Finite Group Databasebeta

← #30648535

Subgroups — #30648535

Order 16256.

376 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 153 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#22550433C211yesmaximal, characteristic
2#22550183C214yesmaximal, complemented
3#22550171C212yesmaximal, complemented
4#22550161C212yesmaximal, complemented
5#22549916C212yesmaximal
6#22549900C211yesmaximal, characteristic
7#22549885C212yesmaximal
8C2×C4×Q8×C127C211yesmaximal, characteristic
9C2×#14523232C2212yes—
10#16861209C2211yescharacteristic
11#16861209C2211yescharacteristic
12#16861201C2212yes—
13#16861188—21no—
14#16861188—21no—
15C4×Dic254—21no—
16C4×Dic254—21no—
17#16861168C2212yes—
18#16861142C2214yes—
19#16861140C2214yes—
20C4⋊(C127⋊C8)C2212yes—
21C4⋊(C127⋊C8)C2212yes—
22#16861133C2214yes—
23#16861133C2214yes—
24C22×Q8×C127C2211yescharacteristic
25C4×Q8×C127—21no—
26C4×Q8×C127—21no—
27C4×Q8×C127C2212yes—
28C4×Q8×C127C2212yes—
29C2×C4⋊C4×C127—21no—
30C2×C4⋊C4×C127C2211yescharacteristic
31C2×C42×C127—21no—
32C2×C42×C127C2211yescharacteristic
33C2×Dic254—21no—
34C2×Dic254—21no—
35#14523232—24no—
36C2×C127⋊C8C2314yes—
37C4⋊Dic127—21no—
38C4⋊Dic127—21no—
39C4⋊Dic127C2312yes—
40C4⋊Dic127C2312yes—
41Dic127⋊C4—22no—
42Dic127⋊C4—22no—
43C22⋊Dic127—22no—
44C4×Dic127—22no—
45C2×Q8×C127—21no—
46C2×Q8×C127—22no—
47C2×Q8×C127Q814yescomplemented
48C2×Q8×C127C2312yes—
49C22×C4×C127—21no—
50C22×C4×C127—21no—
51C22×C4×C127D411yescharacteristic
52C22×C4×C127D411yescharacteristic
53C22×C4×C127C2311yescharacteristic
54C4⋊C4×C127—22no—
55C4⋊C4×C127—22no—
56C4⋊C4×C127—21no—
57C4⋊C4×C127C2312yes—
58C42×C127—22no—
59C42×C127—22no—
60C42×C127D411yescharacteristic
61C42×C127D411yescharacteristic
62C42×C127C2311yescharacteristic
63C42×C127C2311yescharacteristic
64Dic254—42no—
65Dic254—42no—
66C2×Dic127—22no—
67C2×Dic127—22no—
68C127⋊C8—24no—
69C23×C127C2×D411yescharacteristic
70Q8×C127—24no—
71Q8×C127—22no—
72Q8×C127C2×Q814yes—
73C2×C4×C127—24no—
74C2×C4×C127—22no—
75C2×C4×C127—22no—
76C2×C4×C127—22no—
77C2×C4×C127—41no—
78C2×C4×C127—42no—
79C2×C4×C127C2×D411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×Q812yes—
85C2×C4×C127#4112yes—
86C2×C4×C127C2411yescharacteristic
87C22×C127—21no—
88C22×C127—21no—
89C22×C127C22⋊Q812yes—
90C22×C127C22×D411yescharacteristic
91C4×C127—24no—
92C4×C127—21no—
93C4×C127—21no—
94C4×C127—21no—
95C4×C127—42no—
96C4×C127C22⋊Q812yes—
97C4×C127#13712yes—
98C4×C127C22×Q811yescharacteristic
99C4×C127C2×#4111yescharacteristic
100Dic127—42no—
101Dic127—42no—
102C2×C127—22no—
103C2×C127C2×C22⋊Q811yescharacteristic
104C2×C127C2×#13711yescharacteristic
105C2×C127#57511yescharacteristic
106#3078—1271nomaximal, Sylow
107C127#307811yescharacteristic, complemented, Sylow
108#533—1271no—
