Finite Group Databasebeta

← #30639761

Subgroups — #30639761

Order 16256.

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

normal subgroups only

This group has 394 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×#16861242C211yesmaximal, characteristic, complemented
2#22550443C211yesmaximal, characteristic
3C2×#16861196C211yesmaximal, characteristic, complemented
4C2×#16861135C211yesmaximal, characteristic, complemented
5#22549886C211yesmaximal, characteristic, complemented
6#22549637C211yesmaximal, characteristic, complemented
7C2×#100×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C22×Dic254C2211yescharacteristic
10#16861242—22no—
11#16861242—22no—
12C2×Dic508—21no—
13C2×#14523237—21no—
14#16861205—21no—
15#16861196—21no—
16#16861196—22no—
17#16861196—22no—
18#16861180—22no—
19#16861178—22no—
20#16861168C2211yescharacteristic
21C2×C22⋊Dic127—22no—
22C2×C4×Dic127C2211yescharacteristic
23#16861135—22no—
24#16861134—22no—
25C22×D4×C127C2211yescharacteristic
26C2×M16×C127C2211yescharacteristic
27C2×M16×C127C2211yescharacteristic
28#100×C127—22no—
29#14523248—21no—
30#14523248—21no—
31#14523248—21no—
32#14523248—21no—
33C2×Dic254—21no—
34C2×Dic254—21no—
35C2×Dic254—21no—
36C2×Dic254—21no—
37C2×Dic254—21no—
38C2×Dic254D411yescharacteristic
39C2×Dic254D412yescomplemented
40C2×D508D411yescharacteristic
41C2×C22⋊D127—24no—
42C2×C22⋊D127—21no—
43C22×Dic127—21no—
44C22×Dic127—21no—
45C2×C4×D127—21no—
46Dic508—24no—
47#14523237—24no—
48C4⋊Dic127D412yes—
49C22⋊Dic127—28no—
50C22⋊Dic127—24no—
51C22⋊Dic127—21no—
52C4×Dic127—21no—
53C4×Dic127—21no—
54C4×Dic127D412yescomplemented
55C24×C127D412yes—
56C2×D4×C127—21no—
57C2×D4×C127—21no—
58C2×D4×C127—21no—
59C2×D4×C127—21no—
60C2×D4×C127—21no—
61C2×D4×C127—21no—
62C22×C4×C127C2311yescharacteristic
63M16×C127—21no—
64M16×C127—21no—
65M16×C127—21no—
66M16×C127D412yescomplemented
67C2×C8×C127—21no—
68C2×C8×C127—21no—
69C22×D127—21no—
70Dic254—22no—
71Dic254—21no—
72Dic254—22no—
73Dic254—21no—
74Dic254—41no—
75Dic254—41no—
76D508—21no—
77D508—21no—
78C22⋊D127—41no—
79C22⋊D127—41no—
80C22⋊D127—84no—
81C2×Dic127—22no—
82C2×Dic127—22no—
83C2×Dic127—21no—
84C2×Dic127—22no—
85C2×Dic127—42no—
86C2×Dic127—41no—
87C2×Dic127—41no—
88C4×D127—41no—
89C4×D127—41no—
90C23×C127—22no—
91C23×C127—22no—
92C23×C127—22no—
93C23×C127—44no—
94C23×C127C2×D411yescharacteristic
95D4×C127—41no—
96D4×C127—41no—
97D4×C127—41no—
98D4×C127—41no—
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127#4111yescharacteristic
105C8×C127—22no—
106C8×C127—41no—
107C22×C127—22no—
108C22×C127—44no—
109C22×C127—42no—
110C22×C127—44no—
111C22×C127—82no—
112C22×C127#12011yescharacteristic
113C22×C127#12011yescharacteristic
114C22×C127#12111yescharacteristic
115C2×D127—41no—
116C2×D127—42no—
117C4×C127#12011yescharacteristic
118C4×C127#12111yescharacteristic
119C4×C127#12111yescharacteristic
120C4×C127#12711yescharacteristic
121Dic127—42no—
122Dic127—41no—
123Dic127—42no—
124Dic127—42no—
125C2×C127—21no—
126C2×C127—21no—
127C2×C127—82no—
128C2×C127#39211yescharacteristic
129C2×C127#45512yes—
130D127—81no—
131#2004—1271nomaximal, Sylow
