Finite Group Databasebeta

← #29708894

Subgroups — #29708894

Order 15616.

1014 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839730C211yesmaximal, characteristic
2#21448412C211yesmaximal, characteristic
3#21448407C211yesmaximal, characteristic
4#21447253C212yesmaximal, complemented
5#21447253C212yesmaximal, complemented
6#21447248C212yesmaximal, complemented
7#21447243C212yesmaximal, complemented
8#21445539C211yesmaximal, characteristic, complemented
9#21445535C211yesmaximal, characteristic, complemented
10C2×#16839524C211yesmaximal, characteristic, complemented
11C2×C61×#428C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14C2×#14519736C2211yescharacteristic
15C22×C4⋊Dic61C2211yescharacteristic
16#16839730—24no—
17C2×#14519689C2212yes—
18C2×#14519689C2212yes—
19#16839668—21no—
20#16839667—21no—
21#16839664—21no—
22#16839663—21no—
23#16839641—22no—
24#16839639—22no—
25#16839619C2212yes—
26#16839619C2212yes—
27#16839619C2212yes—
28#16839619C2212yes—
29#16839525C2212yes—
30#16839525C2212yes—
31#16839524C2211yescharacteristic
32#16839524C2212yes—
33#16839524C2211yescharacteristic
34#16839524C2212yes—
35#16839524C2211yescharacteristic
36#16839524C2211yescharacteristic
37C22×M16×C61C2211yescharacteristic
38C2×#117×C61C2211yescharacteristic
39C22×C4⋊C4×C61C2211yescharacteristic
40C61×#428—24no—
41C2×#107×C61C2212yes—
42C2×#106×C61C2212yes—
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×#92418—22no—
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#14519689—24no—
74#14519689—24no—
75#14519684—22no—
76#14519684—22no—
77#14519684—22no—
78#14519684—22no—
79C23×C4×C61—21no—
80C23×C4×C61C2311yescharacteristic
81C2×M16×C61—22no—
82C2×M16×C61—22no—
83C2×M16×C61D414yes—
84C22×C8×C61C2312yes—
85#117×C61—22no—
86#117×C61—22no—
87C2×C4⋊C4×C61—22no—
88C2×C4⋊C4×C61—22no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C4⋊C4×C61C2312yes—
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C22⋊C4×C61—21no—
94C2×C42×C61—21no—
95#107×C61—24no—
96#106×C61—24no—
97C24×C61C2×D411yescharacteristic
98C23×D61C2×D412yes—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—22no—
102C22×C4×C61—42no—
103C22×C4×C61—81no—
104C22×C4×C61—81no—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D412yes—
107C22×C4×C61#4112yes—
108C22×C4×C61#4112yes—
109C22×C4×C61#4112yes—
110C22×C4×C61#4112yes—
111C22×C4×C61#4112yes—
112C22×C4×C61C2411yescharacteristic
113C2×C22⋊D61—24no—
114C2×C22⋊D61—24no—
115C2×C22⋊D61—24no—
116C22×Dic61—22no—
117C22×Dic61—21no—
118C22×Dic61—21no—
119C22×Dic61—42no—
120C22×Dic61—81no—
121C22×Dic61—81no—
122C22×Dic61C2×D412yes—
123#11758222—44no—
124#11758222—44no—
125#11758222—44no—
126#11758222—44no—
127C2×D244—22no—
128C2×D244—22no—
129C2×D244—22no—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139M16×C61—28no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C2×C8×C61C2×D414yes—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—42no—
150C22⋊C4×C61—42no—
151C42×C61—42no—
152C22⋊Dic61—42no—
153#11758198—82no—
154#11758198—82no—
155#11758198—82no—
156#11758198—82no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61—42no—
164C4×Dic61—42no—
165C2×#92418—21no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418—44no—
172C23×C61—21no—
173C23×C61—21no—
174C23×C61—21no—
175C23×C61—21no—
176C23×C61#12112yes—
177C23×C61#12312yes—
178C23×C61#12312yes—
179C23×C61C22×D411yescharacteristic
180C22×D61—22no—
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—22no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—42no—
190C2×C4×C61—41no—
191C2×C4×C61—42no—
192C2×C4×C61—41no—
193C2×C4×C61—42no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—84no—
197C2×C4×C61#12112yes—
198C2×C4×C61#12314yes—
199C2×C4×C61#12312yes—
200C2×C4×C61#12312yes—
201C2×C4×C61#12414yes—
202C2×C4×C61#12614yes—
