Finite Group Databasebeta

← #29647867

Subgroups — #29647867

Order 15616.

1066 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459673C211yesmaximal, characteristic
2#21454859C211yesmaximal, characteristic, complemented
3#21448730C211yesmaximal, characteristic, complemented
4#21444383C211yesmaximal, characteristic, complemented
5#21444380C211yesmaximal, characteristic, complemented
6#21444361C214yesmaximal, complemented
7#21444322C214yesmaximal, complemented
8#21444313C211yesmaximal, characteristic, complemented
9C61×#1503C211yesmaximal, characteristic, complemented
10#16840497—22no—
11#16840485C412yescomplemented
12#16840478C412yescomplemented
13#16840465—22no—
14#16840434C412yescomplemented
15#16840415C412yescomplemented
16#16840352—22no—
17#16840331—21no—
18#16840312C2211yescharacteristic
19#16840275C2214yes—
20#16840242—21no—
21C2×C4×D244—21no—
22C2×C4×D244C2211yescharacteristic
23#16839841—22no—
24#16839839—22no—
25#16839815C2211yescharacteristic
26#16839724C2214yes—
27#16839660—21no—
28#16839633—22no—
29#16839628C2211yescharacteristic
30#16839621C2212yes—
31#95×D61C2212yes—
32#16839499C2214yes—
33#16839497C2214yes—
34#16839495C2212yes—
35#16839495C2212yes—
36C61×#533C2211yescharacteristic
37C61×#431C2211yescharacteristic
38C4×C4⋊C4×C61C2211yescharacteristic
39C61×#328C2212yes—
40C61×#328C2212yes—
41C2×#11758226C2×C414yescomplemented
42C2×#11758222—21no—
43C22×D244D411yescharacteristic
44C22×C4×D61—22no—
45C22×C4×D61C2312yes—
46C2×#11758189—24no—
47#14519777C2×C411yescharacteristic, complemented
48#14519774—22no—
49#14519774C2×C411yescharacteristic, complemented
50#14519772—24no—
51#14519766—24no—
52#14519765—22no—
53#14519765—21no—
54#14519762—24no—
55#14519762—24no—
56#14519753C2×C414yescomplemented
57#14519752—22no—
58#14519752—22no—
59#14519752—24no—
60#14519752C2×C414yescomplemented
61#14519751—21no—
62#14519751C2312yes—
63#14519747—24no—
64C4×Dic122—22no—
65C4×D244—21no—
66C4×D244—22no—
67C4×D244—44no—
68C4×D244C2×C412yes—
69C4×D244C2×C412yes—
70C4×D244C2×C412yescomplemented
71C4×D244D412yes—
72C4×D244D412yes—
73#14519739—24no—
74#14519737—24no—
75#14519735—22no—
76#14519735C2×C414yescomplemented
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61D411yescharacteristic
79C2×#11758196—22no—
80C4⋊C4×D61—22no—
81C2×#11758198—22no—
82C2×#11758198D412yes—
83C4⋊#92418—21no—
84C22⋊#92418C2312yes—
85#14519689—24no—
86C4×#92418—21no—
87#14519684—21no—
88#14519684—22no—
89#14519684C2312yes—
90C4⋊Q8×C61C2×C411yescharacteristic, complemented
91C4⋊Q8×C61C2×C411yescharacteristic, complemented
92#125×C61—21no—
93C22⋊Q8×C61—21no—
94C22⋊Q8×C61C2312yes—
95C4×Q8×C61—21no—
96C4×Q8×C61C2×C412yescomplemented
97#117×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C42×C61—21no—
100C2×C42×C61C2311yescharacteristic
101C4⋊C8×C61—21no—
102C22⋊C8×C61C2312yes—
103C4×C8×C61—21no—
104#95×C61C2311yescharacteristic
