Finite Group Databasebeta

← #30639848

Subgroups — #30639848

Order 16256.

342 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 75 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×#16861239C211yesmaximal, characteristic
2C22×#14523238C211yesmaximal, characteristic, complemented
3C2×#16861128C211yesmaximal, characteristic, complemented
4C2×#16861127C211yesmaximal, characteristic, complemented
5#22549843C211yesmaximal, characteristic
6#22549640C211yesmaximal, characteristic, complemented
7C2×#137×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9#16861239—22no—
10#16861239—22no—
11C2×#14523238—22no—
12C2×#14523238—22no—
13C2×#14523238—22no—
14C2×#14523238—21no—
15C2×#14523236—21no—
16C2×#14523232C2211yescharacteristic
17C2×#14523232C2211yescharacteristic
18C22×C127⋊C8C2211yescharacteristic
19#16861128—22no—
20#16861127—22no—
21C22⋊(C127⋊C8)—22no—
22C2×#41×C127C2211yescharacteristic
23C22×Q8×C127C2211yescharacteristic
24#137×C127—22no—
25#137×C127—22no—
26C2×Q16×C127—21no—
27C2×SD16×C127—21no—
28C2×M16×C127C2211yescharacteristic
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254—21no—
32C2×Dic254—21no—
33C2×Dic254—21no—
34C2×Dic254D412yes—
35C22×Dic127—21no—
36#14523238—28no—
37#14523238—24no—
38#14523236—24no—
39#14523232—21no—
40#14523232—21no—
41#14523232—21no—
42#14523232D412yes—
43C2×C127⋊C8—21no—
44C2×C127⋊C8—21no—
45C2×C127⋊C8—21no—
46C2×C127⋊C8—21no—
47C2×C127⋊C8D412yes—
48#41×C127—21no—
49#41×C127—21no—
50#41×C127—21no—
51#41×C127—21no—
52C2×Q8×C127—21no—
53C2×Q8×C127—21no—
54C2×Q8×C127—21no—
55C2×Q8×C127—21no—
56C2×Q8×C127—21no—
57C2×Q8×C127D411yescharacteristic
58C2×Q8×C127D412yes—
59C2×D4×C127—21no—
60C2×D4×C127D411yescharacteristic
61C22×C4×C127—21no—
62C22×C4×C127—21no—
63C22×C4×C127C2311yescharacteristic
64Q16×C127—24no—
65SD16×C127—24no—
66M16×C127—21no—
67M16×C127D412yes—
68C2×C8×C127—21no—
69Dic254—22no—
70Dic254—22no—
71Dic254—41no—
72Dic254—41no—
73C2×Dic127—22no—
74C2×Dic127—41no—
75C2×Dic127—41no—
76C127⋊C8—22no—
77C127⋊C8—22no—
78C127⋊C8—41no—
79C23×C127—21no—
80C23×C127#4111yescharacteristic
81Q8×C127—22no—
82Q8×C127—21no—
83Q8×C127—22no—
84Q8×C127—21no—
85Q8×C127—41no—
86Q8×C127—41no—
87D4×C127—21no—
88D4×C127—21no—
89D4×C127—41no—
90D4×C127—41no—
91C2×C4×C127—21no—
92C2×C4×C127—22no—
93C2×C4×C127—41no—
94C2×C4×C127—41no—
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×C127C2×D411yescharacteristic
103C8×C127—22no—
104C22×C127—22no—
105C22×C127—41no—
106C22×C127—42no—
107C22×C127#12111yescharacteristic
108C22×C127#12111yescharacteristic
109C22×C127#12111yescharacteristic
110C4×C127—41no—
111C4×C127—42no—
112C4×C127#12011yescharacteristic
113C4×C127#12011yescharacteristic
114C4×C127#12011yescharacteristic
115C4×C127#12711yescharacteristic
116Dic127—42no—
117C2×C127—21no—
118C2×C127—21no—
119C2×C127—81no—
120C2×C127#39211yescharacteristic
