Finite Group Databasebeta

← #30608452

Subgroups — #30608452

Order 15616.

348 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 56 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#21462100C211yesmaximal, characteristic, complemented
2#21462096C211yesmaximal, characteristic, complemented
3#21460958C212yesmaximal, complemented
4#21460950C212yesmaximal, complemented
5#21456863C211yesmaximal, characteristic
6#16840741—21no—
7C2×#14519836—21no—
8#16840669—21no—
9#16840667—21no—
10#16840667C2212yes—
11#16840667C2212yes—
12#16840331C2211yescharacteristic
13#16840149C411yescharacteristic, complemented
14#16840138C2211yescharacteristic
15#16840128C2211yescharacteristic
16#16840119C411yescharacteristic, complemented
17#16839977—21no—
18#16839957—21no—
19#16839905—21no—
20#16839895—21no—
21#16839823C411yescharacteristic, complemented
22C61×#540C411yescharacteristic, complemented
23#14519856—41no—
24C2×C22⋊#6923—41no—
25#14519836—22no—
26#14519836—22no—
27#14519836—41no—
28#14519835—41no—
29C2×#11758222D411yescharacteristic
30C22×D244D411yescharacteristic
31C2×#11758200—21no—
32C2×#11758199—21no—
33C2×#11758199—21no—
34C2×#11758199—21no—
35#14519774D411yescharacteristic
36#14519774C2311yescharacteristic
37#14519765C2×C411yescharacteristic
38#14519765D411yescharacteristic
39#14519752—41no—
40#14519747—41no—
41C4⋊#92418—21no—
42#14519700—21no—
43#14519697—21no—
44#14519697—21no—
45#14519697—21no—
46C22⋊#92418—21no—
47#14519688C2×C411yescharacteristic, complemented
48C4×#92418C2×C411yescharacteristic, complemented
49#127×C61C2×C411yescharacteristic, complemented
50#125×C61C2×C411yescharacteristic, complemented
51#123×C61—41no—
52#121×C61—21no—
53#121×C61—41no—
54C4×D4×C61—21no—
55#117×C61C2×C411yescharacteristic
56C23×D61—21no—
57C22×#6923—81no—
58C2×D4×C61—21no—
59C2×D4×C61—41no—
60C2×D4×C61—41no—
61C22×C4×C61—41no—
62C22×C4×C61C22⋊C411yescharacteristic
63C2×C22⋊D61—21no—
64#11758222—81no—
65C2×D244—21no—
66C2×D244—41no—
67C2×D244—41no—
68C2×D244—81no—
69C2×D244C22⋊C411yescharacteristic
70C2×D244C2×D411yescharacteristic
71C2×D244C2×D411yescharacteristic
72C2×C4×D61—21no—
73C2×Dic122C22⋊C411yescharacteristic
74C22⋊#6923—42no—
75C22⋊#6923—42no—
76C22⋊#6923—81no—
77C4⋊#6923—81no—
78C4×#6923—81no—
79C4⋊C4×C61—21no—
80C4⋊C4×C61—21no—
81C4⋊C4×C61—41no—
82C22⋊C4×C61—21no—
83C22⋊C4×C61—41no—
84C22⋊C4×C61—41no—
85C42×C61C22⋊C411yescharacteristic
86C42×C61C22×C411yescharacteristic
87#11758200—81no—
88#11758199—81no—
89#11758199—81no—
90#11758199—81no—
91#11758198—41no—
92#11758198—41no—
93C2×#92418—21no—
94C2×#92418—21no—
95C23×C61—41no—
96C23×C61#9911yescharacteristic
97C22×D61—21no—
98C22×D61—21no—
99C22×D61—21no—
100C22×D61—41no—
101C22×D61—41no—
102C22×D61—161no—
103C2×#6923—82no—
104C2×#6923—82no—
105C2×#6923—162no—
106D4×C61—41no—
107D4×C61—41no—
108D4×C61—161no—
109D4×C61—161no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—161no—
115C2×C4×C61#9911yescharacteristic
116C2×C4×C61C2×C22⋊C411yescharacteristic
117C22⋊D61—81no—
118C2×Dic61—21no—
119D244—41no—
120D244—41no—
121D244—41no—
122D244—81no—
123D244—161no—
124D244—161no—
125C4×D61—41no—
126C4×D61—41no—
127Dic122—41no—
128#92418—41no—
129#92418—41no—
130#13609—611nomaximal, Sylow
131C22×C61—21no—
132C22×C61—41no—
133C22×C61—81no—
134C22×C61—81no—
135C22×C61—81no—
136C22×C61C2×#9911yescharacteristic
137C2×D61—41no—
138C2×D61—41no—
139C2×D61—41no—
140C2×D61—82no—
141C2×D61—81no—
142C2×D61—81no—
143C2×D61—81no—
144C2×D61—161no—
145#6923—162no—
146#6923—162no—
147C4×C61—21no—
148C4×C61—21no—
149C4×C61—41no—
150C4×C61—41no—
151C4×C61—81no—
152Dic61—41no—
153#3220—611no—
154#2128—611no—
155#2127—611no—