132C127#200411yescharacteristic, complemented, Sylow
133C2×#137—1271no—
134#535—1271no—
135C2×#124—1271no—
136C2×#104—1271no—
137#419—1271no—
138C2×#100D12711yescharacteristic, complemented
139#339—1271no—
140C2×#41—1271no—
141C22×Q8—1271no—
142C22×D4C2×D12711yescharacteristic
143#137—2542no—
144#137—2542no—
145C2×Q16—2541no—
146C2×SD16—2541no—
147C2×M16C2×D12711yescharacteristic
148C2×M16C2×D12711yescharacteristic
149#127—2541no—
150#124—2542no—
151#124—2542no—
152#124—2541no—
153#121—2542no—
154#120—2542no—
155#117—1271no—
156C2×C22⋊C4—2542no—
157C2×C42—1271no—
158#104—2542no—
159#103—2542no—
160#100—22no—
161C24D50812yes—
162#41—2541no—
163#41—2541no—
164#41—2541no—
165#41—2541no—
166C2×Q8—1271no—
167C2×Q8—1272no—
168C2×Q8—2541no—
169C2×Q8—2541no—
170C2×Q8—2541no—
171C2×Q8—2541no—
172C2×Q8—2541no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—1271no—
180C2×D4—2544no—
181C2×D4—2541no—
182C22×C4—2541no—
183C22×C4—2541no—
184C22×C4—2541no—
185C22×C4C22×D12711yescharacteristic
186Q16—2544no—
187SD16—2544no—
188M16—21no—
189M16—21no—
190M16—21no—
191M16C22⋊D12712yescomplemented
192C2×C8—21no—
193C2×C8—21no—
194C4⋊C4—1272no—
195C22⋊C4—2548no—
196C22⋊C4—2544no—
197C22⋊C4—2541no—
198C42—1272no—
199C42—2541no—
200C42—2541no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—44no—
205C23—2541no—
206C23C2×D50811yescharacteristic
207Q8—5081no—
208Q8—5081no—
209Q8—2542no—
210Q8—2541no—
211Q8—2542no—
212Q8—2541no—
213D4—41no—
214D4—41no—
215D4—41no—
216D4—41no—
217D4—10164no—
218D4—5081no—
219D4—5081no—
220D4—2541no—
221D4—2541no—
222C2×C4—5082no—
223C2×C4—5081no—
224C2×C4—5081no—
225C2×C4—5081no—
226C2×C4—5081no—
227C2×C4—2542no—
228C2×C4—2542no—
229C2×C4—2542no—
230C2×C4—2541no—
231C2×C4D4×D12711yescharacteristic
232C2×C4D4×D12711yescharacteristic
233C2×C4D4×D12711yescharacteristic
234C2×C4D4×D12711yescharacteristic
235C2×C4C2×C22⋊D12711yescharacteristic
236C2×C4#1452324811yescharacteristic
237C8—22no—
238C8—41no—
239C22—22no—
240C22—44no—
241C22—42no—
242C22—44no—
243C22—82no—
244C22—5081no—
245C22—5082no—
246C22#1686117911yescharacteristic
247C22#1686117911yescharacteristic
248C22#1686118311yescharacteristic
249C4—5082no—
250C4—5081no—
251C4—5082no—
252C4—5082no—
253C4#1686117711yescharacteristic
254C4#1686118211yescharacteristic
255C4#1686118211yescharacteristic
256C4#1686120511yescharacteristic
257C2—21no—
258C2—21no—
259C2—82no—
260C2—10161no—
261C2#2254978511yescharacteristic
262C2#2255005612yes—
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#100×C127C2
socleC22×C127#120
prime radical#22550443C2
hypercenterC2×#100D127
hypocenterC127#2004
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. C2×#16861242C2semidirect
3. C2×#16861196C2semidirect
4. C2×#16861135C2semidirect
5. #22549886C2semidirect
6. #22549637C2semidirect
7. C2×#100×C127C2semidirect
39. C2×Dic254D4semidirect
54. C4×Dic127D4semidirect
66. M16×C127D4semidirect
132. C127#2004semidirect
138. C2×#100D127semidirect
191. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612422
#225504432
C2×#168611962
C2×#168611352
#225498862
#225496372
C2×#100×C1272
#2004127