Finite Group Databasebeta

← #16857512

Subgroups — #16857512

Order 3968.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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#14521865C212yesmaximal, complemented
2#14521864C212yesmaximal, complemented
3#14521428C211yesmaximal, characteristic, complemented
4#14521323C212yesmaximal, complemented
5#14521322C212yesmaximal, complemented
6#14521233C212yesmaximal, complemented
7#14521231C211yesmaximal, characteristic, complemented
8#14521163C212yesmaximal, complemented
9C31×#456C211yesmaximal, characteristic, complemented
10C31×2−1+4C2211yescharacteristic
11#11758567C2212yes—
12#11758564C2211yescharacteristic
13C31×#137C2212yes—
14C31×C4⋊Q8C2211yescharacteristic
15#11758528—22no—
16#11758528C2212yes—
17#11758527—22no—
18#11758527C2212yes—
19#11758525C2212yes—
20C2×#92482—22no—
21C2×#92482C2212yes—
22C2×#92482C2212yes—
23#11758519—22no—
24#11758519C2212yes—
25#11758495—22no—
26Q16×D31—22no—
27#11758493—22no—
28#11758492—22no—
29#11758486C2212yes—
30#11758484C2212yes—
31#11758446C2211yescharacteristic
32C31×#104C2212yes—
33C31×#101C2211yescharacteristic
34#11758424C2211yescharacteristic
35#11758412C2212yes—
36#11758410C2212yes—
37#11758393C2212yes—
38C4×C31⋊C8C2211yescharacteristic
39#41×C31—22no—
40#41×C31—41no—
41#41×C31C2312yes—
42C2×Q8×C31—22no—
43C2×Q8×C31C2311yescharacteristic
44C2×Q8×C31C2311yescharacteristic
45#92499—22no—
46Q8×D31—22no—
47Q8×D31—22no—
48#92497—22no—
49#92497—42no—
50#92495—22no—
51#92495—41no—
52#92495C2312yes—
53C2×Dic62—22no—
54C2×Dic62C2312yes—
55Q16×C31—22no—
56SD16×C31—22no—
57M16×C31C2312yes—
58C4⋊C4×C31—41no—
59C42×C31C2311yescharacteristic
60#92482—22no—
61#92482—22no—
62#92482—22no—
63#92482—22no—
64#92482—22no—
65#92482—22no—
66#92482D412yes—
67#92481—22no—
68#92481D412yes—
69#92480—22no—
70#92480D412yes—
71#92479D412yes—
72C4⋊Dic31—41no—
73#92474—22no—
74#92474C2312yes—
75C2×C31⋊C8—22no—
76C2×C31⋊C8C2312yes—
77Dic124—22no—
78#92470—22no—
79#92469—22no—
80C8×D31—22no—
81Q8×C31—22no—
82Q8×C31—22no—
83Q8×C31—41no—
84Q8×C31C2×D412yes—
85D4×C31—22no—
86D4×C31—41no—
87D4×C31C2×D412yes—
88C2×C4×C31—21no—
89C2×C4×C31—22no—
90C2×C4×C31—21no—
91C2×C4×C31—21no—
92C2×C4×C31—42no—
93C2×C4×C31C2411yescharacteristic
94C22⋊D31—22no—
95C22⋊D31—42no—
96C2×Dic31—22no—
97C2×Dic31—42no—
98D124—41no—
99D124C2×D412yes—
100C4×D31—22no—
101C4×D31—42no—
102C4×D31—42no—
103Dic62—22no—
104Dic62—22no—
105Dic62—41no—
106Dic62—42no—
107Dic62C2×D412yes—
108C8×C31—22no—
109C31⋊C8—22no—
110C31⋊C8C2×D412yes—
111C31⋊C8C2×D412yes—
112#3293—311nomaximal, Sylow
113C22×C31—22no—
114C22×C31H(5,𝔽2)11yescharacteristic
115C2×D31—22no—
116C4×C31—22no—
117C4×C31—22no—
118C4×C31—41no—
119C4×C31—41no—
120C4×C31C22×D412yes—
121Dic31—22no—
122Dic31—42no—
123#578—312no—
124#578—312no—
