Finite Group Databasebeta

← #30639560

Subgroups — #30639560

Order 16256.

320 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 83 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#22550422C211yesmaximal, characteristic
2#22549899C211yesmaximal, characteristic, complemented
3C2×#16861137C211yesmaximal, characteristic, complemented
4C2×#16861136C211yesmaximal, characteristic, complemented
5C2×#16861135C211yesmaximal, characteristic, complemented
6#22549722C211yesmaximal, characteristic, complemented
7C2×#104×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10#16861207C411yescharacteristic, complemented
11#16861205C411yescharacteristic, complemented
12#16861198C411yescharacteristic, complemented
13#16861196C411yescharacteristic, complemented
14#16861181—22no—
15C4×C22⋊D127—22no—
16C2×C4×Dic127C2211yescharacteristic
17C2×C4×Dic127C2211yescharacteristic
18Dic127⋊C8—22no—
19#16861137—22no—
20#16861136—22no—
21#16861135—22no—
22C127⋊#95—22no—
23C2×#41×C127C2211yescharacteristic
24C2×M16×C127C2211yescharacteristic
25C2×C42×C127C2211yescharacteristic
26#104×C127—22no—
27#14523248—21no—
28#14523248—21no—
29#14523248—21no—
30#14523248—21no—
31C2×Dic254C2×C411yescharacteristic
32C2×D508C2×C411yescharacteristic
33C2×C22⋊D127—22no—
34C2×C22⋊D127—21no—
35C22×Dic127—21no—
36C22×Dic127D412yes—
37C2×C4×D127—21no—
38#14523232—21no—
39#14523232—21no—
40C2×C127⋊C8—21no—
41Dic127⋊C4—22no—
42C22⋊Dic127—22no—
43(C2×D127)⋊C4—22no—
44C4×Dic127—21no—
45C4×Dic127C2×C411yescharacteristic
46C4×Dic127C2×C411yescharacteristic
47C4×Dic127D412yes—
48C4×Dic127D412yescomplemented
49#41×C127—21no—
50#41×C127—21no—
51#41×C127—21no—
52#41×C127—21no—
53C2×Q8×C127C2×C411yescharacteristic
54C2×D4×C127—21no—
55C2×D4×C127C2×C411yescharacteristic
56C22×C4×C127—21no—
57C22×C4×C127—21no—
58C22×C4×C127C2311yescharacteristic
59M16×C127—21no—
60M16×C127—21no—
61C2×C8×C127—21no—
62C42×C127—21no—
63C42×C127D412yescomplemented
64C22×D127—21no—
65Dic254—21no—
66Dic254—21no—
67D508—21no—
68D508—21no—
69C22⋊D127—44no—
70C22⋊D127—41no—
71C22⋊D127—41no—
72C2×Dic127—21no—
73C2×Dic127—22no—
74C2×Dic127—42no—
75C2×Dic127—42no—
76C2×Dic127C22⋊C412yes—
77C2×Dic127C22⋊C412yes—
78C4×D127—41no—
79C4×D127—41no—
80C127⋊C8—41no—
81C23×C127—21no—
82C23×C127C2×D411yescharacteristic
83Q8×C127—21no—
84Q8×C127—21no—
85D4×C127—21no—
86D4×C127—21no—
87D4×C127—41no—
88D4×C127—41no—
89C2×C4×C127—21no—
90C2×C4×C127—22no—
91C2×C4×C127—42no—
92C2×C4×C127—41no—
93C2×C4×C127—41no—
94C2×C4×C127C22×C411yescharacteristic
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127#4111yescharacteristic
99C2×C4×C127#4111yescharacteristic
100C8×C127—41no—
101C22×C127—22no—
102C22×C127—41no—
103C22×C127—42no—
104C22×C127C2×C22⋊C411yescharacteristic
105C22×C127#12111yescharacteristic
106C22×C127#12111yescharacteristic
107C2×D127—41no—
108C2×D127—42no—
109C4×C127—42no—
110C4×C127—41no—
111C4×C127C4×D411yescharacteristic
112C4×C127C4×D411yescharacteristic
113C4×C127#12411yescharacteristic
114C4×C127#12711yescharacteristic
115Dic127—22no—
116Dic127—22no—
117Dic127—42no—
118Dic127—41no—
119C2×C127—21no—
120C2×C127—21no—
121C2×C127—81no—
122C2×C127#39011yescharacteristic
123C2×C127#42112yes—
124D127—81no—
