Finite Group Databasebeta

← #30640046

Subgroups — #30640046

Order 16256.

324 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 85 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
2C2×Dic127⋊C8C211yesmaximal, characteristic, complemented
3C2×#16861134C211yesmaximal, characteristic, complemented
4#22549781C211yesmaximal, characteristic, complemented
5#22549779C211yesmaximal, characteristic
6#22549638C211yesmaximal, characteristic, complemented
7C2×#102×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523236—22no—
10C2×#14523236—22no—
11C2×#14523236—22no—
12C22×C127⋊C8C2211yescharacteristic
13C2×C4⋊Dic127C2211yescharacteristic
14C2×C22⋊Dic127—21no—
15C2×C22⋊Dic127—21no—
16C2×C4×Dic127C2211yescharacteristic
17Dic127⋊C8—22no—
18#16861134—22no—
19C127⋊#95—21no—
20C127⋊#95—21no—
21C22×D4×C127C2211yescharacteristic
22C22×C8×C127C2211yescharacteristic
23C2×C4⋊C4×C127C2211yescharacteristic
24#102×C127—22no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254D412yes—
31C22×Dic127—21no—
32C22×Dic127—21no—
33C22×Dic127D411yescharacteristic
34C22×Dic127D411yescharacteristic
35#14523236—28no—
36C2×C127⋊C8—21no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8D412yes—
39C4⋊Dic127—21no—
40C4⋊Dic127—21no—
41C22⋊Dic127—42no—
42C22⋊Dic127—42no—
43C4×Dic127—21no—
44C4×Dic127—21no—
45C24×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127—21no—
48C2×D4×C127—21no—
49C2×D4×C127—21no—
50C2×D4×C127—21no—
51C2×D4×C127D412yes—
52C22×C4×C127—21no—
53C22×C4×C127C2311yescharacteristic
54C2×C8×C127—21no—
55C2×C8×C127—21no—
56C2×C8×C127—21no—
57C4⋊C4×C127—21no—
58C4⋊C4×C127—21no—
59Dic254—22no—
60Dic254—22no—
61Dic254—41no—
62Dic254—41no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—21no—
66C2×Dic127—22no—
67C2×Dic127—21no—
68C2×Dic127—21no—
69C2×Dic127—41no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C2×Dic127—41no—
73C2×Dic127—41no—
74C2×Dic127SD1612yescomplemented
75C127⋊C8—22no—
76C23×C127—22no—
77C23×C127—41no—
78C23×C127—41no—
79C23×C127—41no—
80C23×C127—41no—
81C23×C127—41no—
82C23×C127—41no—
83C23×C127C2×D411yescharacteristic
84D4×C127—22no—
85D4×C127—22no—
86D4×C127—41no—
87D4×C127—41no—
88C2×C4×C127—41no—
89C2×C4×C127—41no—
90C2×C4×C127—41no—
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127#4111yescharacteristic
95C2×C4×C127#4111yescharacteristic
96C2×C4×C127#4111yescharacteristic
97C8×C127—41no—
98C22×C127—42no—
99C22×C127—42no—
100C22×C127—42no—
101C22×C127—81no—
102C22×C127—81no—
103C22×C127—81no—
104C22×C127—81no—
105C22×C127#12111yescharacteristic
106C22×C127#12111yescharacteristic
107C22×C127#12111yescharacteristic
108C22×C127C2×SD1611yescharacteristic
109C22×C127#13511yescharacteristic
110C22×C127AGL(1,ℤ/8)11yescharacteristic
111C22×C127#13711yescharacteristic
112C4×C127—81no—
113C4×C127#12011yescharacteristic
114C4×C127#12311yescharacteristic
115C4×C127#12311yescharacteristic
116C4×C127#12411yescharacteristic
117Dic127—22no—
118Dic127—42no—
119Dic127—41no—
120Dic127—81no—
121C2×C127—82no—
122C2×C127#39411yescharacteristic
123C2×C127#46011yescharacteristic
124C2×C127#46111yescharacteristic
125C2×C127#46311yescharacteristic
126C2×C127#46411yescharacteristic
