Finite Group Databasebeta

← #29575111

Subgroups — #29575111

Order 15616.

612 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454381C211yesmaximal, characteristic, complemented
2#21449059C211yesmaximal, characteristic
3#21444245C211yesmaximal, characteristic, complemented
4#21444243C211yesmaximal, characteristic, complemented
5#21444131C211yesmaximal, characteristic, complemented
6#21443722C211yesmaximal, characteristic, complemented
7C61×#1303C211yesmaximal, characteristic, complemented
8C2×M16×D61—21no—
9C2×C4×D244—21no—
10C2×C4×D244C2211yescharacteristic
11C2×#14519734C2211yescharacteristic
12#16839815C2211yescharacteristic
13#16839787—22no—
14#16839787—22no—
15#16839786C2211yescharacteristic
16#16839785—21no—
17#16839694C411yescharacteristic
18#16839689C411yescharacteristic
19#16839684—21no—
20#16839680C411yescharacteristic
21#16839679—21no—
22#16839672C411yescharacteristic
23C4×#11758198—21no—
24#16839491—22no—
25#16839489—22no—
26#16839487—21no—
27#16839485—21no—
28C61×#431C2211yescharacteristic
29C61×#423C2211yescharacteristic
30C4×C4⋊C4×C61C2211yescharacteristic
31C22×D244—21no—
32C22×D244C2×C411yescharacteristic
33C22×C4×D61—21no—
34C22×C4×D61D411yescharacteristic
35C2×#11758194—21no—
36C2×#11758189—22no—
37M16×D61—42no—
38M16×D61—42no—
39C2×C8×D61—22no—
40C4×D244—21no—
41C4×D244—21no—
42C4×D244—21no—
43C4×D244—21no—
44C4×D244—42no—
45C4×D244—42no—
46#14519734—21no—
47#14519734—21no—
48#14519734—21no—
49#14519734—21no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61C2×C411yescharacteristic
52C2×#11758196C2×C411yescharacteristic
53C2×#11758196D411yescharacteristic
54C2×#11758198—21no—
55C2×#11758198—21no—
56C2×#11758198—21no—
57C2×#11758198—21no—
58C2×#11758198—21no—
59C2×#11758198C2×C411yescharacteristic
60C2×#11758198D411yescharacteristic
61C2×C4×Dic61D411yescharacteristic
62C4⋊#92418—21no—
63C4⋊#92418—21no—
64C4⋊#92418D412yescomplemented
65#14519706—24no—
66C22⋊#92418—21no—
67C4×#92418—21no—
68#14519684—21no—
69#14519684—21no—
70C2×M16×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C42×C61—21no—
73C2×C42×C61C2311yescharacteristic
74C4⋊C8×C61—21no—
75C4⋊C8×C61—21no—
76C4⋊C8×C61D412yescomplemented
77C22⋊C8×C61—21no—
78C4×C8×C61—21no—
79#95×C61C2×C411yescharacteristic
80#95×C61C2×C411yescharacteristic
81C23×D61—21no—
82C23×D61C22⋊C411yescharacteristic
83C22×C4×C61—21no—
84C22×C4×C61—21no—
85C22×C4×C61C22×C411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×Dic61—21no—
89C22×Dic61C22⋊C411yescharacteristic
90C22×Dic61C22⋊C411yescharacteristic
91C22×Dic61C22⋊C411yescharacteristic
92C2×D244—21no—
93C2×D244—21no—
94C2×D244—42no—
95C2×D244—42no—
96C2×D244—42no—
97C2×D244—42no—
98C2×C4×D61—21no—
99C2×C4×D61—21no—
100C2×C4×D61—21no—
101C2×C4×D61—21no—
102C2×C4×D61—41no—
103C2×C4×D61—41no—
104C2×C4×D61—41no—
105C2×C4×D61—41no—
106C2×C4×D61—41no—
107C2×C4×D61—41no—
108C2×C4×D61D812yes—
109C2×C4×D61SD1612yes—
110M16×C61—41no—
111M16×C61—41no—
112C2×C8×C61—22no—
113C2×C8×C61—41no—
114C4⋊C4×C61—42no—
115C42×C61—41no—
116C42×C61—41no—
117C42×C61C2×D411yescharacteristic
118C42×C61C2×D411yescharacteristic
119C42×C61#4111yescharacteristic
120C42×C61#4111yescharacteristic
121#11758198—21no—
122#11758198—21no—
123#11758198—21no—
124#11758198—21no—
125#11758198—42no—
126#11758198—42no—
127#11758198—42no—
128#11758198—41no—
129#11758198—41no—
130#11758198—42no—
131C4⋊Dic61—21no—
132C4⋊Dic61—21no—
133C4⋊Dic61—42no—
134#11758196—21no—
135#11758196—21no—
136#11758196—21no—
137#11758196—21no—
138C4×Dic61—21no—
139C4×Dic61—21no—
140#11758194—41no—
141#11758194—41no—
142C2×#92418—22no—
143C2×#92418—41no—
144#11758189—44no—
