Finite Group Databasebeta

← #29592584

Subgroups — #29592584

Order 15616.

472 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 63 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#21447043C211yesmaximal, characteristic
2#21444787C212yesmaximal
3#21443849C212yesmaximal, complemented
4#21443765C211yesmaximal, characteristic, complemented
5C61×#2551C211yesmaximal, characteristic, complemented
6#16840343C2211yescharacteristic
7#16839901—22no—
8C2×#14519705—21no—
9#16839761—21no—
10#16839531C2211yescharacteristic
11#16839495—22no—
12#16839488C2212yes—
13C2×C4⋊Q8×C61—22no—
14C61×#395—21no—
15C61×#395C2212yes—
16C61×#389—22no—
17C61×#384C2211yescharacteristic
18C4×C4⋊C4×C61—21no—
19C2×#11758222—21no—
20C2×#11758201—42no—
21C2×#11758194—21no—
22#14519757—41no—
23#14519752—41no—
24#14519751—22no—
25#14519705—42no—
26#14519701—42no—
27#14519700—41no—
28#14519697—41no—
29C22⋊#92418—22no—
30C22×Q8×C61—21no—
31C22×Q8×C61D412yes—
32C4⋊Q8×C61—48no—
33C2×C4⋊C4×C61—21no—
34C2×C4⋊C4×C61—21no—
35C2×C4⋊C4×C61—22no—
36C2×C4⋊C4×C61—22no—
37C2×C4⋊C4×C61—22no—
38C2×C4⋊C4×C61D411yescharacteristic
39C2×C4⋊C4×C61C2311yescharacteristic
40C2×C42×C61—22no—
41C2×C42×C61D411yescharacteristic
42#95×C61—21no—
43#95×C61—22no—
44#95×C61D412yes—
45C2×Q8×C61—44no—
46C2×Q8×C61—44no—
47C2×Q8×C61—42no—
48C2×Q8×C61—81no—
49C22×C4×C61—21no—
50C22×C4×C61—22no—
51C22×C4×C61—22no—
52C22×C4×C61—21no—
53C22×C4×C61C2×D411yescharacteristic
54C22×C4×C61C2×D412yes—
55C2×C22⋊D61—41no—
56#11758222—41no—
57#11758222—41no—
58#11758222—41no—
59#11758222—41no—
60C2×D244—21no—
61C2×D244—22no—
62C2×C4×D61—41no—
63C2×Dic122—21no—
64C4⋊C4×C61—24no—
65C4⋊C4×C61—24no—
66C4⋊C4×C61—41no—
67C4⋊C4×C61—44no—
68C4⋊C4×C61—44no—
69C4⋊C4×C61—44no—
70C4⋊C4×C61—41no—
71C42×C61—21no—
72C42×C61—44no—
73C42×C61SD1612yescomplemented
74#11758201—84no—
75#11758198—42no—
76C4⋊Dic61—22no—
77#11758194—41no—
78#11758194—41no—
79C2×#92418—42no—
80C2×#92418—41no—
81C23×C61#12011yescharacteristic
82C22×D61—41no—
83Q8×C61—44no—
84Q8×C61—82no—
85Q8×C61—84no—
86C2×C4×C61—22no—
87C2×C4×C61—22no—
88C2×C4×C61—22no—
89C2×C4×C61—21no—
90C2×C4×C61—21no—
91C2×C4×C61—24no—
92C2×C4×C61—24no—
93C2×C4×C61—22no—
94C2×C4×C61—42no—
95C2×C4×C61—42no—
96C2×C4×C61—42no—
97C2×C4×C61—42no—
98C2×C4×C61—42no—
99C2×C4×C61—42no—
100C2×C4×C61—41no—
101C2×C4×C61C2×SD1611yescharacteristic
102C2×C4×C61#13711yescharacteristic
103C22⋊D61—81no—
104C22⋊D61—81no—
105C2×Dic61—41no—
106D244—41no—
107D244—41no—
108D244—82no—
109C4×D61—81no—
110C4×D61—81no—
111Dic122—41no—
112Dic122—41no—
113#92418—81no—
114#92418—82no—
115#12180—611nomaximal, Sylow
116C22×C61—21no—
117C22×C61—21no—
118C22×C61#44811yescharacteristic
119C22×C61#45411yescharacteristic
120C22×C61B(4,𝔽2)11yescharacteristic
121C2×D61—81no—
122C2×D61—81no—
123C2×D61—81no—
124C4×C61—24no—
125C4×C61—21no—
126C4×C61—21no—
127C4×C61—44no—
128C4×C61—42no—
129C4×C61—44no—
130C4×C61—81no—
131Dic61—81no—
