Finite Group Databasebeta

← #30639834

Subgroups — #30639834

Order 16256.

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

normal subgroups only

This group has 350 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#22550676C211yesmaximal, characteristic, complemented
2#22550674C212yesmaximal, complemented
3#22549842C212yesmaximal, complemented
4C127⋊#338C211yesmaximal, characteristic, complemented
5#408×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C2212yes—
7#16861242—22no—
8#16861240—22no—
9#16861239—22no—
10#16861237—22no—
11#16861235—22no—
12#16861234—22no—
13#16861231—22no—
14#16861229—24no—
15C2×#14523232C2212yes—
16(C2×D127)⋊C8—24no—
17C2×#41×C127C2211yescharacteristic
18C2×M16×C127C2211yescharacteristic
19C22×C8×C127C2211yescharacteristic
20C22⋊C8×C127—22no—
21#14523248—22no—
22#14523248—22no—
23#14523248—22no—
24#14523248—22no—
25C2×Dic254D412yes—
26C2×D508D412yes—
27C2×C22⋊D127—22no—
28C2×C4×D127—22no—
29Dic508—22no—
30#14523238—24no—
31#14523237—24no—
32#14523236—24no—
33#14523235—24no—
34D1016—22no—
35#14523233—24no—
36#14523232—22no—
37#14523232—22no—
38C2×C127⋊C8D414yes—
39#41×C127—22no—
40#41×C127—22no—
41C2×Q8×C127D411yescharacteristic
42C2×D4×C127—21no—
43C2×D4×C127D411yescharacteristic
44C22×C4×C127—21no—
45C22×C4×C127C2311yescharacteristic
46M16×C127—22no—
47C2×C8×C127—21no—
48C2×C8×C127—22no—
49C2×C8×C127D412yes—
50C22×D127—22no—
51Dic254—22no—
52Dic254—22no—
53D508—22no—
54D508—22no—
55C22⋊D127—42no—
56C22⋊D127—42no—
57C2×Dic127—22no—
58C4×D127—42no—
59C4×D127—42no—
60C127⋊C8—24no—
61C23×C127—21no—
62C23×C127#4111yescharacteristic
63Q8×C127—22no—
64D4×C127—22no—
65D4×C127—42no—
66C2×C4×C127—21no—
67C2×C4×C127—41no—
68C2×C4×C127—41no—
69C2×C4×C127C2×D412yes—
70C2×C4×C127C2×D412yes—
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127C2×D411yescharacteristic
73C8×C127—22no—
74C8×C127—41no—
75C22×C127—41no—
76C22×C127—42no—
77C22×C127—41no—
78C22×C127#12112yes—
79C22×C127#12111yescharacteristic
80C2×D127—44no—
81C2×D127—42no—
82C4×C127—41no—
83C4×C127#12011yescharacteristic
84C4×C127#12012yes—
85C4×C127#12711yescharacteristic
86Dic127—42no—
87C2×C127—22no—
88C2×C127—21no—
89C2×C127—81no—
90C2×C127#39211yescharacteristic
91D127—82no—
92#2017—1271nomaximal, Sylow
93C127#201711yescharacteristic, complemented, Sylow
94#575—1272no—
95#575—1271no—
96#408—1272no—
97#408D12711yescharacteristic, complemented
98#338—1271no—
99C2×#41—1272no—
100C2×#41C2×D12711yescharacteristic
101#137—2542no—
102#137—2542no—
103#137—2542no—
104AGL(1,ℤ/8)—2542no—
105AGL(1,ℤ/8)—2542no—
106AGL(1,ℤ/8)—2542no—
107#135—2544no—
108#135—2542no—
109C2×M16—1272no—
110C2×M16C2×D12711yescharacteristic
111C22×C8C2×D12711yescharacteristic
112C22⋊C8—22no—
113C22⋊C8—2544no—
114#41—22no—
115#41—22no—
116#41—2542no—
117#41—2542no—
118#41—2542no—
119#41—2542no—
120C2×Q8—1272no—
121C2×Q8D50811yescharacteristic
122C2×D4—21no—
123C2×D4—1272no—
124C2×D4—2542no—
125C2×D4D50811yescharacteristic
126C22×C4—21no—
127C22×C4—2542no—
128C22×C4C22×D12711yescharacteristic
129Q16—2542no—
130Q16—2544no—
131SD16—2544no—
132SD16—2544no—
133SD16—2544no—
134D8—2542no—
135D8—2544no—
136M16—22no—
137M16—2542no—
138M16—2542no—
139C2×C8—22no—
140C2×C8—21no—
141C2×C8—1274no—
142C2×C8C22⋊D12712yes—
143C23—21no—
144C23—2542no—
145C23#1452324811yescharacteristic
146Q8—22no—
147Q8—2542no—
148Q8—2542no—
149D4—22no—
150D4—42no—
151D4—5082no—
152D4—5082no—
153D4—2542no—
154D4—2542no—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—5082no—
159C2×C4—5082no—
160C2×C4—2542no—
161C2×C4D4×D12712yes—
162C2×C4D4×D12712yes—
163C2×C4C2×C22⋊D12711yescharacteristic
164C2×C4C2×D50811yescharacteristic
165C8—22no—
166C8—41no—
167C8—2544no—
168C22—41no—
169C22—42no—
170C22—41no—
171C22—5084no—
172C22—5082no—
173C22#1686118212yes—
174C22#1686118311yescharacteristic
175C4—41no—
176C4—5082no—
177C4#1686117711yescharacteristic
178C4#1686117912yes—
179C4#1686120511yescharacteristic
180C2—21no—
181C2—22no—
182C2—81no—
183C2—10162no—
184C2#2254978511yescharacteristic
subgroupG/H
centerC4#16861205
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#408×C127C2
socleC2×C127#392
hypercenter#408D127
hypocenterC127#2017
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #22550676C2semidirect
2. #22550674C2semidirect
3. #22549842C2semidirect
4. C127⋊#338C2semidirect
5. #408×C127C2semidirect
93. C127#2017semidirect
97. #408D127semidirect

Maximal subgroups

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

subgroupindex
#225506762
#225506742
#225498422
C127⋊#3382
#408×C1272
#2017127