Finite Group Databasebeta

← #29649758

Subgroups — #29649758

Order 15616.

682 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454786C211yesmaximal, characteristic, complemented
2#21454780C211yesmaximal, characteristic, complemented
3C4×#14519797C211yesmaximal, characteristic
4#21444713C212yesmaximal, complemented
5#21444712C212yesmaximal, complemented
6#21444706C212yesmaximal, complemented
7#21444701C212yesmaximal, complemented
8#21444565C211yesmaximal, characteristic, complemented
9#21444202C211yesmaximal, characteristic, complemented
10C2×C8×#92418C211yesmaximal, characteristic, complemented
11C61×#1476C211yesmaximal, characteristic, complemented
12C2×#14519797C411yescharacteristic
13C4×#11758222C2211yescharacteristic
14#16839893C2212yes—
15#16839889C2212yes—
16C4×#11758202C411yescharacteristic
17C4×#11758202C2211yescharacteristic
18C4×#11758201C411yescharacteristic
19C4×#11758201C2211yescharacteristic
20C4×#11758200C411yescharacteristic
21C4×#11758200C2211yescharacteristic
22C4×#11758199C411yescharacteristic
23C4×#11758199C2211yescharacteristic
24#16839840C2212yes—
25#16839839C2212yes—
26#16839834—22no—
27#16839830—22no—
28#16839815—21no—
29#16839815C2211yescharacteristic
30#16839813—21no—
31#16839798C411yescharacteristic
32#16839758C411yescharacteristic
33#16839756C411yescharacteristic
34C4×#11758194—21no—
35C4×#11758189—21no—
36C2×C4×#92418C2211yescharacteristic
37C2×C4×#92418C2211yescharacteristic
38#16839514C2212yes—
39#16839512C2212yes—
40#16839494C2212yes—
41#16839491C2212yes—
42C8×#92418C2212yes—
43C8×#92418C2212yes—
44C4×#41×C61C2211yescharacteristic
45C61×#431C2211yescharacteristic
46C2×C4×C8×C61C2211yescharacteristic
47C61×Q8⋊C8C2212yes—
48C61×D4⋊C8C2212yes—
49C2×#11758222C2×C411yescharacteristic
50#14519797—24no—
51C2×#11758202C2×C411yescharacteristic
52C2×#11758200C2×C411yescharacteristic
53C2×#11758201C2×C411yescharacteristic
54C2×#11758199C2×C411yescharacteristic
55C2×#11758194—22no—
56C2×#11758189—22no—
57C22×#92418C2×C411yescharacteristic
58C22×#92418C2×C411yescharacteristic
59C22×#92418D412yes—
60C4×Dic122C2×C411yescharacteristic
61C4×Dic122C2311yescharacteristic
62C4×D244C2×C411yescharacteristic
63C4×D244C2311yescharacteristic
64C4×C22⋊D61—21no—
65C4×C22⋊D61—21no—
66#14519736C2×C411yescharacteristic
67#14519734—21no—
68C42×D61—21no—
69#14519710C2×C411yescharacteristic
70#14519710C2×C411yescharacteristic
71#14519708C2×C411yescharacteristic
72#14519708C2×C411yescharacteristic
73C4⋊#92418—22no—
74C4⋊#92418C2312yes—
75#14519706—22no—
76#14519701C2×C411yescharacteristic
77#14519701C2×C411yescharacteristic
78#14519700C2×C411yescharacteristic
79#14519700C2×C411yescharacteristic
80#14519698C2×C411yescharacteristic
81#14519698C2×C411yescharacteristic
82#14519697C2×C411yescharacteristic
83#14519697C2×C411yescharacteristic
84C22⋊#92418—21no—
85C22⋊#92418—22no—
86#14519689—22no—
87#14519688—22no—
88#14519687—22no—
89C4×#92418—21no—
90C4×#92418C2×C411yescharacteristic
91C4×#92418C2×C411yescharacteristic
92C4×#92418D411yescharacteristic