132#2551D6111yescharacteristic, complemented
133#2039—611no—
134#1604—612no—
135#1354—612no—
136#1325—611no—
137C2×C61—21no—
138C2×C61—21no—
139C2×C61#159911yescharacteristic
140C2×C61#219611yescharacteristic
141C2×C61#220511yescharacteristic
142D61—161no—
143#537—611no—
144C2×C4⋊Q8—22no—
145#448—1222no—
146C2×#104—1221no—
147#417—1221no—
148#395—21no—
149#395C2×D6112yes—
150#389—22no—
151#384C2×D6111yescharacteristic
152C4×C4⋊C4—21no—
153#343—611no—
154#327—1222no—
155#324—612no—
156C61#1218011yescharacteristic, complemented, Sylow
157C2×#41—1221no—
158C22×Q8—21no—
159C22×Q8C22⋊D6112yes—
160C2×SD16—2442no—
161C2×M16—1221no—
162C4⋊Q8—48no—
163C22⋊Q8—2441no—
164#121—1222no—
165#121—2441no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4C22⋊D6111yescharacteristic
172C2×C4⋊C4C22×D6111yescharacteristic
173C2×C42—22no—
174C2×C42C22⋊D6111yescharacteristic
175#104—2442no—
176#103—2441no—
177#103—2442no—
178#102—2441no—
179C22⋊C8—1222no—
180#95—21no—
181#95—22no—
182#95C22⋊D6112yes—
183#41—2441no—
184#41—2441no—
185#41—2441no—
186#41—2441no—
187C2×Q8—44no—
188C2×Q8—44no—
189C2×Q8—42no—
190C2×Q8—81no—
191C2×Q8—1221no—
192C2×D4—1222no—
193C2×D4—1221no—
194C2×D4—2441no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—2441no—
200C22×C4C2×C22⋊D6111yescharacteristic
201C22×C4C2×C22⋊D6112yes—
202SD16—4884no—
203M16—2441no—
204M16—2441no—
205C2×C8—2442no—
206C2×C8—2441no—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—1222no—
215C22⋊C4—2442no—
216C42—21no—
217C42—44no—
218C42#1175820112yescomplemented
219C23—2441no—
220C23#1451974611yescharacteristic
221Q8—44no—
222Q8—82no—
223Q8—84no—
224Q8—2441no—
225Q8—2441no—
226D4—4882no—
227D4—4881no—
228D4—4881no—
229D4—2441no—
230D4—2441no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—4881no—
247C2×C4—4881no—
248C2×C4—2441no—
249C2×C4C2×#1175820111yescharacteristic
250C2×C4#1451980811yescharacteristic
251C8—4881no—
252C8—4882no—
253C22—21no—
254C22—21no—
255C22—4881no—
256C22—4881no—
257C22—4881no—
258C22#1683990111yescharacteristic
259C22#1683993511yescharacteristic
260C22#1683994911yescharacteristic
261C4—24no—
262C4—21no—
263C4—21no—
264C4—44no—
265C4—42no—
266C4—44no—
267C4—81no—
268C4—4881no—
269C2—21no—
270C2—21no—
271C2—9761no—
272C2#2144475911yescharacteristic
273C2#2144780311yescharacteristic
274C2#2144784911yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2551C2
socleC22×C61B(4,𝔽2)
prime radical#16840343C22
hypercenter#2551D61
hypocenterC61#12180
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2551

Complements

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

NH ≅ G/Ntype
3. #21443849C2semidirect
4. #21443765C2semidirect
5. C61×#2551C2semidirect
73. C42×C61SD16semidirect
132. #2551D61semidirect
156. C61#12180semidirect
218. C42#11758201semidirect

Maximal subgroups

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

subgroupindex
#214470432
#214447872
#214438492
#214437652
C61×#25512
#1218061