Finite Group Databasebeta

← #16856109

Subgroups — #16856109

Order 3968.

392 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#14521595C211yesmaximal, characteristic
2C2×#11758467C211yesmaximal, characteristic, complemented
3#14521416C212yesmaximal, complemented
4#14521412C212yesmaximal, complemented
5#14521405C212yesmaximal, complemented
6#14521402C212yesmaximal, complemented
7#14521051C211yesmaximal, characteristic, complemented
8#14521050C211yesmaximal, characteristic, complemented
9#14521048C211yesmaximal, characteristic, complemented
10M16×Dic31C211yesmaximal, characteristic, complemented
11C31×#417C211yesmaximal, characteristic, complemented
12C2×#41×C31C2211yescharacteristic
13C2×#92495C2211yescharacteristic
14C2×M16×C31C2211yescharacteristic
15C2×C4⋊C4×C31C2211yescharacteristic
16#11758524C2211yescharacteristic
17#11758521C2211yescharacteristic
18#11758517C2211yescharacteristic
19#11758516—21no—
20#11758516—21no—
21#11758513C2211yescharacteristic
22C4×C22⋊D31—21no—
23C4×C22⋊D31—21no—
24C2×C4⋊Dic31—21no—
25C2×C4⋊Dic31C2211yescharacteristic
26C2×Dic31⋊C4—21no—
27C2×C4×Dic31C2211yescharacteristic
28C2×#92474C2211yescharacteristic
29#11758467—21no—
30#11758467—21no—
31#11758467C2212yes—
32#11758467C2212yes—
33C31×#103C2212yes—
34C31×#102C2212yes—
35#11758423C2212yes—
36#11758420C2212yes—
37#11758409C2212yes—
38#11758404C2212yes—
39#11758403C2212yes—
40C8×Dic31C2212yes—
41#11758398C2212yes—
42#11758397C2212yes—
43#41×C31—22no—
44#41×C31—22no—
45C2×Q8×C31C2311yescharacteristic
46C2×D4×C31—21no—
47C2×D4×C31C2311yescharacteristic
48C22×C4×C31—21no—
49C22×C4×C31—21no—
50C22×C4×C31C2311yescharacteristic
51C2×C22⋊D31—21no—
52C2×C22⋊D31—21no—
53C2×C22⋊D31—21no—
54C22×Dic31—21no—
55C22×Dic31D411yescharacteristic
56C22×Dic31D411yescharacteristic
57#92495—22no—
58#92495—22no—
59C2×D124C2311yescharacteristic
60C2×C4×D31—21no—
61C2×Dic62C2311yescharacteristic
62M16×C31—22no—
63C2×C8×C31C2312yes—
64C4⋊C4×C31—21no—
65C4⋊C4×C31C2312yes—
66C22⋊Dic31—21no—
67C22⋊Dic31—21no—
68(C2×D31)⋊C4—21no—
69(C2×D31)⋊C4—21no—
70C4⋊Dic31—21no—
71C4⋊Dic31—22no—
72C4⋊Dic31—22no—
73C4⋊Dic31C2312yes—
74Dic31⋊C4—22no—
75Dic31⋊C4—22no—
76Dic31⋊C4—21no—
77Dic31⋊C4—21no—
78C4×Dic31D411yescharacteristic
79C4×Dic31D411yescharacteristic
80C4×Dic31C2311yescharacteristic
81C4×Dic31C2311yescharacteristic
82#92474—22no—
83C2×C31⋊C8C2312yes—
84C23×C31—21no—
85C23×C31C2×D411yescharacteristic
86C22×D31—21no—
87Q8×C31—21no—
88Q8×C31—21no—
89D4×C31—21no—
90D4×C31—21no—
91D4×C31—42no—
92C2×C4×C31—22no—
93C2×C4×C31—21no—
94C2×C4×C31—42no—
95C2×C4×C31—42no—
96C2×C4×C31C2×D411yescharacteristic
97C2×C4×C31#4112yes—
98C2×C4×C31#4112yes—
99C2×C4×C31C2411yescharacteristic
100C22⋊D31—42no—
101C22⋊D31—42no—
102C22⋊D31—42no—
103C2×Dic31—22no—
104C2×Dic31—21no—
105C2×Dic31—41no—
106C2×Dic31—41no—
107C2×Dic31—42no—
108C2×Dic31C2×D411yescharacteristic
109C2×Dic31C2×D411yescharacteristic
110C2×Dic31C2×D411yescharacteristic
111C2×Dic31C2×D411yescharacteristic
112D124—21no—
113D124—21no—
114C4×D31—42no—
115Dic62—21no—
116Dic62—21no—
117C8×C31—22no—
118C31⋊C8—22no—
119#3142—311nomaximal, Sylow
120C22×C31—21no—
121C22×C31—21no—
122C22×C31—41no—
123C22×C31—41no—
124C22×C31—41no—
125C22×C31#12412yes—
126C22×C31C22×D411yescharacteristic
127C2×D31—41no—
128C2×D31—41no—
129C2×D31—41no—
130C4×C31—41no—
131C4×C31—42no—
132C4×C31#12412yes—
133C4×C31C2×#4111yescharacteristic
134C4×C31C2×#4111yescharacteristic
135Dic31—21no—
136Dic31—21no—
137Dic31—21no—
