Finite Group Databasebeta

← #29692168

Subgroups — #29692168

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
2#21448374C211yesmaximal, characteristic, complemented
3#21448280C211yesmaximal, characteristic
4#21446776C211yesmaximal, characteristic, complemented
5#21446560C211yesmaximal, characteristic, complemented
6#21446555C211yesmaximal, characteristic, complemented
7#21446514C211yesmaximal, characteristic
8#21446514C211yesmaximal, characteristic
9#21446445C211yesmaximal, characteristic
10#21446438C211yesmaximal, characteristic
11#21446396C211yesmaximal, characteristic, complemented
12#21446389C211yesmaximal, characteristic, complemented
13#21445826C211yesmaximal, characteristic
14#21445674C211yesmaximal, characteristic, complemented
15C61×#1949C211yesmaximal, characteristic, complemented
16C2×#14519769C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×#14519734—21no—
19#16840063C418yescomplemented
20C2×#14519711C2211yescharacteristic
21C2×#14519709C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23C2×#14519699C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519697C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519687C2211yescharacteristic
28C2×#14519687C2211yescharacteristic
29#16839663C2211yescharacteristic
30#16839663C2211yescharacteristic
31#16839658C2211yescharacteristic
32#16839658C2211yescharacteristic
33#16839658C2211yescharacteristic
34#16839658C2211yescharacteristic
35#16839651—21no—
36#16839651—21no—
37#16839644—21no—
38#16839644—21no—
39C4⋊C4×Dic61C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41#16839613—21no—
42#16839613—21no—
43C4×#11758198—21no—
44C42×Dic61C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#106×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C61×#382C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519769—22no—
65#14519769C2×C414yes—
66C4×D244—22no—
67C4×D244C2×C414yes—
68#14519734—44no—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61D411yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519711C2×C414yes—
92#14519709C2×C414yes—
93C4⋊#92418—22no—
94#14519699C2×C414yes—
95#14519699C2×C414yes—
96#14519697—41no—
97#14519697—41no—
98#14519687—41no—
99#14519687—41no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C42×C61—21no—
113C2×C42×C61C2311yescharacteristic
114#107×C61—41no—
115#106×C61—41no—
116C4⋊C8×C61C2×C414yes—
117#95×C61—21no—
118#95×C61—21no—
119#95×C61—21no—
120C23×D61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
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×C61C22×C412yes—
149C2×C8×C61C22×C412yes—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C42×C61—42no—
159C42×C61C22×C412yes—
160C42×C61#4112yes—
161#11758198—24no—
162#11758198—42no—
163#11758198—42no—
164#11758198—42no—
165#11758198—42no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61C22×C412yes—
170C4⋊Dic61C22×C412yes—
171C4⋊Dic61C22×C412yes—
172#11758196—24no—
173#11758196—24no—
174#11758196—42no—
175#11758196—42no—
176#11758196—42no—
177#11758196—42no—
178C4×Dic61—21no—
179C4×Dic61—21no—
180C4×Dic61—21no—
181C4×Dic61—21no—
182C4×Dic61—22no—
183C4×Dic61—41no—
184C4×Dic61—41no—
185C4×Dic61—41no—
186C4×Dic61—42no—
187C4×Dic61—42no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
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—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61C4×D412yes—
219C2×C4×C61C4×D412yes—
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—21no—
227C2×Dic61—21no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61AGL(1,ℤ/8)12yes—
247D244—22no—
248D244—22no—
249D244—41no—
250D244—41no—
251C4×D61—48no—
252C8×C61—22no—
253C8×C61—22no—
254#92418—41no—
255#92418—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—81no—
274C4×C61—81no—
275C4×C61—81no—
276C4×C61—81no—
277C4×C61#51911yescharacteristic
278C4×C61#52011yescharacteristic
279C4×C61#54111yescharacteristic
280C4×C61#54211yescharacteristic
281Dic61—22no—
282Dic61—42no—
283Dic61—42no—
284Dic61—42no—
285Dic61—41no—
286Dic61—41no—
287Dic61—41no—
288C2×#478—611no—
289C4×#125—611no—
290#2293—611no—
291#1989—611no—
292#1949D6111yescharacteristic, complemented
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#1767—611no—
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×#107C2×D6111yescharacteristic
317C2×#106—611no—
318C2×#106C2×D6111yescharacteristic
319C2×C4⋊C8—611no—
320C2×C4⋊C8C2×D6111yescharacteristic
321C2×#102—611no—
