Finite Group Databasebeta

← #16854942

Subgroups — #16854942

Order 3968.

738 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 143 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×Q8×D31C211yesmaximal, characteristic, complemented
2C2×#11758492C212yesmaximal, complemented
3#14521559C211yesmaximal, characteristic
4#14521207C211yesmaximal, characteristic, complemented
5#14521206C212yesmaximal, complemented
6#14521205C211yesmaximal, characteristic, complemented
7#14521187C211yesmaximal, characteristic, complemented
8#14521186C212yesmaximal, complemented
9#14521056C212yesmaximal, complemented
10#14521011C211yesmaximal, characteristic, complemented
11C31×#448C211yesmaximal, characteristic, complemented
12C22×Q8×C31C2211yescharacteristic
13C2×Q8×D31—28no—
14C2×Q8×D31—21no—
15C2×Q8×D31—21no—
16C2×Q8×D31C2212yes—
17C2×Q8×D31C2212yes—
18C2×#92497C2211yescharacteristic
19C22×C4×D31—21no—
20C22×C4×D31C2211yescharacteristic
21C22×Dic62—21no—
22C22×Dic62C2211yescharacteristic
23C2×SD16×C31C2212yescomplemented
24C31×#121C2211yescharacteristic
25C2×#92482C2212yescomplemented
26#11758515C2211yescharacteristic
27#11758514—21no—
28D4×Dic31—21no—
29C2×#92480C2212yescomplemented
30#11758501C2211yescharacteristic
31#11758492—28no—
32C2×Dic124C2212yescomplemented
33C2×#92469C2212yes—
34#11758474C2211yescharacteristic
35#11758469C2211yescharacteristic
36#11758460—21no—
37(C22⋊D31)⋊C4—21no—
38C31×#103C2212yescomplemented
39C31×C22⋊C8C2211yescharacteristic
40#11758423C2212yescomplemented
41C22⋊(C31⋊C8)C2211yescharacteristic
42(C2×D31)⋊C8C2212yes—
43#11758404C2212yescomplemented
44#11758398C2212yescomplemented
45C23×D31D411yescharacteristic
46C2×Q8×C31—24no—
47C2×Q8×C31—21no—
48C2×Q8×C31C2312yes—
49C2×D4×C31—21no—
50C2×D4×C31C2311yescharacteristic
51C22×C4×C31—21no—
52C22×C4×C31C2311yescharacteristic
53C2×C22⋊D31—21no—
54C2×C22⋊D31—21no—
55C22×Dic31—21no—
56C22×Dic31—21no—
57C22×Dic31D411yescharacteristic
58Q8×D31—28no—
59Q8×D31—28no—
60Q8×D31—24no—
61Q8×D31—24no—
62Q8×D31—24no—
63Q8×D31D418yescomplemented
64#92497—24no—
65C2×C4×D31—28no—
66C2×C4×D31—22no—
67C2×C4×D31—22no—
68C2×C4×D31—24no—
69C2×C4×D31D411yescharacteristic
70C2×C4×D31D411yescharacteristic
71C2×C4×D31C2311yescharacteristic
72C2×C4×D31C2311yescharacteristic
73C2×Dic62—28no—
74C2×Dic62—21no—
75C2×Dic62—22no—
76C2×Dic62—22no—
77C2×Dic62—24no—
78C2×Dic62C2311yescharacteristic
79C2×Dic62C2312yes—
80SD16×C31—24no—
81C2×C8×C31C2312yes—
82C4⋊C4×C31C2311yescharacteristic
83C22⋊C4×C31—21no—
84C22⋊Dic31—21no—
85C22⋊Dic31—21no—
86#92482—24no—
87#92480—24no—
88(C2×D31)⋊C4—21no—
89(C2×D31)⋊C4—21no—
90C4⋊Dic31—21no—
91C4⋊Dic31C2311yescharacteristic
92Dic31⋊C4—21no—
93C4×Dic31—21no—
94C2×C31⋊C8C2312yes—
95Dic124—24no—
96#92469—24no—