145C8×D61—44no—
146C23×C61C2×C22⋊C411yescharacteristic
147C22×D61—21no—
148C22×D61—21no—
149C22×D61—22no—
150C22×D61—41no—
151C22×D61—41no—
152C22×D61—41no—
153C22×D61—41no—
154C22×D61—41no—
155C22×D61—41no—
156C22×D61—41no—
157C22×D61—41no—
158C22×D61#10212yes—
159C2×C4×C61—22no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61C4×D411yescharacteristic
168C2×C4×C61C4×D411yescharacteristic
169C2×C4×C61#12011yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61#12311yescharacteristic
173C2×C4×C61C2×D811yescharacteristic
174C2×C4×C61C2×SD1611yescharacteristic
175C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C61#13711yescharacteristic
177C2×Dic61—21no—
178C2×Dic61—21no—
179C2×Dic61—21no—
180C2×Dic61—21no—
181C2×Dic61—41no—
182C2×Dic61—41no—
183C2×Dic61—41no—
184C2×Dic61—41no—
185C2×Dic61—41no—
186C2×Dic61#10212yes—
187D244—22no—
188D244—22no—
189D244—41no—
190D244—41no—
191D244—48no—
192C4×D61—22no—
193C4×D61—22no—
194C4×D61—22no—
195C4×D61—22no—
196C4×D61—81no—
197C4×D61—81no—
198C4×D61—81no—
199C4×D61—81no—
200C8×C61—42no—
201C8×C61—81no—
202#92418—42no—
203#92418—81no—
204#9035—611nomaximal, Sylow
205C22×C61—21no—
206C22×C61—21no—
207C22×C61#38611yescharacteristic
208C22×C61C2×#10211yescharacteristic
209C22×C61#41711yescharacteristic
210C2×D61—22no—
211C2×D61—41no—
212C2×D61—41no—
213C2×D61—42no—
214C2×D61—42no—
215C2×D61—42no—
216C2×D61—42no—
217C2×D61—81no—
218C2×D61—81no—
219C2×D61—81no—
220C2×D61—81no—
221C2×D61—81no—
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—21no—
226C4×C61—42no—
227C4×C61—81no—
228C4×C61#44711yescharacteristic
229C4×C61#44811yescharacteristic
230C4×C61#48011yescharacteristic
231C4×C61#48111yescharacteristic
232Dic61—22no—
233Dic61—41no—
234Dic61—41no—
235Dic61—81no—
236#2905—611no—
237#2392—611no—
238#1486—611no—
239#1486—611no—
240#1458—611no—
241#1303—611no—
242#1303D6111yescharacteristic, complemented
243C2×C61—21no—
244C2×C61—21no—
245C2×C61#187411yescharacteristic
246C2×C61#188411yescharacteristic
247C2×C61#189111yescharacteristic
248D61—42no—
249D61—82no—
250C22×M16—1221no—
251C2×C4×D4—611no—
252C2×C4×D4—1221no—
253C2×#117—611no—
254#431—611no—
255#431C2×D6111yescharacteristic
256#424—1222no—
257#424—1222no—
258#423—611no—
259#423—1221no—
260#423C2×D6111yescharacteristic
261#398—611no—
262#397—611no—
263#396—1221no—
264#394—611no—
265#394—1221no—
266#392—611no—
267C4×C4⋊C4C2×D6111yescharacteristic
268C4×C22⋊C4—1221no—
269D4⋊C8—1222no—
270D4⋊C8—1222no—
271#324—1221no—
272#323—1221no—
273C61#903511yescharacteristic, complemented, Sylow
274C22×D4—611no—
275C22×D4—1221no—
276C23×C4—611no—
277C23×C4—1221no—
278C2×M16—21no—
279C2×M16—1222no—
280C2×M16—1221no—
281C2×M16—2442no—
282C2×M16—2442no—
283C22×C8—1222no—
284C4×D4—1221no—
285C4×D4—1221no—
286C4×D4—1221no—
287C4×D4—1221no—
288C4×D4—2442no—
289C4×D4—2442no—
290#117—1221no—
291#117—1221no—
292#117—1221no—
293#117—1221no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—611no—
296C2×C4⋊C4—611no—
297C2×C4⋊C4—611no—
298C2×C4⋊C4—1221no—
299C2×C22⋊C4—611no—
300C2×C22⋊C4—611no—
301C2×C22⋊C4—1221no—
302C2×C22⋊C4—1221no—
303C2×C22⋊C4—1221no—
304C2×C22⋊C4—1221no—
305C2×C22⋊C4—1221no—
306C2×C42—21no—
307C2×C42—611no—
308C2×C42C22×D6111yescharacteristic
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—612no—
312C4⋊C8—1224no—
313C4⋊C8—1221no—
314C4⋊C8—1221no—
315C4⋊C8D24412yescomplemented
316C22⋊C8—21no—
317C22⋊C8—1221no—
318C4×C8—21no—
319C4×C8—1221no—
320#95—1221no—
321#95—1221no—
322#95C4×D6111yescharacteristic
323#95C4×D6111yescharacteristic
324C24—611no—
325C24—1221no—
326C2×D4—1221no—
327C2×D4—1221no—
328C2×D4—2442no—
