Finite Group Databasebeta

← #27170300

Subgroups — #27170300

Order 12928.

228 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×Dic202C211yesmaximal, characteristic
2#20265528C211yesmaximal, characteristic, complemented
3#20265473C211yesmaximal, characteristic, complemented
4#20265467C211yesmaximal, characteristic
5#20264924C211yesmaximal, characteristic, complemented
6#20264909C211yesmaximal, characteristic, complemented
7D4⋊C8×C101C211yesmaximal, characteristic
8C22×Dic202—21no—
9#15692255C2211yescharacteristic
10#15692255C2211yescharacteristic
11#15692254—21no—
12#15692234—21no—
13#15692232—22no—
14C4×Dic202C2211yescharacteristic
15D4×Dic101—21no—
16#15692189—21no—
17#15692186—21no—
18C4⋊#11717579C2211yescharacteristic
19#15692180—21no—
20#15692180—22no—
21#15692176—21no—
22C4×D4×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24C4×C8×C101C2211yescharacteristic
25C2×Dic202—22no—
26C2×Dic202—21no—
27C2×Dic202—21no—
28C2×Dic202—42no—
29C2×Dic202—42no—
30C2×Dic202D411yescharacteristic
31C22×Dic101—22no—
32C2×#11717579—21no—
33C4⋊Dic101—21no—
34C4⋊Dic101—21no—
35C4⋊Dic101—21no—
36C4⋊Dic101D412yes—
37C4⋊Dic101D411yescharacteristic
38#13108053—22no—
39#13108053—21no—
40C22⋊Dic101—22no—
41C4×Dic101—21no—
42C2×D4×C101D411yescharacteristic
43C22×C4×C101—21no—
44C2×C8×C101—21no—
45C2×C8×C101—21no—
46C4⋊C4×C101D411yescharacteristic
47C22⋊C4×C101—21no—
48C42×C101C2311yescharacteristic
49C23×C101—21no—
50D4×C101—21no—
51D4×C101Q1612yes—
52C2×C4×C101—21no—
53C2×C4×C101—41no—
54C2×C4×C101—41no—
55C2×C4×C101C2×D411yescharacteristic
56C2×C4×C101C2×D411yescharacteristic
57C2×C4×C101C2×D411yescharacteristic
58C2×Dic101—21no—
59C2×Dic101—21no—
60C2×Dic101—22no—
61C2×Dic101—44no—
62C2×Dic101—41no—
63Dic202—21no—
64Dic202—44no—
65Dic202—41no—
66Dic202—41no—
67Dic202—42no—
68Dic202SD1612yes—
69C8×C101—41no—
70C8×C101—41no—
71#11717579—41no—
72C22×C101—22no—
73C22×C101—41no—
74C22×C101—41no—
75C22×C101#12011yescharacteristic
76C4×C101—21no—
77C4×C101—41no—
78C4×C101C2×SD1611yescharacteristic
79C4×C101C2×Q1611yescharacteristic
80C4×C101AGL(1,ℤ/8)11yescharacteristic
81C4×C101#13711yescharacteristic
82Dic101—22no—
83Dic101—42no—
84Dic101—41no—
85Dic101—81no—
86C2×C101—42no—
87C2×C101#45011yescharacteristic
88C2×C101#45111yescharacteristic
89C2×C101#45411yescharacteristic
90#1637—1011nomaximal, Sylow
91C101#163711yescharacteristic, complemented, Sylow
92D4×Q8—1011no—
93#487—1011no—
94#477—1011no—
95#476—1011no—
96#332—1011no—
97Q8⋊C8—1011no—
98D4⋊C8D10111yescharacteristic
99C22×Q8—2021no—
100C4⋊Q8—1011no—
101C4⋊Q8—1011no—
102C4⋊Q8—2021no—
103C22⋊Q8—2022no—
104C22⋊Q8—2021no—
105C4×Q8—1011no—
106C4×D4—2021no—
107C4×D4C2×D10111yescharacteristic
108#107—2021no—
109#106—2021no—
110C4⋊C8—1011no—
111C4⋊C8C2×D10111yescharacteristic
112#103—2022no—
113#103—2021no—
114#102—2021no—
115C4×C8C2×D10111yescharacteristic
116C2×Q8—1011no—
117C2×Q8—4042no—
118C2×Q8—4042no—
119C2×Q8—2022no—
120C2×Q8—2021no—
121C2×Q8—2021no—
122C2×D4D40411yescharacteristic
123C22×C4—21no—
124C22×C4—2022no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—2021no—
128C4⋊C4—1012no—
129C4⋊C4—1011no—
130C4⋊C4—2022no—
131C4⋊C4—2021no—
132C4⋊C4—2021no—
133C4⋊C4—2021no—
134C4⋊C4—2021no—
135C4⋊C4D40411yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—2022no—
138C42—2021no—
139C42C22×D10111yescharacteristic
140C23—21no—
141Q8—1012no—
142Q8—4042no—
143Q8—4044no—
144Q8—4041no—
145Q8—4041no—
146Q8—2021no—
147D4—21no—
148D4Dic40412yes—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—4041no—
153C2×C4—4044no—
154C2×C4—2021no—
155C2×C4—2022no—
156C2×C4—2021no—
157C2×C4D4×D10111yescharacteristic
158C2×C4D4×D10111yescharacteristic
159C2×C4C2×D40411yescharacteristic
160C8—41no—
161C8—41no—
162C8—4041no—
163C22—22no—
164C22—41no—
165C22—41no—
166C22#1569222511yescharacteristic
167C4—21no—
168C4—41no—
169C4—8081no—
170C4—4042no—
171C4—4041no—
172C4—2022no—
173C4SD16×D10111yescharacteristic
174C4C2×Dic40411yescharacteristic
175C4#1569227911yescharacteristic
176C4#1569228811yescharacteristic
177C2—42no—
178C2#2026532911yescharacteristic
179C2#2026533511yescharacteristic
180C2#2026535111yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
FittingD4⋊C8×C101C2
socleC22×C101#120
prime radicalC2×D4×C101D4
hypercenterD4⋊C8D101
hypocenterC101#1637
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. #20265473C2semidirect
5. #20264924C2semidirect
6. #20264909C2semidirect
91. C101#1637semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2022
#202655282
#202654732
#202654672
#202649242
#202649092
D4⋊C8×C1012
#1637101