Finite Group Databasebeta

← #30639971

Subgroups — #30639971

Order 16256.

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

normal subgroups only

This group has 318 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
1C2×#16861197C211yesmaximal, characteristic
2C2×#16861132C211yesmaximal, characteristic, complemented
3C2×#16861132C211yesmaximal, characteristic, complemented
4C2×#16861129C211yesmaximal, characteristic
5C2×#16861129C211yesmaximal, characteristic
6#22549631C211yesmaximal, characteristic, complemented
7#389×C127C211yesmaximal, characteristic, complemented
8C22×Dic254C2211yescharacteristic
9C22×D508C2211yescharacteristic
10C22×C127⋊C8C2211yescharacteristic
11C22×C127⋊C8C2211yescharacteristic
12#16861197—22no—
13#16861197—22no—
14C2×(C2×D127)⋊C4—21no—
15C2×(C2×D127)⋊C4—21no—
16#16861132—22no—
17#16861132—22no—
18#16861129—22no—
19#16861129—22no—
20C2×C4⋊C4×C127—21no—
21C2×C4⋊C4×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23C2×C42×C127C2211yescharacteristic
24#95×C127—21no—
25C23×D127—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254D412yes—
32C2×D508—21no—
33C2×D508—21no—
34C2×D508—21no—
35C2×D508—21no—
36C2×D508—21no—
37C2×D508D412yes—
38C22×Dic127—21no—
39C2×C127⋊C8—21no—
40C2×C127⋊C8—21no—
41C2×C127⋊C8—21no—
42C2×C127⋊C8—21no—
43C2×C127⋊C8—21no—
44C2×C127⋊C8—21no—
45(C2×D127)⋊C4—24no—
46(C2×D127)⋊C4—24no—
47C22×C4×C127—21no—
48C22×C4×C127—21no—
49C22×C4×C127D411yescharacteristic
50C22×C4×C127D411yescharacteristic
51C22×C4×C127C2311yescharacteristic
52C4⋊C4×C127—21no—
53C4⋊C4×C127—21no—
54C4⋊C4×C127—21no—
55C4⋊C4×C127—42no—
56C4⋊C4×C127D412yes—
57C42×C127—21no—
58C42×C127—21no—
59C22×D127—22no—
60C22×D127—41no—
61C22×D127—41no—
62C22×D127—41no—
63C22×D127—41no—
64C22×D127—41no—
65C22×D127—41no—
66Dic254—22no—
67Dic254—22no—
68Dic254—41no—
69Dic254—41no—
70D508—22no—
71D508—22no—
72D508—41no—
73D508—41no—
74C2×Dic127—22no—
75C2×Dic127—41no—
76C2×Dic127—41no—
77C127⋊C8—41no—
78C127⋊C8—41no—
79C23×C127C2×D411yescharacteristic
80C2×C4×C127—21no—
81C2×C4×C127—21no—
82C2×C4×C127—22no—
83C2×C4×C127—21no—
84C2×C4×C127—21no—
85C2×C4×C127—21no—
86C2×C4×C127—21no—
87C2×C4×C127—41no—
88C2×C4×C127—41no—
89C2×C4×C127—41no—
90C2×C4×C127—41no—
91C2×C4×C127—41no—
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127#4111yescharacteristic
96C2×C4×C127#4111yescharacteristic
97C2×C4×C127#4111yescharacteristic
98C22×C127#12011yescharacteristic
99C22×C127#12111yescharacteristic
100C22×C127#12411yescharacteristic
101C22×C127#13511yescharacteristic
102C22×C127#13511yescharacteristic
103C22×C127AGL(1,ℤ/8)11yescharacteristic
104C22×C127#13711yescharacteristic
105C2×D127—42no—
106C2×D127—42no—
107C2×D127—42no—
108C2×D127—81no—
109C2×D127—81no—
110C2×D127—81no—
111C2×D127—81no—
112C4×C127—41no—
113C4×C127—41no—
114C4×C127—42no—
115C4×C127—81no—
116C4×C127#12111yescharacteristic
117C4×C127#12111yescharacteristic
118C4×C127#12311yescharacteristic
119C4×C127#12311yescharacteristic
120Dic127—42no—
121C2×C127#39411yescharacteristic
122C2×C127#44911yescharacteristic
123C2×C127#45211yescharacteristic
124C2×C127#46311yescharacteristic
125C2×C127#46411yescharacteristic
126C2×C127#48811yescharacteristic
