Finite Group Databasebeta

← #30639845

Subgroups — #30639845

Order 16256.

362 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 73 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×#16861239C211yesmaximal, characteristic, complemented
2C2×#16861234C211yesmaximal, characteristic
3C22×#14523237C211yesmaximal, characteristic, complemented
4C2×#16861127C211yesmaximal, characteristic, complemented
5C2×#16861125C211yesmaximal, characteristic, complemented
6#22549641C211yesmaximal, characteristic, complemented
7#408×C127C211yesmaximal, characteristic, complemented
8C22×Dic254C2211yescharacteristic
9C22×D508C2211yescharacteristic
10#16861239—22no—
11#16861239—22no—
12#16861234—22no—
13#16861234—22no—
14C2×#14523238—21no—
15C2×#14523237—22no—
16C2×#14523237—22no—
17C2×#14523237—22no—
18C2×#14523236—21no—
19C2×#14523235—21no—
20C2×#14523233—21no—
21C2×#14523232C2211yescharacteristic
22C2×#14523232C2211yescharacteristic
23#16861127—22no—
24#16861125—22no—
25C2×#41×C127C2211yescharacteristic
26C2×M16×C127C2211yescharacteristic
27C22×C8×C127C2211yescharacteristic
28C22⋊C8×C127—22no—
29C23×D127—21no—
30C2×Dic254—21no—
31C2×Dic254—21no—
32C2×Dic254—21no—
33C2×Dic254—21no—
34C2×Dic254—21no—
35C2×Dic254D412yes—
36C2×D508—21no—
37C2×D508—21no—
38C2×D508—21no—
39C2×D508—21no—
40C2×D508—21no—
41C2×D508D412yes—
42C22×Dic127—21no—
43#14523238—24no—
44#14523237—28no—
45#14523236—24no—
46#14523235—24no—
47#14523233—24no—
48#14523232—21no—
49#14523232—21no—
50#14523232D412yes—
51#14523232D412yes—
52C2×C127⋊C8—21no—
53C2×C127⋊C8—21no—
54#41×C127—21no—
55#41×C127—21no—
56#41×C127—21no—
57#41×C127—21no—
58C2×Q8×C127D411yescharacteristic
59C2×D4×C127—21no—
60C2×D4×C127D411yescharacteristic
61C22×C4×C127—21no—
62C22×C4×C127C2311yescharacteristic
63M16×C127—21no—
64M16×C127—21no—
65C2×C8×C127—21no—
66C2×C8×C127—21no—
67C2×C8×C127—21no—
68C2×C8×C127D412yes—
69C22×D127—22no—
70C22×D127—41no—
71C22×D127—41no—
72C22×D127—82no—
73Dic254—22no—
74Dic254—22no—
75Dic254—41no—
76Dic254—41no—
77D508—22no—
78D508—22no—
79D508—41no—
80D508—41no—
81C2×Dic127—22no—
82C2×Dic127—41no—
83C2×Dic127—41no—
84C127⋊C8—22no—
85C127⋊C8—22no—
86C23×C127—21no—
87C23×C127#4111yescharacteristic
88Q8×C127—21no—
89Q8×C127—21no—
90D4×C127—21no—
91D4×C127—21no—
92D4×C127—41no—
93D4×C127—41no—
94C2×C4×C127—21no—
95C2×C4×C127—41no—
96C2×C4×C127—41no—
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2×D411yescharacteristic
103C8×C127—22no—
104C8×C127—41no—
105C22×C127—22no—
106C22×C127—41no—
107C22×C127—42no—
108C22×C127#12111yescharacteristic
109C22×C127#12111yescharacteristic
110C22×C127#12111yescharacteristic
111C2×D127—42no—
112C2×D127—82no—
113C2×D127—82no—
114C2×D127—82no—
115C4×C127—41no—
116C4×C127#12011yescharacteristic
117C4×C127#12011yescharacteristic
118C4×C127#12011yescharacteristic
119C4×C127#12711yescharacteristic
120Dic127—42no—
121C2×C127—21no—
122C2×C127—21no—
123C2×C127—81no—
124C2×C127#39211yescharacteristic
125C2×C127#47112yes—
126D127—82no—
127#2019—1271nomaximal, Sylow