322C2×#102—611no—
323C2×#102—611no—
324C2×#102—611no—
325C2×#97—611no—
326C2×#97—611no—
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#382C2×D6111yescharacteristic
338C4×C4⋊C4—611no—
339C4×C4⋊C4—611no—
340C4×C4⋊C4—1221no—
341C4×C4⋊C4—1221no—
342C4×C4⋊C4C2×D6111yescharacteristic
343C4×C22⋊C4—1221no—
344C43—611no—
345#340—611no—
346#340—611no—
347#340—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340C2×D6111yescharacteristic
352#340C2×D6111yescharacteristic
353#336—611no—
354#336—611no—
355C61#2212011yescharacteristic, complemented, Sylow
356C22×D4—611no—
357C23×C4—1221no—
358C22×C8—611no—
359C22×C8—611no—
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—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4C22×D6111yescharacteristic
377C2×C4⋊C4C22×D6111yescharacteristic
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C4⋊C4C22×D6111yescharacteristic
380C2×C22⋊C4—1221no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C42—21no—
386C2×C42—611no—
387C2×C42—611no—
388C2×C42—611no—
389C2×C42—611no—
390C2×C42—611no—
391C2×C42—611no—
392C2×C42—1221no—
393C2×C42—1221no—
394C2×C42C22×D6111yescharacteristic
395#107—41no—
396#107—614no—
397#106—41no—
398#106—614no—
399C4⋊C8—1222no—
400C4⋊C8C4×D6114yes—
401#102—614no—
402#102—614no—
403#102—2441no—
404#102—2441no—
405#97—2441no—
406#97—2441no—
407#95—21no—
408#95—21no—
409#95—21no—
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—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1224no—
435C22×C4—2442no—
436C22×C4—2442no—
437C22×C4D4×D6111yescharacteristic
438C22×C4D4×D6111yescharacteristic
439C22×C4C23×D6111yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8C2×C4×D6112yes—
451C2×C8C2×C4×D6112yes—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—612no—
461C4⋊C4—612no—
462C4⋊C4—612no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1224no—
467C4⋊C4—1224no—
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—42no—
478C42—1222no—
479C42—1221no—
480C42—1221no—
481C42—1221no—
482C42—1221no—
483C42—2441no—
484C42—2441no—
485C42—2441no—
486C42—2442no—
487C42—2442no—
488C42C2×C4×D6112yes—
489C42#1175822612yes—
490C23—1222no—
491C23—2441no—
492C23—2441no—
493C23—2441no—
494C23—2441no—
495C23—2441no—
496C23—2441no—
497C23C2×D4×D6111yescharacteristic
498D4—1222no—
499D4—1222no—
500D4—2441no—
501D4—2441no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—612no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—2448no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4#1451973912yes—
541C2×C4#1451973912yes—
542C2×C4C22×C4×D6111yescharacteristic
543C2×C4C2×#1175822611yescharacteristic
544C2×C4#1451982211yescharacteristic
545C2×C4#1451982411yescharacteristic
546C2×C4#1451982411yescharacteristic
547C2×C4#1451982511yescharacteristic
548C8—22no—
549C8—22no—
550C8—2441no—
551C8—2441no—
552C22—4881no—
553C22—4881no—
554C22—4881no—
555C22—4881no—
556C22—2442no—
557C22—2442no—
558C22—2442no—
559C22C2×#1451973911yescharacteristic
560C22#1684033511yescharacteristic
561C22#1684034111yescharacteristic
562C22AGL(1,ℤ/8)×D6111yescharacteristic
563C22#1684055911yescharacteristic
564C22#1684055911yescharacteristic
565C22#1684056111yescharacteristic
566C4—22no—
567C4—22no—
568C4—81no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4—1222no—
573C4—2441no—
574C4—2442no—
575C4—2442no—
576C4—2442no—
577C4—2441no—
578C4—2441no—
579C4#1684023811yescharacteristic
580C4#1684024111yescharacteristic
581C4#1684037511yescharacteristic
582C4#1684038611yescharacteristic
583C2—4882no—
584C2#2144849911yescharacteristic
585C2#2145461911yescharacteristic
586C2#2145462611yescharacteristic
587C2#2145700711yescharacteristic
588C2#2145706411yescharacteristic
589C2#2145717811yescharacteristic
590C2#2145721811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1949C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1949D61
hypocenterC61#22120
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
2. #21448374C2semidirect
4. #21446776C2semidirect
5. #21446560C2semidirect
6. #21446555C2semidirect
11. #21446396C2semidirect
12. #21446389C2semidirect
14. #21445674C2semidirect
15. C61×#1949C2semidirect
19. #16840063C4semidirect
292. #1949D61semidirect
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
#214483742
#214482802
#214467762
#214465602
#214465552
#214465142
#214464452
#214464382
#214463962
#214463892
#214458262
#214456742
C61×#19492
#2212061