Finite Group Databasebeta

← #27170393

Subgroups — #27170393

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
1D4×Dic202C211yesmaximal, characteristic
2#20265573C211yesmaximal, characteristic, complemented
3#20265459C211yesmaximal, characteristic, complemented
4#20264922C211yesmaximal, characteristic, complemented
5#20264907C211yesmaximal, characteristic, complemented
6D4⋊#11717579C211yesmaximal, characteristic
7#461×C101C211yesmaximal, characteristic
8C22×Dic202—21no—
9#15692255C2211yescharacteristic
10#15692254—21no—
11#15692234—21no—
12#15692232—22no—
13C4×Dic202C2211yescharacteristic
14D4×Dic101—21no—
15#15692186—22no—
16#15692186—21no—
17C4⋊#11717579C2211yescharacteristic
18#15692178—21no—
19C4×#11717579C2211yescharacteristic
20C2×SD16×C101—21no—
21C4⋊Q8×C101C2211yescharacteristic
22C4×D4×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24#103×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⋊Dic101D411yescharacteristic
35C4⋊Dic101D412yes—
36#13108053—22no—
37#13108053—21no—
38C22⋊Dic101—22no—
39C4×Dic101—21no—
40C2×Q8×C101—21no—
41C2×D4×C101D411yescharacteristic
42C22×C4×C101—21no—
43SD16×C101—42no—
44C2×C8×C101—21no—
45C4⋊C4×C101—21no—
46C4⋊C4×C101—21no—
47C4⋊C4×C101D411yescharacteristic
48C22⋊C4×C101—21no—
49C42×C101C2311yescharacteristic
50C23×C101—21no—
51Q8×C101—41no—
52Q8×C101—41no—
53D4×C101—21no—
54D4×C101SD1612yes—
55C2×C4×C101—21no—
56C2×C4×C101—41no—
57C2×C4×C101—41no—
58C2×C4×C101—41no—
59C2×C4×C101C2×D411yescharacteristic
60C2×C4×C101C2×D411yescharacteristic
61C2×C4×C101C2×D411yescharacteristic
62C2×Dic101—21no—
63C2×Dic101—21no—
64C2×Dic101—22no—
65C2×Dic101—44no—
66Dic202—21no—
67Dic202—44no—
68Dic202—41no—
69Dic202—41no—
70Dic202SD1612yescomplemented
71C8×C101—41no—
72#11717579—22no—
73#11717579—41no—
74C22×C101—22no—
75C22×C101—41no—
76C22×C101—41no—
77C22×C101#12011yescharacteristic
78C4×C101—21no—
79C4×C101—41no—
80C4×C101—81no—
81C4×C101C2×SD1611yescharacteristic
82C4×C101C2×SD1611yescharacteristic
83C4×C101AGL(1,ℤ/8)11yescharacteristic
84C4×C101#13711yescharacteristic
85Dic101—22no—
86Dic101—42no—
87Dic101—41no—
88C2×C101—42no—
89C2×C101#44811yescharacteristic
90C2×C101#45011yescharacteristic
91C2×C101#45611yescharacteristic
92#1651—1011nomaximal, Sylow
93C101#165111yescharacteristic, complemented, Sylow
94D4×Q8—1011no—
95#498—1011no—
96#475—1011no—
97#461D10111yescharacteristic
98#332—1011no—
99Q8⋊C8—1011no—
100D4⋊C8—1011no—
101C22×Q8—2021no—
102C2×SD16—21no—
103C4⋊Q8—1011no—
104C4⋊Q8—2021no—
105C4⋊Q8C2×D10111yescharacteristic
106C22⋊Q8—2022no—
107C22⋊Q8—2021no—
108C4×Q8—1011no—
109C4×D4—2021no—
110C4×D4C2×D10111yescharacteristic
111#106—2021no—
112#106—2022no—
113C4⋊C8—1011no—
114C4⋊C8C2×D10111yescharacteristic
115#103—21no—
116#103—2021no—
117C4×C8—1011no—
118C2×Q8—21no—
119C2×Q8—1011no—
120C2×Q8—4042no—
121C2×Q8—4042no—
122C2×Q8—2022no—
123C2×Q8—2021no—
124C2×D4C22⋊D10111yescharacteristic
125C22×C4—21no—
126C22×C4—2022no—
127SD16—42no—
128C2×C8—21no—
129C2×C8—2021no—
130C2×C8—2021no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—1011no—
134C4⋊C4—1012no—
135C4⋊C4—2022no—
136C4⋊C4—2021no—
137C4⋊C4—2021no—
138C4⋊C4C22⋊D10111yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—2022no—
141C42—2021no—
142C42C22×D10111yescharacteristic
143C23—21no—
144Q8—41no—
145Q8—41no—
146Q8—1012no—
147Q8—4044no—
148Q8—4041no—
149Q8—4041no—
150Q8—2021no—
151D4—21no—
152D4#1310806212yes—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—4044no—
158C2×C4—2021no—
159C2×C4—2022no—
160C2×C4—2021no—
161C2×C4D4×D10111yescharacteristic
162C2×C4D4×D10111yescharacteristic
163C2×C4C2×C22⋊D10111yescharacteristic
164C8—41no—
165C8—4041no—
166C8—2022no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#1569222311yescharacteristic
171C4—21no—
172C4—41no—
173C4—81no—
174C4—4042no—
175C4—4041no—
176C4—2022no—
177C4SD16×D10111yescharacteristic
178C4C2×#1310806211yescharacteristic
179C4#1569228211yescharacteristic
180C4#1569228511yescharacteristic
181C2—42no—
182C2#2026531511yescharacteristic
183C2#2026532611yescharacteristic
184C2#2026535811yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#461×C101C2
socleC22×C101#120
prime radicalC2×D4×C101D4
hypercenter#461D101
hypocenterC101#1651
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #20265573C2semidirect
3. #20265459C2semidirect
4. #20264922C2semidirect
5. #20264907C2semidirect
70. Dic202SD16semidirect
93. C101#1651semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2022
#202655732
#202654592
#202649222
#202649072
D4⋊#117175792
#461×C1012
#1651101