Finite Group Databasebeta

← #30639064

Subgroups — #30639064

Order 16256.

402 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 113 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×#16861132C211yesmaximal, characteristic, complemented
3C2×#16861130C211yesmaximal, characteristic
4#22549782C211yesmaximal, characteristic, complemented
5#22549638C211yesmaximal, characteristic, complemented
6#22549631C211yesmaximal, characteristic, complemented
7C2×C4×D4×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523236—22no—
10C2×#14523236—22no—
11C2×#14523236C414yescomplemented
12C22×C127⋊C8C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14C2×C4⋊Dic127C2211yescharacteristic
15#16861132—22no—
16#16861130—22no—
17C127⋊#95—21no—
18C127⋊#95—21no—
19C22×D4×C127C2211yescharacteristic
20C23×C4×C127—21no—
21C4×D4×C127—22no—
22C4×D4×C127—22no—
23C2×C4⋊C4×C127C2211yescharacteristic
24C2×C22⋊C4×C127—21no—
25C2×C42×C127C2211yescharacteristic
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254C2×C412yescomplemented
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×C412yescomplemented
41C4⋊Dic127—21no—
42C4⋊Dic127—21no—
43C24×C127—21no—
44C2×D4×C127—21no—
45C2×D4×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127C2×C412yes—
48C2×D4×C127D412yes—
49C2×D4×C127D412yes—
50C22×C4×C127—24no—
51C22×C4×C127—21no—
52C22×C4×C127—42no—
53C22×C4×C127—42no—
54C22×C4×C127D411yescharacteristic
55C22×C4×C127D411yescharacteristic
56C22×C4×C127C2311yescharacteristic
57C4⋊C4×C127—21no—
58C4⋊C4×C127D412yes—
59C22⋊C4×C127—24no—
60C42×C127—21no—
61C42×C127—21no—
62Dic254—22no—
63Dic254—22no—
64Dic254—41no—
65Dic254—41no—
66C2×Dic127—22no—
67C2×Dic127—41no—
68C2×Dic127—41no—
69C2×Dic127—41no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C127⋊C8—22no—
73C127⋊C8—41no—
74C23×C127—22no—
75C23×C127—22no—
76C23×C127—22no—
77C23×C127—41no—
78C23×C127—41no—
79C23×C127—41no—
80C23×C127—41no—
81C23×C127C2×D411yescharacteristic
82D4×C127—22no—
83D4×C127—22no—
84D4×C127—22no—
85D4×C127C22⋊C414yes—
86C2×C4×C127—21no—
87C2×C4×C127—21no—
88C2×C4×C127—21no—
89C2×C4×C127—22no—
90C2×C4×C127—21no—
91C2×C4×C127—21no—
92C2×C4×C127—48no—
93C2×C4×C127—41no—
94C2×C4×C127—41no—
95C2×C4×C127SD1612yescomplemented
96C2×C4×C127C22×C411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127#4111yescharacteristic
101C2×C4×C127#4111yescharacteristic
102C22×C127—24no—
103C22×C127—42no—
104C22×C127—42no—
105C22×C127—42no—
106C22×C127—42no—
107C22×C127—42no—
108C22×C127—42no—
109C22×C127C4×D411yescharacteristic
110C22×C127#12011yescharacteristic
111C22×C127#12111yescharacteristic
112C22×C127C2×SD1611yescharacteristic
113C22×C127#13511yescharacteristic
114C22×C127AGL(1,ℤ/8)11yescharacteristic
115C22×C127#13711yescharacteristic
116C4×C127—22no—
117C4×C127—42no—
118C4×C127—41no—
119C4×C127C2×C22⋊C411yescharacteristic
120C4×C127C4×D411yescharacteristic
121C4×C127#12111yescharacteristic
122C4×C127#12311yescharacteristic
123Dic127—42no—
124Dic127—81no—
125C2×C127—44no—
126C2×C127#38611yescharacteristic
127C2×C127C4×SD1611yescharacteristic
128C2×C127#44011yescharacteristic
129C2×C127#45011yescharacteristic
130C2×C127#45211yescharacteristic
131C2×C127#46111yescharacteristic
132C2×C127#46311yescharacteristic
133#1877—1271nomaximal, Sylow
