Finite Group Databasebeta

← #27170202

Subgroups — #27170202

Order 12928.

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

normal subgroups only

This group has 236 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#20265884C211yesmaximal, characteristic, complemented
2#20265549C211yesmaximal, characteristic, complemented
3#20265374C211yesmaximal, characteristic, complemented
4#20264925C211yesmaximal, characteristic, complemented
5#20264907C211yesmaximal, characteristic, complemented
6D4⋊#11717579C211yesmaximal, characteristic, complemented
7#459×C101C211yesmaximal, characteristic
8C2×#13108062—22no—
9C2×#13108062—21no—
10#15692255C2211yescharacteristic
11#15692246—21no—
12#15692234—21no—
13#15692232—22no—
14C4×Dic202C2211yescharacteristic
15D4×Dic101—21no—
16C2×C4⋊Dic101—21no—
17C4⋊#11717579C2211yescharacteristic
18#15692176—21no—
19C4×#11717579C2211yescharacteristic
20C2×D8×C101—21no—
21#127×C101C2211yescharacteristic
22C4×D4×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24#102×C101—21no—
25C2×Dic202D411yescharacteristic
26C2×Dic202D412yes—
27C22×Dic101—22no—
28#13108062—44no—
29#13108062—42no—
30C2×#11717579—21no—
31C2×#11717579—21no—
32C4⋊Dic101—22no—
33C4⋊Dic101—21no—
34C4⋊Dic101—21no—
35C4⋊Dic101D411yescharacteristic
36#13108053—22no—
37#13108053—21no—
38C22⋊Dic101—22no—
39C4×Dic101—21no—
40C2×D4×C101—21no—
41C2×D4×C101—21no—
42C2×D4×C101—21no—
43C2×D4×C101D411yescharacteristic
44C22×C4×C101—21no—
45D8×C101—42no—
46C2×C8×C101—21no—
47C4⋊C4×C101D411yescharacteristic
48C22⋊C4×C101—21no—
49C42×C101C2311yescharacteristic
50C23×C101—21no—
51C23×C101—41no—
52D4×C101—21no—
53D4×C101—42no—
54D4×C101—41no—
55D4×C101—41no—
56D4×C101—41no—
57D4×C101—41no—
58D4×C101SD1612yescomplemented
59C2×C4×C101—21no—
60C2×C4×C101—41no—
61C2×C4×C101—41no—
62C2×C4×C101C2×D411yescharacteristic
63C2×C4×C101C2×D411yescharacteristic
64C2×C4×C101C2×D411yescharacteristic
65C2×Dic101—21no—
66C2×Dic101—21no—
67C2×Dic101—22no—
68C2×Dic101—82no—
69Dic202—21no—
70Dic202—21no—
71Dic202—42no—
72C8×C101—41no—
73#11717579—22no—
74#11717579—41no—
75C22×C101—22no—
76C22×C101—41no—
77C22×C101—41no—
78C22×C101—81no—
79C22×C101—81no—
80C22×C101—81no—
81C22×C101#12011yescharacteristic
82C4×C101—21no—
83C4×C101—41no—
84C4×C101C2×SD1611yescharacteristic
85C4×C101#13511yescharacteristic
86C4×C101AGL(1,ℤ/8)11yescharacteristic
87C4×C101#13711yescharacteristic
88Dic101—41no—
89Dic101—42no—
90Dic101—41no—
91C2×C101—42no—
92C2×C101—161no—
93C2×C101#44811yescharacteristic
94C2×C101#44911yescharacteristic
95C2×C101#45311yescharacteristic
96#1622—1011nomaximal, Sylow
97C101#162211yescharacteristic, complemented, Sylow
98#554—1011no—
99#492—1011no—
100#460—1011no—
101#459D10111yescharacteristic
102#333—1011no—
103Q8⋊C8—1011no—
104D4⋊C8—1011no—
105C2×SD16—2021no—
106C2×SD16—2022no—
107C2×D8—21no—
108C4⋊Q8—1011no—
109#127C2×D10111yescharacteristic
110#125—2021no—
111C22⋊Q8—2022no—
112C22⋊Q8—2021no—
113C4×Q8—1011no—
114C4×D4—2021no—
115C4×D4C2×D10111yescharacteristic
116C2×C4⋊C4—2021no—
117C4⋊C8—1011no—
118C4⋊C8C2×D10111yescharacteristic
119#102—21no—
120#102—2021no—
121C4×C8—1011no—
122C2×Q8—1012no—
123C2×Q8—1011no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4C22⋊D10111yescharacteristic
128C22×C4—21no—
129C22×C4—2022no—
130SD16—4042no—
131SD16—4044no—
132D8—42no—
133C2×C8—21no—
134C2×C8—2021no—
135C2×C8—2021no—
136C4⋊C4—1011no—
137C4⋊C4—2022no—
138C4⋊C4—2022no—
139C4⋊C4—2021no—
140C4⋊C4—2021no—
141C4⋊C4—2021no—
142C4⋊C4C22⋊D10111yescharacteristic
143C22⋊C4—21no—
144C22⋊C4—2022no—
145C42—2021no—
146C42C22×D10111yescharacteristic
147C23—21no—
148C23—41no—
149Q8—4042no—
150Q8—2021no—
151Q8—2021no—
152D4—21no—
153D4—42no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4#1310806212yescomplemented
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—8082no—
163C2×C4—2021no—
164C2×C4—2021no—
165C2×C4—2022no—
166C2×C4D4×D10111yescharacteristic
167C2×C4D4×D10111yescharacteristic
168C2×C4C2×C22⋊D10111yescharacteristic
169C8—41no—
170C8—4041no—
171C8—2022no—
172C22—22no—
173C22—41no—
174C22—41no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22#1569222311yescharacteristic
179C4—21no—
180C4—41no—
181C4—4041no—
182C4—4041no—
183C4—4042no—
184C4C2×#1310806211yescharacteristic
185C4#1569227411yescharacteristic
186C4#1569227911yescharacteristic
187C4#1569228511yescharacteristic
188C2—42no—
189C2—161no—
190C2#2026531511yescharacteristic
191C2#2026531911yescharacteristic
192C2#2026534311yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting, prime radical#459×C101C2
socleC22×C101#120
hypercenter#459D101
hypocenterC101#1622
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #20265884C2semidirect
2. #20265549C2semidirect
3. #20265374C2semidirect
4. #20264925C2semidirect
5. #20264907C2semidirect
6. D4⋊#11717579C2semidirect
58. D4×C101SD16semidirect
97. C101#1622semidirect
158. D4#13108062semidirect

Maximal subgroups

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

subgroupindex
#202658842
#202655492
#202653742
#202649252
#202649072
D4⋊#117175792
#459×C1012
#1622101