93C4×#92418D411yescharacteristic
94C4×#92418C2311yescharacteristic
95C4×#92418C2311yescharacteristic
96C4×#92418C2312yes—
97C2×#41×C61C2×C411yescharacteristic
98C22×C8×C61—21no—
99C4×Q8×C61C2×C411yescharacteristic
100C4×Q8×C61C2311yescharacteristic
101C4×D4×C61—21no—
102C4×D4×C61—21no—
103C4×D4×C61C2×C411yescharacteristic
104C4×D4×C61C2311yescharacteristic
105#117×C61—21no—
106#117×C61C2×C411yescharacteristic
107C2×C42×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C4⋊C8×C61C2312yes—
110C22⋊C8×C61—21no—
111C4×C8×C61—21no—
112C4×C8×C61—21no—
113C4×C8×C61C2312yes—
114#41×C61—22no—
115#41×C61—22no—
116C2×Q8×C61C22×C411yescharacteristic
117C2×D4×C61—21no—
118C2×D4×C61C22×C411yescharacteristic
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61#4111yescharacteristic
125C2×C22⋊D61—21no—
126#11758222—22no—
127#11758222—22no—
128C2×D244C22×C411yescharacteristic
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×Dic122C22×C411yescharacteristic
133C2×C8×C61—24no—
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61C22×C411yescharacteristic
139C4⋊C4×C61C22×C411yescharacteristic
140C22⋊C4×C61—21no—
141C22⋊C4×C61—21no—
142C22⋊C4×C61—21no—
143C42×C61—21no—
144C42×C61—21no—
145C42×C61—21no—
146C42×C61C22×C411yescharacteristic
147C42×C61C2×D411yescharacteristic
148C42×C61#4111yescharacteristic
149C42×C61C2411yescharacteristic
150C22⋊Dic61—21no—
151#11758202—22no—
152#11758201—22no—
153#11758200—22no—
154#11758199—22no—
155#11758198—21no—
156#11758198—21no—
157C4⋊Dic61C22×C411yescharacteristic
158C4⋊Dic61C22×C411yescharacteristic
159#11758196—21no—
160#11758196—21no—
161C4×Dic61—21no—
162C4×Dic61—21no—
163C4×Dic61—21no—
164#11758194—44no—
165C2×#92418—22no—
166C2×#92418—24no—
167C2×#92418—22no—
168C2×#92418M1614yescomplemented
169C2×#92418C22×C411yescharacteristic
170C2×#92418C22×C411yescharacteristic
171C2×#92418C22×C411yescharacteristic
172C2×#92418C22×C411yescharacteristic
173C2×#92418C2×D412yes—
174C2×#92418C2×D412yes—
175#11758189—44no—
176C23×C61—21no—
177C23×C61C4×D411yescharacteristic
178C22×D61—21no—
179Q8×C61—21no—
180Q8×C61—21no—
181D4×C61—21no—
182D4×C61—21no—
183D4×C61—42no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61C4×D411yescharacteristic
193C2×C4×C61C4×D411yescharacteristic
194C2×C4×C61C4×D411yescharacteristic
195C2×C4×C61C2×M1612yes—
196C2×C4×C61#13112yes—
197C2×C4×C61C23×C411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C2×#4111yescharacteristic
200C22⋊D61—42no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204D244—21no—
205D244—21no—
206C4×D61—42no—
207C4×D61—42no—
208C4×D61—42no—
209Dic122—21no—
210Dic122—21no—
211C8×C61—24no—
212C8×C61—42no—
213#92418—21no—
214#92418—21no—
215#92418—22no—
216#92418—42no—
217#92418C2×M1612yes—
218#92418C2×M1612yes—
219#92418#13514yescomplemented
220#17886—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—41no—
224C22×C61—41no—
225C22×C61—41no—
226C22×C61#43612yes—
227C22×C61C2×C4×D411yescharacteristic
228C2×D61—41no—
229C2×D61—41no—
