Finite Group Databasebeta

← #16857579

Subgroups — #16857579

Order 3968.

420 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 123 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#14521714C211yesmaximal, characteristic, complemented
2#14521704C211yesmaximal, characteristic
3#14521399C211yesmaximal, characteristic, complemented
4#14521378C212yesmaximal, complemented
5#14521335C211yesmaximal, characteristic, complemented
6#14521275C212yesmaximal, complemented
7#14521186C212yesmaximal, complemented
8C4×#92480C211yesmaximal, characteristic, complemented
9#14521114C211yesmaximal, characteristic, complemented
10#103×D31C212yesmaximal, complemented
11C31×#461C211yesmaximal, characteristic, complemented
12C2×Q8×D31C2212yes—
13C2×#92495—21no—
14C2×SD16×C31C2212yes—
15C31×C4⋊Q8C2211yescharacteristic
16C4×D4×C31C2211yescharacteristic
17#11758523C2212yes—
18#11758515C2212yes—
19C2×#92480C2211yescharacteristic
20#11758501—21no—
21C2×C8×D31C2212yes—
22#11758473—22no—
23#11758470—21no—
24C4⋊C4×D31C2212yes—
25#11758465—21no—
26#11758460—22no—
27C4×D124C2211yescharacteristic
28#11758446C2211yescharacteristic
29C4×Dic62C2211yescharacteristic
30C31×C4⋊C8C2211yescharacteristic
31C31×#103C2212yes—
32#11758423C2212yes—
33#11758423C2212yes—
34#11758420C2211yescharacteristic
35(C2×D31)⋊C8C2212yes—
36#11758405C2212yes—
37#11758404C2212yes—
38#11758398C2211yescharacteristic
39#11758396C2211yescharacteristic
40C4×C31⋊C8C2211yescharacteristic
41C2×Q8×C31C2312yes—
42C2×D4×C31C2311yescharacteristic
43C22×C4×C31—21no—
44C2×C22⋊D31—22no—
45Q8×D31—24no—
46Q8×D31—42no—
47#92495—44no—
48C2×D124D411yescharacteristic
49C2×C4×D31—22no—
50C2×C4×D31—22no—
51C2×C4×D31C2312yes—
52C2×Dic62—21no—
53C2×Dic62—22no—
54C2×Dic62C2311yescharacteristic
55SD16×C31—42no—
56C2×C8×C31C2312yes—
57C4⋊C4×C31—21no—
58C4⋊C4×C31—21no—
59C4⋊C4×C31C2311yescharacteristic
60C22⋊C4×C31—21no—
61C42×C31C2311yescharacteristic
62C22⋊Dic31—22no—
63#92480—22no—
64(C2×D31)⋊C4—22no—
65(C2×D31)⋊C4D412yes—
66C4⋊Dic31—21no—
67C4⋊Dic31D411yescharacteristic
68C4⋊Dic31C2311yescharacteristic
69C4⋊Dic31C2312yes—
70Dic31⋊C4—21no—
71Dic31⋊C4—22no—
72Dic31⋊C4—22no—
73C4×Dic31—21no—
74C2×C31⋊C8C2311yescharacteristic
75C2×C31⋊C8C2311yescharacteristic
76C8×D31—42no—
77C23×C31—21no—
78C22×D31C2×D412yes—
79Q8×C31—22no—
80Q8×C31—22no—
81D4×C31—21no—
82D4×C31—21no—
83C2×C4×C31—21no—
84C2×C4×C31—22no—
85C2×C4×C31—42no—
86C2×C4×C31C2×D411yescharacteristic
87C2×C4×C31C2×D411yescharacteristic
88C2×C4×C31C2411yescharacteristic
89C22⋊D31—44no—
90C2×Dic31—21no—
91C2×Dic31—21no—
92C2×Dic31—22no—
93C2×Dic31C2×D412yes—
94D124SD1614yescomplemented
95C4×D31—22no—
96C4×D31—22no—
97C4×D31—44no—
98C4×D31—44no—
99Dic62—21no—
100Dic62—44no—
101Dic62—42no—
102Dic62#4112yescomplemented
103C8×C31—22no—
104C31⋊C8—21no—
105C31⋊C8—21no—
106#3298—311nomaximal, Sylow
107C22×C31—41no—
108C22×C31—41no—
109C22×C31—41no—
110C22×C31C22×D411yescharacteristic
111C2×D31—22no—
112C2×D31—22no—
113C2×D31C2×SD1614yes—
114C4×C31—21no—
115C4×C31—42no—
116C4×C31—41no—
117C4×C31C2×SD1612yes—
118C4×C31C2×#4111yescharacteristic
119C4×C31H(5,𝔽2)11yescharacteristic
120Dic31—22no—
121Dic31—22no—
122Dic31—42no—
123Dic31—41no—
124D4×Q8—311no—
