Finite Group Databasebeta

← #30604779

Subgroups — #30604779

Order 15616.

504 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462047C211yesmaximal, characteristic, complemented
2C2×#16840636C211yesmaximal, characteristic, complemented
3#21460807C212yesmaximal, complemented
4#21460806C212yesmaximal, complemented
5#21455402C211yesmaximal, characteristic
6C22×C4⋊#6923—21no—
7C2×#14519852—21no—
8#16840657C2212yes—
9C22⋊#11758187—21no—
10#16840636—21no—
11#16840636C2212yes—
12C2×#14519800C411yescharacteristic, complemented
13C22×C8×D61C2211yescharacteristic
14#16840275—21no—
15#16840000C2211yescharacteristic
16#16839999C2211yescharacteristic
17#16839998C411yescharacteristic, complemented
18#16839994—21no—
19#16839993—21no—
20#16839799C411yescharacteristic, complemented
21#16839757—21no—
22C61×#467C411yescharacteristic, complemented
23C23×#6923—41no—
24C2×C4⋊#6923—24no—
25C2×C4⋊#6923—22no—
26C2×C4⋊#6923—42no—
27C2×C4⋊#6923—41no—
28#14519852—42no—
29C2×#11758187—22no—
30C2×#11758226—21no—
31C22×C4×D61D411yescharacteristic
32#14519800—42no—
33C2×#11758201—21no—
34C2×D488C2×C411yescharacteristic, complemented
35C22×#92418D411yescharacteristic
36C2×C8×D61—42no—
37C2×C8×D61C2×C411yescharacteristic
38C2×C8×D61D411yescharacteristic
39C2×C8×D61D411yescharacteristic
40C2×C8×D61C2311yescharacteristic
41#14519762—41no—
42#14519753—21no—
43#14519752—21no—
44#14519751—21no—
45#14519747—41no—
46#14519739—41no—
47#14519737—41no—
48#14519735—21no—
49#14519711C2×C411yescharacteristic, complemented
50#14519708—21no—
51#14519701—21no—
52#14519699—21no—
53#14519697—21no—
54C2×Q16×C61C2×C411yescharacteristic, complemented
55C22×C8×C61C2×C411yescharacteristic
56C22⋊Q8×C61—21no—
57#107×C61C2×C411yescharacteristic, complemented
58#103×C61—21no—
59C23×D61D811yescharacteristic
60C22×#6923—44no—
61C22×#6923—42no—
62C22×#6923—82no—
63C22×#6923—82no—
64C2×Q8×C61—21no—
65C22×C4×C61C22⋊C411yescharacteristic
66C2×C22⋊D61—41no—
67C22×Dic61SD1611yescharacteristic
68#11758226—42no—
69#11758226—42no—
70C2×D244—21no—
71C2×D244—41no—
72C2×D244—41no—
73C2×C4×D61—22no—
74C2×C4×D61—22no—
75C2×C4×D61—41no—
76C2×C4×D61C22⋊C411yescharacteristic
77C2×C4×D61D811yescharacteristic
78C2×C4×D61SD1611yescharacteristic
79C2×C4×D61C2×D411yescharacteristic
80C4⋊#6923—24no—
81C4⋊#6923—42no—
82C4⋊#6923—44no—
83Q16×C61—41no—
84C2×C8×C61—41no—
85C2×C8×C61C22⋊C411yescharacteristic
86C2×C8×C61C22×C411yescharacteristic
87C4⋊C4×C61—21no—
88C4⋊C4×C61—41no—
89C22⋊C4×C61—41no—
90#11758201—42no—
91#11758198—41no—
92#11758198—41no—
93#11758198—41no—
94#11758198—41no—
95C4⋊Dic61—21no—
96#11758196—41no—
97C4×Dic61—41no—
98C2×#92418—41no—
99C2×#92418C22⋊C411yescharacteristic
100C2×#92418C2×D411yescharacteristic
101D488—41no—
102C8×D61—22no—
103C8×D61—22no—
104C8×D61—41no—
105C8×D61—41no—