329C2×D4—2442no—
330C2×D4—2442no—
331C2×D4—2442no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—611no—
335C22×C4—612no—
336C22×C4—612no—
337C22×C4—611no—
338C22×C4—611no—
339C22×C4—1221no—
340C22×C4—1221no—
341C22×C4—1221no—
342C22×C4—1221no—
343C22×C4—1221no—
344C22×C4—2441no—
345C22×C4—2441no—
346C22×C4—2441no—
347C22×C4—2441no—
348C22×C4—2441no—
349C22×C4—2441no—
350C22×C4C2×C4×D6111yescharacteristic
351C22×C4D4×D6111yescharacteristic
352C22×C4D4×D6111yescharacteristic
353M16—41no—
354M16—41no—
355M16—2444no—
356M16—2441no—
357M16—2441no—
358C2×C8—22no—
359C2×C8—41no—
360C2×C8—1222no—
361C2×C8—2444no—
362C2×C8—2441no—
363C4⋊C4—42no—
364C4⋊C4—1221no—
365C4⋊C4—1221no—
366C4⋊C4—1221no—
367C4⋊C4—1221no—
368C4⋊C4—1221no—
369C4⋊C4—1221no—
370C4⋊C4—2442no—
371C22⋊C4—1221no—
372C22⋊C4—1221no—
373C22⋊C4—1221no—
374C22⋊C4—1221no—
375C22⋊C4—2442no—
376C22⋊C4—2442no—
377C22⋊C4—2442no—
378C22⋊C4—2442no—
379C22⋊C4—2441no—
380C22⋊C4—2441no—
381C42—41no—
382C42—41no—
383C42—1221no—
384C42—1221no—
385C42C2×D24411yescharacteristic
386C42#1175822211yescharacteristic
387C42D4×D6111yescharacteristic
388C42#1175822611yescharacteristic
389C23—612no—
390C23—1222no—
391C23—1221no—
392C23—1221no—
393C23—2441no—
394C23—2441no—
395C23—2441no—
396C23—2441no—
397C23—2441no—
398C23—2441no—
399C23—2441no—
400C23—2441no—
401C23C22⋊C4×D6111yescharacteristic
402D4—1222no—
403D4—1222no—
404D4—2441no—
405D4—2441no—
406D4—2448no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—612no—
416C2×C4—1221no—
417C2×C4—1221no—
418C2×C4—1222no—
419C2×C4—1222no—
420C2×C4—1222no—
421C2×C4—1222no—
422C2×C4—1221no—
423C2×C4—1221no—
424C2×C4—4881no—
425C2×C4—4881no—
426C2×C4—4881no—
427C2×C4—4881no—
428C2×C4—2441no—
429C2×C4—2441no—
430C2×C4—2441no—
431C2×C4—2441no—
432C2×C4—2441no—
433C2×C4#1451973911yescharacteristic
434C2×C4C4×D24411yescharacteristic
435C2×C4#1451974711yescharacteristic
436C2×C4#1451975011yescharacteristic
437C2×C4#1451975211yescharacteristic
438C2×C4#1451976211yescharacteristic
439C2×C4D8×D6111yescharacteristic
440C2×C4SD16×D6111yescharacteristic
441C2×C4#1451980411yescharacteristic
442C2×C4#1451980911yescharacteristic
443C8—42no—
444C8—81no—
445C8—4881no—
446C8—2442no—
447C22—21no—
448C22—21no—
449C22—1222no—
450C22—4881no—
451C22—4881no—
452C22—4881no—
453C22—4881no—
454C22—4881no—
455C22—2442no—
456C22—2442no—
457C22—2442no—
458C22—2442no—
459C22—2441no—
460C22—2441no—
461C22#1683964011yescharacteristic
462C22#102×D6111yescharacteristic
463C22#1683976211yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—42no—
469C4—81no—
470C4—1222no—
471C4—4881no—
472C4—2441no—
473C4—2441no—
474C4#1683989711yescharacteristic
475C4#1683990211yescharacteristic
476C4#1684007711yescharacteristic
477C4#1684008211yescharacteristic
478C2—21no—
479C2—21no—
480C2—4882no—
481C2—2442no—
482C2#2144613511yescharacteristic
483C2#2144620111yescharacteristic
484C2#2144623611yescharacteristic
subgroupG/H
centerC2×C4#14519747
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1303C2
socleC22×C61#386
prime radical#16839672C4
cosocleC2×C42×C61C23
hypercenter#1303D61
hypocenterC61#9035
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
1. #21454381C2semidirect
3. #21444245C2semidirect
4. #21444243C2semidirect
5. #21444131C2semidirect
6. #21443722C2semidirect
7. C61×#1303C2semidirect
64. C4⋊#92418D4semidirect
76. C4⋊C8×C61D4semidirect
242. #1303D61semidirect
273. C61#9035semidirect
315. C4⋊C8D244semidirect

Maximal subgroups

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

subgroupindex
#214543812
#214490592
#214442452
#214442432
#214441312
#214437222
C61×#13032
#903561