105#95×C61C2311yescharacteristic
106C23×D61C2×D412yes—
107C2×Q8×C61—21no—
108C2×Q8×C61C22×C412yes—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2411yescharacteristic
114C2×C22⋊D61—22no—
115C22×Dic61—22no—
116C22×Dic61C2×D412yes—
117#11758226—84no—
118#11758222—42no—
119#11758222—42no—
120C2×D244—22no—
121C2×D244—22no—
122C2×D244—21no—
123C2×D244—24no—
124C2×D244—24no—
125C2×D244—21no—
126C2×D244C22⋊C412yes—
127C2×D244C22⋊C412yes—
128C2×D244C22×C412yes—
129C2×C4×D61—22no—
130C2×C4×D61—24no—
131C2×C4×D61—44no—
132C2×C4×D61—44no—
133C2×C4×D61—82no—
134C2×C4×D61—82no—
135C2×C4×D61C22×C414yes—
136C2×C4×D61C2×D414yes—
137C2×Dic122—21no—
138C2×C8×C61—22no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—42no—
145C4⋊C4×C61C22×C412yes—
146C22⋊C4×C61—22no—
147C42×C61—22no—
148C42×C61—42no—
149C42×C61C22×C411yescharacteristic
150C42×C61C22×C411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61C2×D411yescharacteristic
153#11758198—24no—
154#11758198—24no—
155#11758198—22no—
156#11758198—22no—
157#11758198—24no—
158#11758198—44no—
159#11758198C22⋊C414yes—
160#11758198C22⋊C414yes—
161C4⋊Dic61—21no—
162C4⋊Dic61—42no—
163C4⋊Dic61C22⋊C412yes—
164C4⋊Dic61C22⋊C412yes—
165C4⋊Dic61C22×C412yes—
166#11758196—24no—
167#11758196—82no—
168C4×Dic61—24no—
169C2×#92418—22no—
170#11758189—84no—
171C23×C61C22×D411yescharacteristic
172C22×D61—22no—
173C22×D61—22no—
174C22×D61—22no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61C2×C22⋊C414yes—
178Q8×C61—42no—
179Q8×C61—81no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61C2×C22⋊C411yescharacteristic
191C2×C4×C61C2×C22⋊C411yescharacteristic
192C2×C4×C61C2×C22⋊C411yescharacteristic
193C2×C4×C61C2×C22⋊C411yescharacteristic
194C2×C4×C61AGL(1,ℤ/8)12yes—
195C2×C4×C61#13712yes—
196C2×C4×C61C23×C411yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C22⋊D61—42no—
199C22⋊D61—42no—
200C2×Dic61—22no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205C2×Dic61C2×C22⋊C414yes—
206D244—24no—
207D244—24no—
208D244—42no—
209D244—42no—
210D244—81no—
211D244—84no—
212D244—84no—
213C4×D61—44no—
214C4×D61—44no—
215C4×D61—44no—
216C4×D61—84no—
217C4×D61—88no—
218Dic122—42no—
219C8×C61—42no—
220#92418—42no—
221#17630—611nomaximal, Sylow
222C22×C61—21no—
223C22×C61—21no—
224C22×C61C22×C22⋊C411yescharacteristic
225C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
226C22×C61C2×#13711yescharacteristic
227C2×D61—24no—
228C2×D61—24no—
229C2×D61—24no—
230C2×D61—44no—
231C2×D61—44no—
232C2×D61—42no—
233C2×D61—42no—
234C2×D61—42no—
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—21no—
238C4×C61—21no—
239C4×C61—42no—
240C4×C61—42no—
241C4×C61—42no—
242C4×C61—81no—
243C4×C61#41712yes—
244C4×C61#41712yes—
245Dic61—24no—
246Dic61—42no—
247Dic61—82no—
248#3506—611no—
249#2975—611no—
250#2349—611no—
251#1518—611no—
252#1517—611no—
253#1513—614no—
254#1504—614no—