230C2×D61—41no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—41no—
234C4×C61—41no—
235C4×C61—41no—
236C4×C61#43612yes—
237C4×C61#44312yes—
238C4×C61C22×M1611yescharacteristic
239C4×C61C2×#13111yescharacteristic
240C4×C61C2×#13512yes—
241Dic61—41no—
242Dic61—41no—
243Dic61—41no—
244#2965—611no—
245#2965—611no—
246C4×#135—611no—
247#1591—612no—
248#1590—612no—
249#1590—612no—
250#1589—612no—
251#1564—611no—
252#1476—611no—
253#1476D6111yescharacteristic, complemented
254C2×C82—611no—
255C2×C61—22no—
256C2×C61—81no—
257C2×C61C2×#43611yescharacteristic
258C2×C61C4×#13511yescharacteristic
259C2×C61C2×#44311yescharacteristic
260D61—81no—
261C2×#135—611no—
262C4×#41—611no—
263C4×#41C2×D6111yescharacteristic
264#446—612no—
265#446—612no—
266C4×Q16—611no—
267C4×Q16—611no—
268C4×SD16—611no—
269C4×SD16—611no—
270C4×SD16—611no—
271C4×SD16—611no—
272C4×D8—611no—
273C4×D8—611no—
274#436—612no—
275#436—612no—
276#435—1222no—
277#435—1222no—
278#431—611no—
279#431—1221no—
280#431—1221no—
281#431C2×D6111yescharacteristic
282#427—611no—
283#416—611no—
284#416—611no—
285C4×M16—1221no—
286C4×M16—1221no—
287C2×C4×C8—611no—
288C2×C4×C8—611no—
289C2×C4×C8C2×D6111yescharacteristic
290#335—612no—
291#334—612no—
292Q8⋊C8—612no—
293Q8⋊C8C2×D6112yes—
294D4⋊C8—612no—
295D4⋊C8C2×D6112yes—
296C82—612no—
297C82—612no—
298C61#1788611yescharacteristic, complemented, Sylow
299C2×#41—611no—
300C2×#41C4×D6111yescharacteristic
301#135—1224no—
302C2×Q16—611no—
303C2×SD16—611no—
304C2×SD16—611no—
305C2×D8—611no—
306C2×M16—1222no—
307C2×M16—1222no—
308C22×C8—21no—
309C22×C8—611no—
310C22×C8—611no—
311C22×C8—612no—
312C4×Q8—611no—
313C4×Q8—611no—
314C4×Q8C4×D6111yescharacteristic
315C4×Q8C22×D6111yescharacteristic
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—611no—
319C4×D4—611no—
320C4×D4—1221no—
321C4×D4—1221no—
322C4×D4C4×D6111yescharacteristic
323C4×D4C22×D6111yescharacteristic
324#117—21no—
325#117—611no—
326#117—1221no—
327#117C4×D6111yescharacteristic
328C2×C42—21no—
329C2×C42—1221no—
330C2×C42C22×D6111yescharacteristic
331#107—611no—
332#107—611no—
333#106—611no—
334#106—611no—
335C4⋊C8—612no—
336C4⋊C8—1222no—
337C4⋊C8—1222no—
338C4⋊C8C22×D6112yes—
339#103—611no—
340#103—611no—
341#103—611no—
342#103—611no—
343#102—611no—
344#102—611no—
345#102—611no—
346#102—611no—
347C22⋊C8—21no—
348C22⋊C8—1222no—
349C22⋊C8—1222no—
350C22⋊C8—1221no—
351#97—1222no—
352#97—1222no—
353C4×C8—21no—
354C4×C8—21no—
355C4×C8—611no—
356C4×C8—611no—
357C4×C8—611no—
358C4×C8—611no—
359C4×C8—611no—
360C4×C8—611no—
361C4×C8—612no—
362C4×C8—1221no—
363C4×C8C22×D6112yes—
364#41—22no—
365#41—22no—
366#41—1222no—
367#41—1222no—
368C2×Q8—611no—
369C2×Q8C2×C4×D6111yescharacteristic
370C2×D4—21no—
371C2×D4—611no—
372C2×D4—1221no—
373C2×D4C2×C4×D6111yescharacteristic
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—1221no—
378C22×C4—1221no—
379C22×C4—1221no—
380C22×C4C2×C4×D6111yescharacteristic
381C22×C4D4×D6111yescharacteristic
382C22×C4#1175822411yescharacteristic
383Q16—1222no—
384SD16—1222no—
385SD16—1222no—
