Finite Group Databasebeta

← #30639824

Subgroups — #30639824

Order 16256.

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

normal subgroups only

This group has 390 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×#14523238C211yesmaximal, characteristic, complemented
2C22×#14523236C211yesmaximal, characteristic
3C2×#16861133C211yesmaximal, characteristic, complemented
4C2×#16861130C211yesmaximal, characteristic
5#22549782C211yesmaximal, characteristic, complemented
6#22549631C211yesmaximal, characteristic, complemented
7C2×#124×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523238—22no—
10C2×#14523238—22no—
11C2×#14523238—22no—
12C2×#14523236—22no—
13C2×#14523236—22no—
14C2×#14523236—22no—
15C22×C127⋊C8C2211yescharacteristic
16C22×C127⋊C8C2211yescharacteristic
17C2×C4⋊Dic127C2211yescharacteristic
18#16861133—22no—
19#16861130—22no—
20C127⋊#95—22no—
21C22×Q8×C127C2211yescharacteristic
22C22×D4×C127C2211yescharacteristic
23#124×C127—22no—
24#124×C127—22no—
25C2×C22⋊C4×C127—22no—
26C2×C42×C127C2211yescharacteristic
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254—21no—
32C2×Dic254D412yes—
33C22×Dic127—21no—
34C22×Dic127—21no—
35#14523238—28no—
36#14523236—28no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8—21no—
40C2×C127⋊C8—21no—
41C2×C127⋊C8D412yes—
42C2×C127⋊C8D412yes—
43C4⋊Dic127—21no—
44C4⋊Dic127—21no—
45C24×C127—21no—
46C2×Q8×C127—21no—
47C2×Q8×C127—21no—
48C2×Q8×C127—21no—
49C2×Q8×C127—21no—
50C2×Q8×C127—21no—
51C2×Q8×C127D412yes—
52C2×D4×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127—21no—
56C2×D4×C127—21no—
57C2×D4×C127D412yes—
58C22×C4×C127—21no—
59C22×C4×C127D412yes—
60C22×C4×C127C2311yescharacteristic
61C22⋊C4×C127—28no—
62C42×C127—21no—
63C42×C127—21no—
64Dic254—22no—
65Dic254—22no—
66Dic254—41no—
67Dic254—41no—
68C2×Dic127—22no—
69C2×Dic127—41no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C2×Dic127—42no—
73C127⋊C8—22no—
74C127⋊C8—22no—
75C23×C127—22no—
76C23×C127—41no—
77C23×C127—41no—
78C23×C127—42no—
79C23×C127—42no—
80C23×C127C2×D411yescharacteristic
81Q8×C127—22no—
82Q8×C127—22no—
83Q8×C127—41no—
84Q8×C127—41no—
85D4×C127—22no—
86D4×C127—22no—
87D4×C127—41no—
88D4×C127—41no—
89C2×C4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127—22no—
92C2×C4×C127—22no—
93C2×C4×C127—41no—
94C2×C4×C127—41no—
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127#4111yescharacteristic
101C22×C127—42no—
102C22×C127—44no—
103C22×C127—82no—
104C22×C127—81no—
105C22×C127—81no—
106C22×C127#12011yescharacteristic
107C22×C127#12111yescharacteristic
108C22×C127#12711yescharacteristic
109C22×C127#13512yes—
110C22×C127#13712yes—
111C4×C127—42no—
112C4×C127—42no—
113C4×C127#12011yescharacteristic
114C4×C127#12011yescharacteristic
115C4×C127#12111yescharacteristic
116C4×C127#12111yescharacteristic
117Dic127—42no—
118Dic127—81no—
119C2×C127—82no—
120C2×C127#39211yescharacteristic
121C2×C127#45212yes—
122C2×C127#46412yes—
123C2×C127#49511yescharacteristic
124C2×C127#49711yescharacteristic
125#2015—1271nomaximal, Sylow
126C127#201511yescharacteristic, complemented, Sylow
127C22×Q16—1271no—
128C22×SD16—1271no—
129C2×#124D12711yescharacteristic
130C2×#103—1271no—
131C2×#102—1271no—
132#391—1271no—
133#336—1271no—
134C22×Q8—1271no—
135C22×Q8C2×D12711yescharacteristic
136C22×D4C2×D12711yescharacteristic
137C2×Q16—2542no—
138C2×Q16—2542no—
139C2×Q16—2542no—
140C2×SD16—2542no—
141C2×SD16—2542no—
142C2×SD16—2542no—
143C22×C8—1271no—
144C22×C8—1271no—
145#124—22no—
146#124—22no—
147C2×C4⋊C4—1271no—
148C2×C22⋊C4—22no—
149C2×C42C2×D12711yescharacteristic
150#103—2542no—
151#102—2542no—
152#95—2542no—
153C24—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—1272no—
160C2×Q8—2541no—
161C2×Q8—2541no—
162C2×Q8—2541no—
163C2×Q8—2541no—
164C2×Q8—2541no—
165C2×Q8C22⋊D12712yes—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4C22⋊D12712yes—
172C22×C4—21no—
173C22×C4—2541no—
174C22×C4—2541no—
175C22×C4C22⋊D12712yes—
176C22×C4C22×D12711yescharacteristic
177Q16—2548no—
178SD16—2548no—
179C2×C8—1272no—
180C2×C8—1272no—
181C2×C8—2541no—
182C2×C8—2541no—
183C2×C8—2541no—
184C2×C8—2541no—
185C4⋊C4—2541no—
186C4⋊C4—2541no—
187C22⋊C4—28no—
188C42—21no—
189C42—21no—
190C23—22no—
191C23—41no—
192C23—42no—
193C23—42no—
194C23—41no—
195C23C2×C22⋊D12711yescharacteristic
196Q8—22no—
197Q8—22no—
198Q8—41no—
199Q8—41no—
200Q8—5081no—
201Q8—5081no—
202Q8—2542no—
203Q8—2542no—
204D4—22no—
205D4—22no—
206D4—41no—
207D4—41no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—5081no—
215C2×C4—5082no—
216C2×C4—5081no—
217C2×C4—5081no—
218C2×C4—2542no—
219C2×C4D4×D12711yescharacteristic
220C2×C4D4×D12711yescharacteristic
221C2×C4D4×D12711yescharacteristic
222C2×C4C2×C22⋊D12711yescharacteristic
223C2×C4C2×C22⋊D12711yescharacteristic
224C2×C4#1452324711yescharacteristic
225C8—2542no—
226C8—2542no—
227C22—44no—
228C22—42no—
229C22—82no—
230C22—81no—
231C22—81no—
232C22#1686117811yescharacteristic
233C22#1686118011yescharacteristic
234C22#1686120511yescharacteristic
235C22#1686122912yes—
236C22#1686123912yes—
237C4—42no—
238C4—42no—
239C4—10161no—
240C4—5082no—
241C4#1686117711yescharacteristic
242C4#1686117711yescharacteristic
243C4#1686118111yescharacteristic
244C4#1686118111yescharacteristic
245C2—82no—
246C2#2254978411yescharacteristic
247C2#2255003812yes—
248C2#2255009512yes—
249C2#2255025811yescharacteristic
250C2#2255027111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#124×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
hypercenterC2×#124D127
hypocenterC127#2015
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C22×#14523238C2semidirect
3. C2×#16861133C2semidirect
5. #22549782C2semidirect
6. #22549631C2semidirect
126. C127#2015semidirect

Maximal subgroups

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

subgroupindex
C22×#145232382
C22×#145232362
C2×#168611332
C2×#168611302
#225497822
#225496312
C2×#124×C1272
#2015127