Finite Group Databasebeta

← #16847716

Subgroups — #16847716

Order 3968.

260 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 41 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#14521788C211yesmaximal, characteristic, complemented
2#14521239C211yesmaximal, characteristic, complemented
3#14521217C211yesmaximal, characteristic, complemented
4#14520841C211yesmaximal, characteristic, complemented
5#14520838C211yesmaximal, characteristic, complemented
6#14520836C211yesmaximal, characteristic, complemented
7C31×B(4,𝔽2)C211yesmaximal, characteristic, complemented
8C31×H(5,𝔽2)C2211yescharacteristic
9C31×#120—21no—
10C31×#120C2211yescharacteristic
11#11758530—21no—
12#11758526—21no—
13#11758517C2211yescharacteristic
14#11758516—21no—
15#11758515—21no—
16#11758512—21no—
17#11758512C2211yescharacteristic
18#11758461—21no—
19#11758460—21no—
20#11758456—21no—
21C31×#99—21no—
22C31×#99C2211yescharacteristic
23#11758425—21no—
24#11758422—21no—
25#11758422C2211yescharacteristic
26#11758421C2211yescharacteristic
27C24×C31D411yescharacteristic
28#41×C31—21no—
29#41×C31—41no—
30C2×D4×C31—21no—
31C2×D4×C31—21no—
32C2×D4×C31—21no—
33C2×D4×C31—41no—
34C2×D4×C31—41no—
35C2×D4×C31C2311yescharacteristic
36C2×C22⋊D31—21no—
37C2×C22⋊D31—21no—
38C2×C22⋊D31—21no—
39C2×C22⋊D31—21no—
40C22×Dic31D411yescharacteristic, complemented
41C2×D124D411yescharacteristic, complemented
42C2×C4×D31—21no—
43C22⋊C4×C31—21no—
44C22⋊C4×C31—21no—
45C22⋊C4×C31—41no—
46C22⋊C4×C31D411yescharacteristic
47C22⋊Dic31—21no—
48C22⋊Dic31—21no—
49C22⋊Dic31—21no—
50C22⋊Dic31—21no—
51C22⋊Dic31—41no—
52C22⋊Dic31D411yescharacteristic, complemented
53#92481—41no—
54#92479—41no—
55(C2×D31)⋊C4—21no—
56C4⋊Dic31—21no—
57Dic31⋊C4—21no—
58Dic31⋊C4—21no—
59Dic31⋊C4—41no—
60C4×Dic31D411yescharacteristic, complemented
61#92474—21no—
62C23×C31—21no—
63C23×C31—21no—
64C23×C31—21no—
65C23×C31—41no—
66C23×C31—81no—
67C23×C31C2×D411yescharacteristic
68C23×C31C2×D411yescharacteristic
69C22×D31—21no—
70Q8×C31—21no—
71D4×C31—21no—
72D4×C31—41no—
73D4×C31—41no—
74D4×C31—41no—
75D4×C31—41no—
76D4×C31—81no—
77D4×C31—82no—
78C2×C4×C31—21no—
79C2×C4×C31—21no—
80C2×C4×C31—21no—
81C2×C4×C31—41no—
82C2×C4×C31—41no—
83C2×C4×C31C2×D411yescharacteristic
84C22⋊D31—81no—
85C22⋊D31—81no—
86C22⋊D31—81no—
87C22⋊D31—81no—
88C2×Dic31—21no—
89C2×Dic31—21no—
90C2×Dic31—21no—
91C2×Dic31—41no—
92C2×Dic31—41no—
93D124—41no—
94C4×D31—81no—
95C31⋊C8—41no—
96#2200—311nomaximal, Sylow
97C22×C31—21no—
98C22×C31—21no—
99C22×C31—41no—
100C22×C31—41no—
101C22×C31—41no—
102C22×C31—41no—
103C22×C31—42no—
104C22×C31—41no—
105C22×C31—41no—
106C22×C31—81no—
107C22×C31—81no—
108C22×C31—81no—
109C22×C31#12011yescharacteristic
110C2×D31—41no—
111C2×D31—41no—
112C2×D31—41no—
113C4×C31—21no—
114C4×C31—41no—
115C4×C31—41no—
116C4×C31—81no—
117Dic31—41no—
118Dic31—41no—
119Dic31—41no—
120Dic31—81no—
121#560—311no—
122#458—311no—
123B(4,𝔽2)D3111yescharacteristic, complemented
