Finite Group Databasebeta

← #30639804

Subgroups — #30639804

Order 16256.

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

normal subgroups only

This group has 426 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
1C22×#14523233C211yesmaximal, characteristic
2C2×#16861198C211yesmaximal, characteristic, complemented
3C2×#16861133C211yesmaximal, characteristic, complemented
4C2×#16861131C211yesmaximal, characteristic, complemented
5#22549779C211yesmaximal, characteristic, complemented
6#22549630C211yesmaximal, characteristic, complemented
7C22×SD16×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C2×#14523233—22no—
10C2×#14523233—22no—
11C2×#14523233—22no—
12C22×C127⋊C8C2211yescharacteristic
13#16861198—22no—
14#16861198—22no—
15C2×C4⋊Dic127C2211yescharacteristic
16C2×C22⋊Dic127—22no—
17C2×(C2×D127)⋊C4—22no—
18C2×C4×Dic127C2211yescharacteristic
19#16861133—22no—
20#16861131—22no—
21C22×Q8×C127C2211yescharacteristic
22C22×D4×C127C2211yescharacteristic
23C2×SD16×C127—22no—
24C2×SD16×C127—22no—
25C2×SD16×C127—22no—
26C22×C8×C127C2211yescharacteristic
27C23×D127—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508—21no—
32C2×D508—21no—
33C2×D508D412yes—
34C22×Dic127—21no—
35C22×Dic127D412yes—
36#14523233—28no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8D412yes—
40C4⋊Dic127—21no—
41C4⋊Dic127—21no—
42C22⋊Dic127—44no—
43(C2×D127)⋊C4—28no—
44C4×Dic127—21no—
45C4×Dic127—21no—
46C24×C127—21no—
47C2×Q8×C127—21no—
48C2×Q8×C127—21no—
49C2×Q8×C127—21no—
50C2×Q8×C127—21no—
51C2×Q8×C127—21no—
52C2×Q8×C127D412yes—
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127—21no—
56C2×D4×C127—21no—
57C2×D4×C127—21no—
58C2×D4×C127D412yes—
59C22×C4×C127—21no—
60C22×C4×C127C2311yescharacteristic
61SD16×C127—28no—
62C2×C8×C127—21no—
63C2×C8×C127—21no—
64C2×C8×C127D412yes—
65C22×D127—22no—
66C22×D127—41no—
67C22×D127—41no—
68C22×D127—42no—
69C22×D127—42no—
70D508—22no—
71D508—22no—
72D508—41no—
73D508—41no—
74C2×Dic127—22no—
75C2×Dic127—22no—
76C2×Dic127—22no—
77C2×Dic127—41no—
78C2×Dic127—42no—
79C127⋊C8—22no—
80C23×C127—22no—
81C23×C127—41no—
82C23×C127—41no—
83C23×C127—42no—
84C23×C127—42no—
85C23×C127C2×D411yescharacteristic
86Q8×C127—22no—
87Q8×C127—22no—
88Q8×C127—41no—
89Q8×C127—41no—
90D4×C127—22no—
91D4×C127—22no—
92D4×C127—41no—
93D4×C127—41no—
94C2×C4×C127—22no—
95C2×C4×C127—41no—
96C2×C4×C127—41no—
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127#4111yescharacteristic
103C8×C127—22no—
104C22×C127—42no—
105C22×C127—44no—
106C22×C127—82no—
107C22×C127—81no—
108C22×C127—81no—
109C22×C127#12011yescharacteristic
110C22×C127#12111yescharacteristic
111C22×C127#12711yescharacteristic
112C22×C127#13512yes—
113C22×C127AGL(1,ℤ/8)12yes—
114C2×D127—42no—
115C2×D127—44no—
116C2×D127—82no—
117C2×D127—81no—
118C2×D127—81no—
119C4×C127—42no—
120C4×C127#12011yescharacteristic
121C4×C127#12011yescharacteristic
122C4×C127#12111yescharacteristic
123C4×C127#12111yescharacteristic
124Dic127—42no—
125Dic127—81no—
126C2×C127—82no—
127C2×C127#39211yescharacteristic
128C2×C127#44912yes—
129C2×C127#46312yes—
130C2×C127#49511yescharacteristic
131C2×C127#49611yescharacteristic