125#494—311no—
126#473—312no—
127#473—312no—
128#456—311no—
129#456—312no—
130#456D3111yescharacteristic, complemented
131#443—312no—
132C2×C31—21no—
133C2×C31—42no—
134C2×C31D4211yescharacteristic
135D31—42no—
1362−1+4—312no—
1372−1+4—311no—
1382−1+4C2×D3111yescharacteristic
139#137—312no—
140#137—312no—
141#137—312no—
142#137—622no—
143#137—622no—
144#137—622no—
145#137—622no—
146#137C2×D3112yes—
147#135—312no—
148#135—622no—
149#135—622no—
150C2×Q16—312no—
151C2×Q16—312no—
152C2×Q16—622no—
153C2×Q16—622no—
154#131—312no—
155#131—312no—
156C4⋊Q8—311no—
157C4⋊Q8C2×D3111yescharacteristic
158#108—312no—
159#104—312no—
160#104C2×D3112yes—
161#101—311no—
162#101—312no—
163#101C2×D3111yescharacteristic
164C4×C8—311no—
165C31#329311yescharacteristic, complemented, Sylow
166#41—22no—
167#41—41no—
168#41—312no—
169#41—622no—
170#41—622no—
171#41—622no—
172#41—1241no—
173#41—1242no—
174#41C22×D3112yes—
175C2×Q8—22no—
176C2×Q8—312no—
177C2×Q8—622no—
178C2×Q8—622no—
179C2×Q8—622no—
180C2×Q8C22×D3111yescharacteristic
181C2×Q8C22×D3111yescharacteristic
182Q16—22no—
183Q16—312no—
184Q16—622no—
185Q16—622no—
186Q16—622no—
187Q16—622no—
188Q16—622no—
189Q16—622no—
190Q16—622no—
191SD16—22no—
192SD16—312no—
193SD16—312no—
194SD16—622no—
195SD16—622no—
196SD16—622no—
197D8—312no—
198M16—312no—
199M16—622no—
200M16—622no—
201M16C22×D3112yes—
202C2×C8—312no—
203C2×C8—622no—
204C2×C8—622no—
205C4⋊C4—41no—
206C4⋊C4—1241no—
207C42C22×D3111yescharacteristic
208Q8—22no—
209Q8—22no—
210Q8—41no—
211Q8—312no—
212Q8—622no—
213Q8—622no—
214Q8—1242no—
215Q8—1241no—
216Q8D4×D3112yes—
217D4—22no—
218D4—41no—
219D4—312no—
220D4—622no—
221D4—1241no—
222D4—1242no—
223D4D4×D3112yes—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—622no—
230C2×C4—622no—
231C2×C4—1242no—
232C2×C4—1242no—
233C2×C4—1242no—
234C2×C4C23×D3111yescharacteristic
235C8—22no—
236C8—312no—
237C8—312no—
238C8—622no—
239C22—22no—
240C22—622no—
241C22#1175856111yescharacteristic
242C4—22no—
243C4—22no—
244C4—41no—
245C4—41no—
246C4—622no—
247C4—1242no—
248C4C2×D4×D3112yes—
249C2—21no—
250C2—42no—
251C2—1242no—
252C2#1452168111yescharacteristic
subgroupG/H
centerC2#14521681
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#456C2
socleC2×C31D42
hypercenter#456D31
hypocenterC31#3293
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #14521865C2semidirect
2. #14521864C2semidirect
3. #14521428C2semidirect
4. #14521323C2semidirect
5. #14521322C2semidirect
6. #14521233C2semidirect
7. #14521231C2semidirect
8. #14521163C2semidirect
9. C31×#456C2semidirect
130. #456D31semidirect
165. C31#3293semidirect

Maximal subgroups

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

subgroupindex
#145218652
#145218642
#145214282
#145213232
#145213222
#145212332
#145212312
#145211632
C31×#4562
#329331