124#453—311no—
125#353—311no—
126#352—311no—
127AΣL(1,𝔽16)—311no—
128C2×C31—21no—
129C2×C31—41no—
130C2×C31—41no—
131C2×C31—81no—
132C2×C31—81no—
133C2×C31B(4,𝔽2)11yescharacteristic
134D31—81no—
135H(5,𝔽2)C2×D3111yescharacteristic
136AGL(1,ℤ/8)—621no—
137#127—311no—
138#126—621no—
139#123—311no—
140#123—621no—
141#123—621no—
142#121—621no—
143#121—621no—
144#121—621no—
145#120—21no—
146#120—621no—
147#120C2×D3111yescharacteristic
148#104—621no—
149#100—311no—
150#99—21no—
151#99—311no—
152#99—621no—
153#99C2×D3111yescharacteristic
154C31#220011yescharacteristic, complemented, Sylow
155C24C22⋊D3111yescharacteristic
156#41—21no—
157#41—41no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—311no—
164C2×D4—621no—
165C2×D4—621no—
166C2×D4—621no—
167C2×D4—621no—
168C2×D4C22×D3111yescharacteristic
169C22×C4—311no—
170C22×C4—621no—
171SD16—1241no—
172D8—1241no—
173M16—621no—
174C4⋊C4—621no—
175C4⋊C4—621no—
176C4⋊C4—621no—
177C4⋊C4—1241no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—41no—
181C22⋊C4—311no—
182C22⋊C4—621no—
183C22⋊C4—621no—
184C22⋊C4—621no—
185C22⋊C4—621no—
186C22⋊C4—621no—
187C22⋊C4—1241no—
188C22⋊C4C22⋊D3111yescharacteristic
189C42—311no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—41no—
194C23—81no—
195C23—621no—
196C23D4×D3111yescharacteristic
197C23C2×C22⋊D3111yescharacteristic
198Q8—21no—
199D4—21no—
200D4—41no—
201D4—41no—
202D4—41no—
203D4—41no—
204D4—81no—
205D4—82no—
206D4—1241no—
207D4—2481no—
208D4—2481no—
209D4—2481no—
210D4—2481no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—621no—
217C2×C4—621no—
218C2×C4—621no—
219C2×C4—1241no—
220C2×C4—1241no—
221C2×C4—2481no—
222C2×C4D4×D3111yescharacteristic
223C8—1241no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—41no—
232C22—41no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22—1241no—
237C22—1241no—
238C22—1241no—
239C22#1175851411yescharacteristic
240C4—21no—
241C4—41no—
242C4—41no—
243C4—81no—
244C4—1241no—
245C4—1241no—
246C4—1241no—
247C4—2481no—
248C2—21no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2—81no—
253C2—2481no—
254C2#1452123511yescharacteristic
subgroupG/H
centerC2#14521235
commutator, cosocleC2×D4×C31C23
FrattiniC2×D4C22×D31
FittingC31×B(4,𝔽2)C2
socleC2×C31B(4,𝔽2)
hypercenterB(4,𝔽2)D31
hypocenterC31#2200
LCSG ▷ C2×D4×C31 ▷ C22×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ B(4,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #14521788C2semidirect
2. #14521239C2semidirect
3. #14521217C2semidirect
4. #14520841C2semidirect
5. #14520838C2semidirect
6. #14520836C2semidirect
7. C31×B(4,𝔽2)C2semidirect
40. C22×Dic31D4semidirect
41. C2×D124D4semidirect
52. C22⋊Dic31D4semidirect
60. C4×Dic31D4semidirect
123. B(4,𝔽2)D31semidirect
154. C31#2200semidirect

Maximal subgroups

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

subgroupindex
#145217882
#145212392
#145212172
#145208412
#145208382
#145208362
C31×B(4,𝔽2)2
#220031