132D127—82no—
133#2012—1271nomaximal, Sylow
134C127#201211yescharacteristic, complemented, Sylow
135C22×SD16D12711yescharacteristic, complemented
136C22×D8—1271no—
137C2×#124—1271no—
138C2×#103—1271no—
139C2×#102—1271no—
140#390—1271no—
141#336—1271no—
142C22×Q8C2×D12711yescharacteristic
143C22×D4—1271no—
144C22×D4C2×D12711yescharacteristic
145C2×SD16—22no—
146C2×SD16—22no—
147C2×SD16—22no—
148C2×D8—2542no—
149C2×D8—2542no—
150C2×D8—2542no—
151C22×C8—1271no—
152C22×C8C2×D12711yescharacteristic
153#124—2542no—
154#124—2542no—
155C2×C4⋊C4—1271no—
156C2×C22⋊C4—2542no—
157C2×C22⋊C4—2542no—
158C2×C42—1271no—
159#103—2542no—
160#102—2542no—
161C24—21no—
162C24—2541no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8C22⋊D12712yes—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—1272no—
175C2×D4—2541no—
176C2×D4—2541no—
177C2×D4—2541no—
178C2×D4—2541no—
179C2×D4—2541no—
180C2×D4C22⋊D12712yes—
181C22×C4—21no—
182C22×C4—1272no—
183C22×C4—2541no—
184C22×C4C22×D12711yescharacteristic
185SD16—28no—
186D8—2548no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—1272no—
190C2×C8—2541no—
191C2×C8—2541no—
192C2×C8C22⋊D12712yes—
193C4⋊C4—2541no—
194C4⋊C4—2541no—
195C22⋊C4—5084no—
196C22⋊C4—2548no—
197C42—2541no—
198C42—2541no—
199C23—22no—
200C23—41no—
201C23—42no—
202C23—42no—
203C23—41no—
204C23—5081no—
205C23—5082no—
206C23—5082no—
207C23—5081no—
208C23—2542no—
209C23D4×D12711yescharacteristic
210Q8—22no—
211Q8—22no—
212Q8—41no—
213Q8—41no—
214D4—22no—
215D4—22no—
216D4—41no—
217D4—41no—
218D4—5081no—
219D4—5081no—
220D4—2542no—
221D4—2542no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—5081no—
226C2×C4—5082no—
227C2×C4—2542no—
228C2×C4—2542no—
229C2×C4—2542no—
230C2×C4D4×D12711yescharacteristic
231C2×C4D4×D12711yescharacteristic
232C2×C4C2×C22⋊D12711yescharacteristic
233C2×C4C2×C22⋊D12711yescharacteristic
234C2×C4C2×C22⋊D12711yescharacteristic
235C2×C4#1452324711yescharacteristic
236C8—22no—
237C8—2542no—
238C22—42no—
239C22—44no—
240C22—82no—
241C22—81no—
242C22—81no—
243C22—10162no—
244C22—10161no—
245C22—10161no—
246C22—5084no—
247C22—5082no—
248C22#1686117711yescharacteristic
249C22#1686118111yescharacteristic
250C22#1686120511yescharacteristic
251C22#1686123012yes—
252C22#1686123612yes—
253C4—42no—
254C4—10161no—
255C4—5082no—
256C4#1686117711yescharacteristic
257C4#1686117811yescharacteristic
258C4#1686118011yescharacteristic
259C4#1686118111yescharacteristic
260C2—82no—
261C2—10162no—
262C2#2254978411yescharacteristic
263C2#2255002012yes—
264C2#2255009312yes—
265C2#2255025711yescharacteristic
266C2#2255026211yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC22×SD16×C127C2
socleC23×C127C2×D4
prime radicalC22×#14523233C2
hypercenterC22×SD16D127
hypocenterC127#2012
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. C2×#16861198C2semidirect
3. C2×#16861133C2semidirect
4. C2×#16861131C2semidirect
5. #22549779C2semidirect
6. #22549630C2semidirect
7. C22×SD16×C127C2semidirect
134. C127#2012semidirect
135. C22×SD16D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232332
C2×#168611982
C2×#168611332
C2×#168611312
#225497792
#225496302
C22×SD16×C1272
#2012127