Finite Group Databasebeta

← #30639081

Subgroups — #30639081

Order 16256.

438 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 103 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×C4×D508C211yesmaximal, characteristic, complemented
3C2×#16861130C211yesmaximal, characteristic, complemented
4C2×#16861129C211yesmaximal, characteristic, complemented
5#22549638C211yesmaximal, characteristic, complemented
6#22549631C211yesmaximal, characteristic, complemented
7#390×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C22×C4×D127—21no—
10C2×#14523233—22no—
11C2×#14523233—22no—
12C2×#14523233C414yescomplemented
13C22×C127⋊C8C2211yescharacteristic
14C22×C127⋊C8C2211yescharacteristic
15C4×D508—22no—
16C4×D508—22no—
17C2×C4⋊Dic127C2211yescharacteristic
18C2×(C2×D127)⋊C4—21no—
19#16861130—22no—
20#16861129—22no—
21C22×D4×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23C2×C22⋊C4×C127—21no—
24C2×C22⋊C4×C127—21no—
25C2×C42×C127C2211yescharacteristic
26C23×D127—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508C2×C412yes—
31C2×D508D412yes—
32C2×D508D412yes—
33C22×Dic127—21no—
34C2×C4×D127—24no—
35C2×C4×D127—42no—
36C2×C4×D127—42no—
37#14523233—28no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8—21no—
40C2×C127⋊C8—21no—
41C2×C127⋊C8—21no—
42C2×C127⋊C8—21no—
43C2×C127⋊C8C2×C412yescomplemented
44C4⋊Dic127—21no—
45C4⋊Dic127D412yes—
46(C2×D127)⋊C4—24no—
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×C127C2×C412yescomplemented
54C22×C4×C127—21no—
55C22×C4×C127D411yescharacteristic
56C22×C4×C127D411yescharacteristic
57C22×C4×C127C2311yescharacteristic
58C4⋊C4×C127—21no—
59C4⋊C4×C127—21no—
60C22⋊C4×C127—42no—
61C22⋊C4×C127—42no—
62C42×C127—21no—
63C42×C127—21no—
64C22×D127—22no—
65C22×D127—22no—
66C22×D127—22no—
67C22×D127—41no—
68C22×D127—41no—
69C22×D127—41no—
70C22×D127—41no—
71D508—22no—
72D508—22no—
73D508—22no—
74D508C22⋊C414yescomplemented
75C2×Dic127—22no—
76C2×Dic127—41no—
77C2×Dic127—41no—
78C4×D127—48no—
79C127⋊C8—22no—
80C127⋊C8—41no—
81C23×C127—22no—
82C23×C127—41no—
83C23×C127—41no—
84C23×C127—41no—
85C23×C127—41no—
86C23×C127—41no—
87C23×C127—41no—
88C23×C127C2×D411yescharacteristic
89D4×C127—22no—
90D4×C127—22no—
91D4×C127—41no—
92D4×C127—41no—
93C2×C4×C127—21no—
94C2×C4×C127—21no—
95C2×C4×C127—21no—
96C2×C4×C127—21no—
97C2×C4×C127—21no—
98C2×C4×C127—41no—
99C2×C4×C127—41no—
100C2×C4×C127—41no—
101C2×C4×C127D812yescomplemented
102C2×C4×C127C22×C411yescharacteristic
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127#4111yescharacteristic
107C2×C4×C127#4111yescharacteristic
108C22×C127—42no—
109C22×C127—42no—
110C22×C127—42no—
111C22×C127—81no—
112C22×C127—81no—
113C22×C127—81no—
114C22×C127—81no—
115C22×C127C4×D411yescharacteristic
116C22×C127#12011yescharacteristic
117C22×C127#12111yescharacteristic
118C22×C127C2×D811yescharacteristic
119C22×C127#13511yescharacteristic
120C22×C127AGL(1,ℤ/8)11yescharacteristic
121C22×C127AGL(1,ℤ/8)11yescharacteristic
122C2×D127—24no—
123C2×D127—42no—
124C2×D127—42no—
125C2×D127—42no—
126C2×D127—42no—
127C2×D127—42no—
128C2×D127—42no—
129C4×C127—22no—
130C4×C127—41no—
131C4×C127—81no—
132C4×C127C2×C22⋊C411yescharacteristic
133C4×C127C4×D411yescharacteristic
134C4×C127#12111yescharacteristic
135C4×C127#12311yescharacteristic
136Dic127—42no—
137C2×C127—82no—
138C2×C127#38611yescharacteristic
139C2×C127C4×D811yescharacteristic
140C2×C127#44211yescharacteristic
141C2×C127#44711yescharacteristic
142C2×C127#44911yescharacteristic
143C2×C127#45911yescharacteristic
144C2×C127#46311yescharacteristic
145D127—44no—
146#1879—1271nomaximal, Sylow
147C127#187911yescharacteristic, complemented, Sylow