125#1963—1271nomaximal, Sylow
126C127#196311yescharacteristic, complemented, Sylow
127#532—1271no—
128#423—1271no—
129C2×#104—1271no—
130C2×#104—1271no—
131C2×#104—1271no—
132C2×#104D12711yescharacteristic, complemented
133#376—1271no—
134C2×#41—1271no—
135C2×#41C2×D12711yescharacteristic
136C2×M16—1271no—
137C2×M16C2×D12711yescharacteristic
138C4⋊Q8—1271no—
139#127—1271no—
140#124—1271no—
141#124—1271no—
142#121—2542no—
143C4×D4—2542no—
144C2×C42—1271no—
145C2×C42—1271no—
146C2×C42C2×D12711yescharacteristic
147C4⋊C8—2542no—
148#104—22no—
149#104—2542no—
150#104—2542no—
151#104—2542no—
152#95—2542no—
153#41—21no—
154#41—21no—
155#41—21no—
156#41—21no—
157#41—2541no—
158#41—2541no—
159#41—2541no—
160#41—2541no—
161C2×Q8—1271no—
162C2×Q8C4×D12711yescharacteristic
163C2×D4—21no—
164C2×D4—1271no—
165C2×D4—2542no—
166C2×D4—2541no—
167C2×D4C4×D12711yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—1272no—
171C22×C4—2541no—
172C22×C4—2541no—
173C22×C4C22×D12711yescharacteristic
174M16—21no—
175M16—21no—
176M16—2541no—
177M16—2541no—
178C2×C8—21no—
179C2×C8—2541no—
180C4⋊C4—2542no—
181C22⋊C4—2542no—
182C22⋊C4—2542no—
183C42—21no—
184C42—1271no—
185C42—1271no—
186C42—1272no—
187C42—1272no—
188C42—2541no—
189C42C22⋊D12712yescomplemented
190C23—21no—
191C23—2541no—
192C23D4×D12711yescharacteristic
193Q8—21no—
194Q8—21no—
195Q8—2541no—
196Q8—2541no—
197D4—21no—
198D4—21no—
199D4—41no—
200D4—41no—
201D4—5081no—
202D4—5081no—
203D4—5084no—
204D4—2541no—
205D4—2541no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—1272no—
212C2×C4—1272no—
213C2×C4—5082no—
214C2×C4—5081no—
215C2×C4—5081no—
216C2×C4—5082no—
217C2×C4—2541no—
218C2×C4—2542no—
219C2×C4C2×C4×D12711yescharacteristic
220C2×C4D4×D12711yescharacteristic
221C2×C4D4×D12711yescharacteristic
222C2×C4C2×C22⋊D12711yescharacteristic
223C2×C4#1452324711yescharacteristic
224C2×C4#1452324811yescharacteristic
225C8—41no—
226C8—5081no—
227C22—22no—
228C22—41no—
229C22—42no—
230C22—5081no—
231C22—5082no—
232C22C22⋊C4×D12711yescharacteristic
233C22#1686118111yescharacteristic
234C22#1686118211yescharacteristic
235C4—42no—
236C4—41no—
237C4—5082no—
238C4—5081no—
239C4—2542no—
240C4—2542no—
241C4(C22⋊D127)⋊C411yescharacteristic
242C4C4×C22⋊D12711yescharacteristic
243C4#1686119511yescharacteristic
244C4#1686120511yescharacteristic
245C2—21no—
246C2—21no—
247C2—81no—
248C2—10161no—
249C2#2254977611yescharacteristic
250C2#2254989112yes—
subgroupG/H
centerC2×C4D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#104×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550422C2
cosocleC22×C4×C127C23
hypercenterC2×#104D127
hypocenterC127#1963
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. #22549899C2semidirect
3. C2×#16861137C2semidirect
4. C2×#16861136C2semidirect
5. C2×#16861135C2semidirect
6. #22549722C2semidirect
7. C2×#104×C127C2semidirect
10. #16861207C4semidirect
11. #16861205C4semidirect
12. #16861198C4semidirect
13. #16861196C4semidirect
48. C4×Dic127D4semidirect
63. C42×C127D4semidirect
126. C127#1963semidirect
132. C2×#104D127semidirect
189. C42C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225504222
#225498992
C2×#168611372
C2×#168611362
C2×#168611352
#225497222
C2×#104×C1272
#1963127