Finite Group Databasebeta

← #29600311

Subgroups — #29600311

Order 15616.

454 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 71 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#21456272C211yesmaximal, characteristic
2#21447484C211yesmaximal, characteristic, complemented
3#21443977C214yesmaximal, complemented
4C61×#2132C211yesmaximal, characteristic, complemented
5C2×#14519802—21no—
6C2×#14519802—21no—
7C2×#14519774C411yescharacteristic, complemented
8#16840152C412yescomplemented
9#16840152C2212yes—
10#16840138—21no—
11#16840136—21no—
12#16839736—21no—
13C4×#11758194C411yescharacteristic, complemented
14#16839557C2212yescomplemented
15#16839555—21no—
16C61×#532C2211yescharacteristic
17C61×#413—21no—
18C61×#356—21no—
19C61×#355C2212yescomplemented
20C22×D244—21no—
21C22×D244—41no—
22#14519802—42no—
23#14519802—42no—
24#14519802—42no—
25#14519802—42no—
26C2×#11758201—22no—
27C2×#11758201—22no—
28C2×#11758199—22no—
29C2×#11758199—22no—
30C2×#11758194—21no—
31#14519774—21no—
32#14519774—41no—
33#14519774C2×C412yes—
34#14519736—41no—
35#14519692—22no—
36#14519692—41no—
37#14519688C2×C412yes—
38C4×#92418—21no—
39C2×#41×C61D411yescharacteristic
40C2×#41×C61D411yescharacteristic
41C2×M16×C61—41no—
42C4⋊Q8×C61D411yescharacteristic
43#127×C61D411yescharacteristic
44#124×C61C2×C411yescharacteristic
45#124×C61C2311yescharacteristic
46#121×C61—41no—
47C4×D4×C61—41no—
48C2×C42×C61C2×C411yescharacteristic
49#101×C61—22no—
50#100×C61—41no—
51C23×D61—41no—
52#41×C61—41no—
53#41×C61—41no—
54#41×C61—41no—
55#41×C61—41no—
56C2×Q8×C61C22⋊C411yescharacteristic
57C2×Q8×C61C2×D411yescharacteristic
58C2×D4×C61—21no—
59C2×D4×C61—21no—
60C2×D4×C61—41no—
61C2×D4×C61C22⋊C411yescharacteristic
62C2×D4×C61C2×D411yescharacteristic
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61C22⋊C411yescharacteristic
67C2×D244—22no—
68C2×D244—41no—
69C2×D244—41no—
70C2×D244—41no—
71C2×D244—41no—
72C2×D244—42no—
73C2×D244—42no—
74C2×D244—41no—
75C2×D244—84no—
76M16×C61—42no—
77M16×C61—81no—
78C2×C8×C61—81no—
79C4⋊C4×C61—41no—
80C22⋊C4×C61—41no—
81C22⋊C4×C61—41no—
82C42×C61—21no—
83C42×C61C22⋊C411yescharacteristic
84C42×C61C22×C411yescharacteristic
85C22⋊Dic61—42no—
86#11758201—44no—
87#11758201—44no—
88#11758199—44no—
89#11758199—44no—
90C4⋊Dic61—81no—
91C4×Dic61—81no—
92#11758194—22no—
93#11758194—22no—
94C2×#92418—22no—
95C23×C61—21no—
96C23×C61—21no—
97C23×C61#9911yescharacteristic
98C22×D61—42no—
99C22×D61—81no—
100C22×D61—81no—
101C22×D61—84no—
102Q8×C61—41no—
103Q8×C61—41no—
104D4×C61—41no—
105D4×C61—41no—
106D4×C61—82no—
107D4×C61—81no—
108D4×C61—81no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—42no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61#9911yescharacteristic
121C2×C4×C61C2×C22⋊C411yescharacteristic
122C2×Dic61—82no—
123D244—42no—
124D244—42no—
125D244—84no—
126D244—84no—
127D244—81no—
128D244—81no—
129C8×C61—82no—
130#92418—24no—
131#13579—611nomaximal, Sylow
132C22×C61—21no—
133C22×C61—41no—
134C22×C61—41no—
135C22×C61—41no—
136C22×C61—81no—
137C22×C61—81no—
138C22×C61C2×#9911yescharacteristic
139C2×D61—84no—
140C2×D61—82no—
141C2×D61—162no—
142C2×D61—162no—
143C4×C61—22no—
144C4×C61—21no—
145C4×C61—41no—
146C4×C61—41no—
147C4×C61—41no—
148C4×C61#35311yescharacteristic
149C4×C61#35311yescharacteristic
150Dic61—162no—
151#3151—611no—
152#2132D6111yescharacteristic, complemented
153#2126—611no—
154#1406—614no—
155C2×C61—21no—
156C2×C61—41no—
157C2×C61—81no—
158C2×C61—81no—
159C2×C61C2×#35311yescharacteristic
