Finite Group Databasebeta

← #29721311

Subgroups — #29721311

Order 15616.

912 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840043C211yesmaximal, characteristic, complemented
2#21454171C211yesmaximal, characteristic, complemented
3C4×#14519751C211yesmaximal, characteristic
4#21447247C212yesmaximal, complemented
5#21447009C212yesmaximal
6#21446770C211yesmaximal, characteristic, complemented
7#21446583C211yesmaximal, characteristic, complemented
8#21446562C212yesmaximal, complemented
9#21446516C212yesmaximal, complemented
10#21445820C211yesmaximal, characteristic, complemented
11C2×C61×#432C211yesmaximal, characteristic, complemented
12C2×#14519777C2211yescharacteristic
13C2×#14519756—21no—
14C2×#14519751C2211yescharacteristic
15C2×C4×D244C2211yescharacteristic
16C2×C4×C22⋊D61—21no—
17#16840043—24no—
18C2×#14519711C2212yes—
19C2×#14519709C2212yes—
20C2×#14519699C2212yes—
21C2×#14519699C2212yes—
22#16839668—21no—
23#16839664C2211yescharacteristic
24#16839660—21no—
25#16839660C2212yes—
26#16839660C2211yescharacteristic
27#16839653—21no—
28#16839641—21no—
29#16839632C2211yescharacteristic
30#16839622—21no—
31#16839622C2211yescharacteristic
32#16839622C2212yes—
33C4×C4⋊Dic61C2211yescharacteristic
34C4×#11758198—21no—
35#16839525C2212yes—
36#16839525C2212yes—
37#16839525C2212yes—
38#16839525C2212yes—
39C22×C42×C61C2211yescharacteristic
40C61×#432—24no—
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×C22⋊C8×C61C2212yes—
44C2×#97×C61C2212yes—
45C22×Dic122—21no—
46C22×D244C2311yescharacteristic
47C22×C22⋊D61—21no—
48C22×C4×D61—21no—
49#14519777—22no—
50#14519777—22no—
51#14519756—44no—
52#14519751—22no—
53#14519751—22no—
54C4×D244—22no—
55C4×D244—22no—
56C4×C22⋊D61—44no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61C2311yescharacteristic
61C2×C4⋊Dic61C2312yes—
62C2×C4⋊Dic61C2312yes—
63C2×C4⋊Dic61C2311yescharacteristic
64C2×C4⋊Dic61C2311yescharacteristic
65C2×#11758196—21no—
66C2×#11758196—22no—
67C2×C22⋊Dic61—21no—
68C2×C22⋊Dic61—22no—
69C2×#11758198—21no—
70C2×#11758198—21no—
71C2×#11758198—21no—
72C2×#11758198—21no—
73C2×C4×Dic61—21no—
74#14519711—24no—
75#14519709—24no—
76#14519699—24no—
77#14519699—24no—
78#14519684—21no—
79#14519684—21no—
80#14519684—21no—
81#14519684—22no—
82#14519684—22no—
83C23×C4×C61D411yescharacteristic
84C23×C4×C61D411yescharacteristic
85C23×C4×C61C2311yescharacteristic
86C22×C8×C61C2312yes—
87C22×C8×C61C2312yes—
88C2×C42×C61—22no—
89C2×C42×C61—22no—
90C2×C42×C61D411yescharacteristic
91C2×C42×C61D411yescharacteristic
92C2×C42×C61C2311yescharacteristic
93C2×C42×C61C2311yescharacteristic
94C4⋊C8×C61—22no—
95C4⋊C8×C61D414yescomplemented
96C22⋊C8×C61—24no—
97#97×C61—24no—
98C24×C61C2×D411yescharacteristic
99C23×D61—21no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61—21no—
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—41no—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2411yescharacteristic
118C2×C22⋊D61—24no—
119C2×C22⋊D61—42no—
120C2×C22⋊D61—42no—
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61—22no—
124C22×Dic61—21no—
125C22×Dic61—22no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244#4112yes—
132C2×C4×D61—42no—
133C2×C4×D61—42no—
134C2×C4×D61—42no—
135C2×Dic122—42no—
136C2×Dic122—42no—
137C2×Dic122—42no—
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C2×C8×C61C2×D414yes—
144C42×C61—22no—
145C42×C61—22no—
146C42×C61—21no—
147C42×C61—21no—
148C42×C61C2×D412yes—
149C42×C61#4112yes—
150C22⋊Dic61—42no—
151C22⋊Dic61—44no—
152#11758198—24no—
153#11758198—42no—
154#11758198—42no—
155#11758198—42no—
156C4⋊Dic61—21no—
157C4⋊Dic61—22no—
158C4⋊Dic61—21no—
159C4⋊Dic61—22no—
160C4⋊Dic61—22no—
161C4⋊Dic61—21no—
162C4⋊Dic61—22no—
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165C4⋊Dic61—42no—
166C4⋊Dic61—42no—
167C4⋊Dic61—42no—
168C4⋊Dic61#4112yes—
169#11758196—42no—
170#11758196—44no—
171C4×Dic61—42no—
172C23×C61—21no—
173C23×C61—21no—
174C23×C61—21no—
175C23×C61—21no—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61#13712yes—
