Finite Group Databasebeta

← #16856202

Subgroups — #16856202

Order 3968.

532 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 149 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×#11758519C211yesmaximal, characteristic, complemented
2C2×#11758528C211yesmaximal, characteristic, complemented
3C2×#11758527C211yesmaximal, characteristic
4C22×#92482C211yesmaximal, characteristic, complemented
5#14521625C211yesmaximal, characteristic, complemented
6#14521444C212yesmaximal, complemented
7#14521441C212yesmaximal, complemented
8#14521431C212yesmaximal, complemented
9#14521428C212yesmaximal, complemented
10#14521005C211yesmaximal, characteristic, complemented
11C31×#536C211yesmaximal, characteristic, complemented
12C2×#41×C31C2211yescharacteristic
13C22×Q8×C31C2211yescharacteristic
14C2×#92495C2211yescharacteristic
15C22×Dic62C2211yescharacteristic
16C31×C4⋊Q8C2212yes—
17C31×#124C2212yes—
18C31×#123—22no—
19C31×C22⋊Q8—22no—
20C31×#117C2211yescharacteristic
21#11758528—24no—
22#11758527—24no—
23C2×#92482—24no—
24C2×#92482C2211yescharacteristic
25C2×#92482C2212yes—
26C2×#92482C2212yes—
27C2×#92482C2212yes—
28C2×#92482C2212yes—
29#11758519—24no—
30C2×#92481C2212yes—
31C2×#92481C2211yescharacteristic
32C2×#92480C2212yes—
33C2×#92480C2211yescharacteristic
34C2×#92479C2211yescharacteristic
35C2×#92474C2211yescharacteristic
36C22×C31⋊C8C2211yescharacteristic
37#11758474—22no—
38#11758463—22no—
39#11758452C2212yes—
40#11758446C2212yes—
41#11758391C2212yes—
42C4×C31⋊C8C2212yes—
43#41×C31—22no—
44#41×C31—22no—
45C2×Q8×C31—21no—
46C2×Q8×C31—22no—
47C2×Q8×C31—22no—
48C2×Q8×C31C2312yes—
49C2×Q8×C31C2311yescharacteristic
50C2×D4×C31—21no—
51C2×D4×C31C2311yescharacteristic
52C22×C4×C31—21no—
53C22×C4×C31—21no—
54C22×C4×C31C2311yescharacteristic
55C2×C22⋊D31—21no—
56C22×Dic31—21no—
57#92495—22no—
58#92495—22no—
59C2×D124C2311yescharacteristic
60C2×C4×D31—21no—
61C2×Dic62—22no—
62C2×Dic62—22no—
63C2×Dic62—21no—
64C2×Dic62C2311yescharacteristic
65C2×Dic62C2312yes—
66C4⋊C4×C31—24no—
67C4⋊C4×C31D412yes—
68C22⋊C4×C31—24no—
69C22⋊C4×C31D412yes—
70C42×C31C2312yes—
71#92482—24no—
72#92482—24no—
73#92482—24no—
74#92482—22no—
75#92482—24no—
76#92481—24no—
77#92481—22no—
78#92480—22no—
79#92480—24no—
80#92479—22no—
81(C2×D31)⋊C4—24no—
82C4⋊Dic31—24no—
83#92474D414yes—
84C2×C31⋊C8D414yes—
85C2×C31⋊C8C2311yescharacteristic
86C2×C31⋊C8C2312yes—
87C2×C31⋊C8C2311yescharacteristic
88C23×C31—21no—
89C23×C31C2×D411yescharacteristic
90C22×D31—21no—
91Q8×C31—22no—
92Q8×C31—21no—
93Q8×C31—22no—
94Q8×C31—21no—
95Q8×C31—42no—
96D4×C31—21no—
97D4×C31—21no—
98D4×C31—42no—
99C2×C4×C31—22no—
100C2×C4×C31—21no—
101C2×C4×C31—42no—
102C2×C4×C31—42no—
103C2×C4×C31C2×D412yes—
104C2×C4×C31C2×D412yes—
105C2×C4×C31C2×D414yes—
106C2×C4×C31C2×D411yescharacteristic
107C2×C4×C31C2411yescharacteristic
108C22⋊D31—42no—
109C2×Dic31—21no—
110C2×Dic31—22no—
111C2×Dic31—42no—
112D124—21no—
113D124—21no—
114C4×D31—42no—
115Dic62—22no—
116Dic62—21no—
117Dic62—22no—
118Dic62—21no—
119Dic62—42no—
120C31⋊C8C2×D412yes—
121C31⋊C8C2×D412yes—
122C31⋊C8C2×D414yes—
123#3156—311nomaximal, Sylow
124C22×C31—22no—
125C22×C31—41no—
126C22×C31—42no—
127C22×C31#12712yes—
128C22×C31C22×D411yescharacteristic
129C2×D31—41no—
130C2×D31—42no—
131C4×C31—24no—
132C4×C31—41no—
133C4×C31—42no—
134C4×C31#12712yes—
135C4×C31C22×D411yescharacteristic
136C4×C31C22×D411yescharacteristic
137Dic31—42no—
