Finite Group Databasebeta

← #30639805

Subgroups — #30639805

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×#14523236C211yesmaximal, characteristic, complemented
2C2×#16861196C211yesmaximal, characteristic, complemented
3C2×#16861134C211yesmaximal, characteristic, complemented
4C2×#16861130C211yesmaximal, characteristic, complemented
5#22549779C211yesmaximal, characteristic, complemented
6#22549630C211yesmaximal, characteristic, complemented
7C22×D8×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523236—22no—
10C2×#14523236—22no—
11C2×#14523236—22no—
12C22×C127⋊C8C2211yescharacteristic
13#16861196—22no—
14#16861196—22no—
15C2×C4⋊Dic127C2211yescharacteristic
16C2×C22⋊Dic127—22no—
17C2×C22⋊Dic127—22no—
18C2×C4×Dic127C2211yescharacteristic
19#16861134—22no—
20#16861130—22no—
21C22×D4×C127C2211yescharacteristic
22C22×D4×C127C2211yescharacteristic
23C2×D8×C127—22no—
24C2×D8×C127—22no—
25C2×D8×C127—22no—
26C22×C8×C127C2211yescharacteristic
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254—21no—
32C2×Dic254D412yes—
33C22×Dic127—21no—
34C22×Dic127—21no—
35C22×Dic127D412yes—
36#14523236—28no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8D412yes—
40C4⋊Dic127—21no—
41C4⋊Dic127—21no—
42C22⋊Dic127—28no—
43C22⋊Dic127—44no—
44C4×Dic127—21no—
45C4×Dic127—21no—
46C24×C127—21no—
47C24×C127—21no—
48C2×D4×C127—21no—
49C2×D4×C127—21no—
50C2×D4×C127—21no—
51C2×D4×C127—21no—
52C2×D4×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127—21no—
56C2×D4×C127—21no—
57C2×D4×C127—21no—
58C2×D4×C127D412yes—
59C2×D4×C127D412yes—
60C22×C4×C127C2311yescharacteristic
61D8×C127—28no—
62C2×C8×C127—21no—
63C2×C8×C127—21no—
64C2×C8×C127D412yes—
65Dic254—22no—
66Dic254—22no—
67Dic254—41no—
68Dic254—41no—
69C2×Dic127—22no—
70C2×Dic127—22no—
71C2×Dic127—22no—
72C2×Dic127—22no—
73C2×Dic127—41no—
74C2×Dic127—41no—
75C2×Dic127—41no—
76C2×Dic127—42no—
77C127⋊C8—22no—
78C23×C127—22no—
79C23×C127—22no—
80C23×C127—41no—
81C23×C127—41no—
82C23×C127—41no—
83C23×C127—41no—
84C23×C127—42no—
85C23×C127—42no—
86C23×C127—42no—
87C23×C127—42no—
88C23×C127C2×D411yescharacteristic
89D4×C127—22no—
90D4×C127—22no—
91D4×C127—22no—
92D4×C127—22no—
93D4×C127—41no—
94D4×C127—41no—
95D4×C127—41no—
96D4×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—42no—
106C22×C127—44no—
107C22×C127—44no—
108C22×C127—82no—
109C22×C127—82no—
110C22×C127—81no—
111C22×C127—81no—
112C22×C127—81no—
113C22×C127—81no—
114C22×C127#12011yescharacteristic
115C22×C127#12111yescharacteristic
116C22×C127#12711yescharacteristic
117C22×C127#13512yes—
118C22×C127AGL(1,ℤ/8)12yes—
119C4×C127#12011yescharacteristic
120C4×C127#12011yescharacteristic
121C4×C127#12111yescharacteristic
122C4×C127#12111yescharacteristic
123Dic127—42no—
124Dic127—42no—
125Dic127—81no—
126C2×C127—82no—
127C2×C127—82no—
128C2×C127#39211yescharacteristic
129C2×C127#44912yes—
130C2×C127#46312yes—
131C2×C127#49511yescharacteristic
132C2×C127#49611yescharacteristic
133#2012—1271nomaximal, Sylow
134C127#201211yescharacteristic, complemented, Sylow
135C22×SD16—1271no—
136C22×D8D12711yescharacteristic
137C2×#124—1271no—
138C2×#103—1271no—
139C2×#102—1271no—
140#390—1271no—
141#336—1271no—
142C22×Q8—1271no—
143C22×D4C2×D12711yescharacteristic
144C22×D4C2×D12711yescharacteristic
145C2×SD16—2542no—
146C2×SD16—2542no—
147C2×SD16—2542no—
148C2×D8—22no—
149C2×D8—22no—
150C2×D8—22no—
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—21no—
163C2×Q8—1272no—
164C2×Q8—2541no—
165C2×Q8—2541no—
166C2×Q8—2541no—
167C2×Q8—2541no—
168C2×Q8—2541no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4C22⋊D12712yes—
180C2×D4C22⋊D12712yes—
181C22×C4—1272no—
182C22×C4—2541no—
183C22×C4—2541no—
184C22×C4C22×D12711yescharacteristic
185SD16—2548no—
186D8—28no—
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—22no—
201C23—41no—
202C23—41no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—41no—
208C23—41no—
209C23D4×D12711yescharacteristic
210Q8—5081no—
211Q8—5081no—
212Q8—2542no—
213Q8—2542no—
214D4—22no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—41no—
219D4—41no—
220D4—41no—
221D4—41no—
222C2×C4—5081no—
223C2×C4—5082no—
224C2×C4—5081no—
225C2×C4—5081no—
226C2×C4—2542no—
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—44no—
239C22—42no—
240C22—42no—
241C22—44no—
242C22—82no—
243C22—82no—
244C22—81no—
245C22—81no—
246C22—81no—
247C22—81no—
248C22#1686117711yescharacteristic
249C22#1686118111yescharacteristic
250C22#1686120511yescharacteristic
251C22#1686122812yes—
252C22#1686123312yes—
253C4—10161no—
254C4—5082no—
255C4—5082no—
256C4#1686117711yescharacteristic
257C4#1686117811yescharacteristic
258C4#1686118011yescharacteristic
259C4#1686118111yescharacteristic
260C2—82no—
261C2—82no—
262C2#2254978411yescharacteristic
263C2#2255001812yes—
264C2#2255008912yes—
265C2#2255025911yescharacteristic
266C2#2255026411yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting, prime radicalC22×D8×C127C2
socleC23×C127C2×D4
hypercenterC22×D8D127
hypocenterC127#2012
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×#14523236C2semidirect
2. C2×#16861196C2semidirect
3. C2×#16861134C2semidirect
4. C2×#16861130C2semidirect
5. #22549779C2semidirect
6. #22549630C2semidirect
134. C127#2012semidirect

Maximal subgroups

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

subgroupindex
C22×#145232362
C2×#168611962
C2×#168611342
C2×#168611302
#225497792
#225496302
C22×D8×C1272
#2012127