127C2×C127#48911yescharacteristic
128D127—82no—
129#2042—1271nomaximal, Sylow
130C127#204211yescharacteristic, complemented, Sylow
131C2×#124—1271no—
132C2×#103—1271no—
133C2×#103—1271no—
134C2×#102—1271no—
135C2×#102—1271no—
136#389D12711yescharacteristic, complemented
137#336—1271no—
138C22×Q8—1271no—
139C22×D4—1271no—
140C22×C8—1271no—
141C22×C8—1271no—
142#124—2542no—
143#124—2542no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4C2×D12711yescharacteristic
146C2×C4⋊C4C2×D12711yescharacteristic
147C2×C22⋊C4—2541no—
148C2×C22⋊C4—2541no—
149C2×C42C2×D12711yescharacteristic
150#103—2542no—
151#103—2542no—
152#102—2542no—
153#102—2542no—
154#95—21no—
155C24—2541no—
156C2×Q8—1272no—
157C2×Q8—2541no—
158C2×Q8—2541no—
159C2×Q8—2541no—
160C2×Q8—2541no—
161C2×Q8—2541no—
162C2×D4—1272no—
163C2×D4—2541no—
164C2×D4—2541no—
165C2×D4—2541no—
166C2×D4—2541no—
167C2×D4—2541no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—2541no—
171C22×C4C22⋊D12711yescharacteristic
172C22×C4C22⋊D12711yescharacteristic
173C22×C4C22×D12711yescharacteristic
174C2×C8—2541no—
175C2×C8—2541no—
176C2×C8—2541no—
177C2×C8—2541no—
178C2×C8—2541no—
179C2×C8—2541no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—42no—
184C4⋊C4D50812yes—
185C22⋊C4—2544no—
186C22⋊C4—2544no—
187C42—21no—
188C42—21no—
189C23—5081no—
190C23—5081no—
191C23—5081no—
192C23—5081no—
193C23—5081no—
194C23—5081no—
195C23—2542no—
196C23C2×C22⋊D12711yescharacteristic
197Q8—5081no—
198Q8—5081no—
199Q8—2542no—
200Q8—2542no—
201D4—5081no—
202D4—5081no—
203D4—2542no—
204D4—2542no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—5081no—
218C2×C4—5081no—
219C2×C4—2542no—
220C2×C4D4×D12711yescharacteristic
221C2×C4D4×D12711yescharacteristic
222C2×C4C2×D50811yescharacteristic
223C2×C4#1452324811yescharacteristic
224C2×C4#1452324911yescharacteristic
225C2×C4#1452324911yescharacteristic
226C8—5081no—
227C8—5081no—
228C22—10161no—
229C22—10161no—
230C22—10161no—
231C22—10161no—
232C22—5082no—
233C22—5082no—
234C22—5082no—
235C22#1686117711yescharacteristic
236C22#1686118311yescharacteristic
237C22#1686119811yescharacteristic
238C22#1686122911yescharacteristic
239C22#1686122911yescharacteristic
240C22#1686123511yescharacteristic
241C22#1686124011yescharacteristic
242C4—41no—
243C4—41no—
244C4—42no—
245C4—81no—
246C4—5082no—
247C4#1686118411yescharacteristic
248C4#1686118411yescharacteristic
249C4#1686119411yescharacteristic
250C4#1686119411yescharacteristic
251C2—10162no—
252C2#2254979311yescharacteristic
253C2#2255001911yescharacteristic
254C2#2255003611yescharacteristic
255C2#2255009111yescharacteristic
256C2#2255009811yescharacteristic
257C2#2255023311yescharacteristic
258C2#2255023811yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#389×C127C2
socleC23×C127C2×D4
prime radicalC22×D508C22
hypercenter#389D127
hypocenterC127#2042
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
2. C2×#16861132C2semidirect
3. C2×#16861132C2semidirect
6. #22549631C2semidirect
7. #389×C127C2semidirect
130. C127#2042semidirect
136. #389D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611972
C2×#168611322
C2×#168611292
#225496312
#389×C1272
#2042127