Finite Group Databasebeta

← #30639071

Subgroups — #30639071

Order 16256.

338 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 113 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×#14523238C211yesmaximal, characteristic
2C2×#16861133C211yesmaximal, characteristic
3C2×#16861132C211yesmaximal, characteristic
4#22549782C211yesmaximal, characteristic
5#22549638C211yesmaximal, characteristic
6#22549631C211yesmaximal, characteristic
7C2×C4×Q8×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523238—22no—
10C2×#14523238—22no—
11C2×#14523238C414yescomplemented
12C22×C127⋊C8C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14C2×C4⋊Dic127C2211yescharacteristic
15#16861133—22no—
16#16861132—22no—
17C127⋊#95—21no—
18C127⋊#95—21no—
19C22×Q8×C127C2211yescharacteristic
20C4×Q8×C127—22no—
21C4×Q8×C127—22no—
22C2×C4⋊C4×C127—21no—
23C2×C4⋊C4×C127C2211yescharacteristic
24C2×C42×C127—21no—
25C2×C42×C127C2211yescharacteristic
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254C2×C412yes—
32C22×Dic127—21no—
33C22×Dic127—21no—
34#14523238—28no—
35C2×C127⋊C8—21no—
36C2×C127⋊C8—21no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8—21no—
40C2×C127⋊C8C2×C412yes—
41C4⋊Dic127—21no—
42C4⋊Dic127—21no—
43C2×Q8×C127—21no—
44C2×Q8×C127—21no—
45C2×Q8×C127—21no—
46C2×Q8×C127C2×C412yes—
47C2×Q8×C127D412yes—
48C2×Q8×C127D412yes—
49C22×C4×C127—21no—
50C22×C4×C127—21no—
51C22×C4×C127D411yescharacteristic
52C22×C4×C127D411yescharacteristic
53C22×C4×C127C2311yescharacteristic
54C4⋊C4×C127—24no—
55C4⋊C4×C127—21no—
56C4⋊C4×C127D412yes—
57C42×C127—24no—
58C42×C127—21no—
59C42×C127—21no—
60Dic254—22no—
61Dic254—22no—
62Dic254—41no—
63Dic254—41no—
64C2×Dic127—22no—
65C2×Dic127—41no—
66C2×Dic127—41no—
67C2×Dic127—41no—
68C2×Dic127—41no—
69C2×Dic127—41no—
70C127⋊C8—22no—
71C127⋊C8—41no—
72C23×C127C2×D411yescharacteristic
73Q8×C127—22no—
74Q8×C127—22no—
75Q8×C127—22no—
76Q8×C127C22⋊C414yes—
77C2×C4×C127—22no—
78C2×C4×C127—21no—
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—21no—
82C2×C4×C127—21no—
83C2×C4×C127—22no—
84C2×C4×C127—21no—
85C2×C4×C127—21no—
86C2×C4×C127—41no—
87C2×C4×C127—41no—
88C2×C4×C127Q1612yescomplemented
89C2×C4×C127C22×C411yescharacteristic
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127#4111yescharacteristic
94C2×C4×C127#4111yescharacteristic
95C22×C127C4×D411yescharacteristic
96C22×C127#12011yescharacteristic
97C22×C127#12111yescharacteristic
98C22×C127C2×Q1611yescharacteristic
99C22×C127#13511yescharacteristic
100C22×C127#13711yescharacteristic
101C22×C127#13711yescharacteristic
102C4×C127—24no—
103C4×C127—22no—
104C4×C127—41no—
105C4×C127—42no—
106C4×C127C2×C22⋊C411yescharacteristic
107C4×C127C4×D411yescharacteristic
108C4×C127#12111yescharacteristic
109C4×C127#12311yescharacteristic
110Dic127—42no—
111Dic127—81no—
112C2×C127#38611yescharacteristic
113C2×C127C4×Q1611yescharacteristic
114C2×C127#44111yescharacteristic
115C2×C127#45111yescharacteristic
116C2×C127#45211yescharacteristic
117C2×C127#46211yescharacteristic
118C2×C127#46411yescharacteristic
