Finite Group Databasebeta

← #16854988

Subgroups — #16854988

Order 3968.

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

normal subgroups only

This group has 752 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×#92480C211yesmaximal, characteristic, complemented
3#14521620C211yesmaximal, characteristic, complemented
4#14521211C214yesmaximal, complemented
5#14521200C214yesmaximal, complemented
6#14521054C211yesmaximal, characteristic, complemented
7#14521045C211yesmaximal, characteristic, complemented
8#14521016C211yesmaximal, characteristic, complemented
9C2×C31×H(5,𝔽2)C211yesmaximal, characteristic
10C31×H(5,𝔽2)—24no—
11C31×H(5,𝔽2)—24no—
12C2×#41×C31—21no—
13C2×#41×C31C2212yes—
14C2×#41×C31C2211yescharacteristic
15C2×#41×C31C2211yescharacteristic
16C22×D4×C31—22no—
17C22×D4×C31—21no—
18C22×D4×C31C2212yes—
19C22×D4×C31C2211yescharacteristic
20C22×Dic62C2211yescharacteristic, complemented
21#11758528—24no—
22#11758521—21no—
23C2×#92482C2212yescomplemented
24#11758513—21no—
25#11758512—22no—
26C2×#92480—24no—
27C2×#92480C2212yescomplemented
28C2×#92480C2214yescomplemented
29#11758503C2211yescharacteristic, complemented
30#11758501C2214yescomplemented
31C2×#92474C2211yescharacteristic, complemented
32C22×C31⋊C8C2211yescharacteristic, complemented
33#11758423C2212yescomplemented
34#11758420C2212yescomplemented
35#11758420C2214yescomplemented
36C22⋊(C31⋊C8)C2214yescomplemented
37C24×C31—22no—
38C24×C31—21no—
39#41×C31—28no—
40#41×C31—24no—
41#41×C31—24no—
42#41×C31—22no—
43#41×C31—22no—
44#41×C31—22no—
45#41×C31D414yes—
46C2×Q8×C31D411yescharacteristic
47C2×Q8×C31C2311yescharacteristic
48C2×D4×C31—216no—
49C2×D4×C31—28no—
50C2×D4×C31—22no—
51C2×D4×C31—21no—
52C2×D4×C31—24no—
53C2×D4×C31—24no—
54C2×D4×C31—22no—
55C2×D4×C31—21no—
56C2×D4×C31—22no—
57C2×D4×C31D412yes—
58C2×D4×C31D411yescharacteristic
59C2×D4×C31D414yes—
60C2×D4×C31C2311yescharacteristic
61C2×D4×C31C2312yes—
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—24no—
74#92480—28no—
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—28no—
84C23×C31—22no—
85C23×C31—21no—
86C23×C31—24no—
87C23×C31—82no—
88C23×C31—84no—
89C23×C31C2×D412yes—
90C23×C31C2×D411yescharacteristic
91Q8×C31—22no—
92Q8×C31—21no—
93Q8×C31C2×D412yes—
94D4×C31—28no—
95D4×C31—28no—
96D4×C31—24no—
97D4×C31—24no—
98D4×C31—22no—
99D4×C31—24no—
100D4×C31—22no—
101D4×C31—21no—
102D4×C31C2×D412yes—
103D4×C31C2×D414yes—
104C2×C4×C31—28no—
105C2×C4×C31—21no—
106C2×C4×C31—24no—
107C2×C4×C31—22no—
108C2×C4×C31—22no—
109C2×C4×C31C2×D412yes—
110C2×C4×C31C2×D411yescharacteristic
111C2×C4×C31C2×D412yes—
112C2×C4×C31C2×D412yes—
113C2×C4×C31C2411yescharacteristic
114C2×Dic31—22no—
115C2×Dic31—22no—
116C2×Dic31—42no—
117Dic62—22no—
118Dic62—22no—
119Dic62—42no—
120C31⋊C8—22no—
121C31⋊C8—22no—
122#3009—311nomaximal, Sylow
123C22×C31—22no—
124C22×C31—22no—
125C22×C31—24no—
126C22×C31—22no—
127C22×C31—24no—
128C22×C31—44no—
129C22×C31—42no—
130C22×C31—84no—
131C22×C31—88no—
132C22×C31#12012yes—
133C22×C31C22×D411yescharacteristic
134C4×C31—22no—
135C4×C31—22no—
136C4×C31#12012yes—
137C4×C31C22×D411yescharacteristic
138C4×C31C22×D411yescharacteristic
139Dic31—42no—
140Dic31—42no—
141C2×H(5,𝔽2)D3111yescharacteristic
142C2×#137—311no—
143C22×SD16—311no—
144#536—311no—
145#452—314no—
146#450—314no—
147#418—311no—
148#416—311no—
149#408—311no—
150C2×C31—22no—
151C2×C31—42no—
152C2×C31—44no—
153C2×C31—42no—
154C2×C31C2×#12011yescharacteristic
155C2×C31#57712yes—
