Finite Group Databasebeta

← #16855836

Subgroups — #16855836

Order 3968.

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

normal subgroups only

This group has 380 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#14521601C211yesmaximal, characteristic
2C2×Q8×Dic31C211yesmaximal, characteristic
3#14521380C211yesmaximal, characteristic
4#14521377C211yesmaximal, characteristic
5#14521332C212yesmaximal, complemented
6#14521257C212yesmaximal, complemented
7#14521205C211yesmaximal, characteristic
8#14521156C212yesmaximal, complemented
9#14521153C212yesmaximal, complemented
10#14521103C211yesmaximal, characteristic
11C31×#451C211yesmaximal, characteristic
12C22×Q8×C31C2211yescharacteristic
13C2×Q16×C31C2212yes—
14C31×C22⋊Q8C2211yescharacteristic
15Q8×Dic31—21no—
16Q8×Dic31—21no—
17Q8×Dic31C2211yescharacteristic
18Q8×Dic31C2212yes—
19Q8×Dic31C2212yes—
20#11758521C2211yescharacteristic
21C2×#92482C2212yes—
22C2×C4⋊Dic31—21no—
23C2×C4⋊Dic31C2211yescharacteristic
24#11758501C2211yescharacteristic
25C2×C4×Dic31—21no—
26C2×C4×Dic31C2211yescharacteristic
27#11758466—21no—
28#11758465C2211yescharacteristic
29#11758464—21no—
30#11758464C2211yescharacteristic
31#11758455—21no—
32#11758454—21no—
33C31×#103C2212yes—
34C31×C22⋊C8C2211yescharacteristic
35#11758423C2212yes—
36#11758423C2212yes—
37C22⋊(C31⋊C8)C2211yescharacteristic
38#11758405C2212yes—
39#11758404C2212yes—
40#11758403C2212yes—
41Dic31⋊C8C2212yes—
42#11758396C2212yes—
43C2×Q8×C31—24no—
44C2×Q8×C31—21no—
45C2×Q8×C31C2311yescharacteristic
46C2×Q8×C31C2312yes—
47C22×C4×C31—21no—
48C22×C4×C31C2311yescharacteristic
49C22×Dic31—21no—
50C22×Dic31D411yescharacteristic
51C22×Dic31D411yescharacteristic
52C2×Dic62—21no—
53C2×Dic62C2311yescharacteristic
54Q16×C31—24no—
55C2×C8×C31C2312yes—
56C4⋊C4×C31—21no—
57C4⋊C4×C31C2311yescharacteristic
58C22⋊C4×C31—21no—
59C22⋊Dic31—21no—
60#92482—24no—
61C4⋊Dic31—21no—
62C4⋊Dic31—21no—
63C4⋊Dic31—22no—
64C4⋊Dic31—22no—
65C4⋊Dic31C2312yes—
66C4⋊Dic31C2311yescharacteristic
67Dic31⋊C4—21no—
68Dic31⋊C4—21no—
69Dic31⋊C4—21no—
70Dic31⋊C4—21no—
71Dic31⋊C4—21no—
72C4×Dic31—21no—
73C4×Dic31—21no—
74C4×Dic31—22no—
75C4×Dic31—22no—
76C4×Dic31D411yescharacteristic
77C4×Dic31D411yescharacteristic
78C4×Dic31C2311yescharacteristic
79C4×Dic31C2311yescharacteristic
80C2×C31⋊C8C2312yes—
81C23×C31C2×D411yescharacteristic
82Q8×C31—24no—
83Q8×C31—22no—
84Q8×C31—22no—
85Q8×C31#4114yes—
86C2×C4×C31—24no—
87C2×C4×C31—22no—
88C2×C4×C31—22no—
89C2×C4×C31—21no—
90C2×C4×C31—21no—
91C2×C4×C31C2×D411yescharacteristic
92C2×C4×C31C2411yescharacteristic
93C2×Dic31—22no—
94C2×Dic31—22no—
95C2×Dic31—22no—
96C2×Dic31—21no—
97C2×Dic31—21no—
98C2×Dic31—42no—
99C2×Dic31C2×D411yescharacteristic
100C2×Dic31C2×D411yescharacteristic
101C2×Dic31C2×D411yescharacteristic
102C2×Dic31C2×D411yescharacteristic
103Dic62—22no—
104Dic62—42no—
105C8×C31—22no—
106C31⋊C8—22no—
107#3113—311nomaximal, Sylow
108C22×C31—21no—
109C22×C31—21no—
110C22×C31#13712yes—
111C22×C31C22×D411yescharacteristic
112C4×C31—24no—
113C4×C31—21no—
114C4×C31—41no—