134C127#187711yescharacteristic, complemented, Sylow
135C22×SD16—1271no—
136C2×C4×D4D12711yescharacteristic
137C2×#103—1271no—
138C2×#102—1271no—
139#391—1271no—
140#340—1271no—
141#336—1271no—
142C22×Q8—1271no—
143C22×D4C2×D12711yescharacteristic
144C23×C4—21no—
145C2×SD16—1274no—
146C2×SD16—2542no—
147C2×SD16—2542no—
148C22×C8—1271no—
149C22×C8—1271no—
150C4×D4—22no—
151C4×D4—22no—
152C2×C4⋊C4—1271no—
153C2×C4⋊C4C2×D12711yescharacteristic
154C2×C22⋊C4—21no—
155C2×C42C2×D12711yescharacteristic
156#103—2542no—
157#102—2542no—
158#95—2541no—
159#95—2541no—
160C24—21no—
161C2×Q8—1272no—
162C2×Q8—2541no—
163C2×Q8—2541no—
164C2×Q8—2541no—
165C2×Q8—2541no—
166C2×Q8—2541no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4C4×D12712yes—
171C2×D4C22⋊D12712yes—
172C2×D4D50812yes—
173C22×C4—24no—
174C22×C4—21no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—2541no—
178C22×C4—2541no—
179C22×C4C22⋊D12711yescharacteristic
180C22×C4C22⋊D12711yescharacteristic
181C22×C4C22×D12711yescharacteristic
182SD16—2548no—
183C2×C8—1272no—
184C2×C8—2541no—
185C2×C8—2541no—
186C2×C8—2541no—
187C2×C8—2541no—
188C2×C8—2541no—
189C4⋊C4—21no—
190C4⋊C4—2541no—
191C4⋊C4—2541no—
192C4⋊C4C22⋊D12712yes—
193C22⋊C4—24no—
194C42—21no—
195C42—21no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23C2×C22⋊D12711yescharacteristic
204Q8—5081no—
205Q8—5081no—
206Q8—2542no—
207Q8—2542no—
208D4—22no—
209D4—22no—
210D4—22no—
211D4(C2×D127)⋊C414yes—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—48no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—5081no—
222C2×C4—5081no—
223C2×C4—5081no—
224C2×C4—5081no—
225C2×C4—5081no—
226C2×C4—2542no—
227C2×C4#1452323612yescomplemented
228C2×C4C2×C4×D12711yescharacteristic
229C2×C4C2×C22⋊D12711yescharacteristic
230C2×C4C2×C22⋊D12711yescharacteristic
231C2×C4C2×D50811yescharacteristic
232C2×C4#1452324811yescharacteristic
233C2×C4#1452324811yescharacteristic
234C8—5081no—
235C8—2542no—
236C22—24no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22C4×C22⋊D12711yescharacteristic
244C22#1686117811yescharacteristic
245C22#1686118311yescharacteristic
246C22C2×#1452323611yescharacteristic
247C22#1686122911yescharacteristic
248C22#1686123511yescharacteristic
249C22#1686123911yescharacteristic
250C4—22no—
251C4—42no—
252C4—41no—
253C4—10161no—
254C4—5082no—
255C4C2×(C2×D127)⋊C411yescharacteristic
256C4C4×C22⋊D12711yescharacteristic
257C4#1686118311yescharacteristic
258C4#1686119211yescharacteristic
259C2—44no—
260C2#2254975511yescharacteristic
261C2C4×#1452323611yescharacteristic
262C2#2254997611yescharacteristic
263C2#2255002711yescharacteristic
264C2#2255003811yescharacteristic
265C2#2255008111yescharacteristic
266C2#2255009111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×C4×D4×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
cosocleC22×C4×C127C23
hypercenterC2×C4×D4D127
hypocenterC127#1877
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#16861132C2semidirect
4. #22549782C2semidirect
5. #22549638C2semidirect
6. #22549631C2semidirect
11. C2×#14523236C4semidirect
31. C2×Dic254C2×C4semidirect
40. C2×C127⋊C8C2×C4semidirect
95. C2×C4×C127SD16semidirect
134. C127#1877semidirect
227. 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×#168611322
C2×#168611302
#225497822
#225496382
#225496312
C2×C4×D4×C1272
#1877127