Finite Group Databasebeta

← #29649727

Subgroups — #29649727

Order 15616.

660 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454937C211yesmaximal, characteristic, complemented
2#21454930C211yesmaximal, characteristic, complemented
3C4×#14519797C211yesmaximal, characteristic
4#21444687C212yesmaximal, complemented
5#21444680C212yesmaximal, complemented
6#21444659C212yesmaximal, complemented
7#21444652C212yesmaximal, complemented
8C2×#16839512C211yesmaximal, characteristic, complemented
9#21444207C211yesmaximal, characteristic, complemented
10#21444112C211yesmaximal, characteristic, complemented
11C61×#1477C211yesmaximal, characteristic, complemented
12C2×#14519797C411yescharacteristic
13C4×#11758222C2211yescharacteristic
14#16839889C2212yes—
15Q8×#92418C2212yes—
16C4×#11758202C411yescharacteristic
17C4×#11758202C2211yescharacteristic
18C4×#11758201C411yescharacteristic
19C4×#11758201C2211yescharacteristic
20C4×#11758200C411yescharacteristic
21C4×#11758200C2211yescharacteristic
22C4×#11758199C411yescharacteristic
23C4×#11758199C2211yescharacteristic
24#16839839C2212yes—
25#16839833—22no—
26D4×#92418—22no—
27D4×#92418C2212yes—
28#16839818—21no—
29#16839815—21no—
30#16839815C2211yescharacteristic
31#16839798C411yescharacteristic
32#16839784—21no—
33C2×C4⋊#92418—21no—
34C2×C4⋊#92418C2211yescharacteristic
35#16839758C411yescharacteristic
36#16839756C411yescharacteristic
37C2×C4×#92418C2211yescharacteristic
38#16839512C2212yes—
39#16839512C2212yes—
40#16839494C2212yes—
41#16839491C2212yes—
42#16839484C2212yes—
43#16839483C2212yes—
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×#11758189—22no—
56C22×#92418—21no—
57C22×#92418—22no—
58C22×#92418C2×C411yescharacteristic
59C22×#92418C2×C411yescharacteristic
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—21no—
74C4⋊#92418—22no—
75C4⋊#92418D411yescharacteristic
76C4⋊#92418D411yescharacteristic
77C4⋊#92418C2312yes—
78#14519706—22no—
79#14519701C2×C411yescharacteristic
80#14519701C2×C411yescharacteristic
81#14519700C2×C411yescharacteristic
82#14519700C2×C411yescharacteristic
83#14519698C2×C411yescharacteristic
84#14519698C2×C411yescharacteristic
85#14519697C2×C411yescharacteristic
86#14519697C2×C411yescharacteristic
87C22⋊#92418—22no—
88C22⋊#92418D412yes—
89#14519689—22no—
90#14519687—22no—
91C4×#92418—22no—
92C4×#92418C2×C411yescharacteristic
93C4×#92418C2×C411yescharacteristic
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×C61—21no—
110C4⋊C8×C61—21no—
111C4⋊C8×C61C2312yes—
112C22⋊C8×C61—21no—
113C4×C8×C61C2312yes—
114#41×C61—22no—
115#41×C61M1614yescomplemented
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—22no—
134C2×C8×C61—22no—
135C2×C8×C61—81no—
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—
164C2×#92418—22no—
165C2×#92418—22no—
166C2×#92418—22no—
167C2×#92418—82no—
168C2×#92418—81no—
169C2×#92418C22×C411yescharacteristic
170C2×#92418C22×C411yescharacteristic
171C2×#92418C22×C411yescharacteristic
172C2×#92418C22×C411yescharacteristic
173C2×#92418C2×D412yes—
174C2×#92418C2×D412yes—
175#11758189—82no—
176C23×C61—21no—
177C23×C61C4×D411yescharacteristic
178C22×D61—21no—
179Q8×C61—21no—
180Q8×C61C2×M1612yes—
181D4×C61—24no—
182D4×C61—21no—
183D4×C61C2×M1612yes—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—24no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—81no—
194C2×C4×C61C4×D411yescharacteristic
195C2×C4×C61C4×D411yescharacteristic
196C2×C4×C61C4×D411yescharacteristic