386D8—1222no—
387M16—2444no—
388M16—2444no—
389C2×C8—24no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—611no—
393C2×C8—611no—
394C2×C8—612no—
395C2×C8—614no—
396C2×C8—611no—
397C2×C8—611no—
398C2×C8—612no—
399C2×C8—1224no—
400C2×C8—1222no—
401C2×C8—1222no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—611no—
405C4⋊C4—611no—
406C4⋊C4—1221no—
407C4⋊C4—1221no—
408C4⋊C4C2×C4×D6111yescharacteristic
409C4⋊C4C2×C4×D6111yescharacteristic
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—1221no—
414C22⋊C4—1221no—
415C22⋊C4—1221no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—1221no—
420C42—1221no—
421C42—1221no—
422C42C2×C4×D6111yescharacteristic
423C42D4×D6111yescharacteristic
424C42#1175822411yescharacteristic
425C42C23×D6111yescharacteristic
426C23—21no—
427C23—1221no—
428C23#1451973711yescharacteristic
429Q8—21no—
430Q8—21no—
431Q8—1221no—
432Q8—1221no—
433D4—21no—
434D4—21no—
435D4—42no—
436D4—1221no—
437D4—1221no—
438D4—2442no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—1221no—
448C2×C4—1221no—
449C2×C4—1221no—
450C2×C4—2442no—
451C2×C4—2442no—
452C2×C4—2442no—
453C2×C4#1451973711yescharacteristic
454C2×C4#1451973711yescharacteristic
455C2×C4#1451973711yescharacteristic
456C2×C4M16×D6112yes—
457C2×C4#1451978312yes—
458C2×C4C22×C4×D6111yescharacteristic
459C2×C4C2×D4×D6111yescharacteristic
460C2×C4C2×#1175822411yescharacteristic
461C8—24no—
462C8—42no—
463C8—612no—
464C8—612no—
465C8—614no—
466C8—1221no—
467C8—1221no—
468C8—1222no—
469C8—2442no—
470C22—21no—
471C22—21no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—2441no—
476C22—2441no—
477C22—2441no—
478C22#1683983712yes—
479C22C2×#1451973711yescharacteristic
480C4—21no—
481C4—21no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—2441no—
486C4—2441no—
487C4—2441no—
488C4#1683983712yes—
489C4#1683987612yes—
490C4C2×M16×D6111yescharacteristic
491C4C2×#1451978311yescharacteristic
492C4#135×D6112yes—
493C2—22no—
494C2—81no—
495C2—4881no—
496C2C2×#1683983711yescharacteristic
497C2#2145458711yescharacteristic
498C2C2×#1683987611yescharacteristic
subgroupG/H
centerC42D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1476C2
socleC22×C61C2×C4×D4
prime radicalC2×#14519797C4
cosocleC42×C61C24
hypercenter#1476D61
hypocenterC61#17886
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C42 ◁ C2×C42 ◁ #1476

Complements

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

NH ≅ G/Ntype
1. #21454786C2semidirect
2. #21454780C2semidirect
4. #21444713C2semidirect
5. #21444712C2semidirect
6. #21444706C2semidirect
7. #21444701C2semidirect
8. #21444565C2semidirect
9. #21444202C2semidirect
10. C2×C8×#92418C2semidirect
11. C61×#1476C2semidirect
168. C2×#92418M16semidirect
219. #92418#135semidirect
253. #1476D61semidirect
298. C61#17886semidirect

Maximal subgroups

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

subgroupindex
#214547862
#214547802
C4×#145197972
#214447132
#214447122
#214447062
#214447012
#214445652
#214442022
C2×C8×#924182
C61×#14762
#1788661