Finite Group Databasebeta

← #30639088

Subgroups — #30639088

Order 16256.

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

normal subgroups only

This group has 374 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×#14523236C211yesmaximal, characteristic
2C2×C4×Dic254C211yesmaximal, characteristic, complemented
3C2×#16861132C211yesmaximal, characteristic, complemented
4C2×#16861130C211yesmaximal, characteristic
5#22549638C211yesmaximal, characteristic, complemented
6#22549631C211yesmaximal, characteristic, complemented
7#390×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523236—22no—
10C2×#14523236—22no—
11C2×#14523236C414yescomplemented
12C22×C127⋊C8C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14C4×Dic254—22no—
15C4×Dic254—22no—
16C2×C4⋊Dic127C2211yescharacteristic
17C2×Dic127⋊C4—21no—
18C2×C4×Dic127—21no—
19#16861132—22no—
20#16861130—22no—
21C22×D4×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23C2×C22⋊C4×C127—21no—
24C2×C22⋊C4×C127—21no—
25C2×C42×C127C2211yescharacteristic
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254C2×C412yes—
30C2×Dic254D412yes—
31C2×Dic254D412yes—
32C22×Dic127—21no—
33C22×Dic127—21no—
34#14523236—28no—
35C2×C127⋊C8—21no—
36C2×C127⋊C8—21no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8—21no—
40C2×C127⋊C8C2×C412yes—
41C4⋊Dic127—21no—
42C4⋊Dic127D412yes—
43Dic127⋊C4—24no—
44C4×Dic127—24no—
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×C127C2×C412yes—
52C22×C4×C127—21no—
53C22×C4×C127D411yescharacteristic
54C22×C4×C127D411yescharacteristic
55C22×C4×C127C2311yescharacteristic
56C4⋊C4×C127—21no—
57C4⋊C4×C127—21no—
58C22⋊C4×C127—42no—
59C22⋊C4×C127—42no—
60C42×C127—21no—
61C42×C127—21no—
62Dic254—22no—
63Dic254—22no—
64Dic254—22no—
65Dic254C22⋊C414yescomplemented
66C2×Dic127—22no—
67C2×Dic127—22no—
68C2×Dic127—22no—
69C2×Dic127—22no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C127⋊C8—22no—
73C127⋊C8—41no—
74C23×C127—22no—
75C23×C127—41no—
76C23×C127—41no—
77C23×C127—41no—
78C23×C127—41no—
79C23×C127—41no—
80C23×C127—41no—
81C23×C127C2×D411yescharacteristic
82D4×C127—22no—
83D4×C127—22no—
84D4×C127—41no—
85D4×C127—41no—
86C2×C4×C127—21no—
87C2×C4×C127—21no—
88C2×C4×C127—21no—
89C2×C4×C127—21no—
90C2×C4×C127—21no—
91C2×C4×C127—41no—
92C2×C4×C127—41no—
93C2×C4×C127—41no—
94C2×C4×C127SD1612yescomplemented
95C2×C4×C127C22×C411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127#4111yescharacteristic
100C2×C4×C127#4111yescharacteristic
101C22×C127—42no—
102C22×C127—42no—
103C22×C127—42no—
104C22×C127—81no—
105C22×C127—81no—
106C22×C127—81no—
107C22×C127—81no—
108C22×C127C4×D411yescharacteristic
109C22×C127#12011yescharacteristic
110C22×C127#12111yescharacteristic
111C22×C127C2×SD1611yescharacteristic
112C22×C127#13511yescharacteristic
113C22×C127AGL(1,ℤ/8)11yescharacteristic
114C22×C127#13711yescharacteristic
115C4×C127—22no—
116C4×C127—41no—
117C4×C127—81no—
118C4×C127C2×C22⋊C411yescharacteristic
119C4×C127C4×D411yescharacteristic
120C4×C127#12111yescharacteristic
121C4×C127#12311yescharacteristic
122Dic127—24no—
123Dic127—42no—
124C2×C127—82no—
125C2×C127#38611yescharacteristic
126C2×C127C4×SD1611yescharacteristic
127C2×C127#44011yescharacteristic
128C2×C127#44811yescharacteristic
129C2×C127#44911yescharacteristic
130C2×C127#46011yescharacteristic
