Finite Group Databasebeta

← #30638998

Subgroups — #30638998

Order 16256.

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

normal subgroups only

This group has 298 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
1C2×#16861140C211yesmaximal, characteristic
2C2×#16861133C211yesmaximal, characteristic
3C2×#16861133C211yesmaximal, characteristic
4#22549774C211yesmaximal, characteristic
5#22549638C211yesmaximal, characteristic
6#22549631C211yesmaximal, characteristic
7C2×C4×Q8×C127C211yesmaximal, characteristic
8C22×C127⋊C8C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10C2×C4⋊Dic127C2211yescharacteristic
11C2×C4⋊Dic127C2211yescharacteristic
12C2×Dic127⋊C4—21no—
13#16861140—22no—
14#16861133—22no—
15#16861133C414yescomplemented
16C127⋊#95—21no—
17C22×Q8×C127C2211yescharacteristic
18C4×Q8×C127—22no—
19C4×Q8×C127—22no—
20C2×C4⋊C4×C127—21no—
21C2×C4⋊C4×C127C2211yescharacteristic
22C2×C42×C127—21no—
23C2×C42×C127C2211yescharacteristic
24C22×Dic127—21no—
25C22×Dic127—21no—
26C2×C127⋊C8—21no—
27C2×C127⋊C8—21no—
28C2×C127⋊C8—21no—
29C2×C127⋊C8—21no—
30C2×C127⋊C8—21no—
31C2×C127⋊C8C2×C412yes—
32C4⋊Dic127—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127C2×C412yes—
36Dic127⋊C4—42no—
37C2×Q8×C127—21no—
38C2×Q8×C127—21no—
39C2×Q8×C127—21no—
40C2×Q8×C127C2×C412yes—
41C2×Q8×C127D412yes—
42C2×Q8×C127Q812yes—
43C22×C4×C127—21no—
44C22×C4×C127—21no—
45C22×C4×C127D411yescharacteristic
46C22×C4×C127D411yescharacteristic
47C22×C4×C127C2311yescharacteristic
48C4⋊C4×C127—24no—
49C4⋊C4×C127—21no—
50C4⋊C4×C127D412yes—
51C42×C127—24no—
52C42×C127—21no—
53C42×C127—21no—
54C2×Dic127—22no—
55C2×Dic127—41no—
56C2×Dic127—41no—
57C2×Dic127—41no—
58C2×Dic127—41no—
59C2×Dic127—41no—
60C127⋊C8—22no—
61C127⋊C8—41no—
62C23×C127C2×D411yescharacteristic
63Q8×C127—22no—
64Q8×C127—22no—
65Q8×C127—22no—
66Q8×C127C4⋊C414yescomplemented
67C2×C4×C127—22no—
68C2×C4×C127—21no—
69C2×C4×C127—22no—
70C2×C4×C127—22no—
71C2×C4×C127—21no—
72C2×C4×C127—21no—
73C2×C4×C127—22no—
74C2×C4×C127—21no—
75C2×C4×C127—21no—
76C2×C4×C127—41no—
77C2×C4×C127—41no—
78C2×C4×C127SD1612yes—
79C2×C4×C127C22×C411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×Q811yescharacteristic
83C2×C4×C127#4111yescharacteristic
84C2×C4×C127#4111yescharacteristic
85C22×C127C4×D411yescharacteristic
86C22×C127#12011yescharacteristic
87C22×C127C22⋊Q811yescharacteristic
88C22×C127C2×SD1611yescharacteristic
89C22×C127#13511yescharacteristic
90C22×C127#13711yescharacteristic
91C22×C127#13711yescharacteristic
92C4×C127—24no—
93C4×C127—22no—
94C4×C127—41no—
95C4×C127—42no—
96C4×C127C2×C4⋊C411yescharacteristic
97C4×C127C4×D411yescharacteristic
98C4×C127C22⋊Q811yescharacteristic
99C4×C127#12311yescharacteristic
100Dic127—42no—
101Dic127—81no—
102C2×C127#38511yescharacteristic
103C2×C127C4×SD1611yescharacteristic
104C2×C127#44111yescharacteristic
105C2×C127#44811yescharacteristic
106C2×C127#45211yescharacteristic