255#1503D6111yescharacteristic, complemented
256#1502—611no—
257C2×C61—21no—
258C2×C61—21no—
259C2×C61#288311yescharacteristic
260C2×C61C2×#41711yescharacteristic
261C2×C61#290111yescharacteristic
262D61—44no—
263D61—82no—
264#559—1222no—
265#556—612no—
266#555—612no—
267#553—1222no—
268#550—612no—
269#547—612no—
270#538—1222no—
271#535—1221no—
272#533C2×D6111yescharacteristic
273#532—611no—
274#525—614no—
275#520—1221no—
276C2×C4×D4—611no—
277C2×C4×D4—1221no—
278#436—1222no—
279#436—1222no—
280#431—611no—
281#431C2×D6111yescharacteristic
282#407—614no—
283#389—1221no—
284#385—1222no—
285#382—611no—
286#380—612no—
287C4×C4⋊C4C2×D6111yescharacteristic
288C2×#95—612no—
289#328—614no—
290#328C2×D6112yes—
291#328C2×D6112yes—
292#327—614no—
293#327—612no—
294#327—612no—
295C61#1763011yescharacteristic, complemented, Sylow
296C2×#41—614no—
297C2×#41—1221no—
298C22×D4—611no—
299C23×C4—612no—
300C23×C4—1222no—
301C2×M16—1224no—
302C4⋊Q8—611no—
303C4⋊Q8C4×D6111yescharacteristic, complemented
304C4⋊Q8C4×D6111yescharacteristic, complemented
305#127—611no—
306#127—1222no—
307#126—1224no—
308#125—21no—
309#124—1222no—
310#124—1224no—
311#124—1221no—
312#123—1224no—
313#123—1224no—
314C22⋊Q8—21no—
315C22⋊Q8—614no—
316C22⋊Q8C22×D6112yes—
317#121—612no—
318#121—614no—
319#121—1224no—
320#121—1222no—
321#121—1222no—
322#121—1221no—
323#120—1224no—
324C4×Q8—21no—
325C4×Q8—1222no—
326C4×Q8C4×D6112yescomplemented
327C4×D4—612no—
328C4×D4—612no—
329C4×D4—612no—
330C4×D4—612no—
331C4×D4—612no—
332C4×D4—1222no—
333C4×D4—1224no—
334C4×D4—1224no—
335C4×D4—1221no—
336C4×D4—2444no—
337#117—21no—
338#117—614no—
339#117—1222no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—611no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—1222no—
344C2×C4⋊C4—1221no—
345C2×C22⋊C4—612no—
346C2×C22⋊C4—1222no—
347C2×C42—21no—
348C2×C42C22×D6111yescharacteristic
349C4⋊C8—21no—
350C4⋊C8—1221no—
351C22⋊C8—612no—
352C22⋊C8—1224no—
353C22⋊C8C22×D6112yes—
354C4×C8—21no—
355C4×C8—1221no—
356#95—612no—
357#95—1222no—
358#95—1221no—
359#95C22×D6111yescharacteristic
360#95C22×D6111yescharacteristic
361C24—612no—
362#41—4884no—
363#41—2442no—
364#41—2442no—
365C2×Q8—21no—
366C2×Q8—1221no—
367C2×Q8C2×C4×D6112yes—
368C2×D4—612no—
369C2×D4—612no—
370C2×D4—612no—
371C2×D4—1224no—
372C2×D4—1224no—
373C2×D4—1222no—
374C2×D4—1222no—
375C2×D4—1221no—
376C2×D4—1221no—
377C2×D4—1222no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—612no—
381C22×C4—614no—
382C22×C4—614no—
383C22×C4—1224no—
384C22×C4—1222no—
385C22×C4—1222no—
386C22×C4—4882no—
387C22×C4—4882no—
388C22×C4—2444no—
389C22×C4—2444no—
390C22×C4D4×D6111yescharacteristic
391C22×C4D4×D6111yescharacteristic
392C22×C4C23×D6111yescharacteristic
393M16—4884no—
394C2×C8—22no—
395C2×C8—1222no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—42no—
402C4⋊C4—612no—
403C4⋊C4—612no—
404C4⋊C4—612no—
405C4⋊C4—1224no—
