Finite Group Databasebeta

← #30648215

Subgroups — #30648215

Order 16256.

480 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 145 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×#16861239C211yesmaximal, characteristic
2C2×#16861207C211yesmaximal, characteristic, complemented
3#41×Dic127C211yesmaximal, characteristic
4#22550084C214yesmaximal, complemented
5#22550079C214yesmaximal, complemented
6#22549899C211yesmaximal, characteristic, complemented
7#22549886C211yesmaximal, characteristic, complemented
8#22549885C211yesmaximal, characteristic, complemented
9C2×#137×C127C211yesmaximal, characteristic
10C22×Dic254C2211yescharacteristic
11#16861239—24no—
12C2×#14523238C2212yes—
13C2×#14523236C2212yes—
14C2×#14523232C2211yescharacteristic
15#16861207—21no—
16#16861207—21no—
17#16861207C2214yes—
18Q8×Dic127C2212yes—
19D4×Dic127—22no—
20D4×Dic127C2212yes—
21#16861168—21no—
22#16861168C2211yescharacteristic
23C2×Dic127⋊C4—22no—
24C2×C4×Dic127—21no—
25C2×C4×Dic127C2211yescharacteristic
26Dic127⋊C8C2214yes—
27#16861134C2214yes—
28#16861133C2214yes—
29C2×#41×C127C2211yescharacteristic
30C22×Q8×C127C2211yescharacteristic
31#137×C127—24no—
32C2×Q16×C127C2212yes—
33C2×SD16×C127C2212yes—
34C2×M16×C127C2211yescharacteristic
35C2×Dic254—21no—
36C2×Dic254—22no—
37C2×Dic254—22no—
38C2×Dic254C2312yes—
39C22×Dic127—22no—
40C22×Dic127—21no—
41C22×Dic127D412yes—
42#14523238—24no—
43#14523236—24no—
44#14523232—22no—
45C2×C127⋊C8C2312yes—
46C4⋊Dic127—22no—
47C4⋊Dic127C2312yes—
48Dic127⋊C4—24no—
49Dic127⋊C4—24no—
50C22⋊Dic127—22no—
51C22⋊Dic127—21no—
52C4×Dic127—22no—
53C4×Dic127—21no—
54C4×Dic127D411yescharacteristic
55C4×Dic127D411yescharacteristic
56C4×Dic127C2312yes—
57#41×C127—22no—
58#41×C127D414yes—
59C2×Q8×C127—21no—
60C2×Q8×C127—22no—
61C2×Q8×C127—22no—
62C2×Q8×C127C2312yes—
63C2×Q8×C127C2311yescharacteristic
64C2×D4×C127—21no—
65C2×D4×C127C2311yescharacteristic
66C22×C4×C127—21no—
67C22×C4×C127—21no—
68C22×C4×C127C2311yescharacteristic
69Q16×C127—24no—
70SD16×C127—24no—
71M16×C127—22no—
72C2×C8×C127C2312yes—
73Dic254—22no—
74Dic254—22no—
75Dic254—42no—
76C2×Dic127—24no—
77C2×Dic127—22no—
78C2×Dic127—22no—
79C2×Dic127—42no—
80C2×Dic127—44no—
81C2×Dic127C2×D412yes—
82C2×Dic127C2×D412yes—
83C127⋊C8—22no—
84C23×C127—21no—
85C23×C127C2×D411yescharacteristic
86Q8×C127—21no—
87Q8×C127—22no—
88Q8×C127—22no—
89Q8×C127—42no—
90Q8×C127C2×D412yes—
91D4×C127—24no—
92D4×C127—21no—
93D4×C127C2×D412yes—
94C2×C4×C127—21no—
95C2×C4×C127—24no—
96C2×C4×C127—22no—
97C2×C4×C127—42no—
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D412yes—
100C2×C4×C127#4112yes—
101C2×C4×C127C2411yescharacteristic
102C8×C127—22no—
103C22×C127—22no—
104C22×C127—22no—
105C22×C127—42no—
106C22×C127#12112yes—
107C22×C127C22×D411yescharacteristic
108C4×C127—22no—
109C4×C127—42no—
110C4×C127#12112yes—
111C4×C127C22×D411yescharacteristic
112C4×C127C2×#4111yescharacteristic
113Dic127—22no—
114Dic127—42no—
115Dic127—42no—
116Dic127#13714yescomplemented
