Finite Group Databasebeta

← #30639593

Subgroups — #30639593

Order 16256.

384 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 103 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×#14523233C211yesmaximal, characteristic
2C2×#16861142C211yesmaximal, characteristic, complemented
3C2×#16861131C211yesmaximal, characteristic, complemented
4C2×C8×Dic127C211yesmaximal, characteristic, complemented
5#22549779C211yesmaximal, characteristic, complemented
6#22549777C211yesmaximal, characteristic, complemented
7C2×#102×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C2×#14523233—22no—
10C2×#14523233—22no—
11C2×#14523233C414yescomplemented
12C22×C127⋊C8C2211yescharacteristic
13C2×C4⋊Dic127C2211yescharacteristic
14C2×C22⋊Dic127—22no—
15C2×(C2×D127)⋊C4—22no—
16C2×C4×Dic127C2211yescharacteristic
17#16861142—22no—
18#16861131—22no—
19C8×Dic127—22no—
20C22×D4×C127C2211yescharacteristic
21C22×C8×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23#102×C127—22no—
24C23×D127—21no—
25C2×D508—21no—
26C2×D508—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508C2×C412yes—
31C22×Dic127—21no—
32C22×Dic127D412yes—
33#14523233—28no—
34C2×C127⋊C8C2×C412yes—
35C2×C127⋊C8D412yes—
36C2×C127⋊C8D412yes—
37C4⋊Dic127—21no—
38C4⋊Dic127—21no—
39C22⋊Dic127—44no—
40(C2×D127)⋊C4—44no—
41C4×Dic127—21no—
42C4×Dic127—21no—
43C24×C127—21no—
44C2×D4×C127—21no—
45C2×D4×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127—21no—
48C2×D4×C127—21no—
49C2×D4×C127C2×C412yes—
50C22×C4×C127—21no—
51C22×C4×C127C2311yescharacteristic
52C2×C8×C127—21no—
53C2×C8×C127—21no—
54C2×C8×C127D412yes—
55C4⋊C4×C127—21no—
56C4⋊C4×C127—21no—
57C22×D127—22no—
58C22×D127—41no—
59C22×D127—41no—
60C22×D127—42no—
61C22×D127—42no—
62D508—22no—
63D508—22no—
64D508—41no—
65D508—41no—
66C2×Dic127—22no—
67C2×Dic127—22no—
68C2×Dic127—41no—
69C2×Dic127—42no—
70C2×Dic127D814yescomplemented
71C127⋊C8C22⋊C414yescomplemented
72C23×C127—22no—
73C23×C127—41no—
74C23×C127—41no—
75C23×C127—42no—
76C23×C127—42no—
77C23×C127C2×D411yescharacteristic
78D4×C127—22no—
79D4×C127—22no—
80D4×C127—41no—
81D4×C127—41no—
82C2×C4×C127—41no—
83C2×C4×C127—42no—
84C2×C4×C127C22×C411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127#4111yescharacteristic
89C2×C4×C127#4111yescharacteristic
90C8×C127—22no—
91C22×C127—42no—
92C22×C127—44no—
93C22×C127—82no—
94C22×C127—81no—
95C22×C127—81no—
96C22×C127C4×D411yescharacteristic
97C22×C127#12111yescharacteristic
98C22×C127#12711yescharacteristic
99C22×C127C2×D812yes—
100C22×C127#13512yes—
101C2×D127—42no—
102C2×D127—44no—
103C2×D127—82no—
104C2×D127—81no—
105C2×D127—81no—
106C4×C127—81no—
107C4×C127C2×C22⋊C411yescharacteristic
108C4×C127C4×D411yescharacteristic
109C4×C127#12111yescharacteristic
110C4×C127#12411yescharacteristic
111Dic127—24no—
112Dic127—81no—
113C2×C127—82no—
114C2×C127#39011yescharacteristic
115C2×C127C4×D812yes—
116C2×C127#46612yes—
117C2×C127#49311yescharacteristic
118C2×C127#49511yescharacteristic
119D127—82no—
120#1970—1271nomaximal, Sylow