197C2×C4×C61C2×M1612yes—
198C2×C4×C61#13112yes—
199C2×C4×C61C23×C411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C2×#4111yescharacteristic
202C22⋊D61—42no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206D244—21no—
207D244—21no—
208C4×D61—41no—
209C4×D61—41no—
210C4×D61—42no—
211C4×D61—81no—
212Dic122—21no—
213Dic122—21no—
214C8×C61—42no—
215C8×C61—42no—
216#92418—21no—
217#92418—21no—
218#92418—21no—
219#92418—21no—
220#92418—22no—
221#92418—22no—
222#92418—42no—
223#17882—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—22no—
227C22×C61—41no—
228C22×C61—41no—
229C22×C61#43512yes—
230C22×C61C2×C4×D411yescharacteristic
231C2×D61—41no—
232C2×D61—41no—
233C2×D61—41no—
234C4×C61—22no—
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—21no—
238C4×C61—41no—
239C4×C61—41no—
240C4×C61#43512yes—
241C4×C61C22×M1611yescharacteristic
242C4×C61C2×#13111yescharacteristic
243C4×C61#57711yescharacteristic
244C4×C61#57711yescharacteristic
245Dic61—41no—
246Dic61—41no—
247Dic61—41no—
248#2989—611no—
249#2988—611no—
250C4×#135—611no—
251#1587—612no—
252#1586—612no—
253#1583—612no—
254#1582—612no—
255C2×#334—611no—
256#1477—611no—
257#1477D6111yescharacteristic, complemented
258#1452—611no—
259C2×C61—22no—
260C2×C61—42no—
261C2×C61C2×#43511yescharacteristic
262C2×C61#295211yescharacteristic
263C2×C61#295711yescharacteristic
264D61—81no—
265C2×#135—611no—
266C4×#41—611no—
267C4×#41C2×D6111yescharacteristic
268#446—612no—
269C8×Q8—612no—
270C4×Q16—611no—
271C4×Q16—611no—
272C4×SD16—611no—
273C4×SD16—611no—
274C4×SD16—611no—
275C4×SD16—611no—
276C4×D8—611no—
277C4×D8—611no—
278#436—612no—
279#435—1222no—
280C8×D4—612no—
281C8×D4—1222no—
282#432—1221no—
283#431—611no—
284#431—1221no—
285#431C2×D6111yescharacteristic
286#427—611no—
287#423—1221no—
288C2×C4⋊C8—611no—
289C2×C4⋊C8—1221no—
290C2×C4⋊C8C2×D6111yescharacteristic
291#416—611no—
292#416—611no—
293C2×C4×C8—611no—
294#334—612no—
295#334—612no—
296Q8⋊C8—612no—
297Q8⋊C8C2×D6112yes—
298D4⋊C8—612no—
299D4⋊C8C2×D6112yes—
300#322—612no—
301#322—612no—
302C61#1788211yescharacteristic, complemented, Sylow
303C2×#41—611no—
304C2×#41C4×D6111yescharacteristic
305#135—1224no—
306C2×Q16—611no—
307C2×SD16—611no—
308C2×SD16—611no—
309C2×D8—611no—
310C2×M16—1222no—
311C22×C8—21no—
312C22×C8—611no—
313C22×C8—611no—
314C22×C8—1222no—
315C22×C8—1221no—
316C4×Q8—611no—
317C4×Q8—611no—
318C4×Q8C4×D6111yescharacteristic
319C4×Q8C22×D6111yescharacteristic
320C4×D4—21no—
321C4×D4—21no—
322C4×D4—611no—
323C4×D4—611no—
324C4×D4—1221no—
325C4×D4—1221no—
326C4×D4C4×D6111yescharacteristic
327C4×D4C22×D6111yescharacteristic
328#117—21no—
329#117—611no—
330#117—1221no—
331#117C4×D6111yescharacteristic
332C2×C42—21no—
333C2×C42—1221no—
334C2×C42C22×D6111yescharacteristic
335#107—611no—
336#107—611no—
337#106—611no—
338#106—611no—
339C4⋊C8—21no—
340C4⋊C8—21no—
341C4⋊C8—611no—
342C4⋊C8—611no—
343C4⋊C8—612no—
344C4⋊C8—1222no—
345C4⋊C8—1222no—
346C4⋊C8—1221no—
347C4⋊C8C22×D6112yes—
348#103—611no—
349#103—611no—
350#103—611no—
351#103—611no—
352#102—611no—
353#102—611no—
354#102—611no—
355#102—611no—
356C22⋊C8—21no—
357C22⋊C8—612no—
358C22⋊C8—1222no—
359C22⋊C8—1222no—
360#97—1222no—
361C4×C8—611no—
362C4×C8—611no—
363C4×C8—611no—
364C4×C8—611no—
365C4×C8—612no—
366C4×C8—1222no—
367C4×C8C22×D6112yes—
368#41—22no—
369#41—1222no—
370#41—1222no—
371#41#1175818914yescomplemented
372C2×Q8—611no—
373C2×Q8C2×C4×D6111yescharacteristic
374C2×D4—21no—
