Finite Group Databasebeta

← #29708879

Subgroups — #29708879

Order 15616.

1014 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839798C211yesmaximal, characteristic, complemented
2C2×#16839730C211yesmaximal, characteristic
3#21448412C211yesmaximal, characteristic
4#21448407C211yesmaximal, characteristic
5#21447258C212yesmaximal, complemented
6#21447257C212yesmaximal, complemented
7#21447251C212yesmaximal, complemented
8#21447246C212yesmaximal, complemented
9#21445539C211yesmaximal, characteristic, complemented
10C2×#16839524C211yesmaximal, characteristic, complemented
11C61×#1736C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14C2×#14519736C2211yescharacteristic
15C22×C4⋊Dic61C2211yescharacteristic
16#16839798—24no—
17C2×#14519710C2211yescharacteristic
18C2×#14519710C2211yescharacteristic
19C2×#14519708C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21#16839730—24no—
22C2×#14519689C2212yes—
23C2×#14519689C2212yes—
24#16839668—21no—
25#16839667—21no—
26#16839664—21no—
27#16839663—21no—
28#16839641—22no—
29#16839639—22no—
30#16839619C2212yes—
31#16839619C2212yes—
32#16839619C2212yes—
33#16839619C2212yes—
34#16839525C2212yes—
35#16839525C2212yes—
36#16839524C2212yes—
37#16839524C2212yes—
38C22×M16×C61C2211yescharacteristic
39C2×#117×C61C2211yescharacteristic
40C22×C4⋊C4×C61C2211yescharacteristic
41C61×#340C2212yes—
42C61×#340C2212yes—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C22×Dic122D411yescharacteristic
50C22×D244D411yescharacteristic
51C22×C22⋊D61D412yes—
52C23×Dic61—21no—
53C22×C4×D61C2312yes—
54C22×#92418—22no—
55C22×#92418—22no—
56C22×#92418D414yes—
57C22×#92418C2311yescharacteristic
58C22×#92418C2311yescharacteristic
59#14519736—22no—
60#14519736—22no—
61C2×C4⋊Dic61—22no—
62C2×C4⋊Dic61—22no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61C2312yes—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C22⋊Dic61—21no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×C4×Dic61—21no—
73#14519710—22no—
74#14519710—22no—
75#14519708—22no—
76#14519708—22no—
77#14519689—24no—
78#14519689—24no—
79#14519684—22no—
80#14519684—22no—
81#14519684—22no—
82#14519684—22no—
83C23×C4×C61—21no—
84C23×C4×C61C2311yescharacteristic
85C2×M16×C61—22no—
86C2×M16×C61—22no—
87C2×M16×C61—22no—
88C22×C8×C61C2312yes—
89#117×C61—22no—
90#117×C61—22no—
91C2×C4⋊C4×C61—22no—
92C2×C4⋊C4×C61—22no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61C2312yes—
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C22⋊C4×C61—21no—
98C2×C42×C61—21no—
99C24×C61C2×D411yescharacteristic
100C23×D61C2×D412yes—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—22no—
104C22×C4×C61—42no—
105C22×C4×C61—81no—
106C22×C4×C61—81no—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D412yes—
109C22×C4×C61#4112yes—
110C22×C4×C61#4112yes—
111C22×C4×C61#4112yes—
112C22×C4×C61#4112yes—
113C22×C4×C61#4112yes—
114C22×C4×C61C2411yescharacteristic
115C2×C22⋊D61—24no—
116C2×C22⋊D61—24no—
117C2×C22⋊D61—24no—
118C22×Dic61—22no—
119C22×Dic61—21no—
120C22×Dic61—21no—
121C22×Dic61—42no—
122C22×Dic61—81no—
123C22×Dic61—81no—
124C22×Dic61C2×D412yes—
125#11758222—44no—
126#11758222—44no—
127#11758222—44no—
128#11758222—44no—
129C2×D244—22no—
130C2×D244—22no—
131C2×D244—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×Dic122—22no—
139C2×Dic122—22no—
140C2×Dic122—22no—
141M16×C61—44no—
142C2×C8×C61—22no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—42no—
152C22⋊C4×C61—42no—
153C42×C61—42no—
154C22⋊Dic61—42no—
155#11758198—82no—
156#11758198—82no—
157#11758198—82no—
158#11758198—82no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165C4⋊Dic61—42no—
166C4×Dic61—42no—
167C2×#92418—28no—
168C2×#92418—21no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418C2×D412yes—
173C2×#92418C2×D412yes—
174C23×C61—21no—
175C23×C61—21no—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61#12112yes—
179C23×C61#12312yes—
180C23×C61#12312yes—