107C2×C127#47511yescharacteristic
108C2×C127#47911yescharacteristic
109#1865—1271nomaximal, Sylow
110C127#186511yescharacteristic, complemented, Sylow
111C2×C4×Q8D12711yescharacteristic
112C2×#106—1271no—
113C2×#103—1271no—
114C2×#103—1271no—
115#389—1271no—
116#340—1271no—
117#336—1271no—
118C22×Q8C2×D12711yescharacteristic
119C22×C8—1271no—
120C22×C8—1271no—
121C4×Q8—22no—
122C4×Q8—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—1271no—
125C2×C4⋊C4—1271no—
126C2×C4⋊C4—2541no—
127C2×C4⋊C4C2×D12711yescharacteristic
128C2×C42—21no—
129C2×C42C2×D12711yescharacteristic
130#106—2542no—
131#103—1274no—
132#103—2542no—
133#95—2541no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8C4×D12712yes—
138C2×Q8C22⋊D12712yes—
139C2×Q8Dic25412yes—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—2541no—
143C22×C4—2541no—
144C22×C4C22⋊D12711yescharacteristic
145C22×C4C22⋊D12711yescharacteristic
146C22×C4C22×D12711yescharacteristic
147C2×C8—1272no—
148C2×C8—2541no—
149C2×C8—2541no—
150C2×C8—2541no—
151C2×C8—2541no—
152C2×C8—2541no—
153C4⋊C4—24no—
154C4⋊C4—21no—
155C4⋊C4—1272no—
156C4⋊C4—5082no—
157C4⋊C4—2541no—
158C4⋊C4—2541no—
159C4⋊C4—2541no—
160C4⋊C4C22⋊D12712yes—
161C42—24no—
162C42—21no—
163C42—21no—
164C23C2×C22⋊D12711yescharacteristic
165Q8—22no—
166Q8—22no—
167Q8—22no—
168Q8Dic127⋊C414yescomplemented
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—5081no—
181C2×C4—5081no—
182C2×C4—5081no—
183C2×C4—5081no—
184C2×C4—5081no—
185C2×C4—2542no—
186C2×C4#1452323512yes—
187C2×C4C2×C4×D12711yescharacteristic
188C2×C4C2×C22⋊D12711yescharacteristic
189C2×C4C2×C22⋊D12711yescharacteristic
190C2×C4C2×Dic25411yescharacteristic
191C2×C4#1452324811yescharacteristic
192C2×C4#1452324811yescharacteristic
193C8—5081no—
194C8—2542no—
195C22C4×C22⋊D12711yescharacteristic
196C22#1686117811yescharacteristic
197C22#1686118811yescharacteristic
198C22C2×#1452323511yescharacteristic
199C22#1686122911yescharacteristic
200C22#1686123911yescharacteristic
201C22#1686124011yescharacteristic
202C4—24no—
203C4—22no—
204C4—41no—
205C4—42no—
206C4—10161no—
207C4—5082no—
208C4C2×Dic127⋊C411yescharacteristic
209C4C4×C22⋊D12711yescharacteristic
210C4#1686118811yescharacteristic
211C4#1686119211yescharacteristic
212C2#2254974911yescharacteristic
213C2C4×#1452323511yescharacteristic
214C2#2254998111yescharacteristic
215C2#2255001511yescharacteristic
216C2#2255003811yescharacteristic
217C2#2255016111yescharacteristic
218C2#2255018311yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×C4×Q8×C127C2
socle, prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenterC2×C4×Q8D127
hypocenterC127#1865
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
15. #16861133C4semidirect
66. Q8×C127C4⋊C4semidirect
110. C127#1865semidirect
168. Q8Dic127⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#168611402
C2×#168611332
#225497742
#225496382
#225496312
C2×C4×Q8×C1272
#1865127