160D61—162no—
161C2×AGL(1,ℤ/8)—1221no—
162C2×AGL(1,ℤ/8)—1221no—
163#532C2×D6111yescharacteristic
164C2×#127—611no—
165#496—612no—
166#496—612no—
167#493—1221no—
168#492—1221no—
169#413—21no—
170#410—1221no—
171C4×M16—611no—
172#356—21no—
173#355C2×D6112yescomplemented
174#354—612no—
175#353—1221no—
176C61#1357911yescharacteristic, complemented, Sylow
177C2×#41D24411yescharacteristic
178C2×#41C22⋊D6111yescharacteristic
179C22×D4—1221no—
180C22×D4—2441no—
181AGL(1,ℤ/8)—2442no—
182AGL(1,ℤ/8)—2442no—
183AGL(1,ℤ/8)—2442no—
184AGL(1,ℤ/8)—2442no—
185C2×SD16—1222no—
186C2×SD16—1222no—
187C2×D8—1222no—
188C2×D8—1222no—
189C2×M16—41no—
190C2×M16—1221no—
191C4⋊Q8C22⋊D6111yescharacteristic
192#127—612no—
193#127—1221no—
194#127—2441no—
195#127D24411yescharacteristic
196#124C4×D6111yescharacteristic
197#124C22×D6111yescharacteristic
198#121—41no—
199C4×D4—41no—
200#117—2441no—
201C2×C42C4×D6111yescharacteristic
202#101—22no—
203#100—41no—
204#99—1222no—
205#99—2441no—
206#97—612no—
207C4×C8—1221no—
208C24—2441no—
209#41—41no—
210#41—41no—
211#41—41no—
212#41—41no—
213C2×Q8#1175819811yescharacteristic
214C2×Q8C2×C22⋊D6111yescharacteristic
215C2×D4—21no—
216C2×D4—21no—
217C2×D4—41no—
218C2×D4—1222no—
219C2×D4—4884no—
220C2×D4—2442no—
221C2×D4—2442no—
222C2×D4—2441no—
223C2×D4—2441no—
224C2×D4—2441no—
225C2×D4—2441no—
226C2×D4—2441no—
227C2×D4#1175819811yescharacteristic
228C2×D4C2×D24411yescharacteristic
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4#1175819811yescharacteristic
233SD16—2444no—
234SD16—2444no—
235D8—2444no—
236D8—2444no—
237M16—42no—
238M16—81no—
239M16—1222no—
240M16—1222no—
241C2×C8—81no—
242C2×C8—1222no—
243C4⋊C4—41no—
244C4⋊C4—4881no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—2442no—
248C42—21no—
249C42—4881no—
250C42#1175819811yescharacteristic
251C42C2×C4×D6111yescharacteristic
252C23—21no—
253C23—21no—
254C23—4881no—
255C23—4881no—
256C23—4884no—
257C23—2442no—
258C23#1451969111yescharacteristic
259Q8—41no—
260Q8—41no—
261D4—41no—
262D4—41no—
263D4—82no—
264D4—81no—
265D4—81no—
266D4—4884no—
267D4—4884no—
268D4—4881no—
269D4—4881no—
270D4—2442no—
271D4—2442no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—4882no—
284C2×C4#1451969111yescharacteristic
285C2×C4C2×#1175819811yescharacteristic
286C8—82no—
287C8—1224no—
288C22—21no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22—81no—
294C22—9762no—
295C22—9762no—
296C22—4884no—
297C22—4882no—
298C22C2×#1451969111yescharacteristic
299C4—22no—
300C4—21no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—9762no—
305C4#1683955411yescharacteristic
306C4#1683955411yescharacteristic
307C2—21no—
308C2—41no—
309C2—81no—
310C2—81no—
311C2—9762no—
312C2C2×#1683955411yescharacteristic
subgroupG/H
centerC2C2×#16839554
commutatorC42×C61C22×C4
Frattini#124C22×D61
FittingC61×#2132C2
socleC2×C61C2×#353
prime radical#21456272C2
cosocle#124×C61C23
hypercenter#2132D61
hypocenterC61#13579
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #2132

Complements

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

NH ≅ G/Ntype
2. #21447484C2semidirect
3. #21443977C2semidirect
4. C61×#2132C2semidirect
7. C2×#14519774C4semidirect
8. #16840152C4semidirect
13. C4×#11758194C4semidirect
14. #16839557C22semidirect
19. C61×#355C22semidirect
152. #2132D61semidirect
173. #355C2×D61semidirect
176. C61#13579semidirect

Maximal subgroups

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

subgroupindex
#214562722
#214474842
#214439772
C61×#21322
#1357961