Finite Group Databasebeta

← #30640039

Subgroups — #30640039

Order 16256.

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

normal subgroups only

This group has 360 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×#14523233C211yesmaximal, characteristic
2C2×Dic127⋊C8C211yesmaximal, characteristic, complemented
3C2×#16861131C211yesmaximal, characteristic, complemented
4#22549779C211yesmaximal, characteristic, complemented
5#22549777C211yesmaximal, characteristic, complemented
6#22549638C211yesmaximal, characteristic, complemented
7C2×#102×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C2×#14523233—22no—
10C2×#14523233—22no—
11C2×#14523233—22no—
12C22×C127⋊C8C2211yescharacteristic
13C2×C4⋊Dic127C2211yescharacteristic
14C2×C22⋊Dic127—21no—
15C2×C22⋊Dic127—21no—
16C2×(C2×D127)⋊C4—21no—
17C2×(C2×D127)⋊C4—21no—
18C2×C4×Dic127C2211yescharacteristic
19Dic127⋊C8—22no—
20#16861131—22no—
21C22×D4×C127C2211yescharacteristic
22C22×C8×C127C2211yescharacteristic
23C2×C4⋊C4×C127C2211yescharacteristic
24#102×C127—22no—
25C23×D127—21no—
26C2×D508—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508D412yes—
32C22×Dic127—21no—
33C22×Dic127D411yescharacteristic
34C22×Dic127D411yescharacteristic
35#14523233—28no—
36C2×C127⋊C8—21no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8D412yes—
39C4⋊Dic127—21no—
40C4⋊Dic127—21no—
41C22⋊Dic127—42no—
42C22⋊Dic127—42no—
43(C2×D127)⋊C4—42no—
44(C2×D127)⋊C4—42no—
45C4×Dic127—21no—
46C4×Dic127—21no—
47C24×C127—21no—
48C2×D4×C127—21no—
49C2×D4×C127—21no—
50C2×D4×C127—21no—
51C2×D4×C127—21no—
52C2×D4×C127—21no—
53C2×D4×C127D412yes—
54C22×C4×C127—21no—
55C22×C4×C127C2311yescharacteristic
56C2×C8×C127—21no—
57C2×C8×C127—21no—
58C2×C8×C127—21no—
59C4⋊C4×C127—21no—
60C4⋊C4×C127—21no—
61C22×D127—22no—
62C22×D127—41no—
63C22×D127—41no—
64C22×D127—41no—
65C22×D127—41no—
66C22×D127—41no—
67C22×D127—41no—
68D508—22no—
69D508—22no—
70D508—41no—
71D508—41no—
72C2×Dic127—21no—
73C2×Dic127—21no—
74C2×Dic127—21no—
75C2×Dic127—21no—
76C2×Dic127—21no—
77C2×Dic127—41no—
78C2×Dic127—41no—
79C2×Dic127—41no—
80C2×Dic127D812yescomplemented
81C127⋊C8—22no—
82C23×C127—22no—
83C23×C127—41no—
84C23×C127—41no—
85C23×C127—41no—
86C23×C127—41no—
87C23×C127—41no—
88C23×C127—41no—
89C23×C127C2×D411yescharacteristic
90D4×C127—22no—
91D4×C127—22no—
92D4×C127—41no—
93D4×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×C127#4111yescharacteristic
101C2×C4×C127#4111yescharacteristic
102C2×C4×C127#4111yescharacteristic
103C8×C127—41no—
104C22×C127—42no—
105C22×C127—42no—
106C22×C127—42no—
107C22×C127—81no—
108C22×C127—81no—
109C22×C127—81no—
110C22×C127—81no—
111C22×C127#12111yescharacteristic
112C22×C127#12111yescharacteristic
113C22×C127#12111yescharacteristic
114C22×C127C2×D811yescharacteristic
115C22×C127#13511yescharacteristic
116C22×C127AGL(1,ℤ/8)11yescharacteristic
117C22×C127AGL(1,ℤ/8)11yescharacteristic
118C2×D127—42no—
119C2×D127—42no—
120C2×D127—42no—
121C2×D127—81no—
122C2×D127—81no—
123C2×D127—81no—
124C2×D127—81no—
125C4×C127—81no—
126C4×C127#12011yescharacteristic
127C4×C127#12311yescharacteristic
128C4×C127#12311yescharacteristic
129C4×C127#12411yescharacteristic
130Dic127—22no—
131Dic127—41no—
132Dic127—81no—
133C2×C127—82no—
134C2×C127#39411yescharacteristic
135C2×C127#45911yescharacteristic
136C2×C127#45911yescharacteristic
137C2×C127#46311yescharacteristic
138C2×C127#46311yescharacteristic
