Finite Group Databasebeta

← #27170394

Subgroups — #27170394

Order 12928.

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

normal subgroups only

This group has 234 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
1Q8×D404C211yesmaximal, characteristic
2#20265573C211yesmaximal, characteristic, complemented
3#20265379C211yesmaximal, characteristic
4#20264924C211yesmaximal, characteristic, complemented
5Q8⋊#11717579C211yesmaximal, characteristic, complemented
6#20264904C211yesmaximal, characteristic
7#475×C101C211yesmaximal, characteristic, complemented
8C2×Q8×D101—21no—
9C2×#13108061—21no—
10#15692255—21no—
11#15692255C2211yescharacteristic
12#15692235—22no—
13#15692235—21no—
14C4×D404—21no—
15C4×D404C2211yescharacteristic
16#15692186—22no—
17C4⋊#11717579C2211yescharacteristic
18#15692179—21no—
19C4×#11717579C2211yescharacteristic
20C4⋊Q8×C101C2211yescharacteristic
21C4×Q8×C101C2211yescharacteristic
22#106×C101—21no—
23C4⋊C8×C101C2211yescharacteristic
24#103×C101—21no—
25C2×Dic202—22no—
26C2×Dic202—21no—
27Q8×D101—42no—
28Q8×D101—42no—
29C2×D404D411yescharacteristic
30C2×C4×D101—22no—
31C2×C4×D101—21no—
32#13108061—42no—
33C2×#11717579—21no—
34C2×#11717579—21no—
35C4⋊Dic101—22no—
36C4⋊Dic101—21no—
37C4⋊Dic101D411yescharacteristic
38C4⋊Dic101D412yes—
39#13108051—22no—
40#13108051—21no—
41C2×Q8×C101—21no—
42C2×Q8×C101D411yescharacteristic
43C2×C8×C101—21no—
44C4⋊C4×C101—21no—
45C4⋊C4×C101—21no—
46C4⋊C4×C101—21no—
47C4⋊C4×C101D411yescharacteristic
48C42×C101—21no—
49C42×C101C2311yescharacteristic
50C22×D101—21no—
51Q8×C101—21no—
52Q8×C101—41no—
53Q8×C101—41no—
54Q8×C101SD1612yescomplemented
55C2×C4×C101—21no—
56C2×C4×C101—21no—
57C2×C4×C101—41no—
58C2×C4×C101C2×D411yescharacteristic
59C2×C4×C101C2×D411yescharacteristic
60C2×C4×C101C2×D411yescharacteristic
61C2×Dic101—22no—
62C2×Dic101—21no—
63D404—21no—
64D404SD1612yes—
65C4×D101—44no—
66C4×D101—41no—
67C4×D101—41no—
68Dic202—44no—
69Dic202—41no—
70Dic202—41no—
71C8×C101—41no—
72#11717579—22no—
73#11717579—41no—
74C22×C101#12011yescharacteristic
75C2×D101—22no—
76C2×D101—41no—
77C2×D101—41no—
78C4×C101—22no—
79C4×C101—21no—
80C4×C101—41no—
81C4×C101—81no—
82C4×C101C2×SD1611yescharacteristic
83C4×C101C2×SD1611yescharacteristic
84C4×C101AGL(1,ℤ/8)11yescharacteristic
85C4×C101#13711yescharacteristic
86Dic101—41no—
87Dic101—42no—
88C2×C101#44811yescharacteristic
89C2×C101#45011yescharacteristic
90C2×C101#45611yescharacteristic
91D101—42no—
92#1651—1011nomaximal, Sylow
93C101#165111yescharacteristic, complemented, Sylow
94D4×Q8—1011no—
95#498—1011no—
96#475D10111yescharacteristic, complemented
97#461—1011no—
98#332—1011no—
99Q8⋊C8—1011no—
100D4⋊C8—1011no—
101C22×Q8—2021no—
102C2×SD16—2021no—
103C4⋊Q8—1011no—
104C4⋊Q8—2021no—
105C4⋊Q8C2×D10111yescharacteristic
106C22⋊Q8—2022no—
107C22⋊Q8—2021no—
108C4×Q8C2×D10111yescharacteristic
109C4×D4—1011no—
110C4×D4—2021no—
111#106—21no—
112#106—2022no—
113C4⋊C8—1011no—
114C4⋊C8C2×D10111yescharacteristic
115#103—21no—
116#103—2021no—
117C4×C8—1011no—
118C2×Q8—21no—
119C2×Q8—4042no—
120C2×Q8—4042no—
121C2×Q8—2022no—
122C2×Q8—2021no—
123C2×Q8C22⋊D10111yescharacteristic
124C2×D4—1011no—
125C22×C4—2022no—
126C22×C4—2021no—
127SD16—4042no—
128C2×C8—21no—
129C2×C8—2021no—
130C2×C8—2021no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—1011no—
135C4⋊C4—1012no—
136C4⋊C4—2022no—
137C4⋊C4—2021no—
138C4⋊C4C22⋊D10111yescharacteristic
139C22⋊C4—2022no—
140C22⋊C4—2021no—
141C42—21no—
142C42C22×D10111yescharacteristic
143C23—2021no—
144Q8—21no—
145Q8—41no—
146Q8—41no—
147Q8—4044no—
148Q8—4041no—
149Q8—4041no—
150Q8#1310806112yescomplemented
151D4—1012no—
152D4—2021no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—4044no—
157C2×C4—4041no—
158C2×C4—4041no—
159C2×C4—2022no—
160C2×C4—2021no—
161C2×C4D4×D10111yescharacteristic
162C2×C4D4×D10111yescharacteristic
163C2×C4C2×C22⋊D10111yescharacteristic
164C8—41no—
165C8—4041no—
166C8—2022no—
167C22—4041no—
168C22—4041no—
169C22—2022no—
170C22#1569222311yescharacteristic
171C4—22no—
172C4—21no—
173C4—41no—
174C4—81no—
175C4—4042no—
176C4—4041no—
177C4SD16×D10111yescharacteristic
178C4C2×#1310806111yescharacteristic
179C4#1569228011yescharacteristic
180C4#1569228411yescharacteristic
181C2—4042no—
182C2#2026531411yescharacteristic
183C2#2026532711yescharacteristic
184C2#2026535811yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#475×C101C2
socleC22×C101#120
prime radicalC2×D404D4
hypercenter#475D101
hypocenterC101#1651
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #20265573C2semidirect
4. #20264924C2semidirect
5. Q8⋊#11717579C2semidirect
7. #475×C101C2semidirect
54. Q8×C101SD16semidirect
93. C101#1651semidirect
96. #475D101semidirect
150. Q8#13108061semidirect

Maximal subgroups

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

subgroupindex
Q8×D4042
#202655732
#202653792
#202649242
Q8⋊#117175792
#202649042
#475×C1012
#1651101