⋊Q8—1228no—
353C22⋊Q8—2444no—
354C4×Q8—1222no—
355C4×Q8—1222no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—614no—
359C4×D4—1224no—
360C4×D4—1224no—
361C4×D4—1222no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—611no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—612no—
366C2×C4⋊C4—612no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4C22×D6111yescharacteristic
372C2×C4⋊C4C22×D6112yes—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1221no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4—1222no—
378C2×C42—612no—
379C2×C42C22×D6111yescharacteristic
380#107—24no—
381#107—1222no—
382C4⋊C8—22no—
383#102—24no—
384#102—1222no—
385#102—1222no—
386#102—1224no—
387C4×C8—1222no—
388#95—1222no—
389C24—21no—
390C24—1221no—
391C2×Q8—24no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—614no—
395C2×Q8—1228no—
396C2×Q8—1222no—
397C2×Q8—1222no—
398C2×Q8—2444no—
399C2×Q8—2444no—
400C2×Q8—24432no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—612no—
407C2×D4—612no—
408C2×D4—612no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4Q8×D6112yescomplemented
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—612no—
419C22×C4—612no—
420C22×C4—1228no—
421C22×C4—1221no—
422C22×C4—1224no—
423C22×C4—1222no—
424C22×C4—2444no—
425C22×C4—2444no—
426C22×C4—2442no—
427C22×C4—2442no—
428C22×C4D4×D6111yescharacteristic
429C22×C4D4×D6111yescharacteristic
430C22×C4C23×D6111yescharacteristic
431D8—1228no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—612no—
436C2×C8—1221no—
437C2×C8—1221no—
438C2×C8—1221no—
439C2×C8—1221no—
440C2×C8—1221no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—612no—
446C4⋊C4—1228no—
447C4⋊C4—1221no—
448C4⋊C4—1224no—
449C4⋊C4—1222no—
450C4⋊C4—1222no—
451C4⋊C4—1222no—
452C4⋊C4—4881no—
453C4⋊C4—2444no—
454C4⋊C4—2442no—
455C4⋊C4C2×C22⋊D6114yes—
456C22⋊C4—42no—
457C22⋊C4—1228no—
458C22⋊C4—1224no—
459C22⋊C4—4882no—
460C22⋊C4—2444no—
461C42—1222no—
462C42—1222no—
463C42D4×D6112yes—
464C42C2×C22⋊D6112yes—
465C23—22no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—1222no—
473C23—1222no—
474C23—1222no—
475C23—2441no—
476C23—2441no—
477C23—2441no—
478C23—2441no—
479C23C2×D4×D6111yescharacteristic
480Q8—48no—
481Q8—1228no—
482Q8—24416no—
483D4—22no—
484D4—22no—
485D4—41no—
486D4—41no—
487D4—614no—
488D4—1222no—
489D4—1222no—
490D4—1222no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—84no—
500C2×C4—614no—
501C2×C4—1222no—
502C2×C4—1222no—
503C2×C4—1222no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—24416no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2448no—
514C2×C4#1451974512yes—
515C2×C4#1451974512yes—
516C2×C4D8×D6112yes—
517C2×C4#1451980112yes—
518C2×C4C2×D4×D6111yescharacteristic
519C2×C4C22×C22⋊D6111yescharacteristic
520C2×C4C2×Q8×D6111yescharacteristic
521C2×C4#41×D6111yescharacteristic
522C2×C4#1451982411yescharacteristic
523C2×C4#1451982511yescharacteristic
524C8—42no—
525C8—1222no—
526C8—2441no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—1224no—
535C22—2442no—
536C22—2442no—
537C22—2442no—
538C22—2442no—
539C22—2442no—
540C22—2442no—
541C22C2×#1451974511yescharacteristic
542C22#1684042211yescharacteristic
543C22#1684042511yescharacteristic
544C22C2×D8×D6111yescharacteristic
545C22C2×#1451980111yescharacteristic
546C22#1684057111yescharacteristic
547C22#1684057811yescharacteristic
548C4—22no—
549C4—44no—
550C4—81no—
551C4—1224no—
552C4—4882no—
553C4—2442no—
554C4—2442no—
555C4#1683989414yes—
556C4C22⋊Q8×D6111yescharacteristic
557C4#1684033511yescharacteristic
558C4#1684041911yescharacteristic
559C4Q8×C22⋊D6111yescharacteristic
560C2—82no—
561C2—2444no—
562C2#2144918011yescharacteristic
563C2C2×#1683989411yescharacteristic
564C2#2145513311yescharacteristic
565C2#2145845311yescharacteristic
566C2#2145853611yescharacteristic
567C2#2145866911yescharacteristic
568C2#2145872211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#474C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenterC2×#474D61
hypocenterC61#24326
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×#16840427C2semidirect
3. #21450741C2semidirect
4. #21447145C2semidirect
5. #21446953C2semidirect
6. #21446781C2semidirect
7. #21446754C2semidirect
9. #21446063C2semidirect
10. #21445820C2semidirect
11. C2×C61×#474C2semidirect
55. C2×#11758199Q8semidirect
114. C2×D4×C61C2×Q8semidirect
145. C2×Dic122D8semidirect
179. C2×#92418C2×Q8semidirect
234. D244C22⋊Q8semidirect
277. C2×#474D61semidirect
329. C61#24326semidirect
412. C2×D4Q8×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168404272
C2×C4×#117581992
#214507412
#214471452
#214469532
#214467812
#214467542
#214461642
#214460632
#214458202
C2×C61×#4742
#2432661