156H(5,𝔽2)—24no—
157H(5,𝔽2)—24no—
158C2×#41—21no—
159C2×#41C2×D3112yes—
160C2×#41C2×D3111yescharacteristic
161C2×#41C2×D3111yescharacteristic
162C22×Q8—311no—
163C22×D4—22no—
164C22×D4—21no—
165C22×D4C2×D3112yes—
166C22×D4C2×D3111yescharacteristic
167#137—624no—
168C2×Q16—312no—
169C2×SD16—314no—
170C2×SD16—312no—
171C2×SD16—624no—
172C2×M16—311no—
173C22×C8—311no—
174C4⋊Q8—621no—
175#124—621no—
176#123—622no—
177C22⋊Q8—314no—
178#117—311no—
179#103—312no—
180#102—312no—
181#102—314no—
182C22⋊C8—314no—
183C31#300911yescharacteristic, complemented, Sylow
184C24—22no—
185C24—21no—
186#41—28no—
187#41—24no—
188#41—24no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41C22⋊D3114yes—
193C2×Q8—312no—
194C2×Q8—621no—
195C2×Q8—622no—
196C2×Q8—622no—
197C2×Q8C22⋊D3111yescharacteristic
198C2×Q8C22×D3111yescharacteristic
199C2×D4—216no—
200C2×D4—28no—
201C2×D4—22no—
202C2×D4—21no—
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4C22⋊D3112yes—
209C2×D4C22⋊D3111yescharacteristic
210C2×D4C22⋊D3114yes—
211C2×D4C22×D3111yescharacteristic
212C2×D4C22×D3112yes—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—621no—
216C22×C4C22×D3112yes—
217C22×C4C22×D3111yescharacteristic
218Q16—624no—
219SD16—628no—
220SD16—624no—
221M16—622no—
222C2×C8—312no—
223C2×C8—312no—
224C2×C8—622no—
225C4⋊C4—312no—
226C4⋊C4—624no—
227C22⋊C4—621no—
228C22⋊C4—624no—
229C42—621no—
230C23—28no—
231C23—22no—
232C23—21no—
233C23—24no—
234C23—82no—
235C23—84no—
236C23C2×C22⋊D3112yes—
237C23C2×C22⋊D3111yescharacteristic
238Q8—22no—
239Q8—21no—
240Q8—622no—
241Q8—622no—
242Q8—1242no—
243Q8C2×C22⋊D3112yes—
244D4—28no—
245D4—28no—
246D4—24no—
247D4—24no—
248D4—22no—
249D4—24no—
250D4—22no—
251D4—21no—
252D4C2×C22⋊D3112yes—
253D4C2×C22⋊D3114yes—
254C2×C4—28no—
255C2×C4—21no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—622no—
260C2×C4—622no—
261C2×C4—1242no—
262C2×C4C2×C22⋊D3112yes—
263C2×C4C2×C22⋊D3111yescharacteristic
264C2×C4C2×C22⋊D3112yes—
265C2×C4C2×C22⋊D3112yes—
266C2×C4C23×D3111yescharacteristic
267C8—622no—
268C8—622no—
269C22—22no—
270C22—24no—
271C22—22no—
272C22—24no—
273C22—22no—
274C22—44no—
275C22—42no—
276C22—84no—
277C22—88no—
278C22#1175853012yes—
279C22C22×C22⋊D3111yescharacteristic
280C4—22no—
281C4—22no—
282C4—1242no—
283C4—1242no—
284C4#1175853012yes—
285C4C22×C22⋊D3111yescharacteristic
286C4C22×C22⋊D3111yescharacteristic
287C2—22no—
288C2—42no—
289C2—44no—
290C2—42no—
291C2C2×#1175853011yescharacteristic
292C2#1452186012yes—
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
Fitting, prime radicalC2×C31×H(5,𝔽2)C2
socleC22×C31C22×D4
hypercenterC2×H(5,𝔽2)D31
hypocenterC31#3009
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#11758528C2semidirect
2. C22×#92480C2semidirect
3. #14521620C2semidirect
4. #14521211C2semidirect
5. #14521200C2semidirect
6. #14521054C2semidirect
7. #14521045C2semidirect
8. #14521016C2semidirect
20. C22×Dic62C22semidirect
23. C2×#92482C22semidirect
27. C2×#92480C22semidirect
28. C2×#92480C22semidirect
29. #11758503C22semidirect
30. #11758501C22semidirect
31. C2×#92474C22semidirect
32. C22×C31⋊C8C22semidirect
33. #11758423C22semidirect
34. #11758420C22semidirect
35. #11758420C22semidirect
36. C22⋊(C31⋊C8)C22semidirect
183. C31#3009semidirect

Maximal subgroups

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

subgroupindex
C2×#117585282
C22×#924802
#145216202
#145212112
#145212002
#145210542
#145210452
#145210162
C2×C31×H(5,𝔽2)2
#300931