128C127#201911yescharacteristic, complemented, Sylow
129C2×#137—1271no—
130C2×AGL(1,ℤ/8)—1271no—
131C22×SD16—1271no—
132C2×#101—1271no—
133C2×#100—1271no—
134#408D12711yescharacteristic, complemented
135#341—1271no—
136C2×#41C2×D12711yescharacteristic
137C22×Q8—1271no—
138C22×D4—1271no—
139#137—2542no—
140#137—2542no—
141AGL(1,ℤ/8)—2542no—
142AGL(1,ℤ/8)—2542no—
143C2×Q16—2541no—
144C2×SD16—2542no—
145C2×SD16—2542no—
146C2×SD16—2541no—
147C2×SD16—2541no—
148C2×SD16—2542no—
149C2×D8—2541no—
150C2×M16—1271no—
151C2×M16—1271no—
152C2×M16C2×D12711yescharacteristic
153C22×C8C2×D12711yescharacteristic
154#101—2542no—
155#100—2542no—
156C22⋊C8—22no—
157C24—2541no—
158#41—21no—
159#41—21no—
160#41—21no—
161#41—21no—
162C2×Q8—1272no—
163C2×Q8—2541no—
164C2×Q8—2541no—
165C2×Q8—2541no—
166C2×Q8—2541no—
167C2×Q8—2541no—
168C2×Q8D50811yescharacteristic
169C2×D4—21no—
170C2×D4—1272no—
171C2×D4—2541no—
172C2×D4—2541no—
173C2×D4—2541no—
174C2×D4—2541no—
175C2×D4—2541no—
176C2×D4D50811yescharacteristic
177C22×C4—21no—
178C22×C4—2541no—
179C22×C4C22×D12711yescharacteristic
180Q16—2544no—
181SD16—2548no—
182SD16—2544no—
183SD16—2544no—
184D8—2544no—
185M16—21no—
186M16—21no—
187M16—1272no—
188M16—1272no—
189M16—2541no—
190M16—2541no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—2541no—
195C2×C8—2541no—
196C2×C8C22⋊D12712yes—
197C23—21no—
198C23—10162no—
199C23—5081no—
200C23—5081no—
201C23—2542no—
202C23#1452324811yescharacteristic
203Q8—21no—
204Q8—21no—
205Q8—5081no—
206Q8—5081no—
207Q8—2542no—
208Q8—2542no—
209D4—21no—
210D4—21no—
211D4—41no—
212D4—41no—
213D4—5081no—
214D4—5081no—
215D4—2542no—
216D4—2542no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—5081no—
221C2×C4—5081no—
222C2×C4—2542no—
223C2×C4D4×D12711yescharacteristic
224C2×C4D4×D12711yescharacteristic
225C2×C4D4×D12711yescharacteristic
226C2×C4D4×D12711yescharacteristic
227C2×C4C2×C22⋊D12711yescharacteristic
228C2×C4C2×D50811yescharacteristic
229C8—22no—
230C8—41no—
231C8—2542no—
232C8—2542no—
233C22—22no—
234C22—41no—
235C22—42no—
236C22—10162no—
237C22—10162no—
238C22—10162no—
239C22—5082no—
240C22#1686118211yescharacteristic
241C22#1686118211yescharacteristic
242C22#1686118311yescharacteristic
243C4—41no—
244C4—5082no—
245C4#1686117711yescharacteristic
246C4#1686117911yescharacteristic
247C4#1686117911yescharacteristic
248C4#1686120511yescharacteristic
249C2—21no—
250C2—21no—
251C2—81no—
252C2—10162no—
253C2#2254978511yescharacteristic
254C2#2255013912yes—
subgroupG/H
centerC22#16861183
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#408×C127C2
socleC22×C127#121
prime radicalC2×#16861234C2
hypercenter#408D127
hypocenterC127#2019
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
3. C22×#14523237C2semidirect
4. C2×#16861127C2semidirect
5. C2×#16861125C2semidirect
6. #22549641C2semidirect
7. #408×C127C2semidirect
128. C127#2019semidirect
134. #408D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
C2×#168612342
C22×#145232372
C2×#168611272
C2×#168611252
#225496412
#408×C1272
#2019127