Finite Group Databasebeta

← #16855006

Subgroups — #16855006

Order 3968.

700 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C2×#11758528C211yesmaximal, characteristic, complemented
2C22×#92482C211yesmaximal, characteristic, complemented
3#14521620C211yesmaximal, characteristic, complemented
4#14521211C214yesmaximal, complemented
5#14521205C214yesmaximal, complemented
6#14521058C211yesmaximal, characteristic, complemented
7#14521045C211yesmaximal, characteristic, complemented
8#14521016C211yesmaximal, characteristic, complemented
9C2×C31×2−1+4C211yesmaximal, characteristic
10C31×2−1+4—24no—
11C31×2−1+4—24no—
12C2×#41×C31—22no—
13C2×#41×C31—21no—
14C2×#41×C31C2212yes—
15C2×#41×C31C2211yescharacteristic
16C2×#41×C31C2211yescharacteristic
17C22×Q8×C31—21no—
18C22×Q8×C31C2212yes—
19C22×Q8×C31C2211yescharacteristic
20C22×Dic62C2211yescharacteristic
21#11758528—24no—
22#11758521—21no—
23C2×#92482—24no—
24C2×#92482C2212yes—
25C2×#92482C2214yes—
26#11758513—21no—
27#11758512—22no—
28C2×#92480C2212yes—
29#11758503C2211yescharacteristic
30#11758501C2214yes—
31C2×#92474C2211yescharacteristic
32C22×C31⋊C8C2211yescharacteristic
33#11758423C2212yes—
34#11758423C2214yes—
35#11758420C2212yes—
36C22⋊(C31⋊C8)C2214yes—
37#41×C31—216no—
38#41×C31—28no—
39#41×C31—24no—
40#41×C31—24no—
41#41×C31—22no—
42#41×C31—22no—
43#41×C31—22no—
44#41×C31D414yes—
45C2×Q8×C31—28no—
46C2×Q8×C31—24no—
47C2×Q8×C31—24no—
48C2×Q8×C31—22no—
49C2×Q8×C31—21no—
50C2×Q8×C31—22no—
51C2×Q8×C31D414yes—
52C2×Q8×C31D411yescharacteristic
53C2×Q8×C31C2311yescharacteristic
54C2×Q8×C31C2312yes—
55C2×D4×C31—22no—
56C2×D4×C31—21no—
57C2×D4×C31D412yes—
58C2×D4×C31D411yescharacteristic
59C2×D4×C31C2311yescharacteristic
60C22×C4×C31—22no—
61C22×C4×C31—21no—
62C22×C4×C31—22no—
63C22×C4×C31—21no—
64C22×C4×C31C2312yes—
65C22×C4×C31C2311yescharacteristic
66C22×Dic31—21no—
67C2×Dic62—22no—
68C2×Dic62—22no—
69C2×Dic62—21no—
70C2×Dic62C2312yes—
71C22⋊Dic31—24no—
72C22⋊Dic31—21no—
73#92482—28no—
74#92482—24no—
75#92480—24no—
76C4⋊Dic31C2312yes—
77Dic31⋊C4—24no—
78C4×Dic31—21no—
79#92474—22no—
80C2×C31⋊C8—22no—
81C2×C31⋊C8C2312yes—
82C2×C31⋊C8C2312yes—
83C23×C31—21no—
84C23×C31C2×D412yes—
85C23×C31C2×D411yescharacteristic
86Q8×C31—28no—
87Q8×C31—24no—
88Q8×C31—22no—
89Q8×C31—21no—
90Q8×C31—22no—
91Q8×C31C2×D412yes—
92Q8×C31C2×D414yes—
93D4×C31—28no—
94D4×C31—24no—
95D4×C31—24no—
96D4×C31—22no—
97D4×C31—21no—
98D4×C31C2×D412yes—
99C2×C4×C31—28no—
100C2×C4×C31—28no—
101C2×C4×C31—24no—
102C2×C4×C31—22no—
103C2×C4×C31—21no—
104C2×C4×C31—24no—
105C2×C4×C31—22no—
106C2×C4×C31—22no—
107C2×C4×C31C2×D412yes—
108C2×C4×C31C2×D411yescharacteristic
109C2×C4×C31C2×D412yes—
110C2×C4×C31C2×D412yes—
111C2×C4×C31C2411yescharacteristic
112C2×Dic31—22no—
113C2×Dic31—22no—
114C2×Dic31—42no—
115Dic62—22no—
116Dic62—22no—
117Dic62—42no—
118C31⋊C8—22no—
119C31⋊C8—22no—
120#3011—311nomaximal, Sylow
121C22×C31—22no—
122C22×C31—22no—
123C22×C31—22no—
124C22×C31—24no—
125C22×C31—42no—
126C22×C31#12012yes—
127C22×C31C22×D411yescharacteristic
128C4×C31—24no—
129C4×C31—22no—
130C4×C31—22no—
131C4×C31#12012yes—
132C4×C31C22×D411yescharacteristic
133C4×C31C22×D411yescharacteristic
134Dic31—42no—
135Dic31—42no—
136C2×2−1+4D3111yescharacteristic
