Finite Group Databasebeta

← #30608422

Subgroups — #30608422

Order 15616.

334 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C2×#16840671C211yesmaximal, characteristic
2C2×#16840669C211yesmaximal, characteristic
3#21460967C212yesmaximal, complemented
4#21460966C212yesmaximal, complemented
5#21456206C211yesmaximal, characteristic
6C2×#14519838—21no—
7C2×#14519836—21no—
8#16840671—21no—
9#16840671C2212yes—
10#16840669—21no—
11#16840669C2212yes—
12#16840319C411yescharacteristic, complemented
13C2×#14519765C2211yescharacteristic
14#16840122—21no—
15#16840122—21no—
16#16840117C411yescharacteristic, complemented
17#16840117C411yescharacteristic, complemented
18#16840117C2211yescharacteristic
19#16840117C2211yescharacteristic
20#16839772—21no—
21#16839768—21no—
22C4×M16×C61C411yescharacteristic, complemented
23C2×#11758217—41no—
24C2×C22⋊#6923—41no—
25#14519838—22no—
26#14519838—41no—
27#14519836—22no—
28#14519836—41no—
29C22×Dic122D411yescharacteristic
30C22×D244D411yescharacteristic
31#14519777—21no—
32#14519769C2×C411yescharacteristic, complemented
33#14519769C2×C411yescharacteristic, complemented
34#14519765—41no—
35#14519765C2×C411yescharacteristic
36#14519765D411yescharacteristic
37#14519765D411yescharacteristic
38#14519765C2311yescharacteristic
39#14519756—41no—
40C4×Dic122—41no—
41#14519736—21no—
42C2×#11758198—41no—
43#14519702—21no—
44#14519702—21no—
45#14519702—21no—
46#14519702—21no—
47#14519699—21no—
48#14519699—21no—
49#14519699—21no—
50#14519699—21no—
51C2×M16×C61—21no—
52C2×C42×C61C2×C411yescharacteristic
53#97×C61—21no—
54C4×C8×C61C2×C411yescharacteristic, complemented
55C4×C8×C61C2×C411yescharacteristic, complemented
56C23×D61—21no—
57C22×#6923—81no—
58C22×C4×C61—21no—
59C22×C4×C61C22⋊C411yescharacteristic
60C22×Dic61—21no—
61C2×D244—21no—
62C2×D244—41no—
63C2×D244—41no—
64C2×D244C22⋊C411yescharacteristic
65C2×D244C2×D411yescharacteristic
66C2×Dic122—21no—
67C2×Dic122—41no—
68C2×Dic122—41no—
69C2×Dic122—41no—
70C2×Dic122C22⋊C411yescharacteristic
71C2×Dic122C2×D411yescharacteristic
72C22⋊#6923—42no—
73C22⋊#6923—81no—
74#11758217—42no—
75#11758217—81no—
76C2×#92420—81no—
77M16×C61—41no—
78M16×C61—41no—
79C2×C8×C61—21no—
80C2×C8×C61—21no—
81C42×C61—21no—
82C42×C61C22⋊C411yescharacteristic
83C42×C61C22×C411yescharacteristic
84C22⋊Dic61—41no—
85#11758198—41no—
86#11758198—41no—
87#11758198—41no—
88#11758198—41no—
89C4⋊Dic61—21no—
90C4⋊Dic61—21no—
91C4⋊Dic61—41no—
92#11758196—41no—
93#11758196—41no—
94C4×Dic61—41no—
95C23×C61#9911yescharacteristic
96C22×D61—21no—
97C22×D61—21no—
98C22×D61—41no—
99C22×D61—41no—
100C22×D61—82no—
101C2×#6923—82no—
102C2×#6923—162no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—81no—
108C2×C4×C61#9911yescharacteristic
109C2×C4×C61C2×C22⋊C411yescharacteristic
110C2×Dic61—21no—
111C2×Dic61—21no—
112C2×Dic61—41no—
113C2×Dic61—41no—
114C2×Dic61—41no—
115C2×Dic61—41no—
116D244—41no—
117D244—41no—
118D244—81no—
119Dic122—41no—
120Dic122—41no—
121Dic122—81no—
122Dic122—82no—
123#92420—82no—
124C8×C61—41no—
125C8×C61—41no—
126#13603—611nomaximal, Sylow
127C22×C61—41no—
128C22×C61#35411yescharacteristic
129C22×C61#35411yescharacteristic
130C22×C61C2×#9911yescharacteristic
131C2×D61—41no—
132C2×D61—41no—
133C2×D61—82no—
134C2×D61—81no—
135C2×D61—81no—
136C2×D61—161no—
137#6923—162no—
138C4×C61—21no—
139C4×C61—21no—
140C4×C61—41no—
141C4×C61—41no—
142Dic61—41no—
143Dic61—41no—
144Dic61—81no—
145Dic61—81no—
146#3140—611no—
147C2×#355—611no—
148C2×#354—611no—
149#1414—612no—
150#1414—612no—