375C2×D4—611no—
376C2×D4—1221no—
377C2×D4C2×C4×D6111yescharacteristic
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—1221no—
382C22×C4—1221no—
383C22×C4—1221no—
384C22×C4C2×C4×D6111yescharacteristic
385C22×C4D4×D6111yescharacteristic
386C22×C4#1175822411yescharacteristic
387Q16—1222no—
388SD16—1222no—
389SD16—1222no—
390D8—1222no—
391M16—4882no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—81no—
395C2×C8—611no—
396C2×C8—611no—
397C2×C8—611no—
398C2×C8—611no—
399C2×C8—612no—
400C2×C8—612no—
401C2×C8—1222no—
402C2×C8—1222no—
403C2×C8—1222no—
404C2×C8—4882no—
405C2×C8—4881no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—611no—
409C4⋊C4—611no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4C2×C4×D6111yescharacteristic
413C4⋊C4C2×C4×D6111yescharacteristic
414C22⋊C4—21no—
415C22⋊C4—21no—
416C22⋊C4—21no—
417C22⋊C4—1221no—
418C22⋊C4—1221no—
419C22⋊C4—1221no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—1221no—
424C42—1221no—
425C42—1221no—
426C42C2×C4×D6111yescharacteristic
427C42D4×D6111yescharacteristic
428C42#1175822411yescharacteristic
429C42C23×D6111yescharacteristic
430C23—21no—
431C23—1221no—
432C23#1451973711yescharacteristic
433Q8—21no—
434Q8—1221no—
435Q8—1221no—
436Q8C2×#1175818912yes—
437D4—24no—
438D4—21no—
439D4—1221no—
440D4—1221no—
441D4—2442no—
442D4C2×#1175818912yes—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—24no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—81no—
453C2×C4—1221no—
454C2×C4—1221no—
455C2×C4—1221no—
456C2×C4—4881no—
457C2×C4—2441no—
458C2×C4—2441no—
459C2×C4—2442no—
460C2×C4#1451973711yescharacteristic
461C2×C4#1451973711yescharacteristic
462C2×C4#1451973711yescharacteristic
463C2×C4C2×#1175818912yes—
464C2×C4#1451978512yes—
465C2×C4C22×C4×D6111yescharacteristic
466C2×C4C2×D4×D6111yescharacteristic
467C2×C4C2×#1175822411yescharacteristic
468C8—42no—
469C8—42no—
470C8—1221no—
471C8—1221no—
472C8—1221no—
473C8—1221no—
474C8—1222no—
475C8—1222no—
476C8—2442no—
477C22—21no—
478C22—21no—
479C22—22no—
480C22—41no—
481C22—41no—
482C22—2441no—
483C22—2441no—
484C22—2441no—
485C22#1683983012yes—
486C22C2×#1451973711yescharacteristic
487C4—21no—
488C4—21no—
489C4—21no—
490C4—22no—
491C4—41no—
492C4—41no—
493C4—2441no—
494C4—2441no—
495C4—2441no—
496C4#1683983012yes—
497C4C22×#1175818911yescharacteristic
498C4C2×#1451978511yescharacteristic
499C4#1684057111yescharacteristic
500C4#1684057211yescharacteristic
501C2—22no—
502C2—42no—
503C2—4881no—
504C2C2×#1683983011yescharacteristic
505C2#2145469811yescharacteristic
506C2#2145473211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1477C2
socleC22×C61C2×C4×D4
prime radicalC2×#14519797C4
cosocleC42×C61C24
hypercenter#1477D61
hypocenterC61#17882
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1477

Complements

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

NH ≅ G/Ntype
1. #21454937C2semidirect
2. #21454930C2semidirect
4. #21444687C2semidirect
5. #21444680C2semidirect
6. #21444659C2semidirect
7. #21444652C2semidirect
8. C2×#16839512C2semidirect
9. #21444207C2semidirect
10. #21444112C2semidirect
11. C61×#1477C2semidirect
115. #41×C61M16semidirect
257. #1477D61semidirect
302. C61#17882semidirect
371. #41#11758189semidirect

Maximal subgroups

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

subgroupindex
#214549372
#214549302
C4×#145197972
#214446872
#214446802
#214446592
#214446522
C2×#168395122
#214442072
#214441122
C61×#14772
#1788261