179C23×C61C22×D411yescharacteristic
180C22×D61—22no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—44no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—44no—
204C2×C4×C61#12112yes—
205C2×C4×C61#12112yes—
206C2×C4×C61AGL(1,ℤ/8)12yes—
207C2×C4×C61AGL(1,ℤ/8)12yes—
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C612−1+411yescharacteristic
213C2×C4×C612−1+411yescharacteristic
214C22⋊D61—48no—
215C2×Dic61—22no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—42no—
219C2×Dic61—41no—
220C2×Dic61—42no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—41no—
226C2×Dic61—42no—
227C2×Dic61—41no—
228C2×Dic61—42no—
229C2×Dic61—41no—
230D244—22no—
231D244—22no—
232D244—41no—
233D244—41no—
234C4×D61—84no—
235Dic122—84no—
236C8×C61—24no—
237C8×C61—42no—
238#24709—611nomaximal, Sylow
239C22×C61—22no—
240C22×C61—22no—
241C22×C61—22no—
242C22×C61—41no—
243C22×C61—41no—
244C22×C61—41no—
245C22×C61—41no—
246C22×C61C2×#12111yescharacteristic
247C22×C61#52511yescharacteristic
248C22×C61#53611yescharacteristic
249C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
250C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
251C22×C61C2×#13711yescharacteristic
252C22×C61#57511yescharacteristic
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—42no—
256C2×D61—81no—
257C2×D61—81no—
258C2×D61—81no—
259C2×D61—81no—
260C4×C61—22no—
261C4×C61—41no—
262C4×C61—41no—
263C4×C61—41no—
264C4×C61#46914yes—
265C4×C61#54711yescharacteristic
266C4×C61#54811yescharacteristic
267C4×C61#55211yescharacteristic
268C4×C61#55611yescharacteristic
269Dic61—42no—
270Dic61—82no—
271Dic61—81no—
272Dic61—82no—
273Dic61—81no—
274C2×#474—611no—
275C2×#432D6111yescharacteristic, complemented
276#2857—611no—
277C4×#121—611no—
278#2076—612no—
279#2034—612no—
280#1988—611no—
281#1948—611no—
282#1944—612no—
283#1936—612no—
284#1802—611no—
285C2×C61—42no—
286C2×C61#241511yescharacteristic
287C2×C61C2×#46911yescharacteristic
288C2×C61#305611yescharacteristic
289C2×C61#311011yescharacteristic
290C2×C61#311511yescharacteristic
291C2×C61#323511yescharacteristic
292C2×C61#323911yescharacteristic
293D61—82no—
294C2×C4⋊Q8—611no—
295C2×C22⋊Q8—1221no—
296C2×#121—611no—
297C2×C4×D4—611no—
298C2×C4×D4—1221no—
299C22×C42C2×D6111yescharacteristic
300#474—1224no—
301#432—24no—
302C2×#107—612no—
303C2×#106—612no—
304C2×C4⋊C8C2×D6111yescharacteristic
305C2×C4⋊C8C2×D6111yescharacteristic
306C2×#102—612no—
307C2×#102—612no—
308C2×C22⋊C8C2×D6112yes—
309C2×#97C2×D6112yes—
310#391—1221no—
311#390—611no—
312#389—612no—
313#389—611no—
314#389—1221no—
315#388—1221no—
316#386—1221no—
317#384—611no—
318#380—611no—
319#380—612no—
320#380—1221no—
321C4×C4⋊C4—611no—
322C4×C22⋊C4—1221no—
323#340—612no—
324#340—612no—
325#340—612no—
326#340—612no—
327C61#2470911yescharacteristic, complemented, Sylow
328C22×Q8—1221no—
329C22×D4—611no—
330C22×D4—1221no—
331C23×C4—1221no—
332C23×C4D24411yescharacteristic
333C23×C4D24411yescharacteristic
334C23×C4C22×D6111yescharacteristic
335C22×C8C22×D6112yes—
336C22×C8C22×D6112yes—
337C4⋊Q8—1222no—
338C4⋊Q8—1222no—
339C22⋊Q8—2444no—
340#121—1222no—
341#121—1222no—
342C4×D4—1222no—
343C4×D4—1222no—
344C4×D4—2444no—
345C2×C4⋊C4—611no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—612no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—612no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4—1222no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1221no—
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1221no—
357C2×C22⋊C4—1222no—
358C2×C22⋊C4—1221no—
359C2×C22⋊C4—1221no—
360C2×C22⋊C4—1221no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—1221no—
364C2×C42D24411yescharacteristic
365C2×C42D24411yescharacteristic
366C2×C42C22×D6111yescharacteristic
367C2×C42C22×D6111yescharacteristic
368#107—1224no—
369#106—1224no—
370C4⋊C8—22no—
371C4⋊C8C22⋊D6114yescomplemented
372#102—1224no—
373#102—1224no—
374C22⋊C8—24no—
375#97—24no—
376#95—1221no—
377#95—1221no—
378#95—1221no—
379#95—1222no—
380#95—1222no—