97C23×C31—21no—
98C23×C31C2×D411yescharacteristic
99C22×D31—22no—
100C22×D31—22no—
101C22×D31—22no—
102C22×D31C2×D411yescharacteristic
103C22×D31C2×D411yescharacteristic
104Q8×C31—24no—
105Q8×C31—22no—
106Q8×C31C2×D414yes—
107D4×C31—22no—
108D4×C31—42no—
109C2×C4×C31—24no—
110C2×C4×C31—22no—
111C2×C4×C31—22no—
112C2×C4×C31—21no—
113C2×C4×C31C2×D411yescharacteristic
114C2×C4×C31C2411yescharacteristic
115C22⋊D31—42no—
116C22⋊D31—42no—
117C2×Dic31—24no—
118C2×Dic31—22no—
119C2×Dic31—22no—
120C2×Dic31—21no—
121C2×Dic31—21no—
122C2×Dic31—42no—
123C2×Dic31C2×D411yescharacteristic
124C2×Dic31C2×D411yescharacteristic
125C4×D31—28no—
126C4×D31—28no—
127C4×D31—22no—
128C4×D31—22no—
129C4×D31—22no—
130C4×D31C2×D414yes—
131Dic62—28no—
132Dic62—24no—
133Dic62—22no—
134Dic62—28no—
135Dic62—22no—
136Dic62C2×D414yes—
137C8×C31—22no—
138C31⋊C8—22no—
139#3004—311nomaximal, Sylow
140C22×C31—21no—
141C22×C31—21no—
142C22×C31—41no—
143C22×C31—41no—
144C22×C31—41no—
145C22×C31#13712yescomplemented
146C22×C31C22×D411yescharacteristic
147C2×D31—28no—
148C2×D31—21no—
149C2×D31—21no—
150C2×D31—21no—
151C2×D31—21no—
152C2×D31#12012yes—
153C2×D31#13712yescomplemented
154C4×C31—24no—
155C4×C31—21no—
156C4×C31—41no—
157C4×C31C22×D412yes—
158Dic31—21no—
159Dic31—24no—
160Dic31—41no—
161Dic31—41no—
162Dic31#12012yes—
163C23×Q8—311no—
164C2×#137—312no—
165#525—311no—
166#451—311no—
167#451—312no—
168#451—311no—
169#448—312no—
170#448—311no—
171#448D3111yescharacteristic, complemented
172#419—312no—
173#407—311no—
174C2×C31—22no—
175C2×C31—81no—
176C2×C31C2×#12011yescharacteristic
177C2×C31C2×#13711yescharacteristic
178C2×C31C2×#13711yescharacteristic
179D31—22no—
180D31—22no—
181C2×#41—311no—
182C22×Q8—312no—
183C22×Q8—312no—
184C22×Q8—311no—
185C22×Q8—621no—
186C22×Q8—621no—
187C22×Q8—621no—
188C22×Q8—628no—
189C22×Q8C2×D3111yescharacteristic
190C23×C4—311no—
191C23×C4—621no—
192#137—628no—
193C2×Q16—312no—
194C2×Q16—312no—
195C2×SD16—312no—
196C2×SD16C2×D3112yescomplemented
197C2×M16—312no—
198#123—621no—
199C22⋊Q8—311no—
200C22⋊Q8—311no—
201#121—311no—
202#121C2×D3111yescharacteristic
203#120—621no—
204C4×D4—621no—
205C4×D4—621no—
206#117—311no—
207#103—312no—
208#103—312no—
209#103—312no—
210#103C2×D3112yescomplemented
211C22⋊C8—311no—
212C22⋊C8—312no—
213C22⋊C8C2×D3111yescharacteristic
214C31#300411yescharacteristic, complemented, Sylow
215C24—311no—
216#41—624no—
217C2×Q8—24no—
218C2×Q8—21no—
219C2×Q8—318no—
220C2×Q8—312no—
221C2×Q8—311no—
222C2×Q8—622no—
223C2×Q8—622no—
224C2×Q8—624no—
225C2×Q8—628no—
226C2×Q8—624no—
227C2×Q8—624no—
228C2×Q8—628no—
229C2×Q8—628no—
230C2×Q8—624no—
231C2×Q8—621no—
232C2×Q8C22×D3112yes—
233C2×D4—21no—
234C2×D4—621no—
235C2×D4—621no—
236C2×D4C22×D3111yescharacteristic