127C2×C127#46511yescharacteristic
128C2×C127#46811yescharacteristic
129#2055—1271nomaximal, Sylow
130C127#205511yescharacteristic, complemented, Sylow
131C22×SD16—1271no—
132C2×C4⋊C8—1271no—
133C2×#103—1271no—
134C2×#102D12711yescharacteristic
135#391—1271no—
136#390—1271no—
137#340—1271no—
138C22×Q8—1271no—
139C22×D4C2×D12711yescharacteristic
140C2×SD16—2542no—
141C2×SD16—2542no—
142C2×SD16—2542no—
143C22×C8—1271no—
144C22×C8C2×D12711yescharacteristic
145C2×C4⋊C4—1271no—
146C2×C4⋊C4C2×D12711yescharacteristic
147C2×C22⋊C4—2541no—
148C2×C22⋊C4—2541no—
149C2×C42—1271no—
150C4⋊C8—2542no—
151#103—2542no—
152#102—22no—
153#95—2541no—
154#95—2541no—
155C24—21no—
156C2×Q8—1272no—
157C2×Q8—2541no—
158C2×Q8—2541no—
159C2×Q8—2541no—
160C2×Q8—2541no—
161C2×Q8—2541no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4C22⋊D12712yes—
168C22×C4—21no—
169C22×C4—1271no—
170C22×C4—1271no—
171C22×C4—2541no—
172C22×C4—2541no—
173C22×C4C22×D12711yescharacteristic
174SD16—2548no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—1272no—
179C2×C8—2541no—
180C2×C8—2541no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—2541no—
184C4⋊C4—2541no—
185C22⋊C4—5082no—
186C22⋊C4—5082no—
187C42—2541no—
188C42—2541no—
189C23—22no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23D4×D12711yescharacteristic
197Q8—5081no—
198Q8—5081no—
199Q8—2542no—
200Q8—2542no—
201D4—22no—
202D4—22no—
203D4—41no—
204D4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—1272no—
209C2×C4—5081no—
210C2×C4—5081no—
211C2×C4—5081no—
212C2×C4—5081no—
213C2×C4—5081no—
214C2×C4—2541no—
215C2×C4—2541no—
216C2×C4—2541no—
217C2×C4—2542no—
218C2×C4—2541no—
219C2×C4—2541no—
220C2×C4D4×D12711yescharacteristic
221C2×C4D4×D12711yescharacteristic
222C2×C4C2×C22⋊D12711yescharacteristic
223C2×C4#1452324711yescharacteristic
224C2×C4#1452324811yescharacteristic
225C2×C4#1452324811yescharacteristic
226C8—41no—
227C8—2542no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—81no—
232C22—81no—
233C22—81no—
234C22—81no—
235C22#1686118111yescharacteristic
236C22#1686118211yescharacteristic
237C22#1686118211yescharacteristic
238C22SD16×D12711yescharacteristic
239C22#1686122811yescharacteristic
240C22#1686123311yescharacteristic
241C22#1686123811yescharacteristic
242C4—81no—
243C4—10161no—
244C4—5082no—
245C4—5081no—
246C4—2542no—
247C4#1686117711yescharacteristic
248C4#1686119111yescharacteristic
249C4#1686119211yescharacteristic
250C4#1686119511yescharacteristic
251C2—82no—
252C2#2254979111yescharacteristic
253C2#2255007511yescharacteristic
254C2#2255007911yescharacteristic
255C2#2255008911yescharacteristic
256C2#2255009711yescharacteristic
257C2#2255010411yescharacteristic
258C2#2255012111yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
hypercenterC2×#102D127
hypocenterC127#2055
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×Dic127⋊C8C2semidirect
3. C2×#16861134C2semidirect
4. #22549781C2semidirect
6. #22549638C2semidirect
74. C2×Dic127SD16semidirect
130. C127#2055semidirect

Maximal subgroups

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

subgroupindex
C22×#145232362
C2×Dic127⋊C82
C2×#168611342
#225497812
#225497792
#225496382
C2×#102×C1272
#2055127