381C24—1221no—
382C24C2×D24411yescharacteristic
383C2×Q8—2442no—
384C2×Q8—2442no—
385C2×Q8—2442no—
386C2×D4—612no—
387C2×D4—1224no—
388C2×D4—1221no—
389C2×D4—1221no—
390C2×D4—1221no—
391C2×D4—1221no—
392C2×D4—1221no—
393C2×D4—2442no—
394C2×D4—2442no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—41no—
407C22×C4—1221no—
408C22×C4—1221no—
409C22×C4—1221no—
410C22×C4—1222no—
411C22×C4—1222no—
412C22×C4—2442no—
413C22×C4—2442no—
414C22×C4—2442no—
415C22×C4C2×D24411yescharacteristic
416C22×C4C2×D24411yescharacteristic
417C22×C4C2×D24411yescharacteristic
418C22×C4C2×D24411yescharacteristic
419C22×C4C2×D24411yescharacteristic
420C22×C4C23×D6111yescharacteristic
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8C2×C22⋊D6114yes—
427C4⋊C4—612no—
428C4⋊C4—1221no—
429C4⋊C4—1222no—
430C4⋊C4—1221no—
431C4⋊C4—1222no—
432C4⋊C4—1222no—
433C4⋊C4—1221no—
434C4⋊C4—1222no—
435C4⋊C4—1221no—
436C4⋊C4—1221no—
437C4⋊C4—2442no—
438C4⋊C4—2444no—
439C4⋊C4—2442no—
440C4⋊C4—2442no—
441C4⋊C4—2442no—
442C22⋊C4—1224no—
443C22⋊C4—2442no—
444C22⋊C4—2442no—
445C22⋊C4—2442no—
446C22⋊C4—2444no—
447C22⋊C4—2442no—
448C42—21no—
449C42—21no—
450C42—22no—
451C42—22no—
452C42—2442no—
453C42#1175822212yes—
454C42C2×C22⋊D6112yes—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—1222no—
462C23—2441no—
463C23—2441no—
464C23—2441no—
465C23—2441no—
466C23—2441no—
467C23—2441no—
468C23#1451981012yes—
469C23C22×D24411yescharacteristic
470Q8—4884no—
471D4—1222no—
472D4—1222no—
473D4—2448no—
474D4—2441no—
475D4—2441no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—44no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—1222no—
494C2×C4—4884no—
495C2×C4—2441no—
496C2×C4—2442no—
497C2×C4—2442no—
498C2×C4—2441no—
499C2×C4—2442no—
500C2×C4—2441no—
501C2×C4—2442no—
502C2×C4—2441no—
503C2×C4—2442no—
504C2×C4—2441no—
505C2×C4—2442no—
506C2×C4—2441no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4#1451975112yes—
510C2×C4#1451975112yes—
511C2×C4#1451980512yes—
512C2×C4#1451980512yes—
513C2×C4C22×C22⋊D6111yescharacteristic
514C2×C4#41×D6111yescharacteristic
515C2×C4#41×D6111yescharacteristic
516C2×C4C2×#1175822211yescharacteristic
517C2×C4#1451982411yescharacteristic
518C2×C4#1451982411yescharacteristic
519C8—24no—
520C8—42no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—4881no—
529C22—4881no—
530C22—4881no—
531C22—4881no—
532C22—2442no—
533C22—2442no—
534C22—2442no—
535C22C2×#1451975111yescharacteristic
536C22#1684027411yescharacteristic
537C22#1684033811yescharacteristic
538C22C2×#1451980511yescharacteristic
539C22C2×#1451980511yescharacteristic
540C22C2×#1451981011yescharacteristic
541C22#1684056211yescharacteristic
542C4—22no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—4881no—
547C4—4882no—
548C4—4882no—
549C4—4881no—
550C4—2442no—
551C4#1684001514yes—
552C4#1684041111yescharacteristic
553C4#1684041911yescharacteristic
554C4#1684045111yescharacteristic
555C4#1684048011yescharacteristic
556C2—42no—
557C2—4882no—
558C2#2144924211yescharacteristic
559C2C2×#1684001511yescharacteristic
560C2#2145544911yescharacteristic
561C2#2145591911yescharacteristic
562C2#2145597111yescharacteristic
563C2#2145701911yescharacteristic
564C2#2145705111yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#432C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenterC2×#432D61
hypocenterC61#24709
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#432

Complements

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

NH ≅ G/Ntype
1. C2×#16840043C2semidirect
2. #21454171C2semidirect
4. #21447247C2semidirect
6. #21446770C2semidirect
7. #21446583C2semidirect
8. #21446562C2semidirect
9. #21446516C2semidirect
10. #21445820C2semidirect
11. C2×C61×#432C2semidirect
95. C4⋊C8×C61D4semidirect
275. C2×#432D61semidirect
327. C61#24709semidirect
371. C4⋊C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400432
#214541712
C4×#145197512
#214472472
#214470092
#214467702
#214465832
#214465622
#214465162
#214458202
C2×C61×#4322
#2470961