Finite Group Databasebeta

← #29692172

Subgroups — #29692172

Order 15616.

784 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840063C211yesmaximal, characteristic
2C4×#14519769C211yesmaximal, characteristic, complemented
3#21448283C211yesmaximal, characteristic
4#21446774C211yesmaximal, characteristic, complemented
5#21446589C211yesmaximal, characteristic, complemented
6#21446562C211yesmaximal, characteristic, complemented
7#21446557C211yesmaximal, characteristic, complemented
8#21446516C211yesmaximal, characteristic
9#21446516C211yesmaximal, characteristic
10#21446447C211yesmaximal, characteristic
11#21446440C211yesmaximal, characteristic
12#21446398C211yesmaximal, characteristic, complemented
13#21446391C211yesmaximal, characteristic, complemented
14#21445830C211yesmaximal, characteristic
15C61×#1767C211yesmaximal, characteristic, complemented
16C2×#14519769C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×#14519734—21no—
19#16840063C418yescomplemented
20C2×#14519711C2211yescharacteristic
21C2×#14519711C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28#16839664C2211yescharacteristic
29#16839664C2211yescharacteristic
30#16839660C2211yescharacteristic
31#16839660C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839653—21no—
35#16839653—21no—
36#16839646—21no—
37#16839646—21no—
38#16839628C2211yescharacteristic
39C4×#11758196—21no—
40C4×#11758196—21no—
41C4×C4⋊Dic61C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43C4×C4⋊Dic61C2211yescharacteristic
44C4×#11758198—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53C2×C4⋊C8×C61C2211yescharacteristic
54C2×C4⋊C8×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C2×#97×C61C2211yescharacteristic
57C43×C61C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62#14519769—22no—
63#14519769C2×C414yes—
64C4×D244—22no—
65C4×D244C2×C414yes—
66#14519734—44no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88#14519711—41no—
89#14519711C2×C414yes—
90#14519709—41no—
91#14519709C2×C414yes—
92#14519699—41no—
93#14519699—41no—
94#14519699C2×C414yes—
95#14519699C2×C414yes—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105C22×C8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C2×C42×C61D411yescharacteristic
110C2×C42×C61D411yescharacteristic
111C2×C42×C61D411yescharacteristic
112C2×C42×C61D411yescharacteristic
113C2×C42×C61C2311yescharacteristic
114C2×C42×C61C2311yescharacteristic
115C2×C42×C61C2311yescharacteristic
116C4⋊C8×C61—22no—
117C4⋊C8×C61C2×C414yes—
118#97×C61—41no—
119#97×C61—41no—
120C23×D61—21no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244C22×C412yes—
141C2×C4×D61—24no—
142C2×C4×D61—42no—
143C2×C4×D61—42no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61C22×C412yes—
155C2×C8×C61C22×C412yes—
156C42×C61—22no—
157C42×C61—21no—
158C42×C61—21no—
159C42×C61—21no—
160C42×C61—21no—
161C42×C61—41no—
162C42×C61—41no—
163C42×C61—41no—
164C42×C61C22×C412yes—
165C42×C61#4112yes—
166#11758198—24no—
167#11758198—42no—
168#11758198—42no—
169#11758198—42no—
170#11758198—42no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61C22×C412yes—
183C4⋊Dic61C22×C412yes—
184C4⋊Dic61C22×C412yes—
185#11758196—24no—
186#11758196—24no—
187#11758196—42no—
188#11758196—42no—
189#11758196—42no—
190#11758196—42no—
191C4×Dic61—42no—
192C4×Dic61—42no—
193C4×Dic61—42no—
194C23×C61C22×D411yescharacteristic
195C22×D61—22no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—41no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61C4×D412yes—
