Finite Group Databasebeta

← #27170296

Subgroups — #27170296

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#20265527C211yesmaximal, characteristic, complemented
3#20265469C211yesmaximal, characteristic, complemented
4#20264922C211yesmaximal, characteristic, complemented
5#20264907C211yesmaximal, characteristic, complemented
6D4⋊#11717579C211yesmaximal, characteristic
7#476×C101C211yesmaximal, characteristic
8C22×Dic202—21no—
9#15692255C2211yescharacteristic
10#15692254—21no—
11#15692234—21no—
12#15692232—22no—
13C4×Dic202C2211yescharacteristic
14D4×Dic101—21no—
15#15692188—21no—
16C4⋊#11717579C2211yescharacteristic
17#15692179—22no—
18#15692178—21no—
19C4×#11717579C2211yescharacteristic
20C4⋊Q8×C101C2211yescharacteristic
21C4×D4×C101C2211yescharacteristic
22#106×C101—21no—
23C4⋊C8×C101C2211yescharacteristic
24#102×C101—21no—
25C2×Dic202—22no—
26C2×Dic202—21no—
27C2×Dic202—42no—
28C2×Dic202—42no—
29C2×Dic202D411yescharacteristic
30C22×Dic101—22no—
31C2×#11717579—21no—
32C2×#11717579—21no—
33C4⋊Dic101—21no—
34C4⋊Dic101D412yes—
35C4⋊Dic101D411yescharacteristic
36#13108053—22no—
37#13108053—21no—
38C22⋊Dic101—22no—
39C4×Dic101—21no—
40C2×Q8×C101—21no—
41C2×D4×C101D411yescharacteristic
42C22×C4×C101—21no—
43C2×C8×C101—21no—
44C4⋊C4×C101—21no—
45C4⋊C4×C101—21no—
46C4⋊C4×C101D411yescharacteristic
47C22⋊C4×C101—21no—
48C42×C101C2311yescharacteristic
49C23×C101—21no—
50Q8×C101—42no—
51D4×C101—21no—
52D4×C101Q1612yes—
53C2×C4×C101—21no—
54C2×C4×C101—41no—
55C2×C4×C101—41no—
56C2×C4×C101—41no—
57C2×C4×C101C2×D411yescharacteristic
58C2×C4×C101C2×D411yescharacteristic
59C2×C4×C101C2×D411yescharacteristic
60C2×Dic101—21no—
61C2×Dic101—21no—
62C2×Dic101—22no—
63C2×Dic101—44no—
64Dic202—21no—
65Dic202—44no—
66Dic202—41no—
67Dic202—41no—
68Dic202SD1612yes—
69C8×C101—41no—
70#11717579—41no—
71#11717579—41no—
72C22×C101—22no—
73C22×C101—41no—
74C22×C101—41no—
75C22×C101#12011yescharacteristic
76C4×C101—21no—
77C4×C101—41no—
78C4×C101—81no—
79C4×C101C2×SD1611yescharacteristic
80C4×C101C2×Q1611yescharacteristic
81C4×C101AGL(1,ℤ/8)11yescharacteristic
82C4×C101#13711yescharacteristic
83Dic101—22no—
84Dic101—42no—
85Dic101—41no—
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#476D10111yescharacteristic
96#332—1011no—
97Q8⋊C8—1011no—
98D4⋊C8—1011no—
99C22×Q8—2021no—
100C4⋊Q8—1011no—
101C4⋊Q8—2021no—
102C4⋊Q8C2×D10111yescharacteristic
103C22⋊Q8—2022no—
104C22⋊Q8—2021no—
105C4×Q8—1011no—
106C4×D4—2021no—
107C4×D4C2×D10111yescharacteristic
108#107—2021no—
109#106—21no—
110C4⋊C8—1011no—
111C4⋊C8C2×D10111yescharacteristic
112#103—2021no—
113#103—2022no—
114#102—21no—
115C4×C8—1011no—
116C2×Q8—21no—
117C2×Q8—1011no—
118C2×Q8—4042no—
119C2×Q8—4042no—
120C2×Q8—2022no—
121C2×Q8—2021no—
122C2×D4C22⋊D10111yescharacteristic
123C22×C4—21no—
124C22×C4—2022no—
125C2×C8—21no—
126C2×C8—2021no—
127C2×C8—2021no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—1012no—
131C4⋊C4—1011no—
132C4⋊C4—2022no—
133C4⋊C4—2021no—
134C4⋊C4—2021no—
135C4⋊C4C22⋊D10111yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—2022no—
138C42—2021no—
139C42C22×D10111yescharacteristic
140C23—21no—
141Q8—42no—
142Q8—1012no—
143Q8—4044no—
144Q8—4041no—
145Q8—4041no—
146Q8—2021no—
147D4—21no—
148D4#1310806412yes—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—4044no—
154C2×C4—2021no—
155C2×C4—2022no—
156C2×C4—2021no—
157C2×C4D4×D10111yescharacteristic
158C2×C4D4×D10111yescharacteristic
159C2×C4C2×C22⋊D10111yescharacteristic
160C8—41no—
161C8—4041no—
162C8—4041no—
163C22—22no—
164C22—41no—
165C22—41no—
166C22#1569222311yescharacteristic
167C4—21no—
168C4—41no—
169C4—81no—
170C4—4042no—
171C4—4041no—
172C4—2022no—
173C4SD16×D10111yescharacteristic
174C4C2×#1310806411yescharacteristic
175C4#1569228211yescharacteristic
176C4#1569228511yescharacteristic
177C2—42no—
178C2#2026532611yescharacteristic
179C2#2026533211yescharacteristic
180C2#2026534811yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#476×C101C2
socleC22×C101#120
prime radicalC2×D4×C101D4
hypercenter#476D101
hypocenterC101#1637
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #20265527C2semidirect
3. #20265469C2semidirect
4. #20264922C2semidirect
5. #20264907C2semidirect
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
#202655272
#202654692
#202649222
#202649072
D4⋊#117175792
#476×C1012
#1637101