Finite Group Databasebeta

← #30639889

Subgroups — #30639889

Order 16256.

246 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 69 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
1#22550504C211yesmaximal, characteristic
2#22549885C211yesmaximal, characteristic
3#22549783C211yesmaximal, characteristic
4#22549644C211yesmaximal, characteristic
5#22549637C211yesmaximal, characteristic
6#22549635C211yesmaximal, characteristic
7#419×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9#16861209—21no—
10#16861208—21no—
11#16861201—21no—
12#16861200—21no—
13#16861186—22no—
14#16861168C2211yescharacteristic
15C2×C4⋊Dic127C2211yescharacteristic
16C2×C4×Dic127C2211yescharacteristic
17#16861133—21no—
18#16861133—21no—
19C127⋊#95—22no—
20C22×Q8×C127C2211yescharacteristic
21C2×M16×C127C2211yescharacteristic
22#117×C127C2211yescharacteristic
23#103×C127—21no—
24#103×C127—21no—
25C2×Dic254—22no—
26C22×Dic127—21no—
27C22×Dic127D412yes—
28#14523232—21no—
29#14523232—21no—
30C2×C127⋊C8—21no—
31C4⋊Dic127—22no—
32C4⋊Dic127—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127D411yescharacteristic
35C4⋊Dic127D411yescharacteristic
36Dic127⋊C4—22no—
37Dic127⋊C4—22no—
38C22⋊Dic127—21no—
39C4×Dic127—21no—
40C4×Dic127—21no—
41C4×Dic127—21no—
42C2×Q8×C127—21no—
43C2×Q8×C127—21no—
44C2×Q8×C127—21no—
45C2×Q8×C127—21no—
46C2×Q8×C127—21no—
47C2×Q8×C127Q812yescomplemented
48C22×C4×C127—21no—
49C22×C4×C127C2311yescharacteristic
50M16×C127—21no—
51M16×C127—21no—
52C2×C8×C127—21no—
53C4⋊C4×C127D411yescharacteristic
54C4⋊C4×C127D411yescharacteristic
55C22⋊C4×C127—21no—
56C42×C127—21no—
57Dic254—82no—
58C2×Dic127—22no—
59C2×Dic127—21no—
60C2×Dic127—21no—
61C2×Dic127—22no—
62C2×Dic127—41no—
63C2×Dic127—41no—
64C2×Dic127—42no—
65C127⋊C8—41no—
66C23×C127C2×D411yescharacteristic
67Q8×C127—22no—
68Q8×C127—22no—
69Q8×C127—41no—
70Q8×C127—41no—
71C2×C4×C127—21no—
72C2×C4×C127—21no—
73C2×C4×C127—22no—
74C2×C4×C127—41no—
75C2×C4×C127—41no—
76C2×C4×C127C2×D411yescharacteristic
77C2×C4×C127C2×D411yescharacteristic
78C2×C4×C127C2×Q811yescharacteristic
79C2×C4×C127#4111yescharacteristic
80C2×C4×C127#4111yescharacteristic
81C2×C4×C127#4111yescharacteristic
82C8×C127—41no—
83C22×C127—21no—
84C22×C127—21no—
85C22×C127#12011yescharacteristic
86C22×C127C22⋊Q811yescharacteristic
87C22×C127#12411yescharacteristic
88C4×C127—42no—
89C4×C127—41no—
90C4×C127—41no—
91C4×C127C22⋊Q811yescharacteristic
92C4×C127C22⋊Q811yescharacteristic
93C4×C127#12411yescharacteristic
94C4×C127#12411yescharacteristic
95Dic127—42no—
96Dic127—42no—
97Dic127—41no—
98Dic127—41no—
99C2×C127—21no—
100C2×C127—21no—
101C2×C127#39311yescharacteristic
102C2×C127#45511yescharacteristic
103C2×C127#45611yescharacteristic
104#2028—1271nomaximal, Sylow
105C127#202811yescharacteristic, complemented, Sylow
106#544—1271no—
107#419—1271no—
108#419D12711yescharacteristic
109#391—1271no—
110#343—1271no—
111#339—1271no—
112#339—1271no—
113C22×Q8C2×D12711yescharacteristic
114C2×M16—1271no—
115C2×M16C2×D12711yescharacteristic
116C4⋊Q8—2541no—
117C4⋊Q8—2541no—
118#125—2541no—
119#125—2541no—
120C22⋊Q8—2542no—
121#117—1271no—
122#117C2×D12711yescharacteristic
123C2×C4⋊C4—1271no—
124C2×C42—1271no—
125#103—21no—
126#103—21no—
127#103—2541no—
128#103—2541no—
129#95—2542no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—2542no—
136C2×Q8Dic25412yescomplemented
137C22×C4—21no—
138C22×C4—1272no—
139C22×C4—2541no—
140C22×C4C22×D12711yescharacteristic
141M16—21no—
142M16—21no—
143M16—2541no—
144M16—2541no—
145C2×C8—21no—
146C2×C8—2541no—
147C4⋊C4—1271no—
148C4⋊C4—1271no—
149C4⋊C4—2542no—
150C4⋊C4—2542no—
151C4⋊C4—2542no—
152C4⋊C4—2541no—
153C4⋊C4—2541no—
154C4⋊C4C22⋊D12711yescharacteristic
155C4⋊C4C22⋊D12711yescharacteristic
156C22⋊C4—21no—
157C22⋊C4—2541no—
158C42—21no—
159C42—2541no—
160C42—2541no—
161C42—2541no—
162C23D4×D12711yescharacteristic
163Q8—22no—
164Q8—22no—
165Q8—41no—
166Q8—41no—
167Q8—10162no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—5081no—
174C2×C4—5081no—
175C2×C4—5082no—
176C2×C4—2542no—
177C2×C4—2541no—
178C2×C4—2541no—
179C2×C4—2542no—
180C2×C4D4×D12711yescharacteristic
181C2×C4C2×C22⋊D12711yescharacteristic
182C2×C4C2×Dic25411yescharacteristic
183C2×C4#1452324711yescharacteristic
184C2×C4#1452324711yescharacteristic
185C2×C4#1452324811yescharacteristic
186C8—41no—
187C8—5081no—
188C22—21no—
189C22—21no—
190C22#1686117711yescharacteristic
191C22#1686118611yescharacteristic
192C22#1686119511yescharacteristic
193C4—42no—
194C4—41no—
195C4—41no—
196C4—5082no—
197C4—5082no—
198C4—5081no—
199C4—5081no—
200C4#1686118611yescharacteristic
201C4#1686118811yescharacteristic
202C4#1686119511yescharacteristic
203C4#1686119611yescharacteristic
204C2—21no—
205C2—21no—
206C2#2254978811yescharacteristic
207C2#2255005211yescharacteristic
208C2#2255005711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#419×C127C2
socleC22×C127#120
prime radicalC23×C127C2×D4
hypercenter#419D127
hypocenterC127#2028
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
47. C2×Q8×C127Q8semidirect
105. C127#2028semidirect
136. C2×Q8Dic254semidirect

Maximal subgroups

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

subgroupindex
#225505042
#225498852
#225497832
#225496442
#225496372
#225496352
#419×C1272
#2028127