Finite Group Databasebeta

← #29594158

Subgroups — #29594158

Order 15616.

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

normal subgroups only

This group has 234 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#21447967C211yesmaximal, characteristic
2#21447865C211yesmaximal, characteristic
3#21445416C211yesmaximal, characteristic, complemented
4C8×D488C211yesmaximal, characteristic
5#21443876C211yesmaximal, characteristic, complemented
6#21443869C211yesmaximal, characteristic, complemented
7C61×#1367C211yesmaximal, characteristic, complemented
8C2×#14519722—21no—
9#16840167C2211yescharacteristic
10#16840160—21no—
11#16840142C2211yescharacteristic
12#16840113—21no—
13C4×D488C2211yescharacteristic
14C8×D244—21no—
15#16839590—21no—
16#16839567—21no—
17#16839564—21no—
18#16839515C2211yescharacteristic
19#16839491—21no—
20C61×C4⋊C16C2211yescharacteristic
21C61×C4⋊C16C2211yescharacteristic
22C82×C61C2211yescharacteristic
23C2×Dic244—21no—
24C2×Dic244—21no—
25C2×D488D411yescharacteristic
26C2×C8×D61—41no—
27#14519777—21no—
28#14519777—21no—
29C4×D244—21no—
30#14519722—42no—
31#14519711—21no—
32#14519711—21no—
33#14519711D411yescharacteristic
34#14519709—41no—
35#14519709—41no—
36C4⋊#92418—21no—
37#14519702—42no—
38#14519699—21no—
39#14519689—41no—
40C2×C16×C61—21no—
41C2×C16×C61—21no—
42C4×C8×C61D411yescharacteristic
43C4×C8×C61D411yescharacteristic
44C4×C8×C61C2311yescharacteristic
45C2×D244—21no—
46C2×C4×D61—41no—
47C2×Dic122—41no—
48C2×Dic122—41no—
49C2×C8×C61—21no—
50C2×C8×C61D811yescharacteristic
51C2×C8×C61D811yescharacteristic
52C2×C8×C61C2×D411yescharacteristic
53C2×C8×C61#4111yescharacteristic
54C42×C61C2×D411yescharacteristic
55#11758198—41no—
56C4⋊Dic61—21no—
57C4⋊Dic61—41no—
58C4⋊Dic61—41no—
59C4⋊Dic61—41no—
60C4⋊Dic61—41no—
61C2×#92418—41no—
62Dic244—41no—
63Dic244—41no—
64Dic244—81no—
65D488—21no—
66D488SD1612yes—
67C8×D61—82no—
68C16×C61—41no—
69C16×C61—41no—
70C22×D61—41no—
71C2×C4×C61#12111yescharacteristic
72C2×C4×C61C2×D811yescharacteristic
73C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
74C2×Dic61—41no—
75C2×Dic61—81no—
76C2×Dic61—81no—
77D244—22no—
78D244—41no—
79C4×D61—81no—
80C4×D61—81no—
81Dic122—82no—
82Dic122—81no—
83Dic122—81no—
84C8×C61—21no—
85C8×C61—21no—
86C8×C61—41no—
87C8×C61SD3212yescomplemented
88C8×C61C2×SD1611yescharacteristic
89C8×C61#13511yescharacteristic
90#92418—81no—
91#12420—611nomaximal, Sylow
92C22×C61#45911yescharacteristic
93C2×D61—42no—
94C2×D61—81no—
95C2×D61—81no—
96C4×C61—21no—
97C4×C61#46511yescharacteristic
98C4×C61#47111yescharacteristic
99C4×C61C2×SD3211yescharacteristic
100C4×C61#51011yescharacteristic
101Dic61—81no—
102Dic61—161no—
103Dic61—161no—
104#2226—611no—
105#2208—611no—
106#1707—611no—
107C8×D8—611no—
108#1367D6111yescharacteristic, complemented
109#1365—611no—
110#1362—611no—
111C2×C61#166511yescharacteristic