131C2×C127#46411yescharacteristic
132#1880—1271nomaximal, Sylow
133C127#188011yescharacteristic, complemented, Sylow
134C22×SD16—1271no—
135C2×C4×Q8—1271no—
136C2×#103—1271no—
137C2×#102—1271no—
138#390D12711yescharacteristic
139#340—1271no—
140#336—1271no—
141C22×Q8—1271no—
142C22×D4C2×D12711yescharacteristic
143C2×SD16—1274no—
144C2×SD16—2542no—
145C2×SD16—2542no—
146C22×C8—1271no—
147C22×C8—1271no—
148C4×Q8—2542no—
149C4×Q8—2542no—
150C2×C4⋊C4—1271no—
151C2×C4⋊C4—2541no—
152C2×C4⋊C4C2×D12711yescharacteristic
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C42—2541no—
156C2×C42C2×D12711yescharacteristic
157#103—2542no—
158#102—2542no—
159C24—21no—
160C2×Q8—1272no—
161C2×Q8—1272no—
162C2×Q8—1272no—
163C2×Q8—2541no—
164C2×Q8—2541no—
165C2×Q8—2541no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4C4×D12712yes—
172C22×C4—21no—
173C22×C4—2541no—
174C22×C4—2541no—
175C22×C4C22⋊D12711yescharacteristic
176C22×C4C22⋊D12711yescharacteristic
177C22×C4C22×D12711yescharacteristic
178SD16—2548no—
179C2×C8—1272no—
180C2×C8—2541no—
181C2×C8—2541no—
182C2×C8—2541no—
183C2×C8—2541no—
184C2×C8—2541no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—1272no—
188C4⋊C4—2544no—
189C4⋊C4—2541no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C42—21no—
193C42—21no—
194C42—2544no—
195C23—22no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23C2×C22⋊D12711yescharacteristic
203Q8—1274no—
204Q8—2542no—
205Q8—2542no—
206Q8—2542no—
207D4—22no—
208D4—22no—
209D4—41no—
210D4—41no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—5081no—
220C2×C4—5081no—
221C2×C4—2542no—
222C2×C4—2542no—
223C2×C4—2542no—
224C2×C4—2542no—
225C2×C4#1452323612yescomplemented
226C2×C4C2×C4×D12711yescharacteristic
227C2×C4D4×D12711yescharacteristic
228C2×C4D4×D12711yescharacteristic
229C2×C4D4×D12711yescharacteristic
230C2×C4#1452324711yescharacteristic
231C2×C4#1452324811yescharacteristic
232C8—5081no—
233C8—2542no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22C4×C22⋊D12711yescharacteristic
242C22#1686117711yescharacteristic
243C22#1686118011yescharacteristic
244C22C2×#1452323611yescharacteristic
245C22#1686122911yescharacteristic
246C22#1686123511yescharacteristic
247C22#1686123911yescharacteristic
248C4—22no—
249C4—41no—
250C4—81no—
251C4—5082no—
252C4—2544no—
253C4C22⋊C4×D12711yescharacteristic
254C4(C22⋊D127)⋊C411yescharacteristic
255C4#1686118211yescharacteristic
256C4#1686119111yescharacteristic
257C2—82no—
258C2#2254975211yescharacteristic
259C2C4×#1452323611yescharacteristic
260C2#2254997611yescharacteristic
261C2#2255001411yescharacteristic
262C2#2255001911yescharacteristic
263C2#2255007311yescharacteristic
264C2#2255009511yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#390×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
cosocleC22×C4×C127C23
hypercenter#390D127
hypocenterC127#1880
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×Dic254C2semidirect
3. C2×#16861132C2semidirect
5. #22549638C2semidirect
6. #22549631C2semidirect
11. C2×#14523236C4semidirect
65. Dic254C22⋊C4semidirect
94. C2×C4×C127SD16semidirect
133. C127#1880semidirect
225. C2×C4#14523236semidirect

Maximal subgroups

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

subgroupindex
C22×#145232362
C2×C4×Dic2542
C2×#168611322
C2×#168611302
#225496382
#225496312
#390×C1272
#1880127