Finite Group Databasebeta

← #30639969

Subgroups — #30639969

Order 16256.

318 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16861196C211yesmaximal, characteristic
2C2×#16861134C211yesmaximal, characteristic, complemented
3C2×#16861132C211yesmaximal, characteristic, complemented
4C2×#16861130C211yesmaximal, characteristic
5#22549772C211yesmaximal, characteristic, complemented
6#22549630C211yesmaximal, characteristic, complemented
7C2×#102×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10#16861196—22no—
11#16861196—22no—
12C2×C4⋊Dic127C2211yescharacteristic
13C2×Dic127⋊C4—21no—
14C2×C22⋊Dic127—21no—
15C2×C22⋊Dic127—21no—
16C2×C4×Dic127C2211yescharacteristic
17#16861134—22no—
18#16861132—22no—
19#16861130—22no—
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
35C2×C127⋊C8—21no—
36C2×C127⋊C8—21no—
37C2×C127⋊C8—21no—
38C4⋊Dic127—21no—
39C4⋊Dic127D412yes—
40Dic127⋊C4—42no—
41C22⋊Dic127—24no—
42C22⋊Dic127—24no—
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—22no—
66C2×Dic127—22no—
67C2×Dic127—21no—
68C2×Dic127—21no—
69C2×Dic127—21no—
70C2×Dic127—21no—
71C2×Dic127—41no—
72C2×Dic127—41no—
73C2×Dic127—41no—
74C2×Dic127—41no—
75C127⋊C8—41no—
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#12011yescharacteristic
106C22×C127#12111yescharacteristic
107C22×C127#12411yescharacteristic
108C22×C127#13511yescharacteristic
109C22×C127#13511yescharacteristic
110C22×C127AGL(1,ℤ/8)11yescharacteristic
111C22×C127#13711yescharacteristic
112C4×C127—81no—
113C4×C127#12111yescharacteristic
114C4×C127#12111yescharacteristic
115C4×C127#12311yescharacteristic
116C4×C127#12311yescharacteristic
117Dic127—42no—
118Dic127—41no—
119Dic127—41no—
120Dic127—42no—
121C2×C127—82no—
122C2×C127#39411yescharacteristic
123C2×C127#44911yescharacteristic
124C2×C127#45211yescharacteristic
125C2×C127#46311yescharacteristic
126C2×C127#46411yescharacteristic
127C2×C127#48811yescharacteristic
128C2×C127#48911yescharacteristic
129#2042—1271nomaximal, Sylow
130C127#204211yescharacteristic, complemented, Sylow
131C2×#124—1271no—
132C2×#103—1271no—
133C2×#103—1271no—
134C2×#102—1271no—
135C2×#102D12711yescharacteristic
136#389—1271no—
137#336—1271no—
138C22×Q8—1271no—
139C22×D4C2×D12711yescharacteristic
140C22×C8—1271no—
141C22×C8C2×D12711yescharacteristic
142#124—2542no—
143#124—2542no—
144C2×C4⋊C4—1271no—
145C2×C4⋊C4—2541no—
146C2×C4⋊C4C2×D12711yescharacteristic
147C2×C22⋊C4—2541no—
148C2×C22⋊C4—2541no—
149C2×C42—1271no—
150#103—2542no—
151#103—2542no—
152#102—22no—
153#102—2542no—
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
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—2541no—
178C2×C8—2541no—
179C2×C8—2541no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—1272no—
183C4⋊C4—5082no—
184C4⋊C4—2541no—
185C22⋊C4—2544no—
186C22⋊C4—2544no—
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—5081no—
209C2×C4—5081no—
210C2×C4—5081no—
211C2×C4—5081no—
212C2×C4—2541no—
213C2×C4—2541no—
214C2×C4—2541no—
215C2×C4—2541no—
216C2×C4—2541no—
217C2×C4—2541no—
218C2×C4—2542no—
219C2×C4—2542no—
220C2×C4D4×D12711yescharacteristic
221C2×C4D4×D12711yescharacteristic
222C2×C4C2×C22⋊D12711yescharacteristic
223C2×C4#1452324711yescharacteristic
224C2×C4#1452324811yescharacteristic
225C2×C4#1452324811yescharacteristic
226C8—41no—
227C8—5081no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—81no—
232C22—81no—
233C22—81no—
234C22—81no—
235C22#1686117711yescharacteristic
236C22#1686118211yescharacteristic
237C22#1686119511yescharacteristic
238C22#1686122811yescharacteristic
239C22#1686123011yescharacteristic
240C22#1686123311yescharacteristic
241C22#1686123811yescharacteristic
242C4—81no—
243C4—5081no—
244C4—5081no—
245C4—5082no—
246C4—5082no—
247C4#1686118111yescharacteristic
248C4#1686118211yescharacteristic
249C4#1686119111yescharacteristic
250C4#1686119211yescharacteristic
251C2—82no—
252C2#2254979111yescharacteristic
253C2#2255001811yescharacteristic
254C2#2255003511yescharacteristic
255C2#2255009011yescharacteristic
256C2#2255009711yescharacteristic
257C2#2255022911yescharacteristic
258C2#2255023411yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
hypercenterC2×#102D127
hypocenterC127#2042
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×#16861134C2semidirect
3. C2×#16861132C2semidirect
5. #22549772C2semidirect
6. #22549630C2semidirect
130. C127#2042semidirect

Maximal subgroups

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

subgroupindex
C2×#168611962
C2×#168611342
C2×#168611322
C2×#168611302
#225497722
#225496302
C2×#102×C1272
#2042127