Finite Group Databasebeta

← #27170242

Subgroups — #27170242

Order 12928.

216 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 57 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#20265525C211yesmaximal, characteristic, complemented
2#20265377C211yesmaximal, characteristic, complemented
3#20265372C211yesmaximal, characteristic
4#20264923C211yesmaximal, characteristic, complemented
5Q8⋊#11717579C211yesmaximal, characteristic, complemented
6D4⋊#11717579C211yesmaximal, characteristic, complemented
7#556×C101C211yesmaximal, characteristic, complemented
8C2×#13108061—21no—
9C2×#13108059—21no—
10#15692252C2211yescharacteristic
11C4⋊#11717579C2211yescharacteristic
12C4⋊#11717579C2211yescharacteristic
13#15692175—22no—
14#15692175—21no—
15#15692175—21no—
16C4×#11717579C2211yescharacteristic
17C4⋊Q8×C101C2211yescharacteristic
18#126×C101—22no—
19#124×C101—21no—
20C22⋊Q8×C101—21no—
21C4×Q8×C101—21no—
22C4×Q8×C101C2211yescharacteristic
23C4×D4×C101C2211yescharacteristic
24#117×C101—21no—
25C2×D404—21no—
26C2×D404—21no—
27C2×D404—21no—
28#13108061—42no—
29#13108059—42no—
30C2×#11717579—21no—
31C2×#11717579—21no—
32C2×#11717579—21no—
33C2×Q8×C101—21no—
34C2×Q8×C101D411yescharacteristic
35C2×D4×C101D411yescharacteristic
36C22×C4×C101—21no—
37C4⋊C4×C101—22no—
38C4⋊C4×C101—21no—
39C4⋊C4×C101—21no—
40C4⋊C4×C101D411yescharacteristic
41C4⋊C4×C101D411yescharacteristic
42C4⋊C4×C101D412yes—
43C22⋊C4×C101—22no—
44C22⋊C4×C101—22no—
45C22⋊C4×C101—21no—
46C42×C101—22no—
47C42×C101—21no—
48C42×C101C2311yescharacteristic
49C23×C101—21no—
50C22×D101—41no—
51Q8×C101—21no—
52Q8×C101—21no—
53Q8×C101—41no—
54Q8×C101—41no—
55D4×C101—21no—
56D4×C101—21no—
57C2×C4×C101—21no—
58C2×C4×C101—21no—
59C2×C4×C101—21no—
60C2×C4×C101—22no—
61C2×C4×C101—41no—
62C2×C4×C101—41no—
63C2×C4×C101C2×D411yescharacteristic
64C2×C4×C101C2×D411yescharacteristic
65C2×C4×C101C2×D411yescharacteristic
66D404—42no—
67D404—41no—
68D404—41no—
69D404—41no—
70D404—41no—
71#11717579—41no—
72#11717579—41no—
73#11717579—41no—
74C22×C101—41no—
75C22×C101—41no—
76C22×C101—41no—
77C22×C101#12011yescharacteristic
78C2×D101—81no—
79C2×D101—81no—
80C2×D101—81no—
81C4×C101—21no—
82C4×C101—41no—
83C4×C101—41no—
84C4×C101—42no—
85C4×C101—41no—
86C4×C101#13511yescharacteristic
87C4×C101#13511yescharacteristic
88C4×C101AGL(1,ℤ/8)11yescharacteristic
89C4×C101AGL(1,ℤ/8)11yescharacteristic
90C2×C101—81no—
91C2×C101#44911yescharacteristic
92C2×C101#44911yescharacteristic
93C2×C101#45411yescharacteristic
94D101—161no—
95#1628—1011nomaximal, Sylow
96C101#162811yescharacteristic, complemented, Sylow
97#556D10111yescharacteristic, complemented
98#486—1011no—
99#460—1011no—
100#459—1011no—
101#332—1011no—
102Q8⋊C8—1011no—
103D4⋊C8—1011no—
104C2×SD16—2021no—
105C2×D8—2021no—
106C4⋊Q8C2×D10111yescharacteristic
107#127—1011no—
108#126—22no—
109#124—21no—
110C22⋊Q8—21no—
111C4×Q8—21no—
112C4×Q8C2×D10111yescharacteristic
113C4×D4C2×D10111yescharacteristic
114#117—21no—
115C4⋊C8—1011no—
116C4⋊C8—1011no—
117#102—2021no—
118#102—2021no—
119#102—2022no—
120C4×C8—1011no—
121C2×Q8—21no—
122C2×Q8C22⋊D10111yescharacteristic
123C2×D4—2021no—
124C2×D4—2021no—
125C2×D4—2021no—
126C2×D4C22⋊D10111yescharacteristic
127C22×C4—21no—
128SD16—4042no—
129D8—4042no—
130C2×C8—2021no—
131C2×C8—2021no—
132C2×C8—2021no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4C22⋊D10111yescharacteristic
137C4⋊C4C22⋊D10111yescharacteristic
138C4⋊C4C22⋊D10112yes—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—21no—
142C42—22no—
143C42—21no—
144C42C22×D10111yescharacteristic
145C23—21no—
146C23—4041no—
147Q8—21no—
148Q8—21no—
149Q8—41no—
150Q8—41no—
151D4—21no—
152D4—21no—
153D4—4042no—
154D4—4041no—
155D4—4041no—
156D4—4041no—
157D4—4041no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4C2×C22⋊D10111yescharacteristic
165C2×C4C2×C22⋊D10111yescharacteristic
166C2×C4C2×C22⋊D10111yescharacteristic
167C8—4041no—
168C8—4041no—
169C8—4041no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—8081no—
174C22—8081no—
175C22—8081no—
176C22#1569222411yescharacteristic
177C4—21no—
178C4—41no—
179C4—41no—
180C4—42no—
181C4—41no—
182C4#1569227511yescharacteristic
183C4#1569227511yescharacteristic
184C4#1569228011yescharacteristic
185C4#1569228011yescharacteristic
186C2—81no—
187C2—16161no—
188C2#2026532211yescharacteristic
189C2#2026532211yescharacteristic
190C2#2026535011yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#556×C101C2
socleC22×C101#120
prime radical#20265372C2
hypercenter#556D101
hypocenterC101#1628
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #20265525C2semidirect
2. #20265377C2semidirect
4. #20264923C2semidirect
5. Q8⋊#11717579C2semidirect
6. D4⋊#11717579C2semidirect
7. #556×C101C2semidirect
96. C101#1628semidirect
97. #556D101semidirect

Maximal subgroups

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

subgroupindex
#202655252
#202653772
#202653722
#202649232
Q8⋊#117175792
D4⋊#117175792
#556×C1012
#1628101