112C2×C61#220811yescharacteristic
113C2×C61#221211yescharacteristic
114D61—82no—
115C2×SD32—1221no—
116#500—611no—
117#498—1221no—
118#494—611no—
119#487—1221no—
120C4×D8—611no—
121C8×D4—1221no—
122#367—1221no—
123C4⋊C16C2×D6111yescharacteristic
124C4⋊C16C2×D6111yescharacteristic
125#358—1221no—
126#357—1221no—
127#335—611no—
128D4⋊C8—1221no—
129C82C2×D6111yescharacteristic
130C61#1242011yescharacteristic, complemented, Sylow
131C2×Q16—1221no—
132C2×Q16—1221no—
133C2×D8—611no—
134C22×C8—2441no—
135C4⋊Q8—1221no—
136C4⋊Q8—1221no—
137C4×D4—1221no—
138SD32—2442no—
139C2×C16—21no—
140C2×C16—21no—
141#107—611no—
142#107—1221no—
143#107—1221no—
144#106—2441no—
145#106—2441no—
146C4⋊C8—1221no—
147#103—2442no—
148#102—1221no—
149C22⋊C8—2441no—
150C4×C8D24411yescharacteristic
151C4×C8D24411yescharacteristic
152C4×C8C22×D6111yescharacteristic
153C2×Q8—2441no—
154C2×Q8—2441no—
155C2×D4—1221no—
156C22×C4—2441no—
157Q16—4881no—
158Q16—2441no—
159Q16—2441no—
160D8—612no—
161D8—1221no—
162C2×C8—21no—
163C2×C8—4882no—
164C2×C8—2441no—
165C2×C8D48811yescharacteristic
166C2×C8D48811yescharacteristic
167C2×C8D4×D6111yescharacteristic
168C2×C8#1175822611yescharacteristic
169C4⋊C4—1221no—
170C4⋊C4—2441no—
171C4⋊C4—2441no—
172C4⋊C4—2441no—
173C4⋊C4—2441no—
174C22⋊C4—2441no—
175C42C2×D24411yescharacteristic
176C16—41no—
177C16—41no—
178C23—2441no—
179Q8—4882no—
180Q8—4881no—
181Q8—4881no—
182D4—1222no—
183D4—2441no—
184C2×C4—4881no—
185C2×C4—4881no—
186C2×C4—4881no—
187C2×C4—4881no—
188C2×C4—2441no—
189C2×C4#1451975211yescharacteristic
190C2×C4C2×D48811yescharacteristic
191C2×C4#1451980511yescharacteristic
192C8—21no—
193C8—21no—
194C8—41no—
195C8—4881no—
196C8#1451972212yescomplemented
197C8SD16×D6111yescharacteristic
198C8#1451979811yescharacteristic
199C22—4881no—
200C22—4881no—
201C22—2442no—
202C22#1683995811yescharacteristic
203C4—21no—
204C4—9761no—
205C4—9761no—
206C4—4881no—
207C4#1683999211yescharacteristic
208C4#1684002611yescharacteristic
209C4C2×#1451972211yescharacteristic
210C4#1684020511yescharacteristic
211C2—4882no—
212C2#2144516111yescharacteristic
213C2#2144786511yescharacteristic
214C2#2144788911yescharacteristic
subgroupG/H
centerC22#16839958
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#1367C2
socleC22×C61#459
prime radicalC2×D488D4
hypercenter#1367D61
hypocenterC61#12420
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C82 ◁ #1367

Complements

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

NH ≅ G/Ntype
3. #21445416C2semidirect
5. #21443876C2semidirect
6. #21443869C2semidirect
7. C61×#1367C2semidirect
87. C8×C61SD32semidirect
108. #1367D61semidirect
130. C61#12420semidirect
196. C8#14519722semidirect

Maximal subgroups

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

subgroupindex
#214479672
#214478652
#214454162
C8×D4882
#214438762
#214438692
C61×#13672
#1242061