119#1878—1271nomaximal, Sylow
120C127#187811yescharacteristic, complemented, Sylow
121C22×Q16—1271no—
122C2×C4×Q8D12711yescharacteristic
123C2×#103—1271no—
124C2×#103—1271no—
125#391—1271no—
126#340—1271no—
127#336—1271no—
128C22×Q8—1271no—
129C22×Q8C2×D12711yescharacteristic
130C2×Q16—1274no—
131C2×Q16—2542no—
132C2×Q16—2542no—
133C22×C8—1271no—
134C22×C8—1271no—
135C4×Q8—22no—
136C4×Q8—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—1271no—
139C2×C4⋊C4C2×D12711yescharacteristic
140C2×C42—21no—
141C2×C42C2×D12711yescharacteristic
142#103—2542no—
143#103—2542no—
144#95—2541no—
145#95—2541no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—1272no—
150C2×Q8—2541no—
151C2×Q8—2541no—
152C2×Q8—2541no—
153C2×Q8—2541no—
154C2×Q8—2541no—
155C2×Q8C4×D12712yes—
156C2×Q8C22⋊D12712yes—
157C2×Q8D50812yes—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—2541no—
161C22×C4—2541no—
162C22×C4C22⋊D12711yescharacteristic
163C22×C4C22⋊D12711yescharacteristic
164C22×C4C22×D12711yescharacteristic
165Q16—2548no—
166C2×C8—1272no—
167C2×C8—2541no—
168C2×C8—2541no—
169C2×C8—2541no—
170C2×C8—2541no—
171C2×C8—2541no—
172C4⋊C4—24no—
173C4⋊C4—21no—
174C4⋊C4—2541no—
175C4⋊C4—2541no—
176C4⋊C4C22⋊D12712yes—
177C42—24no—
178C42—21no—
179C42—21no—
180C23C2×C22⋊D12711yescharacteristic
181Q8—22no—
182Q8—22no—
183Q8—22no—
184Q8—5081no—
185Q8—5081no—
186Q8—2542no—
187Q8—2542no—
188Q8(C2×D127)⋊C414yes—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—5081no—
201C2×C4—5081no—
202C2×C4—5081no—
203C2×C4—5081no—
204C2×C4—5081no—
205C2×C4—2542no—
206C2×C4#1452323812yescomplemented
207C2×C4C2×C4×D12711yescharacteristic
208C2×C4C2×C22⋊D12711yescharacteristic
209C2×C4C2×C22⋊D12711yescharacteristic
210C2×C4C2×D50811yescharacteristic
211C2×C4#1452324811yescharacteristic
212C2×C4#1452324811yescharacteristic
213C8—5081no—
214C8—2542no—
215C22C4×C22⋊D12711yescharacteristic
216C22#1686117811yescharacteristic
217C22#1686118311yescharacteristic
218C22C2×#1452323811yescharacteristic
219C22#1686122911yescharacteristic
220C22#1686123911yescharacteristic
221C22#1686124011yescharacteristic
222C4—22no—
223C4—24no—
224C4—41no—
225C4—42no—
226C4—10161no—
227C4—5082no—
228C4C2×(C2×D127)⋊C411yescharacteristic
229C4C4×C22⋊D12711yescharacteristic
230C4#1686118311yescharacteristic
231C4#1686119211yescharacteristic
232C2#2254975511yescharacteristic
233C2C4×#1452323811yescharacteristic
234C2#2254998111yescharacteristic
235C2#2255003211yescharacteristic
236C2#2255003811yescharacteristic
237C2#2255008611yescharacteristic
238C2#2255009811yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×C4×Q8×C127C2
socle, prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenterC2×C4×Q8D127
hypocenterC127#1878
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
11. C2×#14523238C4semidirect
88. C2×C4×C127Q16semidirect
120. C127#1878semidirect
206. C2×C4#14523238semidirect

Maximal subgroups

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

subgroupindex
C22×#145232382
C2×#168611332
C2×#168611322
#225497822
#225496382
#225496312
C2×C4×Q8×C1272
#1878127