Finite Group Databasebeta

← #30640094

Subgroups — #30640094

Order 16256.

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

normal subgroups only

This group has 208 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
1C2×#16861144C211yesmaximal, characteristic
2#22549903C211yesmaximal, characteristic
3#22549884C211yesmaximal, characteristic
4C2×#16861132C211yesmaximal, characteristic
5C2×#16861127C211yesmaximal, characteristic
6#22549645C211yesmaximal, characteristic
7#428×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C2×#14523232C2211yescharacteristic
11C22×C127⋊C8C2211yescharacteristic
12#16861144—22no—
13C4⋊(C127⋊C8)—22no—
14#16861132—22no—
15#16861132—21no—
16#16861132—21no—
17#16861127—22no—
18C2×M16×C127C2211yescharacteristic
19#117×C127C2211yescharacteristic
20C2×C4⋊C4×C127C2211yescharacteristic
21#107×C127—21no—
22#106×C127—21no—
23C2×Dic254—21no—
24C2×Dic254—21no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254Q812yes—
29C22×Dic127—21no—
30#14523232—21no—
31#14523232—21no—
32#14523232—21no—
33#14523232—21no—
34C2×C127⋊C8—21no—
35C2×C127⋊C8—21no—
36C2×C127⋊C8—21no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8D412yes—
39C22×C4×C127—21no—
40C22×C4×C127C2311yescharacteristic
41M16×C127—21no—
42M16×C127D412yes—
43C2×C8×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127D411yescharacteristic
47C4⋊C4×C127D411yescharacteristic
48C22⋊C4×C127—21no—
49C42×C127—21no—
50Dic254—22no—
51Dic254—22no—
52Dic254—41no—
53Dic254—41no—
54C2×Dic127—22no—
55C2×Dic127—41no—
56C2×Dic127—41no—
57C127⋊C8—22no—
58C127⋊C8—41no—
59C127⋊C8—41no—
60C23×C127#4111yescharacteristic
61C2×C4×C127—21no—
62C2×C4×C127—21no—
63C2×C4×C127—41no—
64C2×C4×C127—81no—
65C2×C4×C127C2×D411yescharacteristic
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127C2×Q811yescharacteristic
69C2×C4×C127#4111yescharacteristic
70C2×C4×C127#4111yescharacteristic
71C8×C127—22no—
72C22×C127—21no—
73C22×C127—21no—
74C22×C127#12111yescharacteristic
75C22×C127C22⋊Q811yescharacteristic
76C22×C127#12611yescharacteristic
77C4×C127—41no—
78C4×C127—41no—
79C4×C127—81no—
80C4×C127#12111yescharacteristic
81C4×C127#12111yescharacteristic
82C4×C127C22⋊Q811yescharacteristic
83C4×C127C22⋊Q811yescharacteristic
84Dic127—42no—
85C2×C127—21no—
86C2×C127—21no—
87C2×C127#39611yescharacteristic
88C2×C127#47111yescharacteristic
89C2×C127#47311yescharacteristic
90#2064—1271nomaximal, Sylow
91C127#206411yescharacteristic, complemented, Sylow
92C2×#108—1271no—
93#428D12711yescharacteristic
94#424—1271no—
95#419—1271no—
96C2×#103—1271no—
97C2×#101—1271no—
98#343—1271no—
99C22×Q8—1271no—
100C2×M16—1271no—
101C2×M16—1271no—
102C2×M16C2×D12711yescharacteristic
103C22×C8—1271no—
104#117C2×D12711yescharacteristic
105C2×C4⋊C4C2×D12711yescharacteristic
106#108—2542no—
107#107—21no—
108#106—21no—
109C4⋊C8—2542no—
110#103—2542no—
111#103—2541no—
112#103—2541no—
113#101—2542no—
114C2×Q8—1272no—
115C2×Q8—2541no—
116C2×Q8—2541no—
117C2×Q8—2541no—
118C2×Q8—2541no—
119C2×Q8—2541no—
120C22×C4—21no—
121C22×C4—2541no—
122C22×C4C22×D12711yescharacteristic
123M16—21no—
124M16—2541no—
125M16—2541no—
126M16—2541no—
127M16—2541no—
128M16D50812yes—
129C2×C8—21no—
130C2×C8—1272no—
131C2×C8—2541no—
132C2×C8—2541no—
133C2×C8—2541no—
134C2×C8—2541no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4C22⋊D12711yescharacteristic
138C4⋊C4C22⋊D12711yescharacteristic
139C22⋊C4—21no—
140C42—21no—
141C23#1452324711yescharacteristic
142Q8—5081no—
143Q8—5081no—
144Q8—2542no—
145Q8—2542no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—81no—
150C2×C4—5081no—
151C2×C4—5081no—
152C2×C4—2542no—
153C2×C4D4×D12711yescharacteristic
154C2×C4C2×C22⋊D12711yescharacteristic
155C2×C4C2×D50811yescharacteristic
156C2×C4Q8×D12711yescharacteristic
157C2×C4#1452324811yescharacteristic
158C2×C4#1452324911yescharacteristic
159C8—22no—
160C8—5081no—
161C8—5081no—
162C8—2542no—
163C22—21no—
164C22—21no—
165C22#1686118111yescharacteristic
166C22#1686118911yescharacteristic
167C22#1686120411yescharacteristic
168C4—41no—
169C4—41no—
170C4—81no—
171C4—5082no—
172C4#1686118311yescharacteristic
173C4#1686118411yescharacteristic
174C4#1686118511yescharacteristic
175C4#1686118711yescharacteristic
176C2—21no—
177C2—21no—
178C2#2254979811yescharacteristic
179C2#2255013711yescharacteristic
180C2#2255014911yescharacteristic
subgroupG/H
centerC22#16861181
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#428×C127C2
socleC22×C127#121
prime radicalC23×C127#41
hypercenter#428D127
hypocenterC127#2064
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
91. C127#2064semidirect

Maximal subgroups

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

subgroupindex
C2×#168611442
#225499032
#225498842
C2×#168611322
C2×#168611272
#225496452
#428×C1272
#2064127