115C4×C31—41no—
116C4×C31C2×#4112yes—
117Dic31—21no—
118Dic31—21no—
119Dic31—21no—
120Dic31—41no—
121Dic31—41no—
122Dic31—42no—
123Dic31#13712yes—
124#533—311no—
125C2×C4×Q8—311no—
126#483—311no—
127#483—311no—
128#475—312no—
129#462—312no—
130#451—311no—
131#451D3111yescharacteristic
132#441—312no—
133#441—312no—
134#432—311no—
135C2×C31—22no—
136C2×C31#52511yescharacteristic
137C2×C31C2×#13711yescharacteristic
138C2×C31C2×#13711yescharacteristic
139C22×Q8C2×D3111yescharacteristic
140C2×Q16—312no—
141C2×Q16C2×D3112yes—
142C4⋊Q8—311no—
143C4⋊Q8—311no—
144#125—621no—
145C22⋊Q8—311no—
146C22⋊Q8—621no—
147C22⋊Q8C2×D3111yescharacteristic
148C4×Q8—312no—
149C4×Q8—312no—
150C4×Q8—311no—
151C4×Q8—311no—
152C4×Q8—621no—
153C4×Q8—621no—
154C4×Q8—621no—
155#117—621no—
156C2×C4⋊C4—311no—
157C2×C4⋊C4—621no—
158C2×C42—311no—
159C2×C42—621no—
160#106—312no—
161#106—312no—
162C4⋊C8—312no—
163#103—312no—
164#103—312no—
165#103—312no—
166#103C2×D3112yes—
167C22⋊C8—311no—
168C22⋊C8C2×D3111yescharacteristic
169#97—312no—
170C31#311311yescharacteristic, complemented, Sylow
171C2×Q8—24no—
172C2×Q8—21no—
173C2×Q8—311no—
174C2×Q8—621no—
175C2×Q8C22×D3111yescharacteristic
176C2×Q8C22×D3112yes—
177C22×C4—21no—
178C22×C4—311no—
179C22×C4—311no—
180C22×C4—621no—
181C22×C4C22×D3111yescharacteristic
182Q16—24no—
183Q16—624no—
184C2×C8—312no—
185C2×C8C22×D3112yes—
186C4⋊C4—21no—
187C4⋊C4—312no—
188C4⋊C4—311no—
189C4⋊C4—621no—
190C4⋊C4—621no—
191C4⋊C4—621no—
192C4⋊C4—621no—
193C4⋊C4—622no—
194C4⋊C4—622no—
195C4⋊C4—621no—
196C4⋊C4—621no—
197C4⋊C4—621no—
198C4⋊C4C22×D3111yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—621no—
201C42—311no—
202C42—311no—
203C42—311no—
204C42—311no—
205C42—622no—
206C42—622no—
207C42—621no—
208C42—621no—
209C23D4×D3111yescharacteristic
210Q8—24no—
211Q8—22no—
212Q8—22no—
213Q8—622no—
214Q8—1242no—
215Q8#9249714yes—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—311no—
222C2×C4—311no—
223C2×C4—311no—
224C2×C4—311no—
225C2×C4—622no—
226C2×C4—622no—
227C2×C4—622no—
228C2×C4—621no—
229C2×C4—621no—
230C2×C4—1242no—
231C2×C4D4×D3111yescharacteristic
232C2×C4C23×D3111yescharacteristic
233C8—22no—
234C8—622no—
235C22—21no—
236C22—21no—
237C22#1175849512yes—
238C22C2×D4×D3111yescharacteristic
239C4—24no—
240C4—21no—
241C4—41no—
242C4—41no—
243C4—312no—
244C4—621no—
245C4—621no—
246C4—621no—
247C4—1241no—
248C4—1241no—
249C4—1242no—
250C4C2×#9249712yes—
251C2—22no—
252C2#1452155511yescharacteristic
253C2#137×D3111yescharacteristic
254C2C2×#1175849511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#451C2
socleC22×C31C22×D4
prime radicalC23×C31C2×D4
hypercenter#451D31
hypocenterC31#3113
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
5. #14521332C2semidirect
6. #14521257C2semidirect
8. #14521156C2semidirect
9. #14521153C2semidirect
170. C31#3113semidirect

Maximal subgroups

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

subgroupindex
#145216012
C2×Q8×Dic312
#145213802
#145213772
#145213322
#145212572
#145212052
#145211562
#145211532
#145211032
C31×#4512
#311331