156#1412—612no—
157#1408—612no—
158C2×C61—21no—
159C2×C61—41no—
160C2×C61—161no—
161C2×C61#212711yescharacteristic
162D61—81no—
163D61—81no—
164D61—81no—
165#540#692311yescharacteristic, complemented
166#535—611no—
167#496—611no—
168#493—611no—
169#490—611no—
170#488—611no—
171#463—1221no—
172#459—1221no—
173#449—1221no—
174#447—1221no—
175#433—611no—
176#410—1221no—
177C2×#99—1221no—
178#354—1221no—
179#353—612no—
180#353—612no—
181#353—1221no—
182C61#1360911yescharacteristic, complemented, Sylow
183C2×#41—611no—
184C22×D4—611no—
185C2×SD16—1221no—
186C2×D8—1221no—
187C2×D8—1221no—
188C2×D8—1221no—
189#127—611no—
190#127—611no—
191#127C2×#692311yescharacteristic, complemented
192#125C2×#692311yescharacteristic, complemented
193#124—611no—
194#124—611no—
195#123—41no—
196#121—21no—
197#121—41no—
198#121—2441no—
199#120—2441no—
200C4×D4—21no—
201#117—2441no—
202#117C2×#692311yescharacteristic
203C2×C22⋊C4—2441no—
204C4⋊C8—1221no—
205#103—1221no—
206#102—1221no—
207#102—1221no—
208#102—1221no—
209#99—1222no—
210#99—1222no—
211#99—2441no—
212#99—2441no—
213C22⋊C8—1221no—
214#97—611no—
215C4×C8—611no—
216C24—1221no—
217#41—4881no—
218C2×Q8—611no—
219C2×D4—21no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—611no—
223C2×D4—611no—
224C2×D4—611no—
225C2×D4—1221no—
226C2×D4—1221no—
227C2×D4—4881no—
228C2×D4—2441no—
229C2×D4—2441no—
230C22×C4—41no—
231C22×C4—1221no—
232C22×C4—4881no—
233C22×C4C22⋊#692311yescharacteristic
234SD16—4881no—
235D8—4881no—
236D8—4881no—
237D8—4881no—
238C2×C8—1221no—
239C2×C8—1221no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—41no—
243C4⋊C4—4881no—
244C22⋊C4—21no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—4881no—
248C22⋊C4—2441no—
249C22⋊C4—2441no—
250C22⋊C4—2442no—
251C22⋊C4—2442no—
252C42—4881no—
253C42C22⋊#692311yescharacteristic
254C42C22×#692311yescharacteristic
255C23—41no—
256C23—1221no—
257C23—1221no—
258C23—1221no—
259C23—9761no—
260C23—2441no—
261C23—2441no—
262C23#1451983511yescharacteristic
263Q8—2441no—
264D4—41no—
265D4—41no—
266D4—161no—
267D4—161no—
268D4—9761no—
269D4—9761no—
270D4—4881no—
271D4—4881no—
272D4—2441no—
273D4—2441no—
274D4—2441no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—161no—
280C2×C4—1221no—
281C2×C4—9762no—
282C2×C4—4882no—
283C2×C4—4882no—
284C2×C4—2441no—
285C2×C4—2441no—
286C2×C4#1451983511yescharacteristic
287C2×C4C2×C22⋊#692311yescharacteristic
288C8—2441no—
289C8—2441no—
290C22—21no—
291C22—41no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—9761no—
296C22—4881no—
297C22—4881no—
298C22—4881no—
299C22—4882no—
300C22—2441no—
301C22—2441no—
302C22—2441no—
303C22C2×#1451983511yescharacteristic
304C4—21no—
305C4—21no—
306C4—41no—
307C4—41no—
308C4—81no—
309C4—9762no—
310C4—9762no—
311C4—2441no—
312C2—21no—
313C2—41no—
314C2—161no—
315C2—4881no—
316C2—4881no—
317C2—4881no—
318C2#2146209411yescharacteristic
subgroupG/H
centerC2#21462094
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#540C4
socleC2×C61#2127
prime radical#21456863C2
cosocle#14519774C23
hypercenter#540#6923
hypocenterC61#13609
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #21462100C2semidirect
2. #21462096C2semidirect
3. #21460958C2semidirect
4. #21460950C2semidirect
13. #16840149C4semidirect
16. #16840119C4semidirect
21. #16839823C4semidirect
22. C61×#540C4semidirect
47. #14519688C2×C4semidirect
48. C4×#92418C2×C4semidirect
49. #127×C61C2×C4semidirect
50. #125×C61C2×C4semidirect
165. #540#6923semidirect
182. C61#13609semidirect
191. #127C2×#6923semidirect
192. #125C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214621002
#214620962
#214609582
#214609502
#214568632
#1360961