109C2×C4×Q8D12711yescharacteristic
110#479—1274no—
111#477—1272no—
112#475—1272no—
113#428—1272no—
114#423—1271no—
115#419—1272no—
116C22×Q8C2×D12711yescharacteristic
117C2×M16—1272no—
118C4⋊Q8—1271no—
119C4⋊Q8—1271no—
120#125—1272no—
121C22⋊Q8—2541no—
122C22⋊Q8—2541no—
123C4×Q8—21no—
124C4×Q8—21no—
125C4×Q8—2541no—
126C4×Q8—2541no—
127C4×Q8C2×D12712yes—
128C4×Q8C2×D12712yes—
129#117—1272no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4C2×D12711yescharacteristic
132C2×C42—21no—
133C2×C42C2×D12711yescharacteristic
134#107—1274no—
135#106—1274no—
136C4⋊C8—1272no—
137C4⋊C8—1272no—
138#103—1274no—
139#103—1274no—
140C2×Q8—21no—
141C2×Q8—22no—
142C2×Q8—2541no—
143C2×Q8—2541no—
144C2×Q8Dic25414yescomplemented
145C2×Q8C22×D12712yes—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4C22⋊D12711yescharacteristic
149C22×C4C22⋊D12711yescharacteristic
150C22×C4C22×D12711yescharacteristic
151M16—2544no—
152C2×C8—1274no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—1272no—
157C4⋊C4—1272no—
158C4⋊C4—2542no—
159C4⋊C4—2542no—
160C4⋊C4—2541no—
161C4⋊C4—2541no—
162C4⋊C4C22×D12712yes—
163C22⋊C4—2542no—
164C42—22no—
165C42—22no—
166C42—2542no—
167C42C22⋊D12711yescharacteristic
168C42C22⋊D12711yescharacteristic
169C42C22×D12711yescharacteristic
170C42C22×D12711yescharacteristic
171C23C2×C22⋊D12711yescharacteristic
172Q8—24no—
173Q8—22no—
174Q8—5082no—
175Q8—5082no—
176Q8C2×Dic25414yes—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—2542no—
184C2×C4—2542no—
185C2×C4C2×C22⋊D12711yescharacteristic
186C2×C4C2×C22⋊D12711yescharacteristic
187C2×C4C2×C22⋊D12711yescharacteristic
188C2×C4C2×C22⋊D12711yescharacteristic
189C2×C4C2×C22⋊D12711yescharacteristic
190C2×C4C2×Dic25412yes—
191C2×C4#1452324812yes—
192C2×C4C23×D12711yescharacteristic
193C8—2544no—
194C22—21no—
195C22—21no—
196C22#1686118812yes—
197C22C22×C22⋊D12711yescharacteristic
198C4—24no—
199C4—21no—
200C4—21no—
201C4—21no—
202C4—42no—
203C4—5082no—
204C4—5082no—
205C4#1686118812yes—
206C4#1686124012yes—
207C4C22×Dic25411yescharacteristic
208C4C2×#1452324811yescharacteristic
209C2—22no—
210C2C2×#1686118811yescharacteristic
211C2C2×#1686124011yescharacteristic
212C2#2255067411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×C4×Q8×C127C2
socleC22×C127C22×D4
prime radicalC23×C127C2×D4
hypercenterC2×C4×Q8D127
hypocenterC127#3078
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #22550183C2semidirect
3. #22550171C2semidirect
4. #22550161C2semidirect
47. C2×Q8×C127Q8semidirect
107. C127#3078semidirect
144. C2×Q8Dic254semidirect

Maximal subgroups

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

subgroupindex
#225504332
#225501832
#225501712
#225501612
#225499162
#225499002
#225498852
C2×C4×Q8×C1272
#3078127