181C23×C61C22×D411yescharacteristic
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—22no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—42no—
192C2×C4×C61—41no—
193C2×C4×C61—42no—
194C2×C4×C61—41no—
195C2×C4×C61—42no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—84no—
199C2×C4×C61#12112yes—
200C2×C4×C61#12314yes—
201C2×C4×C61#12312yes—
202C2×C4×C61#12312yes—
203C2×C4×C61#12414yes—
204C2×C4×C61#12614yes—
205C2×C4×C61#12614yes—
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C22⋊D61—48no—
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—42no—
219C2×Dic61—41no—
220C2×Dic61—42no—
221C2×Dic61—41no—
222C2×Dic61—42no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—84no—
226D244—44no—
227C4×D61—42no—
228C4×D61—42no—
229C4×D61—42no—
230C4×D61—42no—
231Dic122—44no—
232C8×C61—42no—
233#92418—22no—
234#92418—22no—
235#23641—611nomaximal, Sylow
236C22×C61—22no—
237C22×C61—22no—
238C22×C61—22no—
239C22×C61—22no—
240C22×C61—22no—
241C22×C61—22no—
242C22×C61#39714yes—
243C22×C61C2×#12111yescharacteristic
244C22×C61C2×#12311yescharacteristic
245C22×C61C2×#12311yescharacteristic
246C22×C61#57611yescharacteristic
247C22×C61#57711yescharacteristic
248C22×C61#57711yescharacteristic
249C22×C61#57811yescharacteristic
250C2×D61—42no—
251C2×D61—42no—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—42no—
256C2×D61—42no—
257C4×C61—82no—
258C4×C61—81no—
259C4×C61—81no—
260C4×C61#39714yes—
261C4×C61C2×#12311yescharacteristic
262C4×C61C2×#12411yescharacteristic
263C4×C61C2×#12611yescharacteristic
264C4×C61C2×#12611yescharacteristic
265Dic61—42no—
266Dic61—82no—
267Dic61—81no—
268Dic61—81no—
269C2×#427—611no—
270C2×#408—611no—
271#2311—611no—
272#2311—611no—
273#2077—612no—
274#2077—612no—
275#2076—612no—
276#2075—612no—
277#1736—611no—
278#1736D6111yescharacteristic, complemented
279C2×#340—611no—
280C2×C61—24no—
281C2×C61C2×#39711yescharacteristic
282C2×C61#306011yescharacteristic
283C2×C61#306111yescharacteristic
284C2×C61#309611yescharacteristic
285C2×C61#309611yescharacteristic
286C2×C61#309711yescharacteristic
287C2×C61#309711yescharacteristic
288D61—82no—
289C22×#41—611no—
290C22×M16C2×D6111yescharacteristic
291C23×C8—611no—
292C2×#117—611no—
293C2×#117C2×D6111yescharacteristic
294C22×C4⋊C4—611no—
295C22×C4⋊C4C2×D6111yescharacteristic
296#427—1224no—
297C2×#107—611no—
298C2×#107—611no—
299C2×#106—611no—
300C2×#106—611no—
301#408—1224no—
302C2×C22⋊C8—612no—
303C2×C22⋊C8—612no—
304#391—1221no—
305#391—1221no—
306#390—1221no—
307#390—1221no—
308#386—1222no—
309#386—1222no—
310#380—612no—
311#380—612no—
312#380—612no—
313#380—612no—
314#340—612no—
315#340—612no—
316#340—612no—
317#340—612no—
318#340C2×D6112yes—
319#340C2×D6112yes—
320C61#2364111yescharacteristic, complemented, Sylow
321C2×#41—1222no—
322C2×#41—1222no—
323C2×#41—1222no—
324C2×#41—1222no—
325C2×#41—1222no—
326C2×#41—1222no—
327C22×Q8—611no—
328C22×D4—612no—
329C22×D4—611no—
330C23×C4—21no—
331C23×C4—612no—
332C23×C4—1221no—
333C23×C4C22×D6111yescharacteristic
334C2×M16—22no—
335C2×M16—22no—
336C2×M16—22no—
337C22×C8—614no—
338C22×C8—611no—
339C22×C8—611no—
340C22×C8—1222no—
341C22×C8—1222no—
342C22×C8C22×D6112yes—
343#117—22no—
344#117—22no—
345#117—1222no—
346#117—1222no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—612no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—611no—
353C2×C4⋊C4—1222no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4C22×D6112yes—
357C2×C4⋊C4C22×D6111yescharacteristic
358C2×C4⋊C4C22×D6111yescharacteristic
359C2×C22⋊C4—21no—
360C2×C22⋊C4—1222no—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4—1222no—
363C2×C22⋊C4—1222no—
364C2×C22⋊C4—1221no—
365C2×C42—21no—
366C2×C42—1221no—
367#107—1222no—
368#107—1222no—
369#106—1222no—
370#106—1222no—
371C22⋊C8—1224no—
372C22⋊C8—1224no—
373#95—1222no—
374#95—1222no—
375#95—1222no—
376#95—1222no—
377C24—612no—
378C24D4×D6111yescharacteristic
379#41—2444no—
380#41—2444no—