203C2×C4×C61#12614yes—
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C2×#4111yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C22⋊D61—48no—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—22no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—42no—
217C2×Dic61—41no—
218C2×Dic61—42no—
219C2×Dic61—41no—
220C2×Dic61—42no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—84no—
224D244—44no—
225C4×D61—42no—
226C4×D61—42no—
227C4×D61—42no—
228C4×D61—42no—
229Dic122—44no—
230C8×C61—24no—
231#92418—41no—
232#92418—41no—
233#23643—611nomaximal, Sylow
234C22×C61—22no—
235C22×C61—22no—
236C22×C61—22no—
237C22×C61—22no—
238C22×C61—22no—
239C22×C61—22no—
240C22×C61#39714yes—
241C22×C61C2×#12111yescharacteristic
242C22×C61C2×#12311yescharacteristic
243C22×C61C2×#12311yescharacteristic
244C22×C61#57611yescharacteristic
245C22×C61#57711yescharacteristic
246C22×C61#57711yescharacteristic
247C22×C61#57811yescharacteristic
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—42no—
251C2×D61—42no—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—42no—
255C4×C61—81no—
256C4×C61—82no—
257C4×C61—81no—
258C4×C61#39714yes—
259C4×C61C2×#12311yescharacteristic
260C4×C61C2×#12411yescharacteristic
261C4×C61C2×#12611yescharacteristic
262C4×C61C2×#12611yescharacteristic
263Dic61—42no—
264Dic61—82no—
265Dic61—81no—
266Dic61—81no—
267C2×#428D6111yescharacteristic, complemented
268C2×#408—611no—
269#2311—611no—
270#2311—611no—
271#2077—612no—
272#2077—612no—
273#2076—612no—
274#2075—612no—
275#1736—611no—
276#1735—611no—
277C2×#340—611no—
278C2×C61—24no—
279C2×C61C2×#39711yescharacteristic
280C2×C61#306211yescharacteristic
281C2×C61#306311yescharacteristic
282C2×C61#309611yescharacteristic
283C2×C61#309711yescharacteristic
284C2×C61#309811yescharacteristic
285C2×C61#309911yescharacteristic
286D61—82no—
287C22×#41—611no—
288C22×M16C2×D6111yescharacteristic
289C23×C8—611no—
290C2×#117—611no—
291C2×#117C2×D6111yescharacteristic
292C22×C4⋊C4—611no—
293C22×C4⋊C4C2×D6111yescharacteristic
294#428—24no—
295C2×#107C2×D6112yes—
296C2×#106C2×D6112yes—
297#408—1224no—
298C2×C22⋊C8—612no—
299C2×C22⋊C8—612no—
300#391—1221no—
301#391—1221no—
302#390—1221no—
303#390—1221no—
304#386—1222no—
305#386—1222no—
306#380—612no—
307#380—612no—
308#380—612no—
309#380—612no—
310#340—612no—
311#340—612no—
312#340—611no—
313#340—612no—
314#340—611no—
315#340—612no—
316#340—611no—
317#340—611no—
318C61#2364311yescharacteristic, complemented, Sylow
319C2×#41—1222no—
320C2×#41—1222no—
321C2×#41—1222no—
322C2×#41—1222no—
323C2×#41—1222no—
324C2×#41—1222no—
325C22×Q8—611no—
326C22×D4—612no—
327C22×D4—611no—
328C23×C4—21no—
329C23×C4—612no—
330C23×C4—1221no—
331C23×C4C22×D6111yescharacteristic
332C2×M16—22no—
333C2×M16—22no—
334C2×M16C22⋊D6114yes—
335C22×C8—611no—
336C22×C8—611no—
337C22×C8—1222no—
338C22×C8—1222no—
339C22×C8—1222no—
340C22×C8C22×D6112yes—
341#117—22no—
342#117—22no—
343#117—1222no—
344#117—1222no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—612no—
349C2×C4⋊C4—611no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—1222no—
352C2×C4⋊C4—1222no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4C22×D6112yes—
355C2×C4⋊C4C22×D6111yescharacteristic
356C2×C4⋊C4C22×D6111yescharacteristic
357C2×C22⋊C4—21no—
358C2×C22⋊C4—1222no—
359C2×C22⋊C4—1222no—
360C2×C22⋊C4—1222no—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4—1221no—
363C2×C42—21no—
364C2×C42—1221no—
365#107—24no—
366#106—24no—
367C22⋊C8—1224no—
368C22⋊C8—1224no—
369#95—1222no—
370#95—1222no—
371#95—1222no—
372#95—1222no—
373C24—612no—
374C24D4×D6111yescharacteristic
375#41—2444no—
376#41—2444no—
377#41—2444no—
378#41—2444no—
379C2×Q8—1222no—
380C2×Q8—1222no—
381C2×Q8—1222no—
382C2×D4—1224no—
383C2×D4—1224no—
384C2×D4—1224no—
385C2×D4—1222no—
386C2×D4—1222no—
