Finite Group Databasebeta

← #27170084

Subgroups — #27170084

Order 12928.

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

normal subgroups only

This group has 308 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#20265315C212yesmaximal, complemented
2#20265089C211yesmaximal, characteristic, complemented
3#20264911C212yesmaximal, complemented
4#20264901C211yesmaximal, characteristic, complemented
5#534×C101C211yesmaximal, characteristic
6C22×Dic202C2211yescharacteristic, complemented
7C2×#13108062—22no—
8C2×C22⋊Dic101—21no—
9#15692178—22no—
10C22⋊#11717579C2212yescomplemented
11#15692162—21no—
12#15692162C2211yescharacteristic, complemented
13C22×D4×C101—21no—
14#123×C101—22no—
15#121×C101—21no—
16#121×C101C2212yes—
17#120×C101—22no—
18C2×C22⋊C4×C101C2211yescharacteristic
19C2×Dic202—22no—
20C2×Dic202—42no—
21C22×Dic101—21no—
22C22×Dic101—21no—
23#13108062—44no—
24C2×#11717579—22no—
25C22⋊Dic101—42no—
26C24×C101—21no—
27C24×C101D411yescharacteristic
28C2×D4×C101—22no—
29C2×D4×C101—22no—
30C2×D4×C101—21no—
31C2×D4×C101—42no—
32C2×D4×C101—42no—
33C2×D4×C101D412yes—
34C22×C4×C101—21no—
35C22×C4×C101D411yescharacteristic
36C22×C4×C101C2311yescharacteristic
37C4⋊C4×C101—21no—
38C4⋊C4×C101D412yes—
39C22⋊C4×C101—22no—
40C22⋊C4×C101—22no—
41C22⋊C4×C101—22no—
42C23×C101—21no—
43C23×C101—21no—
44C23×C101—22no—
45C23×C101—41no—
46C23×C101—41no—
47C23×C101—42no—
48C23×C101—81no—
49C23×C101—81no—
50C23×C101SD1612yescomplemented
51C23×C101C2×D411yescharacteristic
52D4×C101—42no—
53D4×C101—44no—
54D4×C101—42no—
55D4×C101—42no—
56D4×C101—81no—
57C2×C4×C101—21no—
58C2×C4×C101—22no—
59C2×C4×C101—41no—
60C2×C4×C101—42no—
61C2×C4×C101—41no—
62C2×C4×C101C2×D412yes—
63C2×Dic101—22no—
64C2×Dic101—41no—
65C2×Dic101—41no—
66C2×Dic101—41no—
67C2×Dic101—41no—
68C2×Dic101—41no—
69Dic202—24no—
70Dic202—42no—
71#11717579—42no—
72C22×C101—21no—
73C22×C101—21no—
74C22×C101—22no—
75C22×C101—42no—
76C22×C101—41no—
77C22×C101—41no—
78C22×C101—42no—
79C22×C101—42no—
80C22×C101—42no—
81C22×C101—81no—
82C22×C101—84no—
83C22×C101#12011yescharacteristic
84C22×C101C2×SD1611yescharacteristic
85C22×C101AGL(1,ℤ/8)11yescharacteristic
86C4×C101—22no—
87C4×C101—42no—
88C4×C101—41no—
89Dic101—42no—
90Dic101—81no—
91C2×C101—21no—
92C2×C101—21no—
93C2×C101—42no—
94C2×C101—82no—
95C2×C101#45011yescharacteristic
96C2×C101#45511yescharacteristic
97C2×C101B(4,𝔽2)11yescharacteristic
98#1601—1011nomaximal, Sylow
99C101#160111yescharacteristic, complemented, Sylow
100#534D10111yescharacteristic
101#448—1012no—
102#393—1011no—
103#327—1012no—
104#323—1011no—
105C22×Q8—1011no—
106C22×D4—21no—
107C2×SD16—2022no—
108#123—22no—
109#121—21no—
110#121C2×D10112yes—
111#120—22no—
112C2×C22⋊C4—2021no—
113C2×C22⋊C4C2×D10111yescharacteristic
114#103—2022no—
115C22⋊C8—1012no—
116#95—1011no—
117#95—2021no—
118C24—21no—
119C24C22⋊D10111yescharacteristic
120C2×Q8—4042no—
121C2×Q8—2022no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—21no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4C22⋊D10112yes—
128C22×C4—21no—
129C22×C4—2021no—
130C22×C4—2021no—
131C22×C4C22⋊D10111yescharacteristic
132C22×C4C22×D10111yescharacteristic
133SD16—4044no—
134C2×C8—2022no—
135C4⋊C4—21no—
136C4⋊C4C22⋊D10112yes—
137C22⋊C4—22no—
138C22⋊C4—22no—
139C22⋊C4—22no—
140C22⋊C4—4042no—
141C23—21no—
142C23—21no—
143C23—22no—
144C23—41no—
145C23—41no—
146C23—42no—
147C23—81no—
148C23—81no—
149C23#1310806212yescomplemented
150C23C2×C22⋊D10111yescharacteristic
151Q8—4042no—
152Q8—2024no—
153D4—42no—
154D4—44no—
155D4—42no—
156D4—42no—
157D4—81no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—4041no—
164C2×C4—4041no—
165C2×C4—4041no—
166C2×C4—4041no—
167C2×C4—4041no—
168C2×C4—2022no—
169C2×C4C2×C22⋊D10112yes—
170C8—4042no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—42no—
178C22—42no—
179C22—42no—
180C22—81no—
181C22—84no—
182C22#1569222411yescharacteristic
183C22C2×#1310806211yescharacteristic
184C22#1569228111yescharacteristic
185C4—22no—
186C4—42no—
187C4—41no—
188C4—8081no—
189C4—4042no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—82no—
194C2#2026532811yescharacteristic
195C2#2026535511yescharacteristic
196C2#2026536411yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting, prime radical#534×C101C2
socleC22×C101#120
hypercenter#534D101
hypocenterC101#1601
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #20265315C2semidirect
2. #20265089C2semidirect
3. #20264911C2semidirect
4. #20264901C2semidirect
6. C22×Dic202C22semidirect
10. C22⋊#11717579C22semidirect
12. #15692162C22semidirect
50. C23×C101SD16semidirect
99. C101#1601semidirect
149. C23#13108062semidirect

Maximal subgroups

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

subgroupindex
#202653152
#202650892
#202649112
#202649012
#534×C1012
#1601101