138Dic31—21no—
139Dic31—41no—
140Dic31—42no—
141#532—311no—
142C2×#125—311no—
143#491—312no—
144#490—312no—
145#488—312no—
146#488—312no—
147#417—311no—
148#417—311no—
149#417—311no—
150#417D3111yescharacteristic, complemented
151C4×M16—311no—
152C2×C31—22no—
153C2×C31—81no—
154C2×C31C2×#12411yescharacteristic
155C2×C31#57511yescharacteristic
156C2×C31#57511yescharacteristic
157D31—81no—
158C2×#41—311no—
159C2×#41C2×D3111yescharacteristic
160C2×M16—311no—
161C2×M16C2×D3111yescharacteristic
162C4⋊Q8—311no—
163#127—311no—
164#125—312no—
165#125—312no—
166#125—621no—
167#125—621no—
168#124—311no—
169#124—311no—
170#121—621no—
171#121—621no—
172C4×D4—621no—
173C4×D4—621no—
174C2×C4⋊C4—311no—
175C2×C4⋊C4—621no—
176C2×C4⋊C4—621no—
177C2×C4⋊C4C2×D3111yescharacteristic
178C2×C42—311no—
179#103—312no—
180#103—312no—
181#103—312no—
182#103C2×D3112yes—
183#102—312no—
184#102—312no—
185#102—312no—
186#102C2×D3112yes—
187#97—312no—
188C4×C8—312no—
189C31#314211yescharacteristic, complemented, Sylow
190#41—22no—
191#41—22no—
192#41—622no—
193#41—622no—
194C2×Q8—311no—
195C2×Q8C22×D3111yescharacteristic
196C2×D4—21no—
197C2×D4—311no—
198C2×D4—621no—
199C2×D4—621no—
200C2×D4—621no—
201C2×D4C22×D3111yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—311no—
205C22×C4—311no—
206C22×C4—621no—
207C22×C4—621no—
208C22×C4C22×D3111yescharacteristic
209M16—22no—
210M16—622no—
211C2×C8—312no—
212C2×C8C22×D3112yes—
213C4⋊C4—21no—
214C4⋊C4—312no—
215C4⋊C4—622no—
216C4⋊C4—622no—
217C4⋊C4—621no—
218C4⋊C4—621no—
219C4⋊C4—622no—
220C4⋊C4—622no—
221C4⋊C4—621no—
222C4⋊C4C22×D3112yes—
223C22⋊C4—621no—
224C22⋊C4—621no—
225C22⋊C4—621no—
226C22⋊C4—621no—
227C42—311no—
228C42—311no—
229C42—311no—
230C42—311no—
231C23—21no—
232C23—621no—
233C23D4×D3111yescharacteristic
234Q8—21no—
235Q8—21no—
236Q8—621no—
237Q8—621no—
238D4—21no—
239D4—21no—
240D4—42no—
241D4—621no—
242D4—621no—
243D4—1242no—
244D4—1242no—
245D4—1242no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—311no—
251C2×C4—311no—
252C2×C4—311no—
253C2×C4—311no—
254C2×C4—621no—
255C2×C4—622no—
256C2×C4—1241no—
257C2×C4—1241no—
258C2×C4—1242no—
259C2×C4—1242no—
260C2×C4#9249512yes—
261C2×C4D4×D3111yescharacteristic
262C2×C4#9249712yes—
263C2×C4C23×D3111yescharacteristic
264C8—22no—
265C8—622no—
266C22—21no—
267C22—21no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—1241no—
272C22—1241no—
273C22—1241no—
274C22#1175846212yes—
275C22C2×D4×D3111yescharacteristic
276C4—41no—
277C4—42no—
278C4—621no—
279C4—621no—
280C4—621no—
281C4—621no—
282C4—1241no—
283C4—1242no—
284C4#1175846212yes—
285C4C2×#9249511yescharacteristic
286C4C2×#9249711yescharacteristic
287C2—22no—
288C2—81no—
289C2—2481no—
290C2C2×#1175846211yescharacteristic
291C2#1452184511yescharacteristic
292C2#1452184811yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#417C2
socleC22×C31C22×D4
prime radical#14521595C2
hypercenter#417D31
hypocenterC31#3142
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#11758467C2semidirect
3. #14521416C2semidirect
4. #14521412C2semidirect
5. #14521405C2semidirect
6. #14521402C2semidirect
7. #14521051C2semidirect
8. #14521050C2semidirect
9. #14521048C2semidirect
10. M16×Dic31C2semidirect
11. C31×#417C2semidirect
150. #417D31semidirect
189. C31#3142semidirect

Maximal subgroups

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

subgroupindex
#145215952
C2×#117584672
#145214162
#145214122
#145214052
#145214022
#145210512
#145210502
#145210482
M16×Dic312
C31×#4172
#314231