121C2×C127#47312yes—
122#2020—1271nomaximal, Sylow
123C127#202011yescharacteristic, complemented, Sylow
124C2×#137—1271no—
125C2×#137D12711yescharacteristic
126C22×Q16—1271no—
127C2×#101—1271no—
128C2×#101—1271no—
129#408—1271no—
130#341—1271no—
131C2×#41C2×D12711yescharacteristic
132C22×Q8—1271no—
133C22×Q8C2×D12711yescharacteristic
134#137—22no—
135#137—22no—
136#137—2542no—
137#137—2542no—
138C2×Q16—21no—
139C2×Q16—2542no—
140C2×Q16—2542no—
141C2×Q16—2542no—
142C2×Q16—2541no—
143C2×SD16—21no—
144C2×SD16—2541no—
145C2×M16—1271no—
146C2×M16—1271no—
147C2×M16C2×D12711yescharacteristic
148C22×C8—1271no—
149#101—2542no—
150#101—2542no—
151C22⋊C8—2542no—
152#41—21no—
153#41—21no—
154#41—21no—
155#41—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—1272no—
162C2×Q8—2541no—
163C2×Q8—2541no—
164C2×Q8—2541no—
165C2×Q8—2541no—
166C2×Q8—2541no—
167C2×Q8C22⋊D12711yescharacteristic
168C2×Q8C22⋊D12712yes—
169C2×D4—21no—
170C2×D4C22⋊D12711yescharacteristic
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—2541no—
174C22×C4C22×D12711yescharacteristic
175Q16—24no—
176Q16—2548no—
177Q16—2544no—
178SD16—24no—
179SD16—2544no—
180M16—21no—
181M16—1272no—
182M16—2541no—
183M16—2541no—
184M16—2541no—
185M16C22⋊D12712yes—
186C2×C8—21no—
187C2×C8—1272no—
188C2×C8—2541no—
189C2×C8—2541no—
190C2×C8—2541no—
191C2×C8—2541no—
192C23—21no—
193C23#1452324711yescharacteristic
194Q8—22no—
195Q8—21no—
196Q8—22no—
197Q8—21no—
198Q8—41no—
199Q8—41no—
200Q8—5081no—
201Q8—5081no—
202Q8—2542no—
203Q8—2542no—
204D4—21no—
205D4—21no—
206D4—41no—
207D4—41no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—5081no—
215C2×C4—5081no—
216C2×C4—2542no—
217C2×C4D4×D12711yescharacteristic
218C2×C4D4×D12711yescharacteristic
219C2×C4D4×D12711yescharacteristic
220C2×C4C2×C22⋊D12711yescharacteristic
221C2×C4C2×C22⋊D12711yescharacteristic
222C2×C4C2×C22⋊D12711yescharacteristic
223C8—22no—
224C8—5081no—
225C8—2542no—
226C8—2542no—
227C22—22no—
228C22—41no—
229C22—42no—
230C22#1686118011yescharacteristic
231C22#1686118111yescharacteristic
232C22#1686118111yescharacteristic
233C4—41no—
234C4—42no—
235C4—5082no—
236C4#1686117711yescharacteristic
237C4#1686117711yescharacteristic
238C4#1686117811yescharacteristic
239C4#1686120511yescharacteristic
240C2—21no—
241C2—21no—
242C2—81no—
243C2#2254978411yescharacteristic
244C2#2255014912yes—
subgroupG/H
centerC22#16861181
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#137×C127C2
socleC22×C127#121
prime radicalC2×#41×C127C22
hypercenterC2×#137D127
hypocenterC127#2020
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#14523238C2semidirect
3. C2×#16861128C2semidirect
4. C2×#16861127C2semidirect
6. #22549640C2semidirect
123. C127#2020semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
C22×#145232382
C2×#168611282
C2×#168611272
#225498432
#225496402
C2×#137×C1272
#2020127