151C2×C61—21no—
152C2×C61—21no—
153C2×C61—21no—
154C2×C61C2×#35411yescharacteristic
155D61—81no—
156D61—81no—
157#533—611no—
158C2×#124—611no—
159#489—1221no—
160#489—1221no—
161#488—611no—
162#488—611no—
163#488—611no—
164#488—611no—
165#419—1221no—
166#418—1221no—
167C2×#101—1221no—
168C2×#99—1221no—
169C4×M16#692311yescharacteristic, complemented
170#355—612no—
171#355—1221no—
172#354—612no—
173#354—1221no—
174C61#1360311yescharacteristic, complemented, Sylow
175C22×Q8—611no—
176C22×D4—611no—
177C2×M16—21no—
178C2×M16—2441no—
179C4⋊Q8—1221no—
180#125—611no—
181#125—611no—
182#124—611no—
183#124—611no—
184#124—611no—
185#124—611no—
186#124—2441no—
187C22⋊Q8—2441no—
188C4×Q8—2441no—
189#117—1221no—
190C2×C22⋊C4—2441no—
191C2×C22⋊C4—2441no—
192C2×C42C2×#692311yescharacteristic
193#103—1221no—
194#103—1221no—
195#103—1221no—
196#103—1221no—
197#102—1221no—
198#102—1221no—
199#102—1221no—
200#102—1221no—
201#101—1222no—
202#101—2441no—
203#99—1222no—
204#99—2441no—
205#97—21no—
206C4×C8C2×#692311yescharacteristic, complemented
207C4×C8C2×#692311yescharacteristic, complemented
208C24—1221no—
209C2×Q8—611no—
210C2×Q8—611no—
211C2×Q8—1221no—
212C2×Q8—2441no—
213C2×Q8—2441no—
214C2×Q8—2441no—
215C2×D4—611no—
216C2×D4—611no—
217C2×D4—1221no—
218C2×D4—2441no—
219C2×D4—2441no—
220C22×C4—21no—
221C22×C4—1221no—
222C22×C4—4881no—
223C22×C4C22⋊#692311yescharacteristic
224M16—41no—
225M16—41no—
226M16—4881no—
227M16—2442no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—4881no—
231C4⋊C4—1221no—
232C4⋊C4—1221no—
233C4⋊C4—2441no—
234C4⋊C4—2441no—
235C4⋊C4—2441no—
236C22⋊C4—4881no—
237C22⋊C4—2441no—
238C22⋊C4—2441no—
239C22⋊C4—2441no—
240C22⋊C4—2441no—
241C22⋊C4—2442no—
242C22⋊C4—2441no—
243C42—21no—
244C42—2441no—
245C42C22⋊#692311yescharacteristic
246C42C22×#692311yescharacteristic
247C23—1221no—
248C23—1221no—
249C23—4882no—
250C23—2441no—
251C23—2441no—
252C23#1451983611yescharacteristic
253Q8—4881no—
254Q8—4882no—
255Q8—2441no—
256Q8—2441no—
257D4—4881no—
258D4—2441no—
259D4—2441no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—81no—
265C2×C4—1221no—
266C2×C4—1221no—
267C2×C4—9762no—
268C2×C4—4882no—
269C2×C4—2441no—
270C2×C4—2441no—
271C2×C4—2441no—
272C2×C4—2441no—
273C2×C4#1451983611yescharacteristic
274C2×C4C2×C22⋊#692311yescharacteristic
275C8—41no—
276C8—41no—
277C8—4882no—
278C22—41no—
279C22—9761no—
280C22—4881no—
281C22—4881no—
282C22—4882no—
283C22—2441no—
284C22—2441no—
285C22#1684066911yescharacteristic
286C22#1684066911yescharacteristic
287C22C2×#1451983611yescharacteristic
288C4—21no—
289C4—21no—
290C4—41no—
291C4—41no—
292C4—9762no—
293C4—4881no—
294C4—4881no—
295C4—2441no—
296C4—2441no—
297C2—21no—
298C2—21no—
299C2—21no—
300C2—4881no—
301C2—4881no—
302C2C2×#1684066911yescharacteristic
subgroupG/H
centerC2C2×#16840669
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC4×M16×C61C4
socleC2×C61C2×#354
prime radicalC22×D244D4
cosocle#14519765C23
hypercenterC4×M16#6923
hypocenterC61#13603
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
3. #21460967C2semidirect
4. #21460966C2semidirect
12. #16840319C4semidirect
16. #16840117C4semidirect
17. #16840117C4semidirect
22. C4×M16×C61C4semidirect
32. #14519769C2×C4semidirect
33. #14519769C2×C4semidirect
54. C4×C8×C61C2×C4semidirect
55. C4×C8×C61C2×C4semidirect
169. C4×M16#6923semidirect
174. C61#13603semidirect
206. C4×C8C2×#6923semidirect
207. C4×C8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168406712
C2×#168406692
#214609672
#214609662
#214562062
#1360361