387C2×D4—1222no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—42no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4—612no—
395C22×C4—1222no—
396C22×C4—1222no—
397C22×C4—1222no—
398C22×C4—1222no—
399C22×C4—1222no—
400C22×C4—1222no—
401C22×C4—1221no—
402C22×C4—1221no—
403C22×C4—1222no—
404C22×C4—4881no—
405C22×C4—4881no—
406C22×C4—2442no—
407C22×C4#1175822212yes—
408C22×C4#1175822212yes—
409C22×C4D4×D6111yescharacteristic
410C22×C4#1175822412yes—
411C22×C4#1175822612yes—
412C22×C4#1175822612yes—
413C22×C4C2×C22⋊D6112yes—
414C22×C4C23×D6111yescharacteristic
415M16—28no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—1221no—
419C2×C8—1221no—
420C2×C8—1221no—
421C2×C8—1221no—
422C2×C8—1221no—
423C2×C8—1221no—
424C2×C8—2444no—
425C2×C8C2×C22⋊D6114yes—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—42no—
433C4⋊C4—1222no—
434C4⋊C4—1221no—
435C4⋊C4—1221no—
436C4⋊C4—1222no—
437C4⋊C4—1221no—
438C4⋊C4—1221no—
439C4⋊C4—2442no—
440C22⋊C4—42no—
441C22⋊C4—4882no—
442C22⋊C4—4882no—
443C22⋊C4—4882no—
444C22⋊C4—4882no—
445C22⋊C4—2442no—
446C42—42no—
447C42—2442no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—1222no—
453C23—1222no—
454C23—1222no—
455C23—1222no—
456C23—1222no—
457C23—1222no—
458C23—1222no—
459C23#1451974812yes—
460C23#1451976212yes—
461C23#1451976212yes—
462C23C2×D4×D6111yescharacteristic
463Q8—2444no—
464D4—2448no—
465D4—2444no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—84no—
476C2×C4—1222no—
477C2×C4—1222no—
478C2×C4—1222no—
479C2×C4—4884no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4—2442no—
483C2×C4—2441no—
484C2×C4—2442no—
485C2×C4—2442no—
486C2×C4—2441no—
487C2×C4—2441no—
488C2×C4—2441no—
489C2×C4—2442no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4—2442no—
493C2×C4#1451974812yes—
494C2×C4#1451976014yes—
495C2×C4#1451976212yes—
496C2×C4#1451976212yes—
497C2×C4#1451976614yes—
498C2×C4#1451977214yes—
499C2×C4#1451977214yes—
500C2×C4C22×C22⋊D6111yescharacteristic
501C2×C4C2×#1175822411yescharacteristic
502C2×C4C2×#1175822211yescharacteristic
503C2×C4C2×#1175822211yescharacteristic
504C2×C4C2×#1175822611yescharacteristic
505C2×C4C2×#1175822611yescharacteristic
506C8—24no—
507C8—2441no—
508C8—2441no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—2442no—
516C22—2442no—
517C22—2442no—
518C22—2442no—
519C22—2442no—
520C22—2442no—
521C22—2442no—
522C22#1683968814yes—
523C22C2×#1451974811yescharacteristic
524C22C2×#1451976211yescharacteristic
525C22C2×#1451976211yescharacteristic
526C22#1684056611yescharacteristic
527C22#1684057111yescharacteristic
528C22#1684057211yescharacteristic
529C22#1684057811yescharacteristic
530C4—81no—
531C4—82no—
532C4—81no—
533C4—4882no—
534C4—4881no—
535C4—4881no—
536C4—2442no—
537C4#1683968814yes—
538C4C2×#1451976011yescharacteristic
539C4C2×#1451976611yescharacteristic
540C4C2×#1451977211yescharacteristic
541C4C2×#1451977211yescharacteristic
542C2—24no—
543C2—4882no—
544C2C2×#1683968811yescharacteristic
545C2#2145549311yescharacteristic
546C2#2145550411yescharacteristic
547C2#2145577911yescharacteristic
548C2#2145579011yescharacteristic
549C2#2145580111yescharacteristic
550C2#2145581211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#428C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenterC2×#428D61
hypocenterC61#23643
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. #21447253C2semidirect
5. #21447253C2semidirect
6. #21447248C2semidirect
7. #21447243C2semidirect
8. #21445539C2semidirect
9. #21445535C2semidirect
10. C2×#16839524C2semidirect
11. C2×C61×#428C2semidirect
267. C2×#428D61semidirect
318. C61#23643semidirect

Maximal subgroups

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

subgroupindex
C2×#168397302
#214484122
#214484072
#214472532
#214472482
#214472432
#214455392
#214455352
C2×#168395242
C2×C61×#4282
#2364361