117C2×C127—22no—
118C2×C127—42no—
119C2×C127C2×#12111yescharacteristic
120C2×C127C2×#13712yes—
121#3039—1271nomaximal, Sylow
122C127#303911yescharacteristic, complemented, Sylow
123C2×#137—1271no—
124C2×#137D12711yescharacteristic
125C2×C4⋊Q8—1271no—
126C4×#41—1271no—
127#462—1274no—
128#461—1274no—
129#423—1271no—
130#419—1271no—
131#419—1271no—
132C2×#41C2×D12711yescharacteristic
133C22×Q8—1271no—
134C22×Q8C2×D12711yescharacteristic
135#137—24no—
136#137—2544no—
137C2×Q16—1272no—
138C2×Q16C2×D12712yes—
139C2×SD16—1272no—
140C2×SD16C2×D12712yes—
141C2×M16—1271no—
142C2×M16C2×D12711yescharacteristic
143C4⋊Q8—1274no—
144C4⋊Q8—2541no—
145C4⋊Q8—2541no—
146C4×Q8—1272no—
147C4×D4—1272no—
148C4×D4—2542no—
149#117—1271no—
150#117—2541no—
151C2×C4⋊C4—2542no—
152C2×C42—1271no—
153C2×C42—2541no—
154C4⋊C8—1274no—
155#103—1274no—
156#103—1274no—
157#41—22no—
158#41C22⋊D12714yes—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—1272no—
163C2×Q8—2542no—
164C2×Q8—2542no—
165C2×Q8—2541no—
166C2×Q8C22×D12712yes—
167C2×Q8C22×D12711yescharacteristic
168C2×D4—21no—
169C2×D4C22×D12711yescharacteristic
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—1272no—
173C22×C4—2542no—
174C22×C4—2541no—
175C22×C4C22×D12711yescharacteristic
176Q16—24no—
177Q16—2544no—
178SD16—24no—
179SD16—2544no—
180M16—22no—
181M16—2542no—
182C2×C8—1272no—
183C2×C8C22×D12712yes—
184C4⋊C4—1272no—
185C4⋊C4—2544no—
186C4⋊C4—2544no—
187C4⋊C4—2542no—
188C22⋊C4—2542no—
189C22⋊C4—2541no—
190C42—1271no—
191C42—1271no—
192C42—1272no—
193C42—2542no—
194C42—2541no—
195C23—21no—
196C23D4×D12711yescharacteristic
197Q8—21no—
198Q8—22no—
199Q8—22no—
200Q8—42no—
201Q8—5082no—
202Q8—2542no—
203Q8—2542no—
204Q8C2×C22⋊D12712yes—
205D4—24no—
206D4—21no—
207D4C2×C22⋊D12712yes—
208C2×C4—21no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—1272no—
213C2×C4—1272no—
214C2×C4—5084no—
215C2×C4—5082no—
216C2×C4—2544no—
217C2×C4—2542no—
218C2×C4—2542no—
219C2×C4D4×D12711yescharacteristic
220C2×C4C2×C22⋊D12712yes—
221C2×C4#1452324712yes—
222C2×C4C23×D12711yescharacteristic
223C8—22no—
224C8—2542no—
225C22—22no—
226C22—22no—
227C22—42no—
228C22#1686118112yes—
229C22C2×D4×D12711yescharacteristic
230C4—22no—
231C4—42no—
232C4—1274no—
233C4—5082no—
234C4—5082no—
235C4—2542no—
236C4#1686118112yes—
237C4C22×C22⋊D12711yescharacteristic
238C4C2×#1452324711yescharacteristic
239C2—22no—
240C2—42no—
241C2C2×#1686118111yescharacteristic
242C2#137×D12712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#137×C127C2
socleC22×C127C22×D4
prime radicalC2×#41×C127C22
hypercenterC2×#137D127
hypocenterC127#3039
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#16861207C2semidirect
4. #22550084C2semidirect
5. #22550079C2semidirect
6. #22549899C2semidirect
7. #22549886C2semidirect
8. #22549885C2semidirect
116. Dic127#137semidirect
122. C127#3039semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
C2×#168612072
#41×Dic1272
#225500842
#225500792
#225498992
#225498862
#225498852
C2×#137×C1272
#3039127