Finite Group Databasebeta

← #30639806

Subgroups — #30639806

Order 16256.

426 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 89 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, complemented
2C22×#14523233C211yesmaximal, characteristic
3C2×#16861197C211yesmaximal, characteristic, complemented
4C2×#16861132C211yesmaximal, characteristic, complemented
5C2×#16861129C211yesmaximal, characteristic, complemented
6#22549631C211yesmaximal, characteristic, complemented
7#390×C127C211yesmaximal, characteristic, complemented
8C22×Dic254C2211yescharacteristic
9C22×D508C2211yescharacteristic
10C2×#14523236—22no—
11C2×#14523236—22no—
12C2×#14523236—22no—
13C2×#14523233—22no—
14C2×#14523233—22no—
15C2×#14523233—22no—
16C22×C127⋊C8C2211yescharacteristic
17C22×C127⋊C8C2211yescharacteristic
18#16861197—22no—
19#16861197—22no—
20C2×(C2×D127)⋊C4—22no—
21#16861132—22no—
22#16861129—22no—
23C22×D4×C127C2211yescharacteristic
24C2×C4⋊C4×C127C2211yescharacteristic
25C2×C22⋊C4×C127—22no—
26C2×C42×C127C2211yescharacteristic
27C23×D127—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254—21no—
32C2×Dic254—21no—
33C2×Dic254D412yes—
34C2×D508—21no—
35C2×D508—21no—
36C2×D508—21no—
37C2×D508—21no—
38C2×D508—21no—
39C2×D508D412yes—
40C22×Dic127—21no—
41#14523236—28no—
42#14523233—28no—
43C2×C127⋊C8—21no—
44C2×C127⋊C8—21no—
45C2×C127⋊C8—21no—
46C2×C127⋊C8—21no—
47C2×C127⋊C8D412yes—
48C2×C127⋊C8D412yes—
49(C2×D127)⋊C4—28no—
50C24×C127—21no—
51C2×D4×C127—21no—
52C2×D4×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127—21no—
56C2×D4×C127D412yes—
57C22×C4×C127—21no—
58C22×C4×C127D412yes—
59C22×C4×C127C2311yescharacteristic
60C4⋊C4×C127—21no—
61C4⋊C4×C127—21no—
62C22⋊C4×C127—44no—
63C42×C127—21no—
64C42×C127—21no—
65C22×D127—22no—
66C22×D127—41no—
67C22×D127—41no—
68C22×D127—42no—
69C22×D127—42no—
70Dic254—22no—
71Dic254—22no—
72Dic254—41no—
73Dic254—41no—
74D508—22no—
75D508—22no—
76D508—41no—
77D508—41no—
78C2×Dic127—22no—
79C2×Dic127—41no—
80C2×Dic127—41no—
81C127⋊C8—22no—
82C127⋊C8—22no—
83C23×C127—22no—
84C23×C127—41no—
85C23×C127—41no—
86C23×C127—42no—
87C23×C127—42no—
88C23×C127C2×D411yescharacteristic
89D4×C127—22no—
90D4×C127—22no—
91D4×C127—41no—
92D4×C127—41no—
93C2×C4×C127—22no—
94C2×C4×C127—22no—
95C2×C4×C127—22no—
96C2×C4×C127—41no—
97C2×C4×C127—42no—
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127#4111yescharacteristic
104C22×C127—42no—
105C22×C127—44no—
106C22×C127—82no—
107C22×C127—81no—
108C22×C127—81no—
109C22×C127#12011yescharacteristic
110C22×C127#12111yescharacteristic
111C22×C127#12711yescharacteristic
112C22×C127#13512yes—
113C22×C127AGL(1,ℤ/8)12yes—
114C2×D127—42no—
115C2×D127—44no—
116C2×D127—82no—
117C2×D127—81no—
118C2×D127—81no—
119C4×C127—42no—
120C4×C127—81no—
121C4×C127#12011yescharacteristic
122C4×C127#12011yescharacteristic
123C4×C127#12111yescharacteristic
124C4×C127#12111yescharacteristic
125Dic127—42no—
126C2×C127—82no—
127C2×C127#39211yescharacteristic
128C2×C127#44912yes—
129C2×C127#46312yes—
130C2×C127#49511yescharacteristic
131C2×C127#49611yescharacteristic
132D127—82no—
133#2012—1271nomaximal, Sylow
