Finite Group Databasebeta

← #27170318

Subgroups — #27170318

Order 12928.

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

normal subgroups only

This group has 212 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#20265523C211yesmaximal, characteristic
2#20265462C211yesmaximal, characteristic, complemented
3#20265387C211yesmaximal, characteristic, complemented
4#20264921C211yesmaximal, characteristic
5Q8⋊#11717579C211yesmaximal, characteristic, complemented
6Q8⋊#11717579C211yesmaximal, characteristic, complemented
7#559×C101C211yesmaximal, characteristic
8C2×#13108064—21no—
9#15692255C2211yescharacteristic
10#15692186—21no—
11C4⋊#11717579C2211yescharacteristic
12C4⋊#11717579C2211yescharacteristic
13#15692179—21no—
14#15692178—21no—
15#15692176—22no—
16C4×#11717579C2211yescharacteristic
17#127×C101C2211yescharacteristic
18#124×C101—21no—
19#124×C101—21no—
20#123×C101—22no—
21C4×Q8×C101C2211yescharacteristic
22C4×Q8×C101C2211yescharacteristic
23C4×D4×C101—21no—
24C4×D4×C101—21no—
25C2×Dic202—21no—
26#13108064—42no—
27C2×#11717579—21no—
28C2×#11717579—21no—
29C2×#11717579—21no—
30C4⋊Dic101—21no—
31C4⋊Dic101—21no—
32C2×Q8×C101D411yescharacteristic
33C2×Q8×C101D411yescharacteristic
34C2×D4×C101—21no—
35C2×D4×C101—21no—
36C2×D4×C101D412yes—
37C22×C4×C101—22no—
38C4⋊C4×C101—21no—
39C4⋊C4×C101—21no—
40C4⋊C4×C101D411yescharacteristic
41C4⋊C4×C101D411yescharacteristic
42C22⋊C4×C101—22no—
43C22⋊C4×C101—22no—
44C22⋊C4×C101—22no—
45C42×C101—21no—
46C42×C101—21no—
47C42×C101C2311yescharacteristic
48C23×C101—22no—
49Q8×C101—21no—
50Q8×C101—21no—
51Q8×C101—21no—
52Q8×C101—21no—
53D4×C101—42no—
54D4×C101—41no—
55D4×C101—41no—
56D4×C101—41no—
57D4×C101—41no—
58C2×C4×C101—21no—
59C2×C4×C101—21no—
60C2×C4×C101—21no—
61C2×C4×C101—21no—
62C2×C4×C101—82no—
63C2×C4×C101C2×D411yescharacteristic
64C2×C4×C101C2×D411yescharacteristic
65C2×C4×C101C2×D411yescharacteristic
66C2×Dic101—41no—
67Dic202—41no—
68Dic202—41no—
69#11717579—41no—
70#11717579—41no—
71#11717579—41no—
72C22×C101—42no—
73C22×C101—42no—
74C22×C101—42no—
75C22×C101#12011yescharacteristic
76C4×C101—21no—
77C4×C101—41no—
78C4×C101—41no—
79C4×C101—41no—
80C4×C101—41no—
81C4×C101#13511yescharacteristic
82C4×C101#13511yescharacteristic
83C4×C101AGL(1,ℤ/8)11yescharacteristic
84C4×C101#13711yescharacteristic
85Dic101—81no—
86C2×C101—82no—
87C2×C101#44911yescharacteristic
88C2×C101#45211yescharacteristic
89C2×C101#45511yescharacteristic
90#1640—1011nomaximal, Sylow
91C101#164011yescharacteristic, complemented, Sylow
92#559D10111yescharacteristic
93#486—1011no—
94#475—1011no—
95#462—1011no—
96#331—1011no—
97Q8⋊C8—1011no—
98Q8⋊C8—1011no—
99C2×Q16—2021no—
100C4⋊Q8—1011no—
101#127C2×D10111yescharacteristic
102#124—21no—
103#124—21no—
104#123—22no—
105C4×Q8C2×D10111yescharacteristic
106C4×Q8C2×D10111yescharacteristic
107C4×D4—21no—
108C4×D4—21no—
109#106—2021no—
110C4⋊C8—1011no—
111C4⋊C8—1011no—
112#103—2021no—
113#103—2021no—
114#102—2022no—
115C4×C8—1011no—
116C2×Q8—2021no—
117C2×Q8C22⋊D10111yescharacteristic
118C2×Q8C22⋊D10111yescharacteristic
119C2×D4—21no—
120C2×D4—21no—
121C2×D4C22⋊D10112yes—
122C22×C4—22no—
123Q16—4042no—
124C2×C8—2021no—
125C2×C8—2021no—
126C2×C8—2021no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—2021no—
130C4⋊C4—2021no—
131C4⋊C4C22⋊D10111yescharacteristic
132C4⋊C4C22⋊D10111yescharacteristic
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C42—21no—
137C42—21no—
138C42C22×D10111yescharacteristic
139C23—22no—
140Q8—21no—
141Q8—21no—
142Q8—21no—
143Q8—21no—
144Q8—4041no—
145Q8—4041no—
146D4—42no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—41no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—82no—
156C2×C4—4041no—
157C2×C4C2×C22⋊D10111yescharacteristic
158C2×C4C2×C22⋊D10111yescharacteristic
159C2×C4C2×C22⋊D10111yescharacteristic
160C8—4041no—
161C8—4041no—
162C8—4041no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22#1569222411yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—41no—
171C4—41no—
172C4—8081no—
173C4#1569227511yescharacteristic
174C4#1569227511yescharacteristic
175C4#1569228011yescharacteristic
176C4#1569228511yescharacteristic
177C2—82no—
178C2#2026532211yescharacteristic
179C2#2026533911yescharacteristic
180C2#2026535511yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#559×C101C2
socleC22×C101#120
prime radical#127×C101C22
hypercenter#559D101
hypocenterC101#1640
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #559

Complements

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

NH ≅ G/Ntype
2. #20265462C2semidirect
3. #20265387C2semidirect
5. Q8⋊#11717579C2semidirect
6. Q8⋊#11717579C2semidirect
91. C101#1640semidirect

Maximal subgroups

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

subgroupindex
#202655232
#202654622
#202653872
#202649212
Q8⋊#117175792
#559×C1012
#1640101