106#11758187—42no—
107C23×C61#10211yescharacteristic
108C22×D61—22no—
109C22×D61—22no—
110C22×D61—21no—
111C22×D61—21no—
112C22×D61—41no—
113C22×D61#10211yescharacteristic
114C22×D61C2×D811yescharacteristic
115C2×#6923—44no—
116C2×#6923—88no—
117C2×#6923—82no—
118C2×#6923—82no—
119Q8×C61—41no—
120Q8×C61—41no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61#10211yescharacteristic
126C2×C4×C61C2×C22⋊C411yescharacteristic
127C22⋊D61—82no—
128C2×Dic61—22no—
129C2×Dic61—41no—
130C2×Dic61#10211yescharacteristic
131C2×Dic61C2×SD1611yescharacteristic
132D244—41no—
133D244—41no—
134D244—82no—
135D244—81no—
136D244—81no—
137C4×D61—22no—
138C4×D61—21no—
139C4×D61—21no—
140C4×D61—22no—
141C4×D61—82no—
142C4×D61#9912yes—
143C4×D61#10412yes—
144C8×C61—21no—
145C8×C61—21no—
146#92418—21no—
147#92418—21no—
148#11704—611nomaximal, Sylow
149C22×C61—21no—
150C22×C61—21no—
151C22×C61#36012yes—
152C22×C61C2×#10211yescharacteristic
153C2×D61—24no—
154C2×D61—24no—
155C2×D61—21no—
156C2×D61—21no—
157C2×D61—21no—
158C2×D61—21no—
159C2×D61—81no—
160C2×D61—81no—
161C2×D61—81no—
162C2×D61#32712yes—
163C2×D61#50911yescharacteristic
164C2×D61#51011yescharacteristic
165#6923—84no—
166C4×C61—21no—
167C4×C61—81no—
168C4×C61—81no—
169C4×C61C2×#9911yescharacteristic
170C4×C61C2×#10411yescharacteristic
171Dic61—21no—
172Dic61—81no—
173Dic61#32712yes—
174#3050—611no—
175#2112—611no—
176C2×#340—611no—
177#1348—612no—
178#1347—612no—
179C2×C61—22no—
180C2×C61C2×#32711yescharacteristic
181C2×C61#213911yescharacteristic
182C2×C61C2×#36011yescharacteristic
183D61—21no—
184D61—22no—
185D61—21no—
186D61—161no—
187C2×#135—611no—
188C23×C8—611no—
189#525—1221no—
190C22×C4⋊C4—1221no—
191C2×M32—1221no—
192#467—611no—
193#467#692311yescharacteristic, complemented
194#466—611no—
195#466—611no—
196#465—1221no—
197#465—1221no—
198#427—611no—
199#416—1221no—
200C22⋊C16—612no—
201C22⋊C16—1221no—
202#340—612no—
203#340—1221no—
204C61#1170411yescharacteristic, complemented, Sylow
205C2×#41—1221no—
206C23×C4—611no—
207C23×C4—2441no—
208#135—2442no—
209C2×Q16C2×#692311yescharacteristic, complemented
210C2×SD16—1221no—
211C2×D8—611no—
212C22×C8—611no—
213C22×C8—611no—
214C22×C8—611no—
215C22×C8—611no—
216C22×C8—611no—
217C22×C8—2442no—
218C22×C8C2×#692311yescharacteristic
219#123—2441no—
220C22⋊Q8—21no—
221C22⋊Q8—1221no—
222#121—1221no—
223#121—1221no—
224#120—2441no—
225C4×D4—2441no—
226C4×D4—2441no—
227#117—1221no—
228C2×C4⋊C4—1222no—
229C2×C4⋊C4—1224no—
230C2×C4⋊C4—2441no—
231C2×C4⋊C4—2442no—
232M32—2442no—
233C2×C16—1222no—
234#107—611no—
235#107C2×#692311yescharacteristic, complemented
236#106—1221no—
237#103—21no—
238#103—1221no—
239#102—1221no—
240#102—1221no—
241C24—611no—
242#41—2442no—
243#41—2442no—
244C2×Q8—21no—
245C2×D4—1221no—
246C2×D4—2441no—
247C2×D4—2441no—
248C2×D4—2441no—