121C127#197011yescharacteristic, complemented, Sylow
122C22×D8—1271no—
123C2×#107—1271no—
124C2×#102—1271no—
125C2×#102D12711yescharacteristic, complemented
126C2×C4×C8—1271no—
127#390—1271no—
128#390—1271no—
129C22×D4—1271no—
130C22×D4C2×D12711yescharacteristic
131C2×D8—1274no—
132C2×D8—2542no—
133C2×D8—2542no—
134C22×C8—1271no—
135C22×C8C2×D12711yescharacteristic
136C2×C4⋊C4—1271no—
137C2×C4⋊C4C2×D12711yescharacteristic
138C2×C22⋊C4—2542no—
139C2×C22⋊C4—2542no—
140C2×C42—1271no—
141#107—2542no—
142#102—22no—
143#102—2542no—
144C4×C8—2542no—
145C24—21no—
146C24—2541no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—1272no—
153C2×D4—2541no—
154C2×D4—2541no—
155C2×D4—2541no—
156C2×D4—2541no—
157C2×D4—2541no—
158C2×D4C4×D12712yes—
159C22×C4—21no—
160C22×C4—1272no—
161C22×C4—2541no—
162C22×C4C22×D12711yescharacteristic
163D8—2548no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—1272no—
167C2×C8—1272no—
168C2×C8—1272no—
169C2×C8C22⋊D12712yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—2541no—
173C4⋊C4—2541no—
174C22⋊C4—5084no—
175C22⋊C4—5084no—
176C42—2541no—
177C42—2541no—
178C23—22no—
179C23—41no—
180C23—42no—
181C23—42no—
182C23—41no—
183C23—5081no—
184C23—5082no—
185C23—5082no—
186C23—5081no—
187C23—2542no—
188C23D4×D12711yescharacteristic
189D4—22no—
190D4—22no—
191D4—41no—
192D4—41no—
193D4—5081no—
194D4—5081no—
195D4—2542no—
196D4—2542no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—1274no—
200C2×C4—5081no—
201C2×C4—5082no—
202C2×C4—2542no—
203C2×C4—2542no—
204C2×C4C2×C4×D12711yescharacteristic
205C2×C4D4×D12711yescharacteristic
206C2×C4D4×D12711yescharacteristic
207C2×C4C2×C22⋊D12711yescharacteristic
208C2×C4#1452324711yescharacteristic
209C2×C4#1452324811yescharacteristic
210C8—22no—
211C8—1274no—
212C22—42no—
213C22—44no—
214C22—82no—
215C22—81no—
216C22—81no—
217C22—10162no—
218C22—10161no—
219C22—10161no—
220C22—5082no—
221C22—5084no—
222C22(C22⋊D127)⋊C411yescharacteristic
223C22#1686118211yescharacteristic
224C22#1686120511yescharacteristic
225C22D8×D12712yes—
226C22#1686123012yes—
227C4—81no—
228C4—10161no—
229C4—2544no—
230C4C22⋊C4×D12711yescharacteristic
231C4C4×C22⋊D12711yescharacteristic
232C4#1686118111yescharacteristic
233C4#1686119511yescharacteristic
234C2—82no—
235C2—10162no—
236C2#2254977611yescharacteristic
237C2#2254995612yes—
238C2#2255011112yes—
239C2#2255025111yescharacteristic
240C2#2255025711yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC22×#14523233C2
cosocleC22×C4×C127C23
hypercenterC2×#102D127
hypocenterC127#1970
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×#16861142C2semidirect
3. C2×#16861131C2semidirect
4. C2×C8×Dic127C2semidirect
5. #22549779C2semidirect
6. #22549777C2semidirect
7. C2×#102×C127C2semidirect
11. C2×#14523233C4semidirect
70. C2×Dic127D8semidirect
71. C127⋊C8C22⋊C4semidirect
121. C127#1970semidirect
125. C2×#102D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232332
C2×#168611422
C2×#168611312
C2×C8×Dic1272
#225497792
#225497772
C2×#102×C1272
#1970127