134C127#201211yescharacteristic, complemented, Sylow
135C22×SD16—1271no—
136C22×D8—1271no—
137C2×#124—1271no—
138C2×#103—1271no—
139C2×#102—1271no—
140#390D12711yescharacteristic, complemented
141#336—1271no—
142C22×Q8—1271no—
143C22×D4—1271no—
144C22×D4C2×D12711yescharacteristic
145C2×SD16—2542no—
146C2×SD16—2542no—
147C2×SD16—2542no—
148C2×D8—2542no—
149C2×D8—2542no—
150C2×D8—2542no—
151C22×C8—1271no—
152C22×C8—1271no—
153#124—2542no—
154#124—2542no—
155C2×C4⋊C4C2×D12711yescharacteristic
156C2×C22⋊C4—22no—
157C2×C22⋊C4—2542no—
158C2×C42C2×D12711yescharacteristic
159#103—2542no—
160#102—2542no—
161C24—21no—
162C24—2541no—
163C2×Q8—1272no—
164C2×Q8—2541no—
165C2×Q8—2541no—
166C2×Q8—2541no—
167C2×Q8—2541no—
168C2×Q8—2541no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—1272no—
175C2×D4—2541no—
176C2×D4—2541no—
177C2×D4—2541no—
178C2×D4—2541no—
179C2×D4—2541no—
180C2×D4D50812yes—
181C22×C4—21no—
182C22×C4—2541no—
183C22×C4C22⋊D12712yes—
184C22×C4C22×D12711yescharacteristic
185SD16—2548no—
186D8—2548no—
187C2×C8—1272no—
188C2×C8—1272no—
189C2×C8—2541no—
190C2×C8—2541no—
191C2×C8—2541no—
192C2×C8—2541no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C22⋊C4—44no—
196C22⋊C4—2548no—
197C42—21no—
198C42—21no—
199C23—22no—
200C23—41no—
201C23—42no—
202C23—42no—
203C23—41no—
204C23—5081no—
205C23—5082no—
206C23—5082no—
207C23—5081no—
208C23—2542no—
209C23C2×C22⋊D12711yescharacteristic
210Q8—5081no—
211Q8—5081no—
212Q8—2542no—
213Q8—2542no—
214D4—22no—
215D4—22no—
216D4—41no—
217D4—41no—
218D4—5081no—
219D4—5081no—
220D4—2542no—
221D4—2542no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—5081no—
228C2×C4—5081no—
229C2×C4—2542no—
230C2×C4D4×D12711yescharacteristic
231C2×C4D4×D12711yescharacteristic
232C2×C4D4×D12711yescharacteristic
233C2×C4D4×D12711yescharacteristic
234C2×C4C2×D50811yescharacteristic
235C2×C4#1452324811yescharacteristic
236C8—2542no—
237C8—2542no—
238C22—42no—
239C22—44no—
240C22—82no—
241C22—81no—
242C22—81no—
243C22—10162no—
244C22—10161no—
245C22—10161no—
246C22—5084no—
247C22—5082no—
248C22#1686117711yescharacteristic
249C22#1686118311yescharacteristic
250C22#1686120511yescharacteristic
251C22#1686122912yes—
252C22#1686123512yes—
253C4—42no—
254C4—81no—
255C4—5082no—
256C4#1686117911yescharacteristic
257C4#1686117911yescharacteristic
258C4#1686118211yescharacteristic
259C4#1686118211yescharacteristic
260C2—82no—
261C2—10162no—
262C2#2254978511yescharacteristic
263C2#2255001912yes—
264C2#2255009112yes—
265C2#2255025811yescharacteristic
266C2#2255026311yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#390×C127C2
socleC23×C127C2×D4
prime radicalC22×#14523233C2
hypercenter#390D127
hypocenterC127#2012
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C22×#14523236C2semidirect
3. C2×#16861197C2semidirect
4. C2×#16861132C2semidirect
5. C2×#16861129C2semidirect
6. #22549631C2semidirect
7. #390×C127C2semidirect
134. C127#2012semidirect
140. #390D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232362
C22×#145232332
C2×#168611972
C2×#168611322
C2×#168611292
#225496312
#390×C1272
#2012127