237C22×C4—21no—
238C22×C4—311no—
239C22×C4—311no—
240C22×C4—311no—
241C22×C4—311no—
242C22×C4—311no—
243C22×C4—624no—
244C22×C4—622no—
245C22×C4—622no—
246C22×C4—628no—
247C22×C4—621no—
248C22×C4—621no—
249C22×C4C22×D3111yescharacteristic
250Q16—624no—
251Q16—624no—
252SD16—24no—
253SD16—624no—
254M16—624no—
255C2×C8—312no—
256C2×C8C22×D3112yes—
257C4⋊C4—311no—
258C4⋊C4—621no—
259C4⋊C4—621no—
260C4⋊C4C22×D3111yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—621no—
263C22⋊C4—621no—
264C22⋊C4—621no—
265C22⋊C4—621no—
266C42—621no—
267C23—21no—
268C23—311no—
269C23—311no—
270C23—622no—
271C23—622no—
272C23—622no—
273C23D4×D3111yescharacteristic
274Q8—24no—
275Q8—22no—
276Q8—314no—
277Q8—628no—
278Q8—628no—
279Q8—624no—
280Q8—622no—
281Q8—622no—
282Q8D4×D3114yes—
283D4—22no—
284D4—42no—
285D4—1242no—
286D4—1242no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—311no—
292C2×C4—314no—
293C2×C4—311no—
294C2×C4—622no—
295C2×C4—622no—
296C2×C4—628no—
297C2×C4—628no—
298C2×C4—622no—
299C2×C4—624no—
300C2×C4—622no—
301C2×C4—622no—
302C2×C4—621no—
303C2×C4—621no—
304C2×C4—1242no—
305C2×C4D4×D3111yescharacteristic
306C2×C4C23×D3111yescharacteristic
307C8—22no—
308C8—622no—
309C22—21no—
310C22—21no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—312no—
315C22—312no—
316C22—628no—
317C22—621no—
318C22—621no—
319C22—621no—
320C22—621no—
321C22#1175849212yescomplemented
322C22C2×D4×D3111yescharacteristic
323C4—24no—
324C4—21no—
325C4—41no—
326C4—312no—
327C4—621no—
328C4—624no—
329C4—1241no—
330C4—1241no—
331C4C2×D4×D3112yes—
332C2—22no—
333C2—81no—
334C2—622no—
335C2—622no—
336C2#120×D3111yescharacteristic
337C2#137×D3111yescharacteristic
338C2C2×#1175849211yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#448C2
socleC22×C31C22×D4
prime radical#14521559C2
hypercenter#448D31
hypocenterC31#3004
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D31C2semidirect
2. C2×#11758492C2semidirect
4. #14521207C2semidirect
5. #14521206C2semidirect
6. #14521205C2semidirect
7. #14521187C2semidirect
8. #14521186C2semidirect
9. #14521056C2semidirect
10. #14521011C2semidirect
11. C31×#448C2semidirect
23. C2×SD16×C31C22semidirect
25. C2×#92482C22semidirect
29. C2×#92480C22semidirect
32. C2×Dic124C22semidirect
38. C31×#103C22semidirect
40. #11758423C22semidirect
43. #11758404C22semidirect
44. #11758398C22semidirect
63. Q8×D31D4semidirect
145. C22×C31#137semidirect
153. C2×D31#137semidirect
171. #448D31semidirect
196. C2×SD16C2×D31semidirect
210. #103C2×D31semidirect
214. C31#3004semidirect
321. C22#11758492semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D312
C2×#117584922
#145215592
#145212072
#145212062
#145212052
#145211872
#145211862
#145210562
#145210112
C31×#4482
#300431