138Dic31—41no—
139C2×#137—311no—
140C2×#137—311no—
141C2×#135—311no—
142C22×Q16—311no—
143#536—311no—
144#536D3111yescharacteristic, complemented
145#497—312no—
146#497—312no—
147#495—312no—
148#494—312no—
149#405—311no—
150C2×C31—22no—
151C2×C31—81no—
152C2×C31C2×#12711yescharacteristic
153C2×C31#57812yes—
154D31—81no—
155C2×#41—311no—
156C2×#41C2×D3111yescharacteristic
157C22×Q8—311no—
158C22×Q8C2×D3111yescharacteristic
159#137—624no—
160#137—624no—
161#135—624no—
162C2×Q16—312no—
163C2×Q16—312no—
164C2×Q16—311no—
165C2×Q16—312no—
166C2×Q16—312no—
167C2×Q16—624no—
168C2×SD16—311no—
169C2×SD16—312no—
170C2×SD16—311no—
171C2×SD16—312no—
172C2×D8—311no—
173C2×M16—311no—
174C22×C8—311no—
175C4⋊Q8—312no—
176C4⋊Q8C2×D3112yes—
177#124—312no—
178#124C2×D3112yes—
179#123—22no—
180#123—622no—
181C22⋊Q8—22no—
182C22⋊Q8—622no—
183#117C2×D3111yescharacteristic
184#97—312no—
185C4×C8—312no—
186C31#315611yescharacteristic, complemented, Sylow
187#41—22no—
188#41—22no—
189#41—622no—
190#41—622no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—312no—
195C2×Q8—311no—
196C2×Q8—621no—
197C2×Q8—622no—
198C2×Q8—622no—
199C2×Q8C22×D3112yes—
200C2×Q8C22×D3111yescharacteristic
201C2×D4—21no—
202C2×D4—311no—
203C2×D4—621no—
204C2×D4C22×D3111yescharacteristic
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—621no—
208C22×C4—621no—
209C22×C4C22×D3111yescharacteristic
210Q16—624no—
211Q16—624no—
212Q16—622no—
213Q16—624no—
214Q16—624no—
215SD16—622no—
216SD16—624no—
217SD16—622no—
218SD16—624no—
219D8—622no—
220M16—314no—
221C2×C8—314no—
222C2×C8—311no—
223C2×C8—312no—
224C2×C8—311no—
225C4⋊C4—24no—
226C4⋊C4—624no—
227C4⋊C4C22⋊D3112yes—
228C22⋊C4—24no—
229C22⋊C4—624no—
230C22⋊C4C22⋊D3112yes—
231C42C22×D3112yes—
232C23—21no—
233C23—621no—
234C23C2×C22⋊D3111yescharacteristic
235Q8—22no—
236Q8—21no—
237Q8—22no—
238Q8—21no—
239Q8—42no—
240Q8—622no—
241Q8—621no—
242Q8—622no—
243Q8—621no—
244Q8—1242no—
245D4—21no—
246D4—21no—
247D4—42no—
248D4—621no—
249D4—621no—
250D4—1242no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—622no—
256C2×C4—621no—
257C2×C4—1242no—
258C2×C4—1242no—
259C2×C4D4×D3112yes—
260C2×C4D4×D3112yes—
261C2×C4C2×C22⋊D3114yes—
262C2×C4C2×C22⋊D3111yescharacteristic
263C2×C4C23×D3111yescharacteristic
264C8—312no—
265C8—312no—
266C8—314no—
267C22—22no—
268C22—42no—
269C22—41no—
270C22—1242no—
271C22—1241no—
272C22#1175851712yes—
273C22C22×C22⋊D3111yescharacteristic
274C4—24no—
275C4—41no—
276C4—42no—
277C4—1241no—
278C4—1242no—
279C4#1175851712yes—
280C4C2×D4×D3111yescharacteristic
281C4C2×D4×D3111yescharacteristic
282C2—22no—
283C2—81no—
284C2—2481no—
285C2C2×#1175851711yescharacteristic
286C2#1452186412yes—
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#536C2
socleC22×C31C22×D4
prime radicalC2×#11758527C2
hypercenter#536D31
hypocenterC31#3156
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#11758519C2semidirect
2. C2×#11758528C2semidirect
4. C22×#92482C2semidirect
5. #14521625C2semidirect
6. #14521444C2semidirect
7. #14521441C2semidirect
8. #14521431C2semidirect
9. #14521428C2semidirect
10. #14521005C2semidirect
11. C31×#536C2semidirect
144. #536D31semidirect
186. C31#3156semidirect

Maximal subgroups

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

subgroupindex
C2×#117585192
C2×#117585282
C2×#117585272
C22×#924822
#145216252
#145214442
#145214412
#145214312
#145214282
#145210052
C31×#5362
#315631