Finite Group Databasebeta

← #30639426

Subgroups — #30639426

Order 16256.

284 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 91 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×#16861129C211yesmaximal, characteristic
2C2×#16861129C211yesmaximal, characteristic
3C2×#16861120C211yesmaximal, characteristic, complemented
4#22549778C211yesmaximal, characteristic
5#22549638C211yesmaximal, characteristic, complemented
6#22549638C211yesmaximal, characteristic, complemented
7#389×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10C22×C127⋊C8C2211yescharacteristic
11C2×C4⋊Dic127C2211yescharacteristic
12C2×(C2×D127)⋊C4—21no—
13C2×(C2×D127)⋊C4—21no—
14#16861129—22no—
15#16861129C414yescomplemented
16#16861120—22no—
17C2×C4⋊C4×C127—21no—
18C2×C4⋊C4×C127C2211yescharacteristic
19C2×C4⋊C4×C127C2211yescharacteristic
20C2×C42×C127C2211yescharacteristic
21#95×C127—21no—
22C23×D127—21no—
23C2×D508—21no—
24C2×D508—21no—
25C2×D508—21no—
26C2×D508—21no—
27C2×D508—21no—
28C2×D508C2×C412yes—
29C22×Dic127—21no—
30C2×C127⋊C8—21no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8C2×C412yes—
35C2×C127⋊C8D412yes—
36C4⋊Dic127—21no—
37C4⋊Dic127—21no—
38(C2×D127)⋊C4—42no—
39(C2×D127)⋊C4—42no—
40C22×C4×C127—21no—
41C22×C4×C127—21no—
42C22×C4×C127D411yescharacteristic
43C22×C4×C127D411yescharacteristic
44C22×C4×C127C2311yescharacteristic
45C4⋊C4×C127—21no—
46C4⋊C4×C127—21no—
47C4⋊C4×C127—21no—
48C4⋊C4×C127—42no—
49C4⋊C4×C127C2×C412yes—
50C42×C127—21no—
51C42×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—41no—
64C2×Dic127—41no—
65C2×Dic127—41no—
66C127⋊C8—22no—
67C127⋊C8—22no—
68C23×C127C2×D411yescharacteristic
69C2×C4×C127—22no—
70C2×C4×C127—21no—
71C2×C4×C127—21no—
72C2×C4×C127—21no—
73C2×C4×C127—21no—
74C2×C4×C127—21no—
75C2×C4×C127—21no—
76C2×C4×C127—41no—
77C2×C4×C127—41no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127—41no—
81C2×C4×C127C22×C411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127#4111yescharacteristic
84C2×C4×C127#4111yescharacteristic
85C2×C4×C127#4111yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C22×C127C4×D411yescharacteristic
88C22×C127#12111yescharacteristic
89C22×C127#12411yescharacteristic
90C22×C127AGL(1,ℤ/8)11yescharacteristic
91C22×C127AGL(1,ℤ/8)11yescharacteristic
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—42no—
102C4×C127—41no—
103C4×C127—41no—
104C4×C127—81no—
105C4×C127#11711yescharacteristic
106C4×C127C4×D411yescharacteristic
107C4×C127#12311yescharacteristic
108C4×C127#12611yescharacteristic
109Dic127—81no—
110C2×C127#38811yescharacteristic
111C2×C127#44011yescharacteristic
112C2×C127#44211yescharacteristic
113C2×C127#46811yescharacteristic
114C2×C127#46911yescharacteristic
115C2×C127#48911yescharacteristic
116C2×C127#49011yescharacteristic
117D127—82no—
118#1936—1271nomaximal, Sylow
119C127#193611yescharacteristic, complemented, Sylow
120C2×#102—1271no—
121C2×#102—1271no—
122C2×#97—1271no—
123#390—1271no—
124#389D12711yescharacteristic, complemented
125#340—1271no—
126#340—1271no—
127C22×D4—1271no—
128C22×C8—1271no—
129C22×C8—1271no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—1271no—
132C2×C4⋊C4C2×D12711yescharacteristic
133C2×C4⋊C4C2×D12711yescharacteristic
134C2×C22⋊C4—2541no—
135C2×C22⋊C4—2541no—
136C2×C42C2×D12711yescharacteristic
137#102—1274no—
138#102—2542no—
139#97—2542no—
140#95—21no—
141C24—2541no—
142C2×D4—1272no—
143C2×D4—2541no—
144C2×D4—2541no—
145C2×D4—2541no—
146C2×D4—2541no—
147C2×D4—2541no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—2541no—
151C22×C4C22⋊D12711yescharacteristic
152C22×C4C22⋊D12711yescharacteristic
153C22×C4C22×D12711yescharacteristic
154C2×C8—1272no—
155C2×C8—1272no—
156C2×C8—2541no—
157C2×C8—2541no—
158C2×C8—2541no—
159C2×C8—2541no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—42no—
164C4⋊C4—2541no—
165C4⋊C4—2541no—
166C4⋊C4C4×D12712yes—
167C22⋊C4—5082no—
168C22⋊C4—5082no—
169C42—21no—
170C42—21no—
171C23—5081no—
172C23—5081no—
173C23—5081no—
174C23—5081no—
175C23—5081no—
176C23—5081no—
177C23—2542no—
178C23C2×C22⋊D12711yescharacteristic
179D4—5081no—
180D4—5081no—
181D4—2542no—
182D4—2542no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—5081no—
196C2×C4—5081no—
197C2×C4—5081no—
198C2×C4C2×C4×D12711yescharacteristic
199C2×C4D4×D12711yescharacteristic
200C2×C4#1452324711yescharacteristic
201C2×C4#1452324811yescharacteristic
202C2×C4#1452324911yescharacteristic
203C2×C4#1452324911yescharacteristic
204C8—2542no—
205C8—2542no—
206C22—10161no—
207C22—10161no—
208C22—10161no—
209C22—10161no—
210C22—5082no—
211C22—5082no—
212C22—5082no—
213C22C4×C22⋊D12711yescharacteristic
214C22#1686118111yescharacteristic
215C22#1686119811yescharacteristic
216C22#1686123411yescharacteristic
217C22#1686123411yescharacteristic
218C22#1686123511yescharacteristic
219C22#1686124011yescharacteristic
220C4—41no—
221C4—41no—
222C4—42no—
223C4—81no—
224C4—10161no—
225C4#1686116711yescharacteristic
226C4#1686117111yescharacteristic
227C4#1686119411yescharacteristic
228C4#1686120411yescharacteristic
229C2—10162no—
230C2#2254976611yescharacteristic
231C2#2254997411yescharacteristic
232C2#2254998611yescharacteristic
233C2#2255012411yescharacteristic
234C2#2255012711yescharacteristic
235C2#2255023811yescharacteristic
236C2#2255024211yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#389×C127C2
socleC23×C127C2×D4
prime radicalC22×D508C22
cosocleC22×C4×C127C23
hypercenter#389D127
hypocenterC127#1936
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
3. C2×#16861120C2semidirect
5. #22549638C2semidirect
6. #22549638C2semidirect
7. #389×C127C2semidirect
15. #16861129C4semidirect
119. C127#1936semidirect
124. #389D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611292
C2×#168611202
#225497782
#225496382
#389×C1272
#1936127