Finite Group Databasebeta

← #27170362

Subgroups — #27170362

Order 12928.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 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#20265892C211yesmaximal, characteristic
2#20265528C211yesmaximal, characteristic, complemented
3#20265458C211yesmaximal, characteristic, complemented
4#20265382C211yesmaximal, characteristic, complemented
5#20264925C211yesmaximal, characteristic, complemented
6#20264906C211yesmaximal, characteristic, complemented
7D4⋊C8×C101C211yesmaximal, characteristic, complemented
8C2×#13108062—21no—
9#15692255C2211yescharacteristic
10#15692255C2211yescharacteristic
11#15692241—22no—
12#15692229—21no—
13#15692226—21no—
14C4×D404C2211yescharacteristic
15#15692214—21no—
16#15692213—21no—
17#15692189—21no—
18C4⋊#11717579C2211yescharacteristic
19#15692180—22no—
20#15692178—21no—
21#15692177—21no—
22C4×D4×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24C4×C8×C101C2211yescharacteristic
25C2×Dic202—21no—
26C2×Dic202D412yes—
27C2×D404D411yescharacteristic
28C2×C22⋊D101—22no—
29C2×C4×D101—21no—
30#13108062—42no—
31C2×#11717579—21no—
32C4⋊Dic101—21no—
33C4⋊Dic101—21no—
34C4⋊Dic101—21no—
35C4⋊Dic101—21no—
36C4⋊Dic101D411yescharacteristic
37C22⋊Dic101—22no—
38#13108051—22no—
39#13108051—21no—
40C4×Dic101—22no—
41C2×D4×C101D411yescharacteristic
42C22×C4×C101—21no—
43C2×C8×C101—21no—
44C2×C8×C101—21no—
45C4⋊C4×C101D411yescharacteristic
46C22⋊C4×C101—21no—
47C42×C101C2311yescharacteristic
48C23×C101—21no—
49C22×D101—21no—
50D4×C101—21no—
51D4×C101—21no—
52C2×C4×C101—21no—
53C2×C4×C101—41no—
54C2×C4×C101—41no—
55C2×C4×C101C2×D411yescharacteristic
56C2×C4×C101C2×D411yescharacteristic
57C2×C4×C101C2×D411yescharacteristic
58C22⋊D101—82no—
59C2×Dic101—21no—
60C2×Dic101—22no—
61C2×Dic101—41no—
62D404—21no—
63D404—21no—
64C4×D101—41no—
65C4×D101—41no—
66Dic202—42no—
67Dic202—41no—
68Dic202—41no—
69C8×C101—41no—
70C8×C101—41no—
71#11717579—41no—
72C22×C101—41no—
73C22×C101—41no—
74C22×C101—41no—
75C22×C101#12011yescharacteristic
76C2×D101—41no—
77C2×D101—41no—
78C2×D101—41no—
79C4×C101—21no—
80C4×C101—41no—
81C4×C101#13511yescharacteristic
82C4×C101#13511yescharacteristic
83C4×C101AGL(1,ℤ/8)11yescharacteristic
84C4×C101#13711yescharacteristic
85Dic101—41no—
86Dic101—42no—
87Dic101—81no—
88C2×C101—81no—
89C2×C101#44911yescharacteristic
90C2×C101#45211yescharacteristic
91C2×C101#45511yescharacteristic
92D101—81no—
93#1646—1011nomaximal, Sylow
94C101#164611yescharacteristic, complemented, Sylow
95#555—1011no—
96#487—1011no—
97#474—1011no—
98#461—1011no—
99#333—1011no—
100D4⋊C8—1011no—
101D4⋊C8D10111yescharacteristic, complemented
102C2×SD16—2021no—
103C4⋊Q8—1011no—
104C4⋊Q8—1011no—
105#124—2022no—
106#121—2021no—
107#121—2021no—
108C4×D4—1011no—
109C4×D4C2×D10111yescharacteristic
110#117—2021no—
111#117—2021no—
112#107—2021no—
113C4⋊C8—1011no—
114C4⋊C8C2×D10111yescharacteristic
115#103—2022no—
116#103—2021no—
117#102—2021no—
118C4×C8C2×D10111yescharacteristic
119C2×Q8—1012no—
120C2×Q8—2021no—
121C2×D4—1011no—
122C2×D4—2022no—
123C2×D4D40411yescharacteristic
124C22×C4—21no—
125C22×C4—2021no—
126SD16—4042no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—2021no—
130C4⋊C4—1011no—
131C4⋊C4—2021no—
132C4⋊C4—2021no—
133C4⋊C4—2021no—
134C4⋊C4—2021no—
135C4⋊C4D40411yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—2022no—
138C22⋊C4—2022no—
139C22⋊C4—2021no—
140C42—2022no—
141C42C22×D10111yescharacteristic
142C23—21no—
143C23—2021no—
144Q8—4042no—
145Q8—4041no—
146Q8—4041no—
147D4—21no—
148D4—21no—
149D4—8082no—
150D4—2021no—
151D4—2021no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—4041no—
156C2×C4—4041no—
157C2×C4—4041no—
158C2×C4—2021no—
159C2×C4—2022no—
160C2×C4D4×D10111yescharacteristic
161C2×C4D4×D10111yescharacteristic
162C2×C4C2×D40411yescharacteristic
163C8—41no—
164C8—41no—
165C8—4041no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—4041no—
170C22—4041no—
171C22—4041no—
172C22#1569222511yescharacteristic
173C4—21no—
174C4—41no—
175C4—8081no—
176C4—4041no—
177C4—4042no—
178C4#1569227411yescharacteristic
179C4#1569227711yescharacteristic
180C4#1569228311yescharacteristic
181C4#1569228411yescharacteristic
182C2—81no—
183C2—8081no—
184C2#2026532411yescharacteristic
185C2#2026534011yescharacteristic
186C2#2026535611yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
FittingD4⋊C8×C101C2
socleC22×C101#120
prime radical#20265892C2
hypercenterD4⋊C8D101
hypocenterC101#1646
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20265528C2semidirect
3. #20265458C2semidirect
4. #20265382C2semidirect
5. #20264925C2semidirect
6. #20264906C2semidirect
7. D4⋊C8×C101C2semidirect
94. C101#1646semidirect
101. D4⋊C8D101semidirect

Maximal subgroups

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

subgroupindex
#202658922
#202655282
#202654582
#202653822
#202649252
#202649062
D4⋊C8×C1012
#1646101