218C2×C4×C61C4×D412yes—
219C2×C4×C61AGL(1,ℤ/8)12yes—
220C2×C4×C61C23×C411yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61H(5,𝔽2)11yescharacteristic
223C2×C4×C612−1+411yescharacteristic
224C2×C4×C612−1+411yescharacteristic
225C2×C4×C612−1+411yescharacteristic
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247D244—22no—
248D244—22no—
249D244—41no—
250D244—41no—
251C4×D61—48no—
252C8×C61—22no—
253C8×C61—22no—
254C8×C61—41no—
255C8×C61—41no—
256#22120—611nomaximal, Sylow
257C22×C61C2×C4×D411yescharacteristic
258C22×C61#53611yescharacteristic
259C22×C61#53711yescharacteristic
260C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C61#57511yescharacteristic
262C22×C61#57511yescharacteristic
263C22×C61#57511yescharacteristic
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C4×C61—22no—
272C4×C61—22no—
273C4×C61—22no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61#51911yescharacteristic
278C4×C61#52011yescharacteristic
279C4×C61#54111yescharacteristic
280C4×C61#54211yescharacteristic
281Dic61—42no—
282Dic61—42no—
283Dic61—42no—
284Dic61—81no—
285Dic61—81no—
286Dic61—81no—
287Dic61—81no—
288C2×#478—611no—
289C4×#125—611no—
290#2293—611no—
291#1989—611no—
292#1949—611no—
293#1944—611no—
294#1943—611no—
295#1936—611no—
296#1936—611no—
297#1926—611no—
298#1925—611no—
299#1919—611no—
300#1918—611no—
301#1804—611no—
302#1767D6111yescharacteristic, complemented
303C2×C61#232311yescharacteristic
304C2×C61#294311yescharacteristic
305C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
306C2×C61#323411yescharacteristic
307C2×C61#324011yescharacteristic
308C2×C61#325111yescharacteristic
309C2×C61#325611yescharacteristic
310D61—82no—
311C2×#125—611no—
312C2×C4×D4—611no—
313C2×#117—1221no—
314#478—618no—
315C2×#107—611no—
316C2×#107—611no—
317C2×#106—611no—
318C2×#106—611no—
319C2×C4⋊C8C2×D6111yescharacteristic
320C2×C4⋊C8C2×D6111yescharacteristic
321C2×#102—611no—
322C2×#102—611no—
323C2×#102—611no—
324C2×#102—611no—
325C2×#97C2×D6111yescharacteristic
326C2×#97C2×D6111yescharacteristic
327#390—611no—
328#390—611no—
329#389—611no—
330#389—611no—
331#389—611no—
332#389—611no—
333#388—1221no—
334#388—1221no—
335#387—1221no—
336#387—1221no—
337#382—611no—
338C4×C4⋊C4—611no—
339C4×C4⋊C4—611no—
340C4×C4⋊C4—611no—
341C4×C4⋊C4—1221no—
342C4×C4⋊C4—1221no—
343C4×C22⋊C4—1221no—
344C43C2×D6111yescharacteristic
345#340—611no—
346#340—611no—
347#340—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#336C2×D6111yescharacteristic
354#336C2×D6111yescharacteristic
355C61#2212011yescharacteristic, complemented, Sylow
356C22×D4—611no—
357C23×C4—1221no—
358C22×C8C22×D6111yescharacteristic
359C22×C8C22×D6111yescharacteristic
360C22×C8C22×D6111yescharacteristic
361C22×C8C22×D6111yescharacteristic
362#125—614no—
363#125—1222no—
364C4×D4—614no—
365C4×D4—1222no—
366#117—2444no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C22⋊C4—1221no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C42—1221no—
386C2×C42—1221no—
387C2×C42—1221no—
388C2×C42D24411yescharacteristic
389C2×C42D24411yescharacteristic
390C2×C42D24411yescharacteristic
391C2×C42D24411yescharacteristic
392C2×C42C22×D6111yescharacteristic
393C2×C42C22×D6111yescharacteristic
394C2×C42C22×D6111yescharacteristic
395#107—614no—
396#107—2441no—
397#106—614no—
398#106—2441no—
399C4⋊C8—22no—
400C4⋊C8C4×D6114yes—
401#102—614no—
402#102—614no—
403#102—2441no—
404#102—2441no—
405#97—41no—
406#97—41no—
407#95—1221no—
408#95—1221no—
409#95—1221no—
410#95—1221no—
411#95—1221no—