137C2×#137—311no—
138C22×Q16—311no—
139#536—311no—
140#452—314no—
141#451—314no—
142#419—311no—
143#416—311no—
144#408—311no—
145C2×C31—22no—
146C2×C31—42no—
147C2×C31—42no—
148C2×C31C2×#12011yescharacteristic
149C2×C31#57812yes—
1502−1+4—24no—
1512−1+4—24no—
152C2×#41—22no—
153C2×#41—21no—
154C2×#41C2×D3112yes—
155C2×#41C2×D3111yescharacteristic
156C2×#41C2×D3111yescharacteristic
157C22×Q8—21no—
158C22×Q8—311no—
159C22×Q8C2×D3112yes—
160C22×Q8C2×D3111yescharacteristic
161#137—624no—
162C2×Q16—314no—
163C2×Q16—312no—
164C2×Q16—624no—
165C2×SD16—312no—
166C2×M16—311no—
167C22×C8—311no—
168C4⋊Q8—621no—
169#124—621no—
170#123—622no—
171C22⋊Q8—314no—
172#117—311no—
173#103—312no—
174#103—314no—
175#102—312no—
176C22⋊C8—314no—
177C31#301111yescharacteristic, complemented, Sylow
178#41—216no—
179#41—28no—
180#41—24no—
181#41—24no—
182#41—22no—
183#41—22no—
184#41—22no—
185#41C22⋊D3114yes—
186C2×Q8—28no—
187C2×Q8—24no—
188C2×Q8—24no—
189C2×Q8—22no—
190C2×Q8—21no—
191C2×Q8—22no—
192C2×Q8—312no—
193C2×Q8—621no—
194C2×Q8—622no—
195C2×Q8—622no—
196C2×Q8C22⋊D3114yes—
197C2×Q8C22⋊D3111yescharacteristic
198C2×Q8C22×D3111yescharacteristic
199C2×Q8C22×D3112yes—
200C2×D4—22no—
201C2×D4—21no—
202C2×D4C22⋊D3112yes—
203C2×D4C22⋊D3111yescharacteristic
204C2×D4C22×D3111yescharacteristic
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—621no—
210C22×C4C22×D3112yes—
211C22×C4C22×D3111yescharacteristic
212Q16—628no—
213Q16—624no—
214SD16—624no—
215M16—622no—
216C2×C8—312no—
217C2×C8—312no—
218C2×C8—622no—
219C4⋊C4—312no—
220C4⋊C4—624no—
221C22⋊C4—621no—
222C22⋊C4—624no—
223C42—621no—
224C23—21no—
225C23C2×C22⋊D3112yes—
226C23C2×C22⋊D3111yescharacteristic
227Q8—28no—
228Q8—24no—
229Q8—22no—
230Q8—21no—
231Q8—22no—
232Q8—622no—
233Q8—622no—
234Q8—1242no—
235Q8C2×C22⋊D3112yes—
236Q8C2×C22⋊D3114yes—
237D4—28no—
238D4—24no—
239D4—24no—
240D4—22no—
241D4—21no—
242D4C2×C22⋊D3112yes—
243C2×C4—28no—
244C2×C4—28no—
245C2×C4—24no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—622no—
252C2×C4—622no—
253C2×C4—1242no—
254C2×C4C2×C22⋊D3112yes—
255C2×C4C2×C22⋊D3111yescharacteristic
256C2×C4C2×C22⋊D3112yes—
257C2×C4C2×C22⋊D3112yes—
258C2×C4C23×D3111yescharacteristic
259C8—622no—
260C8—622no—
261C22—22no—
262C22—24no—
263C22—22no—
264C22—22no—
265C22—42no—
266C22#1175853012yes—
267C22C22×C22⋊D3111yescharacteristic
268C4—24no—
269C4—22no—
270C4—22no—
271C4—1242no—
272C4—1242no—
273C4#1175853012yes—
274C4C22×C22⋊D3111yescharacteristic
275C4C22×C22⋊D3111yescharacteristic
276C2—22no—
277C2—42no—
278C2—42no—
279C2C2×#1175853011yescharacteristic
280C2#1452186412yes—
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
Fitting, prime radicalC2×C31×2−1+4C2
socleC22×C31C22×D4
hypercenterC2×2−1+4D31
hypocenterC31#3011
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×#11758528C2semidirect
2. C22×#92482C2semidirect
3. #14521620C2semidirect
4. #14521211C2semidirect
5. #14521205C2semidirect
6. #14521058C2semidirect
7. #14521045C2semidirect
8. #14521016C2semidirect
177. C31#3011semidirect

Maximal subgroups

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

subgroupindex
C2×#117585282
C22×#924822
#145216202
#145212112
#145212052
#145210582
#145210452
#145210162
C2×C31×2−1+42
#301131