Finite Group Databasebeta

← #30639763

Subgroups — #30639763

Order 16256.

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

normal subgroups only

This group has 362 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
1#22550447C211yesmaximal, characteristic
2C2×#16861207C211yesmaximal, characteristic, complemented
3C2×#16861136C211yesmaximal, characteristic
4#22549885C211yesmaximal, characteristic, complemented
5C2×#16861127C211yesmaximal, characteristic, complemented
6#22549637C211yesmaximal, characteristic, complemented
7C2×#137×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10#16861207—22no—
11#16861207—22no—
12#16861207—21no—
13#16861196—21no—
14#16861190—22no—
15#16861188—22no—
16#16861168C2211yescharacteristic
17C2×Dic127⋊C4—22no—
18C2×C4×Dic127C2211yescharacteristic
19#16861136—22no—
20#16861133—22no—
21#16861127—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×Dic254—21no—
35C22×Dic127—21no—
36C22×Dic127D412yes—
37#14523232—21no—
38#14523232—21no—
39C2×C127⋊C8—21no—
40C4⋊Dic127D412yes—
41Dic127⋊C4—24no—
42Dic127⋊C4—28no—
43C22⋊Dic127—24no—
44C22⋊Dic127—21no—
45C4×Dic127—21no—
46C4×Dic127—21no—
47C4×Dic127D412yes—
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—41no—
70Dic254—41no—
71Dic254—41no—
72Dic254—41no—
73C2×Dic127—24no—
74C2×Dic127—22no—
75C2×Dic127—22no—
76C2×Dic127—22no—
77C2×Dic127—22no—
78C2×Dic127—22no—
79C127⋊C8—41no—
80C23×C127—21no—
81C23×C127C2×D411yescharacteristic
82Q8×C127—22no—
83Q8×C127—21no—
84Q8×C127—22no—
85Q8×C127—21no—
86Q8×C127—41no—
87Q8×C127—41no—
88D4×C127—21no—
89D4×C127—21no—
90D4×C127—41no—
91D4×C127—41no—
92C2×C4×C127—21no—
93C2×C4×C127—22no—
94C2×C4×C127—41no—
95C2×C4×C127—41no—
96C2×C4×C127—41no—
97C2×C4×C127—41no—
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127#4111yescharacteristic
104C8×C127—22no—
105C22×C127—22no—
106C22×C127—41no—
107C22×C127—42no—
108C22×C127#12011yescharacteristic
109C22×C127#12011yescharacteristic
110C22×C127#12111yescharacteristic
111C4×C127—41no—
112C4×C127—42no—
113C4×C127#12011yescharacteristic
114C4×C127#12111yescharacteristic
115C4×C127#12111yescharacteristic
116C4×C127#12711yescharacteristic
117Dic127—24no—
118Dic127—42no—
119Dic127—42no—
120C2×C127—21no—
121C2×C127—21no—
122C2×C127—81no—
123C2×C127#39211yescharacteristic
124C2×C127#45612yes—
125#2005—1271nomaximal, Sylow
126C127#200511yescharacteristic, complemented, Sylow
127C2×#137D12711yescharacteristic
128#536—1271no—
129C2×C4⋊Q8—1271no—
130C2×#104—1271no—
131#419—1271no—
132C2×#101—1271no—
133#339—1271no—
134C2×#41C2×D12711yescharacteristic
135C22×Q8—1271no—
136C22×Q8C2×D12711yescharacteristic
137#137—22no—
138#137—22no—
139C2×Q16—21no—
140C2×SD16—21no—
141C2×M16—1271no—
142C2×M16C2×D12711yescharacteristic
143C4⋊Q8—2542no—
144C4⋊Q8—2542no—
145C4⋊Q8—2541no—
146#124—2541no—
147#123—2542no—
148C22⋊Q8—2542no—
149#117—1271no—
150C2×C4⋊C4—2542no—
151C2×C42—1271no—
152#104—2542no—
153#103—2542no—
154#101—2542no—
155#41—21no—
156#41—21no—
157#41—21no—
158#41—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—2541no—
165C2×Q8—2541no—
166C2×Q8—2541no—
167C2×Q8—2541no—
168C2×Q8—2541no—
169C2×Q8—2541no—
170C2×Q8C22⋊D12711yescharacteristic
171C2×Q8C22⋊D12712yes—
172C2×D4—21no—
173C2×D4C22⋊D12711yescharacteristic
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—1272no—
177C22×C4—2541no—
178C22×C4C22×D12711yescharacteristic
179Q16—24no—
180SD16—24no—
181M16—21no—
182M16—2541no—
183M16—2541no—
184M16C22⋊D12712yes—
185C2×C8—21no—
186C2×C8—2541no—
187C4⋊C4—1272no—
188C4⋊C4—2548no—
189C4⋊C4—2544no—
190C22⋊C4—2544no—
191C22⋊C4—2541no—
192C42—1272no—
193C42—2541no—
194C42—2541no—
195C23—21no—
196C23D4×D12711yescharacteristic
197Q8—22no—
198Q8—21no—
199Q8—22no—
200Q8—21no—
201Q8—41no—
202Q8—41no—
203Q8—5081no—
204Q8—5081no—
205Q8—5081no—
206Q8—5081no—
207D4—21no—
208D4—21no—
209D4—41no—
210D4—41no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—2542no—
218C2×C4—2542no—
219C2×C4—2542no—
220C2×C4—2542no—
221C2×C4—2544no—
222C2×C4—2542no—
223C2×C4D4×D12711yescharacteristic
224C2×C4D4×D12711yescharacteristic
225C2×C4C2×C22⋊D12711yescharacteristic
226C2×C4C2×C22⋊D12711yescharacteristic
227C2×C4C2×C22⋊D12711yescharacteristic
228C2×C4#1452324711yescharacteristic
229C8—22no—
230C8—5081no—
231C22—22no—
232C22—41no—
233C22—42no—
234C22#1686117711yescharacteristic
235C22#1686117711yescharacteristic
236C22#1686118011yescharacteristic
237C4—41no—
238C4—42no—
239C4—5082no—
240C4—5082no—
241C4—2544no—
242C4#1686117811yescharacteristic
243C4#1686118111yescharacteristic
244C4#1686118111yescharacteristic
245C4#1686120511yescharacteristic
246C2—21no—
247C2—21no—
248C2—81no—
249C2#2254978411yescharacteristic
250C2#2255005712yes—
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#137×C127C2
socleC22×C127#120
prime radicalC2×#41×C127C22
hypercenterC2×#137D127
hypocenterC127#2005
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. C2×#16861207C2semidirect
4. #22549885C2semidirect
5. C2×#16861127C2semidirect
6. #22549637C2semidirect
126. C127#2005semidirect

Maximal subgroups

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

subgroupindex
#225504472
C2×#168612072
C2×#168611362
#225498852
C2×#168611272
#225496372
C2×#137×C1272
#2005127