Finite Group Databasebeta

← #30640137

Subgroups — #30640137

Order 16256.

274 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 81 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×#16861131C211yesmaximal, characteristic
2C2×#16861129C211yesmaximal, characteristic
3#22549777C211yesmaximal, characteristic
4#22549745C211yesmaximal, characteristic, complemented
5#22549638C211yesmaximal, characteristic, complemented
6#22549630C211yesmaximal, characteristic, complemented
7#340×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10C2×C4⋊Dic127C2211yescharacteristic
11C2×Dic127⋊C4—21no—
12C2×Dic127⋊C4—21no—
13C2×(C2×D127)⋊C4—21no—
14C2×(C2×D127)⋊C4—21no—
15C2×C4×Dic127C2211yescharacteristic
16#16861131—22no—
17#16861129—22no—
18C22×C8×C127C2211yescharacteristic
19C2×C4⋊C4×C127C2211yescharacteristic
20C2×C4⋊C4×C127C2211yescharacteristic
21C23×D127—21no—
22C2×D508—21no—
23C2×D508—21no—
24C2×D508—21no—
25C2×D508—21no—
26C2×D508—21no—
27C2×D508D412yes—
28C22×Dic127—21no—
29C22×Dic127D411yescharacteristic
30C22×Dic127D411yescharacteristic
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33C2×C127⋊C8—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127—21no—
36Dic127⋊C4—42no—
37Dic127⋊C4—42no—
38(C2×D127)⋊C4—42no—
39(C2×D127)⋊C4—42no—
40C4×Dic127—21no—
41C4×Dic127—21no—
42C22×C4×C127—21no—
43C22×C4×C127—21no—
44C22×C4×C127C2311yescharacteristic
45C2×C8×C127—21no—
46C2×C8×C127—21no—
47C2×C8×C127—21no—
48C4⋊C4×C127—21no—
49C4⋊C4×C127—21no—
50C4⋊C4×C127—21no—
51C4⋊C4×C127—21no—
52C22×D127—22no—
53C22×D127—41no—
54C22×D127—41no—
55C22×D127—41no—
56C22×D127—41no—
57C22×D127—41no—
58C22×D127—41no—
59D508—22no—
60D508—22no—
61D508—41no—
62D508—41no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—21no—
66C2×Dic127—21no—
67C2×Dic127—41no—
68C2×Dic127—41no—
69C2×Dic127—41no—
70C2×Dic127D812yes—
71C2×Dic127SD1612yes—
72C127⋊C8—41no—
73C23×C127C2×D411yescharacteristic
74C2×C4×C127—41no—
75C2×C4×C127—41no—
76C2×C4×C127—41no—
77C2×C4×C127—41no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127#4111yescharacteristic
82C2×C4×C127#4111yescharacteristic
83C2×C4×C127#4111yescharacteristic
84C2×C4×C127#4111yescharacteristic
85C2×C4×C127#4111yescharacteristic
86C8×C127—41no—
87C22×C127#12111yescharacteristic
88C22×C127#12311yescharacteristic
89C22×C127#12411yescharacteristic
90C22×C127C2×D811yescharacteristic
91C22×C127C2×SD1611yescharacteristic
92C22×C127AGL(1,ℤ/8)11yescharacteristic
93C22×C127#13711yescharacteristic
94C2×D127—42no—
95C2×D127—42no—
96C2×D127—42no—
97C2×D127—81no—
98C2×D127—81no—
99C2×D127—81no—
100C2×D127—81no—
101C4×C127—81no—
102C4×C127—81no—
103C4×C127#12311yescharacteristic
104C4×C127#12311yescharacteristic
105C4×C127#12611yescharacteristic
106C4×C127#12611yescharacteristic
107Dic127—22no—
108Dic127—22no—
109Dic127—81no—
110C2×C127#39711yescharacteristic
111C2×C127#45911yescharacteristic
112C2×C127#46111yescharacteristic
113C2×C127#48011yescharacteristic
114C2×C127#48111yescharacteristic
115C2×C127#48611yescharacteristic
