Finite Group Databasebeta

← #16854961

Subgroups — #16854961

Order 3968.

806 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#92495C211yesmaximal, characteristic, complemented
2C2×#11758509C212yesmaximal, complemented
3#14521192C214yesmaximal, complemented
4#14521187C212yesmaximal, complemented
5#14521182C212yesmaximal, complemented
6#14521048C212yesmaximal, complemented
7#14521013C211yesmaximal, characteristic, complemented
8C2×C31×#121C211yesmaximal, characteristic, complemented
9C22×D4×C31—21no—
10C22×D4×C31C2211yescharacteristic, complemented
11C23×C4×C31C2211yescharacteristic
12C22×C22⋊D31—21no—
13C2×#92495—21no—
14C2×#92495—21no—
15C2×#92495—28no—
16C2×#92495C2212yes—
17C2×#92495C2212yescomplemented
18C22×D124C2211yescharacteristic, complemented
19C22×C4×D31—21no—
20C22×Dic62C2211yescharacteristic, complemented
21C31×#121—21no—
22C31×#121—21no—
23C31×#121C2212yes—
24C31×#121C2212yescomplemented
25C2×C4⋊C4×C31C2211yescharacteristic, complemented
26C2×C22⋊C4×C31—21no—
27C2×#92480C2214yescomplemented
28#11758509—28no—
29C2×#92479C2214yescomplemented
30C2×#92474C2212yescomplemented
31C22⋊(C31⋊C8)C2212yescomplemented
32C22⋊(C31⋊C8)C2212yescomplemented
33#11758398C2214yescomplemented
34#11758397C2214yescomplemented
35C24×C31—21no—
36C24×C31D411yescharacteristic
37C23×D31—21no—
38C2×D4×C31—22no—
39C2×D4×C31—22no—
40C2×D4×C31—24no—
41C2×D4×C31—21no—
42C2×D4×C31—44no—
43C2×D4×C31C2312yes—
44C22×C4×C31—24no—
45C22×C4×C31—21no—
46C22×C4×C31D411yescharacteristic
47C22×C4×C31D411yescharacteristic
48C22×C4×C31D411yescharacteristic
49C22×C4×C31C2311yescharacteristic
50C22×C4×C31C2311yescharacteristic
51C22×C4×C31C2311yescharacteristic
52C2×C22⋊D31—28no—
53C2×C22⋊D31—22no—
54C2×C22⋊D31—22no—
55C22×Dic31—21no—
56#92495—28no—
57#92495—28no—
58#92495—24no—
59#92495—24no—
60#92495—24no—
61#92495D418yescomplemented
62C2×D124—21no—
63C2×D124—24no—
64C2×D124C2312yes—
65C2×C4×D31—28no—
66C2×C4×D31—22no—
67C2×C4×D31—22no—
68C2×Dic62—21no—
69C2×Dic62—24no—
70C2×Dic62C2312yes—
71C4⋊C4×C31—21no—
72C4⋊C4×C31C2312yes—
73C22⋊C4×C31—22no—
74C22⋊C4×C31—22no—
75#92480—28no—
76#92479—28no—
77#92474—24no—
78C2×C31⋊C8C2314yes—
79C23×C31—22no—
80C23×C31—22no—
81C23×C31—22no—
82C23×C31—41no—
83C23×C31—42no—
84C23×C31—42no—
85C23×C31—42no—
86C23×C31C2×D411yescharacteristic
87C23×C31C2×D411yescharacteristic
88C23×C31C2×D411yescharacteristic
89C22×D31—24no—
90C22×D31—22no—
91C22×D31—42no—
92C22×D31—42no—
93D4×C31—24no—
94D4×C31—44no—
95D4×C31—44no—
96D4×C31—42no—
97C2×C4×C31—22no—
98C2×C4×C31—22no—
99C2×C4×C31—22no—
100C2×C4×C31—22no—
101C2×C4×C31—22no—
102C2×C4×C31—22no—
103C2×C4×C31—42no—
104C2×C4×C31C2×D414yes—
105C2×C4×C31C2×D411yescharacteristic
106C2×C4×C31C2×D411yescharacteristic
107C2×C4×C31C2×D411yescharacteristic
108C2×C4×C31C2411yescharacteristic
109C22⋊D31—28no—
110C22⋊D31—28no—
111C2×Dic31—24no—
112C2×Dic31—22no—
113D124—24no—
114D124—22no—
115D124C2×D414yescomplemented
116C4×D31—28no—
117C4×D31—28no—
118Dic62—24no—
119Dic62—22no—
120Dic62C2×D414yescomplemented
121C31⋊C8—24no—
122#3006—311nomaximal, Sylow
123C22×C31—21no—
124C22×C31—21no—
125C22×C31—21no—
126C22×C31—21no—
127C22×C31—21no—
128C22×C31—21no—
129C22×C31—21no—
130C22×C31—44no—
131C22×C31—42no—
132C22×C31—42no—
133C22×C31—42no—
134C22×C31—84no—
135C22×C31#12012yes—
136C22×C31AGL(1,ℤ/8)12yescomplemented
137C22×C31C22×D411yescharacteristic
138C2×D31—24no—
139C2×D31—48no—
140C2×D31—42no—
141C2×D31—42no—
142C4×C31—21no—
143C4×C31—21no—
144C4×C31—42no—
145C4×C31#12012yes—
146C4×C31C22×D412yes—
147Dic31—24no—
148C22×#41—311no—
149C2×AGL(1,ℤ/8)—312no—
150C2×#121D3111yescharacteristic, complemented
151#449—314no—
152#448—312no—
153#447—312no—
154#417—312no—
155#407—311no—
156C2×C31—22no—
157C2×C31—22no—
158C2×C31—41no—
159C2×C31—82no—
160C2×C31C2×#12011yescharacteristic
161C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
162C2×C31#57511yescharacteristic