381#41—2444no—
382#41—2444no—
383C2×Q8—1222no—
384C2×Q8—1222no—
385C2×Q8—1222no—
386C2×D4—1224no—
387C2×D4—1224no—
388C2×D4—1224no—
389C2×D4—1222no—
390C2×D4—1222no—
391C2×D4—1222no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—42no—
396C22×C4—81no—
397C22×C4—81no—
398C22×C4—612no—
399C22×C4—1222no—
400C22×C4—1222no—
401C22×C4—1222no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1221no—
406C22×C4—1221no—
407C22×C4—1222no—
408C22×C4—4881no—
409C22×C4—4881no—
410C22×C4—2442no—
411C22×C4#1175822212yes—
412C22×C4#1175822212yes—
413C22×C4#1175822212yes—
414C22×C4D4×D6112yes—
415C22×C4D4×D6111yescharacteristic
416C22×C4#1175822412yes—
417C22×C4#1175822612yes—
418C22×C4C23×D6111yescharacteristic
419M16—44no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—612no—
424C2×C8—612no—
425C2×C8—1221no—
426C2×C8—1221no—
427C2×C8—1221no—
428C2×C8—1221no—
429C2×C8—1228no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—42no—
437C4⋊C4—1221no—
438C4⋊C4—1221no—
439C4⋊C4—1222no—
440C4⋊C4—1221no—
441C4⋊C4—1221no—
442C4⋊C4—1222no—
443C4⋊C4—2442no—
444C22⋊C4—42no—
445C22⋊C4—4882no—
446C22⋊C4—4882no—
447C22⋊C4—4882no—
448C22⋊C4—4882no—
449C22⋊C4—2442no—
450C42—42no—
451C42—2442no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—1222no—
457C23—1222no—
458C23—1222no—
459C23—1222no—
460C23—1222no—
461C23—1222no—
462C23—1222no—
463C23#1451975012yes—
464C23#1451975912yes—
465C23#1451976212yes—
466C23C2×D4×D6111yescharacteristic
467Q8—2444no—
468D4—2448no—
469D4—2444no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—84no—
480C2×C4—1222no—
481C2×C4—1222no—
482C2×C4—1222no—
483C2×C4—4884no—
484C2×C4—2441no—
485C2×C4—2441no—
486C2×C4—2442no—
487C2×C4—2441no—
488C2×C4—2442no—
489C2×C4—2442no—
490C2×C4—2441no—
491C2×C4—2441no—
492C2×C4—2441no—
493C2×C4—2442no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4#1451975012yes—
498C2×C4#1451975912yes—
499C2×C4#1451976212yes—
500C2×C4#1451976214yes—
501C2×C4#1451976314yes—
502C2×C4#1451977114yes—
503C2×C4#1451977214yes—
504C2×C4C2×D4×D6111yescharacteristic
505C2×C4C2×#1175822411yescharacteristic
506C2×C4C2×#1175822211yescharacteristic
507C2×C4C2×#1175822211yescharacteristic
508C2×C4C2×#1175822211yescharacteristic
509C2×C4C2×#1175822611yescharacteristic
510C8—42no—
511C8—1222no—
512C8—1222no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—2442no—
520C22—2442no—
521C22—2442no—
522C22—2442no—
523C22—2442no—
524C22—2442no—
525C22—2442no—
526C22#1683969114yes—
527C22C2×#1451975011yescharacteristic
528C22C2×#1451975911yescharacteristic
529C22C2×#1451976211yescharacteristic
530C22#1684056611yescharacteristic
531C22#1684057111yescharacteristic
532C22#1684057211yescharacteristic
533C22#1684057811yescharacteristic
534C4—81no—
535C4—81no—
536C4—82no—
537C4—4881no—
538C4—4881no—
539C4—4882no—
540C4—2442no—
541C4#1683969114yes—
542C4C2×#1451976211yescharacteristic
543C4C2×#1451976311yescharacteristic
544C4C2×#1451977111yescharacteristic
545C4C2×#1451977211yescharacteristic
546C2—24no—
547C2—4882no—
548C2C2×#1683969111yescharacteristic
549C2#2145548011yescharacteristic
550C2#2145548811yescharacteristic
551C2#2145577511yescharacteristic
552C2#2145577911yescharacteristic
553C2#2145578611yescharacteristic
554C2#2145579011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1736C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenter#1736D61
hypocenterC61#23641
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
1. C2×#16839798C2semidirect
5. #21447258C2semidirect
6. #21447257C2semidirect
7. #21447251C2semidirect
8. #21447246C2semidirect
9. #21445539C2semidirect
10. C2×#16839524C2semidirect
11. C61×#1736C2semidirect
278. #1736D61semidirect
320. C61#23641semidirect

Maximal subgroups

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

subgroupindex
C2×#168397982
C2×#168397302
#214484122
#214484072
#214472582
#214472572
#214472512
#214472462
#214455392
C2×#168395242
C61×#17362
#2364161