406C4⋊C4—1221no—
407C4⋊C4—4882no—
408C4⋊C4—2442no—
409C4⋊C4C2×C4×D6112yes—
410C22⋊C4—22no—
411C22⋊C4—614no—
412C22⋊C4—614no—
413C22⋊C4—1224no—
414C22⋊C4—1224no—
415C22⋊C4—1224no—
416C22⋊C4—1222no—
417C22⋊C4—1222no—
418C22⋊C4—2444no—
419C42—22no—
420C42—42no—
421C42—1224no—
422C42C2×C4×D6111yescharacteristic
423C42C2×C4×D6111yescharacteristic
424C42D4×D6111yescharacteristic
425C42D4×D6111yescharacteristic
426C23—614no—
427C23—1222no—
428C23—1222no—
429C23—1222no—
430C23—2442no—
431C23—2442no—
432C23C2×D4×D6111yescharacteristic
433Q8—42no—
434Q8—81no—
435Q8—2442no—
436D4—1224no—
437D4—1224no—
438D4—4881no—
439D4—4884no—
440D4—4884no—
441D4—2442no—
442D4—2442no—
443D4—2442no—
444D4—2442no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—614no—
456C2×C4—1222no—
457C2×C4—4884no—
458C2×C4—4888no—
459C2×C4—2444no—
460C2×C4—2444no—
461C2×C4—2442no—
462C2×C4—2442no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2444no—
466C2×C4C22⋊C4×D6111yescharacteristic
467C2×C4C22⋊C4×D6111yescharacteristic
468C2×C4C22⋊C4×D6111yescharacteristic
469C2×C4C22⋊C4×D6111yescharacteristic
470C2×C4#1451980412yes—
471C2×C4#1451980912yes—
472C2×C4C22×C4×D6111yescharacteristic
473C2×C4C2×D4×D6111yescharacteristic
474C8—42no—
475C8—2442no—
476C22—21no—
477C22—21no—
478C22—1224no—
479C22—1224no—
480C22—1224no—
481C22—2444no—
482C22—2444no—
483C22—2442no—
484C22—2442no—
485C22—2442no—
486C22C2×C22⋊C4×D6111yescharacteristic
487C22C2×#1451980411yescharacteristic
488C22C2×#1451980911yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—81no—
497C4—1224no—
498C4—4882no—
499C4—2442no—
500C4#1683976212yes—
501C4#1683976212yes—
502C2—21no—
503C2—21no—
504C2—4882no—
505C2—2444no—
506C2#2145427811yescharacteristic
507C2C2×#1683976211yescharacteristic
508C2#2145436311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1503C2
socleC22×C61C22×C22⋊C4
prime radical#21459673C2
cosocleC22×C4×C61C24
hypercenter#1503D61
hypocenterC61#17630
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
2. #21454859C2semidirect
3. #21448730C2semidirect
4. #21444383C2semidirect
5. #21444380C2semidirect
6. #21444361C2semidirect
7. #21444322C2semidirect
8. #21444313C2semidirect
9. C61×#1503C2semidirect
11. #16840485C4semidirect
12. #16840478C4semidirect
14. #16840434C4semidirect
15. #16840415C4semidirect
41. C2×#11758226C2×C4semidirect
47. #14519777C2×C4semidirect
49. #14519774C2×C4semidirect
56. #14519753C2×C4semidirect
60. #14519752C2×C4semidirect
70. C4×D244C2×C4semidirect
76. #14519735C2×C4semidirect
90. C4⋊Q8×C61C2×C4semidirect
91. C4⋊Q8×C61C2×C4semidirect
96. C4×Q8×C61C2×C4semidirect
255. #1503D61semidirect
295. C61#17630semidirect
303. C4⋊Q8C4×D61semidirect
304. C4⋊Q8C4×D61semidirect
326. C4×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214596732
#214548592
#214487302
#214443832
#214443802
#214443612
#214443222
#214443132
C61×#15032
#1763061