163D31—44no—
164C2×#41—312no—
165C2×#41—312no—
166C2×#41—628no—
167C2×#41—621no—
168C2×#41—621no—
169C22×Q8—311no—
170C22×D4—21no—
171C22×D4—311no—
172C22×D4—621no—
173C22×D4C2×D3111yescharacteristic, complemented
174C23×C4—621no—
175C23×C4C2×D3111yescharacteristic
176AGL(1,ℤ/8)—628no—
177C2×SD16—314no—
178C2×D8—314no—
179C2×M16—312no—
180#121—21no—
181#121—21no—
182#121C2×D3112yes—
183#121C2×D3112yescomplemented
184C2×C4⋊C4C2×D3111yescharacteristic, complemented
185C2×C22⋊C4—21no—
186#103—314no—
187#102—314no—
188C22⋊C8—312no—
189C22⋊C8—312no—
190C31#300611yescharacteristic, complemented, Sylow
191C24—21no—
192C24—621no—
193C24C22⋊D3111yescharacteristic
194#41—318no—
195#41—624no—
196#41—624no—
197#41—628no—
198#41—628no—
199#41—624no—
200C2×Q8—312no—
201C2×Q8—624no—
202C2×Q8—621no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—24no—
206C2×D4—21no—
207C2×D4—44no—
208C2×D4—312no—
209C2×D4—624no—
210C2×D4—628no—
211C2×D4—622no—
212C2×D4—622no—
213C2×D4—621no—
214C2×D4C22×D3112yes—
215C22×C4—24no—
216C22×C4—21no—
217C22×C4—628no—
218C22×C4—621no—
219C22×C4—622no—
220C22×C4—622no—
221C22×C4C22⋊D3111yescharacteristic
222C22×C4C22⋊D3111yescharacteristic
223C22×C4C22⋊D3111yescharacteristic
224C22×C4C22×D3111yescharacteristic
225C22×C4C22×D3111yescharacteristic
226C22×C4C22×D3111yescharacteristic
227SD16—628no—
228D8—628no—
229M16—624no—
230C2×C8—314no—
231C4⋊C4—21no—
232C4⋊C4C22×D3112yes—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—41no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—624no—
243C23—622no—
244C23—1242no—
245C23—1242no—
246C23C2×C22⋊D3111yescharacteristic
247C23C2×C22⋊D3111yescharacteristic
248C23C2×C22⋊D3111yescharacteristic
249Q8—314no—
250Q8—624no—
251Q8—622no—
252D4—24no—
253D4—44no—
254D4—44no—
255D4—42no—
256D4—314no—
257D4—624no—
258D4—628no—
259D4—628no—
260D4—622no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—42no—
268C2×C4—628no—
269C2×C4—624no—
270C2×C4—622no—
271C2×C4—628no—
272C2×C4D4×D3114yes—
273C2×C4C2×C22⋊D3111yescharacteristic
274C2×C4C2×C22⋊D3111yescharacteristic
275C2×C4C2×C22⋊D3111yescharacteristic
276C2×C4C23×D3111yescharacteristic
277C8—624no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—44no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—84no—
290C22—624no—
291C22—1248no—
292C22—1242no—
293C22—1242no—
294C22#1175850912yescomplemented
295C22#1175851412yes—
296C22C22×C22⋊D3111yescharacteristic
297C4—21no—
298C4—21no—
299C4—42no—
300C4—624no—
301C4#1175851412yes—
302C4C2×D4×D3112yes—
303C2—22no—
304C2—22no—
305C2—41no—
306C2—82no—
307C2—1244no—
308C2C2×#1175851411yescharacteristic
309C2C2×#1175850911yescharacteristic
310C2#1452184711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×#121C2
socleC22×C31C22×D4
hypercenterC2×#121D31
hypocenterC31#3006
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#92495C2semidirect
2. C2×#11758509C2semidirect
3. #14521192C2semidirect
4. #14521187C2semidirect
5. #14521182C2semidirect
6. #14521048C2semidirect
7. #14521013C2semidirect
8. C2×C31×#121C2semidirect
10. C22×D4×C31C22semidirect
17. C2×#92495C22semidirect
18. C22×D124C22semidirect
20. C22×Dic62C22semidirect
24. C31×#121C22semidirect
25. C2×C4⋊C4×C31C22semidirect
27. C2×#92480C22semidirect
29. C2×#92479C22semidirect
30. C2×#92474C22semidirect
31. C22⋊(C31⋊C8)C22semidirect
32. C22⋊(C31⋊C8)C22semidirect
33. #11758398C22semidirect
34. #11758397C22semidirect
61. #92495D4semidirect
115. D124C2×D4semidirect
120. Dic62C2×D4semidirect
136. C22×C31AGL(1,ℤ/8)semidirect
150. C2×#121D31semidirect
173. C22×D4C2×D31semidirect
183. #121C2×D31semidirect
184. C2×C4⋊C4C2×D31semidirect
190. C31#3006semidirect
294. C22#11758509semidirect

Maximal subgroups

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

subgroupindex
C22×#924952
C2×#117585092
#145211922
#145211872
#145211822
#145210482
#145210132
C2×C31×#1212
#300631