148C22×D8—1271no—
149C2×C4×D4—1271no—
150C2×#102—1271no—
151C2×#102—1271no—
152#390D12711yescharacteristic, complemented
153#340—1271no—
154#336—1271no—
155C22×D4—1271no—
156C22×D4C2×D12711yescharacteristic
157C23×C4—2541no—
158C2×D8—1274no—
159C2×D8—2542no—
160C2×D8—2542no—
161C22×C8—1271no—
162C22×C8—1271no—
163C4×D4—2542no—
164C4×D4—2542no—
165C2×C4⋊C4—1271no—
166C2×C4⋊C4C2×D12711yescharacteristic
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—2541no—
170C2×C42C2×D12711yescharacteristic
171#102—2542no—
172#102—2542no—
173C24—21no—
174C24—2541no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—1272no—
181C2×D4—1272no—
182C2×D4—1272no—
183C2×D4—2541no—
184C2×D4—2541no—
185C2×D4—2541no—
186C2×D4C4×D12712yescomplemented
187C22×C4—21no—
188C22×C4—5082no—
189C22×C4—5082no—
190C22×C4—2544no—
191C22×C4—2541no—
192C22×C4C22⋊D12711yescharacteristic
193C22×C4C22⋊D12711yescharacteristic
194C22×C4C22×D12711yescharacteristic
195D8—2548no—
196C2×C8—1272no—
197C2×C8—2541no—
198C2×C8—2541no—
199C2×C8—2541no—
200C2×C8—2541no—
201C2×C8—2541no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—1272no—
205C4⋊C4—2541no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—2544no—
209C42—21no—
210C42—21no—
211C23—22no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—5081no—
219C23—5081no—
220C23—5081no—
221C23—5081no—
222C23—2542no—
223C23—2542no—
224C23—2542no—
225C23C2×C22⋊D12711yescharacteristic
226D4—22no—
227D4—22no—
228D4—41no—
229D4—41no—
230D4—1274no—
231D4—2542no—
232D4—2542no—
233D4—2542no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—5088no—
243C2×C4—5081no—
244C2×C4—5081no—
245C2×C4—2542no—
246C2×C4#1452323312yescomplemented
247C2×C4C2×C4×D12711yescharacteristic
248C2×C4D4×D12711yescharacteristic
249C2×C4D4×D12711yescharacteristic
250C2×C4D4×D12711yescharacteristic
251C2×C4#1452324711yescharacteristic
252C2×C4#1452324811yescharacteristic
253C8—5081no—
254C8—2542no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22—5082no—
263C22—5082no—
264C22—5082no—
265C22—5082no—
266C22—5082no—
267C22—5082no—
268C22—2544no—
269C22C4×C22⋊D12711yescharacteristic
270C22#1686117711yescharacteristic
271C22#1686118011yescharacteristic
272C22C2×#1452323311yescharacteristic
273C22#1686122911yescharacteristic
274C22#1686123411yescharacteristic
275C22#1686123511yescharacteristic
276C4—22no—
277C4—41no—
278C4—81no—
279C4—5082no—
280C4C22⋊C4×D12711yescharacteristic
281C4(C22⋊D127)⋊C411yescharacteristic
282C4#1686118211yescharacteristic
283C4#1686119111yescharacteristic
284C2—82no—
285C2—5084no—
286C2#2254975211yescharacteristic
287C2C4×#1452323311yescharacteristic
288C2#2254998611yescharacteristic
289C2#2255000911yescharacteristic
290C2#2255001911yescharacteristic
291C2#2255006811yescharacteristic
292C2#2255008811yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#390×C127C2
socleC23×C127C2×D4
prime radicalC22×#14523233C2
cosocleC22×C4×C127C23
hypercenter#390D127
hypocenterC127#1879
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×C4×D508C2semidirect
3. C2×#16861130C2semidirect
4. C2×#16861129C2semidirect
5. #22549638C2semidirect
6. #22549631C2semidirect
7. #390×C127C2semidirect
12. C2×#14523233C4semidirect
43. C2×C127⋊C8C2×C4semidirect
53. C2×D4×C127C2×C4semidirect
74. D508C22⋊C4semidirect
101. C2×C4×C127D8semidirect
147. C127#1879semidirect
152. #390D127semidirect
186. C2×D4C4×D127semidirect
246. C2×C4#14523233semidirect

Maximal subgroups

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

subgroupindex
C22×#145232332
C2×C4×D5082
C2×#168611302
C2×#168611292
#225496382
#225496312
#390×C1272
#1879127