125#547—311no—
126#487—311no—
127#483—312no—
128#475—311no—
129#465—312no—
130#461D3111yescharacteristic, complemented
131#448—312no—
132C4×SD16—311no—
133C8×D4—311no—
134C2×#103—312no—
135C2×C31—81no—
136C2×C31#54611yescharacteristic
137C2×C31C22×SD1611yescharacteristic
138C2×C31#57811yescharacteristic
139D31—24no—
140C2×#41—621no—
141C22×Q8—312no—
142C2×SD16—311no—
143C2×SD16C2×D3112yes—
144C22×C8—312no—
145C4⋊Q8—311no—
146C4⋊Q8—621no—
147C4⋊Q8C2×D3111yescharacteristic
148#123—622no—
149C22⋊Q8—312no—
150C22⋊Q8—622no—
151C22⋊Q8—621no—
152#121—312no—
153C4×Q8—311no—
154C4×D4—311no—
155C4×D4—621no—
156C4×D4C2×D3111yescharacteristic
157C2×C4⋊C4—312no—
158#106—312no—
159#106—311no—
160C4⋊C8C2×D3111yescharacteristic
161#103—312no—
162#103—312no—
163#103—312no—
164#103—311no—
165#103C2×D3112yes—
166#102—311no—
167C22⋊C8—312no—
168C4×C8—311no—
169C31#329811yescharacteristic, complemented, Sylow
170#41—1244no—
171C2×Q8—311no—
172C2×Q8—622no—
173C2×Q8—621no—
174C2×Q8—624no—
175C2×Q8—1242no—
176C2×Q8C22×D3112yes—
177C2×D4—311no—
178C2×D4—622no—
179C2×D4C22×D3111yescharacteristic
180C22×C4—21no—
181C22×C4—312no—
182C22×C4—622no—
183C22×C4—622no—
184SD16—42no—
185SD16—622no—
186C2×C8—311no—
187C2×C8—311no—
188C2×C8—1242no—
189C2×C8C22×D3112yes—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—311no—
193C4⋊C4—311no—
194C4⋊C4—312no—
195C4⋊C4—622no—
196C4⋊C4—622no—
197C4⋊C4—621no—
198C4⋊C4—621no—
199C4⋊C4C22×D3111yescharacteristic
200C22⋊C4—21no—
201C22⋊C4—312no—
202C22⋊C4—622no—
203C22⋊C4—622no—
204C42—621no—
205C42C22×D3111yescharacteristic
206C23—21no—
207C23—312no—
208Q8—22no—
209Q8—22no—
210Q8—312no—
211Q8—621no—
212Q8—1244no—
213Q8—1242no—
214D4—21no—
215D4—21no—
216D4—314no—
217D4—1244no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—312no—
222C2×C4—621no—
223C2×C4—622no—
224C2×C4—622no—
225C2×C4—621no—
226C2×C4—622no—
227C2×C4—1244no—
228C2×C4—1244no—
229C2×C4D4×D3111yescharacteristic
230C2×C4D4×D3111yescharacteristic
231C2×C4C23×D3111yescharacteristic
232C8—22no—
233C8—621no—
234C8—621no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—314no—
239C22—622no—
240C22—622no—
241C22C2×D4×D3111yescharacteristic
242C4—21no—
243C4—42no—
244C4—41no—
245C4—622no—
246C4—622no—
247C4—1241no—
248C4—1242no—
249C4SD16×D3112yes—
250C4#1175856111yescharacteristic
251C4#41×D3111yescharacteristic
252C2—81no—
253C2—624no—
254C2#1452169611yescharacteristic
255C2C2×SD16×D3111yescharacteristic
256C2#1452186511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#461C2
socleC22×C31C22×D4
prime radical#14521704C2
hypercenter#461D31
hypocenterC31#3298
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #14521714C2semidirect
3. #14521399C2semidirect
4. #14521378C2semidirect
5. #14521335C2semidirect
6. #14521275C2semidirect
7. #14521186C2semidirect
8. C4×#92480C2semidirect
9. #14521114C2semidirect
10. #103×D31C2semidirect
11. C31×#461C2semidirect
94. D124SD16semidirect
102. Dic62#41semidirect
130. #461D31semidirect
169. C31#3298semidirect

Maximal subgroups

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

subgroupindex
#145217142
#145217042
#145213992
#145213782
#145213352
#145212752
#145211862
C4×#924802
#145211142
#103×D312
C31×#4612
#329831