139C2×C127#46611yescharacteristic
140C2×C127#46911yescharacteristic
141D127—82no—
142#2054—1271nomaximal, Sylow
143C127#205411yescharacteristic, complemented, Sylow
144C22×D8—1271no—
145C2×C4⋊C8—1271no—
146C2×#102—1271no—
147C2×#102D12711yescharacteristic, complemented
148#390—1271no—
149#390—1271no—
150#340—1271no—
151C22×D4—1271no—
152C22×D4C2×D12711yescharacteristic
153C2×D8—2542no—
154C2×D8—2542no—
155C2×D8—2542no—
156C22×C8—1271no—
157C22×C8C2×D12711yescharacteristic
158C2×C4⋊C4—1271no—
159C2×C4⋊C4C2×D12711yescharacteristic
160C2×C22⋊C4—2541no—
161C2×C22⋊C4—2541no—
162C2×C22⋊C4—2541no—
163C2×C22⋊C4—2541no—
164C2×C42—1271no—
165C4⋊C8—2542no—
166#102—22no—
167#102—2542no—
168C24—21no—
169C24—2541no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—1272no—
176C2×D4—2541no—
177C2×D4—2541no—
178C2×D4—2541no—
179C2×D4—2541no—
180C2×D4—2541no—
181C2×D4C22⋊D12712yes—
182C22×C4—21no—
183C22×C4—1271no—
184C22×C4—1271no—
185C22×C4—2541no—
186C22×C4C22×D12711yescharacteristic
187D8—2548no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—1272no—
192C2×C8—2541no—
193C2×C8—2541no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—2541no—
197C4⋊C4—2541no—
198C22⋊C4—5082no—
199C22⋊C4—5082no—
200C22⋊C4—5082no—
201C22⋊C4—5082no—
202C42—2541no—
203C42—2541no—
204C23—22no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—5081no—
212C23—5081no—
213C23—5081no—
214C23—5081no—
215C23—5081no—
216C23—5081no—
217C23—2542no—
218C23D4×D12711yescharacteristic
219D4—22no—
220D4—22no—
221D4—41no—
222D4—41no—
223D4—5081no—
224D4—5081no—
225D4—2542no—
226D4—2542no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—1272no—
231C2×C4—5081no—
232C2×C4—5081no—
233C2×C4—5081no—
234C2×C4—2541no—
235C2×C4—2541no—
236C2×C4—2541no—
237C2×C4—2541no—
238C2×C4—2541no—
239C2×C4D4×D12711yescharacteristic
240C2×C4D4×D12711yescharacteristic
241C2×C4C2×C22⋊D12711yescharacteristic
242C2×C4#1452324711yescharacteristic
243C2×C4#1452324811yescharacteristic
244C2×C4#1452324811yescharacteristic
245C8—41no—
246C8—2542no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—81no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22—10161no—
255C22—10161no—
256C22—10161no—
257C22—10161no—
258C22—5082no—
259C22—5082no—
260C22—5082no—
261C22#1686118111yescharacteristic
262C22#1686118211yescharacteristic
263C22#1686118211yescharacteristic
264C22D8×D12711yescharacteristic
265C22#1686123011yescharacteristic
266C22#1686123311yescharacteristic
267C22#1686123611yescharacteristic
268C4—81no—
269C4—10161no—
270C4—5081no—
271C4—2542no—
272C4#1686117711yescharacteristic
273C4#1686119111yescharacteristic
274C4#1686119211yescharacteristic
275C4#1686119511yescharacteristic
276C2—82no—
277C2—10162no—
278C2#2254979111yescharacteristic
279C2#2255006911yescharacteristic
280C2#2255007011yescharacteristic
281C2#2255009011yescharacteristic
282C2#2255009311yescharacteristic
283C2#2255011111yescharacteristic
284C2#2255012811yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC22×#14523233C2
hypercenterC2×#102D127
hypocenterC127#2054
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×#16861131C2semidirect
4. #22549779C2semidirect
5. #22549777C2semidirect
6. #22549638C2semidirect
7. C2×#102×C127C2semidirect
80. C2×Dic127D8semidirect
143. C127#2054semidirect
147. C2×#102D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232332
C2×Dic127⋊C82
C2×#168611312
#225497792
#225497772
#225496382
C2×#102×C1272
#2054127