412#95—1221no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416C24—1221no—
417C2×D4—612no—
418C2×D4—1221no—
419C2×D4—1221no—
420C2×D4—1221no—
421C2×D4—1221no—
422C2×D4—1221no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—1221no—
426C22×C4—1224no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—2442no—
432C22×C4—2442no—
433C22×C4C2×D24411yescharacteristic
434C22×C4C2×D24411yescharacteristic
435C22×C4C2×D24411yescharacteristic
436C22×C4C2×D24411yescharacteristic
437C22×C4C2×D24411yescharacteristic
438C22×C4C2×D24411yescharacteristic
439C22×C4C23×D6111yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8C2×C4×D6112yes—
451C2×C8C2×C4×D6112yes—
452C4⋊C4—612no—
453C4⋊C4—612no—
454C4⋊C4—612no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—1224no—
459C4⋊C4—1224no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—2442no—
469C4⋊C4—2442no—
470C4⋊C4—2442no—
471C4⋊C4—2442no—
472C22⋊C4—1224no—
473C22⋊C4—2442no—
474C22⋊C4—2442no—
475C22⋊C4—2442no—
476C22⋊C4—2442no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—22no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—2442no—
486C42—2442no—
487C42—2442no—
488C42C2×C4×D6112yes—
489C42#1175822212yes—
490C23—1222no—
491C23—2441no—
492C23—2441no—
493C23—2441no—
494C23—2441no—
495C23—2441no—
496C23—2441no—
497C23C22×D24411yescharacteristic
498D4—1222no—
499D4—1222no—
500D4—2441no—
501D4—2441no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—2448no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4C4×D24412yes—
540C2×C4C4×D24412yes—
541C2×C4#1451980512yes—
542C2×C4C22×C4×D6111yescharacteristic
543C2×C4C2×#1175822211yescharacteristic
544C2×C4#1451982211yescharacteristic
545C2×C4#1451982411yescharacteristic
546C2×C4#1451982411yescharacteristic
547C2×C4#1451982511yescharacteristic
548C8—22no—
549C8—22no—
550C8—41no—
551C8—41no—
552C22—4881no—
553C22—4881no—
554C22—4881no—
555C22—4881no—
556C22—2442no—
557C22—2442no—
558C22—2442no—
559C22C2×C4×D24411yescharacteristic
560C22#1684033811yescharacteristic
561C22#1684034311yescharacteristic
562C22C2×#1451980511yescharacteristic
563C22#1684056211yescharacteristic
564C22#1684056211yescharacteristic
565C22#1684056211yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—4881no—
573C4—4881no—
574C4—4881no—
575C4—4881no—
576C4—2442no—
577C4—2442no—
578C4—2442no—
579C4#1684023811yescharacteristic
580C4#1684024111yescharacteristic
581C4#1684037011yescharacteristic
582C4#1684037711yescharacteristic
583C2—4882no—
584C2#2144850211yescharacteristic
585C2#2145462111yescharacteristic
586C2C4×#1451980511yescharacteristic
587C2#2145701111yescharacteristic
588C2#2145706111yescharacteristic
589C2#2145718211yescharacteristic
590C2#2145722011yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1767C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1767D61
hypocenterC61#22120
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C43 ◁ #1767

Complements

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

NH ≅ G/Ntype
2. C4×#14519769C2semidirect
4. #21446774C2semidirect
5. #21446589C2semidirect
6. #21446562C2semidirect
7. #21446557C2semidirect
12. #21446398C2semidirect
13. #21446391C2semidirect
15. C61×#1767C2semidirect
19. #16840063C4semidirect
302. #1767D61semidirect
355. C61#22120semidirect

Maximal subgroups

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

subgroupindex
C2×#168400632
C4×#145197692
#214482832
#214467742
#214465892
#214465622
#214465572
#214465162
#214464472
#214464402
#214463982
#214463912
#214458302
C61×#17672
#2212061