249C22×C4—611no—
250C22×C4—611no—
251C22×C4—611no—
252C22×C4—611no—
253C22×C4—611no—
254C22×C4—1222no—
255C22×C4—1222no—
256C22×C4—4882no—
257C22×C4—4882no—
258C22×C4—2441no—
259C22×C4—2442no—
260C22×C4—2444no—
261C22×C4C22⋊#692311yescharacteristic
262Q16—41no—
263SD16—2442no—
264D8—2441no—
265C2×C8—41no—
266C2×C8—611no—
267C2×C8—611no—
268C2×C8—1222no—
269C2×C8—1222no—
270C2×C8—2441no—
271C2×C8—2441no—
272C2×C8—2441no—
273C2×C8C22⋊#692311yescharacteristic
274C2×C8C22×#692311yescharacteristic
275C4⋊C4—21no—
276C4⋊C4—41no—
277C4⋊C4—1224no—
278C4⋊C4—1221no—
279C4⋊C4—2441no—
280C4⋊C4—2444no—
281C4⋊C4—2442no—
282C22⋊C4—41no—
283C22⋊C4—2441no—
284C22⋊C4—2441no—
285C22⋊C4—2441no—
286C22⋊C4—2441no—
287C42—2441no—
288C16—2442no—
289C23—611no—
290C23—611no—
291C23—1222no—
292C23—1221no—
293C23—1221no—
294C23—1222no—
295C23—2441no—
296C23#1451983911yescharacteristic
297Q8—41no—
298Q8—41no—
299D4—4881no—
300D4—4881no—
301D4—4882no—
302D4—4882no—
303D4—2441no—
304D4—2441no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—611no—
310C2×C4—612no—
311C2×C4—612no—
312C2×C4—611no—
313C2×C4—1222no—
314C2×C4—1222no—
315C2×C4—1222no—
316C2×C4—1221no—
317C2×C4—1221no—
318C2×C4—4882no—
319C2×C4—4882no—
320C2×C4—4888no—
321C2×C4—4882no—
322C2×C4—2444no—
323C2×C4—2441no—
324C2×C4#1451983911yescharacteristic
325C2×C4C2×C22⋊#692311yescharacteristic
326C8—21no—
327C8—21no—
328C8—1221no—
329C8—1221no—
330C22—21no—
331C22—21no—
332C22—611no—
333C22—611no—
334C22—612no—
335C22—1224no—
336C22—1224no—
337C22—1221no—
338C22—1221no—
339C22—1221no—
340C22—1221no—
341C22—4881no—
342C22—4881no—
343C22—4881no—
344C22#1684067912yes—
345C22C2×#1451983911yescharacteristic
346C4—21no—
347C4—81no—
348C4—81no—
349C4—612no—
350C4—1221no—
351C4—4884no—
352C4—4881no—
353C4D61⋊#9911yescharacteristic
354C4D61⋊#10411yescharacteristic
355C2—22no—
356C2—1221no—
357C2—1221no—
358C2—1222no—
359C2—9761no—
360C2D61⋊#32711yescharacteristic
361C2#2146212711yescharacteristic
362C2C2×#1684067911yescharacteristic
subgroupG/H
centerC22C2×#14519839
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#467C4
socleC22×C61C2×#102
prime radical#21455402C2
cosocleC2×C8×D61C23
hypercenter#467#6923
hypocenterC61#11704
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #21462047C2semidirect
2. C2×#16840636C2semidirect
3. #21460807C2semidirect
4. #21460806C2semidirect
12. C2×#14519800C4semidirect
17. #16839998C4semidirect
20. #16839799C4semidirect
22. C61×#467C4semidirect
34. C2×D488C2×C4semidirect
49. #14519711C2×C4semidirect
54. C2×Q16×C61C2×C4semidirect
57. #107×C61C2×C4semidirect
193. #467#6923semidirect
204. C61#11704semidirect
209. C2×Q16C2×#6923semidirect
235. #107C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214620472
C2×#168406362
#214608072
#214608062
#214554022
#1170461