116C2×C127#48911yescharacteristic
117D127—82no—
118#2071—1271nomaximal, Sylow
119C127#207111yescharacteristic, complemented, Sylow
120C2×#102—1271no—
121C2×#102—1271no—
122#390—1271no—
123#384—1271no—
124#340—1271no—
125#340D12711yescharacteristic, complemented
126#336—1271no—
127C22×D4—1271no—
128C22×C8—1271no—
129C22×C8C2×D12711yescharacteristic
130C2×C4⋊C4—1271no—
131C2×C4⋊C4—2541no—
132C2×C4⋊C4—2541no—
133C2×C4⋊C4C2×D12711yescharacteristic
134C2×C4⋊C4C2×D12711yescharacteristic
135C2×C22⋊C4—2541no—
136C2×C22⋊C4—2541no—
137C2×C42—1271no—
138#102—2542no—
139#102—2542no—
140C24—2541no—
141C2×D4—1272no—
142C2×D4—2541no—
143C2×D4—2541no—
144C2×D4—2541no—
145C2×D4—2541no—
146C2×D4—2541no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—1271no—
150C22×C4—1271no—
151C22×C4—2541no—
152C22×C4C22×D12711yescharacteristic
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—2541no—
157C2×C8—2541no—
158C2×C8—2541no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—5082no—
164C4⋊C4—5082no—
165C4⋊C4—2541no—
166C4⋊C4—2541no—
167C22⋊C4—5082no—
168C22⋊C4—5082no—
169C42—2541no—
170C42—2541no—
171C23—5081no—
172C23—5081no—
173C23—5081no—
174C23—5081no—
175C23—5081no—
176C23—5081no—
177C23—2542no—
178C23D4×D12711yescharacteristic
179D4—5081no—
180D4—5081no—
181D4—2542no—
182D4—2542no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—1272no—
190C2×C4—1272no—
191C2×C4—5081no—
192C2×C4—5081no—
193C2×C4—5081no—
194C2×C4—2541no—
195C2×C4—2541no—
196C2×C4—2541no—
197C2×C4—2541no—
198C2×C4D4×D12711yescharacteristic
199C2×C4#1452324711yescharacteristic
200C2×C4#1452324811yescharacteristic
201C2×C4#1452324811yescharacteristic
202C2×C4#1452324811yescharacteristic
203C2×C4#1452324911yescharacteristic
204C8—41no—
205C8—5081no—
206C22—10161no—
207C22—10161no—
208C22—10161no—
209C22—10161no—
210C22—5082no—
211C22—5082no—
212C22—5082no—
213C22#1686118211yescharacteristic
214C22#1686119411yescharacteristic
215C22#1686119511yescharacteristic
216C22D8×D12711yescharacteristic
217C22SD16×D12711yescharacteristic
218C22#1686123611yescharacteristic
219C22#1686124111yescharacteristic
220C4—81no—
221C4—81no—
222C4—10161no—
223C4—2542no—
224C4—2542no—
225C4#1686119111yescharacteristic
226C4#1686119411yescharacteristic
227C4#1686120311yescharacteristic
228C4#1686120411yescharacteristic
229C2—10162no—
230C2#2254980511yescharacteristic
231C2#2255007011yescharacteristic
232C2#2255008011yescharacteristic
233C2#2255019111yescharacteristic
234C2#2255019611yescharacteristic
235C2#2255022111yescharacteristic
236C2#2255023711yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#340×C127C2
socleC23×C127C2×D4
prime radicalC22×D508C22
hypercenter#340D127
hypocenterC127#2071
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
4. #22549745C2semidirect
5. #22549638C2semidirect
6. #22549630C2semidirect
7. #340×C127C2semidirect
119. C127#2071semidirect
125. #340D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611312
C2×#168611292
#225497772
#225497452
#225496382
#225496302
#340×C1272
#2071127